# Singular FIOs in SAR Imaging, II: Transmitter and Receiver at Different Speeds

## TL;DR

This paper analyzes the mathematical properties of SAR imaging operators with moving transmitter and receiver at different speeds, revealing artifact behaviors and bifurcations in image quality.

## Contribution

It classifies the forward operator as a Fourier integral operator with singularities and characterizes the normal operator's structure in different motion scenarios.

## Key findings

- Artifacts are as strong as the true image in both cases.
- Normal operator belongs to specific classes of Fourier integral operators.
- A bifurcation in artifact types occurs when source and receiver move in opposite directions.

## Abstract

In this article, we consider two bistatic cases arising in synthetic aperture radar imaging: when the transmitter and receiver are both moving with different speeds along a single line parallel to the ground in the same direction or in the opposite directions. In both cases, we classify the forward operator $\Fc$ as a Fourier integral operator with fold/blowdown singularities. Next we analyze the normal operator $\Fc^*\Fc$ in both cases (where $\Fc^{*}$ is the $L^{2}$-adjoint of $\Fc$). When the transmitter and receiver move in the same direction, we prove that $\Fc^*\Fc$ belongs to a class of operators associated to two cleanly intersecting Lagrangians, $I^{p,l} (\Delta, C_1)$. When they move in opposite directions, $\Fc^*\Fc$ is a sum of such operators. In both cases artifacts appear and we show that they are, in general, as strong as the bona-fide part of the image. Moreover, we demonstrate that as soon as the source and receiver start to move in opposite directions, there is an interesting bifurcation in the type of artifact that appears in the image.