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Joint inversion

It is difficult to combine the information from two surveys shot in different directions to take advantage of the different illumination patterns. If we believe that our results are accurately imaged in space, it is tempting to just add them, including some equalization term to account for amplitude differences. The result of adding the common azimuth results of the dip and strike direction data is in Figure [*]. Simply adding the migration results is not wise. Migration operators essentially sum along complex hyperboloids and the migration result is the summation of all of those hyperboloids. Ideally all of the summations will cancel out artifacts. However, in complex areas, we do not have the data necessary to cancel out all of the artifacts. The differences in artifacts and frequency content in the migration results in Figure [*] have degraded the image. The illumination problems visible on the flat reflector at depth=4 km and along the salt base are still present in the added result (Figure [*]).

 
added
added
Figure 3
Result of simply adding the common azimuth results of the dip and strike direction data seen in Figure [*].
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The problems that migration encounters in complex areas are well known. These problems can be solved to some degree by imaging through regularized least-squares inversion Duquet and Marfurt (1999); Nemeth et al. (1999); Ronen and Liner (2000). This inversion process can be described by the fitting goals
\begin{eqnarray}
{\bf 0}\ &\approx&\ {\bf L m}\ -\ {\bf d} \\  \nonumber
{\bf 0}\ &\approx&\ \epsilon {\bf A m}.\end{eqnarray} (1)

These fitting goals relate one dataset, d, to one model, m, using a linear imaging operator L. This first goal is the data fitting goal. The second goal is the model styling goal, wherein a regularization operator, A, acts on the model to help compensate for poor illumination. The regularization parameter $\epsilon$ allows us to balance the strength of the model styling goal against the data fitting goal. We can replicate these fitting goals to obtain two models from two datasets:
\begin{eqnarray}
{\bf 0}\ &\approx&\ \left[
 \begin{array}
{c} 
 {\bf L_{1}} \\ ...
 ...
{c} 
 {\bf m_{1}} \\  
 {\bf m_{2}} \\  
 \end{array} 
 \right]. \end{eqnarray} (2)
(3)

In these fitting goals, ${\bf L_{1}}$ and ${\bf L_{2}}$ are linear imaging operators such as the downward continuation migration operator presented by Prucha and Biondi (2002) that relates the individual models, ${\bf m_{1}}$ and ${\bf m_{2}}$, to the individual datasets ${\bf d_{1}}$ and ${\bf d_{2}}$. The regularization operators ${\bf A_{1}}$ and ${\bf A_{2}}$ are individually applied to the two different models and should be designed to compensate for poor illumination. However, these expanded fitting goals do not take advantage of the fact that we are imaging the same areas of the subsurface using two different datasets. We need an additional fitting goal that will allow us to regularize the models based on each other.

One potential scheme for jointly inverting two datasets shot over the same area is based on regularization between stacks of the models. This inversion can be expressed in terms of three fitting goals that combine the two datasets:
\begin{eqnarray}
{\bf 0}\ &\approx&\ \left[
 \begin{array}
{c} 
 {\bf L_{1}} \\ ...
 ...
{c} 
 {\bf m_{1}} \\  
 {\bf m_{2}} \\  
 \end{array} 
 \right]. \end{eqnarray} (4)
(5)
(6)

The third fitting goal is the key to regularizing the illumination between the two models. ${\bf S_{1}}$ and ${\bf S_{2}}$ are stacking operators for the two models. ${\bf E}$ is a cross-equalization operator that compensates for differences in amplitudes and wavelets between the two models. ${\bf E}$ can be created from the migration result of each dataset Rickett et al. (1997); Rickett and Lumley (1998). The $\epsilon$s are weights that are used to balance the different fitting goals.

These fitting goals will result in two models that have been regularized to help fill in areas that are illuminated differently by the two surveys. Either of these models should have better illumination than the result of migration of the individual datasets. Ideally, if both datasets have the same angular coverage, the stacks of the models should be the same, but the information along the ray parameter axes will be different. This difference is due to the fact that the ray couples for each survey are traveling through different media. The directions of the surveys will cause the rays to have different illumination problems and encounter other problems such as anisotropy, attenuation, and velocity contrasts that may cause evanescence.


next up previous print clean
Next: Future Work Up: M. Clapp: Directions in Previous: Migration results
Stanford Exploration Project
7/8/2003