GLRT-Detection Performance in Subsurface Sounding Massimo Sciotti, Debora Pastina, Pierfrancesco Lombard0

Dept. INFOCOM, University of Rome ‘ l a Sapienza ”, via Eudossiana 18, 00184 Rome, Italy Ph. : +39 0644585860, fax: +39 064873300, e-mail: rn.sciotti@infocom.uniromal. it

Absfrucr- The performance of subsurface deep sounding is pulse processing algorithms. In particular, along-track pulse investigated with reference to the radar sounder MARSIS, integration efficiently accomplished by following the aboard the Mars Express mission, designed to investigate the Generalized Likelihood &tio (GLR) approach. ne GLR presence of water-related interfaces in the subsurface of Mars.

necessary tools for (i) performance prediction and (ii) data the desired detection performance in the considered processor design. To this aim, by using well known models for hhlRsIs condition. the backscattered signal, we compare the expected Signal-to- This paper is devoted to fblly investigate the performance Clutter Ratio values under most of the instrument operative of the introduced GLRT-scheme in several operative conditions. The Generalized Likelihood Ratio approach is conditions. In particular, along-track pulse integration is followed for subsurface interface detection, and along-track

performance. In particular, we address the design of the SCenaiOS, We aim at (i> assessing the necessity of Performing integration window, and the requirements of data along-track integration, and (ii) defining the integration homogeneity. A thorough performance analysis is presented to window size. In this analysis two aspects are considered in cope with the expected MARSIS scenarios- In Particular, we particular: (i) clutter samples correlation, and (ii) data-

model mismatch. Specifically investigate several sources of mismatch between the assumed model and collected data, and derive the performance degradation due to each source. (i) Clutter samples may result correlated along range and

along-track directions. Both conditions are considered 1. INTRODUCTION and the impact on performance is analyzed, with

In [l] we introduced an optimized multi-pulse technique reference to the integration window size, i.e. M. for the processing of subsurface data that will be collected in (ii) Higher values of M require homogeneity over wider the near l ture by the Mars Advanced Radar for Subsurface areas of the surface, and over longer time slots. The and Ionosphere Sounding (MARSIS), [2]. This low- considered scenario may not l lf i l l these requirements, frequency radar sounder and altimeter is presently riding since both clutter and target signals fluctuations are aboard the Mars Express Mission probe on its orbit round expected due to changes in the surface and subsurface Mars. The radar instrument aims at detecting subsurface behavior. These introduce a data-model mismatch, and interfaces, which may indicate the presence of liquid water its impact on the detection performance is hlly on the planet. Specifically, a change in the nature of the investigated. In addition, we address the problem of an pore-filling material (fiom gas or ice to liquid water) is imperfect knowledge of clutter model parameters, and we expected at depths within few kilometers. This would investigate the performance degradation also in this case. introduce an abrupt change in the dielectric behavior of the subsurface, which originates the reflected signal to be sensed by the radar. This “target” signal competes with (i) the random contributions of the subsurface, and (ii) the strong echo that is reflected by the surface (see Fig. 1). Techniques for the extraction of the water-related target signal should cope with such a challenging clutter scenario: adequate Signal-to-Clutter Ratio (SCR) values are required for achieving good detection performance.

In [l], we considered the operative condition that is summarized in Table 1, named condition “zero”, as being representative of a favorable scenario for MARSIS. This yields SCR values of the order of only few a s , and single-

terms of probability of detection (Pd) . This is due to the strong attenuation that is suffered by the signal in the Mars soil, and the large depth to be investigated (&3000 m), A

The analysis proposed in this paper aims at providing the test (GLRT), L1l¶ processes multi-pulse data, thus assuring

integration is introduced in order to achieve the desired addressed in this Paper. By the Of different

pulse detection hardly achieves the desired performance in Table 1. MARSIS Operative Condition “zero”.

11. MARSIS ECHO MODELS

we the Mars to comprise a two-layer porous large performance improvement can be achieved by mae , with pores filled by gas, liquid water or ice,

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depicted in Fig. 1. This results in two well-known models for water-related discontinuities in the Mars subsurface, [2]: the Dry/Ice @/I) interface and the: Icemater 0 interface, named after the pore-filling elements in the two layers. Both these two discontinuities can be hund in basalt or andesite soils, so that four different subsurface conditions are possible, as described in details in [l-21.

The dielectric discontinuity originates a target echo, which largely depends on the interface geometry, as related to the impinging radar wavelength. We describe the backscattering phenomenon in accordance with Kirchhoff assumptions, [l-31, and we assuime a zero-mean Gaussian probability density function (PDF) to describe the random process of interface heights z~(x,J~). We adopt an equivalent rms height oeq for the height random process zss(x,y). This encodes the random reflection and transmission phenomena, which occur in the layered subsurfkce.

The received clutter echo comprises (i) the random contributions fiom the subsurface., and (ii) the echo reflected by the surface. We neglect the former, as being generally well below the latter, and we describe the latter in accordance with [l]. Thus, for a given soil composition, the surface rms height OH and the correlation length Icon filly determine the surface echo level at the antenna. In particular, the clutter incoherent contribution, which constitutes the clutter echo tail, decays with a time constant, a, that is determined by the: two surhce geometric parameters. This only contribution is relevant to our analysis, since it temporally competes to the target signal scattered by the subsurface interface (Fig. 1).

The above assumptions allow us to derive a statistical model for the received signal, which is given by the composition of the surface echo (the clutter) and the subsurface echo (the target). Specifically, we assume a Gaussian PDF for the clutter echo, while the target signal is considered to have a known waveform As mssfdo), [l], which depends on the unknown interface depth do, and unknown amplitude Ass. The clutter signal mean value contribution, ms, is given by the coherent echo, assumed as known, [l]. The incoherent cluttm component gives a zero- mean random contribution with correlation matrix RN(Ac,a), being Ac and a the unknown amplitude and shape parameters, respectively.

At each Pulse Repetition Interval (PRI) we assume to collect N samples, which approximately correspond to N resolution cells. We arrange the N samples in the Nxl received signal vector rk=[r],..,rrlT and we consider the two hypotheses: &, no interface is present, and HI, an interface is present at depth do. For the above considerations, the received signal vector rk can be written as rk =ms + { A s s .mss(do)+nk where FO under &I and el under HI. The vector represents a zero-mean Gaussian disturbance, which encodes the randomness of the

surface scattered field. As in [ 11, we assume a diagonal NxN correlation matrix for the disturbance vector nk:

0 1

...

Under the assumption of homogeneity of the considered scenario, M echoes, due to M subsequent transmitted pulses, share the same statistical properties. This means that the joint PDF ofthe Mvectors rk(k1 , ...,M) canbe written as

(2) Recall that the unknown quantities are Ass, do, Ac, and a under the HI hypothesis, while being Ac and a under the null hypothesis. The statistical description of the M echoes in (2) allows the derivation of the GLRT, reported in [ 11.

nI. GLRT-SCHEME FOR DETECTION The main goal of MARSIS data processing is the

detection and depth estimation of subsurface interfaces. In practice, we first perform detection over the N resolution cells, [dl, ..., dN], covering the investigated range. Then, we refine depth estimation within the cell containing the target.

The GLR criterion, [l], gives the following test to detect an interface at the n-th range cell, d,:

M

lpS(dn)H%l%)-l(rk -%[ 4l V G L R ~ = * M 2

~msS(dn)"%(%)- 'msS(dn~~rk -%)H&(%)-l(rk -%) 4 k=l

(3) If such value exceeds the threshold, we decide for the null

hypothesis, otherwise we have the detection of an interface. The test in (3) is applied to the Nrange cells by specifying the weight vector mss(dn) for each cell, and then the minimum value out of the N outputs is selected and the output is compared to the detection threshold A. As in [l], the detection threshold is set in accordance to the desired Probability of False Alarm (Pya) at the cell under test.

IV. PERFORMANCE ANALYSIS

For MARSIS condition "zero" we demonstrated in [ 11 the necessity of performing along-track integration due to the expected low SCR value. Departing fiom condition "zero", detection performance may drastically change, due to the wide range of values for the key model parameters. Thus, our analysis is first devoted to derive the dependence of the detection performance on radar key parameters, i.e. radar fiequencyf, and orbit height H , and target parameters, i.e.

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target depth do, target model, and target roughness 0,. This analysis allows the design of the integration window, namely the number of pulses to be integrated. Then, we consider non-ideal or non-homogeneous behavior of the collected data within the integration window. The following conditions are investigated (i) correlated clutter samples, (ii) estimation errors, and (iii) target signal fluctuations. The latter two conditions are addressed as sources of data-model mismatch, namely the applied detection scheme assumes clutter and target parameters values, which do not match the true values of the data being processed.

A). MARSIS OPEmnm CONDITIONX Each set of values for system and target parameters yields

an expected SCR value, which determines the detection performance and the number of pulses to be integrated in order to achieve the required Pd. As an example, Figs. 2 and 3 show SCR plots as a function of time delay for different orbit heights and different target models. It is evident that lower orbits are characterized by an improved detection capability if the surface geometry is fixed. This is due to the fact that lower losses are experienced by both clutter and target echoes, but only the surface clutter echo is highly affected by the local grazing angle. Recall that the target signal is considered as mainly determined by the nadir reflected echo fiom the interface at depth d,,. This results in higher SCR values, as evident in Fig. 2. In addition, Fig. 3 shows that the target model of condition “zero”, namely the UW interface in andesite, represents the most detectable target, since other models yield lower values of SCR Recall that the four models determine different propagation speeds for the radar signal, so that echoes from the same target depth are received with different time delays in Fig. 3.

Similar SCR plots have been obtained by varying other key parameters within the expected range of values. Fig. 4 reports the resulting SCR values for all the considered operative conditions, with reference to the SCR value in condition “zero”. This means that the expected SCR value is obtained by adding the SCR value (in dB) obtained in condition “zero”, [l], to the ASCR value reported in Fig. 4.

As apparent condition “zero” represents a favorable case, since most conditions yield lower SCR values, (AscR<O). Higher radar fiequencies and different choices for the target interface roughness, oq, yield reduced detection capability. Only shallower targets are characterized by positive values of Asc~, due to the large increase in target level.

Note that the M axis is also reported in Fig. 4, which indicates the values of M that are required for achieving a Pd close to 0.9 for that SCR value at Pf,=10-3. Such values are obtained by simulating that SCR value in condition “zero”. This is due to the hct that the simulation of all the operative conditions for all values of M is impracticable. Crosschecks via data simulation in sample test conditions yield good correspondence. Thus, the values in Fig. 4 only represent a

~

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rough estimation of the required value of My but they allow the design of the integration window for all the considered operative conditions. It is apparent that in some cases a number of pulses greater than 50 is required. We thus expect a poor detection performance in all these cases, since homogeneity is hardly assured for such long track sectors. Note also that single-pulse detection is feasible just in very few operative conditions, while it generally results that a minimum number of M-5 pulses is required. We assume this number as a good compromise between the detection performance improvement and the scenario homogeneity constraint. Thus, in the following performance analysis, we always consider the condition “zero” with A&l and M-5, in order to investigate the impact of clutter correlation and data-model mismatch. Synthetic data are generated under the HI hypothesis, (see Fig. 5), and Pd plots are derived Note that the detection threshold is set for achieving Pfa=1O3 in the absence of any correlation or mismatch.

B). CLUTTER CORRELATION

The second purpose of our analysis is the investigation of the GLRT behavior with respect to clutter samples correlation. Correlation generally originates fiom overlapping areas of the surface being illuminated at different time delays. Recall that the GLRT in (3) has been derived for uncorrelated clutter samples. If we arrange the integrated vectors rk ( b l ,...,M) into the MN-dimensional vector r, for the model assumptions the correlation matrix of the corresponding MNxl vector n can be written as

Correlation may affect echo samples backscattered by the surface both in the same PRI (range correlation in range) and in adjacent PRIs (along-track correlation). Thus, we first modifir the sub-matrix Ro(~)Nf i , introducing the same

range-correlation properties at each PRI; then, we consider the off-diagonal submatrices of Rmxm as being non-null.

In both cases we adopt a Gaussian shaped autocorrelation function with correlation coefficients: 0.3,0.7, and 0.9.

Fig. 6 shows the GLRT probability of detection as a function of gH in the case of only range-correlated samples. small differences in Pd are reported, and the impact of range correlation can be considered as negligible. This is due to the large number N of pulses that describe the clutter signature, which allows good target discrimination in any condition.

Both range and pulse-to-pulse correlations are considered in Fig. 7 . Again, small Pd variations are reported. In contrast, Fig. 8 shows the ideal Pd plots achieved by the GLRT, by matching the detector to the correlated clutter

samples. It is well known that false alarms control over correlated clutter samples can be efficiently achieved through lower CFAR thresholds than over independent samples. Consequently, lower threshold values introduce a Pd gain, which is apparent in Fig. 8. However, threshold matching implies a good knowledge of the clutter correlation properties via the estimation of the covariance matrix. This is hardly feasible for deep subsurface sounding, due to the large number of coefficients to be estimated and the absence of matrix structure to redulce the estimation cost. Concluding, due to the limited howledge of the MARSIS clutter scenario, we adopt the fully independent samples model for clutter, thus discarding the correlation information. No significant inipact on performance is reported.

C. DATA-MODEL MISMATCH The first source of data-model mismatch here considered

is the imperfect knowledge of the clutter echo time constant, a. This parameter describes the time decay of the clutter incoherent contribution, which directly determines the SCR value. Generally, a priori knowledge is considered available for this parameter, due to rough estimation over wide areas. This may eventually introduce mismatches with local data value of a. Their effect has been derived by processing synthetic data; Figs. 9-10 report the resulting GLRT plots. The performance degradation is impressive: the GLRT scheme suffers fiom both an increase of the number of false alarms (see Fig. 9), and a strong reduction of the detection capability (see Fig. lo), even for mismatches of low entity, i.e. 5 or 10 % the true value of CL. Note that the Pd plots in Fig. 10 are relative to the same detection threshold, as set for Pya=103 in the absence of mismatch. For this reason Pd plots show detection losses increasing with decreasing mismatch: the higher is the mismatch, the higher is the Pfa increase and the lower is the detection loss. Thus, local estimation of a should be performed within reference cells, preceding and following the pulse under test. By performing simple averaging over a small group of pulses, we obtain a good reduction of the performance degradation. However, this is very affected by the synthetic data generation model; performance assessment may come only fiom live data processing or simulations of the received electromagnetic fields. In conclusion, the exploitation of rough estimates and a priori information data seems not to be reliable, being local estimation of a strictly required.

Finally, other two sources of mismatch are investigated. We define the NMdimensional target vector as

k where the N-dimensional vector m, comprises the N

target echo samples received in the PRI. Similarly we can define the NMdimensional clutter vector as

where the N-dimensional vector m ; comprises the N samples of the coherent clutter echo received in the kth PRI. Recall that in the assumed model, the M vectors mis ( mt ) are coincident to m, ( m,) and filly known, being its expression reported in [ 11.

To properly evaluate GLRT robustness to data-model mismatch, we introduce a random phase shift in the target vector and an amplitude factor in the clutter vector. Specifically, we first introduce a zero-mean Gaussian- distributed phase term of variance cq in ST, in order to mismatch the M vectors m ", Note that each target sub-

vector is affected by the same phase shift, while it changes fiom pulse-to-pulse. Then, we introduce errors in the suppression of the clutter vector sc in (3), in terms of a scaled version of Q itself

The resulting Pd plots for target mismatch are reported in Fig. 11. Both mismatched conditions largely affect performance. However, the coherent clutter echo affects only the first range samples in each PRI. At typical target time delays, this contribution has vanished. Therefore, by discarding the first data samples fiom the processing scheme, the coherent clutter mismatch yields negligible effects. In contrast, the performance degradation due to target mismatch is intrinsic to the along-track integration process itself Optimal re-alignment of target samples requires the knowledge of the target pulse-to-pulse behavior. In general, neither reliable models nor reference cells for local estimation of the target behavior are available. Therefore, the Pd losses in Fig. 11 must be limited by reducing the number of pulses Mto be coherently integrated. The design of M thus requires a good compromise between the SCR enhancement and the control of Pd losses in non- homogenous conditions.

IV. CONCLUSIONS

This paper addresses the problem of detecting a subsurface interface, referring in particular to the sounding of the Mars planet which will be performed in the near fitture by the MARSIS instrument. A multi-pulse technique based on the GLR criterion has been introduced. In this context we assessed its performance with respect to the key model parameters and in several operative conditions of MARSIS. Expected SCR values are considered in the light of pulse-to- pulse integration; this allowed data processor design and the definition of the scenario homogeneity requirements. Model mismatch has been considered in order to test the robustness of the GLR-scheme, and to investigate the effects on performance for each model parameter.

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REFERENCES

[l] M. Sciotti, D. Pasha, P. Lombardo, A.M. Pellizzeri, “Subsurface Sounding of Mars: Multi-pulse Detection of Water-related Interfaces”, IEEE Int. Cod. on Radar, 3-5 September 2003, Adelaide (Australia). G. Picardi S. Sorge, R. Seu, G. Fedele, C. Federico, R Orosei, “Mars Advanced Radar for Subsurface and Ionosphere Sounding (MARSIS): models and system analysis”, Infocom TR 007/005/99, May 1999. M.V. Beny, “The statistical properties of echoes difiactedfrom rough surfaces”, Phil. Trans. R Soc., Vol. 273, pp. 61 1-654, Feb 15&, 1973.

[2]

[3]

Fig. 1 Sketch of the received clutter and target power levels, as a fimcfion of bme delay, bemg ongmated by the two-layer structure.

: *

5” *

.. - .

I I I

50 10 1 M

Different target roughness

Different target models

Different target depths

Differeni radar frequencies

Different orbit heights

Dfi in basalt D/I in andesite IIW in basalt

d=500 m +lo00 m h2000 m &=WO m

H . 8 MHZ p 3 . 8 MHz f i 2 . 8 MHz

H=300 Km H400 Km

Fig. 4: A m values for the considered MARSIS operative conditions with reference to the expected SCR value in condition “zero”.

Fig. 2: Signal-tc+Clutter Ratio plots for different platform heights. The other parameters are set as io MARSIS condition “zero”, and aH=15m.

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i 8 9 10 11 12 I3 14 15

. I $ ............ .. ..,... .. ..... .. ..... ...... ..,K .......I...... ~ _..__.__.._,._ .,... ..,... ; ........._....... .i

in”

Fig. 9: GLRT-Probability of f i l s alarm under MARSIS condition “zero” for ~ = 5 pulses are considered for integration.

Fig. 6: GLRT-Probability of detection for range correlated clutter samples under MARSIS condition “zero”.

errOrs in the clutter tirne decay constant

Fig. 7: GLRT-Probability of detection for two-dimensional correlated clutter samples, i.e. in range and along track directions, under condition “Zero”.

Fig. 10: GLRT-Probability of detection under MARSIS condition “zero” for dserent mors in the clutter time decay constant estimation. M=5 pulses are Considered for integration.

I . . .

6 7 B 9 19 11 12 13 14 15 16 Ofi

~ i ~ . 8: GLRT-Probability of detection for two~mensiomi correlated samples under condition ‘.zao93 and of the threshold. for integration.

Fig. 1 1’: GLRT-Probability of detection under MARSIS Condition “zero” for different errors in target and clutter vectors. M=5 pulses are considered

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