Effects of model regularization on the data space

The model regularization schemes I am proposing are designed to compensate for the seismic energy that is lost due to complex subsurfaces. Much of the energy is lost because it is directed outside of the bounds of the seismic survey. The regularized model is in essence expanding the data space through the data fitting goal (). To demonstrate this, I use the simple synthetic dataset with a single reflector below a low velocity lens. The velocity model can be seen in Figure  and the result of downward-continuation migration is in Figure . The actual synthetic data can be seen in Figure . The data is displayed as a flattened cube with a time slice at the top, a common offset panel in the lower left and a common midpoint (CMP) gather in the lower right. The triangular shape of the data as viewed in the time slice is caused by the finite extents of the survey. The missing energy that has left the bounds of the survey is manifested as holes in the CIGs as seen in the migration result (Figure ). Theoretically, if the survey was larger in extent, this energy would be recorded and the migration would not have holes.

 

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Figure 8 The recorded data corresponding to a synthesized seismic survey over the velocity model in Figure . The triangular shaped extent of the data seen in the time slice (top) is caused by the limited survey geometry.

 

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Figure 9 The result of migrating the data in Figure . Note the holes in the CIGs indicating poor illumination caused by seismic energy being directed outside of the survey area.

Regularized inversion with model preconditioning (RIP) tries to create a model in which we compensate for the lost seismic energy. In this case, we are in essence reconstructing the energy that leaves the survey bounds. Performing 3 iterations of geophysical RIP fills in the holes in the CIGs (Figure ). To examine the effects of this on the data space, we apply the adjoint of downward-continuation migration to this result. Figure  shows the data space associated with the result of RIP. Comparing this with Figure , the triangular shape of the original data is still visible, but data has actually been extended beyond its original extents, albeit at a lower amplitude. Hence, model regularization is actually expanding the data space.

 

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Figure 10 Result after 3 iterations of geophysical RIP. The holes in the CIGs seen in Figure  have been largely filled in.

 

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Figure 11 The ``expanded'' data after 3 iterations of geophysical RIP. The regularization of the model space has filled in areas from which the seismic data was directed outside of the survey area. This has ``extended'' the data beyond the limits seen in Figure .