Quantitative synthetic aperture radar inversion

TL;DR

This paper tackles the challenge of quantitatively imaging the dielectric permittivity in a heterogeneous medium from monostatic SAR data by recasting the forward Maxwell problem and introducing a data-driven, reduced-order-model approach. The core idea is to construct an internal wave inside the medium for each antenna position by mapping measurements to snapshot-based representations, then solve an optimization that aligns this internal wave with Maxwell's equations across all antenna locations. The method yields an imaging function with similar computational cost to traditional SAR imaging but with improved target support, and iterative updates provide progressively better estimates of the wave speed. Numerical experiments demonstrate favorable comparisons to full waveform inversion, including robustness to noise, and highlight the potential for efficient, high-resolution SAR quantitative imaging.

Abstract

We study an inverse scattering problem for monostatic synthetic aperture radar (SAR): Estimate the wave speed in a heterogeneous, isotropic and nonmagnetic medium probed by waves emitted and measured by a moving antenna. The forward map, from the wave speed to the measurements, is derived from Maxwell's equations. It is a nonlinear map that accounts for multiple scattering and it is very oscillatory at high frequencies. This makes the standard, nonlinear least squares data fitting formulation of the inverse problem difficult to solve. We introduce an alternative, two-step approach: The first step computes the nonlinear map from the measurements to an approximation of the electric field inside the unknown medium aka, the internal wave. This is done for each antenna location in a non-iterative manner. The internal wave fits the data by construction, but it does not solve Maxwell's equations. The second step uses optimization to minimize the discrepancy between the internal wave and the solution of Maxwell's equations, for all antenna locations. The optimization is iterative. The first step defines an imaging function whose computational cost is comparable to that of standard SAR imaging, but it gives a better estimate of the support of targets. Further iterations improve the quantitative estimation of the wave speed. We assess the performance of the method with numerical simulations and compare the results with those of standard inversion.

Figures (6)

• Figure 1: Antennas at different locations on the flight track emit beams that enter the computational domain surrounded by a PML boundary layer drawn in yellow, designed to absorb the outgoing waves. The domain is divided into the $\Omega_{\varepsilon_o}$ part with $\varepsilon({{\itbf x}}) = \varepsilon_o$ where $u_{s,0}({{\itbf x}})$ is supportend and $\Omega_\varepsilon$ that contains the support of $\varepsilon({{\itbf x}})-\varepsilon_o$.
• Figure 2: Illustration of an initial condition $u_{s,0}({{\itbf x}})$ that defines an incident wave beam as it approaches the disk shaped imaging domain $\Omega_{\rm im}$.
• Figure 3: Left: True permittivity $\varepsilon({{\itbf x}})$. The abscissa and ordinate are in units of $\lambda_o$. The colorbar is in units of $\varepsilon_o$. The probing beams are incident from the left. Right: Evolutions of the objective functions.
• Figure 4: Results of the inversion: The top row shows the estimated permittivities after one iteration. The next two rows show the results after the fifth and tenth iterations, respectively. The left column corresponds to our method, the middle column to FWI, and the right column to our method with noisy data. The white outlines show the true locations of the inclusions. The axes are scaled in multiples of $\lambda_o$. The colorbar is in units of $\varepsilon_o$.
• Figure 5: True permittivity $\varepsilon({{\itbf x}})$ of the second target. The axes are scaled by the center wavelength $\lambda_o$. The probing beams are incident from the left. The colorbar is in units of $\varepsilon_o$.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2