ynl fl∗snlexpjn jnwnl

ABSTRACT

This paper focuses on the estimate of the Subsidence Rate (SR) by repeat-pass SAR interferometry over distributed target with different decorrelation model. The approach, based on two estimation steps, is used to predict the performances achievable by new sensor exploiting conventional differential interferometry, as a function of repeat interval, the time span and the wavelength.

1. INTRODUCTION Let us assume a data-set of N repeat-pass complex SAR surveys over the same area. In particular, we focus on a patch of L pixels on the ground, so small to assume that both statistics and deformations are stationary in space. Let yn (l), be the l-th pixel of the n-th observations, and N the number of images. A suitable model of the data is the following:

(1)

where: - f {} is the linear operator accounting for SAR

acquisition and focusing; - θ is the vector of the unknown parameters,

assmed constant in the L pixels; - φn (θ) is a known function of the unknowns; - sn(l) represents the source in the n −th image,

the observed reflectivity; - αn represents the atmospheric phase noise,

usually referred to as atmospheric phase screen (APS);

- wn(l) is the thermal noise in the n −th image. The source in each image is assumed an

homogenous target, i.e., a fully developed

speckle [1]. Therefore, we model the statistics of the sources as a multivariate normal pdf,

with cross-correlation matrix Γ, whose

elements express the coherence between image pairs.

(2)

The term αn in (1), the APS noise, [2], is assumed as a zero mean normal process normal process, highly correlated in space and uncorrelated in time (the decorrelation time of the APS is about one day).

2. TWO STEP CRAMÉR RAO BOUND

In order to describe the pdf of data it is convenient to represent the problem (1) as follows:

yn Fs n expjn jn where: - yn is a vector correspondent to L pixels out

of the n − th image:

yn yn1 yn2 . . . ynLT

- F is the L×D matrix implementing in the

discrete domain the convolution of a sequence for the acquisition filter f {}, being M the length of the filter and D = L + M −1;

- sn is a vector representing the source in the n-th image:

The joint pdf of N images may be computed by taking the expected value of the pdf of data conditioned w.r.t. the APSs:

(3)

being y the column vector with all the observations stacked, and likewise α the

snl N0, 1

Esnlsm∗ k nm l−k

py; py;|pd

s n sn1 sn2 . . . snDT

OPTIMUM INTERFEROGRAMS STACK VERSUS PS: MULTIFREQUENCY ANALYSIS

Andrea Monti Guarnieri, Stefano Tebaldini, Fabio Rocca

Politecnico di Milano, DEI. P.za Leonardo da Vinci 32, 20133 Milano, Italy Email: monti@elet.polimi.it

_____________________________________________________

Proc. ‘Envisat Symposium 2007’, Montreux, Switzerland 23–27 April 2007 (ESA SP-636, July 2007)

column vector with all the APS contributions, and “*” the Hermitian transposition. Computing the CRB directly from (3) is not practically feasible as it would require the evaluation of an integral on a domain whose dimension equals the number of the images. We propose here a different approach. Let us define the N phases due to the N optical paths as a superposition of the target’s phase (that includes the target motion) and the APS:

(4) We perform the estimate of the parameters vector θ into two steps: (1) we estimate the N phases φn (related to the

optical paths) from the data, (2) we use these phases to estimate ϕn and thus

the subsidence parameters. The purpose here is to derive an expression for the estimator accuracy.

2.1. Step 1 – Phase estimate

Let us introduce the following definition:

The pdf of the observations is circular normal with covariance matrix:

(5) where ⊗ indicates the Kronecker product. Its inverse can be written as follows: (6) that can be exploited for computing the Fisher Information Matrix (FIM) X of the phases φn :

thus we get, from (5) and (6) :

(7) where γnm is the nm−th element of Γ and nm

−1

are the nm−th element of Γ-1. Notice that the rank of X is N−1, since only N−1 phases may be actually estimated. We will then assume all phases to be measured w.r.t. a conventional

master, and define φ1 = 0, φ1 = 0. In the following, we will handle the singularity of X by defining Xε=X+εI and assuming ε≈0. 2.2. Step 2 – Subsidence estimate The output of the first step is the covariance matrix, Xε

−1, of the best unbiased estimator of images phases, φ. We can thus rewrite (4) as follows:

(8) where εn is the estimation error, whose pdf is N(0, Xε

−1). In order to have access to (8), we assume that phase unwrapping has been properly performed, that makes sense since we are deriving a lower bound for the estimate of subsidence parameters, θ. Under this assumption, the linear model (8) leads to the following expression for the parameters FIM: (9) whose inverse provides the lower bound for the covariance of the subsidence parameters. From the point of view of information theory, it may be shown that this bound, which we refer to as Two Step Cramér Rao Bound (TSCRB) is equivalent to the Hybrid Cramér Rao Bound (HCRB), where the unknowns are both deterministic parameters and stochastic variables. For a comprehensive discussion about the HCRB and its applications the reader is referred to [3], [4].

3. EXAMPLES

Let us apply the bound (9) in some case of interest for multi-baseline SAR interferometry. For simplicity’s sake, we focus on the estimation of the sole Line of Sight (LOS) subsidence velocity, and we assume images taken at constant interval Δt. The model for the images phases (4), is the following:

(10)

3.1. The PS case Let us first assume the case of a target with constant coherence in time and baselines, like

( ) ( ) nnnn αϕαφ += θθ,

{ }Njjdiag φφ K,exp( 1=φ

{ } ( )mnnmnmnm LX −− −= δγγ )1(2

nnn εαθφφ ++= )(

{ } ( )mn

T

mnFIMθ

σθ εαθ ∂

∂+

∂∂

= − φφ −1XI 12,

[ ] ( ) ( )*** FFφΓφyyC ⊗== aE

( ) ( ) 1**11 −−− ⊗= FFφφΓC

{ } ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

−=−

nnnm Tr

φφ

1CCX

( ) nn vtn αλπφ +⋅Δ−= 14

in the PS case, or in acquisition on a stable targets with small baselines. The target covariance matrix, represented in Fig. 1, can be expressed as follows: (11) where 1 is a unitary column vector.

Figure 1. Coherence matrix and its inverse for stable targets. The inverse of the coherence matrix has the same expression, but a different constant this means that, in the optimal ML estimate, all the interferogram are to be taken with equal weights. It’s easy to show that for L=1, the estimated phases corresponds to the actual phases of the images (with respect to the master), like in the PS approach. The inverse of the FIM results in: (12) The CRB for the covariance of the phases: generalizes the known expression for N=2. The accuracy in the rate estimate, that results by combining (9),(10) and (12):

(13)

shows the typical N-3 trend of a slope estimate. 3.2 AR(1) model without APS noise Let us now assume that the sources evolve as a stationary AR(1) process [5] so that we may write:

(17) This model has been validated by exploiting many acquisition in the ERS-ICE phase, giving

a time constant τ≈40 days, and has been presented in the same symposium.

Figure 2. Covariance matrix and its inverse for the AR(1) model. Notice that the coherence matrix in (17), represented in Fig. 2 (left), excludes the APS noise. It can be shown that its inverse, represented in Fig. 2 (right), is tridiagonal and achievable in a closed form. This means that the optimal phase estimate is based on the pure combination of interferograms formed by images pair at time step Δt. Following the proposed two step approach, through (9),(10) and (17), and after some matrix manipulations we get:

(18)

the typical N-2 trend that we expect by averaging independent measures. 3.2 AR(1) model and scene decorrelation Let us approach the most realistic case when the target is subject to an exponential temporal decorrelation, with constant τ, an additional long term coherence, γ0 like due to thermal noise, and the APS.

Figure 3. Covariance matrix and its inverse for the AR(1)+noise and APS model. The entries of the covariance matrix, shown in Fig. 3 (left), are given by:

*00 )1( 11IΓ γγ +−=

( )12

11 0020

0 +⎥⎦

⎤⎢⎣

⎡ +−−= −mn

Inm N

NL

X δγγγγ

NN

L11 00

20

02 +−−=

γγγγ

σφ

( )223

22 12

4 ϕα σσπλσ +

−⎟⎠⎞

⎜⎝⎛

Δ=

NNtvAPS

( )τγ /exp; trr mnnm Δ−== −

22

22

11

421

11

tempv NtLrr

Nσ

πλσ

−=⎟

⎠⎞

⎜⎝⎛

Δ−

−=

)())(1(0 mnmnr mnnm −+−−= − δδγγ

Its inverse, shown in Fig. 3 (right), increases its span (the number of non-zero diagonals) according to the level of noise, hence γ0. This means that we have to account also for interferograms taken at larger time lag. There is not a closed form expression for the subsidence rate variance, we can do it numerically. The examples shown in Fig. 4 refers to a C band system with a 12 days revisit time, as for the next Sentinel-1 mission, with γ0=0.6 and APS standard deviation of 1 cm. Notice that when the number of 12-days acquisitions is small, the accuracy improves as N-3, like for the PS case, since the time span is short compared to the target decorrelation, whereas the asymptotical behavior in the long term shows the expected N-1 trend of the AR(1) case. The same configuration has been then exploited for different frequencies, and for N=30 images, the results are shown in Fig. 5. Figure 4. Accuracy in the estimate of the subsidence rate for C-band, as function of the number of images and pixels (L). Notice that performances are quite similar from L to C band with a strong loss at lower frequencies, due to the APS noise and at higher frequencies, due to the thermal effect.

6. CONCLUSIONS As assessment of the accuracy of the estimate of SR parameters has been proposed. The result, based on a two step approach, is shown to be equivalent to the HCRB, hence to provide a lower bound for the variance of the subsidence parameters. In particular, it has been

shown that an accuracy of 7 mm/year can be reasonably achieved in either L or C band, by averaging 12 samples in 1-year observation, with no need of PS processing.

Figure 5. Countour plot of the subsidence rate accuracy as a function of frequency and pixels, L, for N=30 images.

7. REFERENCES [1] Bamler R. and Hartl P., “Synthetic aperture radar interferometry,” Inverse Problems, vol. 14, pp. R1–R54, 1998. [2] Hanssen R. F., Radar Interferometry: Data Interpretation and Error Analysis, Springer Verlag, Heidelberg, 2 edition, 2005. [3] Rockah Y. and Schultheiss P.M., “Array shape calibration using sources in unknown location, Part I: Far field sources,” IEEE Trans. Acoust. Speech, Signal Processing, vol. ASSP-35, pp. 286–299, Mar. 1987. [4] Van Trees H.L., Detection, Estimation and Modulation Theory, Part I, John Wiley and Sons, New York, NY, 1968. [5] Zebker H.A. and Villasenor J., “Decorrelation in interferometric radar echoes,” IEEE Transactions on Geoscience and Remote Sensing, vol. 30, no. 5, pp. 950–959, sept 1992.

N-3/2

L=1

L=10

L=100L=1000

σv std [mm/year]

1

10

100

Number of images1 10010

f0 [GHz]

inde

pend

ents

ampl

es(L

)

2 4 6 8

102030405060708090

100

5 mm/year

7 mm/year

9 mm/year

N=30