When we try to image the subsurface, we are attempting to reverse the way the seismic energy traveled (propagated). Migration methods have difficulty imaging complex areas because migration algorithms cannot accurately reverse the propagation in these areas. Imaging these areas requires an algorithm that is able to invert the propagation.
Chapter 3 explains the new imaging technique that I am using to improve imaging of complex areas. This method is a Regularized Inversion with model Preconditioning (RIP). The inversion uses downward-continuation migration () and its adjoint (matrix transpose) in an iterative conjugate gradient scheme to help extract all of the useful information from the recorded data. In addition, the regularization allows me to introduce a priori knowledge of what the resulting model (image) should look like to the inversion problem. By preconditioning the model, I can reduce the number of iterations needed to get an acceptable result.
I implement the regularization with two different schemes. My geophysical RIP acts to regularize amplitudes along the offset ray parameter (reflection angle) axis of the image. Poor illumination will manifest itself as gaps along the offset ray parameter axis, so the regularization tries to fill these gaps. My geological RIP acts to regularize amplitudes along chosen dips in the common reflection point-depth space of the image. This scheme tries to fill in the shadow zones directly, by imposing an idea of the dips we expect to see in the model upon the inversion problem. Using these regularization schemes in the iterative conjugate gradient inversion will keep us from introducing elements in the model that conflict with the existing data.
Chapter 3 also examines several significant considerations for the use of RIP. First, the use of RIP for imaging is a relatively expensive procedure. Therefore, I compare the result of RIP with the result of model-space weighting, a cheap and commonly used method for trying to improve imaging in shadow zones. Second, geological RIP makes assumptions about the dips that may exist in the shadow zones. I investigate the result of using the wrong dips in a geological RIP scheme. In the geophysical RIP scheme, the regularization assumes that the correct velocity is being used for the downward continuation step. By experimenting with using incorrect velocity models, I test the importance of this assumption on the result of the regularized inversion. Finally, the regularization in the geophysical RIP scheme acts on the amplitudes, so I look at the effect of geophysical RIP on amplitude variation with offset ray parameter (angle).