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In his book,
Claerbout 1994 shows that filters
of the form

| |
(1) |

destroy plane waves *u*(*x*,*t*) = *u*(*t* + *p*_{x} *x*). Letting denote averaging
over time and over space, the value,
| |
(2) |

represents a smoothed least squares estimate of local dip. In the 3-D setting
one can calculate both, *p*_{x}, and,
| |
(3) |

to obtain estimates in the line and cross-line directions so that the vector

defines a local plane at each point in the 3-D volume. Its magnitude,
| |
(5) |

is a scalar which responds in a somewhat dramatic fashion to relatively minor
deviations from local plane wave assumptions. The reasons for this response
may not be completely clear. For time-migrated 3-D volumes, and
are estimates of the post-migration dip. For a given velocity,
they provide the basis for computing the zero offset point from which the local primary-reflection horizon
migrated. In this case, the dip-magnitude is in some sense porportional to the distance to this zero-offset location.
For large dipping events, such as faults, this distance can be quite large.
Abrupt changes in amplitude or wavelet phase can also result in major changes in dip-magnitude.
The quantities *p*_{x} and *p*_{y} are estimates of differential change and so will be significant whenever
discontinuities are present in the data volume.

** Next:** APPLICATION
** Up:** Bednar: Least squares dip
** Previous:** INTRODUCTION
Stanford Exploration Project

10/10/1997