Next: The substitution operator Up: COSMETIC ASPECT OF WAVE Previous: Spatial scaling before migration

## Exponential scaling

Exponential scaling functions have some ideal mathematical properties. Take the Z-transform of a time function at:
 (9)
The exponentially gained time function is defined by
 (10)
The symbol denotes exponential gain. Mathematically, means that Z is replaced by .Polynomial multiplication amounts to convolution of the coefficients:
 (11)
By direct substitution,
 (12)
This means that exponential gain can be done either before or after convolution. You may recall from Fourier transform theory that multiplication of a time function by a decaying exponential is the equivalent of replacing by in the transform domain.

Specialize the downward-continuation operator to some fixed z and some fixed kx. The operator has become a function of that may be expressed in the time domain as a filter at. Hyperbola flanks move upward on migration. So the filter is anticausal. This is denoted by
 (13)
The large negative powers of Z are associated with the hyperbola flanks. Exponentially boosting the coefficients of positive powers of Z is associated with diminishing negative powers--so is A with a weakened tail--and tends to attenuate flanks rather than move them. Thus may be described as viscous.

From a purely physical point of view cosmetic functions like gain control and dip filtering should be done after processing, say, .But is equivalent to ((,and the latter operation amounts to using a viscous operator on exponentially gained data. In practice, it is common to forget the viscosity and create A(.Perhaps this means that dipping events carry more information than flat ones.

Next: The substitution operator Up: COSMETIC ASPECT OF WAVE Previous: Spatial scaling before migration
Stanford Exploration Project
10/31/1997