In this report, the inversion of focusing operators is also used to obtain a velocity model. As in all geophysical inversion problems, the under-determined nature is a problem that should be faced. Normally, this is handled by two methods; 1) choosing a well determined parameterization (global parameterization by user defined layers), or 2) regularizing the optimization (including a priori information and model covariance). A drawback of the first method is that it puts a constraint on the result and it can be laborious. Moreover, incorrect initial parameterization can lead to slow or non-convergence. A drawback of the second method is that the problem of over-parameterization is still not solved (e.g. when a regular grid is used). This might cause problems when the inversion problem is regularized; the model parameters need different levels of regularization, as certain regions in the model might consistently be more under-determined. In the data dependent parameterization shown in this report both methods will be combined; adjustment of the parameterization is based on the covariance after optimization. In the explicit least squares or singular value decomposition (SVD) optimization methods, the a posteriori covariance is generated during optimization. However, in more practical approximate optimization methods, obtaining the covariance becomes more difficult.

In this report, the concept of the CFP method and the focusing operator updating will be explained in the first section. Next, the inversion of the focusing operators, and the data dependent parameterization will be considered. The method will be applied to a synthetic example. The last part will compare the different approximate inversion algorithms for obtaining the inverse and the a posteriori covariance. Finally, some conclusions and plans for future work will be presented.

9/18/2001