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# Irregular-geometry Dip Estimation Methodology

Figure illustrates a simplified conceptual model of seismic data. Given two traces, we assume that a seismic event passes through each trace location at times t1 and t2, and that the event takes the form of a plane in the neighborhood of the two traces.

Imagine that we wanted to measure t2-t1, given the local dip of the plane, px and py. The total time shift is simply the sum of the time shifts along the x and y planes, going first from x1 to x2 and then from y1 to y2. We can write an equation directly:

 t2-t1 = px(x2-x1) + py(y2-y1) (1)

However, in actuality we do not know px and py, but using Claerbout's puck method, we can measure t2-t1. Implemented on a computer, the puck method computes the time shift (in pixels) between two traces that optimally aligns a seismic event on the two traces as a function of time. In other words, the method measures
 (2)
where is the time sampling of the traces. We can now rewrite equation accordingly:
 (3)
Equation (3) describes the linear relationship between the computer dip measured between traces 2 and 1, p21, and the local 3-D reflector dip. We require two equations to obtain a unique estimate of the parameters, but the noise and incoherency inherent to real data make it desirable to use more than two traces and exploit the statistical smoothing'' of least-squares estimation.

If trace 1 is the master'' trace and traces 2 through n its neighbors, let us define the forward modeling operator
 (4)
and the data vector
 (5)
The estimated px and py at trace location 1 are then the solution to the normal equations:
 (6)
Inversion of the matrix is trivial. The result of the process is a pair of dip measurements (px and py) at each trace location. These dip measurements must be interpolated to fill the entire model grid. In this paper, I use an expanding-window smoothing algorithm to accomplish the task. If reflectors are continuous, their dips should be somewhat smooth in space, so to some extent, spatial smoothing is justified.

Next: Review of Inverse Interpolation Up: Brown: Irregular data dip Previous: Introduction
Stanford Exploration Project
10/14/2003