Characterization of geological uncertainties from limited boreholes using copula-based coupled Markov chains for underground construction

Abstract

This paper proposes an efficient method for quantifying the stratigraphic uncertainties and modeling the geological formations based on boreholes. Two Markov chains are used to describe the soil transitions along different directions, and the transition probability matrices (TPMs) of the Markov chains are analytically expressed by copulas. This copula expression is efficient since it can represent a large TPM by a few unknown parameters. Due to the analytical expression of the TPMs, the likelihood function of the Markov chain model is given in an explicit form. The estimation of the TPMs is then re-casted as a multi-objective constrained optimization problem that aims to maximize the likelihoods of two independent Markov chains subject to a set of parameter constraints. Unlike the method which determines the TPMs by counting the number of transitions between soil types, the proposed method is more statistically sound. Moreover, a random path sampling method is presented to avoid the directional effect problem in simulations. The soil type at a location is inferred from its nearest known neighbors along the cardinal directions. A general form of the conditional probability, based on Pickard's theorem and Bayes rule, is presented for the soil type generation. The proposed stratigraphic characterization and simulation method is applied to real borehole data collected from a construction site in Wuhan, China. It is illustrated that the proposed method is accurate in prediction and does not show an inclination during simulation.

Keywords

Geological stratigraphy / Markov chain / Transition probability matrix / Statistical method / Copula

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The study is supported by the National Natural Science Foundation of China (Grant Nos. 71732001 and 52192661), National Key Research & Development Program, China (Grant No. 2021YFF0501001), Shenzhen-Hong Kong-Macau S&T Program (Category C) (Grant No. SGDX20201103095203031) and the Fundamental Research Funds for the Central Universities (Grant No. 2021XXJS079). Moreover, financial sponsorship from Research & Development Program of Shanghai Tunnel Engineering Co. Ltd and Wuhan Metro Group Co. Ltd are also sincerely acknowledged.

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