Velocity inversion from common offset data

Abstract

The authors consider all of three-dimensional (x,y,z) space to be a constant density acoustic medium, thus characterised by a sound speed c(x,y,z). They assume c identical to 1 for z<or=0 and c=(x,z) only for z>0. At each point (xs, 0,0) an impulsive source of acoustic waves is set off, and the resulting reflected waves are measured as a function of time t, for 0<or=t<or=Tmax at receiver location (xR, 0,0) with xR=xs+h for some fixed offset h. From this data they wish to determine the velocity c(x,z) for 0<or=z<or=Zmax=1/2(Tmax2-h2)12/. They study the Born approximation to this problem, i.e. they assume that c-1 is small enough so that the map from velocity to surface response can be adequately approximated by its linearisation at reference velocity c0 identical to 1. They show that this inverse problem is equivalent to an uncoupled system of one-dimensional, second-kind volterra integral equations, indexed by horizontal frequency. Using properties of the kernels in these equations they derive continuous dependence and uniqueness results for the inverse problem, and a constructive solution method is given. The case of multi-offset data is also considered.