Conclusions

The proposed method has shown mixed results. The assumption of local stationarity has far-reaching implications if we estimate the covariance structure and the filter coefficients on an image made up of random features; the method's performance in this case would be similar to fourth order linear interpolation. This happens because the covariance structure of a fairly random image has no directional preference to tune the coefficients, so it takes equal contribution from all directions. On the other hand, if an image has lines dipping in one particular direction, covariance would be lower along the dip than across the dip, and filter coefficients would be tuned accordingly.

Therefore, ideally we would want to divide our image into as many patches as possible and then estimate the covariance structure and filter coefficients of each patch separately, but there are trade-offs. First, the computation cost would go up in direct proportion to the number of patches. More importantly, in small patches the covariance structure is more likely to be affected by local noise rather than global features. In the case of real data examples, where some noise was present, it was observed that use of regularized inversion gave better results.

Another potential application of the covariance-based interpolation filter is to use this filter as a roughener for regularization. This would ensure smoothing not along the cartesian mesh but along edges and discontinuities present in the model, without requiring to explicitly determine them. Though, computing inverse of this filter might not be straight forward. Clapp et al. (1997) discusses the idea of a steering filter where we can steer along the known or estimated geological dips, but with a covariance-based approach, we can achieve the same goal without estimating any dip explicitly, instead using the covariance structure to quantify the same.