Abstract: |
For decades, geostatistics has been invoking the ?maximum spatial entropy? concept, following the intellectual tradition that maximum entropy (MAXENT) models assume less according to Shannon?s interpretation of entropy as an information measure. This paper explores whether or not multi-Gaussian models maximize spatial entropy. Although the answer to this has been assumed to be ?Yes?, there is no specific demonstration of this beyond those that require that the distribution has an analytical mathematical expression. Such demonstrations do not encompass realizations that lack an analytical mathematical expression, the type that arise solely from the application of an algorithm like simulated annealing and whose multivariate distribution function is known only empirically.
If it is possible to produce realizations whose spatial entropy significantly exceeds that of multi-Gaussian models, interesting questions arise. Is MAXENT invoked in geostatistics with a genuine belief in the importance of minimizing added information, or is it invoked as a rationale for multi-Gaussian methods? If spatial entropy can be made even greater by using probability spaces whose distributions lack an analytical expression, should we continue to restrict ourselves to the not-quite-maximum spatial entropy offered by the subset of probability distributions that do have an analytical expression?
With simulated annealing being able to directly maximize any clearly defined metric, this paper uses annealing to explore whether or not a multi-Gaussian model does, indeed, maximize spatial entropy. It begins with an evaluation of what the spatial entropy of a multi-Gaussian model should be, based on a purely analytical calculation. It then confirms that a particular software implementation of sequential Gaussian simulation does, indeed, produce realizations whose spatial entropy conforms to theoretical predictions. The paper then uses simulated annealing to determine whether or not it is possible to produce realizations whose spatial entropy significantly exceeds that of the multi-Gaussian model.
The study establishes that it is possible, in certain situations, to produce realizations whose spatial entropy significantly exceeds that of multi-Gaussian models. It also elucidates the conditions under which non-analytical probability spaces do not offer any distinct benefit, in terms of entropy, that is not already accessible in well-studied and well-understood multi-Gaussian models. |
