Multiposition Synthetic Aperture Radar (MPSAR) 

              Alexander V Ksendzuk 

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4. Multiposition SAR. Optimal signal processing.

Input process in multiposition SAR is an addition of signal (vector of signals scattered from the surface) and noise (vector of additive noises, which may be approximated by  Gaussian process) .

For functionally-determined surface models input process in an stochastic process with mean , for stochastic models input process is an addition of two stochastic vectors  and .

In this section optimal Bayesian and maximum likelihood algorithms for the surface parameter estimation in the multiposition SAR are derived.

4.1. Optimal processing algorithm

for functionally-determined surface models.

For this model mean (vector) of the input process is the function of the parameters to be estimated (- parameters which is function of the vector  - which coordinates are time, and Cartesian coordinates). Correlation function  and its determinant  do not. Probability functional of the input process is

 (4.1)

where  input vector in time moment ,  - mean of the input process, which is the function of the parameters to be estimated.

Signal Groups in Multiposition SAR

With unknown a-priory distribution of the estimated parameters to be estimated , optimal estimations may be found due to maximum likelihood method from the equation , where  means variational derivative.

Common equation for optimal parameter estimation is

.    (4.2)

This equation after some transformations may be simplified to the optimal monostatic or bistatic  SAR processing algorithm.

Exact parameter estimation algorithms may be found only for certain surface models (we still talking about functionally determined surface models). The most common model of the complex scattering coefficient is

,                                (4.3)

where  - complexation function (something �common� in all radiolocation images, for example, humidity of the surface);  - vector of basis function;  - vector of parameters of the complexation function;  - vector of nonessential parameters.

Result of the optimal processing of the input vector

               (4.4)

is equal to  complexation function, smoothed by space ambiguity function of the multiposition system:

,      (4.5)

where  space ambiguity function , may be represented as the weighted total of the space ambiguity functions of different receivers

 , where

.     (4.6)

Simplification of the optimal coprocessing algorithm in multiposition SAR (4.4) allow to achieve clear methods for the parameter estimation:

1. If additive noise vectors coordinates are delta-correlated time processes, which are independent in different receivers, optimal estimation of the  complexation function is a weighted total of matched filtering of all receivers:

 ,                    (4.7)

where dot in uppercase means complex value, sign  means complex conjugation.

If, additionally, energies of the received signals and noise power spectral density are equal ,  rating coefficient before sum simplifies to    and complexation function estimation will be non-weighted total with further normalization

.                                                 (4.8)

4.2. Optimal processing algorithm for stochastic surface models.

With this model input signal vector is the sum of two stochastic processes � scattered signal vector  and additive noise vector .

General optimal processing algorithm is given by

.         (4.9)

Simplification of the (4.6) is based on dependence of input process vector�s mean , correlation function , and determinant of the correlation function  on parameters to be estimated .

The case when only correlation function and its determinant depend on parameters   are considered. Suppose correlation matrix (element of this matrix are time-dependent functions) of the input process may be presented as

,                                                                     (4.10)

where

,                                                                                    (4.11)

.                                                                              (4.12)

In this case optimal processing algorithm will be quite complicated

.     (4.13)

However, in case when we estimate radar cross-section  (exactly vector of the radar cross section for the different receivers) and use simplified surface model optimal algorithm simplifies to

.           (4.14)

Consider surfaces which radar cross-section (as the function of the MPSAR parameters - bistatic angles, carrier frequencies, if we use transmitters with different carriers, polarization etc.) may be represented in a form

,                   (4.15)

where  - complexation function, - basis functions,  - vector of parameters of the complexation function;  - vector of nonessential parameters.

For the surface (4.15)  result of the optimal processing

, (4.16)

is equals to complexation function , smoothed by modified space ambiguity function 

,                                (4.17)

and biased on value

.                                       (4.18)

In monostatic case this algorithm simplifies to modified synthetic aperture

.                                                        (4.19)

4.3 Optimal processing algorithms estimation errors.

In this section we compare results of the optimal processing in multiposition and monostatic SAR. Direct comparison has no sense because in multiposition system quality will be better due to statistical averaging, resulting in lower speckle noise level.

Comparison will be done in case of �disturbances�. Consider estimation of some electrophysical parameter (for example humidity) . Radar cross section of the surface for simplicity will be , where  - non-essential functions, which depend on other electrophysical parameters, Figure 4.1.

Figure 4.1. Complexation function , relief  and local angle of slope of the surface  as spatial functions.

Radar cross-section for arbitrary bistatic pair depends on local incidence angle (or to be exact, for the considered surface model  function depends on local incidence angle).

Figure 4.2. Function  dependence on local incidence angle.

 We will compare estimation of the complexation function  in monostatic SAR and multiposition SAR (MPSAR) for the following cases:

1.      Relief altitude and, correspondingly, local angle of slope of the surface and therefore local incidence angle measured with error.

2.      Presence of the multiplicative atmospheric noise (in multiposition SAR noises in each bistatic pair are independent processes).

3.      Both of 1 and 2 �disturbances�.

Results of the modeling and corresponding estimation errors are shown in Figure 4.3- Figure 4.5.

Figure. 4.3. Estimation of the complexation function (graph 1) in monostatic SAR (graph 2) and multiposition SAR (graph 3) in case when estimated surface�s local angle of slope  (graph 4) differs from its true value  (graph 5).

Figure. 4.4. Estimation of the complexation function (graph 1) in monostatic SAR (graph 2) and multiposition SAR (graph 3) in case when multiplicative noise (caused by atmospheric effects) presence,  realization of this multiplicative noise in one bistatic pair are shown in graph 5.

Figure. 4.5. Estimation of the complexation function (graph 1) in monostatic SAR (graph 2) and multiposition SAR (graph 3) in case when multiplicative noise (caused by atmospheric effects) and  relief altitude estimation errors presence.

Conclusion.

Optimal processing algorithms in multiposition SAR involves coprocessing (simultaneous processing of all data from all bistatic pairs), which allows significantly increase quality of the surface imaging (surface parameters estimation). As was shown in modeling MPSAR provides higher quality even if unknown errors in surface model and in input signal presented.

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