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Uncertainty assessment of a 3D geological model by integrating data errors, spatial variations and cognition bias

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Abstract

A 3D geological structural model is an approximation of an actual geological phenomenon. Various uncertainty factors in modeling reduce the accuracy of the model; hence, it is necessary to assess the uncertainty of the model. To ensure the credibility of an uncertainty assessment, the comprehensive impacts of multi-source uncertainties should be considered. We propose a method to assess the comprehensive uncertainty of a 3D geological model affected by data errors, spatial variations and cognition bias. Based on Bayesian inference, the proposed method utilizes the established model and geostatistics algorithm to construct a likelihood function of modeler’s empirical knowledge. The uncertainties of data error and spatial variation are integrated into the probability distribution of geological interface with Bayesian Maximum Entropy (BME) method and updated with the likelihood function. According to the contact relationships of the strata, the comprehensive uncertainty of the geological structural model is calculated using the probability distribution of each geological interface. Using this approach, we analyze the comprehensive uncertainty of a 3D geological model of the Huangtupo slope in Badong, Hubei, China. The change in the uncertainty of the model during the integration process and the structure of the spatial distribution of the uncertainty in the geological model are visualized. The application shows the ability of this approach to assess the comprehensive uncertainty of 3D geological models.

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Acknowledgements

This research was supported by the “National Key Research and Development Project (2019YFC0605102)”, a project led by China University of Geosciences (Wuhan).

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  1. School of Geography and Information Engineering, China University of Geosciences, Wuhan, 430074, People’s Republic of China

    Dong Liang, WeiHua Hua, Xiuguo Liu, Yabo Zhao & Zhipeng Liu

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  1. Dong Liang
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Appendix: Combining uncertainties according to stratigraphic relations

Appendix: Combining uncertainties according to stratigraphic relations

The contact relationships of strata are divided into two types: depositional and erosional. By assembling the deposition and erosion interfaces on the basis of stratigraphic sequences, the stratigraphic type probability fields can be consistent with the actual geological structure.

First, we define the occurrence probability of each stratigraphic type. Since the (U, V, Z) coordinate system of different interfaces are not the same, to consider multiple interfaces, the calculation of probability field is in (X, Y, Z) coordinate system. Assuming that at location (x, y, z) in the study area, there are n strata sequentially numbered from bottom to top, the occurrence probability of the ith stratum, Li (i = 1, …n), is P(Li). pi(z) is the PDF of the interface of stratum Li and stratum Li + 1. P(Li) can be calculated using the integral of pi(z):

$$ P\left({L}_i\right)=F\left(x,y,z\right)=\underset{L_i}{\int }{p}_i(z) dz. $$
(A1)

In the case of a depositional contact, the stratigraphic type probabilities are calculated from bottom to top, and the probability fields are updated iteratively on the basis of stratigraphic sequence. The iterations depend on the total number of strata. The iterative process in the case of a depositional contact is:

$$ \Big\{{\displaystyle \begin{array}{c}{P}_N\left({L}_i\right)={P}_{N-1}\left({L}_i\right),i=1,2,\dots, N-2\\ {}{P}_N\left({L}_{N-1}\right)={P}_{N-1}\left({L}_{N-1}\right)\left[1-P\left({L}_N\right)\right]\\ {}{P}_N\left({L}_N\right)=P\left({L}_N\right)-\sum \limits_{j=1}^{N-2}P\left({L}_N\right)\cdot {P}_{N-1}\left({L}_j\right)\end{array}} $$
(A2)

In the formula, Li is the ith stratum from the bottom up. PN(Li) is the conditional probability of the stratigraphic type, Li, at the Nth iteration; P(LN) is the probability of stratigraphic type, LN, without considering the impact of other strata. In the Nth iteration, the probabilities of all the strata under LN in the stratigraphic sequence are considered and updated. The iterative update ends after the iteration N = n.

Faults and discordance interfaces appear as discontinuities in the stratigraphic sequence and can be treated as erosion interfaces. Because erosion does not belong to the depositional process, it needs to be treated separately. When the interface of Li and Li + 1 is an erosional interface, the probability field is updated as follows:

$$ \Big\{{\displaystyle \begin{array}{c}{P}_N\left({L}_i\right)={P}_{N-1}\left({L}_i\right)\left[1-P\left({L}_N\right)\right],i=1,2,\dots, N-1\\ {}{P}_N\left({L}_N\right)=P\left({L}_N\right)\end{array}} $$
(A3)

Based on the principle of mutual exclusion of stratigraphic type and the full probability formula, the cumulative probability of all possible strata at each location is equal to 1.

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Liang, D., Hua, W., Liu, X. et al. Uncertainty assessment of a 3D geological model by integrating data errors, spatial variations and cognition bias. Earth Sci Inform 14, 161–178 (2021). https://doi.org/10.1007/s12145-020-00548-4

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