Next: Review of Inverse Interpolation
Up: Brown: Irregular data dip
Previous: Introduction
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 t_{1} and t_{2}, and that the event takes the
form of a plane in the neighborhood of the two traces.
Imagine that we wanted to measure t_{2}t_{1}, given the local dip of the plane,
p_{x} and p_{y}. The total time shift is simply the sum of the time shifts
along the x and y planes, going first from x_{1} to x_{2} and then from
y_{1} to y_{2}. We can write an equation directly:

t_{2}t_{1} = p_{x}(x_{2}x_{1}) + p_{y}(y_{2}y_{1})

(1) 
However, in actuality we do not know p_{x} and p_{y}, but using Claerbout's
puck method, we can measure t_{2}t_{1}. 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, p_{21}, and the local 3D 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 leastsquares 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 p_{x} and p_{y} 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 (p_{x} and p_{y}) at each trace
location. These dip measurements must be interpolated to fill the entire model
grid. In this paper, I use an expandingwindow 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