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PROBLEM SETTING

A reflectivity model in geophysics is a function defined in a physical volume sampled densely enough to fill a computer's random access memory. The volume typically has somewhat fewer than about (103)3 = 109 elements (voxels). The data upon which the model is built is a hundred to a thousand times larger, say 1011 elements and generally requires many magnetic tapes. Thus the matrix relating the data to the model would have 1020 elements (which is a futile way to think about the problem). What we can readily do is manufacture synthetic data from any model by running a program that amounts to a matrix multiply
 (1)
By paying attention to some details it is straight-forward (in all the problems that I have encountered) to write another program that applies the transpose operator to the data giving us a first guess at the model. This first guess is called an image of the model .
 (2)
The method of least squares offers the standard solution
 (3)
which is wholly impractical because of the size of .This suggests we seek a diagonal matrix, say as an approximation for .
 (4)

At my request, Bill Symes offered a suggestion for .His suggestion uses the operator of matrix algebra which simply takes a vector and distributes its components along the diagonal of a matrix:
 (5)

This suggestion seems reasonable because the denominator has two more powers of than does the numerator and the suggestion seems wise because it uses the known data .Unfortunately, the denominator b can easily vanish. To prevent such trouble we rearrange and smooth
 (6)
where is a positive number for experimentation and the numerator and denominator can be locally averaged as indicated by the ''. (Remember that the vectors and are sampled representations of continuous functions of space and time.) Our experience with Symes' suggestion was disappointing, probably for the reason suggested next.

Besides the standard pseudo inverse there is another
 (7)
that arises when a fitting problem is underdetermined. This pseudoinverse solves the problem of minimizing subject to the constraint equations .This pseudoinverse suggests a right side diagonal approximation
 (8)
I don't expect this right-side diagonal to work any better than did the earlier left-side diagonal. Experience with special cases suggests we need both the data-space diagonal and the model-space diagonal, namely
 (9)

Our challange for the mathematicians is to give us some guesses for the diagonal matrices and .This problem is one of a very general nature, far more general than the several applications in which I have found it. It is like the question, What is an all-purpose preconditioner?''

Next: HINTS Up: Claerbout: Diagonal weighting: An Previous: INTRODUCTION
Stanford Exploration Project
11/11/1997