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# A 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 2-D and 3-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:

| |
(143) |

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

Figure 3 The wavefield is emitted at *s* and
recorded at *g*. The multiple bounces somewhere at *g*'.

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Stanford Exploration Project

5/5/2005