Migration by phase shift—An algorithmic description for array processors

Abstract

The phase shift method (Gazdag, 1978) is based on the solution, in the frequency domain, of an approximation (Claerbout, 1976) to the one‐way wave equation with initial conditions defined by a zero‐offset seismic section. Wave velocity is assumed to be constant within each layer of the section grid and is allowed to vary from layer to layer. Under these conditions, the equation written in the frequency domain reduces to a system of independent ordinary differential equations with initial values that can be solved analytically for each layer. The integration process simply amounts to multiplying the initial values by a complex number of unit modulus. The main advantages of this method are simplicity, stability, and high accuracy, since the precision of the Fourier approximation is limited only by the granularity of the seismic section. From a practical viewpoint of computer implementation, the phase shift method offers a great deal of flexibility. Some accuracy can be traded for speed, as needed, by excluding the waves traveling at an angle that exceeds a specified limit, or by ignoring frequencies at the higher end of the spectrum. The migration of a group of reflections can be excluded as well, since the phase shift angles can be accumulated for application to the frequency representation of the section only within a specified depth interval. The most interesting feature of the method may well reside in the fact that the Fourier coefficients of a seismic section can be treated independently of one another. This is of particular importance if the method is implemented for an array processor. In this case, cross‐sections of the frequency data can be sent to the array processor one‐by‐one (or in groups, as may be suitable) for phase shift operations, at minimal storage cost. A formulation of the method for an array processor is outlined here, which takes advantage of the features mentioned above. It is described in a step‐wise algorithmic fashion, using natural language and standard mathematical notations. Functions for discrete Fourier transformations, evaluation of square roots and circular functions (or fast table look‐up or interpolation), and vector reversal are assumed to exist in the array processor, in addition to elementary vector operations. This formulation can also be used for efficient implementation on conventional computers.

On Numerical Methods for Migration in Layered Media*

“…The method described in this section has been analyzed in detail in Gazdag (1978) and Dubrulle and Gazdag (1979). We will thus restrict its discussion to the essentials.…”

“…This equation is the basis for an efficient numerical algorithm applicable to layered media and described in detail in Dubrulle and Gazdag (1979) for a representation of depth by vertical wave traveltime. A few remarks are in order about this method.…”

“…Equation (3.1.10) lends itself to many algorithmic variations. The one we used for this survey follows quite closely that described in Dubrulle and Gazdag (1979), where the Fourier components of the solution are evaluated directly from those of the initial conditions. This algorithm admits variable values of Az and can be used for the migration of selected groups of layers only.…”

Processing and interpretation of time-lapse seismic data from a heavy oil field, Alberta, Canada

“…More details regarding Phase-shift migrations can be found in Gazdag, 1978, Dubrulle, 1979and Bancroft, 2007.…”

“…Through this technique, the subsurface can be approximated by a series of layers of some thickness and velocity. The transition from one layer to the next is computed via a phase shift, dependant on the RMS velocity across the layer of the interest (Gazdag, 1978 (Gazdag, 1978;Dubrulle, 1979;Bancroft, 2007). Although the phase-shift migration assumes no lateral velocity variations, moderate variations can be accommodated, where modifications to the Phase-Shift migration have accommodated for through interpolation (PSPT).…”