Aspects of true amplitude migration

Abstract

Summary Kirchhoff migration has been used successfully in many parts of the world for 3D prestack depth imaging. Despite this success, however, there are areas where Kirchhoff migration exhibits unphysical amplitude behavior, such as beneath salt pinchouts, or where acquisition geometries are complex. These difficulties arise from an interplay between sampling in the acquisition geometry, and distortion of the wavefield as it travels through structural complexity in the subsurface. Here we investigate t he suitability of true amplitude inversion to overcome amplitude irregularities in images obtained in complex areas. In the proce ss we derive a relation between true-amplitude Kirchhoff inversion and dip equalization which allows a simple computation of accurate amplitudes in Kirchhoff inversion. The relation also demonstrates the relative role of geometrical spreading and acquisition sampling in the inversion process, and how both can be effectively removed to give an accurate reflectivity image. Ultimately, the effectiveness of seismic imaging manifests itself in the interplay of acquisition sampling and wavefield distor tion as the sound propagates through the earth. In order for a seismic experiment to be su ccessful in imaging a subsurface reflector, the sound reflected from the reflector must arrive with an appreciable amplitude at the r eceiver. In the high frequency approximation, we can think of the sound propagation and reflection as being approximated roughly by rays that travel through a complex velocity field, and reflect at an imaging point on the reflector via Snell's law. Intuitively, it will only be possibl e to image the reflector if there are source-receiver pairs in the acquisition geometry for which Snell's law at the reflection poin t can be satisfied. Now imagine how this manifests itself in a simple common offset Kirchhoff imaging algorithm that works by simply spreading and superimposing samples along traveltime isochrons (i.e. migration operators). Constructive interference occurs when the tangent to the migration operator matches the dip of the reflector at the imaging point, and destructive interference occurs otherwise, resulting in an image of the reflector. If the tangent matches the dip, then rays from source to image point to receiver for this operator essentially satisfy Snell's law at the image point. Within a simple acoustic approximation that neg lects transmission losses, three effects now conspire to produce the final image amplitude in the summation process. First, the soun d has traveled through a complex velocity field, so its amplitude has dropped according to geometrical spreading. A simple smearing of samples along an isochron followed by a sum may undo much of this effect, but not n ecessarily in a way that is consistent with a true image of reflectivity. Second, if many source-r eceiver pairs in the acquisition geometry nearly satisfy Snell's law at the reflection point, one would expect a large amplitude as a result of the constructive interference process. Conversely, if only few nearly satisfy the law, one would expect a drop in amplitude. The result is an amplitude variation in the image of the reflector which is independent of that due to variations in reflectivity; it is a result of how sampling in the acquisition geometry translates to dip sampling at the reflector via complex ray paths, which we refer to as illumination. Thir d and finally is the reflectivity itself, which has an obvious effect on the seismic amplitude. How to undo the interplay betwee n geometrical spreading and effective sampling at the image point to obtain reflectivity alone is far from obvious, but is a key element in maintaining amplitude fidelity in Kirchhoff migration. Recently much progress has been made in the derivation of explicit Kirc hhoff inversion formulas that solve for reflectivity