Theoretical analysis of the inversion problem

To understand the characteristics of the inversion (accuracy, trade-off between estimates, iterative method to be used), we analyse the singular vectors and associated singular values of the forward modeling matrix ${\bf F}$ (5). We want to analyze whether the problem is well-posed i.e. whether the RMO curves in ADCIGs contain enough information for the anisotropic migration velocities to be accurately resolved.

We analyze the properties of the modeling matrix for a given set of anisotropic Thomsen parameters; we use the Thomsen parameters of the Taylor Sand ($\epsilon=0.110, \delta=-0.035$) described in (11) and a flat reflector. So that the anisotropic migration velocities can be resolved, we also assume that the vertical velocity V_v is known, as is usually assumed in the literature and is sometimes the case in practice (e.g. from check shots or well logs).

In the following, we analyze the accuracy of the inversion as well as the trade-off between the different parameters by looking at the condition number of the matrix ${\bf F}$ and its right-hand singular vectors, respectively.

 

• Accuracy of the inversion - the condition number
• Trade-off between the different parameters - the right-hand singular vectors