AbstractOne considers an elastic halfspace with depth (x1) dependent density Q and Lamé moduli (λ,m̈). Impulsive stresses τli= δ(x2,x3)δ(t) forx1= 0 are applied with displacement responsesui=gi(t,x2,x3) atx1= 0 (i= 1, 2, 3). Letvi(t,x1) = ∫∫uidx2dx3(i= 1, 2 is enough) and setw(t,x1) = ∫∫x2u1dx2dx3. One obtains a system of 3 differential equations forv1,v2, andwto which the spectral techniques of inverse scattering theory are applied as in [25]. The inverse problems for the uncoupledvican then be solved to produce 2 functionsA1andA2involving (Q, λ, μ) as functions of “bound” variablesy1andy2containing (Q, λ, μ), between which a relation then is determined. Analysis of the coupled equation forwthen 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 ofx1. 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