Markov based transition probability geostatistics in groundwater applications: assumptions and limitations

Abstract

Markov based transition probability geostatistics (MTPG) for categorical variables, as implemented by the methodological framework introduced by Carle and Fogg (Math Geol 29(7):891–918, 1997) and extended thereafter, have been extensively applied for the three-dimensional (3D) statistical representation of hydrofacies in real-world aquifers, and the conditional simulation of 3D lithologies for groundwater flow and transport simulations. While conceptually simple and easy to implement, conditional simulation using the MTPG approach is not limitation free. However, to the best of our knowledge, there is no study that raises such concerns in the light of theoretical arguments and numerical findings. That said, the purpose of this study is twofold: (1) present a brief and coherent overview of the basic theory, fundamental assumptions, and limitations of the MTPG methodological framework, and (2) assess its capabilities on the basis of a simple two-dimensional test-case, using large ensembles of stochastic realizations. Contrary to real-world 3D aquifers, where the actual geology is unknown, and the quality of the simulations can be assessed solely on the basis of semi-quantitative arguments using properly selected sets of stochastic realizations, test-cases allow for direct quantitative assessments based on the application of statistical measures to large ensembles of synthetic realizations. Our analysis and obtained results show that stochastic modeling of actual geologies using the MTPG approach of Carle and Fogg (1997), is characterized by simplifying assumptions and theoretical limitations, with the simulated random fields exhibiting statistical structures that strongly depend on the problem under consideration and the modeling assumptions made, leading to increased epistemic uncertainties in the obtained results.

Appendix 1: Spatially averaged statistical measure to assess the performance of conditional simulations of categorical random fields

Suppose a study region D formed by two mutually exclusive and collectively exhaustive geologic categories/states: A and A^c (i.e. the complement of A). Note that for the case of m > 2 mutually exclusive and collectively exhaustive geologic states in D, A is set to a certain category, and A^c to the union of all states other than A. Define now A_act(x) to be the event that material/state A actually occurs at location x ∈ D, and A_sim(x) to be the event that state A is simulated at x ∈ D. Since events A_act(x) and A$_{\text{act}}^{\text{c}}$(x), as well as A_sim(x) and A$_{\text{sim}}^{\text{c}}$(x) are mutually exclusive and collectively exhaustive, one can calculate the total probability of error, p _error (x), at any location x∈D of the conditionally simulated field as:

$$\begin{aligned} p_{error} \left( {\mathbf{x}} \right) \, & = P[\{ {\text{A}}_{\text{sim}}^{\text{c}} \left( {\mathbf{x}} \right) \cap {\text{A}}_{\text{act}} \left( {\mathbf{x}} \right)\} \cup \{ {\text{ A}}_{\text{sim}} \left( {\mathbf{x}} \right) \cap {\text{A}}_{\text{act}}^{\text{c}} \left( {\mathbf{x}} \right)\} ] \\ & = P[{\text{A}}_{\text{sim}}^{\text{c}} \left( {\mathbf{x}} \right) \cap {\text{A}}_{\text{act}} \left( {\mathbf{x}} \right)\left] { + P} \right[{\text{A}}_{\text{sim}} \left( {\mathbf{x}} \right) \cap {\text{A}}_{\text{act}}^{\text{c}} \left( {\mathbf{x}} \right)] \\ & = P[{\text{A}}_{\text{sim}}^{\text{c}} \left( {\mathbf{x}} \right)\left| {{\text{ A}}_{\text{act}} \left( {\mathbf{x}} \right)\left] P \right[{\text{A}}_{\text{act}} \left( {\mathbf{x}} \right)\left] { \, + P} \right[{\text{A}}_{\text{sim}} \left( {\mathbf{x}} \right)} \right|{\text{ A}}_{\text{act}}^{\text{c}} \left( {\mathbf{x}} \right)\left] { P} \right[{\text{A}}_{\text{act}}^{\text{c}} \left( {\mathbf{x}} \right)] \\ \end{aligned}$$

(A.1)

Note now that since either state A_act(x) and A$_{\text{act}}^{\text{c}}$(x) occur with probability 1 at any location x∈D, Eq. (A.1) can also be written in the form:

$$p_{error} \left( {\mathbf{x}} \right) \, = \left\{ {\begin{array}{ll} { 1 { - }q ({\mathbf{x}} ),\quad {\text{if A}}_{\text{act}} {\text{ occurs at }}{\mathbf{x}} \in {\mathbf{D}},} \hfill \\ { \, q ({\mathbf{x}} ) { },\quad {\text{if A}}_{\text{act}}^{\text{c}} {\text{ occurs at }}{\mathbf{x}} \in {\mathbf{D}},} \hfill \\ \end{array} } \right.$$

(A.2)

where q(x) : = P[A_sim(x)] corresponds to the probability that material A is simulated at location x, and can be directly estimated from an ensemble of numerical simulations conditioned on borehole data.

One can now use equation (A.2) to assess the performance of the conditionally simulated fields, by defining the average measure:

$$M = \, 1 - \, 2\bar{p}_{error}$$

(A.3)

where the over-bar denotes spatial averaging of the measure in Eq. (A.2) over the whole study domain D, and 2 is a normalization factor so that measure M varies in the range [0, 1]; i.e. in the case when simulations are independent from the data that are conditioned on, $\bar{p}$ _error  = 1/2 and M = 0. Clearly, for values of M close to 1, the ensemble of conditional simulations closely approximates the actual geology of the region.