Multistatic anisotropic travel-time as a tensor tomography problem

TL;DR

This work reframes multistatic travel-time imaging with anisotropic reflectivity as a tensor tomography problem. By approximating isochrones and leveraging normal Radon transforms of symmetric tensors, it connects travel-time data to tensor field reconstructions, analyzes the null-space and singularity structure, and demonstrates a worked example along with a truncated-SVE numerical inversion strategy. The approach offers a computationally efficient alternative to full wave inversion and provides a principled path to recover singular features of anisotropic objects, with practical implications for radar and sonar imaging. The study also identifies limitations from directional sampling and tensor degeneracy, outlining concrete directions for extending to richer 3D models and more varied data.

Abstract

Travel-time imaging problems seek to reconstruct an image of reflectivity of a scene by measuring travel time (and amplitude, phase) of electromagnetic or acoustic signals, such as radar and sonar. Multistatic, in this context, means that the transmitters and receivers need not be co-located. The reflectivity is anisotropic if it depends on direction, and in the multistatic case this means incoming and outgoing direction. Travel-time problems can be formulated as generalized Radon transforms of integrals over isochrones, in the planar case ellipses with transmitter and receivers at foci. In a simplified case where transmitters and receivers are distant from the scene, isochrones can be approximated by straight lines. We relate this to tensor ray transforms, specifically the longitudinal ray transform of Sharafutdinov, and discuss the implication of its known null-space. In the volumetric case isochrones are spheroids and we relate the problem to the normal Radon transform of tensor fields.

Figures (3)

• Figure 1: Isochrones are ellipses in the 2D case with transmitter ($x_T$) and receiver ($x_R$) at the foci. Bistatic angle indicated by $\beta$.
• Figure 2: Solenoidal components $G$ from solving $\delta d U = \delta F$ with $G_{11}$ on the left, $G_{12}$ in the centre and with $G_{22}$ on the right.
• Figure 3: Reconstructed solenoidal components via the Truncated SVE of an isolated object for $N=50$ and $K=40$, with $a_0$ on the left, $a_1$ in the middle and with $a_2$ on the right