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A General Probabilistic Approach for Inference of Gaussian Model Parameters from Noisy Data of Point and Volume Support

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Abstract

Methods that rely on Gaussian statistics require a choice of a mean and covariance to describe a Gaussian probability distribution. This is the case using for example kriging, sequential Gaussian simulation, least-squares collocation, and least-squares-based inversion, to name a few examples. Here, an approach is presented that provides a general description of a likelihood function that describes the probability that a set of, possibly noisy, data of both point and/or volume support is a realization from a Gaussian probability distribution with a specific set of Gaussian model parameters. Using this likelihood function, the problem of inferring the parameters of a Gaussian model is posed as a non-linear inverse problem using a general probabilistic formulation. The solution to the inverse problem is then the a posteriori probability distribution over the parameters describing a Gaussian model, from which a sample can be obtained using, e.g., the extended Metropolis algorithm. This approach allows detailed uncertainty and resolution analysis of the Gaussian model parameters. The method is tested on noisy data of both point and volume support, mimicking data from remote sensing and cross-hole tomography.

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Acknowledgments

We thank Dong E&P for financial support. Matlab code for inferring properties of three-dimensional anisotropic covariance models from noisy linear average data (and data of point support) using the proposed method is available from http://sippi.sourceforge.net/. We thank professor emeritus Carl Christian Tscherning for illuminating discussion about collocated least-squares.

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Authors and Affiliations

  1. Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark

    Thomas Mejer Hansen, Knud Skou Cordua & Klaus Mosegaard

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  1. Thomas Mejer Hansen
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  2. Knud Skou Cordua
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  3. Klaus Mosegaard
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Correspondence to Thomas Mejer Hansen.

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Hansen, T.M., Cordua, K.S. & Mosegaard, K. A General Probabilistic Approach for Inference of Gaussian Model Parameters from Noisy Data of Point and Volume Support. Math Geosci 47, 843–865 (2015). https://doi.org/10.1007/s11004-014-9567-5

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  • DOI: https://doi.org/10.1007/s11004-014-9567-5

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