Multigrid-Based Inversion for Volumetric Radar Imaging With Asteroid Interior Reconstruction as a Potential Application

Abstract

This study concentrates on advancing mathematical and computational methodology for radar tomography imaging in which the unknown volumetric velocity distribution of a wave within a bounded domain is to be reconstructed. Our goal is to enable effective simulation and inversion of a large amount of full-wave data within a realistic 2D or 3D geometry. For propagating and inverting the wave, we present a rigorous multigrid-based forward approach which utilizes the finite-difference time-domain method and a nested finite element grid structure. Based on the multigrid approach, we introduce and validate a multiresolution algorithm which allows regularization of the unknown distribution through a coarse-to-fine inversion scheme. In this approach, sparse signals can be effectively inverted, as the coarse fluctuations are reconstructed before the finer ones. Furthermore, the number of nonzero entries in the system matrix can be compressed and thus the inversion procedure can be speeded up. As a test scenario we investigate satellite-based asteroid interior reconstruction. We use both full-wave and projected wave data and estimate the accuracy of the inversion under different error sources: noise and positioning inaccuracies. The results suggest that the present full-wave inversion approach allows recovering the interior with a single satellite recording backscattering data. It seems that robust results can be achieved, when the peak-to-peak signal-to-noise ratio is above 10 dB. Furthermore, it seems that reconstructing the deep interior can be enhanced if two satellites can be utilized in the measurements.