Computation of Green's function of laterally varying media by means of a modal expansion

Abstract

In many cases the strongest signals in seismic data arise from scattering at shallow heterogeneities. An important step in modeling and removing this type of deterministic noise is computing Green's function for a 3D laterally varying model. To this extent an efficient method is developed, which is based on the combination of guided-mode expansions and ray-tracing techniques. This method is illustrated for a two layer model. I n t r o d u c t i o n The regions of highest interest in seismic exploration are the deeper ones. However, the strongest signals in seismic data generally arise from heterogeneities in the near-surface regions. These heterogeneities lead to scattering and to waves which contain no information on the deeper regions (e.g. head waves and Rayleigh waves). Many methods have been developed on an ad-hoc basis for removing these effects. Most of these methods are adaptive, and despite the fact that they have been succesful in many cases, they are not always able to deal with the complexity of the scattering process in the shallow subsurface. Examples of these methods are “filtering” , “statics” and “gapped deconvolution” (see for example Yilmaz, 1987). Recently, a method has been developed for removing the effects of heterogeneities in the shallow subsurface by means of linearized inversion and high-frequency asymptotics for the case of a laterally homogeneous model of the earth. This method has been applied succesfully to field data (see Blonk et al., 1994, 1995). The method which is described here is an extension of the former method for the case where the assumption of lateral invariance is no longer valid. Our method is also based on linearized inverse scattering and high frequency asymptotics. The method consists of the following steps: First an estimate of certain characteristics of the background model has to be made. We assume that the background model consists of (possibly heterogeneous) horizontal layers (see figure 1). Due to the layered structure of the background model, the propagation velocity of seismic waves in this background model is frequency-dependent. For this part of the method, concepts form integrated optics are used. The next part of the method consists of estimating the distribution of heterogeneities in the shallow subsurface. This has to be done by using as many data as possible. In this estimation procedure, Green’s function of the background model plays a vital role. The objective of this estimation procedure is to find a distribution which can explain the observed scattering. Due to the ill-posedness of the problem, this does not need to be the actual distribution. The last step in the method consists of determining the scattered field for each shot from the estimated scatterer distribution, and subtracting this field from the original data. In this paper, we concentrate on the computation of the headwave part of Green's function and restrict ourselves to the scalar case. Computa t ion of Green’s func t ion The general structure of the background model we consider here is depicted in figure 1. We have N layers, where for layer i we have < < and = 0. The Nth layer is infinitely thick. The governing equation for Green’s function in the frequency domain is given by: