Erratic time shifts from trace to trace have long been dealt with by
the so-called
surface-consistent statics model.
Using this model you fit the observed time shifts, say, *t*(*s*,*g*),
to a regression model .The statistically determined
functions *t*_{s} (*s*) and *t*_{g} (*g*) can
be interpreted as being derived from altitude or velocity variations
directly under the shot and geophone.
Taner and Coburn [1980] introduced
the closely related idea of a surface-consistent
frequency response model that is part of the statics problem.
We will be interpreting and generalizing that approach.
Our intuitive model for the data is

(5) |

One problem with the split Backus filter is a familiar one--that the time delays and enter the model in a nonlinear way. So to linearize it the model is generalized to

(6) |

Theoretically, taking logarithms gives a linear, additive model:

(7) |

The phase of *P*',
which is the imaginary part of the logarithm, contains the
*
travel-time
*
information in the data.
This information
loses meaning when the data consists of more than one arrival.
The phase function becomes discontinuous, even though the data is well behaved.
In practice, therefore, attention is restricted
to the real part of (7),
which is really a statement about power spectra.
The decomposition (7) is a linear problem, perhaps best solved
by iteration because of the high dimensionality involved.
In reconstructing *S* and *G* from power spectra,
Morley used the Wiener-Levinson technique,
explicitly forcing time-domain zeroes
in the filters *S* and *G* to account for the water path.
He omitted the explicit moveout correction in (5), which may
account for the fact that he only used the inner half of the cable.

10/31/1997