Surface-related multiple prediction theory

This section details how multiples are generated by convolution of shot gathers. For a 1-D earth, the convolution can be done directly in the f-k domain. For a 2-D earth, the convolution becomes nonstationary. In addition, one convolution of the shot gathers tends to overpredict high-order multiples.

If p(g|s) represents a single frequency component of the primary reflected wavefield recorded at g after an impulsive shot at s, then the first-order surface-related multiple, m(g|s), can be computed with a Kirchhoff-style integral over the reflection surface:  

$$m(g\vert s) = \int p(g\vert g') \; p(g'\vert s) \; dg'.$$ (10)

Equation (10) is expensive to evaluate, especially for large 3-D data sets, but nevertheless widely-used for multiples modeling.

 

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Figure 12 The wavefield is emitted at s and recorded at g. The multiple bounces somewhere at g'.