Multifocusing homeomorphic imaging Part 1. Basic concepts and formulas

The decomposition of a total wave field recorded on a set of seismic traces on parts corresponding to different body waves is one of the fundamental p roblems of data processing. The central point of this problem is the correl ation procedure for a seismic event (wave) on a set of recorded traces. In order to implement this procedure, it is necessary to have a local lime cor rection formula for a family of source-receiver pairs arbitrarily distribut ed around a chosen central pair. This formula is derived in the work for a 2D seismic medium of arbitrary structure using a new homeomorphic imaging m ethod called multifocusing. The presentation of multifocusing is divided in to two parts: the basic ideas and concepts of the method, the time correcti on formula and associated geometrical relationships form Part 1. The main c haracteristic of the multifocusing approach is the consideration of the geo metry of all possible wave fronts, which could be formed in the vicinity of a chosen central source-receiver pair. Provided that a target wave exists on a chosen central trace, then there is also a corresponding central ray a nd an infinite family of surrounding wave tubes. The basic idea of the mult ifocusing technique is based on the association of any pair of traces recor ded in the vicinity of the central trace with certain ray tube belonging to the family. This association can be always found. Considering this ray tub e, the local time correction formula is obtained, assuming a spherical appr oximation of two tube cross sections at the end points of central ray. In t he case of a central ray with non-zero offset, the formula consists of the following parameters: two velocities near the source and receiver locations , two angles (departure and arrival) and two pairs of dual curvatures of tu be cross-sections at the ray end points. The first four parameters are comm on for all traces, the pairs of dual curvatures are, as a rule, specific fo r the chosen pair of traces; the formula thus obtained could not be directl y used in practice. The essential part of the first paper is devoted to the parameterization of the family of dual curvatures. The exact formulas deri ved for these curvatures include as parameters, a pair of dual curvatures o f two chosen fundamental ray tubes. Different choices for the fundamental r ay tubes are considered and important relationships between the dual curvat ures and spreading functions for these tubes are established. They are the generalization of the Hubral formula [Hubral, P., 1983. Computing true ampl itude reflections in a laterally inhomogeneous earth. Geophysics 48, 1051-1 062] and known reciprocity relations. In the case of a zero-offset central ray, most important for reflection shooting, the formulas derived are signi ficantly simplified and involve four parameters only. The results obtained can be used not only in the multifocusing method, but also in migration and forward modeling. (C) 1999 Elsevier Science B.V. All rights reserved.