On the constant constitutive parameter (e.g., mass density) assumption in integral equation approaches to (acoustic) wave scattering

Abstract

In 2D acoustic and elastodynamic problems the spatial variability of a constitutive parameter such as the mass density makes it difficult to employ boundary integral and domain integral techniques to solve the forward and inverse wave scattering problems. The oft-employed method for avoiding this problem is to assume this constitutive parameter (which is chosen herein to be the mass density) to be spatially-invariant throughout all space. The reliability of this assumption is evaluated both theoretically and numerically and it is shown, in the example of a canonical-shaped scattering obstacle, that the scattered field can be obtained in the form of a series of powers of the mass density contrast (the latter vanishing for constant mass density). The first term of this series is the solution for the scattered field corresponding to the constant density assumption and it is shown that taking into account only two more terms in the series enables to correct for practically all the errors incurred by the constant mass density assumption for a wide range of the other constitutive parameters and frequencies. It is shown how to apply this result for obstacles of non-canonical shape.

The spurious resonance disease and how to cure it: application to the seismic response of a canyon

“…The material in this section is likewise of general nature (i.e., not dependent on the presence of a scattering body). As shown in [66]…”

“…The material in this section is of general nature (i.e., not dependent on the presence of a scattering body) and can be found in more detail in [40,66] . The free-space Green's function…”

Forward and inverse acoustic scattering problems involving the mass density

“…The task, in forward-scattering problems, is to predict the characteristics of the scattered wavefield for a given solicitation, host medium, and obstacle. This task has occupied researchers for the last two centuries [63,56,65,60,81,3,21,39,40] (and continues to occupy them [51,37,48,80,94]) because the solicitation, and/or the host medium, and/or the obstacle can (and does, in real-world situations) have complicated characteristics. For instance, in the biomedical field, predicting the acoustic wavefield within a human body [71,49], due to an ultrasonic transducer being applied on the surface of the body, is complicated because the body is a complicated medium/structure composed of many closely-spaced heterogeneous organs of different nature and size located within a host medium containing many small-scale heterogeneities, and the ultrasonic pulse-like solicitation wavefield emitted by the transducer is not simple either.…”

“…Our initial intention [94] was to seek answers to this question in relation to the inverse scattering problem of the retrival of one or two constitutive properties of the medium of which an obstacle of arbitrary shape (i.e., potato-like) is composed after having been solicited by a wave. If the related forward problem is difficult to solve (and, as written earlier: now usually carried out numerically), the inverse problem is even harder to deal with for several reasons: 1) it is mathematically non-linear even when the associated forward problem is linear, this being one of the causes of ill-posedness, 2) usually, it cannot be solved in a mathematically-sound (e.g., algebraic manner, as for searching for the roots of a polynomial equation) [88,83], which means that it is treated algorithmically by what resembles a trial and error optimization technique requiring many resolutions of the associated forward-scattering problem [62,86,46], 3) it is not clear what aspects, and what amount, of the scattered-wave data are necessary to treat the inverse problem in the best manner, 4) in real-world situations (such as in geophysical applications [82]), one may dispose of either a very small amount of data (which may be somewhat inappropriate) or a large amount of disparate, unsynchronized data gathered by measurement methods that are not-easily controlled, 5) many of the parameters of the solicitation, the host medium and even of the obstacle (aside, from those that are searched-for) are either not at all, or poorly, known [46,92], and 6) these parameters could also be searched-for by the retrieval scheme, but the latter becomes more difficult as the number of to-be-retrieved parameters increases [67].…”

“…What one usually strives for is a trial model that is as mathematically-simple as possible, while being hopefully-able to account for the most important features of the scattered wavefield. Such simple models are often the outcome of certain assumptions and approximations in what is called the domain integral formulation (treated in depth in [94]) of the forward-scattering problem. These assumptions usually have to do with: 1) treating an elastic wave problem (in a solid or porous medium) as an acoustic wave problem [13,27,41,34,42,8] (in a so-called equivalent fluid), 2) treating a microscopically-inhomogeneous (e.g., porous) medium as a macroscopically-homogeneous (effective) medium [7,18,17,19,5,58,93], 3) treating the bioacoustic, marine acoustic, electromagnetic and geophysical problem as one in which the mass density, or another constitutive property, is constant everywhere (i.e., is the same and spatially-constant within the obstacle as well as in the host) [65,30,64,73,22,23,45,74,66,27,84,55,52,15,34,96,7,29,57,5,78,36,37,86,96], 4) treating a 3D problem as a 2.5D, 2D or even 1D problem [72,…”