Abstract: |
How complex should a spatial or spatial-temporal geostatistical model be in order to suit the purpose for which it will be used? This is a common question to all applications of geostatistical modeling whether it is mining, petroleum, environmental or any other. How many grid-blocks, how many indicator categories should we use? Is a covariance-based model enough or do we need higher order statistics? Surprisingly very few general and flexible tools are available to start addressing this important question. In this paper, we lay out a general framework for determining a suitable spatial model complexity on the basis of a series of 3D model transformations and linear combinations thereof. Our analysis of model complexity is made on the set of geostatistical realizations, or more generally on the posterior pdf, rather than on a single realization. On an initial set of spatially complex realizations, we apply several transformations (or in signal processing language, filters), that map each realization into a new vector space. We then study the variation of these realizations in this vector space and determine the factors (such as variogram range, mean, Boolean parameters, training image parameters) that are most consequential to the variation in this space. Important transformation included in our set are: the spatial entropy transformation, the Hough transformation and the Radon transformation. Each transformation focuses on certain spatial properties often important to real applications, such as spatial randomness, contrast variation, angular and affinity variations of the set of realizations. Next, we evaluate the forward response function, i.e. the function defining the purpose for which the models are used (e.g. mine planning, clean-up operation, production operation), on a few selected complex geostatistical realizations. Using these forward response evaluations we extract by regression in the optimal linear combination of transformations (i.e. the optimal vector space) to use to determine the desired spatial model complexity for that specific purpose. We apply this idea to reduce a large number of lithological categories in a reservoir model to a number suitable for the flow simulation, as well as a case of determining the dimension of the simulation grid in a case of evaluating the spatio-temporal distribution of contaminants. |