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The issue of getting accurate amplitudes in areas of poor illumination
is a well known problem. It is generally difficult to derive and code
even an approximation to the amplitude terms that would compensate for
illumination in our imaging operators Albertin et al. (1999); Beylkin (1985).
Therefore, several schemes have been developed to try to
compensate for our inaccurate amplitude terms. These can be divided into
expensive schemes, like regularized
least-squares inversion, and less expensive schemes, like weighting
operators that can be applied *a posteriori*. Rickett (2001)
described a simple *a posteriori* diagonal weighting operator
.
To approximate , we begin by formulating our imaging
operation as

| |
(1) |

where is an imaging operator, is the data, and
is the model. When an intelligent weighting operator is applied
to the image produced by , it can be a good substitute
for iterative inversion in areas with good signal-to-noise ratios
Ronen and Liner (2000).
The question now is how to get .
Claerbout and Nichols (1994) showed
that by applying a forward operator and its adjoint to a reference model,
we will get a weighting function with the correct physical units.
Rickett (2001) took it a step further to show that for
illumination, we can find the correct weighting function in this manner:

| |
(2) |

Here is the chosen reference model, is a damping parameter related to the signal-to-noise ratio, and the
<> indicate that we take the smoothed analytic signal envelopes.
ensures that we will not be dividing by zero.

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Stanford Exploration Project

7/8/2003