BAYESIAN INFERENCE OF SITE-SPECIFIC PETROPHYSICAL RELATIONS

The growing use of new data sources, such as geophysical tests, for informing hydrologic characterization has brought to the forefront the importance of the development of site-specific petrophysical relations -- or, in other words, empirical formulas that relate measurable geophysical properties (such as seismic velocity or electrical resistivity) to important hydrologic properties such as hydraulic conductivity. There are several interesting facets to this problem. Firstly, absent prior information the petrophysical models must be developed through analyses of sample property correlations, and the form of the underlying relation (e.g. linear, exponential, etc.) must be estimated from data. Secondly, the datapoints used to develop the relation are subject to what we term "omni-directional" noise, i.e. noise in every coordinate axis.

While geophysicists recognize that all sediment properties measured are subject to error, most petrophysical relations to date have been estimated using 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 data under omni-directional noise. 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 curve parameter optimization (i.e. model optimization) and for determining a proper curve model to use (i.e., model selection). We then show that, unlike simpler methods proposed for such problems: 1) the curves obtained through optimization using our objective function are reasonable and do not "over-fit" data; 2) our objective function, which is a probabilistic likelihood measure, can be used to determine whether the data supports an underlying petrophysical relation or whether data variance can be explained exclusively by measurement error; and 3) the developed objective function can be used with probabilistic tests for model comparison, i.e. to gauge whether simpler or more complex relations are justified to fit the data. This method has the interesting feature that it presents a relation between noise magnitude and curve length that is statistically derived and suppresses excursions outside of the region supported by data.