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Statistical Bayesian Inversion of Ultra-deep Electromagnetic LWD Data: Trans-dimensional Markov Chain Monte Carlo with Parallel Tempering
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The interpretation of geological features around the borehole focuses on the search of true earth model parameters. Unlike conventional logging-while-drilling (LWD) technologies and wireline induction and propagation resistivity tools, the azimuthal resistivity measurement has directional sensitivity. Over the past decade, azimuthal resistivity tools have been greatly improved with the depth-of-detection (DoD) now exceeding 100 ft. To resolve the interpretation of geological features within a distance of 100 ft from the borehole, the multiple-layer structure must be considered. However, the traditional methods can not handle the highly nonlinear inverse problems, especially those confronting the complicated earth structure. Meanwhile, no justifiable uncertainty of inverse parameters can be provided. Statistical methods have been adopted to search for the global statistical distribution and uncertainties of earth-model parameters. However, most attempts are still stuck in the nonuniqueness of the solutions, since the structure of earth model has been predefined, where the number of layers is assumed, which introduces another ambiguity into the model interpretation. This paper develops a general transdimensional Bayesian methodology to solve nonlinear inverse problem of earth-layer resistivity and depth to the layer boundaries, from the interpretation of ultradeep azimuthal resistivity measurements. We use a data-driven approach to search out the multidimensional parameter spaces. The simulations indicate that the algorithm can efficiently solve the earth-model interpretation within a 100 ft from the borehole, and examine the uncertainty of earth-model parameters. © 2019 Society of Well Log Analystists Inc. All rights reserved.
Non-linear inverse problems in the geosciences often involve
probabilistic sampling of multimodal density functions or global
optimization and sometimes both. Efficient algorithmic tools for
carrying out sampling or optimization in challenging cases are of major
interest. Here results are presented of some numerical experiments with
a technique, known as Parallel Tempering, which originated in the field
of computational statistics but is finding increasing numbers of
applications in fields ranging from Chemical Physics to Astronomy. To
date, experience in use of Parallel Tempering within earth sciences
problems is very limited. In this paper, we describe Parallel Tempering
and compare it to related methods of Simulated Annealing and Simulated
Tempering for optimization and sampling, respectively. A key feature of
Parallel Tempering is that it satisfies the detailed balance condition
required for convergence of Markov chain Monte Carlo (McMC) algorithms
while improving the efficiency of probabilistic sampling. Numerical
results are presented on use of Parallel Tempering for trans-dimensional
inversion of synthetic seismic receiver functions and also the
simultaneous fitting of multiple receiver functions using global
optimization. These suggest that its use can significantly accelerate
sampling algorithms and improve exploration of parameter space in
optimization. Parallel Tempering is a meta-algorithm which may be used
together with many existing McMC sampling and direct search optimization
techniques. It's generality and demonstrated performance suggests that
there is significant potential for applications to both sampling and
optimization problems in the geosciences.
ian hierarchical model. We suppose given a prior p(k) over models k in a countable set K, and for each k, a prior distribution p( k jk) and a likelihood p(Y jk; k ) for the data Y . For de niteness and simplicity of exposition, we suppose that p( k jk) is a density with respect to n k -dimensional Lebesgue measure, and that there are no other parameters, so that where there are parameters common to all models these are subsumed into each k 2 R . Additional parameters, perhaps in additional layers of a hierarchy, are easily dealt with. Note that in this chapter, all probability distributions are proper. The joint posterior p(k; k jY ) = p(k)p( k jk)p(Y jk; k ) 2K )p( 0 jk )p(Y jk 0 )d can always be factorised as p(k; k jY ) = p(kjY )p( k jk; Y ); that is as the product of posterior model probabilities and model-speci c parameter posteriors. This identity is very often the basis for reporting the inference, and in some of the methods ment
Sensitivity study and uncertainty quantification of azimuthal propagation resistivity measurements
- H Wang
- Q Shen
- J Chen
Sensitivity study and uncertainty quantification of azimuthal propagation resistivity measurements
- wang