On complete recovery of geophysical data

Robert Carroll, Fadil Santosa, L. Payne

Research output: Contribution to journalArticle

12 Citations (Scopus)

Abstract

One considers an elastic halfspace with depth (x1) dependent density Q and Lamé moduli (λ,m̈). Impulsive stresses τli = δ(x2, x3)δ(t) for x1 = 0 are applied with displacement responses ui = gi(t, x2, x3) at x1 = 0 (i = 1, 2, 3). Let vi(t, x1) = ∫∫uidx2dx3 (i = 1, 2 is enough) and set w(t, x1) = ∫∫x2u1dx2dx3. One obtains a system of 3 differential equations for v1, v2, and w to which the spectral techniques of inverse scattering theory are applied as in [25]. The inverse problems for the uncoupled vi can then be solved to produce 2 functions A1 and A2 involving (Q, λ, μ) as functions of “bound” variables y1 and y2 containing (Q, λ, μ), between which a relation then is determined. Analysis of the coupled equation for w then leads to a Fredholm integral equation whose solution provides an additional relation between (Q, λ, μ) from which (Q, λ, μ) can be determined as functions of x1. The integral equation is reduced to a Volterra type equation by results and techniques of transmutation and then solved by a modification of standard techniques. A number of features and results of independent mathematical interest arise from the transmutation theory.

Original languageEnglish (US)
Pages (from-to)33-73
Number of pages41
JournalMathematical Methods in the Applied Sciences
Volume4
Issue number1
DOIs
StatePublished - 1982

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Integral equations
Recovery
Inverse problems
Differential equations
Scattering Theory
Inverse Scattering
Fredholm Integral Equation
Scattering
Volterra
Half-space
Modulus
Integral Equations
Inverse Problem
Differential equation
Dependent
Standards

Cite this

On complete recovery of geophysical data. / Carroll, Robert; Santosa, Fadil; Payne, L.

In: Mathematical Methods in the Applied Sciences, Vol. 4, No. 1, 1982, p. 33-73.

Research output: Contribution to journalArticle

Carroll, Robert ; Santosa, Fadil ; Payne, L. / On complete recovery of geophysical data. In: Mathematical Methods in the Applied Sciences. 1982 ; Vol. 4, No. 1. pp. 33-73.
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