ISOLATION OF A LEADING ORDER DEPTH IMAGING SERIES AND ANALYSIS OF ITS CONVERGENCE PROPERTIES FOR A 1D ACOUSTIC MEDIUM

摘要

The objective of seismic depth imaging is to produce a spatially accurate map of the reflectivity below the earth's surface. Current methods for depth imaging require an adequate velocity model in order to place reflectors at their correct locations. Techniques to derive the velocity model can fail to provide this information with the necessary degree of accuracy, especially in areas that are geologically complex. Wegfein et al. (2000) described an approach that uses the inverse scattering series, a direct non-linear inverse procedure, to image reflectors at depth without needing the exact velocity. The primary objective of this research is to derive and develop a practical inverse series algorithm to perform the task of imaging without accurate velocity information. The strategy employed is to isolate a subseries of the inverse series, a multi-dimensional direct inversion procedure, with the purpose of imaging reflectors in space (Weglein et al., 2002). In this paper, we address the problem of imaging in a constant density acoustic medium where the subsurface velocity is an unknown function of depth. A leading order imaging subseries is isolated and its convergence properties are analyzed. The leading order imaging subseries is an approximation to the full imaging potential of the inverse series in that only the contributions that are leading order in the data are considered. It is shown analytically that this imaging subseries converges for any finite contrast between the actual and reference medium, and for band-limited data with a finite maximum frequency. The rate of convergence is greater for small contrasts and small maximum frequencies. When the convergence criteria are satisfied, a closed form of the leading order imaging series is shown to exist, which has significant computational advantages over the series algorithm. The favorable convergence properties fo this imaging subseries warrant further testing for input data under increasing realistic conditions, such as an analysis for missing low frequencies and extensions to pre-stack data, to a more general multidimensional earth, and to more complex earth model types.