The Inversion of Discrete Systems Using Non-Causal Solutions

Abstract

In all the analysis described in Chapters 1 and 2 we used only causal solutions of the differential equations describing a Goupillaud medium. We recall that a causal solution is one for which the medium is quiescent for t < 0, and for which a ‘cause’ at one time and place can produce an ‘effect’ at another place and some later time only if waves have had time to travel from one to the other. In the methods for solving the inverse problem described in Chapter 2 the crucial property of causal solutions that was used was $${{U}_{{j,j}}} = 0.$$ This states that at time t = j△ the disturbance which started at the surface ξ= 0 has just reachedξ= j△, and nothing has had time to come up from below to reachξ= j △; in other words, there is just a down-wave, no up-wave.