Reconstruction of material properties profiles in one-dimensional macroscopically inhomogeneous rigid frame porous media in the frequency domain

Abstract

The present paper deals with the inverse scattering problem involving macroscopically inhomogeneous rigid frame porous media. It consists of the recovery, from acoustic measurements, of the profiles of spatially varying material parameters by means of an optimization approach. The resolution is based on the modeling of acoustic wave propagation in macroscopically inhomogeneous rigid frame porous materials, which was recently derived from the generalized Biot's theory. In practice, the inverse problem is solved by minimizing an objective function defined in the least-square sense by the comparison of the calculated reflection ͑and transmission͒ coefficient͑s͒ with the measured or synthetic one͑s͒, affected or not by additive Gaussian noise. From an initial guess, the profiles of the x-dependent material parameters are reconstructed iteratively with the help of a standard conjugate gradient method. The convergence rate of the latter and the accuracy of the reconstructions are improved by the availability of an analytical gradient.

Acoustic wave propagation in equivalent fluid macroscopically inhomogeneous materials

“…Equation (13) describes the amplitude of monochromatic acoustical fields in any medium having one-dimensional acoustical inhomogeneity characterized by the wavenumber kðx; xÞ and the impedance zðx; xÞ. The functional form of dependence of both parameters on the spatial coordinate and wave frequency for a given material should be defined based on other considerations or experiments.…”

“…However, often such approximation is too rough and proper comprehension of the measured data requires a description of the wave propagation in inhomogeneous materials, generating increasing interest in the area. 1,2,5,6,[9][10][11][12][13][14] The alternative approach, based on simulation of wave propagation in inhomogeneous materials, is widely applied in the areas of geophysics and biomechanics. [15][16][17][18][19][20] However, it gives rather qualitative insight into physics associated with the wave propagation phenomena in such materials and typically requires a priori information about internal heterogeneity of the material (e.g., from computer tomography), which may be rarely available.…”

Propagation of acoustic waves in a one-dimensional macroscopically inhomogeneous poroelastic material

“…2 These, and other inverse problems, are of great importance in connection with the characterization of the mechanical properties of naturally occurring macroscopically inhomogeneous porous materials, such as bones. The literature on inhomogeneous media is extensive in several fields of physics, from optics and electromagnetism, 3,4 to acoustics, 5,6 and geophysics 7 and granular media.…”

Scattering of acoustic waves by macroscopically inhomogeneous poroelastic tubes

“…It was initially motivated by (i) the design of sound absorbing porous materials with optimal material and geometrical property profiles 1 and (ii) the retrieval of the spatially varying material parameters of porous materials, mainly industrial foams. 2 These, and other inverse problems, are of great importance in connection with the characterization of the mechanical properties of naturally occurring macroscopically inhomogeneous porous materials such as bones or rocks. The wave equation in macroscopically inhomogeneous porous media was derived from the alternative formulation of Biot's theory 3 in De Ryck et al 4 and solved in the case of rigid frame inhomogeneous porous materials via the Wave Splitting method and "transmission" Green's functions approach or via an iterative Born approximation procedure based on the specific Green's function of the configuration.…”

“…5 The recovery of several profiles of spatially varying material parameters by means of an optimization approach, was then achieved in Ref. 2 still in the rigid frame approximation. Recently, the full wave equation in macroscopically inhomogeneous poroelastic media was solved in a planar configuration, by use of the state vector formalism together with a Peano series.…”