The ultimate goal of nearly all geophysical methods is to obtain an accurate model of one or more measurable subsurface properties. Similarly, the goal of this paper is to create a model of a given geological stratum, or horizon, in depth as a function of the spatial coordinates, x and y. This model should be well resolved both vertically and horizontally, densely map a wide area, and of course be obtained cheaply from a computational standpoint. Let us see if any of the current methods at our disposal fit the requirements of this problem.
Singularly, none of these methods meets all the criteria of the ``ideal'' method. If we are to make the most accurate map possible, we must master the art of compromise and design an intelligent hybrid method. Seismic data, though poorly resolved vertically, does a decent job of delineating general horizontal trends over large areas. Well logs provide accurate vertical measurements, but little to no information on horizontal trends and are prohibitively expensive. The hybrid method must account for these strengths and weaknesses.
From the oil industry's perspective, this problem has traditionally been of great importance, since the first source of information on any oilfield is usually seismic data. Seismic data is used to create the first velocity model, upon which many future operations in depth are strongly dependant (van Riel Paul and Mesday (1988), van der Made P. et al. (1990)). Therefore, any procedure which estimates relative seismic misfit should by definition estimate the corresponding error in the velocity model.
The problem of making an accurate depth map of subsurface geology is not unique to exploration geophysics. Hydrogeologists and petroleum engineers interested in reservoir monitoring obtain optimal models by minimizing the error between some unbiased linear estimate of the output model and some known data, a technique known as kriging. Kitanidis (1997)
Previous SEP approaches to solve the same problem have been expressed similarly as ``missing data'' problems: i.e., given the known values of the output model at some points, perform a least squares inversion, subject to other model constraints, to obtain the value of the output model over all space (Claerbout (1997b), Berlioux (1995)). My methodology does not deviate from this tradition, though I make use of some newly developed techniques and tools to obtain a solution.
Assume that in depth, the actual horizon is mathematically representable by a surface, which in general can be discontinuous, multivalued, or both. Compute an approximation to this horizon by picking from seismic data in time the reflection event corresponding to the actual horizon, and then convert it to depth. I iteratively solve a least squares optimization problem to calculate a model which is constrained to match the well log measurements at the well locations, but assumes the general shape of the picked seismic horizon elsewhere - an approach that honors each type of data in its regions of maximum statistical reliability.
The idea of optimized horizon refinement lends itself well to almost any recursive ``layer-stripping'' updating algorithm. As potential future areas of reseach, I discuss application of these ideas to the problems of anisotropic parameter estimation and global velocity update.