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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 (10^{3})^{3} = 10^{9} elements (voxels).
The data upon which the model is built
is a hundred to a thousand times larger, say 10^{11} elements
and generally requires many magnetic tapes.
Thus the matrix relating the data to the model would
have 10^{20} 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?''

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** Up:** Claerbout: Diagonal weighting: An
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Stanford Exploration Project

11/11/1997