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This work deals with the problem of estimating the interferometric phase (IP) of multibaseline InSAR signals in presence of layover for systems with a low number of phase centres and nonuniform linear array (NLA) geometry. In particular, NLA obtained by thinning a full ULA have been considered. A peculiarity of the received signal is the appearance of multiplicative noise (speckle in the radar jargon), which is related to the extended nature of terrain patches.
The classical estimation approaches (Beamforming, Capon and MUSIC) are based on the spatial spectral analysis of the received signal; when a uniform linear array structure (ULA) is available, their performance is satisfactory. Moreover, the ULA structure allows for computationally efficient rooting algorithms such as root-MUSIC, yielding superior estimates as compared with their conventional counterparts. Unfortunately, the choice of the sensing geometry is determined by structural and flight/orbital considerations rather than the estimation requirement. As a consequence, the trade-off between practical realization and estimation accuracy arises the need of investigating different options for the IP estimation when the
acquisition geometry is an NLA. Past works have demonstrated that nonuniform spatial sampling produces anomalous sidelobes in the beampattern and in the functional of spectral estimation methods, resulting in peaks mislocation and spurious peaks.
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This work continues investigating the possibility to use the interpolated array (IA) approach for multibaseline InSAR. This approach consist in the recovery of the output of a virtual ULA structure by means of a linear transformation applied to the data available from the actual NLA, exploiting the a priori information about the overall interferometric phase interval where the backscattering source are supposed to be located, the sector of interpolation (SOI). Two interpolation methods have been presented. The first is the classical deterministic IA
approach, which performs a least square fitting of the steering vectors of the NLA with those of the ULA; the second, named MSE-IA, is an algorithm optimal in the minimum mean square
error (MMSE) sense, which hypothesize a flat power spectral density of the received signal over the SOI.
To carry out the IP estimation it is required the whitening of the noise present in the interpolated data; consequently, the root-MUSIC changes into the IA-root-MUSIC. However, when the number of elements of the virtual ULA is higher than the number of elements of the actual NLA, the interpolated noise autocorrelation matrix is rank-deficient, so the noise can not be whitened. As a consequence, it has been proposed a diagonal loading technique to bypass this difficulty and partially whiten the noise. It is important to precise that the need to reconstruct more elements than those available aims to preserve the IP unambiguous range (UR) after signal deramping; nevertheless, theoretically it could be not always necessary the reconstruction of the full ULA signal.
It is very important to select correctly the position of the SOI. In fact, it must contain the source and reject as much as possible the spurious peaks; this requirement can be met easily after deramping when two sources are present, because their spacing does not exceed half UR. However, when there are more than two sources, the overall spacing can be greater than half UR, thus the spurious peaks could be enclosed into the SOI when the sources are very spaced, badly affecting the estimation. Furthermore, it has been analyzed how the choice of the loading parameter affect the noise whitening. Interestingly, it has been shown that a relatively high value of the loading parameter changes imperceptibly the noise spectrum, but it is sufficient to improve the estimation performance with IA-root-MUSIC algorithm with respect to the case of
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coloured noise. It has also been shown that the diagonal loaded whitening causes a reduction in the power of the noise and of the signal (radiometric loss).
The IP estimation performance of interpolated methods have been analysed in terms of root mean square error (RMSE) by means of extensive Monte Carlo simulations under different InSAR scenarios, with two or three sources present. The interpolated methods demonstrate to be more efficient than the classical estimation methods especially in terms of source resolution and
robustness to the threshold effect at low signal-to-noise ratios, even with the roughest a priori information. More in details, a great attention has been paid to the selection of the number of
elements of the virtual ULA and of the loading parameter. First of all, it has been noticed that the reconstruction of the full ULA improves almost always the accuracy of the estimates. Secondly, the best choice of the value of the loading parameter (δ) depends on the kind of the a
priori information which is conveyed to the whole interpolation process. In particular:
• when the SOI is fixed and independent of the source spacing (but it contains them), the recommended loading is δ =5; as regard the estimation accuracy, DL-IA shows the
best performance on almost all the scenarios analyzed, and a good robustness to the well-known threshold effect of RMSE at low signal-to-noise ratios; the achieved RMSE is nearly 10º for a wide range of source separations;
• when the SOI is partially optimized, i.e. it is selected by means of a rough spectral analysis of the received data or physical information, the suggested choice is δ =0.5,
which corresponds to a partial whitening. DL-MSE-IA is the most efficient method and reaches a RMSE even lower than 10º.
For the sake of completeness, it has also been determined numerically the SOI which minimizes the RMSE for both interpolation approaches as a function of the overall source spacing when
0.05
δ = , i.e. for the case of near maximum whitening. In this case, DL-MSE-IA is again the most efficient method, reaching a RMSE even lower than 1º for some source separations,
considering also that the optimal SOI has proven not to be a function of the number of sources and of the other signal parameters. This does not happen with DL-IA. However, it is worth
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stressing that this case is of least practical relevance, because it requires very sophisticate a priori information.
It has been investigated also the degradation of estimation performance in the presence of array steering vector errors, caused by the array calibration errors or limited accuracy in the positioning instruments or estimator measuring the actual flight path/orbit. It has also been derived the hybrid Cramèr-Rao lower bound (HCRLB) for the IP estimation when the data are
corrupted by such errors. The numerical analysis has demonstrated that the interpolated methods are more robust to array miscalibration than the classical estimation methods; this robustness is
partially due to the limiting effect of the SOI, because the estimates are contained into it. However, for some source separations the RMSE curves of interpolated methods were far from the HCRLB. To improve the estimation efficiency, several robust interpolation techniques based again on least square fitting have been derived and tested in the InSAR case. The numerical analysis has proven the increased efficiency of those methods matched to the statistical model of the calibration error, and showed room for further research.