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One solution to this problem is to introduce a mapping operator that maps from an irregular space to a regular space. This solution holds promise, but the interaction between and becomes confusing.
Another option is to move the data variability problem. As mentioned earlier, our data isn't actually travel time differences, but values calculated by doing SRM. Normally we choose the value corresponding to the maximum semblance at a given location or some smooth version of the maximum Clapp (2003b). The selecting of the value is really where our data uncertainty problem lies. The selection problem has some convenient and some not so convenient properties. On the positive side, we are working with a regular grid and we know that we want some consistency along reflectors. As a result, a steering filter becomes a very obvious choice for our covariance description. On the negative side, the selection problem shares all of the non-linear aspects of the semblance problem Toldi (1985).
To get around these issues I decides to borrow something from both the geostatistics world and the geophysics world. Instead of thinking of the problem in terms of selecting the best value, I am going to think of the problem in terms of selecting a value within a distribution. I am going to construct my distributions in a similar manner to Rothman (1985). Rothman (1985) was trying to solve the non-linear residual statics problem using simulated annealing. He built a distribution based on stack power values from static-shift traces based on the surface locations of the sources and receivers. In this case, my distribution is going to be constructed based on the semblance values at given values.
I do not want the rough solution that () was looking for, instead I am looking for a smooth solution. If I set up the inverse problem
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