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Exponential scaling functions have some ideal mathematical properties.
Take the *Z*-transform of a time function *a*_{t}:

| |
(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 *k*_{x}.
The operator has become a function of that may be expressed
in the time domain as a filter *a*_{t}.
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