Direct Estimation of Faraday Rotationand other system distortion parameters

from polarimetric SAR data

Mariko Burgin1, Mahta Moghaddam1

Anthony Freeman2

1The University of Michigan, Ann Arbor, MI, USA2JPL, California Institute of Technology, Pasadena, CA, USA

IGARSS 2010, Honolulu, HawaiiJuly 25 – 30, 2010

Overview

• Problem statement• Approach:

• Generate synthetic data• Estimate system parameters

• Summary• Future work

Problem statement (1)

Low- frequency radars – radars operating at L-band and lower – must be well calibrated not only for system distortion terms, but also for Faraday rotation.

Polarimetric radar system model:

S = scattering matrixRF = one-way Faraday rotation matrixR = receive distortion matrix (of the radar system)T = transmit distortion matrix (of the radar system)A(r,θ) = real factor representing the overall gain of the radar systemexp(jϕ) = complex factor representing the round-trip phase delay and system

dependent phase effects on a signalN = additive noise terms

With δi (i = 1,2,3,4) = crosstalk values

f1, f2 = channel amplitude and phase imbalance terms

Ω = Faraday rotation angle

2 3

1 1 4 2

1 1cos( ) sin( ) cos( ) sin( )( , ) exp( )

sin( ) cos( ) sin( ) cos( )HH HV HH HV HH HV

VH VV VH VV VH VV

M M S S N NA r j

M M S S N Nf f

δ δθ φ

δ δΩ Ω Ω Ω

= + − Ω Ω − Ω Ω

exp( ) TF FM A j R R SR T Nφ= +

Problem statement (2)

General assumptions:

• A(r,θ), exp(jϕ), N are considered negligible

• Reciprocal crosstalk: δ3 = δ1, δ4 = δ2

• Backscatter reciprocity: SHV = SVH

2 1

1 1 2 2

1 1cos( ) sin( ) cos( ) sin( )

sin( ) cos( ) sin( ) cos( )HH HV HH VH

VH VV VH VV

M M S S

M M S Sf f

δ δδ δ

Ω Ω Ω Ω = − Ω Ω − Ω Ω

KNOWN UNKNOWN

Measured:16 polarimetric cross products from the Muller matrix in the form MxxMxx, x є H, V

Retrieval:15 polarimetric cross products from the scattering matrix in the form SxxSxx, x є H, V plus f1, f2, δ1,δ2, Ω

Approach Overview

• Generate synthetic data= create ground truth for Ω and system parameters

= superimpose on airborne data

• Estimate system parameters• Estimate arg(f1/f2)

• Estimate remaining system parameters from nonlinear system of equations

Generate Ground Truth (1)

AIRSAR data available for P, L, C band

Predict Faraday rotation angle Ω

Faraday rotation ranging from 2.5° (C-band) to 320° (P-band)

Input needed for models:• altitude• frequency • year/date• look/azimuth angle• time• latitude/longitude

Pseudo-spaceborne

Ω

~ 7.7 km data withoutFaraday rotation

~ 1300 km data withFaraday rotation

airborne

2 models needed:• IRI – 2007 (International Reference Ionosphere)• IGRF (International Geomagnetic Reference Field)I gives Ω to within +/- 10°

16 polarimetric cross products of the scattering matrix in the form SxxSxx

Longitude

Faraday rotation angle Ω in degrees March 1, 2001 – 12:00 UTC

100

80

60

40

20

0

-20

-40

-60

-80

Generate Ground Truth (2)

Pseudo-spaceborne “measured” values + ground truth are known

Faraday rotation angle for AIRSAR data sets: L-band: Ω = 1.5° P-band: Ω = 12°

Assumed system distortion parameters (arbitrary):

Build 16 “measured”

Muller matrix polarimetric

cross products

Pseudo-spaceborne

Ω

~ 7.7 km data withoutFaraday rotation

~ 1300 km data withFaraday rotation

airborne

°=<°=<

=

=

5

3

05.1

89.0

2

1

2

1

f

f

f

f

°−=<°=<

−=

−=

10

43

32

35

2

1

2

1

δδ

δ

δ

dB

dB

Estimate system parameters (1)

• Convert system model into a form suggested by Freeman (2008):

Assume Ω is known to within ΔΩ:t = tan(Ω) Ωtruth = Ω + ΔΩΔt = tan(Ω+ΔΩ) – tan(Ω)

Faraday rotation Ω can be assumed known to within +/- 10°,-10° < ΔΩ < 10°

2 3

1 1 4 2

1 1HH HV HH HV

VH VV VH VV

M M S Sx x

M M S Sx F x Fαβ

=

M USVαβ=

1 1 2 1

1 1 21

1 1 2 12

21 1 2 2

1

1 ( ) ( )

(1 ) ( )

(1 ) ( )

( )

f t t t t t

t f t t tx

t f f t t t tx

t f t t t tF

α δ δ δδ δ

αδ δ

αδ δ δ

α

= + + ∆ − + − ∆− − ∆ + − + ∆=

− − ∆ + − + ∆=

+ + ∆ + + + ∆=

2 1 2 1

2 1 23

2 2 2 14

22 1 2 2

2

1 ( ) ( )

( 1) ( )

( 1) ( )

( )

f t t t t t

t f t t t tx

t f f t t tx

t f t t t tF

β δ δ δδ δ

βδ δ

βδ δ δ

β

= + + ∆ + + + ∆− − ∆ + − + ∆=

− + ∆ + − + ∆=

+ + ∆ − + − ∆=

Estimate system parameters (2)

• Invert for SxxSxx polarimetric cross products:

• Find a representative quantity that allows retrieval of additional information:

• A-priori estimates for system distortion parameters available from ground testing + pre-flight calibration

• Sweep values of phase of f1, f2 and calculate A for each (f1, f2) - pair

• For a perfectly calibrated system: A = 0

• The relative phase of f1, f2 can be retrieved by finding the minimum of A

• If a similar relationship can be defined for other parameters, take advantage of them

1 11S U MV

αβ− −=

* *( )HV VH VH HVA abs imag S S imag S S= +

Estimate system parameters: f1, f2 known (1)

2 1

1 1 2 2

1 1cos( ) sin( ) cos( ) sin( )

sin( ) cos( ) sin( ) cos( )HH HV HH VH

VH VV VH VV

M M S S

M M S Sf f

δ δδ δ

Ω Ω Ω Ω = − Ω Ω − Ω Ω

Estimate system parameters: f1, f2 known (2)

• Solve nonlinear equation system with Secant method or Newton-Raphson method (Fletcher-Reeves or Polak-Ribière)

Initialization of unknowns: - 15 SxxSxx polarimetric cross products initialized with respective MxxMxx

- δ1, δ2 good guess

- Ω to within +/- 10°

- f1, f2 assumed known

• Requires first and second derivatives

• Derived derivatives analytically

Excerpt of sample output:

Estimate system parameters: f1, f2 known (3)

With cost function for L-band:

• Ω can be retrieved for an interval Ωtruth = Ωguess +/- 10° for initial conditions in the following range:

• Magnitude of δ1, δ2: |δtruth| +/- 8 dB

• Phase of δ1, δ2: < δtruth +/- 40°

• SxxSxx can be retrieved to within -30 dB

• δ1, δ2 are not well retrieved

Estimate system parameters: f1, f2 known (4)

With cost function for P-band:

• Ω can be retrieved for an interval Ωtruth = Ωguess +/- 10° for the same range of initial conditions

• Observed some sensitivity to initial conditions (estimation sometimes diverges)

• δ1, δ2 are not well retrieved

Estimate system parameters: f1, f2 known (5)

Computation time (Number of iterations = 500)

• Secant: ~ 4 hrs

• Newton-Raphson: ~ 15 hrs

Make cost function sensitive to δ1, δ2:

• Adjust optimization algorithm

• Secant method: adjust α only minor improvement

• Newton-Raphson: algorithm with Fletcher-Reeves converges better than Polak-Ribière

• Create cost function to be sensitive to different parameter

• Retrieval of Faraday rotation angle is opposed to retrieval

of δ1, δ2 split task into two procedures

different cost functions

Estimate system parameters: Sensitive to δ1, δ2 (1)

Original cost function is kept for retrieval of Ω

Find the optimal cost function to retrieve δ1, δ2:

• Investigate individual channels while sweeping different parameters to assess sensitivity

• 32 channels, 20 unknowns:

• allows to discard 12 channels

• weighting of the remaining channels to emphasize desired sensitivity

Estimate system parameters: Sensitive to δ1, δ2 (2)

Plots of magnitude and phase of δ1, δ2 sweeped over respective parameter show:

• Cost function is more sensitive, no more divergence

• Magnitude of δ1 has most sensitivity; magnitude of δ2 and phase of δ1, δ2 are more difficult to retrieve

• Cost function has to be further improved d work in progress

Estimate system parameters: full system model (1)

Full system model: solve for everything in one shot• Setup:

• 32 nonlinear equations, 16 complex polarimetric cross products of the Muller matrix

• 24 unknowns: - 15 SxxSxx poarimetric cross products

- Real and imaginary part of δ1, δ2

- Faraday rotation angle Ω

- Real and imaginary part of f 1, f 2• Cost function without simplifications, all channels contribute

• Initialization of unknowns: - 15 SxxSxx polarimetric cross products initialized with respective MxxMxx

- δ1, δ2, f1, f2 good guess

- Ω to within +/- 10°

Estimate system parameters: full system model (2)

Findings for L-band:• Promising retrieval for Ω, f1, f2 and SxxSxx for

initial conditions in the following range:

• |δtruth| +/- 8 dB

• < δtruth +/- 40°

• To retrieve δ1 and δ2 still need a modified cost function

Summary and Future work

• Developed method to produce pseudo-spaceborne data solid ground truth

• Investigated two different approaches to directly estimate parameters of the system model

• Procedure to improve cost function outlined and enhanced sensitivity established for δ1, δ2

• First results are promising for retrieval of Ω, SxxSxx and f1, f2 as well as δ1, δ2

Find optimal cost function to retrieve δ1, δ2

Integrate the two approaches to make retrieval more robust

Thank you very much for your attention.

Questions?

Retrieval in L-band for different points in AIRSAR data

With cost function in L-band:

Retrieval in location

Pixel: 200/2560

Line: 800/1156

Retrieval in L-band for different points in AIRSAR data

With cost function in L-band:

Retrieval in location

Pixel: 1200/2560

Line: 700/1156

Retrieval in L-band for different points in AIRSAR data

With cost function in L-band:

Retrieval in location

Pixel: 1000/2560

Line: 1200/1156