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

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.