On complete recovery of geophysical data

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 u_i = g_i(t, x₂, x₃) at x₁ = 0 (i = 1, 2, 3). Let v_i(t, x₁) = ∫∫u_idx₂dx₃ (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 v_i 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.