Quantifying and Incorporating Lithological Well Logging as Uncertain Data in Probabilistic Inverse Problems

Abstract Geological models are essential for mapping the accessibility and extent of subsurface resources. Traditionally, these models have been based on deterministic inversion of geophysical data, which provide estimates of geophysical (and not geological) parameters, but fail to capture the full uncertainty. Probabilistic inversion methods have been proposed to address this issue, allowing for, in principle, inference about geological model parameters with detailed uncertainty characterization from geophysical data. In practice, borehole data, often available in the form of lithological descriptions, offer an additional source of information that can further enhance the inference of model parameters. We discuss how to quantify and incorporate uncertainties in lithological borehole data into probabilistic inversion frameworks using a likelihood based on the multinomial distribution. We identify key obstacles in quantifying class probabilities for lithological categories, related to both independent information about all model parameters and information about groups of model parameters. We demonstrate the effect of different strategies for quantifying lithological borehole information. For example, the straightforward assumption of full independence may misrepresent our knowledge and lead to an intractable numerical problem. By explicitly quantifying borehole data, our approach enables integration of borehole data into a probabilistic framework that can be used, for example, in joint probabilistic inversion, with the scope of producing geological models with better description of uncertainties. Lastly, we introduce a flexible method to generate geological prior models and demonstrate how quantified borehole information can be inverted using look-up tables of lithological one-dimensional prior realizations.