the small baseline subset technique: performance ... · the small baseline subset technique:...

Ph.D. in Electronic and Computer Engineering Dept. of Electrical and Electronic Engineering University of Cagliari

The Small BAseline Subset technique: performance assessment and new

developments for surface deformation analysis of very extended areas

Francesco Casu

Advisor: Prof. Giuseppe Mazzarella Co-advisor: Eng. Riccardo Lanari

Curriculum: ING-INF/02 Electromagnetic fields

XXI Cycle February 2009

to Sabrina

Contents Introduction .................................................................................................. 1 Chapter 1 SAR Interferometry: basic concepts and applications.............................. 5

1.1 Synthetic Aperture Radar ................................................................ 6 1.1.1 System geometry............................................................... 6 1.1.2 Geometric resolution and image formation....................... 7 1.1.3 Resolution consideration................................................. 16

1.2 Mapping scene elevation: Digital Elevation Model generation..... 19 1.2.1 Stereometry ..................................................................... 21 1.2.2 SAR Interferometry......................................................... 22 1.2.3 Interferometric Phase Statistics....................................... 26 1.2.4 Decorrelation Effects ...................................................... 30

1.3 Mapping small elevation changes: Differential SAR Interferometry ................................................................................ 34 Table I ..................................................................................................... 40 Appendix A: SAR focusing outline ......................................................... 41 Appendix B: Coherence calculation......................................................... 45 References................................................................................................ 48

Chapter 2 The SBAS technique: key idea and processing chain.............................. 51

2.1 Advanced DInSAR algorithms ...................................................... 51 2.2 Interferograms selection: two different “philosophies”................. 52 2.3 SBAS algorithm: basic rationale.................................................... 55 2.4 SBAS algorithm: processing chain ................................................ 60 2.5 Orbit parameters extraction and Master selection ......................... 62 2.6 Focusing......................................................................................... 63 2.7 DEM conversion and Range files generation ................................ 63 2.8 Data Pairs Selection....................................................................... 68 2.9 Interferometric data pairs co-registration ...................................... 71 2.10 Differential interferogram generation and filtering ....................... 72

ii Contents

2.11 Deformation time series generation............................................... 73 2.11.1 Pixel selection criterion................................................... 74 2.11.2 Phase Unwrapping .......................................................... 75

2.11.2.1 EMCF approach .................................................. 75 2.11.2.2 Region-Growing step .......................................... 76

2.11.3 Mean deformation velocity and residual topography estimation........................................................................ 79

2.11.4 Time series generation .................................................... 80 2.11.5 Reconstruction accuracy evaluation ............................... 80 2.11.6 Atmospheric filtering and orbital ramps estimation ....... 82

2.12 Geocoding...................................................................................... 83 2.13 SBAS technique application: the Long Valley caldera test site..... 84 Appendix A: Phase Unwrapping basic rationale ..................................... 90 Appendix B: MCF algorithm basic rationale........................................... 93 References................................................................................................ 97

Chapter 3 SBAS Performance Assessment .............................................................. 101

3.1 Analysis overview ....................................................................... 101 3.2 Leveling vs. SBAS: the Napoli bay............................................. 103

3.2.1 Napoli bay test site description ..................................... 103 3.2.2 Leveling network .......................................................... 103 3.2.3 SBAS measurements..................................................... 105 3.2.4 Data comparison ........................................................... 107

3.3 GPS vs. SBAS: the Los Angeles area.......................................... 112 3.3.1 Los Angels metropolitan area description .................... 112 3.3.2 GPS network ................................................................. 112 3.3.3 SBAS measurements..................................................... 113 3.3.4 Data comparison ........................................................... 116

3.4 Alignment Array vs. SBAS: the Hayward fault .......................... 116 3.4.1 Hayward fault description............................................. 117 3.4.2 Alignment arrays network............................................. 117 3.4.3 SBAS measurements..................................................... 117 3.4.4 Data comparison ........................................................... 119

3.5 Final remarks ............................................................................... 124 Table I ................................................................................................... 126 Table II................................................................................................... 128 Table III ................................................................................................. 129 Table IV ................................................................................................. 130 Appendix A: Multi-orbit combination ................................................... 131 References.............................................................................................. 133

iii Chapter 4 The SBAS technique for deformation analysis

of very extended areas ......................................................................... 137 4.1 E-SBAS processing chain rationale............................................. 137 4.2 E-SBAS results: Central Nevada case study................................ 140

4.2.1 Site description.............................................................. 140 4.2.2 Data analysis ................................................................. 140 4.2.3 Data validation .............................................................. 143

4.3 Final remarks ............................................................................... 145 References.............................................................................................. 146

Chapter 5 Region-Growing EMCF PhU algorithm................................................. 147

5.1 Region-Growing general concepts............................................... 148 5.2 Region-Growing EMCF strategy................................................. 149

5.2.1 Unwrapped Phase-Gradient vector estimation.............. 155 5.3 Real data results........................................................................... 157

5.3.1 Central Nevada test site................................................. 157 5.3.2 Gardanne test area ......................................................... 160

Appendix A: Non triangulation-driven data pairs in EMCF procedure. 165 Appendix B: Multi-subset EMCF approach........................................... 169 References.............................................................................................. 171

Conclusions ............................................................................................... 173 Acknowledgements ................................................................................... 176

Introduction The Synthetic Aperture Radar (SAR) sensor plays nowadays a

significant role in Remote Sensing scenarios. Indeed, by operating in the microwave spectrum region and providing itself the illumination source (it is a coherent active sensor), it permits to generate high spatial resolution images, independently from wheatear conditions and during both day and night time. Due to these characteristics, SAR sensors have been largely employed in airborne and spaceborne remote sensing to image and monitor the Earth surface and for planetary exploration.

Several techniques have been already developed for an advanced use of SAR data. One of these is represented by Differential SAR Interferometry (DInSAR) that exploits the phase difference (Interferogram) between two distinct SAR images acquired at different times over the same scene, permitting to retrieve the ground deformation (directly related to the interferogram) occurred between the two acquisition times. We stress that the measured interferometric phase is restricted to the ( ),π π− interval (wrapped phase), while its unrestricted value (unwrapped phase) is the one directly linked to the ground deformation. Therefore, a procedure permitting to retrieve unwrapped phase, starting from the wrapped one, is essential in SAR interferometry. This particular operation is generally referred to as Phase Unwrapping (PhU) and is one of the most critical steps within the DInSAR processing chain.

We further remark that the DInSAR technique typically provides high spatial density displacement maps, with accuracy on the order of fraction of the signal wavelength (centimeter/millimeter) and with relatively low costs with respect to “classic” geodetic measurements, such as spirit leveling or GPS. However, DInSAR permits to retrieve only the sensor Line of Sight (LOS) projection of the detected displacement.

Originally developed to investigate single deformation events (for instance, earthquakes or volcanic eruptions), the DInSAR technique has been recently extended to analyze the temporal evolution of the detected displacements. This technique advancement has been possible due to the availability of large SAR data archives acquired, during the last 16 years, by

2 Introduction the European (ERS and ENVISAT) and Canadian (RADARSAT) satellites. Accordingly, several Advanced DInSAR algorithms have been proposed; among these, we concentrate on the one referred to as Small BAseline Subset (SBAS) approach.

The SBAS technique relies on a proper combination of unwrapped interferograms characterized by a small spatial separation (perpendicular baseline) between the acquisition orbits and a short time interval (temporal baseline) between the acquisition epochs. The SBAS approach permits to generate mean deformation velocity maps of the investigated area as well as to detect the temporal evolution of the ongoing displacements. In its original formulation, the SBAS approach allows exploiting averaged (multilook) interferograms relevant to the same area, thus implying (for a typical ERS configuration) to produce deformation maps extending for about 100 by 100 km and with a spatial resolution of the order of 100 by 100 m.

We remark that, although the SBAS algorithm capability to generate mean deformation velocity maps and displacement time series has been already exploited in different works, no extensive analysis on the “quality” of the SBAS results have been performed. Therefore, we have started our analysis by carrying out a quantitative assessment of the SBAS procedure performance by processing a very large archive of SAR data acquired by the ERS sensors and systematically comparing the achieved results with geometric leveling (in Napoli bay zone, Italy), continuous GPS (in Los Angeles area, USA) and alignment arrays (along Hayward fault, San Francisco Bay area, USA) measurements that have been assumed as reference.

Subsequently, we have extended the SBAS processing approach in order to allow the analysis of the ongoing deformation affecting very extended areas. Indeed, although the availability of deformation maps typically extending for 100 by 100 km (corresponding to one SAR data frame) may be very effective to investigate several displacement phenomena, there are cases where it can be crucial to study much larger zones. The extended SBAS approach (E-SBAS) has been applied to analyze a unique large SAR data set including 264 descending ERS SAR data frames, spanning the 1992-2000 time interval and relevant to an area in Central Nevada (USA) extending for about 600 x 100 km. Moreover, by benefiting of the previous quality assessment study, we were able to provide a detailed validation of the achieved results, demonstrating the E-SBAS algorithm capability to detected deformation with an accuracy comparable to that achieved through the conventional SBAS processing.

Finally, we have focused our attention on the Phase Unwrapping operation that, as already stated before, represents a key issue of the DInSAR

3 techniques and, particularly, of the proposed extended SBAS algorithm for large scale data analysis. We underline that in the original SBAS processing chain the unwrapping operation is implemented via a two step approach, consisting of a first “global” PhU step followed by a local Region-Growing (RG) procedure. The latter is based on a linear model of the investigated deformation that is exploited for “propagating” the retrieved solution from already unwrapped pixels to neighboring wrapped points. We remark that the implemented RG approach is strongly limited in regions affected by non linear deformation, where the displacement linearity assumption fails, as well as in highly decorrelated areas, where the Signal-to-Noise Ratio is particularly low. For this reason, we present in this thesis a novel PhU algorithm, focused on multilook interferograms, that properly merges the Region-Growing strategy with a “temporal” PhU approach based on the Extended Minimum Cost Flow (EMCF) technique. The new RG-EMCF algorithm has been applied to the previous analyzed ERS data of the Nevada region, characterized by large non-linear deformation phenomena, as well as to a (particular critical) test site located in Gardanne (France), affected by strong decorrelation phenomena; in both cases the proposed approach has demonstrated its effectiveness.

In synthesis, this thesis is organized as follows: Chapter 1 focuses on the basic SAR concepts and techniques. In

particular, a short analysis of the SAR principles is first provided, followed by a discussion on the basic rationale of SAR Interferometry. Moreover, Differential SAR Interferometry for surface deformation retrieval is also deeply analyzed.

Chapter 2 provides an overview of the Advanced DInSAR algorithms for multi-interferograms analysis, emphasizing the role played by the SBAS approach, whose basic theory, key points and limitations are discussed. Moreover, a detailed analysis of the SBAS processing chain is provided, with particular emphasis given to the available phase unwrapping step. Finally, we also present some key results achieved on real SAR data, which demonstrate the deformation retrieval capability of the SBAS process.

Chapter 3 is focused on a quantitative assessment of the SBAS algorithm performance; this is carried out by fully exploiting large archive of ERS SAR data and systematically comparing the achieved DInSAR measurements with geodetic ones, assumed as reference.

Chapter 4 presents an extension of the SBAS approach, allowing to survey the temporal evolution of the deformation affecting very large areas. The presented results, carried out on a unique large ERS SAR data acquired over a 600 x 100 km wide region, demonstrate that the proposed approach is

4 Introduction suitable to operate as surface deformation survey tool for very extended areas.

Chapter 5 proposes a new Region-Growing PhU algorithm for sequences of multi-temporal, multilook interferograms. After introducing the Region-Growing PhU main concepts, a detailed analysis on the proposed novel algorithm, aimed at increasing DInSAR measurements spatial density by exploiting the EMCF algorithm, is provided. Moreover, experimental results carried out on both very extended and low coherence ERS and ENVISAT interferograms, demonstrate the approach effectiveness.

Chapter 1

SAR Interferometry: basic concepts and applications

Synthetic Aperture Radar (SAR) is a coherent active microwave remote sensing system which permits to reconstruct complex high resolution image of the illuminated scene.

A SAR sensor, usually mounted aboard aircrafts and/or satellites, operates in a side-looking configuration and permits to measure, with high accuracy, the distance between the system moving platform and the scene surface. The received backscattered radar signal (echo) accounts for the physical characteristics of the imaged scene as well as the acquisition geometry and, after a proper processing step, it allows reconstructing high spatial resolution ground images, independently from the sensor altitude. As an active system, the SAR provides its own illumination and is not dependent on light from the sun, thus permitting continuous day/night operation. Furthermore, neither clouds, fog, nor precipitation have a significant effect on microwaves, thus allowing all-weather imaging. As a consequence of its flexibility, SAR technology mostly improved during the last years and several SAR related techniques have been developed in scientific and commercial scenarios.

One of the major applications of the SAR technology is represented by the SAR Interferometry (InSAR) technique which exploits the phase difference, often referred to as Interferogram, between two complex-valued SAR images relevant to an investigated area (acquired from slightly different orbit positions and at different times) in order to retrieve geophysical parameters such as scene topography or ground deformation.

Accordingly, in this Chapter the basic principles of the Synthetic Aperture Radar will be presented. After introducing the SAR Interferometry theory and the processing which leads to the interferometric products, the possible applications of the interferometric technique are explained, with a particular emphasis on Differential InSAR for surface deformation retrieval.

6 SAR Interferometry

1.1 Synthetic Aperture Radar In this section we provide a quick overview of basic SAR principles

focusing on the SAR capability to obtain high spatial resolution images of an illuminated scene.

1.1.1 System geometry Figure 1 depicts the configuration of a side-looking radar. Antenna is

mounted on a platform (usually an aircraft or satellite) moving with a velocity v with respect to the Earth at a constant altitude; flight direction is generally called azimuth. The radar illuminates along the direction perpendicular to the flight path, slant range, with an inclination ϑ (look angle) with respect to the vertical. The transmitted power is partially absorbed by the ground and partially scattered in all directions, the scattering depending on the surface physical properties, such as roughness and slope, but also on its electrical properties, such as the dielectric constant, which varies with temperature, humidity, etc. A certain amount of the incident radiation is scattered back to the antenna, which detects the echoes and registers the corresponding values of intensity and phase, and the round-trip time which, since the pulses travel at the speed of light, is straightforwardly related to the range distance of the antenna from the imaged area. These echoes can be separated from each other as described in the next sessions, and can be therefore ordered as an array along the range direction. The

x

( ), ,P x r ϑ

ϑ

R

r

x′

Flight path

Nadir r

x x′−

Azimuth direction

Range direction

x

( ), ,P x r ϑ

ϑ

R

r

x′

Flight path

Nadir r

x x′−

Azimuth direction

Range direction

Figure 1 – Side-looking radar system geometry.

Chapter 1 7 motion of the sensor along its flight path causes the illuminated area shifts in the azimuth direction. The result is a two-dimensional matrix of the returned echoes, called raw data matrix.

1.1.2 Geometric resolution and image formation Radar self-illuminates an area on the ground by transmitting a series of

electromagnetic pulses and, after an accurate measure of the time delay between the transmitted and the received echoes, is able to figure out the distance (called slant range) between the sensor position along its flight direction and the illuminated targets on the ground. The capability of the sensor to discriminate different echoes returned form spatially separated object is called geometric (or spatial) resolution. More precisely, the resolution length is the minimum distance between two targets to be considered as separate entities, and, therefore, resolved. By the consequence, an image radar system is well-characterized by its attainable spatial resolution.

Let us again refer to Figure 1, where generic radar system geometry is depicted: a cylindrical reference system, whose axis coincides with the sensor flight path, is considered. Let us denote with:

• (x,r) the azimuth and (slant) range coordinates of the scene generic scattering point P;

• ϑ(x,r) the look angle of the generic scattering point P; • R the target-to-antenna distance.

The radar sensor is located on the platform that moves at velocity supposed to be constant and equal to ˆv= ⋅v xr . At times tn-τ/2, the sensor radiates pulses represented, but for an amplitude factor, by:

( ) ( )( ) exp 2 rect ntx n

t tf t j f t p t tπτ−⎡ ⎤= − ⎢ ⎥⎣ ⎦

(1)

where p(t-tn) describes the signal modulation, τ represents the duration of the transmitted pulse, and f is the carrier frequency.

In order to analyze characteristics and properties of the signal backscattered and received onboard, in the following, we first consider an elementary scene consisting of a single scatterer, and then, we move to analyze extended scene.

Accordingly, let us consider the elementary scatterer located at P≡(x,r,ϑ), see Figure 1. The signal backscattered and received onboard is given by:


( )

[ ]2

2 2( , , ) , exp 2

2

rect ,

r n n n

n

n

R Rf x x t t r x r j f p t tc c

Rt tc w x x r

γ π

τ

⎛ ⎞ ⎛ ⎞− − = − − −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎡ ⎤− −⎢ ⎥⋅ −⎢ ⎥

⎢ ⎥⎢ ⎥⎣ ⎦

(2)

where the fast-varying exp(j2πft) term of eq.(1) is cancelled by the heterodyne process, and where:

• xn= vtn is the azimuth coordinate of the antenna phase center; • γ(x,r) is the reflectivity pattern proportional to the ratio between

backscattered and incident field; • c is the light speed; • w(·) is the antenna illumination function, related to the azimuth

antenna footprint over the ground. The latter term is squared in eq. (2) because the same antenna operates

also in receive mode. Equation (2) assumes the platform moving in a stop and go way, in the sense that the system is supposed to transmit and to receive the same pulse at the same position: it can be shown [Franceschetti and Lanari, 1999] that this is a reasonable approximation for all the available SAR systems. From Figure 1 we have:

( ) ( )22,n nR R x x r r x x= − = + − . (3)

Let us now change the time coordinates in range (spatial) coordinates as follows:

2ctr

′′ = (4)

with: nt t t′ = − . (5)

Moreover, we assume the discrete xn-coordinate to be continuous, that is:

xn→ x′. (6)

Accordingly, eq. (2) can be rewritten as follows:

Chapter 1 9

( ) ( )

[ ]2

4 2( , , ) , exp

rect ,2

rf x x r r x r j R p r Rc

r R w x x rc

πγλ

τ

⎛ ⎞ ⎡ ⎤′ ′ ′− = − −⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦′⎡ ⎤− ′⋅ −⎢ ⎥

⎣ ⎦

(7)

where λ=c/f is the carrier wavelength. Equation (7) can be easily rearranged as follows:

( ) [ ]

( ) ( )

2( , , ) , rect ,2

4 2exp

rr r Rf x x r r r x r w x x r

c

j r R p r r Rc

γτ

πλ

′⎡ ⎤− − ∆′ ′ ′− − = −⎢ ⎥⎣ ⎦

⎡ ⎤ ⎡ ⎤′− + ∆ − − ∆⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(8)

where ∆R is given by:

( ) ( )22,R x x r R r r x x r′ ′∆ − = − = + − − (9)

and use of eq. (6) in eq. (3) has been made in the last equation. Some considerations on eq. (8) are now in order. First of all we underline that eq. (8) represents the signal backscattered

by an isolated point target located at fixed range and azimuth coordinates (x,r) and received onboard. In particular, we must note that in eq. (8) x and r are fixed; conversely, x' and r' do vary: they represent the spatial coordinates of the received two-dimensional image fr(·).

As a matter of fact, we must observe that the signal fr(·) collected onboard, referred to as raw data relative to P(x,r) in the following, represents an estimate of the reflectivity pattern γ(x,r) of the illuminated point target. In order to evaluate how much accurate this estimate is, we now focus our attention on the geometric resolution achievable by the received data fr(·). Note that in the two-dimensional data case we have to consider both azimuth and range resolution, hereafter denoted as ∆xraw and ∆rraw, respectively.

Range resolution of the received (i.e., raw) data is now addressed. By considering the two targets labeled as “a” and “b” in Figure 2, we

can observe that they can be effectively discriminated only if the received backscattered pulses (of width τ ) are completely separated. This implies that the distance rδ between the two targets respects the following condition:

2rawcr r τδ ≥ ∆ = (10)


In another way, the range resolution of the raw data is clearly given by the spatial extension of the rect(·) function in eq. (8):

2rawcr τ

∆ = . (11)

that is exactly the eq. (10).

Accordingly, very short pulse durations τ are needed (τ ≈10-8 ÷10-7 s) to achieve a range resolution of some meters (c being approximately 3x108 m/s), see Table I. Improvement of the range resolution requires a reduction of the pulse width, and high peak power for a prescribed mean power operation. A way to circumvent this limitation is to substitute the short pulses by modulated long ones, provided that they are followed by a processing step (usually referred to as pulse compression). To this end, a very popular linear frequency modulation is commonly adopted; in this case p(t) in eq.(1) becomes:

2

( ) exp2tp t j α⎡ ⎤

= −⎢ ⎥⎣ ⎦

(12)

where α is called chirp rate; it can be shown that for large value of ατ2 we

Figure 2 - Raw data geometric resolution.

a

bc

d

Chapter 1 11 have:

2 fατ π= ∆ (13)

being ∆f the transmitted pulse bandwidth (in [Hz]).

In this case, ftx(t) in eq. (1) becomes the so-called chirp pulse, the real

part of which is depicted in Figure 3, and eq. (8) can be rewritten as follows:

[ ]

( ) ( )

2

22

( , , ) ( , )rect ,2

4 2exp exp


c

j r R j r r Rc

γτ

π αλ

′⎡ ⎤− − ∆′ ′ ′− − = −⎢ ⎥⎣ ⎦

⎛ ⎞ ⎡ ⎤′− + ∆ − − ∆⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦

(14)

Pulse compression consists of convolution of eq. (14) with the reference function:

( ) 2

2

2exp rect2

rrf r j rc cα

τ′⎡ ⎤⎡ ⎤′ ′= ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

(15)

It is easy to show [Franceschetti and Lanari, 1999] that this step allows obtaining the following resolution (being τ1≅∆f ):

Figure 3 - Chirp waveform. Arbitrary units (α >0).


2 2c c cr

fτ π

ατ∆ = = =

∆ (16)

whose extent is some meters long, if large pulse duration (τ ≈10-6 s) is coupled with very large chirp rate (α ≈1014 rad·s-2). A more detailed analysis of this processing step, along with an analysis in the Fourier domain, is performed in Appendix A.

Azimuth resolution of the received (i.e., raw) data is now addressed. By considering the two targets labeled as “c” and “d” in Figure 2, it is

clearly visible that along the direction orthogonal to the radar beam, the sensor obtains its resolution through the physical dimensions of antenna aperture. Indeed, according to eqs. (8) and (14), two targets at a given range can be resolved only if they are not within the radar beam at the same time. Accordingly, the azimuth resolution of the raw data coincides with the antenna azimuth footprint X related, in turn, to the antenna beam width λ/L by means of the relation:

rawx X r

Lλ

∆ = = (17)

where L is the (effective) antenna dimension along the azimuth direction, see Figure 4. Equation (17) represents the resolution limit of a conventional side-looking Real Aperture Radar, commonly referred to as RAR. To have an idea of the achievable azimuth resolutions let us apply eq. (17) to the European Remote Sensing (ERS) sensor parameters collected in Table I: the azimuth resolution is of the order of kilometers and this is not acceptable for high resolution image radar. On the other side, airborne sensors (r ≈1÷10 km, and L≈1 m) may achieve azimuth resolution of the order of hundreds of meters, which is neither acceptable. To improve the azimuth resolution we must reduce the wavelength of the carrier frequency and/or increase the antenna dimension. The former is constrained by system characteristics. The latter is not an easy task, unless we implement the synthetic antenna (or aperture): a very large antenna is synthesized by moving along a reference path a real one of limited dimension, see Figure 4; in summary we need to implement a Synthetic Aperture Radar.

The synthesis is carried out by coherently combining the backscattered echoes received and recorded along the flight path. A more detailed analysis of this operation is addressed in Appendix A; however, at this stage we can observe that a second order expansion of the ∆R term in eq. (9) leads to:

Chapter 1 13

( )2

2x x

Rr

′ −∆ ≈ (18)

accordingly, eq.(14) can be rewritten as follows:

[ ]

( ) ( )

2

22

2

( , , ) ( , )rect ,2

4 4 2exp exp exp2


c

x xj r j j r r R

r c

γτ

π π αλ λ

′⎡ ⎤− − ∆′ ′ ′− − ≈ −⎢ ⎥⎣ ⎦

⎛ ⎞′ −⎛ ⎞ ⎡ ⎤′− − − − ∆⎜ ⎟⎜ ⎟ ⎢ ⎥⎜ ⎟⎝ ⎠ ⎣ ⎦⎝ ⎠

(19)

thus exhibiting an interesting characteristic, which is now addressed. The second exponential term in eq. (19) is similar to the chirp term of the third exponential in the same equation; this suggests, also for the azimuth case, a processing procedure, usually called focusing, similar to that considered for the range direction, aimed at improving the azimuth resolution (17).

By properly focusing the raw data, it can be demonstrated that the theoretical achievable azimuth resolution is:

2Lx∆ = (20)

where L is also in this case the azimuth antenna length.

This apparently surprising result (the azimuth resolution is independent from the target-to-sensor distance), leading to an improvement on the

Figure 4 - Synthetic aperture concept.


achievable azimuth resolution, can be explained by observing that the extension of the synthesized array is equal to the antenna azimuth footprint, see eq. (17). Therefore, it leads to a theoretical very short aperture beam for the synthetic antenna, expressed by 2Xλ , where factor 2 comes out by considering that only one array element is present at each time per each received echo. By the consequence, the relevant azimuth resolution can be expressed as:

2 2Lx r

Xλ

∆ = = (21)

which is eq. (20). As a result, the azimuth resolution distance is only a function of the

(effective) antenna dimension along the azimuth direction (that is usually of the order of some meters). Similarly to range pulse compression, also this procedure is deeply investigated in Appendix A.

In summary, a SAR sensor permits to obtain a high resolution radar image by using wideband pulse signals and by properly processing (pulse compression and focusing) the received raw data.

Let us now move to consider SAR raw data collected in the presence of an extended scene.

In the case of a continuous scatterer distribution, described by a reflectivity pattern γ(x,r) proportional to the ratio between backscattered and incident field, the signal collected onboard, i.e., the raw data, can be obtained from eq. (8) by superimposing all the elementary returns from the illuminated surface, hence:

( ) ( )

( ) [ ]2

4 4, , exp exp

2 rect ,2

h x r dx dr x r j r j R

r r Rp r r R w x x rc c

π πγλ λ

τ

⎛ ⎞ ⎛ ⎞′ ′ = − − ∆ ×⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

′⎡ ⎤− − ∆⎡ ⎤′ ′× − − ∆ −⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

∫∫ (22)

Equation (22) can be recast in most compact form:

( ) ( ) ( )4, , exp , ,h x r dx dr x r j r g x x r r rπγλ

⎡ ⎤′ ′ ′ ′= − − −⎢ ⎥⎣ ⎦∫∫ (23)

where:

Chapter 1 15

[ ]

( )

2( , , ) rect ,2

4 2exp

r r Rg x x r r r w x x rc

j R p r r Rc

τ

πλ

′⎡ ⎤− − ∆′ ′ ′− − = −⎢ ⎥⎣ ⎦

⎛ ⎞ ⎡ ⎤′− ∆ ⋅ − − ∆⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦

(24)

is the system impulse response (i.e., the return due to an unitary point target). Equation (23) represents the basic functional form of the SAR raw

signal. It exhibits the relationship between the recorded signal, h(·), the reflectivity pattern, γ(·), and the SAR system impulse response g(·), the latter depending on the physical parameters of the SAR system. Equation (23) clearly shows that the SAR imaging problem, aimed at improving the geometric resolutions (11) and (17), can be managed via an appropriate filter operation that recovers an high resolution estimate of the reflectivity pattern γ(·), starting from the received signal h(·).

By assuming a chirp modulation in the following analysis, which implies the use of eq. (12) in eq. (1), the eq. (24) becomes:

( )

2

22

( , , ) rect2

4 2exp exp

r r R x xg x x r r r wc X

j R j r r Rc

τ

π αλ

′ ′⎡ ⎤− − ∆ −⎡ ⎤′ ′− − = ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦⎛ ⎞ ⎡ ⎤′⋅ − ∆ ⋅ − − − ∆⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦

(25)

where the r-dependence of w(·) is now accounted for through the azimuth footprint X term [Franceschetti and Lanari, 1999], see eq. (17). From eq. (25) we obtain the general expression of the SAR raw data (23) when a chirp pulse is transmitted:

( )

( )

2

22

( , ) , rect2

4 2exp exp

r R x xh x r dx dr x r wc X

j R j r Rc

γτ

π αλ

′ ′⎡ ⎤− −⎡ ⎤′ ′ = ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦⎛ ⎞ ⎡ ⎤′⋅ − ⋅ − −⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦

∫∫

The focusing procedure, concisely described in the Appendix A for a simplified case, leads to the following SAR image (also referred to as Single Look Complex, SLC) expression:


( ) ( )

( ) ( )

4ˆ , , exp

sinc sinc

x r dx dr x r j r

r r x xr x

πγ γλ

π π

⎛ ⎞′ ′ = − ×⎜ ⎟⎝ ⎠

⎡ ⎤ ⎡ ⎤′ ′× − −⎢ ⎥ ⎢ ⎥∆ ∆⎣ ⎦ ⎣ ⎦

∫∫ (26)

where: • the sinc(·) function, whose role will be essential to estimate the

achievable spatial resolution of a SAR system, is the so-defined one:

( ) sin xsinc xx

= and is characterized by a 3dB-main lobe aperture of

about π;

• 2Lx∆ = and

fcr∆

=∆2

are the achievable azimuth and range

resolution expressions, respectively, as already shown before.

1.1.3 Resolution consideration To verify the assertions concerning the achievable resolutions, we may

observe that, if we refer to a scene characterized by a single scatterer located at the SAR coordinates ( )PP rx , , the scene reflectivity function can be expressed as ( ) ( ) ( )PPP rrxxrx −−= '', δδγγ and, the estimated reflectivity value, with respect to the equation (26), will be:

( ) ( ) ( )4ˆ , exp sinc sincP P P Px r j r r r x xr x

π π πγ γλ

⎛ ⎞ ⎡ ⎤ ⎡ ⎤′ ′ ′ ′= − − −⎜ ⎟ ⎢ ⎥ ⎢ ⎥∆ ∆⎝ ⎠ ⎣ ⎦ ⎣ ⎦ (27)

The information associated to the single scatterer is now spread around its true position over an area corresponding to the 3dB main-lobe aperture of each ( )⋅sinc function. Hence, the spatial resolutions can be estimated via the observation that two different targets are, in any case, discernible if their spatial separation is smaller than the spatial main lobe aperture of the relevant ( )⋅sinc function, that is:

1 2 1 2 2

cr r r r rr f

π π− ≥ → − ≥ ∆ =∆ ∆

(28)

1 2 1 2 2

Lx x x x xx

π π− ≥ → − ≥ ∆ =∆

(29)

Chapter 1 17

The range and azimuth resolutions of the focused SAR image are now compared to the corresponding raw resolutions.

Accordingly, by considering eq. (28) and eq. (11) we obtain the so called range compression ratio:

2

2 1

raw

rr f

πατ τ

∆= =

∆ ∆ (30)

therefore, use of chirp rate large enough to have ατ2>>2π, which is the case for all SAR systems, allows significant improvement in the range resolution of the focused SAR image with respect to the unfocused raw data h(x′,r′). By applying eq. (30) to ERS sensor parameters we obtain ∆r/∆rraw= 1/570 (see Table I), and an improvement of the geometric resolution of two-three orders of magnitude is thus achieved in the range direction.

For what concerns the azimuth resolution, by comparing eq. (29) with eq. (17), we obtain the azimuth compression ratio:

2

2 2raw

x L Lx X rλ∆

= =∆

(31)

accordingly, the improvement in the azimuth resolution of ( )ˆ ,x rγ ′ ′ , with respect to h(x′,r′), depends on the ratio between the physical (L) and the synthetic (X) antenna dimension. For all the SAR systems the condition 2λr/L2= 2X/L>>1 is well satisfied. By applying eq. (31) to ERS sensor parameters we obtain ∆x/∆xraw= 1/891 (see Table I), and an improvement of the geometric resolution of two-three orders of magnitude is thus achieved also in the azimuth direction. Note that, according to eq. (29), ∆x can be smaller by adopting airborne sensors, which use smaller antenna dimensions.

As a final remark we highlight that a focused SAR image can be expressed via the equation (26) and managed as a complex-valued matrix, where, by first approximation, each resolution cell corresponds to a pixel on the SAR image. More precisely, as suggested by Nyquist condition, the pixel dimensions will be, generally, smaller than the theoretical resolutions and they will account for the range sampling frequency (fsamp) and the Pulse Repetition Frequency (PRF), respectively, that is

2 2

2

PIXELsamp

PIXELazimuth

c cr rf f

L v vx xB PRF

∆ = ≥ = ∆∆

∆ = = ≥ = ∆∆

(32)


where 2azimuthB v L∆ = is the azimuth bandwidth. For example, if we refer to the ERS parameters collected in Table I, we have 7.91PIXELr m∆ = (which corresponds to a ground pixel spacing of about 20 m) and 4.22PIXELx m∆ = . In Figure 5 a focused ERS SAR amplitude image relevant to the Cagliari bay area (Italy) is shown, as example of a SAR focused image.

Figure 5 – Amplitude of a focused descending ERS SAR image relevant to the Cagliari bay area (Italy).

Chapter 1 19

1.2 Mapping scene elevation: Digital Elevation Model generation

According to the analysis of Section 1.1, a focused SAR acquisition, obtained after the implementation of a proper processing procedure, allows us to obtain a 2-D image (see eq. (26)) whit a resolution along range and azimuth directions expressed by eqs. (28) and (29), respectively. However, it is evident that knowledge of the target range coordinate, say r, is not sufficient to uniquely determine target location and therefore its height above a reference plane.

This is clearly shown in Figure 6a, where the SAR geometry in the plane orthogonal to the azimuth direction is depicted: all the targets within the range beam and located on an equidistance curve are imaged at the same range position r' ≈ r. In other words, a two dimensional focused SAR image does not allow accessing to the third dimension, i. e., ϑ in the usual cylindrical reference system introduced in Section 1.1.

Such a limitation can be overcome if we consider a second image obtained with a sensor which observes the same scene from a different position (see Figure 6b). The spacing b between the two sensors is usually referred to as baseline; the angle β between the vector connecting the first sensor to the second one and the horizontal direction is referred to as tilt angle, see again Figure 6b.

The two images can be either obtained by means of a single bistatic

system with two (one active and one passive) imaging sensors (single-pass imaging system), or with two repeat passes of a single (active) imaging

Figure 6 - Single-imaging (a) and Stereo-imaging (b) sensor geometry in the plane orthogonal to the flight direction.


sensor system (repeat-pass imaging system). In this latter case the properties of the scene relevant to the sensing system must remain unchanged within the time frame of the two passes.

By using the additional information represented by the target range from the second antenna, the ambiguity existing on target location is completely solved. As matter of fact, only one point exists which is located at distance r from the first system and r+δr from the second one.

This is equivalent to say that knowledge of ϑ comes from knowledge of both r and r+δr . To better clarify this point, we observe that (see Figure 6b):

( ) ( )2 2 2 2 sin .r r r b brδ ϑ β+ = + − − (33)

From eq. (33) we get:

( ) ( )2 2 2 sin sinr r b br r bδ ϑ β ϑ β= + − − − ≈ − − (34)

where the last approximation holds when baseline is small compared with the target slant range, as it is always the case; δr thus reduces to the baseline projection onto the look direction (parallel ray assumption). According to the approximation in (34), knowledge of the path difference δr (and not necessary of the distances r and r+δr ) allows us calculating the third dimension ϑ of the considered target, provided that both b and β, describing sensor positions, are known. Moreover, it is easy to show that knowledge of ϑ allows calculating the target height above the reference plane, i.e., its third dimension in a reference system different from that introduced in previous section, by means of the following equation:

cosz H r ϑ= − (35) where H is the height of the first antenna (S1 in Figure 6b) above the reference plane, see Figure 6b.

According to the shown analysis, we can say that knowledge, or better, measurement, of the path difference δr represents a key point to determine target height by using SAR images acquired via a stereo-imaging geometry.

Hence, the height measurement accuracy is limited by the knowledge uncertainty of the path difference δr. To check how this uncertainty generates a corresponding error in the height evaluation, we can write, differentiating eq. (35) and according to eqs. (33) and (34):

( ) ( ) ( )sin

cosz r r rz z rr r bδ δ δ

ϑ ϑσ σ σ σδ ϑ δ ϑ β∂ ∂ ∂

= = ⋅ ≈∂ ∂ ∂ −

(36)

Chapter 1 21

Therefore, the height resolution depends on the parameter r/b, which is very large in all SAR systems. This implies that errors on the evaluation of δr are strongly amplified when they are transferred to height measurements. Moreover, we observe that the larger the baseline b (or better, its projection onto the direction orthogonal to the look angle, ( )cosb b ϑ β⊥ = − ) the

smaller the impact of δr measurement inaccuracies on the height evaluation accuracy. Therefore, in order to generate a Digital Elevation Model (DEM) of an imaged area it is essential to complain with large baseline values.

Let us now focus on different methods to evaluate the δr path difference.

1.2.1 Stereometry A method (stereometry) to calculate the path difference rδ is easily

derived. We consider a point in the first focused SAR image, hereafter referred to as master, with range coordinate1 mr r′ ≈ . Then, we search for it in the second focused SAR (slave) image. In this way we are able to evaluate its range from the second sensor, i.e., sr r rδ′ ≈ + . This method is very sensitive to errors in the knowledge of the range difference. Indeed, also neglecting possible orbital parameters inaccuracies, the range difference measurement precision rδσ essentially depends on the range system resolution, which is on the order of some meters.

If we reasonably suppose to be able to discriminate range displacements of 1/16th of the pixel spacing, which is equal to ( )2 sampc f , we will have for

a C-band sensor 0.5r mδσ = . In turn, by referring to eq. (36) and considering ERS parameters (see Table I), since for satellite-borne SAR systems the ratio r b is very large, for a typical slant range distance of about 800 km, a

23ϑ ≈ ° look angle and 100 m baseline length, the achievable height accuracy will be of about 4 km: this value is clearly unacceptable.

The stereometric accuracy can be improved by increasing the baseline value; however, in this case, the backscattered field from a resolution cell may have large variations when it is imaged from two very different viewing angles, thus having an impact on the identification of the corresponding pixels in the two images. Let us now move to a different method to evaluate

1 Note the important difference between unprimed and primed coordinates. Unprimed coordinates refer to the points on the scene site, whereas primed coordinates refer to the image (i.e., sampled) points.


the path difference, based on considerations on phase difference between two SAR images.

1.2.2 SAR Interferometry An alternative way to measure δr, aimed at improving the accuracy of

target height evaluation, is carried out by analyzing the phase difference (Interferogram) of two focused images. This idea is exploited in interferometric SAR (InSAR) systems [Franceschetti and Lanari, 1999; Bamler and Hartl, 1998; Madsen et al., 1993; Zebker and Goldstein, 1986; Graham, 1974] briefly shown in the following.

According to the analysis of Section 1.1, the first focused SAR image, i.e., the master one, may be represented as follows:

( ) ( )

( ) ( )

14ˆ , , exp

sinc sinc

x r dx dr x r j r

r r x xr x

πγ γλ

π π

⎛ ⎞′ ′ = − ×⎜ ⎟⎝ ⎠

⎡ ⎤ ⎡ ⎤′ ′× − −⎢ ⎥ ⎢ ⎥∆ ∆⎣ ⎦ ⎣ ⎦

∫∫ (37)

Similarly, the slave image is given by:

( ) ( ) ( )

( ) ( )

24ˆ , , exp

sinc sinc

x r dx dr x r j r r

r r r x xr x

πγ γ δλ

π πδ

⎛ ⎞′ ′ = − + ×⎜ ⎟⎝ ⎠

⎡ ⎤ ⎡ ⎤′ ′× − − −⎢ ⎥ ⎢ ⎥∆ ∆⎣ ⎦ ⎣ ⎦

∫∫ (38)

In eqs. (37) and (38) the usual cylindrical reference system, the axis of which coincides with the flight trajectory of the master antenna, is used.

For sake of simplicity, let us assume that system resolutions are infinite, i.e., 0x∆ → and 0r∆ → . In this case, sinc(·) functions in azimuth and range approach the Dirac one, leading to:

( ) ( )14ˆ , , expx r x r j rπγ γλ

⎛ ⎞′ ′ ′ ′ ′≈ −⎜ ⎟⎝ ⎠

(39)

( ) ( )24ˆ , , expx r x r r j rπγ γ δλ

⎛ ⎞′ ′ ′ ′ ′ ′≈ − −⎜ ⎟⎝ ⎠

(40)

where ( )rrrr ′==′ δδ . From eqs. (39) and (40) the range displacement (misalignment) of the

two images is evident: this is direct consequence of the existing path

Chapter 1 23 difference. Accordingly, the first necessary step is a proper image registration [Costantini and Rosen, 1999; Curlander and McDonough, 1991; Bamler and Einder, 2005; Fornaro and Franceschetti, 1995; Lin et al., 1992], implying for the slave image the following new expression:

( ) ( ) ( ) ( )⎥⎦⎤

⎢⎣⎡ ′+′−′′=′+′′→′′ rrjrxrrxrx δ

λπγδγγ 4exp,,ˆ,ˆ 22 (41)

Then, the second step consists of phase extraction, which is implemented via the product between the master image and the complex conjugate of the other:

( ) ( ) ( )2

*1

4ˆ ˆ, Ph , ,x r x r x r r rπϕ γ γ δ δλ

⎡ ⎤′ ′ ′ ′ ′ ′ ′ ′= + =⎣ ⎦ (42)

where Ph[·] is the operator which gives the full phase, i.e., the phase not restricted to the ]-π, π] interval. We must note that complex data allow only to measure phase differences ϕm restricted in the ]-π, π] interval, while the total ϕ variation largely exceeds this range in most (if not in all) practical cases. Techniques, referred to as Phase Unwrapping (PhU) in the literature [Pepe and Lanari, 2006; Costantini and Rosen, 1999; Fornaro et al., 1996; Ghiglia and Romero, 1994; Ghiglia and Pritt, 1998; Goldstein et al., 1988], which allow reconstruction of the true phase (unwrapped), ϕ, starting from knowledge of the wrapped phase, ϕm, are then needed.

Equation (42) provides the desired interferometric phase, i. e., it represents the Interferogram. Accordingly, in InSAR systems, the information on the path difference δr' is obtained by considering phase differences of target responses in the two SAR images.

Let us now address to the obtainable accuracy on the path difference estimation carried out via SAR Interferometry.

From eq. (42) it is clearly visible that rδσ depends on phase accuracy, indeed:

4rϕ δ

πσ σλ

= (43)

By substituting eq. (43) in eq. (36) we obtain:

( ) ( )sin sin

cos cos 4z rr r

b bδ ϕϑ ϑ λσ σ σ

ϑ β ϑ β π≈ =

− − (44)


stating that height accuracy is related to phase error. Note that, in this case factor r b is reduced by the transmitted signal wavelength that is of the order of a few centimeters (in the microwave region).

For a typical ERS configuration, with ' 800r km= , 100b m⊥ = , °= 23'ϑ , cm56.5=λ , 20ϕσ = ° , the achievable accuracy gives an uncertainty on the

height measurement of about 5m, which is significantly less with respect to the stereometry case.

Figure 7 represents an Interferogram relevant to Mt. Etna. By substituting eqs. (34) and (35) in eq. (42) we obtain the simplified phase – topography relation:

4

sinb z

rπϕλ ϑ

⊥≈′ ′

(45)

Figure 7 – ERS interferogram relevant to Mt. Etna (Italy). Perpendicular baseline is of about 125m, temporal baseline 1 day. Each color cycle corresponds to about 70m topography variation.

Chapter 1 25

For ERS configuration shown in Figure 7 ( 125b m⊥ = ) it is clearly visible that a 2π phase variation, i. e., an interferometric fringe, corresponds to about 70m topography variation.

Note that in eq. (45) we assumed to have removed the Flat-Earth phase component expressed by:

( ) ( )04', ' ' '' tan 'flat

bx r r rr

πϕλ ϑ

⊥= − − (46)

imposing the interferometric phase equal to zero in a given pixel of SAR coordinates ( )00 ',' rx . Flat-Earth term is due to the looking configuration and completely depends on the acquisition geometry ( , ,b r ϑ⊥ ′ ′ ). Flat-Earth is an approximation for the (actual) phase variation due to the target-to-sensor distance increase along the range direction, supposing the Earth to be flat (note that an interferometric phase term is present also in absence of topography). It can be demonstrated that a Flat-Earth hypothesis, avoiding to consider Earth curvature, is perfectly acceptable for InSAR applications [Franceschetti and Lanari, 1999]. This term, in the following assumed to be exactly compensated, can be easily computed via geometric considerations. By the consequence, wrong orbit parameters will produce wrong Flat-Earth

Figure 8 – Interferometric Digital Elevation Model of Mt. Etna (courtesy of IRECE/DLR)


estimation, introducing residual phase ramps in interferograms. This possibility will be addressed in Chapter 2.

Finally, after the previous mentioned Phase Unwrapping step we are able to provide the DEM of the analyzed scene, see Figure 8

1.2.3 Interferometric Phase Statistics In the following, in order to establish which values of the phase standard

deviation could be considered, we will discuss how SAR image noise affects phase measurements.

First of all, we address the effects of the thermal noise and, accordingly, we add to the SAR image expressions (see equations (39) and (41)) the noise components which are supposed to be mutually incoherent

( ) ( ) ( )

( ) ( ) ( ) ( )

4 '

1 1

4 ' '

2 2

', ' ', ' ', '

', ' ', ' ', '

j r

j r r

I x r x r e n x r

I x r x r e n x r

πλ

π δλ

γ

γ

−

− +

= +

= +

(47)

To introduce a measure about the phase quality, its is useful to refer to the cross-correlation factor, defined as follows

( )*

1 2

2 21 2

expE I I

E I E Iχ χ ϕ

⎡ ⎤⋅⎣ ⎦= =⎡ ⎤ ⎡ ⎤⋅⎣ ⎦ ⎣ ⎦

(48)

wherein the [ ]E ⋅ is the mean statistical operator. By assuming ergodic conditions, these operations are substituted by spatial averages. In ERS case, the choice of the optimal averaged boxes is obtained taking account for the original asymmetry between azimuth and ground range image resolutions. Therefore, there is a ratio of about 1/5th between the two spatial resolutions, so that a profitably choice would require an averaged spatial box with

20azN = azimuth lines and 4rgN = range samples, leading to an averaged resolution cell, which is approximately squared with a side of about 100m .

The average procedure, usually referred to as multilook operation, leads to an improvement of the interferometric phase standard deviation, as outlined in the following. With regard to the multilooked phases, we also want to stress that, if we want to estimate the phase from N independent interferogram samples, the maximum-likelihood estimator (MLE), which provides the averaged phase difference for distributed, homogenous targets, will be

Chapter 1 27

( ) ( )

( ) ( ) ⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

=Φ

∑

∑

=

=− N

jjj

N

jjj

lookN

PIPI

PIPI

121

121

*Re

*Im

arctan (49)

wherein Pj is the j-th SAR pixel within the averaged box and the operators [ ]Im ⋅ and [ ]Re ⋅ stand for the extraction operations of the imaginary and the

real part of a complex variable, respectively. The basic procedure is pictorially outlined in Figure 9 where both a single look and a az rgN N N= × -multilooked phase interferogram have been shown. Although the evaluation of the statistics of the multilooked SAR images is outside the scope of this work and can be found in Zebker and Villasenor [1992], Just and Bamler [1994] and Lee et al. [1994], the statistics relevant to the interferometric phase pattern will be presented in the following.

We may observe that cross-correlation factor can be decomposed in amplitude and corresponding phase term, respectively. The amplitude, known as coherence, ranges between [0,1] and accounts for the similarity of the two images, while phase corresponds to the multilooked interferometric phase, directly. A zero coherence stands for a completely uncorrelated scene, whereas a coherence close to 1 corresponds to a noise-free interferogram. In the simplified case of equation (47) it is particularly easy to express the coherence as a function of the signal to noise ratio (SNR) [Franceschetti et al., 1995]

1+

=SNR

SNRχ (50)

Being interested on the standard deviation interferometric phase

estimation, we could relate it to the evaluated coherence value. To this end, let us start from the expression of the single-look interferometric probability density function:

( )( )

( ) ( )( )( ) ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−

−−−+⋅

⋅−−

−=−

022

00

02

2

cos1

cosarccoscos1

cos11

21

ϕϕ

ϕϕϕϕ

ϕϕπϕ

k

kk

kkpdf LookSingle

(51)


where 0ϕ is the actual (noise-free) phase and k is the coherence value (see Figure 10a). From the observation of its shape, we note that the phase distribution is less dispersed when the coherence approaches to 1. It is clear that the higher is the coherence, the more concentrated around its expectation value is the phase distribution, thus suggesting lower value of the standard deviation. The latter can be also analytically expressed (unless only for the single-look case) as follows [Tough et al., 1995]

( ) ( ) ( )2222arcsin arcsin

3 2

Li kk kϕ

πσ π= − + − (52)

where 2Li is the Euler’s dilogarithm. In the case of N-multi looked interferograms, the corresponding pdf will

be analytically estimated from Montecarlo simulations or calculated by [Joughin et al., 1994; Bamler and Just, 1993]:

Figure 9 - Pictorial representation of the N-multilook operation. Presented interferograms are relevant to the Napoli (Italy) bay area and have been generated from two ERS descending acquisitions (note that they are differential interferograms thus the fringes are related to the deformation of the considered area).

Chapter 1 29

( )

( ) ( )

( ) ( )( )( ) ( )⎟

⎠⎞

⎜⎝⎛ −

−+

+−−Γ

−−⎟⎠⎞

⎜⎝⎛ +Γ

=+

−

022

12

2

21

022

02

cos;21;1,

21

cos12

cos121

ϕϕπ

ϕϕπ

ϕϕϕ

kNFk

kN

kkNpdf

N

N

N

lookN

(53)

where 2 1F is the hypergeometric function. The important (and expected) result is that the pdf becomes narrower at

the increase of the used number of looks. Unfortunately, an exact, analytical expression for the phase standard deviation, in this case, is not available, unless for the Cramer-Rao bound case (whose validation is nevertheless limited to the high coherence region)

Figure 10 - (a) Single look phase interferogram pdf as a function of the spatial coherence. (b) Single look phase standard deviation vs. coherence and (c) its variation with respect to the look number.


Nk

Nlook2

1 2−=−Φσ (54)

An empirical relation between the phase standard deviation and the coherence value (for different number of looks) can be however obtained numerically as shown in Figure 10 [Franceschetti et al., 1995]. This kind of representation may be effectively used to estimate the achievable measurement accuracy; indeed, for typical values of coherence of about 0.3 the corresponding single-look phase standard deviation will be of about π/2 which, properly substituted in the equation (44) and for typical ERS values

100b m= , ' 800r Km= , cm56.5=λ and °= 23'ϑ , will correspond to a topographic height accuracy of about 22m .

1.2.4 Decorrelation Effects At this stage, we can address the origin for the coherence decrease.

Basically, the cross-correlation factor will depend on the different noise sources and can be profitably factorized as the product of the cross-correlation factors relevant to each single noise source [Zebker and Villasenor, 1992] as follows

4 'j r

thermal temporal spatial doppler misregistration eπ δλχ χ χ χ χ χ= ⋅ ⋅ ⋅ ⋅ ⋅ (55)

We will focus on each single factor, thus exploring the SAR parameters that are responsible for them.

First of all, we have to refer to the general expression of a SAR image (see eqs. (37) and (38)) in order to investigate on the origin of the decorrelation sources and, for sake of simplicity, let us now neglect both the effect of the different reflectivity functions (responsible for the temporal decorrelation effect) and thermal noise contributions, respectively. Following these assumptions, as demonstrated in Franceschetti et al. [1995], the expression of the cross-correlation factor (see eq. (55) and Appendix B), if no mis-registrations have occurred, is:

( )4 '

1 224tan '

j rd d

at

bre r x Lr

π δλ ξ ξχ

λ πϑ⊥

⎧ ⎫⎡ ⎤ ⎧ ⎫−∆⎪ ⎪ ⎡ ⎤= ⋅ ∆ ⋅ Λ ⋅ ∆ ⋅ Λ⎢ ⎥⎨ ⎬ ⎨ ⎬⎢ ⎥′ − Ω ⎣ ⎦⎩ ⎭⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭(56)

wherein the tτ

⎛ ⎞Λ ⎜ ⎟⎝ ⎠

function stands for a triangle pulse of width τ , which is

factorized in two distinct contributions. The former is responsible for the

Chapter 1 31 spatial decorrelation effect that is typical of SAR interferometry. Indeed, the same ground resolution cell is imaged from two slightly different looking directions. Change of look angle leads to a shift between the two SAR images range spectra [Franceschetti and Lanari, 1999]. For this reason, if the perpendicular baseline increases, the corresponding spectral shift could be responsible for a complete non overlap between the two range spectra, thus leading to the complete decorrelation case. Mathematically, it happens when

( ) ( )tct

c rr

br

cfr

bcΩ−

∆=→

∆=∆=

Ω−⋅ ⊥

⊥ 'tan'22'tan'

22

ϑλϑλ

(57)

which corresponds to the value obtained by imposing that the relevant contribution on the cross-correlation factor expression (see equation (56)) is equal to zero. The cb⊥ term is usually referred to as critical baseline, i. e., the maximum baseline value to avoid spatial decorrelation. For the ERS satellites critical baseline value is of about 1200m for a flat terrain, although baselines greater than 600 m are very difficult to be used. As an example, in Figure 11 direct comparison between an uncorrelated differential interferograms (where the fringes are related to the deformation, as explained in the following) and a coherent one has been also presented. It is clearly visible the spatial baseline impact on the spatial decorrelation phenomena.

Last factor of eq. (56) accounts for the Doppler decorrelation effects. It

Figure 11 - Multilook SAR image of the Napoli (Italy) bay area (left), spatially correlated differential interferogram (center) and a decorrelated one (right).


is due to the presence of a difference between the two SAR image Doppler centroids, 1 2,d dξ ξ respectively, which is equivalent to an azimuth spectral shift. No spectral overlap is present when:

1 24

d dLπξ ξ− > (58)

The previous two explored noise sources can be effectively treated in the Fourier transform domain, being both responsible for a corresponding spectral shift, as pictorially represented in Figure 12. Let us now investigate another decorrelation factor.

When two images are not taken simultaneously but at a certain time interval from each other, as usual for a satellite InSAR system, the spatial baseline and the viewing geometry remain the same, spatial decorrelation does not occur. However, the reflectivity function change between two sensor passages over the illuminated area can not be negligible; in particular it introduces another source of decorrelation, called temporal decorrelation. Indeed, the backscattering characteristics of a terrain depend on several factors, such as for example the terrain composition, moisture, roughness,

Figure 12 – Pictorial representation of two-dimensional spectra relevant to two generic SAR images involved in an interferometric data pair.

Chapter 1 33

etc. and if any of these factors is different in the two images, then also the reflection from the terrain will be different. Depending on the amount of backscattering change, the interferometric combination will be more or less decorrelated.

Temporal decorrelation is very difficult to be statistically modeled and is usually associated to weather changes that modify the electromagnetic response of the ground. It can be also due to the human activities (we refer to rural areas, for example) and is strongly dependent on the used wavelength. Such decorrelation phenomenon affects different kind of land coverage on different time scales, depending on how fast they are likely to change (some of) their properties. For example, water decorrelates in fractions of a second, due to its fast surface movements, while vegetated areas decorrelate in few days, due to the vegetation growth, and fields usually in a few months or less, depending on the farming activities. The most slowly decorrelating terrains are deserts in non windy conditions: for these areas, interferograms can be obtained also on time scales of one or more years. Urban zones also preserve coherence along time even with very large temporal baselines. Coherence maps (see Figure 13) are useful to get one estimate of this type of decorrelation and are often used to generate thematic maps, for instance, vegetation coverage and lava flow characterization in volcanic areas.

Finally, let us give a few words on decorrelation due to misregistration. Misregistration errors induce a decrease in the cross-correlation index

Figure 13 – Coherence maps relevant to two interferograms characterized by a similar spatial baseline and by a temporal baseline of about 35 (a) and 1040 (b) days; note that decorrelation increases passing from (a) to (b).


amplitude. Complete decorrelation is generated when range and/or azimuth misalignment equals range and azimuth resolution, respectively.

1.3 Mapping small elevation changes: Differential SAR Interferometry

An interesting extension of repeat-pass InSAR is a relatively new technique, referred to as Differential SAR Interferometry (DInSAR) [Gabriel et al., 1989], which exploits the phase difference between SAR image pairs acquired at different times in order to detect at centimeter scale [Massonet et al., 1993; Massonet et al., 1995] relative displacement occurred in the illuminated area between two different acquisitions. DInSAR basic principles are hereafter outlined.

Let us refer to Figure 14 where the two-pass DInSAR geometry in the plane orthogonal to antennas trajectories is depicted. Suppose that a ground target displacement d takes place in between the two passes.

As in Section 1.2.2, let us assume that system resolutions are infinite, so that image points coordinates are coincident with scene points coordinates; i.e., primed coordinates are coincident with unprimed ones.

The target range in the master and in the slave SAR images are indicated with 1r and 2r , respectively, while r% is the one relevant to the same site, but with absence of surface displacements. The interferometric phase is now

Figure 14 – DInSAR geometry.

Chapter 1 35 given by:

2 1

4 4r rπ πϕλ λ

= − (59)

which can be rewritten as:

( ) ( )2 14 4

d d tr r r r r rπ πϕ δ δ ϕ ϕλ λ

= − + − = + = +% % (60)

The contributions to the path difference due to target displacement, drδ , and to topography height profile, δr , have been distinguished. Note that the latter is the path difference in absence of any ground displacement.

Let us now suppose ideally that the two passes are repeated exactly on the same orbit, i.e., b=0. In this case the topographic contribution is equal to zero and the interferometric phase ϕ is related only to δrd, as follows:

( )4 4 sind d dr lπ πϕ δ ϑ α

λ λ= ≈ − (61)

where the last approximation holds when the displacement d is small compared with the target slant range, see Figure 14. Equation (61) shows that δrd is equal to the displacement component parallel to the look direction, usually referred to as line of sight (LOS) displacement component.

Accordingly, in this ideal situation DInSAR technique allows measuring such a component of the displacement occurred between the different acquisitions, with accuracy on the order of wavelength fractions: indeed a differential phase change of 2π is associated to a LOS displacement of λ/2.

Let us now move to the real situation, which requires to address the topographic contribution δr.

The assumption b=0 is not realistic, so that separation between topographic and displacement contributions in eq. (60) must be carried out. To this end, we can use eqs. (34) and (35) introduced in Section 1.2, provided that r→r' ϑ→ϑ'.

Differently from Section 1.2.2, we now start from given, i.e., known, DEM of the area which provides us topography information z. By assuming knowledge of sensor orbits, i. e., of H, b and β, we first evaluate ϑ′ via eq. (35) by the knowledge of r; then we use this information in eq. (34) to evaluate δr. This allows us reconstructing the so called synthetic interferogram, given by:


4t rπϕ δ

λ= . (62)

Accordingly, subtraction of the synthetic interferogram (62) from the interferometric phase of eq. (60) leads to:

4d dr

πϕ δλ

= (63)

which allows us, after proper phase unwrapping, measuring the component of the displacement occurred in the illuminated surface between the different acquisitions, again with an accuracy of the order of fractions of wavelength. In Figure 15 an ERS differential interferogram relevant to Mt. Etna is provided. Note that each color fringe correspond to a 2.8cm phase variation.

Additional considerations on the DInSAR technique are now in order. First of all, we observe that the accuracy of the surface displacement

Figure 15 – ERS differential interferogram relevant to Mt. Etna (Italy). Perpendicular baseline is of about 30m, temporal baseline is ~1 year. Each color cycle corresponds to about 2.8cm LOS displacement variation.

Chapter 1 37 measurement is strongly related to the achievable accuracy in the generation of the synthetic interferogram of eq. (62), which, in turn, depends on the accuracy of the available DEM. To check how DEM errors generate a corresponding error in the synthetic interferogram generation, we must use again eq. (36). However, in this case, eq. (36) should be reversed because we want to ascertain the error induced on δr by the assumed uncertainty on z.

Accordingly, height measurement errors are strongly reduced when they are transferred to δr. By referring to the ERS parameters of Table I, a perpendicular baseline b of 100m and an accuracy on z of 30m imply an accuracy on δr equal to 1cm. Accordingly, this is the error transferred to δrd once the topographic contribution δr has been subtracted. Moreover, differently from InSAR (requiring large baseline for accurate height profiles generation), in DInSAR technique, the smaller the perpendicular baseline b the better is the accuracy of displacement measurements.

Actually, a more realistic expression for a differential interferogram should take into account other signals, representing the main source of deformation phase mis-interpretation. Accordingly, the deformation phase expression is:

_

_

DInSAR deformation res orbit

res topography atmosphere noise

∆Φ = ∆Φ + ∆Φ +

+ ∆Φ + ∆Φ + ∆Φ (64)

where: • ( )deformation∆Φ ⋅ is deformation signal corresponding to the

displacement occurred between the two flights;

• _ 04tan sin

res orbit r r zb br r

πλ ϑ ϑ⊥ ⊥

−⎛ ⎞∆Φ ≅ ∆ + ∆⎜ ⎟′ ′⎝ ⎠ represents the orbital error

residual fringes due to an inaccurate knowledge of the sensor orbit position along the flight path;

• _ 4sin

res topography bz

rπλ ϑ

⊥⎛ ⎞∆Φ ≅ ∆⎜ ⎟′⎝ ⎠ is the residual topographic signal due

to the error in the knowledge of the scene topography and/or to an incorrect DEM re-sampling into SAR coordinates;

• ( )atmosphere∆Φ ⋅ accounts for troposphere inhomogeneities between the two radar sensor passages over the illuminated area and essentially are due to changed weather conditions (see Figure 16). These phenomena cause phase noise usually referred to as atmospheric artifacts. Moreover, atmospheric influence (i.e. delay) depends not only on the weather conditions, but it also considerably


depends on the range, i.e. on the look angle. Indeed, for an increasing look angle, the path of the ray through the atmosphere gets longer. Also, the atmospheric delay depends strongly on topography, which should be significant in mountain areas. Finally, as explained in chapter two, Atmospheric Phase Screen (APS) can be properly filtered out using a convenient spatial-temporal procedure;

• ( )noise∆Φ ⋅ is additive noise contribution. Another source of mis-interpretation about deformation is intrinsic to the

technique itself and it is related to the fact that phase is known only on the restricted [ [,π π− interval. For this reason, a fundamental non linear operation, allowing us to reconstruct the full phase differences, is requested. This operation, usually referred to as Phase Unwrapping (PhU), represents a crucial point in this context.

Phase unwrapping errors are integer multiples of 2π, however they can propagate within the data inversion process, significantly affecting deformation measurements. A more detailed analysis on this topic will be provided in Chapter 5.

Figure 16 - DInSAR geometry (the same of Figure 14) where has been also introduced the effect of possible atmospheric index refraction inhomogeneities, liable for an incorrect estimation of the deformation signal.

Chapter 1 39

It is clear that in order to isolate only the deformation phase contribution, proper processing and assumptions are needed.

Finally, we underline that different algorithms have been recently proposed (for instance, by Berardino et al. [2002] or Ferretti et al. [2001]) and successfully applied, in order to detect and follow the temporal evolution of deformation via the combination of several differential interferograms relative to different temporal acquisitions, with subsequent generation of surface displacement time series. In the next Chapter an overview of the so-called Advanced DInSAR technique, and in particular of the Small BAseline Subset (SBAS) approach [Berardino et al., 2002] will be presented.


Table I

ERS SAR sensor technical specifications

Parameters ERS/1-2

Carrier wavelength (λ ) 5.656 cm (C-band)

Transmitted bandwidth (∆f ) 15.5 MHz

Pulse duration (τ ) 37.1 µs

Chirp rate (α ) 2.62×1012 rad/s2

Pulse Repetition Frequency

(PRF) 1679 Hz

Antenna dimensions (azimuth/range) 11.1m x 1m

Altitude 785 km

Azimuth footprint (X) 100 km

Look angle (ϑ ) 23°

Sensor velocity 7120 m/sec

Range sampling frequency (fsamp )

18.96 MHz

Chapter 1 41

Appendix A SAR focusing outline In this appendix we show the procedure which leads to eq.(26) (referred

in the following as eq.(A1) for sake of clearness):

( ) ( )

( ) ( )

4ˆ , , exp

sinc sinc

x r dx dr x r j r

r r x xr x

πγ γλ

π π

⎛ ⎞′ ′ = − ×⎜ ⎟⎝ ⎠

⎡ ⎤ ⎡ ⎤′ ′× − −⎢ ⎥ ⎢ ⎥∆ ∆⎣ ⎦ ⎣ ⎦

∫∫ (A1)

The focusing procedure requires a proper filtering operation on the received raw data, which can be easily performed in the spatial frequency domain. To this aim, the SAR Transfer Function (TF) must be firstly evaluated.

Starting from eq.(23):

( ) ( ) ( )4, , exp , ,h x r dx dr x r j r g x x r r rπγλ

⎡ ⎤′ ′ ′ ′= − − −⎢ ⎥⎣ ⎦∫∫ (A2)

we can obtain the following expression for the raw data spectrum:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

, , exp exp

, , , exp exp

, exp exp , ,

H dx dr h x r j x j r

dx dr x r dx dr g x x r r r j x j r

dx dr x r j x j r G r

ξ η ξ η

γ ξ η

γ ξ η ξ η

′ ′ ′ ′ ′ ′= − −

′ ′ ′ ′ ′ ′= − − − −

= − −

∫∫∫∫ ∫∫∫∫

(A3)

where:

( ) ( ) 4, , expx r x r j rπγ γλ

⎛ ⎞= −⎜ ⎟⎝ ⎠

(A4)

and:

( ) ( ) ( ) ( ), , ' ' ', ', exp ' exp 'G r dx dr g x x r r r j x x j r rξ η ξ η= − − − − − −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦∫∫ (A5)

is the SAR system TF, i. e., the Fourier Transform (FT) of the system impulse response g(·).

Note that, when the r-dependence of the function g(·) can be neglected, eq. (A2) reduces to the two-dimensional (2D) convolution


( ) ( )', ' ', 'x r g x rγ ⊗ , and, accordingly, the simplified expression for eq. (A3):

( ) ( ) ( ) ( ) ( )

( ) ( )

, , exp exp ,

, ,

H dx dr x r j x j r G

G

ξ η γ ξ η ξ η

ξ η ξ η

= − −

= Γ

∫∫ (A6)

is obtained. In eq. (A6) ( )Γ ⋅ is the FT of ( )γ ⋅ . Unfortunately, the simplification leading to eq. (A6) is not allowed in general, and the direct r-dependence of the function g(·) requires special care when SAR data processing operations are implemented. Use of Stationary Phase Method [Bleistein and Handelsman, 1986] allows calculating the integral (A5), where nonessential amplitude factors have been neglected, as follows:

( )

( )

2

2 2

, , rect exp4

rect exp

r

x

G r jb

j r

η ηξ η

ξ η ξ η

⎡ ⎤ ⎡ ⎤= ⎢ ⎥ ⎢ ⎥Ω ⎣ ⎦⎣ ⎦

⎡ ⎤ ⎡ ⎤⋅ − − −⎢ ⎥ ⎢ ⎥⎣ ⎦Ω⎣ ⎦

(A7)

where:

4 f fbc

πλ τ

∆= (A8)

22x LπΩ = (A9)

22rfbc

cτ π ∆

Ω = = (A10)

4πη ηλ

= + (A11)

f being the carrier frequency; other symbols in eqs. (A8)-(A11) have already been defined in Section 1.3.

The simplest and most efficient processing scheme is based on the following approximation:

( ) ( ) ( )0 0, , , , ,G r G r Gξ η ξ η ξ η≈ = (A12)

where r0 is the range coordinate of the scene center. The approximation in (A12) leads to the approximated expression of the raw data spectrum of eq.(A6) which suggests the following space invariant filtering:

Chapter 1 43

( ) ( ) ( ) ( ) ( ) ( )0 0 0ˆ , , , , , ,H G G Gξ η ξ η ξ η ξ η ξ η ξ η∗ ∗Γ = ≈ Γ (A13)

carried out in the two dimensional Fourier Domain. The filtering operation of eq. (A13) leads to

( ) ( )ˆ , , rect rect

x r

ξ ηξ η ξ η⎡ ⎤ ⎡ ⎤

Γ = Γ ⎢ ⎥ ⎢ ⎥Ω Ω⎣ ⎦⎣ ⎦ (A14)

which is band limited, according to eq.(A7), and leads, after a 2D inverse Fourier Transform, to the following focused SAR image:

( ) ( ) ( ) ( )ˆ , , sinc sinc2 2

r xx r dx dr x r r r x xγ γ Ω Ω⎡ ⎤ ⎡ ⎤′ ′ ′ ′= − −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∫∫ (A15)

which coincides with eq.(A1), after substituting eqs. (A9) and (A10). The spectral multiplication of eq. (A13) allows efficient focusing of the

SAR image along the scheme depicted in Figure A1.

Azimuth and Range FTs (2D-FT)

Azimuth and Range IFTs (2D-IFT)

Raw signal

( )0

* ,G ξ η

SAR image

Figure A1 - Narrow focus SAR processing block diagram.


Note that the operation implemented by such a processing scheme can be carried out directly in time domain, since it is equivalent to a deconvolution step applied to h(x′,r′). However, the system impulse response generally extends for several hundreds points in both azimuth and range directions; typical dimensions of the raw data are very large too (thousands complex samples for each direction). Hence, it is convenient to carry out the deconvolution operation in the Fourier Domain due to the availability of Fast Fourier Transform (FFT) codes.

The key point of the processing scheme of Figure A1 is represented by the knowledge of the TF of eq. (A7) by assuming r=r0. For this reason we refer to this processing procedure as the narrow focus code, as only the central part of the scene, r=r0, is perfectly focused. Obviously, this is not acceptable in many cases and ways to obtain a fully focused image are desirable. To circumvent this problem, several efficient (from the computational point of view) solutions, carried out in the spectral domain, have been presented in literature [Franceschetti and Lanari, 1999; Bamler, 1992; Rocca, 1987] and successfully applied.

Chapter 1 45

Appendix B Coherence calculation Let us start our analysis from the expression of two SAR image:

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( )

( )

1 1

1

2 2

2

4ˆ ', ' , exp sinc '

sinc ' exp '

4ˆ ' ' , ' ' , exp

sinc ' ' sinc ' ' '

exp '

d

x r

r x

d

x r x r j r r rr

x x j x x dxdrx

x u r u x r j r r

r r u r x x u xr x

j x x dxd

π πγ γλ

π ξ

πγ γ δλ

π πδ δ

ξ

⎛ ⎞ ⎡ ⎤= − − ⋅⎜ ⎟ ⎢ ⎥∆⎝ ⎠ ⎣ ⎦⎡ ⎤⋅ − −⎡ ⎤⎣ ⎦⎢ ⎥∆⎣ ⎦

⎛ ⎞+ + = − + ⋅⎜ ⎟⎝ ⎠

⎡ ⎤ ⎡ ⎤⋅ − + − ⋅ − + − ⋅⎢ ⎥ ⎢ ⎥∆ ∆⎣ ⎦ ⎣ ⎦⋅ −⎡ ⎤⎣ ⎦

∫∫

∫∫

r

(B1)

and let us evaluate the complex interferogram expected value, by supposing the reflectivity pattern is represented by a homogeneous, uncorrelated process.

Hence

( ) ( ) ( ) ( )2*1 1 2 2 1 2 1 2, ,E x y x y x x y yγ γ γ δ δ⎡ ⎤ = − −⎣ ⎦ (B2)

The cross-correlation factor is essentially a normalized mutual correlation between the two complex SAR images, represented by the relations (B1), and can be written as

*

1 2

2 21 2

E

E E

γ γχ

γ γ

⎡ ⎤⎣ ⎦=⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

(B3)

The calculation of the numerator is now in order. For this purpose, we rewrite the expression (B1) by introducing the variable changes

''

p x xq r r

= −= −

Hence, we have


( ) ( )

[ ]

( ) ( ) ( )

( ) ( )

( )

1 1

1

2 2

4ˆ ', ' exp ' ' , '

4exp sinc sinc exp

4ˆ ', ' exp ' ' ' , '

4 4exp ' exp '

sinc '

d

r

x r j r x p r q

j q q p j p dpdqr x

x r j r r x p r q

j q r r j q r r

q u rr

πγ γλ

π π π ξλ

πγ δ γλ

π πδ δ δ δλ λ

π δ

⎛ ⎞= − − − ⋅⎜ ⎟⎝ ⎠

⎛ ⎞ ⎡ ⎤ ⎡ ⎤⋅ ⎜ ⎟ ⎢ ⎥ ⎢ ⎥∆ ∆⎝ ⎠ ⎣ ⎦ ⎣ ⎦

⎛ ⎞= − + − − ⋅⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⋅ − + − + ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎡⋅ + −⎢ ∆⎣

∫∫

∫∫

( ) [ ]2sinc ' ' expx dp u x j p dpdqx

π δ ξ⎤ ⎡ ⎤+ −⎥ ⎢ ⎥∆⎦ ⎣ ⎦

(B4)

and, consequently

( ) ( )

( ) ( )

*1 2 1 2

4 4exp ' exp '

sinc sinc ' ' sinc sinc '

d d

x r

E j r dpdq j r r p

p p u x q q u rx x r r

π πγ γ δ δ δ ξ ξλ λ

π π π πδ δ

⎛ ⎞ ⎛ ⎞⎡ ⎤ = − + − ⋅⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎝ ⎠ ⎝ ⎠⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⋅ + − + −⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥∆ ∆ ∆ ∆⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

∫∫ (B5)

The slant range difference between the cell center and its value within the resolution cell as a function of the local slope of the surface is [Franceschetti and Lanari, 1999]

( ) ( ) ( )1' '

' tan ' ' tan 't t

b br r r r q

r rδ δ

ϑ ϑ⊥ ⊥− ≈ − − = −

− Ω − Ω (B6)

The previous condition and the assertion that the variation on the slant range affects the phase term, but is not relevant for the integrating amplitude term, lead to rearrange the previous expression in this novel form

( )

( )

* 21 2

2

4 4exp ' exp sinc' tan '

exp sinc

t

d

bE j r dq j q q

rr

dp j p px

π π πγ γ δλ λ ϑ

πξ

⊥⎛ ⎞⎛ ⎞ ⎛ ⎞⎡ ⎤ ⎜ ⎟= − ⋅⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎜ ⎟ ∆− Ω⎝ ⎠ ⎝ ⎠⎝ ⎠

⎛ ⎞⋅ ∆ ⎜ ⎟∆⎝ ⎠

∫

∫ (B7)

where the possible mis-registration effects have been neglected. At this stage we can observe that, for example, the second integral can be viewed as the Fourier transform of the ( )2sinc ⋅ function that is equal to the convolution of two rectangular spectra pulse (which is equal to a triangle pulse)

Chapter 1 47

( ) 2exp sinc4d adp j p p L

xπ ξξ

π∆⎛ ⎞ ⎛ ⎞∆ ∝ Λ⎜ ⎟ ⎜ ⎟∆⎝ ⎠ ⎝ ⎠∫ (B8)

22 2exp sinc2c c c

b b brdq j q qr b r r b bπ π π

π⊥ ⊥ ⊥

⊥ ⊥ ⊥

⎛ ⎞ ⎛ ⎞ ⎛ ⎞∆⎛ ⎞− ∝ Λ = Λ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟∆ ∆ ∆⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠∫ (B9)


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Chapter 2

The SBAS technique: key idea and processing chain

Differential Synthetic Aperture Radar Interferometry (DInSAR) is a microwave remote sensing technique that allows us to investigate surface deformation phenomena with centimeter to millimeter accuracy and with large spatial coverage capability. Its basic principles and limitations have been discussed in previous chapter.

DInSAR methodology has been applied first to investigate single deformation events. However, more recently, it has also been exploited to analyze the temporal evolution of the detected displacements via the generation of deformation time series. In this case, a time series of deformation can be retrieved through the inversion of an appropriate sequence of DInSAR interferograms. The interest on the development of this methodology is testified by several approaches which have already been presented or that are under development.

In the following, we will give an overview of these procedures, emphasizing the role played by the different data inversion algorithms up to now proposed. In particular, we will focus on the technique referred to as Small BAseline Subset (SBAS), whose basic theory, principles and limitations will be deeply discussed. Moreover, a detailed analysis on the SBAS processing chain will be also provided, followed by experimental results achieved on Long Valley caldera area (California) data.

2.1 Advanced DInSAR algorithms Since its first description by Gabriel et al. [1989], most DInSAR

applications have been based on exploitation of single or few interferograms. The advantage of these simple configurations is the flexibility to provide spatially dense information on deformation, even with a reduced SAR data availability [Massonnet et al., 1993; Peltzer and Rosen, 1995; Rignot, 1998].

As it was discussed in the previous chapter, standard two pass DInSAR is limited by at least two error sources: the presence of atmospheric inhomogeneities between the acquisition epoch and the inaccuracies of the

52 SBAS technique and processing chain

external DEM involved in the removal of the topography component from the signal interferences.

In addition, time sampling frequency restricted to only two acquisition epoch, in spite of the huge SAR data set collected through the years all over the world by the European (ERS1/2 and ENVISAT) and Canadian (RADARSAT) satellites, is an important limitation to investigate slow and long deformation events.

In order to overcome these limitations several algorithms have been developed. They try to exploit the previous mentioned large SAR data set by generating a large number of DInSAR interferograms and subsequently by combining the relevant deformation information in order to obtain a time series of the detected displacement. Combination is achieved by searching for the solution of a system of linear equations (interferograms), each one represented as:

,B A B At t t tψ ϕ ϕ= −% , (1)

where Bt

ϕ and At

ϕ are the phases associated to two SAR images acquired at

time Bt and At , respectively, while ,B At tψ% is the relevant (unwrapped) interferogram (phase difference).

It is important to note that all these algorithms essentially differentiate each other by interferogram selection criteria, which also implies different ways to solve the system.

Before analyzing in details the selection strategies, it is worth stressing that using large image stacks represents a key factor to achieve a fully quantitative DInSAR monitoring via whatever multipass algorithm. Indeed, main DInSAR limitations such as topography and atmospheric artifacts can be easily overcome by exploiting the time variable.

DInSAR analysis based on large image stacks is suitable to perform in-depth studies of deformation phenomena but, on the other hand, it requires weighty economical and computational resources for image acquisition and data processing. Moreover, the number of available images, the temporal evolution and the spatial extent of the deformation at hand, the SAR images coherence in the area of interest, are some of the technique main limitations.

2.2 Interferograms selection: two different “philosophies”

Distinctive character of DInSAR algorithms known in literature is the interferometric data pairs selection. Let us start our analysis by considering

Chapter 2 53

the Temporal/Perpendicular baseline plane, where we can plot the SAR acquisition distribution, after imposing a selected image as a baseline reference (Master image). Indeed, in this plane, each SAR acquisition is represented by a point while each interferogram corresponds to an arc connecting two points, i. e., two SAR images, see Figure 1. As explained in Chapter 1, the larger are the perpendicular and/or temporal baseline values, the more significant will be the decorrelation on interferogram. Hence, depending on the choice of the interferometric distribution, different phase assumptions would be done.

Let us refer to a set of 1N + SAR images relevant to the same area, the easiest choice would requires the generation of N consecutives interferograms between the 1N + data acquisitions (see Figure 1a). This implies to solve a determined N N× system that, however, includes high decorrelated interferograms, due to the presence of high perpendicular baseline interferograms. Another possibility involves the generation of a set of interferograms between each single SAR image and the Master one (see Figure 1b); also in this case the system to be resolved is well determined. Moreover, we can observe that, another time, some considered interferograms may have a perpendicular baseline value greater than the critical one (and, for this reason, strongly affected by decorrelation noise).

Using large baseline interferograms implies some considerations on signal phase. Indeed, only points maintaining the same scattering characteristics along space and time can preserve interferometric phase information.

This is the key idea of the Permanent Scatterers (PS) algorithm [Ferretti et al., 2001], where only the phases of a properly chosen point-wise scatterer set could be retrieved for interferometry. Within this technique the pixels are selected by studying their amplitude stability along the whole set of images, which requires a minimum of 30 scenes. Hence, a proper radiometric calibration is needed for a reliable pixel selection. Note that the maximum resolution of SAR images is preserved; indeed, no multilook operation is involved.

However, another possible way to choose interferometric pairs is driven by the need to preserve spatial and temporal conventional DInSAR correlation characteristics. This can be done by imposing a threshold on the maxima allowed spatial and temporal baseline values. In other terms, it can be possible to choose only Small Baseline interferograms (see Figure 1c). This criterion permits mitigating the noise effects on the interferograms; however, it implies the possible generation of more than one independent acquisition subset; by the consequence, the latter have to be “linked” each others.


In analytical terms, it means that in general the system of equations (1)

has a rank deficiency and, therefore, has to be solved by exploiting more sophisticated mathematical instruments; one of these could be the Singular Value Decomposition (SVD) method, aimed to give a solution in the Least Square (LS) sense for a not invertible linear system.

This issue represents the core of the Small BAseline Subset (SBAS) algorithm [Berardino et al., 2002], which will be fully described in Section 2.3.

As already mentioned, the adopted pair selection strategy leads to identify two Advanced DInSAR algorithm “families”: one based on the Permanent (or Persistent) Scatterers; one based on the Small Baseline exploitation. Moreover, a third “category” is beside these two and essentially try to jointly exploit different characteristics of the two main groups.

During the last decade several algorithm have been developed; among them: the Permanent Scatterers (PS) technique [Ferretti et al., 2001]; the

Figure 1 - Pictorial representation of the possible interferometric distribution. (a) Set of consecutive interferometric pairs; in this case, a single subset is obtained. (b) Set of interferometric pairs where the reference master image is the first available acquisition; also in this case a single subset is achieved. (c) Set of small baseline interferometric pairs; in this case several subsets can be obtained.

Chapter 2 55

Small BAseline Subset (SBAS) technique [Berardino et al., 2002]; the Coherent Point Target Analysis (CPTA) [Mora et al., 2003]; the Interferometric Point Target Analysis (IPTA) [Werner et al., 2003]; the Stanford Method for Persistent Scatterers (StaMPS) [Hooper et al, 2004]; the Spatio-Temporal Unwrapping Network (STUN) [Kampes and Adam, 2005]; the one proposed by Crosetto et al. [2005]; the Enhanced Spatial Differences (ESD) [Fornaro et al., 2008] and the one presented by Rune et al. [2008].

In particular, PS, StaMPS, IPTA and STUN belong to a processing category that, similarly to the PS analysis, applies to full resolution data, i.e., to single look data and relies on the measurement of deformation on a spatial grid of amplitude stable points - the PS candidates (PSC) grid. To increase the spatial density of the measures, a growth of the spatial grid is achieved by analyzing the phase stability of the Multi-Temporal/Multi-Baseline interferogram stack, after separation, interpolation and compensation of atmospheric contributions via a gradient (spatial variation) analysis carried out on the PSC grid.

Instead, following the lines of the conventional (single-pair) DInSAR analysis, SBAS, CPTA and Crosetto et al. [2005] carry out an analysis based on interferogram stacking and process multilook data, i.e. data that have been spatially averaged, and thus characterized by lower resolution. The monitoring is achieved by separating the deformation contributions from residual topography and atmospheric artifacts on a sparse grid of reliable pixels, selected on a spatial coherence analysis.

Finally, ESD, Rune et al. [2008] and also a StaMPS implementation try to exploit and combine both the advantages of PS and SBAS philosophies.

In the following sections we provide a detailed analysis on the Small BAseline Subset approach [Berardino et al., 2002], focusing our attention first on the basic rationale an subsequently on its processing chain.

2.3 SBAS algorithm: basic rationale Let us start by considering a set of 1N+ SAR images relative to the same

area, acquired at the ordered times ( )0, , Nt tK . We also assume, for sake of simplicity, that all the images are co-registered with respect to a single one (referred to as master) in order to have a common reference grid.

Subsequently, we generate a number, say M , of differential interferograms involving the previously mentioned set of 1N+ SAR acquisitions, properly generated in order to mitigate the decorrelation phenomena. To achieve this task, the SAR image pairs selected for the interferograms generation are characterized by a small spatial and temporal


baseline as well as by a small frequency shift between the Doppler centroids [Franceschetti and Lanari, 1999]. As already stated in Section 2.2, as a consequence of these constraints, the SAR images involved in the interferograms generation could be grouped in several independent small baseline subsets that must be appropriately “linked” in order to retrieve the deformation time series.

We further remark that, since the phase information in the computed interferograms represents the modulo-2π restriction (wrapped) of the original (unwrapped) interferometric signal, each interferogram has to be processed in order to retrieve the original phase. This operation is referred to as Phase Unwrapping (PhU) and can be carried out via different approaches, for example the Minimum Cost Flow (MCF) algorithm presented by Costantini [1998] and Costantini and Rosen [1999].

Let us now refer to a generic pixel of our SAR images with azimuth and range coordinates ( ),x r , respectively, and assume that the phase signal of each unwrapped interferogram is referred to a pixel characterized by high coherence, whose deformation behavior is a priori known (typically, it is located in a non-deforming zone). Note that the selection of this reference SAR pixel must be carefully carried out because any error affecting this point will influence the overall results. In this context, particularly critical can be the impact of the atmospheric phase artifacts that, as already remarked, are spatially correlated, thus can be confused with deformation signals [Beauducel et al., 2000].

The expression of the generic j-th (unwrapped) interferogram computed from the SAR acquisitions at times Bt and At is, according to Berardino et al. [2002], the following:

( ) ( ) ( )

( ) ( )

( ) ( )

, , , , ,

4 4, , , ,sin

4 , , , , , 1, , ,

j

j

atm atm j

B A

B A

B A

x r t x r t x r

b zd t x r d t x r

r

d t x r d t x r n j M

ψ ϕ ϕ

π πλ λ ϑπλ

⊥

= −

∆⎡ ⎤≈ − + +⎣ ⎦

⎡ ⎤+ − +∆ ∀ =⎣ ⎦

%

K

(2)

wherein λ is the transmitted signal central wavelength, ( ), ,Bt x rϕ and

( ), ,At x rϕ represent the phases of the two images involved in the

interferogram generation and ( ), ,Bd t x r and ( ), ,Ad t x r are the radar line of sight projections of the cumulative deformation at times Bt and At , with respect to the instant 0t assumed as a reference, i. e., implying

Chapter 2 57

( ) ( )0, , 0, ,t x r x rϕ = ∀ . Moreover, concerning the right hand side in equation (2), the second term accounts for possible topographic artifacts z∆ that can be present in the Digital Elevation Model (DEM) used for the interferograms generation; it depends on the perpendicular (spatial) baseline component jb⊥ as well as on the sensor-target distance r and on the look angle ϑ . The terms

( ), ,atm Ad t x r and ( ), ,atm Bd t x r account for possible atmospheric phase artifacts [Goldstein, 1995] and the last term jn∆ for the decorrelation effects.

Let us now observe that we can identify the following two index vectors:

[ ] [ ]MM ,IE,IE , IE ,IS,IS IS KK 11 == (3)

corresponding to the acquisition time-indexes associated with the image pairs used for the interferogram generation. Note also that we assume the master (IE) and slave (IS) images to be chronologically ordered, i. e.,

1, ,j jIE IS j M> ∀ = K . By considering these index vectors, we can rewrite the first part of eq.

(2) as:

( ) ( ) 1, ,j jj IE ISt t j Mψ ϕ ϕ= − ∀ =% K (4)

Accordingly, equation (4) defines a system of M equations in N unknowns which can be reorganized, by using matrix formalism, as follows:

⋅A Φ = Ψ% (5)

being A an M N× matrix where 1, ,j M∀ = K we have: ( ), 1jj IS =−A if

0jIS ≠ , ( ), 1jj IE =A and 0 otherwise. For instance, should be 1 4 2ψ ϕ ϕ= −%

and 2 3 0ψ ϕ ϕ= −% , A would have the following form:

0 -1 0 1 ...0 0 1 0 ...... ... ... ... ...... ... ... ... ...

+⎡ ⎤⎢ ⎥+⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

A (6)

Note that A is an incidence-like matrix, directly depending on the set of interferograms generated from the available data. Due to this characteristic, if all the acquisitions belong to a single subset it turns out that NM ≥ and A is an N-rank matrix. Accordingly, the system of equations (5) is a well-


( NM = ) or an over-determined ( NM > ) system and its solution can be obtained, in general, in the LS sense as:

( )( )1−⋅ ⋅ ⋅ ⋅+ T TΦ = A Ψ = A A A Ψ% % (7)

where +A is the left inverse matrix. Unfortunately, the SAR acquisitions are, in general, coupled in several

different subsets; this implies that the matrix A exhibits a rank deficiency and therefore TA A is singular. Its rank is reduced according to the number of subsets; for instance, if we assume to face L different subsets, the rank of A will be N-L+1 and the system will have infinite solutions.

In this case, we can use the Singular Value Decomposition (SVD) method to invert the system [Golub and Van Loan, 1996; Flannery et al., 1988].

The SVD decomposition of a generic matrix A may be written as:

⋅ ⋅ TA = U S V (8)

where U is an M M× orthogonal matrix, whose first N columns are the eigenvectors of the matrix TAA , S is an M M× diagonal matrix, whose elements, the so-called singular values ( iσ ), are the square root of the corresponding eigenvalues of the MM × matrix TAA . and V is an N M× orthogonal matrix, whose columns are the eigenvectors of the matrix TA A . Note that, being generally M N> , then M N− eigenvalues are zero; moreover, due to the singular nature of A there are 1L − additional null eigenvalues, being L the number of the different subset. In summary, only

1N L− + of the diagonal values of the matrix S will be not equal to zero. The minimum norm least squares solution for Φ is then achieved via the equations (7) and (8):

( )1

1 2 N-L+11

1 ,1 ,...,1 ,0,...,0N L

ii

i i

σ σ σσ

+

− +

=

= ⋅ =

⋅⎡ ⎤= ⋅ ⋅ ⋅ =⎣ ⎦ ∑

TT

Φ A ψψ u

V diag U ψ v

%

% (9)

where iu and iv are the column vectors of U and V , respectively. This solution is characterized by a minimum norm constraint on the

phase signal Φ and, accordingly, on the detected deformation, see eq. (2). As a consequence, the method forces the solution, in accordance with the starting equation system (5), to be as close to zero as possible. Unfortunately, this solution method may introduce large discontinuities in the cumulative deformation obtained, thus leading to a physically

Chapter 2 59

meaningless result; a pictorial example is shown in Figure 2 (see orange line).

An effective strategy to overcome this problem can be accomplished by

manipulating the system of equations (5) in such a way to replace the present unknowns with the mean phase velocity between time adjacent acquisitions. Accordingly, the new unknowns become:

1 0 12 11 2

1 0 2 1 1

, ,...,T N Nd d d Nd

N N

v v v vt t t t t tϕ ϕ ϕ ϕϕ ϕ −

−

⎛ ⎞− −−= = = =⎜ ⎟− − −⎝ ⎠

(10)

which, properly substituted in equation (5), lead to the new system of equations:

⋅ =dB v Ψ% (11)

where B is an M N× matrix, whose elements are so-defined:

[ ] 1 , 1, ,,

0k k j jt t IS k IE j M

B j kelsewhere

+ − ≤ ≤ ∀ =⎧= ⎨⎩

K (12)

The unknown mean velocity vector can be again estimated via the application of the SVD technique, but now with reference to the matrix B , and the minimum norm constraint for the velocity vector dv does not imply

Figure 2 - Example describing how the SVD works. Blue and orange points represent SAR acquisitions belonging to two different subsets. Dashed black line is the actual deformation; continuous orange line refers to the time series obtained by solving system (5) (phase minimization); red line represents the estimated deformation obtained by solving system (11) (velocity minimization).


the presence of large discontinuities in the final solution (see the red continuous line in Figure 2). In this case, an additional integration step will be necessary in order to compute the required deformation phases, starting from the estimated vector dv :

( ) ( )

( )

11

0

1,...,

0

i

i kd k kk

t v t t i N

with t

ϕ

ϕ

−=

= − ∀ =

=

∑ (13)

As a further remark we observe that, after resolving the system of equations (11), an estimation of the topographic artifacts z∆ and of the atmospheric signal component ( ).atmd , see equation (2), should be carried out; these issues will be addressed in the following sections.

In summary, the SBAS algorithm satisfies two key requirements:

– to preserve the conventional DInSAR capability to provide spatially dense deformation maps, by exploiting small baseline interferograms;

– to maximize the “temporal sampling rate” of the retrieved displacements signals, by using nearly all the available SAR acquisitions.

Accordingly, the SBAS approach permits us, by exploiting standard multilook DInSAR interferograms, to detect and follow the temporal evolution of surface deformation with a high degree of temporal and spatial coverage [Lanari et al., 2007]. For sake of completeness, we just remember that the SBAS approach has been also extended to work with single look SAR data; more information can be found in Lanari et al. [2004]

2.4 SBAS algorithm: processing chain Let us now focus on the SBAS processing chain (see Figure 3) that can

be decomposed in the following main steps: 1. Evaluation of the orbital parameters associated to each SAR

acquisition (subsequently used within the estimation of the spatial baseline value) and reference Master image selection;

2. Generation of a set of single look SAR images (SLC) from the available raw data files (focusing);

3. DEM conversion into reference Master image SAR coordinates and computation of sensor to target distance files (range files) for each acquisition;

4. Optimal interferometric data pair distribution selection;

Chapter 2 61

5. Co-registration of each SLC data pair with respect to the selected reference Master image;

6. Differential interferograms and corresponding spatial coherence maps generation;

7. Noise-filtering of the generated DInSAR fringes; 8. Generation of the mean deformation velocity map and the

corresponding time series, for each coherent pixel of the investigated area, via the inversion of the computed sequence of DInSAR interferograms. At the same time, an estimate of the possible residual topographic components and the atmospheric contributions is also accomplished;

9. Geocoding of the obtained results and projection onto a universal cartographic grid.

Figure 3 – SBAS processing chain.


2.5 Orbit parameters extraction and Master selection

Let us start our analysis by remarking that, in DInSAR context, a perfect knowledge of the SAR sensor orbital parameters (i.e., of the actual position of the system along its flight path) is necessary, in order to properly estimate (and remove from interferograms) the topography phase component. For this reason, precise satellite orbit state vectors can be available at a global scale and collected into data-base (this is for instance the case of the ERS precise orbits provided by the University of Delft) [Scharoo and Visser, 1998; TU-Delft Web]. Obviously, the state vector errors will correspond to an inaccurate estimation of the actual spatial position of the radar sensor, thus implying a subsequent wrong estimation of the topography fringes.

At this step, by exploiting the orbital information, it is possible to plot each SAR image in the Temporal/Perpendicular Baseline plane, as explained in Section 2.2 and shown in Figure 4.

By benefiting of this graphical representation, it is possible to properly

select the reference Master image. In order to help the subsequent co-

Figure 4 - Example of the SAR image distribution in the Temporal/Perpendicular baseline plane. Each SAR acquisition is represented via a black diamond; the selected reference Master image has been marked via a black box.

Chapter 2 63

registration step, it is useful to select as a reference the SAR image with a baricentrical position in the acquisition distribution plot. Indeed, it implies the baseline values between the images are the smaller as possible. It is worthy to note that, although all the SAR images are co-registered to the Master one, no large baseline interferograms are subsequently computed, thus preserving the SBAS approach key idea.

In summary, the Master image only represents the reference geometry for the whole processing.

2.6 Focusing The focusing procedure has been already analyzed; therefore, analytical

details are forwarded to the previous chapter. However, it is important to note that all the available SAR raw data are focused at this step and only those respecting quality selection criteria can be suitable for the next processing. In particular, we essentially refer to acquisitions corrupted by a large amount of data missing lines, by line counter inconsistencies, etc.

2.7 DEM conversion and Range files generation The evaluation of topographic phase components, which have to be

subsequently subtracted to the interferometric phase to retrieve the deformation phase pattern, is a critical step in the DInSAR methodology.

The strategy we follow is based on the observation that (see Section 1.3) the phase difference associated to the scene topography can be estimated, for each pixel of the two SAR images, via the measure of the master-slave slant range differences, see Figure 5. As a matter of fact, the expression of the topographic interferometric phase components is the following:

( )4synth m sr rπϕ

λ∆ = − (14)

wherein λ is wavelength, mr and sr represent the slant range sensor-to-target distances from the first and the second flight track, respectively.

Therefore, in order to compute eq. (14) it is important to correctly evaluate the two mr and sr quantities. It is clear that these distances are peculiar to each SAR image; hence, by considering a DEM of the area and the orbit information of each acquisition, it is possible to calculate the mentioned range distances, referred to as Range files. This procedure can be summarized as follow:


1) Evaluation of the DEM point positions with respect to the reference Master SAR geometry (this corresponds to a classical inverse-geocoding problem);

2) Estimation of the SAR sensor position along the flight orbit for each acquisition;

3) Evaluation (pixel by pixel) of the slant range distances for each SAR image;

4) Estimation of the topographic phase term, according to equation (14).

The first step of the procedure is accomplished by using the orbital parameters of the reference Master image, which allows individuating the actual sensor position along its flight path. The procedure starts from the conversion of the available DEM points (typically known onto a cartographic reference system) in an ellipsoidical Cartesian reference system (see Figure 6). In this way, each DEM point is transformed in a corresponding point ( ), ,e e eP x y z in the Cartesian reference system, which, in turn, must be associated to a pixel of coordinates ( ),Q i j≡ onto the reference Master image geometry. To this aim, the following system of equations must be solved for:

( )( ) ( ) 0

M i M

M i M i

P s t r

s t P s t

− =

⎡ ⎤⋅ − =⎣ ⎦

r r

rr r (15)

where ( ),i Mt r are the unknowns, Pr

is the position vector of the generic target on the local ellipsoid, illuminated by the radar sensor at the instant it , while ( )M is tr is the radar position vector. The sensor position, depending on the time acquisition it can be, in turn, expressed in the same reference system by using the orbital parameters associated to the reference master image itself, as follows:

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

2 30 1 0 2 0 3 0

2 30 1 0 2 0 3 0

2 30 1 0 2 0 3 0

M

M

M

s i i i

M s i i i

s i i i

x a a t t a t t a t t

s y b b t t b t t b t t

z c c t t c t t c t t

⎧ = + − + − + −⎪⎪= = + − + − + −⎨⎪

= + − + − + −⎪⎩

r (16)

where the coefficients are related to the orbital parameters and 0t is the origin of the temporal reference on the orbit.

Chapter 2 65

Figure 5 - Repeat-pass acquisition geometry used within stereometric/interferometric applications. SAR sensor observes the same scene from slightly different orbital positions thus leading to the reconstruction of the scene topography.

Figure 6 - Acquisition geometry of the reference Master image, represented with respect to an ellipsoidical Cartesian reference system; vector P

r represents the

position of the target of height h, illuminated by the radar sensor at the time ti.


Once the unknowns ( ),i Mt r have been evaluated, the SAR coordinates ( ),i j of the given DEM point can be estimated by solving these equations:

0

0 2

i

Msamp

it tPRF

cr r jf

= +

= + (17)

with respect to the ( ),i j unknown SAR coordinates, wherein 0r is the near range distance, c is the light speed, PRF the pulse repetition frequency and

sampf the range sampling frequency, respectively. In this way, for each pixel of the available DEM, we obtain the corresponding position onto the reference SAR geometry.

Obviously, the regular cartographic grid (see Figure 7) will be deformed in an irregular SAR output grid and, for this reason, a proper regridding operation onto the converted pixels will be necessary. The regridding operation could be also performed for each different orbital position but, since reference geometry has been chosen, it can be more efficient to make one single operation to limit possible errors.

In this way, if we refer to the conversion of the Cartesian coordinates of

the DEM point, we have:

Figure 7 - Pictorially representation of the conversion operation of each DEM point (a) into the SAR reference Master image geometry (b). As shown in this figure, the converted points are not necessarily spaced on a regular grid; hence, a regridding operation is generally required.

Chapter 2 67

( )( )( )

( )( )( )

( , ), ,, ,, ,

e e ei j regr

e e e

e e e

x X i j X i jy Y i j Y i jz Z i j Z i j

⎧ ⎧⎧⎪ ⎪⎪→ →⎨ ⎨ ⎨⎪ ⎪ ⎪

⎩ ⎩⎩

%

%

% (18)

The second step of the previous procedure consists in measuring, for each acquisition, the SAR sensor position with respect to the SAR reference geometry (Master image).

The vector position of the k-th SAR acquisition at time it (which corresponds to the azimuth pixel i onto the SAR geometry), is:

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

2 30 1 0 2 0 3 0

2 30 1 0 2 0 3 0

2 30 1 0 2 0 3 0

k

k

k

s i i i

k s i i i

s i i i

x a a t t a t t a t t

s y b b t t b t t b t t

z c c t t c t t c t t

⎧ = + − + − + −⎪⎪= = + − + − + −⎨⎪

= + − + − + −⎪⎩

% % % %

r % % % %

% % % %

(19)

The third step concerns the estimation of the slant range distances (Range files) performed directly onto the SAR reference system.

This estimate can be easily carried out by computing, via Pitagora’s Law, the distance sensor-to-target for each azimuth line:

( ) ( ) ( )

( )

2 2 2

0

( , ) ( , ) ( ) ( , ) ( ) ( , ) ( )

1,...,k k kk e s e s e s

RG

r i j X i j x i Y i j y i Z i j z i

j Ni t t PRF

= − + − + −

∀ =

= − ⋅

% % %

(20)

wherein NRG is the number of range samples.

Finally, according to equation (14), the topography fringes can be directly evaluated onto the SAR geometry by exploiting the two so-computed slant range distances, associated to the SAR images involved in the considered DInSAR data pair. Note that, the Range files contain not only the topography information but also the Flat-Earth phase component. By the consequence, the use of Range files in the differential interferogram generation implies to contemporary remove both these phase components, jointly included in the so called synthetic interferogram (eq. (14), Figure 8). Operatively, this step is performed during the Differential Interferogram generation, where the synthetic fringes are subtracted to the interferometric phase.


2.8 Data Pairs Selection As already mentioned in Section 2.2, data pairs selection is driven by

imposing a threshold on the maxima temporal and perpendicular baseline allowed values; it means that only small baseline interferograms would be generated. Moreover, as explained in the previous chapter, a large difference between SAR image Doppler centroid implies the generation of highly uncorrelated differential interferograms, due to the relevant azimuth spectral shift (see Section 1.2.4).

Accordingly, the evaluation of the optimal DInSAR data pairs distribution is linked to the constraint that the decorrelation effects are as limited as possible: this task will be easily achieved by imposing that both the (spatial and temporal) baseline separation of the SAR acquisitions and the differences on their Doppler centroid values are small enough.

Basically, the selection of the optimal data pair distribution can be simply obtained by imposing that each data pair satisfies the following constraints:

Figure 8 - Example of synthetic interferogram relevant to the Napoli bay (Italy) area. The topographic fringes related to Mt. Vesuvio volcano are visible in right-lower part of the figure.

Chapter 2 69

max

max

2dopp

b b

t tPRFf h

⊥ ⊥<

∆ < ∆

∆ <

(21)

where b⊥ and maxb⊥ represent the generic and maximum allowed perpendicular baseline values, respectively; t∆ and maxt∆ the temporal and the maximum allowed temporal baseline values, respectively; doppf∆ the Doppler centroid difference between two SAR acquisitions involved in the data pair, PRF the Pulse Repetition Frequency and h a selected fraction of the azimuth spectra.

However, different selection strategy could be performed. Indeed, a

proper data pairs selection could help in the Phase Unwrapping procedure, permitting to apply the so called Extended Minimum Cost Flow (EMCF) algorithm [Pepe and Lanari, 2006].

Figure 9 - Example of the SAR image data pairs distribution in the Temporal/Perpendicular baseline plane corresponding to the SAR image distribution of Figure 4. Note that each interferogram corresponds to an arc connecting two black diamonods, i. e., two SAR acquisitions. Red arcs are relevant to interferograms belonging to the Delaunay triangulation.


In particular, let us suppose the 1N + SAR data orbit information were known; therefore, we can easily evaluate both the perpendicular

( )0 1, ,..., Nb b b⊥ ⊥ ⊥ ⊥=B and the temporal ( )0 1, ,...,m m N mt t t t t t= − − −TB baseline vectors, respectively, of each SAR image with respect to the reference Master one, where ( )0 1, ,..., Nt t t=t is the SAR acquisition epoch vector. Consequently, the involved SAR images can be properly represented in the Temporal/Perpendicular baseline TB B⊥× plane (as already explained in Section 2.2). Indeed, in this plane each SAR acquisition is identified by a point, say ( ),i i iP t b⊥≡ , while an arc connecting two points, say

( ),i i iP t b⊥≡ and ( ),j j jP t b⊥≡ , represents a corresponding DInSAR data

pair. It is clear that, by benefiting of this representation, a TB B⊥× Delaunay triangulation [Delaunay, 1934] can be built (see Figure 9). Note that, in order to generate the latter one, a fixed ratio between the perpendicular and the temporal axis units has to be assumed, say max maxt b⊥∆ ∆ , wherein maxt∆ and maxb⊥∆ represent the interferogram related maximum allowed temporal and spatial baseline value, respectively (see eq. (21)). maxt∆ and maxb⊥∆ also represent the maximum acceptable temporal and spatial decorrelation effect [Zebker and Villasenor, 1992] in the generated interferograms, respectively.

By imposing the mentioned fixed ratio, it is possible that the computed Delaunay triangulation will include interferograms characterized by baseline values exceeding the assumed maxima. Therefore, to avoid considering highly decorrelated data pairs, a further step aimed to remove all the triangles involving large baselines (or large Doppler centroid differences) arcs, i. e., interferograms, has to be accomplished. Note that this triangle removal step may lead to discarding some acquisitions and/or to the generation of more than one independent subset of triangles, i.e., to a data representation consistent with the one described in the SBAS procedure [Berardino et al., 2002]. Therefore, the compatibility between this data organization and the one exploited in the SBAS technique is evident.

Same considerations can be done if two (or more) SAR image subsets even acquired by different (but geometrically compatible) radar systems (for instance, the case of the ERS-1/2 and the ENVISAT ASAR sensors [Pepe et al., 2005]) are present. Also in this case, the identification of DInSAR interferograms can be still achieved by using the previously outlined procedure, but considering two (or more) independent Delaunay triangulations.

Chapter 2 71

It is clear from Figure 9 that not only interferograms belonging to the triangulation are involved in the time series generation, in order to improve the information redundancy. However, triangulation based interferograms represent the core of the whole data pair distribution and, concerning the EMCF algorithm application, they play an important role in the PhU procedure.

2.9 Interferometric data pairs co-registration Once the optimal DInSAR data pairs distribution has been performed,

the registration step must be carried out for each interferometric pair. This procedure is fundamental to subsequently extract the phase difference relevant to corresponding pixels on the different images.

Let us consider a single data pair, the co-registration procedure [Fornaro and Franceschetti, 1995; Franceschetti et al., 1995] concerns the compensation of the azimuth and range pixel displacements of the slave image with respect to the master one. For a generic point P in the master image, the SAR coordinates of the same point in the slave image is given by:

( )( )

' ' ' ', '

' ' ' ', 's m

s m

x x x x r

r r r x r

δ

δ

= +

= + (22)

wherein ( )' , 'm mx r and ( )' , 's sx r are the coordinates of the point P in the master and slave geometry, respectively, and ( )', 'x rδ δ represent the difference in the SAR coordinates to be applied to the master image coordinates to retrieve the corresponding point in the slave one.

The registration process can be therefore defined as the problem of computing the geometric image transformation functions and, subsequently, the problem of re-sampling the second image with respect to the master one, in such a way that each ground point is located at the same position in the two different images. Accuracies of the order of 1/8th of the image pixel dimension are usually considered acceptable [Franceschetti and Lanari, 1999].

In order to improve the registration performances, it can be useful to register the images via a two-step procedure. The former provides the registration of each single SAR image with respect to the chosen reference master one. In this way, it is possible to accommodate the remaining, less significant, registration errors, thus improving the global precision and assuring accuracy at the order of the sub-pixel level.


Moreover, we note that this two-step procedure can be simplified by also exploiting the viewing geometries relevant to the two SAR acquisitions, being these techniques usually referred to as geometric registration approaches [Sansosti et al., 2006]. In particular, the approach presented in Sansosti et al. [2006] effectively exploits the range and azimuth shifts computed via the orbital information. In this approach, shifts needed to correctly register two images are then exactly known by geometry considerations, but for a constant value easily calculated via cross-correlation estimation. This approach is fully implemented in the SBAS procedure and permits to effectively register large baseline data pairs, as in the case of Master registration in order to have the same reference geometry (see Section 2.5).

2.10 Differential interferogram generation and filtering

The generation of the differential interferograms is carried out by subtracting the estimated topographic phase contribution from the interferometric phase term (Sections 1.3 and 2.7). As already highlighted in Chapter 1, the differential interferometric phase ϕ∆ is related to the deformation occurred between the two passes of the SAR sensor by the following relation:

4 rπϕλ

∆ = ∆ (23)

where r∆ represents the slant range sensor-to-target difference (after removing the synthetic fringes) between the two passes and λ is the sensor wavelength.

As stated in Section 1.3, the retrieved phase takes also account of spurious terms depending on the inaccuracies on the knowledge of the real topography, on the errors of the baseline estimation and on the presence of noise contributions. Hence, before to proceed, it is convenient to reduce the noise effect via a proper noise filtering operation. This task can be achieved, for example, by following the approach proposed by Goldstein and Werner [1998], that relies on the use of an adaptive filtering algorithm sensitive to the local phase noise and fringe rate.

A pictorially example of the achievable improvement which can be gained by applying this algorithm to a differential interferograms is shown in Figure 10, wherein non-filtered and the corresponding filtered differential interferogram have been presented.

Chapter 2 73

For each differential interferogram is also generated the spatial

coherence map which is subsequently used in the selection of the pixels reliable for the phase unwrapping step.

2.11 Deformation time series generation Deformation time series generation is the SBAS processing chain main

core. In particular, it can be summarized as follows:

1. In order to extract from each generated differential interferogram an information relevant to the actual deformation occurring between the two sensor flights over the illuminated area, the computed differential phases, representing the modulo-2π restriction (wrapped) of the original interferometric phase signal, must be, first of all, properly unwrapped.

2. The unwrapped interferograms, previously filtered from the residual topography artifacts, are used within the time series generation process via the application of the SVD method [Flannery et al., 1988]. Indeed, the selected DInSAR pairs are characterized by a small spatial and temporal baseline as well as by a small frequency shift between the Doppler centroids [Franceschetti and Lanari, 1999]. As a consequence of these constraints, the SAR images involved in the interferogram generation could be grouped in several independent small baseline subset, thus an appropriate “link” of such subsets is required (see Section 2.2);

3. Finally, the atmospheric phase signal is evaluated and removed from the computed deformation time series;

Figure 10 - DInSAR products relevant to the Napoli bay (Italy) area. (a) SAR amplitude image of the zone. (b-c) Multilook noise unfiltered (b) and filtered (c) differential interferogram computed for the ERS data pair acquired on April 19, 1993, and September 6, 1993.


4. Eventually, an additional orbit parameter correction can be carried out in order to remove possible orbit inaccuracies. In this case another iteration of previous three steps is carried out.

In the following we will deeply analyze the whole procedure depicted in Figure 11.

2.11.1 Pixel selection criterion The first step consists in the selection of the pixels showing a stable

phase screen (i.e., with a relatively small phase standard deviation). This task can be accomplished by considering those pixels that, for a significant percentage of the interferogram sequence, have an estimated spatial coherence value greater than a selected threshold (typically 0.35); this steps leads to the generation of the so-called “spatial coherence mask”.

This pixel selection procedure is aimed to neglect areas with significant decorrelation effects that are particularly difficult to be analyzed, especially in the Phase Unwrapping step.

Since interferometric products are differential measurements, before to start, it is necessary to select one of the identified points as a spatial reference for the whole interferogram set. It would be characterized by high

Figure 11 – Deformation time series processing chain flow chart.

Chapter 2 75

spatial and temporal coherence and its deformation behavior should be a priori known (typically, it is located in a non-deforming zone). Indeed, note that any error affecting this reference point will also affect the overall results; in this context, particularly critical can be the impact of the atmospheric artifacts that are spatially correlated thus to be confused with deformation signals.

2.11.2 Phase Unwrapping The Phase Unwrapping (PhU) step, i.e., the problem to retrieve, starting

from the computed modulo-2π restricted differential phases, the full phase variation signal, the latter associated to the occurred deformation, is probably the most critical task in SAR interferometry (see Appendix A). Phase unwrapping problems occur from aliasing errors due to phase noise caused by low coherence and undersampling phenomena because of locally high fringe rates.

During last years, very efficient procedures have been developed [Goldestein et al., 1988; Ghiglia and Romero, 1994; Fornaro et al., 1996; Ghiglia and Pritt, 1998; Costantini, 1998; Davidson and Bamler, 1999; Chen and Zebker, 2001] to unwrap the phase of single interferogram.

Among these, the one referred to as Minimum Cost Flow (MCF) algorithm [Costantini, 1998; Costantini and Rosen, 1999] permits to unwrap data available in a sparse spatial grid. To achieve this task, it applies a Delaunay triangulation to a defined set of assumed reliable starting points (usually selected via coherence analysis) and reduces the unwrapping to a minimization problem with integer variables (the multiples of 2π). In particular, it searches for the minimum cost flow on a network, for the solution of which there are very efficient algorithms. A more detailed analysis of the MCF basic rationale, for a regular spatial grid, can be found in Appendix B.

However, in a multi-temporal DInSAR scenario, where large interferogram data set are involved, exploitation of temporal relations among the different interferometric phases can help Phase Unwrapping procedures. In particular, the Extended Minimum Cost Flow algorithm presented by Pepe and Lanari [2006] performs a joint analysis of both spatial and temporal constraints on the differential phases, leading to a reduction of unwrapping errors.

2.11.2.1 EMCF approach EMCF procedure exploits the MCF idea and relies on two Delaunay

triangulation, i. e., two networks, to be solved with the minimum cost flow


approach. In particular, we consider a spatial network computed, in the Azimuth/Range ( Az Rg× ) domain, over a set of selected pixels (as in the MCF procedure) common to the whole interferogram data set, i.e., the spatial coherence mask of Section 2.11.1. Moreover, we consider also a “temporal” network computed via a Delaunay triangulation carried out in the Temporal/Perpendicular baseline plain, as presented in Section 2.8, that identifies the set of wrapped interferograms to be properly unwrapped.

Operatively, the EMCF algorithm, for each arc of the spatial network (i. e., connecting spatial neighboring pixels), retrieves unwrapped phase differences for each interferogram, by searching for the minimum cost flow of the “temporal” network. Subsequently, these estimates are used as starting point for the spatial unwrapping of each data pair, implemented again via the MCF approach but carried out this time on the spatial Delaunay triangulation.

By applying this strategy, EMCF permits to efficiently unwrap multi-temporal DInSAR data, with better performances with respect to single interferogram PhU approaches.

Finally, note that not only interferograms belonging to the triangulation are involved in the time series generation, in order to improve the information redundancy (see Section 2.8) and, in some cases, the temporal sampling. In order to manage non-triangulation interferograms, it is possible to consider, for each pixel, a model for the deformation (permitting to reconstruct the unwrapped phase via the application of the SBAS technique). It is clear that the more the model is wrong the more the modulo-2π restriction of the estimated unwrapped phase will differ from the original wrapped one. Therefore, this phase residue can be properly unwrapped for instance by applying the MCF approach.

2.11.2.2 Region-Growing step We explicitly note that, in some cases, also the EMCF algorithm can

operate in critical conditions, thus leading to very sparse grid of unwrapped pixels. For instance, wide spatial scale analysis, where large amount of pixels are involved, can require strong computational efforts as well as highly complexity network solutions, leading to a PhU problem infeasibility. Therefore, relying with a limited number of pixels can overcome this issue.

Another critical condition is represented by highly decorrelated study areas. Indeed, decorrelation reduces the SNR introducing noise in the phase which limits the number of correctly unwrapped pixels.

For these reasons a two pass PhU procedure, configured in a cascade of a global and local PhU algorithm, would be very effective to retrieve

Chapter 2 77

spatially dense unwrapped phase values. Indeed, the first global unwrapping step permits to rely with a limited number of pixels. Subsequently, starting form the latter ones assumed to be correctly unwrapped, the local PhU pass allows increasing spatial density of the retrieved phase.

This two step procedure is implemented in the SBAS processing chain. In particular, the EMCF approach is applied, as global step, to a sparse spatial grid while the local PhU is performed via a Region-Growing [Xu and Cumming, 1999] algorithm.

We now investigate the main rationale of the Region-Growing procedure. Let us start our analysis by identifying two pixel sets. First one involves already unwrapped pixels that will be considered as starting point for the entire procedure; they are usually referred to as Seed Points

( )0 1 1, ,..., QS S S Q O−= ≤S , being O the total number of points in the

considered scene. Second one is relevant to the previously not unwrapped pixels, hereinafter simply referred to as Candidate Points

( )0 1 1, ,..., PC C C P O Q−= ≤ −C . By considering these two sets of pixels, a proper Region-Growing

procedure can be applied, scanning each Candidate Point in order to retrieve its unwrapped phase. Note that, in our case the adopted Candidate Points scanning strategy relies on a spiral path centered at the DInSAR Reference Point location and including all previously selected Candidate Points.

Let be ( )0 1 1, ,..., M −= Φ Φ ΦΦ a sequence of wrapped interferograms,

pC a generic Candidate Point and , 0,..., 1pk pS k K= − a set of pK

neighboring Seed Pixels. For each couple ( ),p pkC S we can estimate the

wrapped phase-gradient vector:

( )( ) ( ) , 0,..., 1pk p pk pWr C S k K= − = −∆Φ Φ Φ (24)

In order to manage eq. (24) we can consider a linear model for the j-th interferogram deformation and topography:

( ) ( )4 4,

sinj

jk k k k j kj j

bm h v h t v

rπ πλ ϑ λ

⊥∆∆ ∆ ≈ ⋅ ⋅∆ + ⋅∆ ⋅∆

⋅ (25)

The first term on the right side of (25) accounts for the inaccuracies on the knowledge of the scene topography h∆ , the second one is related to the deformation velocity difference v∆ over the selected arc. Exploiting (25) we can solve the system:


( )pk k pk kWr= + −∆Ψ m ∆Φ m (26)

where ( ) ( )k p pk kC Wr= −r ∆Φ m is the residual phase gradient (see

Appendix A). Equation (26) permits us to estimate the unwrapped phase gradient of each k-th arc ( ),p pkC S , for a defined value of topography and

deformation velocity. By iteratively varying these values we can calculate the best ( ),k kh v∆ ∆ couple that maximize the periodogram (being 1i = − ):

( )( )1

0exp

M

kj pjperiod

k

i r C

Mγ

−

=

⎡ ⎤⎣ ⎦

=∑

(27)

By integrating, for each k-th arc ( ),p pkC S we obtain the unwrapped

phase value in the selected Candidate Point pC :

( ) ( )k p pk pkC S= + ∆Ψ Ψ Ψ (28)

Note that, ( )k pCr has been assumed to be included in the ( ),π π− interval. In other words, it implies that the residual term is considered already unwrapped; note that, whether this assumption is incorrect the periodogram coherence period

kγ will strongly decrease. Finally, we average different solutions:

( ) ( )1 1

0 0

p pK K

p k k p kk k

C w C w− −

= =

⎛ ⎞ ⎛ ⎞= Ψ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∑ ∑Ψ (29)

where kw are properly selected weights taking account for the k-th solution reliability; kw can be temporal coherence values, as defined in the following eq. (32). Equation (29) represents the final unwrapped phase value for the selected Candidate Point.

We final remark that a novel Region-Growing procedure based on the exploitation of the EMCF concept will be presented in Chapter 5.

In summary, according with Figure 11, we can say that the PhU procedure starts from wrapped phases

1

M

j jϕ

= and generates, for each SAR

pixel, the corresponding unwrapped 1

M

j jψ

= ones.

Chapter 2 79

2.11.3 Mean deformation velocity and residual topography estimation

The unwrapped phase are directly related to the deformation occurred between the corresponding radar sensor flights over the illuminated area. Eq. (2) gives us the expression of the generic j-th differential interferogram computed between two SAR images acquired at Bt and At epochs, here reported for simplicity (with clear meaning of the symbols):

( ) ( ) ( )

( ) ( )

( ) ( )

, , , , ,

4 4, , , ,sin

4 , , , , , 1, , ,

j

j

atm atm j

B A

B A

B A

x r t x r t x r

b zd t x r d t x r

r

d t x r d t x r n j M

ψ ϕ ϕ

π πλ λ ϑπλ

⊥

= −

∆⎡ ⎤≈ − + +⎣ ⎦

⎡ ⎤+ − +∆ ∀ =⎣ ⎦

%

K

We would like just to remark that possible orbital ramps artifacts can be considered as a part of the unknown atmospheric contribution (since orbital errors are image referred as well as atmospheric artifacts) and that the

sinjb r ϑ⊥ term (which may significantly vary over the whole scene) here has been thought, for sake of simplicity, constant.

By considering previous equations, the evaluation of z∆ benefits from the characteristics of the topographic phase component that is correlated

with the perpendicular baseline vector ( )1, ,

Mb b⊥ ⊥K ; note also that the need to

estimate the topographic factor in equation (2) could appear irrelevant due to the hypothesis of small spatial baseline interferograms. However, since the amplitude of local topographic artifacts may significantly exceed the expected DEM accuracy, it can cause, if uncompensated, a remarkable quality degradation of the produced DInSAR fringes.

By the way, we can easily compute the residual topographic term z∆ by imposing that the ( ) ( )atmd d⋅ + ⋅ signal could be approximated with a linear model, which depends on the mean velocity factor v , only. In this way, we can re-arrange equation (2) with respect to the two unknowns ( ),z v∆ , and neglecting noise term, as follows:

4sin

jj

bz v T

rπψλ ϑ

⊥⎡ ⎤≅ ∆ + ∆⎢ ⎥

⎣ ⎦ (30)

which can be solved in the LS sense. As a further remark, we observe that the evaluation of the residual topography z∆ benefits for the presence of


interferometric distribution with significantly large perpendicular baseline values, thus suggesting us to increase the maximum acceptable perpendicular baseline value; on the other hand, the higher are the baseline values the more significant are the noise effects. Therefore, the choice of the optimal maximum baseline separation must be take into account these two opposite requirements.

In summary, the residual topographic pattern can be subtracted to each unwrapped differential interferogram, thus leading to the following expression of the topography-compensated differential interferogram:

( ) ( ) ( )

( ) ( ) ( ) ( )

, , , , ,

4 , , , , , , , ,

1, ,

j

atm atm j

B A

B BA A

x r t x r t x r

d t x r d t x r d t x r d t x r n

j M

ψ ϕ ϕ

πλ

= − =

⎡ ⎤≈ − + − +∆⎣ ⎦

∀ =

%

K

. (31)

2.11.4 Time series generation This step represents the SBAS algorithm key idea and its detailed

rational has been already described in Section 2.3. In this context, we just remark that by exploiting the previously

unwrapped filtered interferograms 1

M

j jψ

= it is possible to obtain the phase

values 1

1

Ni i

ϕ +

= at each SAR epoch, i. e., the temporal evolution of the

deformation for each investigated pixel.

2.11.5 Reconstruction accuracy evaluation Phase unwrapping errors, noisy differential interferograms and

incongruence on the solution of the system of equations (11), may lead to a resulting deformation pattern that is not consistent with the original phase fringes. Accordingly, a procedure allowing us to argue on the correctness of the obtained results has to be introduced.

To this aim, we can first reproduce the original DInSAR phases, starting from the achieved phase terms 1

1

Ni i

ϕ +

= associated to each SAR image;

obviously, within the reconstruction procedure, the previously estimated residual topography must be again taken into account. Then, we introduce the quality index of the deformation retrieval, the so-called temporal coherence factor tempC , defined, for each SAR pixel, as follows:

Chapter 2 81

( )1exp

Mrecj j

jtemp

iC

M

ϕ ϕ=

⎡ ⎤−⎣ ⎦=∑

(32)

where it has been imposed 1i = − to avoid confusion and, 1, ,j M∀ = K ,

jϕ and recjϕ represent the j-th original wrapped interferogram and the

“reconstructed” one, respectively, whose expression is:

( ) ( )4, , , , , ,sin

b jrec x r z x r t x r t x rj IE ISr j j

πϕ ϕ ϕλ ϑ

⎡ ⎤⎛ ⎞ ⎛ ⎞⊥ ⎢ ⎥⎜ ⎟ ⎜ ⎟= ∆ + −⎢ ⎥⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎣ ⎦ (33)

Note that for pixels where 1tempC → , we expect that no errors are present, since a nearly perfect retrieval of the original phase has been obtained, while low values of tempC will correspond to poorly reconstructed data. A pictorial example of this kind of products has been shown in Figure 12.

As a final remark we underline that the availability of the temporal coherence factor tempC leads us to identify the pixels where reliable information have been retrieved. More specifically, they are those characterized by a temporal coherence value greater than a selected

Figure 12 - Interferometric products relevant to the estimate of the obtained result quality. (a) Residual phase term, defined as the difference between the original wrapped and the reconstructed phases. (b) Temporal coherence map represented in a grey scale.


threshold (typically 0.7). Based on the selected threshold, we may identify and discard the pixels that are characterized by significant unwrapping errors and/or that “coherently” contribute only to a portion of the generated interferograms (partially coherent targets) which is not significant for retrieving reliable information.

2.11.6 Atmospheric filtering and orbital ramps estimation

The final step of the SBAS processing chain is aimed on detecting possible atmospheric artifacts. This operation is based on the observation that the atmospheric phase signal component is highly correlated in space but poorly in time [Ferretti et al., 2001; Goldstein, 1995]. Accordingly, the undesired atmospheric phase signal is estimated from the computed time series through the cascade of a lowpass filtering step in the two-dimensional spatial domain followed by a temporal highpass filtering operation. The spatial filtering is implemented by using a box filter with spatial extent of approximately 1 km in azimuth and range, consistent with the spatial correlation length of the atmospheric phase signal [Hanssen, 2001]. This step is followed by a temporal highpass filtering operation performed with respect to the time variable via a triangular window whose temporal extension is typically of 300 days. Following their identification, the atmospheric artifacts are removed and the generation of the final deformation time series is computed; the conversion into a displacement signal is achieved via the multiplication by the correction factor πλ 4/ , see eq. (23).

Moreover, starting from the atmospheric filtered deformation time series 0 1, ,...,filt filt filt

Nϕ ϕ ϕ , it is possible to estimate, for each temporally coherent pixel, the correct mean velocity deformation value. To this aim, we search for the inclination of the line that best fits the deformation time series, i. e, we look for the term α which satisfies the following condition

min filttα ϕ−r r (34)

As an additional remark, we also want to stress that the estimated atmospheric phase signal term could also contain phase terms caused by inaccuracies in the SAR sensors orbit information. Indeed, such errors are also typically not correlated in time but strongly correlated in space and they are often well approximated by spatial ramps usually referred to as orbital ramps. Based on these considerations, we perform in our approach an estimate of these orbital patterns by searching for the best-fit ramp to the

Chapter 2 83

temporal high-pass/spatially low-pass time series signal component; following this step, we remove the detected ramps from each differential interferograms. Once the orbital ramp term have been estimated, they can be effectively subtracted modulo-2π from the original interferograms, thus obtaining a new sequence of differential interferograms that can be subsequently re-processed via the same strategy up to now described.

2.12 Geocoding All the SBAS deformation products are obtained onto the SAR reference

geometry but they can be represented with respect to an easier output reference grid (for example, a cartographic reference system); in this case, we say that a “geocoding” operation has to be performed [Schreirer 1993; Sansosti, 2004]. To do this, the knowledge of the scene topography, eventually achieved directly onto the SAR geometry via the application of InSAR techniques and/or properly corrected via the procedure discussed in Section 2.11.3, is needed.

The problem to locate the target on the ground, that is to individuate its coordinates with respect to the Cartesian reference system we considered (see Section 2.7), can be solved if the height of the target (with respect to the reference ellipsoid) is a priori known. In this way, each SAR pixel can be transformed in a corresponding point of coordinates ( ), ,x y z in the Cartesian reference system, by searching for the solution of the following, non linear system of equations with respect to the P unknowns:

( )

( ) ( )

2 2 2

2 2

( )

( ) ( ) 0

1

r t

t t

x y za h b h

= −

⋅ − =

++ =

+ +

s P

v s P (35)

where the last equation accounts for the fact that the target has a known height h with respect to the reference ellipsoid. Subsequently, the target position P can be referred to a geographic system or transformed into a cartographic projection (for example, the Universal Transverse Mercator (UTM) projection). In Figure 13, the SAR amplitude image of the Napoli bay area (Italy), represented with respect to the SAR reference geometry (Figure 13a) and the UTM cartographic grid (Figure 13b), respectively, is shown.


2.13 SBAS technique application: the Long Valley caldera test site

In this section we show a sketch of the SBAS technique capability to retrieve ground displacements. In particular, we exploit SAR data relevant to the Long Valley caldera (California) area that have been acquired by the ERS-1/2 satellites in the 1992-2000 time interval from descending tracks. This section is part of the work presented by Tizzani et al. [2007].

Figure 13 – SAR amplitude image of the Napoli bay (Italy) area, represented with respect to the SAR reference geometry (Figure 13a) and the UTM cartographic grid (Figure 13b).

Chapter 2 85

The Long Valley caldera, an east-west elongate oval depression with an extent of approximately 17 km by 32 km, was formed about 0.76 Ma B.P. [Hill and Prejean, 2005]. The most recent activity in the area was characterized by a significant unrest period between 1997 and 1998. In this case, an inflation phenomenon slowly started in mid-1997 and then exponentially increased in late 1997, before rapidly returning to quiescence by mid-1998. The effects of this unrest episode were recorded via the exploitation of several geophysical instruments deployed in the area

Figure 14 – LOS mean deformation velocity map of Long Valley caldera superimposed on a 75 x 75 km DEM of the area. Caldera area and resurgent dome have been identified by white and yellow dashed lines, respectively. White triangles identify the location of the plots presented in Figure 15. Black lines represent two leveling paths. Black box encloses the Casa Diablo geothermal area highlighted in the inset. White square is the reference SAR pixel.


(including seismometers, EDM, dilatometers, tiltmeters and GPS) and via a leveling campaign that was carried out in 1997, following those performed in 1995 and 1992 [Battaglia et al., 2003a, b, c; Langbein, 2003].

In order to investigate the ongoing phenomena via the SBAS approach, we processed a set of 21 descending orbit SAR images (Track 485, Frame 2845), acquired by the ERS-1/2 sensors from June 1992 to October 2000, that we used to produce our interferograms. In particular, a Shuttle Radar Topography Mission (SRTM) Digital Elevation Model (DEM) [Rosen et al., 2001] of the study area and precise orbital information [TU-Delft Web] were used for the interferograms generation. Moreover, a complex multilook operation was performed with 4 range looks and 20 azimuth looks, resulting in a final pixel size of approximately 100 m by 100 m.

Let us now illustrate the computed SBAS results. In order to provide an overall picture of the detected deformation pattern, we present the geocoded InSAR mean deformation velocity map superimposed on a 75 km by 75 km shaded relief of the DEM of the study area (Figure 14). The velocity map was computed with respect to a reference pixel represented by a white square and is relevant to coherent pixels only. Note that these pixels have been identified by using the temporal coherence map, see eq. (32); in greater than 0.7, that is a typical value in InSAR applications [Borgia et al., 2005; Casu et al., 2006].

The large uplifting area, clearly visible in Figure 14, has a signal dominated by the deformation occurring in correspondence to the Long Valley caldera. We remark that the applied SBAS approach is able to provide information not only on the mean displacements, but also on the temporal evolution of the detected deformation; accordingly, some additional results are presented.

In particular, in Figure 15a is shown the deformation time series relevant to the pixel labeled as RD in Figure 14, corresponding to the maximum deforming area within the resurgent dome zone. From the analysis of this plot it clearly appears that the deformation pattern is characterized by the sequence of three different effects: a 1992-1997 uplift background, a 1997-1998 unrest phenomenon and a 1998-2000 subsidence phase. Note that these three different effects have been highlighted in Figure 15a where, for each time interval, the best fit linear trend of the InSAR time series has been plotted.

We further remark that, by analyzing the retrieved deformation time series of the Long Valley caldera and surrounding area, it appears that the shape of the deformation time series inside the caldera (passing from the background to the unrest and, finally, to the subsidence phase) is also

Chapter 2 87

retrieved externally to the caldera rims (see for instance the plot in Figure 15b for the pixel labeled as NC in Figure 14).

To better clarify this issue, we have used the available DInSAR results in order to detect the pixels with a temporal deformation behavior that is highly correlated (we considered the linear Pearson correlation coefficient [Stanton, 2001]) with the time series of the pixel RD located within the resurgent dome zone (see Figure 14). The mean deformation velocity map of the pixels with a correlation value greater than 0.95 is shown in Figure 16. It is evident that, even for a rather restrictive correlation value (0.95), there are areas located outside the northern part of the external slopes of the caldera whose deformation behavior is strongly correlated with that of the resurgent dome.

Finally, we note the temporal evolution of deformation in the Casa Diablo geothermal area within the Long Valley caldera (Figure 14, corresponding to the plot labeled as CD in Figure 15c). In this case, it clearly appears that the general subsidence trend, due to the geothermal station

Figure 15 – LOS deformation time series for the pixels marked by the white triangles labeled in Figure 14 as RD (Figure 15a), NC (Figure 15b) and CD (Figure 15c), respectively. Note that in Figure 15a the best fit linear trends of the DInSAR time series relevant to the 1992-1997 uplift background (dashed line), 1997-1998 unrest phenomenon (continuous line) and 1998-2000 subsidence phase (dotted line) have been plotted.


activity, is interrupted by the caldera uplift during the unrest episode. In summary, the presented results clearly underline the displacement

retrieval capability of the SBAS approach. In order to confirm the validity of the achieved results we carried out a

comparison between the DInSAR and leveling network present in the investigated area. We identified (Figure 14) two paths, following the Highways 395 and 203, respectively, for which the data for 103 benchmarks were available to us [Battaglia et al., 2003c]. For each leveling line, we plotted (Figure 17) the cumulative displacements for 1992–1997 evaluated from the SBAS (red triangles) and the leveling (black asterisks) data. The former was computed for coherent pixels only and the latter projected in the radar LOS. We have assumed that the detected deformation was vertical, since the leveling measurements can only reveal the vertical component of the displacements. The above assumption is very likely responsible for some

Figure 16 – Shaded relief of the DEM (grey scale) with superimposed the DInSAR mean LOS deformation velocity map (in color) relevant to the area including the pixels showing a temporal deformation behavior with a correlation value greater than 0.95, with respect to the time series (see Figure 15a) of the pixel RD located within the resurgent dome zone. Note that, at variance of Figure 14, no saturation of the deformation velocity values has been applied. The location of the pixel NC of Figure 14 is also highlighted.

Chapter 2 89

differences between the SAR and leveling data, which appear in the right hand side portion of the plots of Figure 17. Note that in these areas, significant horizontal deformation is present as discussed by Battaglia et al. [2003a] and Langbein [2003]. By the way, a rather good agreement between the deformation profiles available from the leveling data and from the DInSAR measurements is observed. We have also computed for the coherent pixels the standard deviation of the differences between the InSAR and the LOS-projected leveling measurements shown in Figure 17; the obtained standard deviation value is of approximately 1.5 cm. A more detailed analysis on the SBAS accuracies will be carried out in Chapter 3.

Figure 17 – Comparison between the cumulative displacements computed in the 1992–1997 time interval from the DInSAR (red triangles) and the leveling (black asterisks) data, the latter projected in LOS, for the two leveling lines labeled in Figure 14 as HW 395 (Figure 17a) and HW 203 (Figure 17b), respectively. The location of the Casa Diablo site is also highlighted.


Appendix A Phase unwrapping basic rationale Let us now introduce Phase unwrapping key aspects and basic concepts.

First of all, let us refer to a single differential interferogram and denote with ( ),x yϕ the measured interferometric phase related to the point of SAR

coordinates ( ),x y ; ( ),x yϕ is modulo-2π restricted (see Chapter 1). Essentially, Phase Unwrapping consists in the retrieval of the original interferometric phase ( ),x yψ from the modulo-2π restricted one, i. e., it consists in the evaluation of the 2π-multiple integers ( ),H x y to be added to the wrapped phase in order to retrieve the unwrapped one. In analytical terms it means:

( ) ( ) ( ), , 2 ,x y x y H x yψ ϕ π= + . (A1)

Therefore, the aim of the phase unwrapping procedure is to retrieve a measure of the full phase through the observation of its wrapped value. To solve this problem, additional information must be employed. First of all, we may observe that, since the interferograms are sampled, they can be viewed as functions of discrete coordinates ( ),i j (that is, for example, ( ),i jψ ψ= ), being

0

0 2

s

samp

vx x i

PRFcy r j

f

= +

= + (A2)

the expression of the corresponding SAR coordinates, wherein (see Chapter 1) 0x is the azimuth of the first line, sv the sensor velocity, PRF the pulse repetition frequency, 0r is the near range, c is the speed of the light and, finally, sampf is the range sampling frequency.

Basically, existing phase unwrapping techniques relies on the fact that it is possible to determine the discrete counterpart of the partial derivatives of unwrapped phase (i. e., the neighboring pixel differences) when these differences are less than π in absolute value. Consequently, we can introduce the “measurable” phase gradient vector, defined as follows

, ,ˆ ˆ ˆx yπ π π π

ψ ψ− −

∇ = ∆ + ∆ψ x y (A3)

whose components with respect to the two spatial axis x and y are:

Chapter 2 91

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ), , ,

, ,,

, 1, , 1, ,

, , 1 , , 1 ,

x

y

i j i j i j i j i j

i j i j i j i j i jπ π π π π π

π π π ππ π

ψ ψ ψ ϕ ϕ

ψ ψ ψ ϕ ϕ− − −

− −−

∆ = + − = + −

∆ = + − = + − (A4)

being ,π π−

⋅ representative for the modulo-2π operation. In other words, the measurable phase gradient vector has been estimated by wrapping possible phase differences greater than π in absolute value in the ( ),π π− interval, by adding the correct multiples of 2π and implicitly assuming that, in a properly sampled interferogram, the phase differences of adjacent samples are likely to be restricted to the ( ),π π− interval.

However, in some points there is a non null probability that the phase difference exceed π in absolute value. It can depend both on the noise level and, eventually, on the slope of the topography. This can cause the estimated unwrapped phase discrete derivatives to be inconsistent. In other words, they do not form an “irrotational” vector field, that is:

Figure 18 - Representation of the key point of the Phase Unwrapping problem. To connect two points on the space it is possible to follow different paths but, any of these, could cross critical areas where the wrapped phase differences are in module greater than 2π.


( )ˆ 0ψ∇× ∇ ≠ (A5)

Eq. (A5) implies that phase integration will be path-dependent. A pictorial view of the phase gradient inconsistencies is depicted in Figure 18.

The relevant curl (hereafter referred to as residue field) can be expressed as (see eq. (A5) and Figure 19):

( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( ),,

, ,, ,

ˆ, , , ,

, 1, , 1 ,

x y y x

x y x y

r i j i j i j i j

i j i j i j i j

π ππ π

π π π ππ π π π

ψ ψ ψ

ψ ψ ψ ψ

−−

− −− −

⎡ ⎤ ⎡ ⎤= ∇× ∇ = ∆ ∆ −∆ ∆ =⎣ ⎦⎣ ⎦

= ∆ + ∆ + − ∆ + − ∆(A6)

and can assume either zero (no residues) or 2π± (positive or negative residue, respectively) value. Since errors on the phase gradient estimate are localized and come in integer multiples of 2π , the residue field can be profitably used to reconstruct the full phase term.

Figure 19 - Representation of the residue field r(i,j).

Chapter 2 93

Appendix B MCF algorithm basic rationale Among several PhU algorithms, very popular are those referred to as

Branch-cut ones. They are based on the integration of the estimated neighboring pixel differences of the unwrapped phase along conservative paths (branch), the latter being delimited by cuts that avoid integration in regions where the estimated differences are inconsistent (see Figure 18).

The problem of building cuts (in the following referred to as branch-cuts) delimiting these regions is very difficult and the resulting phase unwrapping algorithm is very computationally expensive. However, we may exploit the fact that the neighboring pixel differences of the unwrapped phases are estimated with possibly an error which is an integer multiple of 2π . This circumstance allows formulating the phase unwrapping problem as the one of minimizing the weighted deviations between the estimated and the unknown neighboring pixel differences of the unwrapped phases, with the constraint that the deviations must be integer multiple of 2π . With this constraint, the unwrapping results will not depend critically on the weighting mask we used, and errors are prevented to spread. Equation Section 2

Minimization problems with integer variables are usually computationally very complex. However, by recognizing the network structure underlying the phase unwrapping problem, it is possible to employ very efficient strategies for its solution. Indeed, the problem can be considered equivalent to the one of finding the minimum cost flow on a network, for the solution of which there are very efficient algorithms. This is the key idea of the Minimum Cost Flow (MCF) algorithm [Costantini, 1998; Costantini and Rosen, 1999].

In order to explain its basic principles and clarify how it can be performed, we refer to the unknown, unwrapped phase field and we impose its discrete derivatives to be consistent (see Appendix A), thus requiring the irrotational property of the phase field ψ∇ :

( ) ( )( ) ( )( )( ) ( ) ( ) ( )

, , ,

, 1, , 1 , 0x y y x

x y x y

i j i j i j

i j i j i j i j

ψ ψ ψ

ψ ψ ψ ψ

∇×∇ = ∆ ∆ −∆ ∆ =

= ∆ +∆ + −∆ + −∆ = (B1)

Obviously, we can also express each term of equation (B1) with respect to the wrapped phase derivates by introducing, for each phase term, a corresponding, unknown 2π-multiple term, as follows


( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

,

,

,

,

, , 2 ,

1, 1, 2 1,

, 1 , 1 2 , 1

, , 2 ,

x x x

y y y

x x x

y y y

i j i j K i j

i j i j K i j

i j i j K i j

i j i j K i j

π π

π π

π π

π π

ψ ψ π

ψ ψ π

ψ ψ π

ψ ψ π

−

−

−

−

∆ = ∆ +

∆ + = ∆ + + +

∆ + = ∆ + + +

∆ = ∆ +

(B2)

These relations, properly substituted in the equation (B1) finally lead to the following equation:

( ) ( ) ( ) ( ) ( ),, 1, , 1 ,

2x y x yr i j

K i j K i j K i j K i jπ

+ + − + − = − (B3)

that relates the ( ) [ ], ,qK i j q x y∈ unknown terms to the measurable residues

( ),r i j . At this stage, the phase unwrapping problem can be formulated as the search for the K terms, satisfying the constraints (B2), which solve the following minimization problem:

( ) ( ) ( ) ( )min , , , ,x y

x x y yK K i j i jc i j K i j c i j K i j

⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

+∑∑ ∑∑ (B4)

wherein ( )c ⋅ are the so-called cost functions allowing us to individuate areas where the location of branch-cuts is likely or unlikely. Cost functions are essentially expressed as a function of the estimated local interferogram quality (by exploiting the spatial coherence, or the phase gradient density or other properly identified quality factors). The problem given in (B4) is a non-linear minimization problem with integer variables, and, if the following change of variables is considered:

( ) ( )( ) ( )( ) ( )( ) ( )

, min 0, ,

, max 0, ,

, min 0, ,

, max 0, ,

x x

x x

y y

y y

K i j K i j

K i j K i j

K i j K i j

K i j K i j

−

+

−

+

⎡ ⎤⎣ ⎦⎡ ⎤⎣ ⎦⎡ ⎤⎣ ⎦⎡ ⎤⎣ ⎦

=

=

=

=

(B5)

it can be re-formulated via two different linear problem, as follows

Chapter 2 95

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

min , , , ,

, , , ,

x yx x y yK K i j

x x y yi j

c i j K i j c i j K i j

c i j K i j c i j K i j

+ +

− −

⎧⎪ ⎡ ⎤⎨ ⎣ ⎦⎪⎩⎫⎪⎡ ⎤⎬⎣ ⎦⎪⎭

+ +

+ +

∑∑

∑∑ (B6)

It can be seen that the problem stated in (B6) can be transformed so that it defines a minimum cost flow problem on a network (see Figure 20), the new variables representing the net flow running along the network arcs.

Once the network has been solved, the solutions in terms of the 2π-multiple integer functions will be expressed by:

Figure 20 - The equivalent network associated to phase unwrapping problem. Circles and arrows represent network nodes and arcs, respectively. Note that boundaries arcs are connected to the “ground” node (reference node), by analogy with electrical networks [Costantini, 1998].


( ) ( ) ( )( ) ( ) ( )

, , ,, , ,

x x x

y y y

K i j K i j K i jK i j K i j K i j

+ −

+ −

= −= −

(B7)

and, finally

ˆ 2 Kψ ψ π∇ =∇ + . (B8)

The full phase gradient vector allows us to individuate the phase associated to each pixel because we can choose a generic spatial path to integrate it, starting from the selected reference point position.

The presented algorithm can be easily generalized to investigate a set of sparse grid pixels, as stated by Costantini and Rosen [1999]. In this case a different spatial network connecting the pixels has to be selected, for instance the Delaunay [Delaunay, 1934] triangulation one.

Chapter 2 97

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Chapter 3

SBAS Performance Assessment We now address the performance of the Small BAseline Subset (SBAS)

algorithm, previously described in Chapter 2. In particular, we have carried out a quantitative assessment of the SBAS procedure performance by processing SAR data acquired by the ERS sensors and comparing the achieved results with geodetic measurements that are assumed as reference. The analysis has been focused on the Napoli bay (Italy), Los Angeles (California) and Hayward fault (California) test areas, where different deformation phenomena are present and, at the same time, a large amount of ERS SAR data is available as well as geometric leveling (in Napoli zone), continuous GPS (in Los Angeles area) and alignment arrays (along Hayward fault, San Francisco Bay area) measurements, to be used for our performance analysis. Moreover, due to the presence of large urbanized zones, the selected test sites are also characterized by extended, highly coherent areas in DInSAR maps.

The presented study shows that the SBAS technique provides an estimate of the mean deformation velocity with a standard deviation of about 1 mm/year, for a typical ERS data set including between 40 and 60 images. Moreover, the single displacement measurements, computed with respect to a reference point of known motion, show a sub-centimetric accuracy with a standard deviation of about 5 mm. We final remark that the results provided in this Chapter have been also presented in Casu et al. [2006] and Lanari et al. [2007].

3.1 Analysis overview SBAS algorithm capability to generate mean deformation velocity maps

and displacement time series from ERS SAR data has been already exploited in different works [Berardino et al., 2002; Lanari et al., 2002; Lanari et al., 2004a; Lanari et al., 2004b; Lundgren et al., 2004; Borgia et al., 2005; Manzo et al, 2006]. In these studies some comparisons between the achieved DInSAR products and geodetic measurements have been shown in order to confirm the presented results but no extensive analysis on the quality of the DInSAR measurements have been carried out. Accordingly, the key idea of

102 SBAS Performance Assessment this chapter is to provide a quantitative assessment of the SBAS algorithm performance; in particular, we concentrate on the basic SBAS technique that has been originally developed to investigate large spatial scale displacements by exploiting low-pass filtered (multilook) DInSAR interferograms [Rosen et al., 2000].

Our analysis is focused on three test areas where different deformation phenomena are present and, at the same time, a large amount of SAR data is available as well as ground measurements to be used for an extensive SAR/geodetic data comparison. Note also that, due to the presence of large urbanized zones, the selected test sites are also characterized by extended, highly coherent areas in the DInSAR maps. We stress that the goal of the presented study is not the investigation of the detected deformation processes, which have been already analyzed in previous works exploiting the DInSAR technology. On the contrary, we benefit of the knowledge of these phenomena and of the availability of a large data set of SAR and geodetic measurements, the latter assumed as reference, for carrying out the assessment of the SBAS procedure performance.

In particular, the first test site is located in the Napoli bay (Italy) area which includes the highly urbanized zone of the city of Napoli and three active volcanoes (the Campi Flegrei caldera, the Somma-Vesuvio volcanic complex and the Ischia island) whose overall deformations have been already studied via DInSAR techniques [Avallone et al. 1999, Tesauro et al. 2000, Lundgren et al. 2001, Lanari et al. 2002, Lanari et al. 2004a, Lanari et al. 2004c, Beauducel et al. 2004, Borgia et al. 2005]. In this area a very large spirit leveling network is present, including several hundreds of benchmarks with repeat measurements that are systematically carried out (INGV-OV, 2001, 2002, 2003). Moreover, a huge SAR data set relevant to ascending and descending tracks of the ERS-1 and ERS-2 sensors, spanning the time interval from 1992 until 2003, is available to us. The availability of these SAR data permits to analyze the temporal behavior of the detected deformations and, at the same time, to discriminate vertical and East-West displacement components. The former have been compared with the measurements available from the leveling campaigns in areas where both SAR and geodetic data were available.

The second test area is the Los Angeles (California) metropolitan zone which is a tectonically active region with surface deformation that is a combination of natural and anthropogenic signals. Also in this case a large number of ERS SAR data is available, in particular acquired from descending orbits, that have been already exploited to investigate the ongoing phenomena [Bawden et al. 2001, Watson et al. 2002, Colesanti et al. 2003, Lanari et al. 2004b and Argus et al. 2005]. Moreover, a very large

Chapter 3 103 amount of geodetic measurements has been recorded through the Southern California Integrated GPS Network (SCIGN), see Hudnut et al. [2001]. Accordingly, a detailed comparison between the DInSAR and the GPS measurements has been carried out in this case.

Finally, the third test area is relevant to the Hayward fault which is running beneath the eastern San Francisco Bay Area, California, and it is continuously creeping by varying amounts along most of its length. Fault creep phenomena and in particular those relevant to the Hayward fault have already been the subject of DInSAR studies [Bürgmann et al., 2000; Schmidt et al., 2005; Bürgmann et al., 2006], via the exploitation of large ERS descending data set. Also in this test area, a wide geodetic network is present. In particular, several alignment arrays are placed along the fault in since mid sixties [Lienkaemper et al. 1997; Lienkaemper et al. 2001].

As a result of the overall DInSAR/geodetic measurements comparison, a quantitative assessment of the SBAS procedure performance for surface deformations retrieval has been finally provided.

3.2 Leveling vs. SBAS: the Napoli bay This section is dedicated to present the results of the comparison

between deformation measurements retrieved via the SBAS technique and those available from geometric leveling campaigns, the latter carried out in the Napoli bay area network [Casu et al., 2006].

3.2.1 Napoli bay test site description Neapolitan Volcanic area includes three active volcanoes: the Campi

Flegrei caldera, the Ischia island and the Somma-Vesuvio complex, all characterized by a high hazard degree because 2.5 million people live under their threats; this scenario clearly explains the necessity of providing these areas with monitoring networks in order to reveal the resumption of volcanic phenomena.

3.2.2 Leveling network In our study we focused on the portion of the Napoli bay area extending

from the NW border of the Campi Flegrei caldera up to the SE margin of the Vesuvio volcano, thus including the entire Napoli city area, see Figure 1. In this zone one of the most extended geometric leveling network of the world, with about 600 benchmarks and more than 350 km of linear extension, is present [INGV-OV, 2001, 2002, 2003]. This network (Figure 1) managed by

104 SBAS Performance Assessment the Osservatorio Vesuviano (OV) belonging to the Italian National Institute of Geophysics and Volcanology (INGV), is a very important element of the surveillance system of this area because it allows to achieve measurements of the vertical component of the deformations with a high accuracy [INGV-OV, 2003]. In particular, the spirit leveling network component relevant to the Campi Flegrei area includes 300 benchmarks, extending for about 120 linear km, organized into 11 linked loops; the reference benchmark is located in the Napoli city (more precisely in the Mergellina Harbor zone, marked by LNA001 in Figure 1) and the whole network is periodically surveyed.

The portion of the network dedicated to monitor the Somma-Vesuvio

volcano complex consists of about 290 benchmarks, extending on 220 linear km, organized in 15 closed circuits partially overlapping with the Campi Flegrei network component. Also the whole vesuvian leveling network is periodically measured and all the benchmarks are typically referred to the one located in the Sorrento Peninsula (benchmark LVE078, see Figure 1); however, in our study we have re-referred all the available measurements to the same benchmark LNA001, located in the Napoli city, in order to have a single reference point consistent with what occurs for the DInSAR results.

Figure 1 - SAR amplitude image of the investigated portion of the Napoli bay (Italy) area. The benchmarks of the existing leveling network have been identified by black squares; moreover, the main locations within the area have been also highlighted. Note that the two benchmarks labeled LNA001 and LVE078 represent the reference points for the Campi Flegrei and the Vesuvio leveling network portions; respectively. The inset in upper right corner shows the location of the study area.

Chapter 3 105 3.2.3 SBAS measurements

As far as the SAR measurements is concerned, we used in our study 116 SAR data acquired by the European Space Agency (ESA) ERS-1/2 sensors. In particular, 58 acquisitions are relevant to ascending passes (track: 129, frame: 809) spanning the January 1993 - May 2003 time interval; the remaining 58 data have been acquired from descending orbits (track: 36, frame: 2781) between June 1992 and October 2002. Each interferometric SAR image pair has been chosen with a perpendicular baseline value smaller than 300 m and with a maximum time interval of 4 years; precise satellite orbital information and a Shuttle Radar Topography Mission (SRTM) DEM [Rosen et al., 2001] of the study area have also been used. Starting from the available SAR data set, we have computed 133 interferograms from the acquisitions relevant to the ascending orbits, while 148 interferograms have been produced from the descending orbits data. All DInSAR products have been obtained following a complex multilook operation [Rosen et al., 2000] with 4 looks in the range direction and 20 looks in the azimuth one, with a resulting pixel dimension of the order of 100 x 100 m.

As a first result achieved by applying the SBAS approach, we present in Figure 2a-b the estimated radar line of sight (LOS) mean deformation velocities relevant to the ascending and descending SAR acquisitions, respectively. They have been computed with respect to a reference pixel located in the Napoli Harbor zone, corresponding to the reference leveling benchmark LNA001 of Figure 1. Note also that the DInSAR products shown in Figure 2a-b have been geocoded and superimposed on a SAR amplitude image of the zone; moreover, the noisy areas with low accuracy measurements have been excluded.

By considering Figure 2a-b we remark that reliable information are available only on urbanized zones and rocky areas. Regarding the detected main deformation patterns, we observe that significant LOS displacements are visible on several coherent areas which can be easily identified: the Campi Flegrei caldera, on the left hand side, various deforming zones within the city of Napoli approximately located in the image centre and other displacements occurring on the top and around the base of the Vesuvio as well as in the eastern and south-eastern sectors of the image.

Moreover, we also stress that the availability of both ascending and descending data allows us to detect not only the LOS ground deformations but also to discriminate vertical and east-west displacement components, as already shown by Lundgren et al. [2004] and Borgia et al. [2005]. In particular, the difference between the mean surface velocity maps computed from the ascending and descending orbits (see Figure 2a-b, respectively) on

106 SBAS Performance Assessment

Figure 2- Mean deformation velocity maps: a) LOS deformation map computed in coherent areas of the ascending ERS data set, spanning the January 1993 - May 2003 time interval, and superimposed on the SAR amplitude image of the Napoli bay area; b) LOS deformation map computed from the descending ERS data acquired in the time period between June 1992 and October 2002; c) east–west deformation component computed from the ascending and descending velocity patterns shown in Figures 2a and 2b, respectively, for those pixels which are common to both maps; d) vertical deformation component map. Note that the red color corresponds in a) and b) to an increase of the sensor-pixel distance, in c) to a displacement toward west, in d) to a subsidence effect. Moreover, the black square in Figures 2a-2d identifies the reference SAR pixel for all DInSAR measurements that has been chosen in correspondence of the reference leveling benchmark LNA001 of Figure 1.

Chapter 3 107 pixels that are common to both maps, allows us to get an estimate of the surface deformation velocities in the east-west direction, see Figure 2c. In addition, the sum of the ascending and descending deformation patterns allows us to get a picture that is mostly vertical motion, see Figure 2d. More analytical details can be found in Appendix A as well as in Manzo et al. [2006].

3.2.4 Data comparison The availability of the estimated vertical deformation component shown

in Figure 2d represents a key element of our study because it allows us to carry out a quantitative comparison between DInSAR results and vertical displacement measurements available from the previously described leveling network. The basic idea, in the following analysis, is to benefit of the space-time information available from the SAR data and, for coherent SAR pixels common to both ascending and descending maps and located in correspondence to leveling benchmarks, to carry out the following two operations: first of all comparing the mean vertical displacement velocities estimated from the SAR and the geodetic data; subsequently, in areas characterized by a nearly vertical deformation only, comparing the time series achieved from the SAR data relevant to the ascending and descending orbits with those available from the leveling measurements, the latter projected in the radar LOS 1 to be consistent with the DInSAR data.

In particular, our comparison is focused on the benchmarks of the leveling network identified by the six lines shown in Figure 3a. These lines, for which leveling measurements spanning the 1990-2003 time interval have been considered because of their temporal overlap with the overall available SAR data set, represent the part of the whole leveling network shown in Figure 1 for which measurements were made available to us.

As concerns the first issue of our comparison, the achieved results are presented in the plots of Figure 3b-g for each leveling line. They clearly show the good agreement between the two mean vertical velocity measurements (SAR: red triangles, leveling: black stars). Moreover, a detailed list of the estimated DInSAR E-W and vertical mean velocity displacements is provided in Table I (see the second and the third columns from the left, respectively) for all coherent pixels associated to the leveling benchmarks. In addition, in Table I it is also reported the vertical deformation velocity computed from the leveling measurements within the interval common to the SAR data and the corresponding difference between

1 A 23° mean incidence angle is assumed in this case.

108 SBAS Performance Assessment the SAR and the leveling data (see the fourth and the fifth columns from the left, respectively).

Based on these last results shown in the right hand side column of Table I, we have computed the standard deviation value, say vσ , of the difference

Figure 3 - SAR/leveling vertical mean deformation velocities comparison: a) SAR amplitude image of the investigated zone with superimposed a selection of leveling network lines identified by different colored squares, for which the geodetic measurements are available to us. The location of the reference leveling benchmark LNA001 is highlighted as well as the beginning (square) and the end (triangle) of each line; b-g) plots of the mean vertical deformation velocity computed from the SAR data (red triangles) compared to the corresponding one relevant to the leveling measurements (black stars) along the lines identified in Figure 3a by the green, red, cyan, blue, yellow and violet squares, respectively. The standard deviation value of the difference between SAR and leveling vertical velocities has been reported in each plot.

Chapter 3 109 between SAR and leveling vertical velocities; we obtained vσ = 1.0 mm/year, corresponding in LOS to about 0.9 mm/year. Note also that about 76% and 96% of the differences between SAR and leveling vertical velocities are included within the ( ),v vσ σ− and ( )2 , 2v vσ σ− intervals, respectively.

For what concerns the second issue of our comparison, we have identified all the leveling benchmarks corresponding to coherent pixels in both ascending and descending maps, where the amplitude of the estimated horizontal mean deformation velocity (see the second column from the left of Table I) is significantly small and, in particular, smaller than a selected threshold that in our case is assumed of 2.5 mm/year; the remaining benchmarks are not considered in the following analysis. This implies that, due to the deformation phenomena affecting the Campi Flegrei caldera [Lundgren et al., 2001] and the Vesuvio volcano flank [Borgia et al., 2005], see Figure 2c, several benchmarks characterized by displacements with significant horizontal components have been discarded.

The leveling benchmarks identified after this selection process have

been superimposed to the vertical mean deformation velocity map of Figure 2d, as shown in Figure 4. Note that for these pixels we can now compare the temporal evolution of the detected deformation from both the ascending and

Figure 4 - Locations of the leveling network benchmarks (black and white squares) characterized by a dominantly vertical displacement, superimposed on the mean vertical deformation velocity map of Figure 2d. White squares mark the selected benchmarks relevant to the following plots shown in Figure 5.

110 SBAS Performance Assessment descending orbits with the leveling measurements projected in the radar line of sight. In this case, we have provided first a qualitative comparison on a selection of twelve benchmarks with different spatial locations and features in their time series; they are labeled in Figure 4 as LCF/25A, LCF/030, LCF/087, LCF/092, LCF/101, LCF/234, LCF/236, LCF/061, LVE083/063, LVE083/067C, LVE034 and LVE010, respectively. For each of these benchmarks we have superimposed the line of sight projected leveling measurements and the deformation time series computed from the ascending and descending data. These comparisons are shown in Figure 5.

Figure 5 - Comparison between the DInSAR LOS deformation time series (ERS ascending data: blue triangles, ERS descending data: red triangles) and the corresponding leveling measurements projected on the radar LOS (black stars) for the pixels labeled in Figure 4 as LCF/25A (a), LCF/030 (b), LCF/087 (c), LCF/092 (d), LCF/101 (e), LCF/234 (f), LCF/236 (g), LCF/061 (h), LVE083/063 (i), LVE083/067C (j), LVE034 (k) and LVE010 (l), respectively. The standard deviation value of the differences between the SAR (ascending and descending) and LOS-projected leveling measurements has been reported in each plot.

Chapter 3 111

By observing the presented plots, it is clear the strong similarity between the temporal evolution of the ascending (blue triangles) and descending (red triangles) SAR measurements, supporting the previous assumption of a dominantly vertical displacement, as well as their agreement with the LOS-projected leveling (black stars) deformation time series.

Moreover, we have also carried out a quantitative assessment of the DInSAR time series quality. To achieve this result, we have systematically compared all the DInSAR and LOS-projected leveling time series, the latter interpolated via a linear regression within the interval common to the SAR data, for all the benchmarks identified in Figure 4 by the black and white squares. In particular, we have computed the standard deviation value of the difference between the two time series; the obtained results are shown in Table II. Note that if we compute the average value of the standard deviations relevant to the differences between SAR and LOS-projected leveling measurements, we obtain that they are consistent between ascending and descending products and correspond to

L Adσ = 4.6 mm for the

ascending and L D

dσ = 4.8 mm for the descending data, respectively.

Accordingly, we may finally assume as conclusive value for the standard deviation of the difference between SAR and leveling data the factor

Ldσ = 4.7 mm, obtained by further averaging the standard deviation values

of the ascending and descending results. Note also that 60% and 96% of the differences between ascending SAR and leveling data are included within

the ( ),L LA A

d dσ σ− and ( )2 ,2L LA A

d dσ σ− intervals, respectively; the

homologous values for the descending products are nearly identical, corresponding to 60% and 98%, respectively.

We want to finally stress that, in our deformation time series analysis, by discarding points where significant horizontal deformation components have been detected (in particular, with amplitude greater than 2.5 mm/year), we are not eliminating “bad pixels”, i.e. those affected by significant errors. To verify our statement, we have shown that the discarded pixels have essentially the same accuracy, for what concerns the retrieved deformation measurements, of the remaining investigated points; of course, this assessment could be done for the measured mean deformation velocity analysis only. To achieve this task, we have divided the overall investigated coherent pixels (93 in total) in two portions; the first one, involving 44 pixels, is relevant to the points characterized by an estimated horizontal mean deformation velocity amplitude smaller than 2.5 mm/year. The second portion corresponds to the remaining 49 pixels characterized by an estimated

112 SBAS Performance Assessment horizontal mean deformation velocity amplitude greater than 2.5 mm/year. For these two classes of data, we have computed the standard deviation of the difference between the computed DInSAR and leveling mean vertical deformation velocities. In the former case we obtained a standard deviation

vσ = 0.92 mm/year, while in the latter we got vσ = 1.09 mm/year. These two results are consistent and in very good agreement with the already computed overall standard deviation value. Accordingly, by removing from our deformation time series analysis the pixels characterized by an horizontal velocity amplitude greater than 2.5 mm/year, we may expect that we are not excluding points with large errors (the errors are of the same order of those affecting pixels with small horizontal deformation components), but simply pixels that can not be really investigated. Indeed, due to the horizontal deformation component, these points can not show consistent results among the leveling and the ascending and descending DInSAR time series.

3.3 GPS vs. SBAS: the Los Angeles area The results presented in the following, relevant to Los Angeles

metropolitan area, are focused on the comparison between the SBAS results and the measurements available from continuous GPS network [Casu et al., 2006].

3.3.1 Los Angels metropolitan area description The Los Angeles metropolitan zone, localized in Southern California

(USA), is a tectonically active region with surface deformation that are a combination of fault related tectonics plus a variety of other natural and anthropogenic signals [Fuis et al., 2001; Anderson et al., 2003; Argus et al., 2005]. This zone includes effects relevant to aquifers such as the Santa Ana and Pomona basins, oil related displacement at a number of locations, and motion across the Newport-Inglewood and San Jose faults that, as previously mentioned, have been already investigated by applying the DInSAR technology [Bawden et al. 2001, Watson et al. 2002, Colesanti et al. 2003, Lanari et al. 2004b, Argus et al. 2005].

3.3.2 GPS network A key element for the seismic surveillance on the whole area is

represented by the Southern California Integrated GPS Network (SCIGN)

Chapter 3 113 which is an array of GPS stations, described by Hudnut et al. [2001], that reached in the summer of 2001 its target goal of 250 operational sites spread out across southern California and northern Baja California, Mexico. The map of the investigated area, with highlighted the GPS SCIGN sites, is shown in Figure 6.

3.3.3 SBAS measurements For what concerns the presented DInSAR analysis, we have

reconsidered the results of the study presented by Lanari et al. [2004b], that was originally focused on the Santa Ana basin aquifer. In this case the SBAS algorithm has been applied to a set of 42 SAR data acquired by the ERS satellites from late 1995 into 2002 (track: 170, frame: 2925), coupled in 102 interferograms with a perpendicular baseline smaller than 300 m and a maximum time interval of 4 years; precise satellite orbital information and an SRTM DEM of the study area have also been used. This is consistent

Figure 6 - SAR amplitude image of the investigated portion of the Los Angeles metropolitan area with the black squares marking the SCIGN GPS site locations; the GPS site labeled ELSC is highlighted. The inset in upper right corner shows the location of the study area.

114 SBAS Performance Assessment with the DInSAR products relevant to the previously discussed analysis on the Napoli bay site (see section 3.2); also in this case the results have been obtained following a complex multilook operation, with 4 looks in the range direction and 20 looks in the azimuth one, with a resulting pixel dimension of the order of 100 x 100 m.

As a first result of the SBAS algorithm analysis, we present in Figure 7 the LOS mean displacement velocity map computed via the SBAS processing, superimposed to a shaded relief of the SAR image relevant to the investigated area and with highlighted the location of the major surface deformation features. We further remark that in Figure 7 we have identified via the black and the white squares the positions of the SCIGN GPS sites that are located in coherent areas of the DInSAR map and for which measurements are available before 2000, thus ensuring at least two years of overlap with the available SAR data.

Figure 7 - Mean LOS displacement velocity map computed in coherent areas, with respect to the investigated 1995-2002 time interval, and superimposed to a shaded relief of the SAR image relevant to the Los Angeles zone; the major surface deformation features and the GPS site ELSC, located in correspondence of the SAR reference pixel, have been highlighted. The pixels identified by (black and white) squares mark a selection of the SCIGN GPS sites that are located in coherent areas and for which measurements are available before 2000. In particular, the white squares identify the GPS sites relevant to the plots shown in the following Figure 8.

Chapter 3 115

Figure 8 - Comparison between the DInSAR LOS deformation time series (red triangles) and the GPS measurements projected on the radar LOS (black stars) for the pixels labeled in Figure 7 as CLAR (a), CVHS (b), LONG (c), CIT1 (d), DSHS (e), WHC1 (f), SNHS (g), SACY (h), CCCO (i), TORP (j), LBC2 (k) and FVPK (l), respectively. The standard deviation value of the differences between the SAR and LOS-projected GPS measurements has been reported in each plot.

116 SBAS Performance Assessment 3.3.4 Data comparison

Following GPS selection, an extensive comparison has been carried out between the homologous DInSAR and GPS time series, the latter obtained through the SCIGN Web-Site [SCIGN Web] and projected on the SAR sensor line of sight in order to be consistent with the radar observations. Moreover, all the measurements (SAR and GPS) have been computed with respect to a reference pixel located in correspondence to ELSC station highlighted in Figure 7.

Similarly to section 3.2.4, we have started our analysis by first providing a qualitative comparison between SAR and LOS-projected GPS displacements on a selection of GPS sites with a rather uniform distribution in the test area and characterized by different features in their deformation time series. These twelve stations are those identified by the white squares of Figure 7 and are labeled as CLAR, CVHS, LONG, CIT1, DSHS, WHC1, SNHS, SACY, CCCO, TORP, LBC2 and FVPK, respectively. For each of these stations we have compared the DInSAR results (red triangles) with the corresponding LOS-projected GPS measurements (black stars) available within the SAR acquisition window.

The results of these comparisons are shown in the plots of Figure 8, clearly showing the good agreement between these two measurements. Following these results we focused on a quantitative assessment of the SAR measurements quality. Accordingly, we have compared the DInSAR time series with the corresponding LOS-projected GPS measurements for all the sites identified in Figure 7 by the black and white squares. In particular, we have computed the standard deviation value of the differences between these two time series; the achieved results are summarized in the central column of Table III. Based on these measurements, we have computed the average standard deviation value relevant to the differences between SAR and GPS data which corresponds to

Gdσ = 6.9 mm. We further remark that 50% and

100% of the differences between SAR and GPS data are included within the

( ),G Gd dσ σ− and ( )2 , 2

G Gd dσ σ− intervals, respectively.

3.4 Alignment Array vs. SBAS: the Hayward fault

In this section comparison results between deformation measurements retrieved via the SBAS technique and those available from alignment arrays installed along Hayward fault are presented [Lanari et al., 2007].

Chapter 3 117 3.4.1 Hayward fault description

The Hayward fault, running about 60 km along the eastern San Francisco Bay Area, California, is creeping by varying amounts along most of its length and is capable of moderate size (M 6.8) earthquakes directly beneath an urban area [Bakun, 1999; Yu and Segall, 1996]. The large number of alignment arrays and other geodetic measurements, coupled with extensive seismic and geologic observations, allowed the development of a fairly detailed representation of the fault in three dimensions (3-D). Of particular interest has been the identification of the distribution of locked and creeping patches and the relationship of the geodetically inferred locked and slipping portions of the fault and the background seismicity [Waldhauser and Ellsworth, 2002; Schmidt et al., 2005]. In February 1996, several alignment arrays along the southernmost 10 km of the fault experienced up to 2 cm right-lateral creep. Several studies have looked at this event in terms of the timing of the event and the spatial location of this segment of the Hayward fault in relation to the 1989 Loma Prieta earthquake [Lienkaemper et al., 1997; Lienkaemper et al., 2001].

3.4.2 Alignment arrays network A wide range of instruments are deployed in the San Francisco bay area

to monitor crustal deformation. In particular, they include alignment arrays that provide the most accurate and complete measurements of creep since they are generally wide enough (typically 130 m) to span the entire creeping zone, but narrow enough to exclude significant elastic strain away from the fault. They are deployed along the fault trace and measure fault slip by recording the displacement between 2 piers or monuments located on opposite sides of the fault with a precision of 1-2 mm. The overall alignment array data, used in the following analysis, are available at USGS Web-Site [USGS Web; Lienkaemper, 2006].

3.4.3 SBAS measurements The SBAS algorithm was applied to a data set of 45 SAR images

acquired on descending orbits by the ERS satellites, from mid-1992 up to the end of 2000 (track 70, frame 2853). These images were paired in 109 interferograms with a perpendicular baseline smaller than 300 m and a maximum time interval of 4 years. Precise satellite orbital information and a 3 arc second spacing SRTM DEM of the study area were used. Moreover, all the interferograms were obtained following a complex multilook operation

118 SBAS Performance Assessment (4 looks in the range direction and 20 looks in the azimuth one) with a resulting pixel dimension of about 100 x 100 m.

The study area extends for about 90 x 90 km and covers a large part of the San Francisco Bay Area, including about a 50 km portion of the Hayward fault. The estimated LOS mean deformation velocity map, that was geocoded and superimposed on the SAR amplitude image of the investigated zone, was presented in Figure 9 in order to provide an overall picture of detected displacement. Note also that in our mean deformation velocity map only those pixels with a temporal coherence (see Chapter 2 for the temporal coherence definition) value not smaller than 0.7 were considered. By observing Figure 9 several deforming zones are clearly visible, most of these already highlighted in previous studies [Bürgmann et al., 2000; Schmidt and Bürgmann, 2003; Schmidt et al., 2005; Bürgmann et al., 2006].

Figure 9 - Mean deformation velocity map (in false color) of San Francisco Bay for the 1992-2000 time interval, superimposed on a SAR amplitude image (grey scale) of the area. The investigated part of the Hayward fault has been highlighted as well as the reference SAR pixel, the latter identified by a black square. The inset shows the location of the study area.

Chapter 3 119 3.4.4 Data comparison

As already done in previous sections, the DInSAR measurements were compared to those relevant to the ground measurements, in this case the alignment arrays that are present along the investigated portion of the fault, extending from Oakland to Milpitas (Figure 10). In particular, we have considered and used sixteen of the twenty-one original alignment arrays investigated by Lienkaemper et al. [2001]; these sites are located in zones that were coherent in the DInSAR map, on both sides of the fault.

Figure 10 - Mean deformation velocity map of the portion of Hayward fault highlighted in Figure 9, superimposed on an optical image of the area. The trace of the fault and the locations of the alignment arrays [Lienkaemper, 2006] have been indicated by the black line and the red circles, respectively.


For each analyzed alignment array, the DInSAR deformation time series were considered in correspondence to the coherent pixels within two box areas located on the opposite sides of the fault, each of these extending for about 400 m along the fault and 200 m in the perpendicular direction. Subsequently, we have averaged for each box the deformation time series of the corresponding pixels and then computed the difference between these two averages in order to obtain the relative displacement across the fault.

Note that, different than the typical SBAS procedure, no atmospheric artifacts filtering was carried out on the DInSAR time series in this case. Although this filtering operation can be an important step within the overall DInSAR processing chain, it did not represent a significant issue for the proposed creep measurement analysis. Indeed, when computing the relative displacements across the fault between area pairs, that are close and of limited extension, the impact of atmospheric artifacts differences in the two zones are expected to be minimal because of the fact that these effects are highly correlated in space [Ferretti et al., 2000].

After computing DInSAR relative displacement time series, these DInSAR measurements were finally compared to the LOS-projected alignment array data. The results of these comparisons are shown in Figure 11. The presented plots clearly show the generally good agreement between the two measurements. Moreover, the creep event that occurred in February 1996 on Parkmeadow and Camellia sites was clearly detected by the SBAS results (Figure 11).

In addition to this first qualitative evaluation, a quantitative analysis of DInSAR deformation measurements was also carried out. To achieve this task, all the retrieved DInSAR time series were systematically compared to the LOS-projected displacements relevant to the alignment arrays. The DInSAR time series corresponding to these sites were interpolated via a linear regression in correspondence to the acquisition times relevant to the in situ measurements. In particular, the standard deviation value of the differences between the DInSAR and the LOS-projected alignment array time series was computed for each site. These results are reported in Table IV, see the second column from the left. The maximum value of the computed standard deviation corresponds to the Gilbert site and is approximately 5 mm. Moreover, we have also computed the standard deviation value relevant to the whole alignment array data set, by considering all the differences between the SAR and the geodetic measurements; this value is equal to 2 mm.

Chapter 3 121

Figure 11 - Comparisons between the LOS deformation time series across the fault computed from the DInSAR (red triangles) and the alignment array (black asterisks) data, the latter available at http://quake.wr.usgs.gov/docs/deformation/hfcreep/. These plots correspond to the sixteen sites identified by the red circles shown in Figure 10.


We further remark that the previous analysis has been repeated for the

DInSAR time series generated by applying the atmospheric filtering step. These results, also reported in Table IV (see the third column from the left), are nearly identical to those obtained without any filtering, thus confirming the irrelevance of this operation for this kind of analysis. To validate this

Figure 12 - DInSAR analysis relevant to the area around the Camellia and Parkmeadow sites. a) Blow-up of the zone showing the trace of the fault (black line) and the locations of the alignment arrays (red circles). b-d) Comparisons between the LOS-projected deformation time series relevant to the Camellia alignment array site (black asterisks) and the corresponding DInSAR data (red triangles), the latter relevant to a box of about 400 x 200 m, 400 x 400 m and 400 x 1000 m, respectively. e-g) Comparisons between the LOS-projected deformation time series relevant to the Parkmeadow alignment array site (black asterisks) and the corresponding DInSAR data (red triangles), the latter relevant to a box of about 400 x 200 m, 400 x 400 m and 400 x 1000 m, respectively. The standard deviation of the difference between the DInSAR and the LOS-projected alignment array measurements is reported in each plot.

Chapter 3 123 observation, the (linear Person) correlation coefficient between the filtered and not-filtered DInSAR time series was computed for each site. We obtained values always greater than 0.95 and this is consistent with DInSAR covariance estimates over short spatial scales [Hanssen, 2001].

Figure 13 - DInSAR analysis relevant to the area around the Rose and D_St sites. a) Blow-up of the zone showing the trace of the fault (black line) and the locations of the alignment arrays (red circles). b–d) Comparisons between the LOS-projected deformation time series relevant to the Rose alignment array site (black asterisks) and the corresponding DInSAR data (red triangles), the latter relevant to a box of about 400×200 m, 400×400 m and 400×1000 m, respectively. e–g) Comparisons between the LOS-projected deformation time series relevant to the D_St alignment array site (black asterisks) and the corresponding DInSAR data (red triangles), the latter relevant to a box of about 400×200 m, 400×400 m and 400×1000 m, respectively. The standard deviation of the difference between the DInSAR and the LOS projected alignment array measurements is shown for all the plots.


As additional information, we have included in the fourth column from the left of Table IV the mean values of the LOS-projected standard deviations relevant to the alignment arrays. We remark that they are always smaller than 1 mm, clearly showing the high accuracy of the reference data used in our study.

In addition to the previous analysis, it was also investigated the impact on the results of the box dimensions used to average the SAR pixels on either sides of the fault. To do that, the comparisons between the DInSAR and the alignment array time series were repeated by computing the former with different box dimensions. In particular, the 400 m box length along the fault was maintained but the one relevant to the perpendicular direction was increased first to 400 m and finally to 1 km. As example, in Figure 12 and Figure 13 the blow-ups of the areas around the Camellia and Parkmeadow sites and the Rose and D_St sites, respectively, are presented. We remark that the increase of the box dimensions generally leads to a degradation of the quality of the DInSAR measurements. Indeed, we measured an increase of the standard deviation of the difference between the SAR and the alignment array measurements up to about 50%. On the other hand, we also report that the general deformation trend is still preserved even for the box extending for 1 km in the direction perpendicular to the fault. In spite of the increased box dimension also in this case the DInSAR time series did not show a significant impact due to the atmospheric filtering operation; moreover, these results are consistent for the overall data set.

As final step of our study, the deformation rates retrieved from the DInSAR-SBAS analysis were compared to those measured through the alignment arrays. The amplitude of the differences between these two measurements is shown, for each site, in the fifth column from the left of Table IV; it is evident that an agreement of better than 1 mm/year was obtained.

3.5 Final remarks The presented analysis clearly demonstrates the capability of the SBAS

technique to detect, from ERS data set including between 40 and 60 images, displacements with mean deformation velocities of the order of 1 mm/year. Moreover, the standard deviation values for the SAR/leveling, SAR/GPS and SAR/alignment array deformation time series differences correspond to 4.7 mm, 6.9 mm and 2 mm, respectively. Accordingly, it is evident from these results the capability of the procedure to provide deformation time series having a sub-centimetric accuracy with respect to a common reference point of known motion.

Chapter 3 125

Moreover, we further remark that in our analysis, we have assumed the geodetic data as a reference, thus completely neglecting the errors that may affect these measurements. However, we may reconsider this issue by referring to the typical accuracy of the geometric leveling data which is of about 2 mm [INGV-OV, 2003], and to the estimated LOS-projected GPS errors listed in the right hand side column of Table III. Note that the LOS projected alignment array errors are significantly small (see Table IV); therefore these geodetic data are assumed error free and are neglected in the following considerations. Tacking into account previous values, we may remove the bias due to the estimated errors relevant to the geodetic measurements. Accordingly, we finally obtain from the SAR/leveling and SAR/GPS deformation time series comparisons the standard deviation values

Ldσ = 4.3 mm and Gdσ = 5.6 mm, respectively.

In summary, a value of dσ = 5 mm represents a very realistic number for the standard deviation of the SBAS LOS deformation time series.


Table I

DInSAR MEASUREMENTS

LEVELING MEASUREMENTS

LIN

ES

IDE

NT

IFIE

D

BY

CO

LO

RS

IN

FIG

UR

E 3

a

LEVELING BENCHMARKS

E-W MEAN DEFORMATION

VELOCITY [cm/year]

VERTICAL MEAN

DEFORMATION VELOCITY

[cm/year]

VERTICAL MEAN

DEFORMATION VELOCITY

[cm/year]

DIFFERENCE BETWEEN SAR AND LEVELING

VERTICAL MEAN

DEFORMATION VELOCITIES

[cm/year] LCF/002 0.14 -0.16 0.01 -0.17 LCF/10 -0.53 -0.21 -0.22 0.01

LCF/012A -0.67 -0.28 -0.31 0.03 LCF/016 -1.27 -1.10 -1.10 0.00

LCF/017A -1.41 -1.43 -1.43 0.00 LCF/020 -1.22 -2.58 -2.41 -0.17 LCF/022 -1.03 -2.82 -2.79 -0.03 LCF/25A 0.13 -2.85 -2.91 0.06 LCF/030 0.08 -2.74 -2.69 -0.05 LCF/035 0.97 -1.29 -1.40 0.11 LCF/045 0.74 -0.24 -0.24 0.00 LCF/051 0.43 -0.40 -0.29 -0.11 LCF/166 0.41 -0.17 -0.22 0.05 LCF/170 0.37 -0.08 -0.44 0.36

GR

EE

N L

INE

LCF/172 0.27 -0.08 -0.10 0.02 LCF/073 -0.95 -0.71 -0.77 0.06 LCF/063 -0.25 -2.51 -2.66 0.15 LCF/086 0.14 -0.69 -0.91 0.22 LCF/087 0.23 -0.59 -0.76 0.17 LCF/088 0.15 -0.45 -0.62 0.17 LCF/089 0.07 -0.36 -0.49 0.13 LCF/090 0.07 -0.24 -0.40 0.16 LCF/091 0.11 -0.16 -0.31 0.15 LCF/092 0.01 -0.08 -0.13 0.05 LCF/093 0.08 -0.10 -0.08 -0.02 LCF/100 -0.26 -0.15 -0.33 0.18 LCF/101 -0.19 -0.14 -0.29 0.15 LCF/102 -0.46 -0.28 -0.27 -0.01 LCF/106 -0.38 -0.12 -0.21 0.09 LCF/107 -0.32 -0.15 -0.30 0.15

LCF/114A -0.57 -0.43 -0.46 0.03 LCF/116 -0.82 -0.65 -0.90 0.25

RE

D L

INE

LCF/118 -0.82 -0.61 -0.63 0.02 LCF/233 -0.07 -0.16 -0.26 0.10 LCF/234 -0.16 -0.31 -0.34 0.03 LCF/235 -0.07 -0.17 -0.20 0.03 LCF/236 0.00 -0.55 -0.55 0.00

CY

AN

LIN

E

LCF/237 -0.08 -0.26 -0.17 -0.09 LVE053 0.46 -0.03 -0.17 0.14 LVE054 0.28 -0.10 -0.12 0.02 LVE055 0.45 -0.05 -0.14 0.09 LVE056 0.38 -0.11 -0.17 0.06 V

IOL

ET

L

INE

LVE057 0.43 -0.09 -0.14 0.05

Chapter 3 127

Table I (cont.d) LVE059 0.03 0.04 0.00 0.04

LVE64B18P -0.04 0.04 0.00 0.04 LVE64B19 -0.06 0.06 0.00 0.06

LVE060 0.04 0.04 0.02 0.02 LVE061 -0.08 0.08 0.06 0.02 LVE062 0.21 0.02 0.01 0.01 LVE063 -0.11 0.06 -0.01 0.07 LVE10L 0.01 -0.03 0.00 -0.03

LVE083/068 0.04 -0.04 -0.03 -0.01 LVE083/067C -0.05 0.02 -0.05 0.07

LVE143P -0.10 0.06 -0.03 0.09 LVE083/066 0.00 0.04 -0.17 0.21 LVE083/065 0.10 0.03 -0.22 0.25

LVE083/064P 0.21 0.03 -0.17 0.20 LVE083/064 0.10 -0.01 -0.15 0.14 LVE083/063 0.19 0.04 -0.18 0.22 LVE083/062 0.31 -0.02 -0.19 0.17

LVE083/061C 0.25 0.01 -0.19 0.20 LVE56L 0.30 -0.01 -0.21 0.20 LVE57L 0.35 -0.04 -0.17 0.13

LVE083/060C 0.30 0.03 -0.18 0.21 LVE083/059 0.16 -0.03 -0.16 0.13 LVE083/058 0.26 0.02 -0.23 0.25

LVE083/057N 0.34 -0.01 -0.20 0.19 LVE083/057 0.39 -0.05 -0.19 0.14 LVE083/055 0.35 -0.05 -0.12 0.07

LVE083/054C 0.36 -0.01 -0.16 0.15 LVE064 0.39 -0.08 -0.13 0.05

LVE083/052S 0.39 -0.04 0.13 -0.17 LVE065 0.55 -0.05 -0.07 0.02 LVE066 0.45 0.07 0.03 0.04 LVE067 0.37 0.02 -0.13 0.15 LVE068 0.47 -0.09 0.03 -0.12

LVE069P 0.39 -0.12 -0.19 0.07 LVE070P 0.40 -0.17 -0.19 0.02 LVE071 0.44 -0.14 -0.28 0.14 LVE114 0.53 -0.16 -0.26 0.10

BL

UE

LIN

E

LVE113 0.41 -0.06 -0.20 0.14 LVE079 0.24 -0.03 -0.08 0.05 LVE080 0.11 0.05 -0.06 0.11 LVE20L 0.24 -0.02 -0.08 0.06 LVE21L 0.21 -0.05 -0.02 -0.03 LVE081 0.22 -0.02 -0.04 0.02 LVE082 0.13 0.02 -0.03 0.05 LVE083 0.27 -0.04 -0.16 0.12 LVE034 0.23 -0.11 -0.11 0.00 LVE33P 0.14 -0.09 -0.08 -0.01

LVE027LP 0.31 -0.17 -0.12 -0.05 LVE024 0.27 -0.18 -0.09 -0.09

YE

LL

OW

LIN

E

LVE010 0.15 -0.28 -0.16 -0.12


Table II

STANDARD DEVIATION OF THE DIFFERENCE BETWEEN SAR AND

LOS-PROJECTED LEVELING MEASUREMENTS [cm]

LEVELING BENCHMARKS

ASCENDING DESCENDING LCF/002 0.42 0.31 LCF/25A 0.97 1.09 LCF/030 0.83 0.85 LCF/063 1.09 0.96 LCF/086 0.64 0.78 LCF/087 0.53 0.63 LCF/088 0.54 0.57 LCF/089 0.49 0.50 LCF/090 0.52 0.52 LCF/091 0.39 0.65 LCF/092 0.35 0.42 LCF/093 0.46 0.45 LCF/101 0.78 0.33 LCF/233 0.38 0.36 LCF/234 0.37 0.42 LCF/235 0.32 0.35 LCF/236 0.32 0.40 LCF/237 0.35 0.43 LVE059 0.31 0.23

LVE64B18P 0.30 0.26 LVE64B19 0.25 0.51

LVE060 0.29 0.38 LVE061 0.21 0.37 LVE062 0.33 0.33 LVE063 0.42 0.36 LVE10L 0.35 0.42

LVE083/068 0.27 0.39 LVE083/067C 0.32 0.54

LVE143P 0.39 0.31 LVE083/066 0.59 0.47 LVE083/065 0.69 0.59

LVE083/064P 0.51 0.59 LVE083/064 0.52 0.48 LVE083/063 0.62 0.54 LVE083/059 0.56 0.64

LVE079 0.36 0.46 LVE080 0.40 0.44 LVE20L 0.31 0.39 LVE21L 0.32 0.43 LVE081 0.30 0.44 LVE082 0.33 0.41 LVE034 0.47 0.51 LVE33P 0.48 0.49 LVE010 0.90 0.70

Chapter 3 129

Table III

GPS STATIONS

STANDARD DEVIATION OF THE DIFFERENCE BETWEEN SAR AND

LOS-PROJECTED GPS MEASUREMENTS [cm]

LOS-PROJECTED GPS ERRORS* [cm]

AZU1 0.71 0.43 BGIS 0.48 0.38

BRAN 0.85 0.42 CCCO 0.68 0.37 CCCS 0.73 0.37 CIT1 0.70 0.38

CLAR 1.09 0.37 CRHS 0.41 0.37 CSDH 0.58 0.36 CVHS 0.53 0.38 DSHS 0.54 0.43 DYHS 0.49 0.37 ECCO 0.48 0.38 EWPP 0.61 0.37 FVPK 0.82 0.37 JPLM 0.93 0.41 LASC 0.76 0.37 LBC1 0.79 0.42 LBC2 0.59 0.37 LONG 0.90 0.43 LORS 0.81 0.39 LPHS 0.47 0.40

MHMS 1.13 0.50 NOPK 0.47 0.44 PMHS 0.50 0.38 PVHS 0.77 0.44 PVRS 0.77 0.40 RHCL 0.49 0.39 SACY 0.85 0.43 SNHS 0.72 0.39 SPMS 0.65 0.37 TORP 0.69 0.38 USC1 0.73 0.42 VTIS 0.80 0.40

VYAS 0.48 0.39 WCHS 0.49 0.39 WHC1 0.38 0.39 WHI1 0.20 0.39

* These values have been obtained by projecting along the radar line of sight the information relevant to the errors available from the SCIGN Web-Site [SCIGN Web].


Table IV

Alignment Array Station Name

Standard deviation of the difference between the DInSAR

and the LOS-projected Alignment Array measurements [mm]

Not Filtered Filtered

Mean value of the standard

deviation relevant to the LOS-projected

Alignment Array

measurements [mm]

Difference between the

DInSAR and the LOS-projected

Alignment Array deformation rates

[mm/year]

Lincoln 1.4 1.4 0.5 0.4 39th 1.1 1.1 0.5 0.1 Encina 1.9 1.9 0.4 0.6 Chabot 2.3 2.3 0.4 0.7 167th 2.1 2.1 0.3 0.8 Rose 0.9 0.9 0.3 0.2 D_St 1.3 1.3 0.3 0.4 Palisade 2.8 2.8 0.7 0.8 Appian 2.1 2.1 0.5 0.5 Gilbert 4.8 4.7 0.2 0.8 Rockett 2.0 1.9 0.4 0.1 Hancock 3.8 3.8 0.3 0.5 Union 3.0 3.0 0.4 0.2 Camellia 2.6 2.6 0.3 0.6 Parkmeadow 1.6 1.6 0.5 0.1 Prune 2.1 2.0 0.7 0.3

Chapter 3 131

Appendix A Multi-orbit combination The availability of SAR data acquired from ascending and descending

orbits may allow us to retrieve the east–west and vertical displacement components, by properly combining the radar LOS deformation estimates achieved from both tracks [Manzo et al., 2006; Gourmelen at al., 2007]. To explain the basic rationale of this retrieval step we refer to Figure 14, where a general SAR geometry is sketched.

In this case we consider two different viewing angles (labeled 1ϑ and

2ϑ , respectively), both belonging to the East-z plane. The deformation components measured from the SAR sensors, along the two lines of sight, can be expressed as:

1 1

2 2

1 1

2 2

sin cos

sin cosLOS LOS H V

LOS LOS H V

d d i d d

d d i d d

ϑ ϑ

ϑ ϑ

⎧ = ⋅ = +⎪⎨

= ⋅ = − +⎪⎩

r r

r r (A1)

Figure 14 - SAR geometry in the East-z plane with indicated the displacement vector d

r

(black continuous arrow), its vertical Vd and east-west Hd deformation components

(black dotted arrows) and its LOS projections (1LOSd and

2LOSd ).

132 SBAS Performance Assessment which can be equivalently expressed via the following matrix representation:

1

2

1 1

2 2

sin cossin cos

LOSH

V LOS

ddd d

ϑ ϑϑ ϑ

⎛ ⎞⎛ ⎞⎛ ⎞⋅ = ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(A2)

and easily solved with respect to the two unknown deformation components Hd and Vd , relevant to the east-west and vertical direction, respectively:

( )

1 2

1 1 2

2

2 1

12 11 1

2 2 1 2

cos cos

sin sinsin cossin cos sin

LOS LOS

LOS LOS LOSH

V LOS

d d

d d ddd d

ϑ ϑ

ϑ ϑϑ ϑϑ ϑ ϑ ϑ

−

−⎛ ⎞⎜ ⎟⎜ ⎟+⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎝ ⎠= ⋅ =⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟− +⎝ ⎠⎝ ⎠ ⎝ ⎠

(A3)

In our case, i.e., ascending and descending SAR data combination, the two look-angles, 1ϑ and 2ϑ , are the same, being the investigated area imaged by two perfectly symmetrical viewing angles. Accordingly, say ϑ the common look angle, it is easy to demonstrate that (A3) can be written as follow:

( ) ( )

( ) ( )

1 2

1 2

1 2

1 2

cossin 2 2sin

sinsin 2 2cos

LOS LOSH LOS LOS

LOS LOSV LOS LOS

d dd d d

d dd d d

ϑϑ ϑ

ϑϑ ϑ

−⎧= − =⎪

⎪⎨ +⎪ = + =⎪⎩

(A4)

We underline that the east-west displacement component is related to the difference of the two LOS deformation measurements while the vertical component to the sum of the two deformation measurements.

It is important to stress that the sensibility of the radar measurements along the north-south direction is quite limited. Indeed, the sensor flight trajectories are approximately parallel to the north-south direction and, as already explained, the radar techniques (measuring range distances, only) are not capable to measure displacements with respect to the sensor flight trajectory (i.e., along the azimuth direction).

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Chapter 4

The SBAS technique for deformation analysis of very extended areas

We present first experiment results to survey the temporal evolution of the deformation affecting very large areas, by exploiting and improving the SBAS algorithm. In particular, we have analyzed a set of 264 descending ERS SAR data frames, spanning the 1992-2000 time interval; these data are relevant to an area in Central Nevada (USA) extending for about 600 x 100 km. The starting point of our study has been the generation of an appropriate set of small baseline multilook interferograms computed from long SAR image strips, obtained by jointly focusing six contiguous raw data frames. Following their generation, the selected interferograms, computed on a 160 x 160 m spatial grid, have been inverted via the SBAS technique to retrieve, for each coherent pixel, the displacement time series and the corresponding mean deformation velocity. Moreover, a detailed validation step has been also provided, demonstrating the algorithm capability to detected deformation with accuracy comparable to that achieved through the conventional SBAS processing.

This approach, from now on referred to as Extended-SBAS (E-SBAS), is the first one involving such an extended multi-temporal SAR data set and, due to its capabilities, it can be suitable to operate as surface deformation survey tool for large areas. Note that a more detailed E-SBAS algorithm analysis can be found in Casu et al. [2008].

4.1 E-SBAS processing chain rationale As discussed in Chapter 2, SBAS technique is a DInSAR algorithm

[Berardino et al., 2002] allowing us to detect Earth surface deformation and to analyze their temporal evolution by generating mean deformation velocity maps and time series. The basic SBAS technique has originally been developed to investigate, by using multilook interferograms, sites with a typical extension of 100 by 100 km and with a spatial resolution of the order of 80 by 80 m. However, although the availability of deformation maps extending for 100 by 100 km may be very effective to investigate several

138 SBAS for very extended areas analysis

displacement phenomena, there are cases where it can be crucial to study much more extended zones in order to detect and analyze the ongoing deformation [Gourmelen and Amelung, 2005].

In this chapter we have applied the SBAS technique to a set of relatively low resolution multilook stripmap interferograms (with about 160 x 160 m spatial resolution) that have been used to investigate the algorithm capability to detect and analyze deformation occurring in large areas [Casu et al, 2008]. In particular, we have first combined several contiguous ERS stripmap SAR raw data frames (belonging to the same satellite track), to generate long SAR image strips. After the focusing step, the azimuthal resolution has deliberately been degraded to reduce the amount of data to be processed.

To achieve this task, the Average Doppler Centroid (ADC) of the whole data set has first been identified; subsequently, each SAR image has been low pass filtered in azimuth around the estimated ADC and then downsampled of a properly chosen factor: in our case a factor 5. Note that, to effectively combine the SAR image pairs in the multilook interferogram generation, we have assumed that a spectral overlap of at least 50% of the Doppler bandwidth should be guaranteed; therefore, data presenting a Doppler centroid not included in the (ADC-PRF/2, ADC+PRF/2) interval have been discarded. In any case, in our experiment all the available SAR images satisfied this constraint.

Then, the small baseline SAR data pairs have been selected by assuming a maximum spatial and temporal baseline of about 400 m and 1500 days, respectively; subsequently, the multilook interferograms have been generated. At this stage the overall interferogram sequence has been inverted via the standard SBAS procedure, thus allowing us to retrieve the mean deformation velocity and the corresponding displacement time series for each investigated coherent pixel.

We remark that the SBAS inversion also includes a step for orbital fringes detection and removal [Lanari et al., 2004]; moreover, it is also performed a filtering operation of the atmospheric phase components [Berardino et al., 2002]. To summarize the rationale of the processing chain, its block diagram is shown in Figure 1.

As final remark, we underline that the data frame merging operation could be carried out either at the raw level or following the SAR raw data focusing step. We preferred the former because of the direct availability of a counter associated to each raw data line which makes very easy the data merging process. Note also that the overall estimation of the Doppler parameters is carried out during the focusing stage that, together with the data merging step, requires only a limited amount of the overall computing time (about 10%).

Chapter 4 139

Figure 1 - Block diagram of the E-SBAS processing chain.


4.2 E-SBAS results: Central Nevada case study

4.2.1 Site description E-SBAS processing chain has been applied to a unique set of 264 ERS-

1/2 SAR data frames (track 442, frames: from 2781 to 2871), spanning the 1992-2000 time interval, that are relevant to an area located in Central Nevada (USA) extending for about 600 x 100 km. In particular, we generated 148 DInSAR interferograms, see Figure 2, with a spatial resolution of about 160 x 160 m.

4.2.2 Data analysis To provide an overall picture of the detected deformation, we present in

Figure 3a the computed mean deformation velocity map, geocoded and superimposed on a SRTM DEM of the area. This representation allows us to provide mean information (with respect to time) on the detected deformation; note also that, areas where deformation measurement accuracy is affected by decorrelation noise have been excluded from the color map. Figure 3a clearly shows significant deformation patterns present in the area due to natural and anthropogenic causes. Let us provide some details on the detected displacements and, at the same time, show the deformation time series retrieval capability of the E-SBAS inversion. To achieve this task, we present in Figure 3b-f the computed time series relevant to five selected coherent pixels identified in Figure 3a by the black triangles labeled as B, C, D, E and F, respectively. In particular, plots of Figure 3b-c show subsidence due to water pumping in support of open-pit gold mining at Lone Tree Gold Mine and Cortez Gold Mine, respectively [Gourmelen et al., 2007]; Figure 3d represents subsidence due to the agricultural exploitation of Antelope Valley; Figure 3e illustrates the deformation relevant to the Eureka Valley Earthquake (May, 17 1993); finally, Figure 3f shows deformation associated with geothermal production and intense microseismicity at Coso Geothermal area.

Chapter 4 141

Figure 2 – Computed DInSAR data. a) Histogram of the generated interferograms with respect to the temporal (upper) and perpendicular (lower) baselines. b) SAR data representation in the plane relevant to the interferogram perpendicular baselines and acquisition dates, the latter represented with “ddmmyyyy”.


Figure 3 – E-SBAS results. a) Mean deformation velocity map (in color) superimposed on a SRTM DEM (grey scale) of the zone; the location of the 2799 and 2853 frames have been highlighted as well as the DInSAR reference point (white square). b-f) DInSAR deformation time series for the pixels marked by triangles in Figure 3a and labeled by the letters from B to F. The inset in the upper-right corner shows the study area location.

Chapter 4 143

4.2.3 Data validation Let us now carry out a validation of the presented DInSAR products.

This is not an easy task because of the lack, in the investigated area, of available geodetic data (GPS, leveling, etc.) ensuring a significant temporal overlap with the computed DInSAR measurements. Accordingly, we focused on a cross-comparison between the presented results and those obtained from alternatively computed DInSAR data; in particular, we exploited those achieved by applying the standard SBAS approach to the data relevant to two selected frames involved in the long strip processing but with no degradation of the SAR image resolution. First of all, we focused on the SAR data relevant to the 2799 frame (its location is highlighted in Figure 3A) by considering the same acquisition dates and the same interferogram data pair distribution we used to produce the results of Figure 3. On these data, a standard SBAS inversion was carried out, i. e., without applying any azimuth low pass filtering to SAR images, but considering the same corrections of the orbital errors computed for the large area processing chain. Subsequently, to make the achieved results comparable with those shown in Figure 3, a downsampling operation has also been performed, leading to a 160 x 160 m output grid. It is worthy to note that for these results, assumed as a reference, we may expect standard deviation values of the order of

1vσ = mm/year and 5dσ = mm for the mean deformation velocity and the displacement time series, respectively [Casu et al., 2006].

Our comparison has been carried out by computing the difference between the mean deformation velocities relevant to the two different DInSAR processing, in correspondence to common coherent pixels. In particular, in Figure 4a the mean deformation velocity map relevant to the 2799 frame data is presented, while in Figure 4b it is shown the difference between the map of Figure 4a and the one relevant to the corresponding area in Figure 3a. By observing Figure 4b it clearly appears the good agreement between the two measurements. In addition, we calculated the standard deviation of the difference between the homologous deformation time series, see Figure 4c; also in this case it is evident a generally good agreement between the two retrieved measurements.

In order to provide a quantitative analysis, we also computed for the map shown in Figure 4b, the percentage of pixels whose values are included within the ( ),v vσ σ− and ( )2 , 2v vσ σ− intervals, obtaining values of 87% and 99%, respectively. For what concerns deformation time series quantitative assessment, we performed a similar analysis on the results


presented in Figure 4c; in this case, we obtained that 98% and 99% of the coherent pixels are included within the ( ),d dσ σ− and ( )2 , 2d dσ σ− intervals, respectively.

A similar cross-comparison has been carried out for the SAR data relevant to the 2853 frame (its location being highlighted in Figure 3a). Concerning the mean deformation velocities comparison we obtained that the percentage of pixels included within the ( ),v vσ σ− and ( )2 , 2v vσ σ− intervals are of 98% and 99%, respectively; moreover, for the standard deviation of the difference between the homologous deformation time series, we obtained values of 98% and 99% as amount of pixels in the ( ),d dσ σ−

and ( )2 , 2d dσ σ− intervals, respectively. The consistence of the results obtained for the 2799 and the 2853 frames is evident.

Figure 4 - SBAS results cross-comparison. a) Mean deformation velocity map for the 2799 frame data processed via conventional SBAS algorithm. b) Difference between the deformation velocity map shown in Figure 4a and the one relevant to the corresponding area in Figure 3a; the black arrows indicate the locations of the main discrepancies between the two measurements. c) Standard deviation of the difference between homologous deformation time series.

Chapter 4 145

As a final remark we underline that the main discrepancies between the two DInSAR measurements (highlighted by arrows in Figure 4b for the 2799 frame) are located in the areas affected by the maximum displacements. Our interpretation of these discrepancies is mostly relevant to unwrapping errors due to the spatial resolution degradation following the image low pass filtering and the interferogram multilooking operations. Similar phenomena also appear in the data relevant to the 2853 frame.

4.3 Final remarks The presented results demonstrate the capability of the exploited solution

to provide, in such large areas, space–time information on the detected deformation with accuracy comparable with that achieved through the conventional SBAS processing.

We further remark that the proposed approach can be also extended to analyze the SAR data acquired by the ScanSAR systems [Tomyasu, 1981]. In this context, the availability of burst-synchronized long ScanSAR time series is foreseen in the future because it may allow, based on solutions like the one applied in this study, to exploit the DInSAR technology for continental base deformation analysis. In this sense, the improved quality of the ENVISAT ScanSAR acquisitions may represent a very important issue [Rosich et al., 2007], as well as the DInSAR capabilities of the L-band ALOS sensor [Shimada et al, 2007].

In these scenarios, additional investigation will also require the development of effective tools for filtering possible phase artifacts due to the errors on the sensor orbit information, particularly in the along track direction. It will be certainly beneficial, in this case, the increased availability of continuous GPS time series.

Finally, the development of robust phase unwrapping codes, allowing mitigating errors due to the reduced spatial resolution as well as the high extension of the analyzed area, certainly represents an important issue. For this reason, in the next Chapter a new phase unwrapping approach is presented, aimed to overcome mentioned limitations.


References Berardino, P., Fornaro, G., Lanari, R., Sansosti, E. (2002). A new Algorithm for Surface Deformation Monitoring based on Small Baseline Differential SAR Interferograms. IEEE Trans. Geosci. Remote Sens., 40, 11, pp. 2375-2383.

Casu, F., Manzo, M., Lanari, R. (2006). A quantitative assessment of the SBAS algorithm performance for surface deformation retrieval from DInSAR data. Remote Sensing of Environment, 102(3-4), pp. 195–210, doi:10.1016/j.rse.2006.01.023.

Casu, F., Manzo, M., Pepe, A., Lanari R. (2008). SBAS-DInSAR Analysis of Very Extended Areas: First Results on a 60,000 km2 Test Site. IEEE Geosci. Remote Sens. Lett., 5, 3, doi:10.1109/LGRS.2008.916199.

Gourmelen, N., Amelung, F, (2005). Postseismic Mantle Relaxation in the Central Nevada Seismic Belt, Science, 310, pp. 1473-1476, doi: 10.1126/science.1119798.

Gourmelen, N., Amelung, F. Casu, F., Manzo, M., Lanari, R. (2007). Mining-related ground deformation in Crescent Valley, Nevada: Implications for sparse GPS networks. Geoph. Res. Lett., 34, L09309, doi:10.1029/2007GL029427.

Lanari, R., Lundgren, P., Manzo, M., Casu, F., (2004). Satellite radar interferometry time series analysis of surface deformation for Los Angeles, California. Geoph. Res. Lett., 31, L23613, doi:10.1029/2004GL021294.

Rosich, B., Monti Guarnieri, A., D’Aria, D., Navas, I., Duesmann, B., Cafforio, C., Guccione, P., Vazzana, S., Barois, O., Colin, O., Mathot, E., (2007). ASAR wide swath mode interferometry: optimisation of the scan pattern synchronization, Proc. ‘Envisat Symposium 2007’, Montreux, Switzerland 23–27 April 2007 (ESA SP-636, July 2007).

Shimada, M., Isoguchi, O., Tadono, T., Higuchi, R., Isono, K., (2007). PALSAR CALVAL Summary and Update 2007, Proc. of IGARSS '07.

Tomiyasu, K., (1981). Conceptual performance of a satellite borne, wide swath synthetic aperture radar, IEEE Trans. Geosci. Remote Sens., 19, pp. 108-116.

Chapter 5

Region-Growing EMCF PhU algorithm Phase Unwrapping (PhU) is the problem to retrieve the original

interferometric phase starting from the computed modulo-2π restricted one and is probably the most critical task in SAR interferometry. Indeed, phase unwrapping problems occur from aliasing errors due to phase noise caused by low coherence and undersampling phenomena because of locally high fringe rates.

As already stated in Chapter 2, very efficient PhU procedures have been developed. In particular, they were firstly oriented to unwrap single interferogram (in a conventional DInSAR scenario) and, subsequently, concentrated on the exploitation of the temporal relationship between multiple interferograms (in multi-temporal DInSAR), in order to improve the unwrapping performances. Concerning the second ones, in the SBAS processing chain is implemented the Extended Minimum Cost Flow (EMCF) [Pepe and Lanari, 2006] algorithm which permits to effectively unwrap a set of multilook interferograms relevant to the same scene, performing a joint analysis of both spatial and temporal constraints of the differential phases.

Moreover, in order to increase the unwrapped pixel spatial density, starting from the previous EMCF unwrapped pixels, a further PhU step is implemented. In particular, in the SBAS processing chain, is included a Region-Growing (RG) [Xu and Cumming, 1999] procedure that, based on a linear model of the investigated pixel deformation (see Chapter 2 for more details), try to “propagate” the retrieved solution from already unwrapped pixels to neighboring points. It is clear that pixels with non linear motion will be uncorrectly unwrapped.

Therefore, this approach is strongly limited in region affected by non linear deformation, where the displacement linearity assumption fails, as well as in decorrelated areas, where noise strongly increases the difference between the model and the actual phase, implying the phase residue to be not included in the ( ),π π− interval (see Chapter 2).

For this reason, in this Chapter we present a novel Region-Growing PhU approach aimed at increase the unwrapped pixel spatial density. The new Region-Growing algorithm is derived from the “temporal” PhU strategy at

148 Region-Growing EMCF PhU algorithm

the base of the EMCF algorithm and, therefore, it is highly oriented to the deformation time series retrieval.

The new RG-EMCF algorithm has been applied to sequences of multilook differential interferograms, relevant to the previous analyzed Central Nevada region (see Chapter 4), characterized by large non-linear deformation phenomena, as well as to a test site located in Gardanne area (France), affected by strong decorrelation. Presented results confirm the effectiveness of the proposed approach.

5.1 Region-Growing general concepts Basic idea of the Region-Growing procedure is to retrieve the

unwrapped phase of a set of investigated pixels that are in the neighborhood of an already unwrapped region. In particular, the unwrapped phase of the analyzed pixels is computed by exploiting the phase values of neighboring unwrapped pixels [Xu and Cumming, 1999].

Accordingly, first Region-Growing procedure step concerns in the identification of two pixel sets. High temporally coherent pixels, i. e., already correctly unwrapped, are considered as starting point for the entire procedure. They are usually referred to as Seed Points

( )0 1 1, ,..., QS S S Q O−= ≤S , being O the total number of points in the

considered scene. In order to proceed with their selection, a threshold on the temporal coherence factor is imposed ( maxγ γ≥ ).

Moreover, is also identified the set of previously not unwrapped points, hereinafter simply referred to as Candidate Points

( )0 1 1, ,..., PC C C P O Q−= ≤ −C . This task is achieved via simple considerations on temporal coherence map; in particular, each point whose temporal coherence respects the min maxγ γ γ< < constraint is selected as Candidate one. Note that maxγ is a threshold value setting the limit between high and poor coherent areas (the one used to individuate Seed Points) while

minγ is chosen in order to avoid the scanning process to be highly time consuming (typical value is 0.3).

By considering these two sets of pixels, a proper Region-Growing procedure can be applied, scanning each Candidate Point in order to retrieve its unwrapped phase. Note that, the adopted Candidate Points scanning strategy relies on a spiral path centered at the DInSAR Reference Point location and including all previously selected Candidate Points (Figure 1).

Chapter 5 149

5.2 Region-Growing EMCF strategy Let us now focus on the presentation of a new Region-Growing PhU

approach which relies on the exploitation of temporal constraints among a properly chosen sequence of multi-temporal interferograms.

In particular, this approach is based on the key idea of the EMCF algorithm [Pepe and Lanari, 2006] and, similarly, it requires that the selected DInSAR data-pairs are consistent with a Delaunay [Delaunay, 1934] triangulation in the Temporal/Perpendicular Baseline TB B⊥× plane (see Section 2.8, 2.11.2 and Figure 2), permitting to exploit the temporal relationships among the selected sequence of multi-temporal DInSAR interferograms. Since it directly originate form the EMCF key idea, it is clear that RG-EMCF is a deformation time series oriented PhU approach.

Let us now refer to the previously identified Seed and Candidate Pixels (see Section 5.1). As every Region-Growing algorithm, the new approach starts from the knowledge of unwrapped DInSAR phase values

Figure 1 – Pictorial representation of the Candidate (gray boxes) and Seed (white boxes) pixels. Spiral path Candidate Points scanning strategy has been also highlighted (red arrow).


( )0 1 1, ,..., M −= Ψ Ψ ΨΨ in correspondence to a set of pixels with high temporal coherence values (Seeds). To this purpose, the essential requirement to be satisfied is that the involved DInSAR sequence of wrapped interferograms ( )0 1 1, ,..., M −= Φ Φ ΦΦ is consistent with a

Delaunay triangulation into the TB B⊥× plane. The Region-Growing process flow chart is depicted in Figure 3 and, for

sake of generalization, we directly refer in the following to the p-th iteration, i. e., to the p-th Candidate-Growing pixel (labeled to as pC in Figure 3)

The estimation of the (unwrapped) phase vector relevant to the generic Candidate point ( ),p p pC az rg Az Rg≡ ∈ × is:

Figure 2 – Data pair distribution in the Temporal/Perpendicular baseline plane (see also Section 2.8) relevant to Central Nevada area (see Chapter 4). Note that each interferogram corresponds to an arc connecting two black diamonds, i. e., two SAR acquisitions. Arcs highlighted in red are relevant to interferograms belonging to the Delaunay triangulation.

Chapter 5 151

( ) ( )( ) ( ) ( )( )0 1 1

,

, , , ,..., ,

p p p

p p p p M p p

C az rg

az rg az rg az rg−

≡ ≡

≡ Ψ Ψ Ψ

Ψ Ψ (1)

Figure 3 - RG-EMCF block diagram. Gray box identifies operations performed for each

spatial arc ( ),p pkC S .


being Az Rg× the Azimuth-Range spatial plane, and takes advantage from the (known) unwrapped phase vectors corresponding to its neighboring Seed Pixels. The latter are identified, among all the available Seed Points, through a CW CWaz rg× pixel-wide window centered at the ( ),p paz rg Candidate

Point position, being CWaz and CWrg pixel number values, see Figure 4. In particular, for each , 0,..., 1pk pS k K= − Seed Point located within the

searching window (being pK their number), one prediction of the Candidate Point unwrapped phase vector can be achieved via a trivial integration step of the unwrapped ( ),p pkC S phase-gradient vector pk∆Ψ , see Figure 4, that

is:

( ) ( )k p pk pkC S= + ∆Ψ Ψ Ψ (2)

The estimation of the (unknown) unwrapped phase-gradient vectors

Figure 4 – Candidate-Seed phase gradient pk∆Ψ identification relevant (in the Az Rg×

plain) to the ( )1,p pC S arc (red arrow). Dashed red line indicates the CW CWaz rg× pixel-

wide Seed Point searching window, centered at the generic Cp Candidate Point.

Chapter 5 153

, 0,..., 1pk pk K∆ = −Ψ represents the core of the whole PhU procedure and is obtained by applying the same “temporal” PhU approach at the base of the EMCF algorithm. In particular, the “temporal” PhU operation permits to retrieve unwrapped phase differences for each interferogram, by searching for the minimum cost flow of the Temporal/Perpendicular baseline network of (measured) wrapped phase-gradient:

( )( ) ( ) , 0,..., 1pk p pk pWr C S k K= − = −∆Φ Φ Φ (3)

being ( )Wr ⋅ the wrapped-phase operator. In addition to the unwrapped phase-gradient vectors, also the minimum temporal network costs, representing a good confidence measure of the solution correctness, are available. A comprehensive description of the applied Temporal PhU strategy will be given in Section 5.2.1.

Since different integration paths generally correspond to different Candidate Point unwrapped phase predictions, a weighted average of the pK

individual ( )k pCΨ estimates is considered in order to retrieve a unique

phase value for the selected pC point, thus obtaining the following prediction:

( ) ( )1 1

0 0

p pK K

p k k p kk k

C w C w− −

= =

⎛ ⎞ ⎛ ⎞= Ψ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∑ ∑Ψ (4)

whose weights are chosen as: min

min1pk

kpk

Costw

Costρρ

⎧Γ <⎪= ⎨ >⎪⎩ (5)

wherein Γ represents the maximum allowed weight, minpkCost is the

minimum temporal network cost relevant to the k-th Candidate-Seed spatial arc ( ),p pkC S , and ρ is a threshold value that, according to Pepe and

Lanari [2006] is set not greater than 5% of the interferogram total number M .

Moreover, to avoid that Candidate Point wrapped- and unwrapped- phase vectors do not differ by 2π integer multiple, due to the average


operation (see eq. (4)) involved in unwrapping step, a residual phase vector

( ) ( ) ( )( )p p pC Wr C C= −r Φ Ψ is subsequently added to (4).

Finally, the estimate of the unwrapped phase vector in the selected Candidate Point is:

( ) ( ) ( )ˆp p pC C C= +Ψ Ψ r . (6)

In order to assess the reliability of the retrieved phase vector relevant to the p-th Candidate Point, see eq. (6), two tests have to be performed.

The first one is based on the application of the SBAS algorithm to the unwrapped phase estimates (6) in order to retrieve the corresponding deformation time-series. The computed deformation time series are therefore used to reconstruct the original phases, thus estimating the temporal coherence factor ( )pCγ , as defined in Chapter 2.

The reliability test is verified if:

( )pCγ γ> (7)

where, for instance, a threshold 0.7γ = is a typical value in DInSAR applications [Casu et al., 2006].

Second test is based on the assumption that the larger are the residual phase vector ( )pCr values, the larger is the probability to have got a wrong

estimate. Accordingly, we introduce a novel quality index, named Similarity factor and defined as follows:

( )( )( )

1

0exp

M

j pj

p

i r CC

M

−

=

⎡ ⎤⎣ ⎦

Ω =∑

. (8)

wherein 1i = − . The reliability test is verified if:

( )pCΩ > Ω (9)

where can be convenient a γΩ ≥ selection. When both (7) and (9) conditions are satisfied, Candidate Point is

considered unwrapped and, accordingly, it can be assumed as a Seed Point in the next iterations. Otherwise, when the pC unwrapped phase vector is considered unreliable the relevant point is consequently discarded. In both

Chapter 5 155 cases, Region-Growing algorithm proceeds by analyzing next Candidate Points.

We observe that the presented RG-EMCF algorithm can be generalized and applied when new DInSAR interferograms are added to the original

TB B⊥× triangulation distribution, without belonging to new triangles; this can be done, for instance, to have a more dense interferograms distribution. Anyway, added interferograms are basically unwrapped by taking into account for a deformation model, the latter derived from the displacement time series relevant to the Delaunay triangulation unwrapped interferograms. A more detailed discussion on this topic can be found in Appendix A.

We final remark that the method exploited to unwrap Seed Points, in this case the EMCF PhU algorithm, does not represent a constraint for the subsequent Region-Growing step.

5.2.1 Unwrapped Phase-Gradient vector estimation As already stated, unwrapped phase-gradients of the Candidate-Seed

spatial arcs are accomplished in the TB B⊥× plane by applying the EMCF “temporal” phase unwrapping strategy. Let us now deal with a comprehensive description of this approach.

First of all, we refer to the generic ( ),p pkC S spatial arc considered in

Section 5.2, connecting the pixels ( ),p p pC az rg≡ and ( ),pk pk pkS az rg≡

within the Az Rg× grid plane. Let pk∆Φ and pk∆Ψ be the (measured) wrapped and the (unknown) unwrapped DInSAR phase-gradient vectors, respectively. Clearly, the unwrapped phase-gradient vector elements differ by 2π-multiple integers from the corresponding wrapped ones and, consequently, the two vectors can be related according to the following equation:

2pk pk pkπ= +∆Ψ ∆Φ H (10)

wherein pkH is an integer-valued vector. Generally, the target of a “temporal” phase unwrapping algorithm is to

retrieve an estimate of the (unknown) pkH vector, accounting for the temporal relationships among the DInSAR data pairs. To make easier the whole procedure, similarly to what done in Pepe and Lanari [2006], a profitably model for the (unknown) unwrapped phase-gradient may be


considered, whose j-th component can be expressed, in a more generalized way, as follows:

( ) ( )4 4,

sinj

j jj j

bm h v h t v

rπ πλ ϑ λ

⊥∆∆ ∆ ≈ ⋅ ⋅∆ + ⋅∆ ⋅∆

⋅ (11)

wherein r is the sensor-to-target distance, jλ is the operational wavelength associated to the j-th interferogram and ϑ is the incidence angle.

The first term on the right side of (11) accounts for the inaccuracies on the knowledge of the scene topography h∆ and the second one is related to the deformation velocity difference v∆ over the selected arc.

By taking account for eqs. (10) and (11), the unknown unwrapped phase difference vector pk∆Ψ can be expressed as a function of the considered model m , as follows:

( )( )

2 2

2

pk pk pk pk

pk pk

Wr

Wr

π π

π

= + + − + =

= + − +

∆Ψ m ∆Φ H m G

m ∆Φ m G (12)

where last identity is dictated by the observation that 2π-multiple integer vectors do not affect the wrapped operator ( )Wr ⋅ and that pkG is the unknown integer-valued vector.

For a given ( ),h v∆ ∆ couple, pk∆Ψ (unknown) unwrapped phase-gradient vector can be related to the corresponding (measured) phase vector

( ) ( )pk pk pkWr= + −ξ m,∆Φ m ∆Φ m , (13)

via the following relation:

2pk pk pkπ= +∆Ψ ξ G (14)

The problem can be now viewed as the “search” of the pkG unknown integer-valued vector, and can be efficiently solved by applying the basic Minimum Cost Flow phase unwrapping technique into the TB B⊥× plane.

More precisely, this task is accomplished by searching for the solution of the following minimization problem:

1

,0

minM

pk jj

Cost G−

=

⎡ ⎤=⎢ ⎥

⎣ ⎦∑ (15)

Chapter 5 157 subject to the constraint that loop integrals of the unwrapped phase differences over all the TB B⊥× triangles are equal to zero.

Different values of ( ),h v∆ ∆ will obviously correspond to solutions

characterized by different network costs ( ) ,Cost h v∆ ∆ .

Accordingly, we may assume as optimum unwrapped phase-gradient vector pk∆Ψ

), relevant to the considered k-th spatial arc, the one

corresponding to the smallest network cost, expressed as:

( )( ) min

,min ,pk h v

Cost Cost h v∆ ∆

= ∆ ∆ (16)

and, accordingly, to the optimum ( ),opt opth v∆ ∆ couple.

In summary, multilook unwrapped phase-gradient estimates are:

( )( ), , 2pk pk opt opt pk pkh v π= ∆ ∆ +∆Ψ ξ m ∆Φ G) )

(17)

wherein 2 pkπG)

is the integer-valued vector corresponding to the minimum cost condition.

5.3 Real data results

5.3.1 Central Nevada test site Let us start providing some experimental results achieved for a wide area

in central Nevada (USA) extending for about 600 x 100 km. In particular, we exploit 264 ERS-1/2 SAR data frames (track 442, frames: from 2781 to 2871), spanning the 1992-2000 time interval, and we generate 148 multilook DInSAR interferograms, with a spatial resolution of about 160 x 160 m. Note that SAR data set and interferogram distribution are the same exploited by Casu et al [2008] and already presented in Chapter 4.

First of all, we select the set of Seed Points from which the unwrapped phase will be propagated. To do this, we just impose a threshold value to the temporal coherence achieved by applying the EMCF algorithm (see Section 5.1), in particular we impose max 0.8γ = , obtaining more than 850,000 Seed pixels. In Figure 5a it is shown the selected Seed Point mask.


Subsequently, according to Section 5.1, we also individuate the Candidate Point set, by imposing a 0.2 0.8γ< < threshold to the temporal coherence, and to all of them we finally apply the RG-EMCF algorithm. Note that, according to eqs. (7) and (9), we impose as reliability tests ( ) 0.7pCγ γ> = (as usual in DInSAR applications, see Casu et al. [2006])

for the temporal coherence value and ( ) 0.7pC γΩ > Ω = = for the

Similarity factor, permitting to obtain the final pixel distribution shown in Figure 5b. As a final remark, note that the CW CWaz rg× extension of the Seed Point search window is set to 50 50× pixels.

In order to evaluate the phase unwrapping step performances, we compare the achieved RG-EMCF results with those presented in Chapter 4 (obtained by exploiting the conventional SBAS processing chain, see Chapter 2) and applied to the same Seed and Candidate Point sets. In particular, Figure 5b and Figure 5c show the temporal coherence masks of the reliable pixels (obtained by imposing previous mentioned thresholds for the RG-EMCF algorithm and a ( ) 0.7pCγ γ> = for the conventional one)

relevant to the conventional and new Region-Growing algorithm, respectively. By a first qualitative analysis, it is clear that the RG-EMCF approach permits to obtain a larger number of points at a given coherence threshold. More systematically, we measured an increase of about 88% for the number of “grown” pixels, while the total image increment is of about the 32%.

Moreover, we also compute the temporal coherence histogram, shown in Figure 6, relevant to the common “grown” pixels only. It is clear the coherence improvement achieved by the RG-EMCF results. As mean coherence value we obtain 0.90 for the RG-EMCF approach while 0.85 for the conventional SBAS results. Also in this case, it is clearly visible the improvement of the new proposed algorithm.

As final remark, we present in Figure 7 the mean deformation velocity maps of the two set of unwrapped phases processed via the SBAS approach. Figure 7 clearly demonstrates the RG-EMCF capability to increment reliable pixels not only in non deforming areas, but also in zone affected by strong and non linear deformation (see Figures 3b-f in Chapter 4).

Chapter 5 159

Figure 5 - Temporal coherence masks relevant to Central Nevada area. a) Seed Point mask (~850,000 pixels). b) RG-EMCF results (~1,770,000 pixels). c) Conventional SBAS results (~1,340,000 pixels). Grown points are highlighted in red while in white are represented the Seed Points of Figure 5a.


5.3.2 Gardanne test area In general, Region-Growing is very helpful to unwrap wide areas.

Indeed, manage large set of points can imply PhU infeasibility, due to the complexity of the problem to be solved as well as to the high computational burden. Therefore, a cascade of global and local PhU steps [Reigber and Moreira, 1997] permits to correctly unwrap large areas, by first deliberately reducing the amount of investigated points (Seed Pixels) and, subsequently, reconsidering the removed ones (Candidate Pixels) in a proper Region-Growing step.

However, a Region-Growing approach can be applied also in highly decorrelated region, where the low coherence implies to deal with a reduced amount of Seed Pixels. Therefore, a Region-Growing step can help increasing the unwrapped pixel spatial density.

Accordingly, we applied the proposed RG-EMCF PhU approach to the Gardanne (France) region which is highly vegetated and where an open pit mine is present. Therefore, the zone is characterized by very low temporal coherence and is affected by strongly non linear deformation, implying to be a very critical test area for multi-temporal DInSAR analysis.

Figure 6 – Temporal coherence histogram relevant to common “grown” pixels obtained by exploiting conventional (dashed line) and new (continuous line) Region-Growing algorithm.

Chapter 5 161

In Figure 8, we present mean deformation velocity maps computed by applying the SBAS approach to a set of 75 ERS-1/2 and 8 ENVISAT images

Figure 7 – Mean deformation velocity maps, in SAR coordinates, relevant to Central Nevada area. a) Conventional SBAS results (already presented in Figure 3 of Chapter 4. b) RG-EMCF results.


acquired in the 1992-2004 time period and coupled in 243 interferograms, the latter being unwrapped via both the procedure implemented in the SBAS processing chain and the RG-EMCF algorithm. Note that, DInSAR data have been multilooked, obtaining a final spatial ground resolution of about 80 by 80 meters.

Figure 8 - Mean deformation velocity maps relevant to the Gardanne area (France). a) Conventional SBAS results. b) RG-EMCF results.

Chapter 5 163

We just remark that, simultaneous exploitation of ERS and ENVISAT data implies that no cross sensor interferograms can be generated, due to the different signal wavelength of the two sensors. Therefore, the Temporal/Perpendicular baseline network will result decomposed in several subsets (at least two), corresponding to the different sensor interferograms. A more detailed analysis on this topic is provided in Appendix B.

Anyway, applied algorithm parameters and constraints are the following: – Seed Points are selected by imposing max 0.8γ = (~73,000 pixels); – Candidate Pixels are selected in the coherence range 0.2 0.8γ< < ; – Reliability test for conventional SBAS has been set to

( ) 0.7pCγ γ> = , while for the RG-EMCF it has been imposed

( ) 0.7pCγ γ> = and ( ) 0.7pC γΩ > Ω = = for the temporal

coherence and Similarity factor, respectively. Also in this case, looking at Figure 8, it is clearly visible the strong

increase of the correctly unwrapped pixels. Indeed, “grown” pixel number pass from about 2,500 for the conventional case to more than 87,000 for the RG-EMCF one, with a global improvement (for the whole area, i. e., accounting also for the Seed Points) of about the 113%. It is worthy to remark that the RG approach available in the SBAS processing essentially does not produced any results (only 2,500 “grown” pixels starting from 73,000 Seeds): this is because the strong decorrelation affecting the area.

Concerning the histogram plots (see Figure 9) relevant to the common

Figure 9 – Temporal coherence histogram relevant to common “grown” pixels obtained by exploiting conventional (dashed line) and new (continuous line) Region-Growing algorithm.


“grown” pixels, we note an increase of the computed temporal coherence in the RG-EMCF case; moreover, we calculate a mean coherence value of 0.81 and 0.86 for the conventional and RG-EMCF results, respectively.

Also for this case study, the RG-EMCF demonstrates to be an effective phase unwrapping approach, permitting to significantly expand SBAS deformation measurement density in highly decorrelated areas.

Chapter 5 165

Appendix A Non triangulation-driven data pairs in EMCF procedure

As explained within Section 5.2, application of the RG-EMCF phase unwrapping algorithm explicitly requires a peculiar interferometric distribution, obtained via a Delaunay triangulation on the Temporal/Perpendicular Baseline plane (see Figure 10a). In any case, in order to increase the robustness of the deformation time series computation, it is possible to expand the original interferometric data set (triangulation-driven) by introducing a set of additional interferometric pairs, the latter concerning SAR acquisitions already exploited or not involved jet in the triangulation distribution.

The key idea is to consider, with respect to the triangulation-driven interferometric distribution, a set of additional interferograms which respects the maximum baseline separation constraints (also eventually including the limitations on the common azimuth bandwidth depending on the Doppler centroid differences, see Chapter 2). The whole interferometric distribution, shown in Figure 10b will be subsequently used for the extraction of the corresponding deformation time series.

Unwrap these additional interferograms, in a RG-EMCF PhU scenario, implies to exploit the information coming from the set of already unwrapped interferograms relevant to the Delaunay triangulation.

Let us now consider a set of 1N + independent SAR images of the same area, acquired at the ordered times ( )0 1, ,..., Nt t t and connected by a Delaunay triangulation in the TB B⊥× plane, whose arcs are associated to the set of differential interferogram ( )1

,...,MTr Tr Trϕ ϕ ϕ= . Moreover, a set of L additional

interferograms can be introduced with respect to the original data set, and the overall wrapped phase sequence is finally the following one

[ ] ( )1 1, ,..., , ,...,

M LTr Added Tr Tr A Aϕ ϕ ϕ ϕ ϕ ϕ ϕ= = . For each investigated arc between a Candidate and a Seed Point we can

calculate, via the RG-EMCF, the unwrapped phase values for the triangulation-driven interferograms ( )1

,...,MTr Tr Trψ ψ ψ= in correspondence to

the Candidate pixel. By applying the SBAS approach, we are able to retrieve the deformation time series for the selected Candidate. In particular, it is possible to evaluate the residual topography z∆ and the relevant deformation time series ( )0 1, ,...,Tr Nd t t t . In such a way, the additional phases Addedϕ can be


unwrapped by considering a displacement model, which can be retrieved from the achieved deformation time series and the residual topography.

Figure 10 - Interferogram distribution in the Temporal/Perpendicular Baseline plain. a) Dealunay triangulation. b) Delaunay triangulation (red arcs) and non triangulation-driven interferograms (black arcs).

a

b

Chapter 5 167

To better explain this concept, let us refer to Figure 11 and suppose that the generic added j-th differential interferogram 1,...,

jA j Lϕ = involves as master the image acquired at time , jM At and as slave the one acquired at , jS At

epoch, with ( ) ( ), 0 1 , 0 1, ,..., , , ,...,j jM A N S A Nt t t t t t t t∈ ∈ . Consequently, we can

introduce the following model for the unwrapped phase:

( ) ( ), ,4

sinj

j j j

AA M A S A

bm z d t d t

rπλ ϑ

⊥∆⎛ ⎞= ∆ + −⎜ ⎟⎜ ⎟

⎝ ⎠ (A1)

wherein jAb⊥∆ is the interferogram perpendicular baseline, λ is the signal

wavelength, r is the scene center slant range distance and ϑ is the sensor look angle.

However, when ( ) ( ), 0 1 , 0 1, ,..., , , ,...,j jM A N S A Nt t t t t t t t∈ ∈ condition is not

verified, a model can be also introduced by taking into account for the mean deformation velocity Trv , which can be estimated via a linear fit of the retrieved deformation ( )0 1, ,...,Tr Nd t t t . In this case, the model can be

Figure 11 - Pictorial representation of the strategy applied to unwrap non triangulation-driven interferograms, characterized by the introduction of a linear (red circles) and/or non-linear (blue circles) deformation model, respectively.


expressed as:

( ), ,4

sinj

j j j

AA Tr M A S A

bm z v t t

rπλ ϑ

⊥∆⎛ ⎞= ∆ + −⎜ ⎟⎜ ⎟

⎝ ⎠ (A2)

In absence of noise effects (phase unwrapping and reconstruction errors) the wrapped version of the introduced model, i. e., ( )jAWr m , should

automatically fit (by neglecting the effect of the model inaccuracies) the phase

jAϕ . In general, it will be present a not null residual term, defined as:

( )j jj A Ar Wr mϕ= − (A3)

which accounts for the inaccuracies of the used model. Note that, residual phases are assumed to be included in the ( ),π π− interval. Failing of this assumption will lead to temporal coherence decrease.

From eq. (A3), the unwrapped phase solution will be finally expressed as follows:

1,...,j jA A jm r j Lψ = + = (A4)

Chapter 5 169

Appendix B Multi-subset EMCF approach In the SBAS algorithm application scenario, where large SAR data set,

as those acquired by European or Canadian satellites, are exploited, it is frequent that acquisitions are grouped in several Small Baseline interferogram subsets, the latter being separated each others by large baseline values (see Chapter 2).

In order to apply both the EMCF and RG-EMCF PhU technique to a multi-subset configuration, the Temporal/Perpendicular baseline Delaunay triangulation in the TB B⊥× plain has to be properly generated. Indeed, since no cross-subset interferograms can be computed, the whole data pair distribution will result decomposed in a set of independent triangulations.

The modification to be applied to the algorithm, with respect to the case where a single triangulation is present, concerns the introduction of additional “ground” nodes (see Appendix B of Chapter 2) whose number is equal to the number of identified subsets. In such a way, the equivalent-temporal network to be solved (for each spatial arc) will be composed by

Figure 12 - Pictorial representation of a possible two subsets data pair distribution. The equivalent temporal network to be solved will be composed by two sub networks with two “ground” nodes, respectively.


several sub-networks (whose number corresponds to that of the subset ones). A pictorial view of this issue is represented in Figure 12.

We final remark that a particular multi subset case is represented by the multi-sensor SBAS approach presented by Pepe et al. [2005], for instance the simultaneous exploitation of ERS and ENVISAT data. Indeed, SAR sensors onboard these two satellites transmit with a different carrier wavelength and, accordingly, cross-interferograms are affected by decorrelation, due to absence of spectra overlap. Therefore, avoiding cross-interferogram generation implies that in the data pair distribution are present at least two acquisition subsets, relevant to the ERS and ENVISAT data, respectively.

Chapter 5 171

References Casu, F., Manzo, M., Lanari, R. (2006). A quantitative assessment of the SBAS algorithm performance for surface deformation retrieval from DInSAR data. Remote Sensing of Environment, 102(3-4), pp. 195–210, doi:10.1016/j.rse.2006.01.023.

Casu, F., Manzo, M., Pepe, A., Lanari R. (2008). SBAS-DInSAR Analysis of Very Extended Areas: First Results on a 60,000 km2 Test Site. IEEE Geosci. Remote Sens. Lett., 5, 3, doi:10.1109/LGRS.2008.916199.

Delaunay, B. (1934). Sur la sphere vide. Bulletin of Academy of Sciences of the USSR, pp. 793−800.

Pepe, A., Sansosti, E., Berardino, P., Lanari, R. (2005). On the generation of ERS/ENVISAT DInSAR time-series via the SBAS technique. IEEE Geosci. Remote Sensing Lett., 2, 265-269, doi:10.1109/LGRS.2005.848497.

Pepe, A., Lanari, R. (2006). On the extension of the Minimum Cost Flow Algorithm for Phase Unwrapping of Multi-temporal Differential SAR Interferograms. IEEE Trans. Geosci. Remote Sens., 44, 9.

Reigber, A., Moreira, J. (1997). Phase Unwrapping by Fusion of Local and Global Methods. IEEE IGARSS ’97 Symposium, 2, pp. 869-871.

Xu, W., Cumming, I. (1999). A region-growing algorithm for InSAR phase unwrapping. IEEE Transaction on Geoscience and Remote Sensing, 37(1), pp. 124−134.

Conclusions This thesis is focused on the advanced multi-temporal interferometric

technique referred to as Small BAseline Subset approach, which exploits properly selected sequences of multilook (averaged) DInSAR interferograms to detect surface displacements, in the investigated areas, and to follow their temporal evolution.

Although the surface deformation retrieval capability of the SBAS technique has been already exploited in previous publications, no extensive performance analysis has been carried out yet. Accordingly, this work starts with a quantitative assessment of the SBAS procedure performance by investigating the quality of the retrieved key results, i. e., the mean deformation velocity maps and the displacement time series. This task has been achieved by processing large archives of SAR data acquired by the ERS sensors and comparing the achieved SBAS results with geometric leveling (in Napoli bay zone, Italy), continuous GPS (in Los Angeles area, USA) and alignment arrays (along Hayward fault, San Francisco Bay area, USA) measurements that have been assumed as reference. The presented results show that, for a typical ERS data set including between 40 and 60 images, the SBAS technique provides an estimate of the mean deformation velocity with a standard deviation of about 1 mm/year. Moreover, for the single displacement measurements, it shows a sub-centimeter accuracy with a standard deviation of about 5 mm.

Subsequently, we have proposed an extension of the basic SBAS technique (referred to as E-SBAS) in order to develop an advanced tool for surveying the temporal evolution of the deformation affecting very extended areas. By exploiting the proposed E-SBAS algorithm, we have analyzed a unique large data set of 264 descending ERS SAR data frames, spanning the 1992-2000 time interval, and relevant to an area in Central Nevada (USA) extending for about 600 x 100 km. The achieved results, obtained by exploiting multilook interferograms with a 160 x 160 m spatial grid, have been also validated demonstrating the algorithm capability to detect deformation with an accuracy comparable to that achieved through the conventional SBAS processing

174 Conclusions

Finally, we concentrated on the Phase Unwrapping problem, one of the most critical steps in DInSAR techniques, particularly when analyzing multilook interferograms relevant to very extended or low coherence areas. For this reason, we proposed a novel Region-Growing PhU approach, aimed at increase the unwrapped pixel spatial density. The new Region-Growing algorithm is derived from the “temporal” PhU strategy at the base of the Extended Minimum Cost Flow (EMCF) algorithm and, therefore, it is highly oriented to retrieve deformation time series. The presented results, achieved by applying the developed RG-EMCF PhU algorithm to multilook sequences of differential interferograms, confirm the effectiveness of the proposed approach.

Despite the very encouraging presented results, we want to remark that Phase Unwrapping is still an open issue in the SBAS algorithm framework and more generally in the DInSAR scenario. In particular, the PhU operation still represents a very difficult task for interferograms relevant to areas affected by large, fast and non linear deformation. This is due to the high interferometric phase rate, leading to absolute phase differences exceeding π between neighboring pixels. Deformation models as well as offsets calculated from the SAR amplitude images can be helpful for unwrapping such interferograms.

Other open research topics are also related to the orbital information inaccuracies and to the residual atmospheric phase artifacts, which also represent relevant error sources in the deformation retrieval process; indeed, they may affect the accuracy as well as the coherent pixel density of the retrieved DInSAR results.

In addition to these critical issues, there are also limitations related to the present DInSAR technology that is essentially based on SAR sensors operating at C-band (the ERS, ENVISAT and RADARSAT sensors wavelength is of about 5.6 cm) with relatively long revisit time (35 days for ERS and ENVISAT, 24 days for RADARSAT). In this case, the coherence is essentially maintained only in urbanized and rocky areas, while it is completely vanished in vegetated zones.

In this context, a key role can be played by the next generation sensors like the Japanese ALOS-PALSAR system (launched in January, 24th 2006) which operates within the L-band (about 23.6 cm wavelength). In this case, although a sensitivity reduction in the displacement detection is caused by the wavelength increase, a very significant coherence improvement is present in the generated DInSAR interferograms. Moreover, by benefiting of its ScanSAR operational mode, characterized by an illumination footprint of some hundreds of kilometers, it can be possible to apply the presented large

175 scale SBAS approach to burst-synchronized ALOS ScanSAR acquisitions, to exploit the DInSAR technology for continental based deformation analysis.

To conclude, future scenarios involving lower frequency, multi-LOS and multi-temporal SAR observations on extended areas, coupled with reduced revisit time, seem extremely promising for the development of advanced applications of the SBAS approach and more generally for the DInSAR techniques. This may finally lead to include the DInSAR methodology, exploited for surface deformation survey, within routine surveillance scenarios.

Acknowledgements I would like simply to be grateful to, in random order: Riccardo Lanari,

Giuseppe Mazzarella, Serafino and Lorenza, Paola and Anna, Maria Caterina, Sebastiano, Andrea, Chris, George S. and George J., Stè e Frà, Iole Manunza, Simonetta and Maurizio, Stefania, Lorenzo and Polpettina, Mary and Prof. Perna, Pepper, Michele, Manu, Episcopio, Mimma, Paucio, Diccone, Dicchino and Dicchetta, Vanni, Eug, Paolino, Alfius and Dolcetto, Mega, Silviona, la petite Ivy, Fabius, Carletta, Ornella, Angelo, Patata and Canellu, Silvia Mu, sorella Mu, mamma Mu, Master Natale, the Soul Girl, Silvia and Gianlu(igi!), Vale and Giamma, Anna C and Attila, Kiki, Gu, Stefy, Ninnina, il mezzadro, Giuoia, Sandrino, Ettore, Checchino, Enrico, Quelo, Fico, Daniele, Emi, Fabio Casu, Alessandro, Valeria, Gigi and Barbara, Gaio Valerio Catullo, the AMRA center, the project manager, le assegniste, S. Angelo dei Lombardi, il Tugurio, Mrs. De Luca, Mrs. Vanda, Chicca and Alberto, Carmine and Giovanna, Casteddu, Il Casteddu, Rome, Naples, Venice, Los Angeles, Austin, Sean, Vikas, Dochul, Poba, Noel, Paul, Valeria, Marco and Silvia, Giuliana and Rossella, Prof. Migliaccio, Ciccio, Mister G., Laura and Ciccio, Pablo and Valeria, Leonardo A., Leonardo B., Nenno Lello, le oche, gli ochi, Antonietta, Gianca, Gnagno, Don G.P., Barbie, the Slow Food friends, Bra, gli EELST, the Queen, Carlo, Luca, Stefano, Andreina, Cè, Manu, Giorgio, Rita and, dulcis in fundo, Dany.

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