Wave-Equation Migration: Two Approaches

ABSTRACT A conventional stacked seismic section displays data only as a passing wave-field recorded at selected points on the earth's surface. In regions of complex geology; this display may bear little resemblance to a cross-section of subsurface reflectors. Migration is the technique used to transform the wave-field of a seismic section into a reflectivity display. As such it should be possible to relate any seismic migration method to a solution of the scalar wave equation -- the assumed mathematical description of wave propagation in the earth's subsurface. After discussing fundamental assumptions required in the two migration approaches, this paper focuses primarily on comparison migrations of both synthetic Data and of marine and land profiles. For good data of modest dip, the two approaches produce results which are remarkably similar despite their very different conceptual bases and realizations. This outcome is very encouraging as it increases confidence in the rationale behind migration. For poorer data of modest dip, the solutions based on differential forms of the scalar wave-equation have noticeably superior S/N compared with their integral form counterparts. The seismic trace spacing {receiver group interval) is found to play different, but fundamental, roles in governing the accuracy and quality of both types of migration. INTRODUCTION Migration is the technique used to transform the wave-field of a stacked (or unstacked) seismic section into a cross-sectional reflectivity display. In this process, one wave-field is transformed into another; hence the mechanism of wave propagation in the earth assumes fundamental importance. If the equation is a reasonable mathematical representation of wave propagation in the earth's subsurface, then any seismic migration method should be related to a solution of the wave equation. Such a solution can be derived from either an integral or a differential form of this equation. Over the past 5 or 6 years, a particular digital migration technique has become standard: the conventional approach of summing seismic amplitudes along diffraction hyperbolas. With some refinements this heuristic summation approach is shown to be founded on the integral solution to the scalar wave equation. In more recent years, such integral solutions have been complemented by direct solutions of differential forms of the wave equation. We are referring here to the finite-difference approach pioneered by Jon F. Claerbout of Stanford University. This finite-difference approach has often been referred to as the so-called "wave equation migration". An important point that we hope to make here is that both the integral (summation) approach and the differential (finite-difference) approach represent approximate solutions of the same wave equation. Both require simplifying approximations to arrive at an equation describing wave propagation; both require simplifying assumptions about boundary conditions at the earth's surface and at depth; both require constraints on the variability of the subsurface velocity distribution; and, finally, both involve a discretization approximation that allows us to manipulate digital seismic data.