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Clapp (2004) suggested breaking up the tomography problem
into two portions: creating several realizations
of maps and using them as input to the tomography
problem.
Estimating the field is in itself difficult.
The standard approach is to calculate semblance
over a range of move-out values. The move-out at given point
is then the maximum semblance at the location.
To reduce noise, the semblance field is often smoothed.
This is still far from in ideal solution. We are constantly
fighting a battle between selecting local minima (not enough
smoothing) and missing important move-out features (too
much smoothing).
In Clapp (2004) the various maps were created
by selecting a smooth set of random number and converting
them into values based on a normal score transform Isaaks and Srivastava (1989a).
This approach was somewhat successful, but suffered from the fact
that we don't, and effectively can't scan over an infinite set
of move-outs. Therefore our distribution function is misleading.
Methods to correct for the limited range proved *adhoc*.

Instead I am going to start from the approach
outlined in Clapp (2003b).
My goal is to estimate a smooth set of semblance
values . I begin by
selecting the maximum semblance at each
point .I solve the simple minimization problem

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(6) |

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where is again a steering filter, and is a function of the semblance value at each location.
After estimating I select the
maximum within a range around to form a new ,and repeat the estimation. At each iteration, the
window I search around and the amount of smoothing ()decreases.
To create a series of models I introduce random
noise into scaled by the variance
in the semblance at each location.
With different sets of random noise I get
different realistic models.

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Stanford Exploration Project

10/23/2004