On complete recovery of geophysical data

Abstract

Abstract One considers an elastic halfspace with depth ( x 1 ) dependent density Q and Lamé moduli (λ,m̈). Impulsive stresses τ li = δ( x 2 , x 3 )δ( t ) for x 1 = 0 are applied with displacement responses u i = g i ( t , x 2 , x 3 ) at x 1 = 0 ( i = 1, 2, 3). Let v i ( t , x 1 ) = ∫∫ u i d x 2 d x 3 ( i = 1, 2 is enough) and set w ( t , x 1 ) = ∫∫ x 2 u 1 d x 2 d x 3 . One obtains a system of 3 differential equations for v 1 , v 2 , and w to which the spectral techniques of inverse scattering theory are applied as in [25]. The inverse problems for the uncoupled v i can then be solved to produce 2 functions A 1 and A 2 involving (Q, λ, μ) as functions of “bound” variables y 1 and y 2 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 x 1 . 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.