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Inverse Problems in Geophysics

  • R. Snieder
  • J. Trampert
Part of the International Centre for Mechanical Sciences book series (CISM, volume 398)

Abstract

An important aspect of the physical sciences is to make inferences about physical parameters from data. In general, the laws of physics provide the means for computing the data values given a model. This is called the “forward problem”, see figure 1. In the inverse problem, the aim is to reconstruct the model from a set of measurements. In the ideal case, an exact theory exists that prescribes how the data should be transformed in order to reproduce the model. For some selected examples such a theory exists assuming that the required infinite and noise-free data sets would be available. A quantum mechanical potential in one spatial dimension can be reconstructed when the reflection coefficient is known for all energies [Marchenko, 1955; Burridge, 1980]. This technique can be generalized for the reconstruction of a quantum mechanical potential in three dimensions [Newton, 1989], but in that case a redundant data set is required for reasons that are not well understood. The mass-density in a one-dimensional string can be constructed from the measurements of all eigenfrequencies of that string [Borg,1946], but due to the symmetry of this problem only the even part of the mass-density can be determined. If the seismic velocity in the earth depends only on depth, the velocity can be constructed exactly from the measurement of the arrival time as a function of distance of seismic waves using an Abel transform [Herglotz, 1907; Wiechert, 1907]. Mathematically this problem is identical to the construction of a spherically symmetric quantum mechanical potential in three dimensions [Keller et al., 1956]. However, the construction method of Herglotz-Wiechert only gives an unique result when the velocity increases monotonically with depth [Gerver and Markushevitch, 1966]. This situation is similar in quantum mechanics where a radially symmetric potential can only be constructed uniquely when the potential does not have local minima [Sabatier, 1973].

Keywords

Inverse Problem Rayleigh Wave Love Wave Model Vector Seismic Tomography 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Wien 1999

Authors and Affiliations

  • R. Snieder
    • 1
  • J. Trampert
    • 1
  1. 1.Utrecht UniversityUtrechtThe Netherlands

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