A Fast 3-D Modeling Technique And Fundamentals Of 3-D Frequency Domain Migration
Authors: Herman A.J. , Anania R.M. , Chun J.H. , Jacewitz C.A. , Pepper R.E.F. 
Published: October 2008

ABSTRACT
The frequency domain approach to migration extends to three dimensions. The application of a ray tracing approach to simple seismic events and the principle of superposition leads to the correct equation for the frequency domain migration and a further understanding of the basic migration procedure in three dimensions.
Synthetic examples are used to illustrate the fact that two-dimensional migration of three-dimensional data will not result in a correct section. However, in the frequency domain, three-dimensional migration may be reduced to a series of two-dimensional migrations.
The synthetics for this paper were generated by a particularly fast and straightforward technique. Such a fast three-dimensional seismic model can be generated for structures with certain symmetry. Starting from a two-dimensional synthetic, a three dimensional model can be constructed by rotation about the center of symmetry. Any line across the feature may then be synthesized from the basic data.
Certain more complex three-dimensional models can also be developed using the superposition of simpler models.

INTRODUCTION
In recent years, there has been considerable interest in three-dimensional (3-D) data acquisition and true three-dimensional processing. In particular, J. Hagedorn (1954) and A. Musgrave (1961) discussed methods of three-dimensional migration. Hilterman (1970) and French (1974) published physically recorded sections from three-dimensional models. Hilterman compared the recorded data with calculated data. The calculated data was available for the models with true two-dimensional (2-D) structure. That is, these particular models may be generated by moving a 2-D cross-section in a straight line. French discussed the migration of the physically recorded model data with both 2-D and 3-D techniques.
More recently, W. Schneider (1978) and R. Stolt (1978) have dealt with three-dimensional migration. Schneider considered the problem from an integral standpoint, while Stolt used the frequency domain approach. Stolt arrived at the basic equation for three-dimensional migration.
In this paper, we have included a discussion of some of the fundamentals of three-dimensional migration in both the time and the frequency domain. This is primarily a tutorial approach based on a geometric or ray theory viewpoint. In order to relate these concepts to specifics, several synthetic examples will be considered. These were generated via a fast, simple technique described in this paper.

THREE-DIMENSIONAL MIGRATION IN THE FREQUENCY DOMAIN
Three-dimensional migration in the frequency domain is, at least conceptually, a natural extension of 2-D migration. Prior to discussing 3-D migration, we will review a few fundamental properties of the 3-D Fourier transform. Consider the infinite horizontal plane ABCD depicted in Figure la. In this figure, we will assume that the reflection coefficient is a constant throughout the plane. Suppose the Fourier transform is performed in the × direction. The plane will be transformed into a line as shown in Figure lb. Next, after the transform is performed in the y direction, the line will be transformed to a point as shown in Figure 1c.