The factorization method and Capon’s method for random source identification in experimental aeroacoustics

Abstract

Experimental aeroacoustics is concerned with the estimation of acoustic source power distributions, which are for instance caused by fluid structure interactions on scaled aircraft models inside a wind tunnel, from microphone array measurements of associated sound pressure fluctuations. In the frequency domain aeroacoustic sound propagation can be modelled as a random source problem for a convected Helmholtz equation. This article is concerned with the inverse random source problem to recover the support of an uncorrelated aeroacoustic source from correlations of observed pressure signals. We show that a variant of the factorization method from inverse scattering theory can be used for this purpose. We also discuss a surprising relation between the factorization method and a commonly used beamforming algorithm from experimental aeroacoustics, which is known as Capon’s method or as the minimum variance method. Numerical examples illustrate our theoretical findings.

Regularized factorization method for a perturbed positive compact operator applied to inverse scattering

“…along with the radiation condition (21). Now, to prove the claim we assume that z ∈ D. Then, we have that Φ(z, •) is a smooth solution to Helmholtz equation in R d \{z}.…”

“…In this section, we will provide numerical reconstructions using MATLAB R2022a. To this end, we will consider recovering a scatterer D from the far-field pattern u ∞ corresponding to (20) and (21). In all our examples, for simplicity we will assume that parameters n and η are constants given by n = 4 and η = 2 with wave number k = 2π unless stated otherwise.…”

“…Over the years the factorization method has become a useful analytical and computational tool for shape reconstruction. See the papers [8,9,11,12,14,20,21,25,35,38] and the references therein for applications of the factorization method for solving inverse shape problems for elliptic and hyperbolic partial differential equations(PDEs).…”

Regularization of the factorization method with applications to inverse scattering

“…In order to proceed, we need to synthetically compute the far-field pattern. To this end, recall that the scattered field u s (x, ŷ) is given by (23) and in [15] we see that for scatterers such that |D| 1…”

“…Over the years the factorization method has become a useful analytical and computational tool for shape reconstruction. See the papers [9,11,13,14,16,22,23,27,37,40] and the references therein for applications of the factorization method for solving inverse shape problems for elliptic and hyperbolic PDEs.…”