When solving a missing data problem, geophysicists and geostatisticians have very similar strategies. Each use the known data to characterize the model's covariance. At SEP we often characterize the covariance through Prediction Error Filters (PEFs) Claerbout (1999). Geostatisticians build variograms from the known data to represent the model's covariance Isaaks and Srivastava (1989). Once each has some measure of the model covariance they attempt to fill in the missing data. Here their goals slightly diverge. The geophysicist solves a global estimation problem and attempts to create a model whose covariance is equivalent to the covariance of the known data. The geostatistician performs kriging, solving a series of local estimation problem. Each model estimate is the linear combination of nearby data points that best fits their predetermined covariance estimate. Both of these approaches are in some ways exactly what we want: given a problem give me `the answer'.
The single solution approach however has a couple significant drawbacks. First, the solution tends to have low spatial frequency. Second, it does not provide information on model variability or provide error bars on our model estimate. Geostatisticians have these abilities in their repertoire through what they refer to as `multiple realizations' or `stochastic simulations'. They introduce a random component, based on properties (such as variance) of the data, to their estimation procedure. Each realization's frequency content is more accurate and by comparing and contrasting the equiprobable realizations, model variability can be assessed.
In this paper I present a method to achieve the same goal using a formulation that better fits into geophysical techniques. I modify the model styling goal, replacing the zero vector with a random vector. I show how the resulting models have a more pleasing texture and can provide information on variability.