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Next: Application Up: M. Clapp: Illumination compensation: Previous: Introduction

Methodology

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 ${\bf W}_{\rm m}$.

To approximate ${\bf W}_{\rm m}$, we begin by formulating our imaging operation as  
 \begin{displaymath}
{\bf W}_{\rm m}^{2}{\bf L}'{\bf d}\ \approx\ {\bf m}\end{displaymath} (1)
where ${\bf L}'$ is an imaging operator, ${\bf d}$ is the data, and ${\bf m}$ is the model. When an intelligent weighting operator is applied to the image produced by ${\bf L}'{\bf d}$, 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 ${\bf W}_{\rm m}$. 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:  
 \begin{displaymath}
{\bf W}_{\rm m}^{2} = \frac{ {\rm \bf diag} ( <{\bf m}_{\rm ...
 ... L}'\, {\bf L} \; {\bf m}_{\rm ref}\gt) +
\epsilon^2 {\bf I}}
.\end{displaymath} (2)

Here ${\bf m}_{\rm ref}$ is the chosen reference model, $\epsilon$is a damping parameter related to the signal-to-noise ratio, and the <> indicate that we take the smoothed analytic signal envelopes. $\epsilon$ ensures that we will not be dividing by zero.


next up previous print clean
Next: Application Up: M. Clapp: Illumination compensation: Previous: Introduction
Stanford Exploration Project
7/8/2003