Mikhail Gilman

Reduction of ionospheric distortions for spaceborne synthetic aperture radar with the help of image registration

We propose a robust technique for reducing the ionospheric distortions in spaceborne synthetic aperture radar (SAR) images. It is based on probing the terrain on two distinct carrier frequencies. Compared to our previous work on the subject (Smith and Tsynkov 2011 SIAM J. Imaging Sciences 4 501–42), the increase in robustness is achieved by applying an area-based image registration algorithm to the two images obtained on two frequencies. This enables an accurate evaluation of the shift between the two images, which, in turn, translates into an accurate estimate of the total electron content and its along-track gradient in the ionosphere. These estimates allow one to correct the matched filter and thus improve the quality of the image. Moreover, for the analysis of SAR resolution in the current paper we take into account the Ohm conductivity in the ionosphere (in addition to its temporal dispersion), and also consider the true Kolmogorov spectrum of the ionospheric turbulence, as opposed to its approximate representation that we have used previously.

Single-polarization SAR imaging in the presence of Faraday rotation

We discuss the single-polarization SAR imaging with the Faraday rotation (FR) taken into account. The FR leads to a reduction in the intensity of the received radar signal that varies over the signal length. That, in turn, results in a degradation of the image. In particular, the image of a point target may have its intensity peak split in the range direction. To distinguish between the cases of low reflectivity and those where the low antenna signal is due to the FR, we employ the image autocorrelation analysis. This analysis also helps determine the parameters of the FR, which, in turn, allow us to introduce an approach for correcting the single-polarization SAR images distorted by FR.

A Mathematical Model for SAR Imaging beyond the First Born Approximation

The assumption of weak scattering is standard for the mathematical analysis of synthetic aperture radar (SAR), as it helps linearize the inverse problem via the first Born approximation and thus makes it amenable to solution. Yet it is not consistent with another common assumption, that the interrogating waves do not penetrate into the target material and get scattered off its surface only, which essentially means that the scattering is strong. In the paper, we revisit the foundations of the SAR ambiguity theory in order to address this and other existing inconsistencies, such as the absence of the Bragg scale in scattering. We introduce a new model for radar targets that allows us to compute the scattered field from first principles. This renders the assumption of weak scattering unnecessary yet keeps the overall inverse scattering problem linear. Finally, we show how one can incorporate the Leontovich boundary condition into SAR ambiguity theory.

A linearized inverse scattering problem for the polarized waves and anisotropic targets

We analyze the scattering of a plane transverse linearly polarized electromagnetic wave off a plane interface between the vacuum and a given material. For a variety of predominantly dielectric materials, from isotropic to anisotropic and weakly conductive, we show that when the scattering is weak, the first Born approximation predicts the correct scattered field in the vacuum region. We also formulate and solve the corresponding linearized inverse scattering problem. Specifically, we provide a necessary and sufficient condition under which interpreting the target material as a weakly conductive uniaxial crystal allows one to reconstruct all the degrees of freedom contained in the complex 2 × 2 Sinclair scattering matrix. This development can help construct a full-fledged radar ambiguity theory for polarimetric imaging by means of a synthetic aperture radar (SAR), which is in contrast to the approach that currently dominates the SAR literature and exploits a fully phenomenological scattering matrix. Moreover, the linearized scattering off a material half-space naturally gives rise to the ground reflectivity function in the form of a single layer (i.e. a δ-layer) at the interface. A ground reflectivity function of this type is often introduced in the SAR literature without a rigorous justification. Besides the conventional SAR analysis, we expect that the proposed approach may appear useful for the material identification SAR (miSAR) purposes.

Discussion and outstanding questions

This monograph presents a mathematical perspective on synthetic aperture imaging of the Earth’s surface from satellites. Its main focus is on the accurate quantitative description of the distortions of SAR images due to the ionosphere and on the development and analysis of the means for mitigating those distortions (Chapter 3). The discussion of transionospheric SAR imaging also includes the case of a turbulent ionosphere (Chapter 4) and the case of a gyrotropic ionosphere (Chapter 5).

Inverse scattering off anisotropic targets

The model for radar targets described in Chapter 7 allowed us, in particular, to identify a physically meaningful observable quantity in SAR imaging. It was a slowly varying amplitude of the Bragg harmonic in the spectrum of ground reflectivity (i.e., of the variation of the target permittivity). The derivation of this observable revealed, among other things, one possible mechanism of scattering anisotropy, which is the dependence of the scattered field, including its polarization, on the incident and scattered direction.

Modeling radar targets beyond the first Born approximation

In this chapter, we return to the foundations of the SAR ambiguity theory that we presented in Chapter 2, and address the inconsistencies of the conventional approach outlined in Section 2.7 A standard representation of the image in the SAR ambiguity theory is by the convolution integral ( 2.1) [see also ( 2.31)]:

Conventional SAR imaging

In this chapter, we explain the fundamental principles of SAR data collection and image formation, i.e., inversion of the received data. Synthetic aperture radar uses microwaves for imaging the surface of the Earth from airplanes or satellites. Unlike photography which generates the picture by essentially recoding the intensity of the light reflected off the different parts of the target, SAR imaging exploits the phase information of the interrogating signals and as such can be categorized as a coherent imaging technology.

The effect of ionospheric anisotropy

In Chapter 3, we have shown that the Earth’s ionosphere exerts an adverse effect on SAR imaging. It is due to the mismatch between the actual radar signal affected by the dispersion of radio waves in the ionosphere and the matched filter used for signal processing. Accordingly, to improve the image one should correct the filter. This requires knowledge of the total electron content in the ionosphere, as well as of another parameter that characterizes the azimuthal variation of the electron number density (see Section 3.9). These quantities can be reconstructed by probing the ionosphere on two distinct carrier frequencies and exploiting the resulting redundancy in the data (see Section 3.10).

The effect of ionospheric turbulence

In Chapter 3, we have shown that temporal dispersion of the propagation medium (Earth’s ionosphere) causes distortions of SAR images (see Section 3.8). Moreover, we have identified the key integral characteristics of the ionospheric plasma that allow one to quantify those distortions. They are the zeroth moment of the electron number density N e, i.e., the TEC N H given by (3.66), as well the first moment \(\mathcal{Q}\) of the azimuthal derivative of Ne defined by ( 3.182). We have also demonstrated that one can obtain the unknown quantities N H and \(\mathcal{Q}\) with the help of dual carrier probing (see Section 3.10) and subsequently incorporate them into the SAR matched filter in order to effectively eliminate the distortions (see Section 3.11). This correction of the filter is possible because one and the same pair of values \((N_{H},\mathcal{Q})\) “serves” all antenna signals used for the construction of the image, i.e., all the terms in the azimuthal sum. Once the values of N H and \(\mathcal{Q}\) have been derived, the corrected filter will match the received signals for all antenna positions along the synthetic array.