Apparent Attenuation and Dispersion Arising in Seismic Body-Wave Velocity Retrieval

Abstract

The fact that seismologists often make measurements, using natural seismic solicitations, of properties of the Earth on rather large scales (laterally and in terms of depth) has led to interrogations as to whether attenuation of body waves is dispersive and even significant. The present study, whose aim is to clarify these complicated issues, via a controlled thought measurement, concerns the retrieval of a single, real body wave velocity of a simple geophysical configuration (involving two homogeneous, isotropic, non-dissipative media, one occupying the layer, the other the substratum), from its simulated response to pulsed plane wave probe radiation. This inverse problem is solved, at all frequencies within the bandwidth of the pulse. Due to discordance between the models associated with the assumed and trial responses, the imaginary part of the retrieved velocity turns out to be non-nil even when both the layer and substratum are non-lossy, and, in fact, to be all the greater, the larger is the discordance. The reason for this cannot be due to intrinsic attenuation, scattering, or geometrical spreading since these phenomena are absent in the chosen thought experiment, but rather to uncertainty in the measurement model.

Forward and inverse acoustic scattering problems involving the mass density

“…Thus, to resume: the dispersive nature of the retrievals of the real part of the layer wavespeed is due to (i.e., is induced by) trial model error rather than by what is often thought to be of physical origin [92], and these dispersive effects appear to be exacerbated by certain choices of the priors such as largely-negative ǫ.…”

“…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].…”

“…Nevertheless, it should be underlined that in the area of electromagnetic scattering, the (acoustic) constant-density assumption becomes the oft-employed assumption of constant permeability ( [72,60], p. 2), which is now routinely dropped, notably to account for the exotic properties of metamaterials [10,69,70,25,91]. Also, more and more studies of acoustic (and elastic wave) inverse scattering no longer appeal to the constant-density assumption [64,12,33,52,68,17,43,53,90,92,44,59,94] A less-radical (than the constant-density) assumption is that: a) the density of the host is spatially-constant, b) the density of the obstacle, different from that of the host, is also spatiallyconstant, and c) the mass-density contrast ǫ (involving the difference of the two constant densities) is small. If the inverse problem is to retrieve the obstacle mass-density (which appears, for instance, to be of interest in certain biomedical applications [96,43,95,53,90]), an important question would appear to be: how small must ǫ (which is unknown, like the obstacle mass density, when it is the sought-for-parameter) be for the retrieval of ǫ (or of another constitutive parameter such as the wavespeed in the obstacle) to be reliable if at the outset the small-ǫ assumption is incorporated in the trial model?…”

Dynamical homogenization of a transmission grating

“…In spite of this shortcoming, the layer model deriving from our low-frequency homogenization scheme, appears to give meaningful predictions of the response of the transmission grating well beyond the static limit and can therefore be qualified as 'dynamical'. Moreover, these predictions can be improved either by the technique outlined in [56,57] or by taking into account higher-order iterates in the scheme presented herein. Finally, our homogenization scheme may provide a useful alternative to traditional multiscale and field averaging approaches to homogenization of periodic structures as regards their response to dynamic solicitations.…”

Computational parameter retrieval approach to the dynamic homogenization of a periodic array of rigid rectangular blocks

“…A remarkable feature of these figures is that they show how C [1] D results in much better agreement than does C [1] S between the far-field data and the reconstructed far-field response, which fact again points to the "curative virtue" of dispersion introduced by the retrieval method. However, the curative power of this 'induced dispersion' [56,57,58] has its limits as seen in fig. 28.…”