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The Leading Edge; February 2001; v. 20; no. 2; p. 210; DOI: 10.1190/1.1438920
© 2001 Society of Exploration Geophysicists
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Altering offsets and azimuths—in a computer

Stephen J. Hill, Robert Stolt and Stephen Chiu

Conoco, Ponca City, Oklahoma, U.S.

Corresponding author: Steve.J.Hill@usa.conoco.com

The first 20% of the full text of this article appears below.

For years, seismic processing geophysicists used dip moveout (DMO) to alter the acquisition geometry. Our industry has found many uses for DMO because it converts the timing of nonzero-offset data to the timing of zero-offset data. More recently, theoreticians created data mapping as a generalization of the principle behind DMO. Data mapping converts data obtained at an observed offset and azimuth to data at a new offset and/or azimuth. The precise derivation of the data mapping resides in an arduous solution of the wave equation. Thus, data mapping transformations may appear magical. This article provides a simple, geometric understanding of the data mapping transformation.

Before turning to the more general 3-D case, we first present the 2-D case. For the 2-D case, data mapping transforms data obtained at one offset distance into data "observed" at a second offset distance. To understand this procedure, we will use one principle, one requirement, and two assumptions. The principle is linear superposition and the requirement is physical invariance. For conceptual convenience, we assume reflection coefficients do not change with offset, and we also assume a constant velocity earth.

The principle of linear superposition simplifies our task. As shown in Figure 1, it allows us to construct any input data from a linear superposition of spikes or impulses. In addition, data-mapped output is a linear superposition of data-mapped spike responses. Consequently, we need to understand only the data-mapping operation on a single spike at an arbitrary location.


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  		Figure 1. Linear superposition simplifies data mapping.

 
The physical invariance restriction is obvious—the earth does not change as a result of altering the acquisition offset. Likewise, our image of that earth should not change. Depth migration should image the same subsurface using either the original data or the offset-transformed data.

The constant velocity assumption . . . [Full Text of this Article]






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