Stochastic Systems Group  

A maximum likelihood method for the simultaneous deconvolution and interpolation of geophysical data
Jonathan Kane
Shell E&P
Using geophysical methods to explore for oil requires the fusion of different data sets obtained by different methods at different spatial scales. Combining these data sets for the optimal estimation of geophysical parameters is not trivial. In our application we have a 2D tomographic image of the subsurface along with well data intersecting the same image at sparse locations. The tomographic image is a blurred version of the parameter field being estimated, while the well data are essentially point samples of the same field. The tomographic image covers a large area but is of low resolution; the wells are high resolution, but only provide local information. Optimal estimation of the subsurface parameters requires simultaneous inversion of both data sets.
Three issues must be addressed in order to solve our problem:
1) We require a robust framework that can handle the estimation of
parameters from multiple data sets.
2) In order to use structural information from the tomographic image to
guide the extrapolation of point samples, we require accurate
calculation of spatial gradients of the image.
3) We need an efficient optimization algorithm to minimize our joint
objective function and give a solution.
To address the first issue we propose maximum likelihood estimation. It incorporates any number of data sets in the solution of an estimation problem  along with apriori information. To address the second issue we project our problem into a Bspline basis, which allows for fast and accurate calculation of spatial derivatives. For the third issue we compare two quadratic programming algorithms (conjugate gradients and Jacobi's method) along with 3 different preconditioners. The optimization methods are tested on a simple synthetic 1D problem for optimality, and then applied to the real 2D problem.
Adjustable tradeoff parameters in the objective function allow for 1) simultaneous estimation, 2) "nonlinear diffusion" image deconvolution, or 3) imageguided point sample extrapolation.
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