Applications of Inverse Methods to the Analysis of Refraction and Wide-Angle Seismic Data

Nowack, Robert L.

doi:10.1007/978-1-4612-1878-4_20

Cited by 3 publications

(2 citation statements)

References 32 publications

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“…The inverse problem of recovering c(z) from the traveltime information of diving waves was solved circa 1910 by Herglotz [18], Wiechert and Geiger [39] and Bateman [4] in what is perhaps the first contribution by mathematicians to seismology. Their explicit formula takes the form of an inverse Abel transform and has been textbook material for a long time [2,29,3,25,26]. It will be reviewed in this paper, along with the analysis of its stability.…”

Section: Problem Setup and Contextmentioning

confidence: 99%

Conditioning bounds for traveltime tomography in layered media

Baek¹,

Demanet²

2012

Inverse Problems

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This paper revisits the problem of recovering a smooth, isotropic, layered wave speed profile from surface traveltime information. While it is classic knowledge that the diving (refracted) rays classically determine the wave speed in a weakly well-posed fashion via the Abel transform, we show in this paper that traveltimes of reflected rays do not contain enough information to recover the medium in a well-posed manner, regardless of the discretization. The counterpart of the Abel transform in the case of reflected rays is a Fredholm kernel of the first kind which is shown to have singular values that decay at least root-exponentially. Kinematically equivalent media are characterized in terms of a sequence of matching moments. This severe conditioning issue comes on top of the well-known rearrangement ambiguity due to low velocity zones. Numerical experiments in an ideal scenario show that a waveform-based model inversion code fits data accurately while converging to the wrong wave speed profile.

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Section: Problem Setup and Contextmentioning

confidence: 99%

Conditioning bounds for traveltime tomography in layered media

Baek¹,

Demanet²

2012

Inverse Problems

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“…For example, nondestructive inspection has been carried out to determine the shapes and locations of cavities or cracks in metal by using the data from measurements such as electric potentials on the surface (e.g., Nishimura and Kobayashi, 1991;Tanaka and Yamagiwa, 1988;Kubo et al, 1988). However, there are only a few applications of the domain/boundary inversion to seismology (Nowack and Braile, 1993;White, 1989).…”

Section: Introductionmentioning

confidence: 99%

Boundary shape waveform inversion for two-dimensional basin structure using three-component array data of plane incident wave with an arbitrary azimuth

Aoi

Iwata

Fujiwara

et al. 1997

Bulletin of the Seismological Society of America

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We extend a new waveform inversion scheme (Aoi et al., 1995) for estimating underground structure with an irregular-shaped basement as a target to cases where plane waves with an arbitrary azimuth impinge on the structure, i.e., from a direction not necessarily perpendicular to the major axis of the structure. We proved the validity of this scheme by numerical experiments. We had already achieved the formulation and numerical experiments for the cases where an SH wave impinges on a 2D basin structure and had shown that we could estimate the entire basin structure with seismic waveforms from only a few surface stations by using whole waveforms that include the surface waves. However, when the epicenter is located out of the plane including the observation stations, even the cases of 2D structure cannot be treated as a simple 2D (SH or P-SV) problem because of the wave with an azimuth that is not 0°. Therefore, by dealing with 3D wave fields in the present study, we extend the inversion scheme in order to apply it to incident waves with an arbitrary azimuth. The differential seismograms, which represent the sensitivity of change in the waveform, show different patterns in three components, and we demonstrate that inversion with three components, compared with the inversion with only one of them, leads to a linearized equation system with a smaller condition number and a more stable computation. Furthermore, we detect certain parts that are estimated with much less difficulty than others, depending on the direction from which the incident wave impinged. In the latter case, we can estimate the entire structure by employing simultaneously several data from incident waves arriving from different directions. We thus demonstrate by numerical experiments that the extension of our inversion method to cases where the incident wave with an arbitrary azimuth impinges on the structure enables us to estimate with increased accuracy an underground structure under more general conditions of the epicenter locations.

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H

Howarth¹

2017

Dictionary of Mathematical Geosciences

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Applications of Inverse Methods to the Analysis of Refraction and Wide-Angle Seismic Data

Cited by 3 publications

References 32 publications

Conditioning bounds for traveltime tomography in layered media

Conditioning bounds for traveltime tomography in layered media

Boundary shape waveform inversion for two-dimensional basin structure using three-component array data of plane incident wave with an arbitrary azimuth

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