A Segmentation Approach for Stochastic Geological Modeling Using Hidden Markov Random Fields

Abstract

Stochastic modeling methods and uncertainty quantification are important tools for gaining insight into the geological variability of subsurface structures. Previous attempts at geologic inversion and interpretation can be broadly categorized into geostatistics and process-based modeling. The choice of a suitable modeling technique directly depends on the modeling applications and the available input data. Modern geophysical techniques provide us with regional data sets in two- or three-dimensional spaces with high resolution either directly from sensors or indirectly from geophysical inversion. Existing methods suffer certain drawbacks in producing accurate and precise (with quantified uncertainty) geological models using these data sets. In this work, a stochastic modeling framework is proposed to extract the subsurface heterogeneity from multiple and complementary types of data. Subsurface heterogeneity is considered as the “hidden link” between multiple spatial data sets. Hidden Markov random field models are employed to perform three-dimensional segmentation, which is the representation of the “hidden link”. Finite Gaussian mixture models are adopted to characterize the statistical parameters of multiple data sets. The uncertainties are simulated via a Gibbs sampling process within a Bayesian inference framework. The proposed modeling method is validated and is demonstrated using numerical examples. It is shown that the proposed stochastic modeling framework is a promising tool for three-dimensional segmentation in the field of geological modeling and geophysics.

Appendices

Appendix A: MRF Energy and Likelihood Energy

According to Bayesian theory

$$\begin{aligned} p(\mathbf{x}|\mathbf{y},\Phi )\propto p(\mathbf{x})p(\mathbf{y}|\mathbf{x},\Phi )\propto \exp (-U(\mathbf{x})+\sum _{j\in V} {\log f_{x_j } (y_j ;\theta _{x_j } )} ). \end{aligned}$$

(26)

Although it is not possible to sample a posteriori realizations of x according to Eq. (26) directly, but one may note that the conditional random field $p(\mathbf{x}|\mathbf{y},\Phi )$ is still a Gibbs field if one substitutes $U^{\prime }(\mathbf{x})=U(\mathbf{x})-\sum \nolimits _{j\in V} {\log f_{x_j } (y_j ;\theta _{x_j } )}$ into Eq. (26). Assuming the emission distribution is Gaussian (i.e., $\theta _{x_j } =(\mu _{x_j },\Sigma _{x_j } ))$, the corresponding local conditional distribution is

$$\begin{aligned}&p(x_j |y_j,\mathbf{x}_{\partial _j },\theta _{x_j } )\propto \exp \left[ -U(x_j,\mathbf{x}_{\partial _j } )-\frac{1}{2}(x_j -\mu _{x_j } )^{T}\Sigma _{x_j } ^{-1}(x_j -\mu _{x_j } )\right. \nonumber \\&\left. \quad -\frac{1}{2}\log \left| {\Sigma _{x_j }} \right| \right] \end{aligned}$$

(27)

which can be rewritten as

$$\begin{aligned} p(x_j |y_j,\mathbf{x}_{\partial _j },\theta _{x_j } )\propto \exp [-(U(x_j,\mathbf{x}_{\partial _j } )+U(y_j |x_j,\theta _{x_j } ))] \end{aligned}$$

(28)

with the MRF energy $U(x_j,\mathbf{x}_{\partial _j } )$, and the likelihood energy is calculated as follows

$$\begin{aligned} U(y_j |x_j,\theta _{x_j } )=\frac{1}{2}(x_j -\mu _{x_j } )^{T}\Sigma _{x_j } ^{-1}(x_j -\mu _{x_j } )+\frac{1}{2}\log \left| {\Sigma _{x_j }} \right| . \end{aligned}$$

(29)

Appendix B: Chromatic Sampler

The chromatic sampler applies a classic graph coloring technique to parallel job scheduling, so that a direct parallelization of the sequential scan Gibbs sampler can be achieved. To be more specific, the entire set of voxels is decomposed into k subsets such that adjacent vertices in the corresponding graph will have different colors. The k-coloring of the MRF ensures that within a certain subset, all vertices are conditionally independent given the configuration of all other vertices in the remaining colors. Therefore, all vertices with the same color can be sampled independently and in parallel. According to Gonzalez et al. (2011), it is guaranteed that given p processors and a k-coloring of an MRF with n vertexes, the parallel chromatic sampler is ergodic and generates a single joint sample in running time: $O(n/p+k)$ which results in a p reduction in the mixing time. Given sufficient parallel resource, the running time is mainly dominated by the number of different colors k. A simple example is provided here: a three-dimensional grid system (Fig. 10) equipped with the neighborhood system defined in Sect. 2 is a graph with 8 colors ($k=8$).

Fig. 10

A graph coloring result of a $5 \times 5 \times 5$ grid system according to the neighborhood system defined in Sect. 2.3 (any two neighboring voxels are assigned with different colors, a total number of 8 colors are used)

Appendix C: Calculating Information Entropy

First, calculate the probability $P_l (i)$ of assigning a certain label $l\in L$ to a given voxel $i\in V$ using the following expression

$$\begin{aligned} P_l (X_i )=\frac{1}{n}\sum _{k=1}^n {I_l \left( x_i^{(k)} \right) }, \end{aligned}$$

(30)

where n is a predefined number of realizations after the burn-in period and $I_l (\cdot )$ is an indicator function which is defined as

$$\begin{aligned} I_l (x_i )=\left\{ {\begin{array}{ll} 1&{}\quad \hbox { for }x_i =l \\ 0&{}\quad \hbox { for x}_\mathrm{i} \ne l \\ \end{array}} \right. . \end{aligned}$$

(31)

Second, for voxel i, the information entropy reads

$$\begin{aligned} H(X_i )=-\sum _{l\in L} {P_l (X_i )\log P_l (X_i )}. \end{aligned}$$

(32)

Based on Eq. (32), the average information entropy for the entire physical domain can be calculated as

$$\begin{aligned} H_T (\mathbf{X})=-\frac{1}{\left| V \right| }\sum _{i\in V} {H(X_i )}, \end{aligned}$$

(33)

where $\left| V \right|$ denotes the cardinality of the set V. The average information entropy is used to quantify the uncertainties of the entire physical domain with a single number.

Appendix D: Geometric Separability Index (GSI)

According to Thornton (1998), the Geometric Separability Index (GSI) for a two-cluster complete data set (i.e., both cluster labels x and observed features y are known) is defined as

$$\begin{aligned} \mathrm{GSI}=\frac{\sum _{i=1}^n {(f(x_i )+f(x_i^{\prime } )+1)\bmod 2}}{n}. \end{aligned}$$

(34)

Here f(.) is a binary target function that returns 0 or 1 according to the input label, $x_i \in L$ is the label at site i. $x_i^{\prime } \in L$ is the label of site i’s nearest neighbor and n is the total number of data points. The nearest neighbor is defined using Euclidean distance in the feature space.

For cases with multiple clusters, simply define the binary target function linked to the specific label $l\in L$

$$\begin{aligned} f_l (x_i )=\left\{ {{\begin{array}{ll} 1&{}\quad x_i =l \\ 0&{}\quad x_i \ne l \\ \end{array}}} \right. . \end{aligned}$$

(35)

Then the average GSI is defined as the algebraic mean of $\mathrm{GSI}_l \hbox { }l\in L$

$$\begin{aligned} \overline{\mathrm{GSI}} =\frac{\sum \nolimits _{l\in L} {\mathrm{GSI}_l }}{\left| L \right| }, \end{aligned}$$

(36)

where $\left| L \right|$ is the cardinality of the label set L. The average GSI is used to provide a measure of the overall separability in a single number.

The average GSI intuitively quantifies the degree to which data points mix together and hence indicates how “difficult” the segmentation problem is. If the centroids of the clusters almost coincide or the observed data points are uniformly distributed in the feature space (i.e., highly overlapped), the GSI will be close to 0.5; in contrast, if there is almost no overlap among clusters, the GSI will be close to 1.0.