A MICROLOCAL ANALYSIS OF MIGRATION

The mathematics of the propagation of seismic energy relevant in seism ic reflection experiments is assumed to be governed by the linear acou stic wave equation, in which the coefficient is the speed of sound in the subsurface. If the speed of sound suddenly changes, the seismic si gnal is (partly) reflected. To find the position of these changes one considers first the so-called forward map, which sends the coefficient of the wave equation to its solution at the surface of the earth, whe re the recording equipment is positioned. This map is highly non-linea r. For inversion one therefore usually takes its formal derivative, wh ich leads to a linearized inverse problem. It is well known that this linearized forward map corresponds in a high frequency approximation t o a Fourier integral operator. The construction of a parametrix for th is Fourier integral operator requires the computation of the so-called normal operator, i.e. the composition of the linearized forward map w ith its adjoint. An important result by G. Beylkin [The inversion prob lem and applications of the generalized Random transform, Comm. Pure A ppl. Math. 37 (1984) 579-599] states that, if there are no caustics in the medium, this normal operator is an elliptic pseudo-differential o perator. In many practical situations, however, the no-caustics assump tion is violated. In this paper a microlocal analysis of this more gen eral case will be presented. We will show that for spatial dimensions less than or equal to 3, the normal operator remains a Fourier integra l operator, be it not a pseudo-differential operator anymore. Instead, it is the sum of an elliptic pseudo-differential operator and a more general Fourier integral operator of lower order than the pseudo-diffe rential part. We will also formulate a mild injectivity condition on t he traveltime function under which Beylkin's result remains true, i.e. under which the normal operator is purely pseudo-differential. This i njectivity condition includes various types of multi-valued traveltime s, which occur frequently in practice. Its geometrical interpretation will be discussed. Finally, we derive an approximate explicit formula for the inverse, which is suitable for numerical evaluation. (C) 1998 Elsevier Science B.V. All rights reserved.