A microlocal analysis of migration

Abstract

The mathematics of the propagation of seismic energy relevant in seismic reflection experiments is assumed to be governed by the linear acoustic wave equation, in which the coefficient is the speed of sound in the subsurface. If the speed of sound suddenly changes, the seismic signal 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, where the recording equipment is positioned. This map is highly non-linear. For inversion one therefore usually takes its formal derivative, which leads to a linearized inverse problem. It is well known that this linearized forward map corresponds in a high frequency approximation to a Fourier integral operator. The construction of a parametrix for this Fourier integral operator requires the computation of the so-called normal operator, i.e. the composition of the linearized forward map with its adjoint. An important result by G. Beylkin [The inversion problem and applications of the generalized Random transform, Comm. Pure Appl. Math. 37 (1984) 579–599] states that, if there are no caustics in the medium, this normal operator is an elliptic pseudo-differential operator. In many practical situations, however, the no-caustics assumption is violated. In this paper a microlocal analysis of this more general case will be presented. We will show that for spatial dimensions less than or equal to 3, the normal operator remains a Fourier integral 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-differential part. We will also formulate a mild injectivity condition on the traveltime function under which Beylkin's result remains true, i.e. under which the normal operator is purely pseudo-differential. This injectivity condition includes various types of multi-valued traveltimes, 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.