Fitting Data Under Omnidirectional Noise: A Probabilistic Method for Inferring Petrophysical and Hydrologic Relations

While geophysicists recognize that all measurements of media properties are subject to noise, methods for fitting petrophysical relations commonly employ regression-based formulas, which assume no error in one of the properties being related (the "independent" variable). To derive a more rigorous method for fitting such relations, we take a probabilistic viewpoint on the problem of fitting petrophysical relations to sample correlation data. Unlike prior approaches, we take into account the fact that noise is present in both measured properties. Under basic assumptions, we derive a new objective function for such problems which is proportional to the data likelihood and which can be used for both model parameter optimization and model selection. We present several numerical experiments outlining the utility of our method, and compare results of our method against results of other commonly used methods, such as kriging, regression, and total distance minimization. The results of our applications using the maximum likelihood technique appear visually accurate, and we also provide quantitative comparisons of performance that suggest the method produces more desirable results.