Numerical comparison of iterative and functional-analytical algorithms for inverse acoustic scattering

TL;DR

The paper addresses inverse acoustic scattering in a two-dimensional Helmholtz setting, focusing on reconstructing a scatterer from boundary data. It compares a Novikov-type iterative algorithm with a functional-analytical approach, both operating on scattering amplitudes derived from measurements. Key findings show middle-strength scatterers are recoverable with comparable accuracy by both methods, while strong scatterers benefit from the functional-analytical method; the iterative approach is enhanced by wave-vector filtering and multifrequency data to improve convergence and robustness. The work provides the first numerical comparison of these approaches in this setting and offers guidance on method choice for practical acoustic tomography.

Abstract

In this work the numerical solution of acoustic tomography problem based on the iterative and functional-analytical algorithms is considered. The mathematical properties of these algorithms were previously described in works of R.G.Novikov for the case of the Schrödinger equation. In the present work, for the case of two-dimensional scalar Helmholtz equation, the efficiency of the iterative algorithm in reconstruction of middle strength scatterers and advantages of the functional-analytical approach in recovering strong scatterers are demonstrated. A filtering procedure is considered in the space of wave vectors, which additionally increases the convergence of the iterative algorithm. Reconstruction results of sound speed perturbations demonstrate the comparable noise immunity and resolution of the considered algorithms when reconstructing middle strength scatterers. A comparative numerical investigation of the iterative and functional-analytical algorithms in inverse acoustic scattering problems is implemented in this work for the first time.

Figures (5)

• Figure 1: Tomography area $V_S$ contains the scattering region $\mathfrak{R}$; locations of quasi-point sources and receivers at the boundary $\Upsilon$ are described by vectors $\mathbf{x}$ and $\mathbf{y}$, respectively; wave vectors $\mathbf{k}$ and $\boldsymbol{\ell}$ show the directions of incident and scattered plane waves.
• Figure 2: General view of true scatterer $v$ (a), for which the relative contrast of sound speed $\Delta c(\mathbf{r}) / c_0$ ranges from -0.084 to 0.25, maximum additional phase shift is $\Delta \psi \approx 0.46 \pi$, norm of scattering data is $\| f(\phi, \phi^\prime) \| \approx 11 / (3 \pi)$, dimensionless coefficient is $\text{A}_0 = 0.43$; - central cross sections of true scatterer $v$ (b, thin solid line), reconstruction results obtained by the iterative and functional-analytical algorithms, which are visually identical $\hat{v} \simeq \hat{v}^{(10)}$ (b, dotted line), and the Born estimate $\hat{v}_{\text{born}}$ (b, thick solid line); - dependence of discrepancy for the solution $\delta_v^{(n)}$ on the iteration number $n$ (c).
• Figure 3: General view of the scatterer $\hat{v}^{(58)}$ reconstructed at 58-th iteration (à), for which the relative contrast of sound speed $\Delta c(\mathbf{r}) / c_0$ ranges from -0.1 to 0.36, maximum additional phase shift is $\Delta \psi \approx 0.6 \pi$, norm of scattering data is $\| f(\phi, \phi^\prime) \| \approx 13.4 / (3 \pi)$, dimensionless coefficient is $\text{A}_0 = 0.55$; - central cross sections of true scatterer $v$ (b, thin solid line), reconstruction results obtained by the functional-analytical algorithm $\hat{v}$ (b, dotted line), by the iteration method $\hat{v}^{(58)}$ (b, dash line) and the Born estimate $\hat{v}_{\text{born}}$ (b, thick solid line); - dependence of discrepancy $\delta_v^{(n)}$ (c) and the parameter of filtration $\tau^{(n)}$ (d) on the iteration number $n$.
• Figure 4: Dependence of discrepancy $\delta_v^{(n)}$ (a) and parameter of filtration $\tau^{(n)}$ (b) on the iteration number $n$, when reconstructing a strong scatterer, for which values $\Delta c(\mathbf{r}) / c_0$ ranges from - 0.16 to 1.08, maximum additional phase shift is $\Delta \psi \approx 1.1 \pi$, norm of scattering data is $\| f(\phi, \phi^\prime) \| \approx 19.3 / (3 \pi)$, dimensionless coefficient is $\text{A}_0 = 0.91$. After $n$ = 35 iterations discrepancy is $\delta_v^{(35)} \approx 0.72$, that is smaller than discrepancy of the Born estimate $\delta_v^{(0)} \approx 0.84$, but significantly more than the functional-analytical algorithm result $\delta_v \approx 0.012$; this shows the limitations of the iterative algorithm for the reconstruction of strong scatterers. The central cross sections of true scatterer $v$ and its estimates are shown in (c): thin solid line shows $v$, dotted line – $\hat{v}$, dash line – $\hat{v}^{(35)}$, thick solid line – $\hat{v}_{\text{born}}$.
• Figure 5: General view of middle strength scatterer with small size elements (à), for which the relative contrast of sound speed $\Delta c(\mathbf{r}) / c_0$ ranges from - 0.07 to 0.19, maximum additional phase shift is $\Delta \psi \approx 0.46 \pi$, norm of scattering data is $\| f(\phi, \phi^\prime) \| \approx 12.8 / (3 \pi)$, dimensionless coefficient is $\text{A}_0 = 1.1$; - central cross sections of true scatterer $v$ and its estimates obtained by using scattering data without noise are shown in (b): thin solid line shows $v$, dotted line – $\hat{v}$ (the discrepancy is $\delta_v \approx 0.018$), dash line – $\hat{v}^{(17)}$ ($\delta_v^{(17)} \approx 0.047$), thick solid line – $\hat{v}_{\text{born}}$ ($\delta_v^{(0)} \approx 0.31$); - general view of the Born estimate $\hat{v}^\text{noise}_{\text{born}}(\mathbf{r}, \omega_1)$ (c) obtained by using noisy data at one frequency with noise rms amplitude deviation $\sigma_{\text{ns}}(\omega_j) = 0.15 \bar{G}_{\text{sc}}(\omega_j)$; - central cross sections of true scatterer $v$ and its estimates obtained by using noisy data are shown in (d): thin solid line shows $v$, dotted line – $\hat{v}^\text{noise}$ (the discrepancy is $\delta_v \approx 0.018$), dash line – $\hat{v}^{(17), \, \text{noise}}$ ($\delta_v^{(17)} \approx 0.047$), thick solid line – $\hat{v}^\text{noise}_{\text{born}}$ ($\delta_v^{(0)} \approx 0.31$).