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## Subtractive removal of multiple reflections

Stacking may be thought of as a multiplicative process. Modeling leads to subtractive processes. The subtractive processes are a supplement to stacking, not an alternative: After subtracting, you can stack.

First we try to model the multiple reflections, then we try to subtract them from the data. In general, removal by subtraction is more hazardous than removal by multiplication. To be successful, subtraction requires a correct amplitude as well as a timing error of less than a quarter-wavelength.

Statistically determined empirical constants may be introduced to account for discrepancies between the modeling and reality. In statistics this is known as regression. For example, knowing that a collection of data points should fit a straight line, we can use the method of least-sum-squared-residuals to determine the best parameters for the line. A careful study of the data points might begin by removing the straight line, much as we intend to remove multiple reflections. Naturally an adjustable parameter can help account for the difficulty expected in calculating the precise amplitude for the multiples. An unknown timing error is much harder to model. Because of the nonlinearity of the mathematics, a slightly different, more tractable approach is to take as adjustable parameters the coefficients in a convolution filter. Such a filter could represent any scale factor and time shift. It is tempting to use a time-variable filter to account for time-variable modeling errors. An inescapable difficulty with this is that a filter can represent a lot more than just scaling and amplitude. And the more adjustable parameters you use, the more the model will be able to fit the data, whether or not the model is genuinely related to the data.

The difficulty of subtracting multiple reflections is really just this: If an inadequate job is done of modeling the multiples--say, for example, of modeling the geometry or velocity--then you need many adjustable parameters in the regression. With many adjustable parameters, primary reflections get subtracted as well as multiples. Out goes the baby with the bath water.

Next: Slanted deconvolution and inversion Up: MULTIPLE REFLECTION PROSPECTS Previous: Modeling regimes
Stanford Exploration Project
10/31/1997