The last few years have seen a significant increase in research assessing risk. Several papers deal with assessing risk from a geostatistical framework Gambus-Ordaz et al. (2002); Shanor et al. (2002). The general methodology is to create equi-probable models based on simplified covariance descriptions and probability functions. Each point is visited in turn and a value selected based on a priori probability distribution, surrounding point value, and the covariance description.
In previous work I showed how we accomplish something similar in a global inversion problem. As long as a decorrelator exists (such as a regularization operator that is an inverse noise covariance operator) adding random noise into the residual space will create equi-probable models Clapp (2001a). This methodology can be applied to tomography in two distinct ways. If random noise is added to components in the residual vector corresponding to the regularization operator, we produce models that have not only correct covariance but also a reasonable variance. These models add fine layered features that standard tomography can not resolve. From realization to realization these features change in shape and amplitude. They do not effect the kinematics of the final image, but do have an effect on the amplitudes Clapp (2003a).
The second choice, adding noise to the portion of the residual vector corresponding to the data fitting goal, does have a more significant effect on the velocity and the final image Clapp (2004). Adding noise in this space corresponds to selecting an alternate set of data points. These new data points aren't simply random perturbations from the original model. In the case of tomography, they are similar to not selecting the maximum amplitude of a move-out measure, but a trend of lower or higher move-out. The added complexity is that the migration velocity analysis problem is not linear. We routinely will do several non-linear iterations to come up with the final answer. How to best deal with this non-linearity is unclear.
In this paper I take a slightly different tact from the one taken Clapp (2004). I perform four iterations of non-linear tomography. In the first two iterations I create five equi-probable realizations for the move-out functions. For the last two iterations I choose the minimum-energy move-out function. The resulting twenty-five models provide an interesting and instructive measure of the uncertainty involved in standard migration velocity analysis and its effect on final image.