Far-Field Sensitivity to Local Boundary Perturbations in 2D Wave Scattering

TL;DR

This work investigates how local boundary perturbations of 2D scatterers affect the far-field pattern, framing the inverse-shape problem. It uses a spline-based, locally-supported boundary representation together with a spectral Nyström solution of boundary integral equations to compute the Fréchet derivative of the far-field operator. By assembling a sensitivity matrix from perturbation directions and performing an SVD, the authors identify geometric modes that are most visible in the far-field and how visibility changes with frequency (via $ka$) and incident directions. The results provide quantitative insight into which boundary features are recoverable from far-field data and expose ill-posed aspects of the inverse problem, especially for small or highly localized features.

Abstract

We numerically investigate the sensitivity of the scattered wave field to perturbations in the shape of a scattering body illuminated by an incident plane wave. This study is motivated by recent work on the inverse problem of reconstructing a scatterer shape from measurements of the scattered wave at large distances from the scatterer. For this purpose we consider star-shaped scatterers represented using cubic splines, and our approach is based on a Nyström method-based discretisation of the shape derivative. Using the singular value decomposition, we identify fundamental geometric modes that most strongly influence the scattered wave, providing insight into the most visible boundary features in scattering data.

Figures (5)

• Figure 1: Visualisation of our reference scatterer, a polygonal approximation to the coast of mainland Australia.
• Figure 2: Star-shaped cubic spline scatterers $D$ approximating the shape in Figure \ref{['fig:Aus True Cartoon']}, for $N_\text{spline}=12$, $48$. The "$\boldsymbol{+}$" symbol represents the scatterers center and the "$\boldsymbol{\times}$" visualise the knots and their associated data. The left plot includes reference $(r,\theta^{\partial D})$ boundary values.
• Figure 3: Left: Input function to the Fréchet operator (\ref{['dF/d omega_j']}) for our 2 scatterers at $\theta_i^{\partial D}=\pi/3$. Jacobian matrix $|\mathbf{J}|$ for $\mathbf{\widehat{{d}}}=\boldsymbol{\hat{e}_x}$ for $ka=2\pi$, $N_\text{spline}=12$ (center) and $N_\text{spline}=48$ (right).
• Figure 4: Visualisation of the first 3 singular values and associated right and left singular vectors for $N_\text{spline}=12$ and $|\mathbf{J}|$ comprising of $4$ incident waves with $\mathbf{\widehat{{d}}}= \pm \boldsymbol{\hat{e}_x}, \pm \boldsymbol{\hat{e}_y}$.
• Figure 5: Visualisation of the first 3 singular values and associated right and left singular vectors for $N_\text{spline}=48$ and $|\mathbf{J}|$ comprising of $4$ incident waves with $\mathbf{\widehat{{d}}}= \pm \boldsymbol{\hat{e}_x}, \pm \boldsymbol{\hat{e}_y}$.