Abstract
The geological model plays an important role in geophysics and engineering geology. The data source of geological modeling comes from interpretation data, borehole data, and outcrop data. Due to economic and technical limitations, it is impossible to obtain highly accurate and highdensity data sources. The sparsity and inaccuracy of data sources lead to the uncertainty in geological models. Unlike the problem of probability, there is not enough samples for a geological model. Spatial diffusion model and merging model are introduced, which are more satisfied with the cognition of uncertainty than the existing methods. And then, using conditional information entropy, a quantification method of geological uncertainty, is proposed. Compared with the approaches of information entropy, this method took full account of the constraints of geological laws. Based on the uncertainty models and conditional information entropy, a framework of uncertainty assessment in geological models is established. It is not necessary in our framework to create multiple geological models, which is a timeconsuming and laborious task. The application of Hashan survey located at north of China shows that the method and framework of this study are reasonable and effective.
1 Introduction
The uncertainty of geological models is widely recognized as an important issue [1,2,3,4 5,6,7,8,9,10,11,12,13]. Due to the lack of mathematical systematization of uncertainty theory, it is difficult to quantitatively describe and universally interpret the uncertainty in geological models.
First, different geological models can be established by different geological modeling methods and different modeling parameters. Due to the lack of actual geological models, it is difficult to evaluate the accuracy and the error between the constructed model and the real ones.
Next obtaining dense and highprecision seismic data is costly and often impossible. The basic datasets of geological modeling are outcrop data, remotely sensed seismic imagery, field data, and borehole data. Though the outcrop data and borehole data can be measured by precision instruments, it is too sparse for geological modeling. The field data are denser but the accuracy is very low because of low resolution and unreliable velocity model [14].
Finally, the geological model is composed of multiple geological blocks based on the geological events (stratum deposition, tectonic movement, and fault events). The subsurfaces (horizons or faults) are the boundaries of the geological blocks. The outcrop data, structure interpretation data, and borehole data describe the geometry form of subsurfaces. The uncertainty of data source leads to the uncertainty of geological block boundaries. And then the uncertainty of boundaries cause the uncertainty of geological model.
Therefore, the uncertainty problem in geological modeling has three aspects.
1.1 The description of data source uncertainty
Similar to random variable, uncertain variable can be used to describe the uncertainty of data sources [15]. The uncertainty distribution function and the uncertainty density function are the basic mathematical model of uncertainty variable. Lindsay et al. [16] explored the broader variations in geological uncertainty, introducing the concept of geodiversity analysis. The data point is perturbed in the range. Previous studies [8,9] used dip and slip to describe the geological features of faults. The uncertainty of fault is shown as the uncertainty of dip and slip. From the aspect of uncertainty of fault network, Aydin et al. [17] proposed a method of quantifying structural uncertainty on fault networks using a marked point process within a Bayesian framework.
1.2 The spatial diffusing and merging of uncertainty
Though Bond [14] tried to establish the basic uncertainty theory from the aspect of uncertainty measure, the character of uncertainty spatial diffusion is ignored. Probability theory is suitable for the problem of multiple samples. Unfortunately, there is only one sample or few samples for the problem of uncertainty.
The geological model has spatial relation. The uncertainty originates from the uncertainty data source. The uncertainty will spread from the uncertainty data source to the whole space. Geological blocks have corresponding spatial coordinates, and the position relation of geological blocks reflects the process of sedimentation. Up to now, there is no research on the problem of uncertainty of the spatial diffusion. The perturbation of geological model is used widely in the research on the uncertainty of geological model [18] and the uncertainty of topology [9]. The process of uncertainty diffusion is too simplified, which leads to the fact that uncertainty distribution of geological model is not reasonable.
1.3 The quantification of geological model uncertainty
Olierook et al. [1] proposed the Bayesian geological and geophysical data fusion for the construction and uncertainty quantification of 3D geological models. But this method validated in a datarich area and it is not effective in the case of limited data.
In essence, uncertainty has no specific physical meaning. It is difficult to quantify the uncertainty. Wellmann and RegenauerLieb [18] proposed an approach of uncertainty quantification of geological model based on the information entropy. Jzza et al. [2] proposed an improved coupled Markov chain method for simulating geological uncertainty. Optimization for borehole location and number is presented based on information entropy. Shannon entropy [19] was defined for probability distributions and then its usage was expanded to measure the uncertainty of knowledge for systems with complete information. The expressions of uncertainty and information entropy are similar. Information entropy represents the amount of information, and the value range is from 0 to 1. The information entropy being equal to 0 shows that there is no information. And the bigger the information entropy, the more information the data gives. Similarly, the higher the uncertainty, the bigger the information entropy. The approach of quantification of uncertainty based on information entropy is applied widely in different fields [11,20–24].
Summarily, the problem of geological model uncertainty was studied widely from the mathematical physical model of uncertainty variable, the perturbation geological model based on Monte Carlo simulation and the quantification of geological model uncertainty. But the spatial diffusion of uncertainty has not been discussed in the above approaches.
This study aims to develop an improved framework of uncertainty assessment of complex geological models under limited data. The main improvements and innovations are as follows:
A mathematical model based on uncertainty spatial diffusion is presented. The model contains two aspects: the uncertainty diffusing model and uncertainty merging model. With the distance from the data source getting farther and farther, the uncertainty will become higher and higher. And then, the sparser the data source, the higher the uncertainty. The results are coincident with the cognition of uncertainty spatial diffusing.
An uncertainty quantification approach of geological models based on conditional information entropy is proposed. The information entropy is a valid method to quantify the uncertainty of geological model. With the constraints of geological laws (i.e., sedimentary sequence, erosion relation, fault relation), the conditional information entropy is more suitable to quantify the uncertainty of complex geological models. The rationality of geological model is translated into conditional probability. The result of uncertainty quantification is more reasonable for geological model.
Systematic framework and process of uncertainty analysis in geological model are established. As we know, geological modeling is a timeconsuming and laborious task. Comparing with Monte Carlo simulation based on random perturbation, it is not necessary to establish multiple geological models with this method. Especially, the geological laws (i.e., sedimentary sequence, erosion relation, fault relation) can be integrated into the process of uncertainty analysis. So, this presented framework is reasonable and feasible to assess the uncertainty of geological model.
The arrangement of this article is as follows. In Section II, we introduce the methods. We discuss the uncertain theory at first. Second, we establish the two physical models of uncertainty (spatial diffusion model and merging model). Third, an uncertainty quantification approach of geological models based on conditional information entropy is proposed and discussed. In Section III, we apply our framework and methods to assess the uncertainty of geological model of a survey located at north of China. In Section IV, we draw a conclusion of this study and discuss the future research.
2 Methods
2.1 Uncertain theory
Uncertain variable is a fundamental conception in uncertainty theory, it is used to represent quantities with uncertainty. Roughly speaking, an uncertain variable is a measurable function on an uncertainty space. A formal definition is given in the previous study [15], where µ is the expected value and δ ^{2} is the variance of uncertain variable.
The variance of uncertain variable provides the degree of the spread of the distribution around its expected value. A small value of variance indicates that the uncertain variable is tightly concentrated around its expected value, and a large value of variance indicates that the uncertain variable has a widespread around its expected value. That is to say, the value of variance indicates the degree of the uncertainty of an uncertain variable. A small value of variance indicates that the uncertain variable has low uncertainty, and a large value of variance indicates that the uncertain variable has high uncertainty. Especially, δ ^{2} = 0 indicates that the variable is certain and without uncertainty.
2.2 Physical model of uncertainty diffusion
Assume that there is only one point (borehole) for a subsurface. Now the uncertainty of the surface will be discussed. As shown in Figure 1, let P(x, y, z) represent the data point,
where Px and Py are the world coordinates, Pz (depth or time) is an uncertain variable. Q(x, y, z) is the other point with world coordinates Qx and Qy. The problem of uncertainty diffusion is that how the uncertainty propagates from P(x, y, z) to point Q(x, y, z). That is to say, the uncertainty diffusion model is represented as the relationship between the uncertainty of P and Q.
Theorem of uncertainty spatial diffusion: Assume that the data source with spatial coordinate has a normal distribution N(µ, δ ^{2}), the uncertainty unit distance far away from the data source has a normal distribution N(µ,(√2δ)^{2}).
Proof
As shown in Figure 2, S(x, y) is the uncertainty source point with uncertain depth z, P(x, y) is the unit distance far from S. Because the depth of P is affected by S, Pz is also uncertain. Let uncertain variable ξ represent the uncertainty of S and ζ represents the uncertainty of P. For each possible value ξi of ξ, where ξi is a certain variable, ζi is the uncertain variable after unit distance diffusion, and the expected value of ζi is equal to ξi, and the variance of ζi is equal to the uncertain variance of ξ.
The uncertainty with discrete distribution: let uncertain variable x = {x1, x2,… xn} and
The uncertainty with continuous distribution is given as follows:
If the uncertainty variable x has a normal uncertainty distribution N(µ, δ), Φ(y) is represented as follows:
We can find that Φ(y) is a normal distribution N(µ, (√2δ)^{2}). That is to say, the expected values of Q and P are equal, and the variance of Q is equal to (√2δ)^{2}. The theorem of uncertainty spatial diffusion is proof. Assume P and Q are two points in space. P is the uncertainty source point with normal distribution N(µ, δ ^{2}). D is the distance between P and Q. Then, the uncertainty of Q also has normal distribution, and the uncertainty diffusion model is as follows:
where d is the unit distance.
According to formula (4), the uncertainty variance is increasing far from the uncertainty data source. Simulation result is shown in Figure 3.
The distance varies from 0 to 100, and the uncertainty variance becomes bigger and bigger. d is the unit distance, and with the increase in d, the increase in the velocity of variance becomes smaller and smaller. That is, d can be considered as a scale factor.
Scale factor reflects the intensity of uncertainty diffusion. The larger the intensity of uncertainty diffusion, the larger the region controlled by uncertainty data source. If the subsurface has flat structure, d should be a bigger value. Otherwise, if the subsurface has steep structure, d should be a smaller value. The simulation result is satisfied with the axiom of uncertainty diffusion.
2.3 The physical model of uncertainty merging
When we discuss the uncertainty of geological modeling, we must consider the influence of all uncertainty data sources. The data source of geological modeling comes from outcrop data, geological interpretation data, and borehole data.
At first, we hypothesize a simple model with two uncertainty data sources. The model is shown in Figure 4.
The uncertainty of two data sources, named S1 and S2, respectively are independent and identically distributed. The uncertainty distribution function Φ is normal distribution N(µ, δ2). Now, we discuss the uncertainty distribution of point P. Let d1 be the distance from S1 to P and d2 be the distance from S2 to P.
According to the physical model of uncertainty diffusing in previous section, the uncertainty of S1 diffusing from S1 to P is normal with uncertainty distribution N(µ, δ _{1} ^{2}). Similarly, the uncertainty of S2 diffusing from S2 to P is normal with uncertainty distribution N(µ, δ _{2} ^{2}). Let unit distance be d, then we can achieve the following relationships:
Now, the problem is that how to merge the uncertainty of δ _{1} ^{2} and δ _{2} ^{2}. Then, we establish the following relationships:
where the parameter p is the influence factor of distance. The formulae (5) and (6) are the mathematical model of uncertainty merging with multiple data sources.
We hypothesize that there are two uncertainty data sources located at 0 and 100, respectively. The uncertainty of interspace is affected by the two uncertainty data sources. Figure 5 shows the simulation result with different values of d.
Like the uncertainty diffusion model, with the increase in the value of d, the increase in the velocity of variance becomes smaller and smaller. Figure 6 shows the simulation result with different values of p.
From formula (6), the parameter p is to modify the weight of different uncertainty data sources. That is, p can be considered as a merging factor.
Merging factor reflects the influence degree of different uncertainty data sources with different distances. The larger the merger factor p, the smaller the influence of farther data source. If the structure of the region is simple, p should be a smaller value, and if the structure of region is complex, p should be a bigger value.
According to the uncertainty merging model, the uncertainty distribution of surface with inhomogeneous seed points is shown in Figure 7.
Because more seed points are distributed in the region of lower left, the region of upper right has higher uncertainty, and the region of lower left has lower uncertainty. In addition, the region far away from the seed point has higher uncertainty. The simulation result is satisfied with the cognition of uncertainty.
Based on the model of uncertainty diffusion and the model of uncertainty merging, we can obtain the uncertainty distribution of subsurface. It lays a foundation for analyzing the uncertainty of structural models.
2.4 Uncertainty quantification and information entropy
The quantification of uncertainty is basic for quantitative analysis. Wellmann and RegenauerLieb [18] proposed an approach to quantify the uncertainty using information entropy. As shown in Figure 8, there are two horizons h1 and h2 in the model. h1 is earlier than h2.
The model space is split into three geological blocks by h1, h2. In order to analyze the uncertainty of geological model, we can split the space of geological model into multiple cells. For example, the red cell may belong to block1, block2, or block3. Let p _{1} be the probability of belonging to block1, p _{2} be the probability of belonging to block2 and p _{3} be the probability of belonging to block3.
According to the approach of uncertainty quantification [15], the entropy of the red cell can be represented as follows:
So, the key problem is how to compute the probabilities p _{1}, p _{2}, and p _{3}. The approach of Monte Carlo simulation can create multiple geological models by randomly perturbing horizon data. So, we can get the probability easily. Because of the shortcoming of Monte Carlo simulation, we propose a new method based on the model of uncertainty diffusion and uncertainty merging.
First, we can get the uncertainty distribution of B and C based on the model of uncertainty. If the uncertainty of data source has normal distribution, we can let the uncertainty distribution of B and C be N(µ _{ A }, δ _{ A } ^{2}) and N(µ _{ B }, δ _{ B } ^{2}). We can get the following relationships:
Given the distribution of B and C, we can calculate the probabilities p _{1}, p _{2}, and p _{3} with the condition of independent distribution.
2.5 Geological laws and conditional entropy
A rational geological model must be satisfied with the geological laws. The information entropy model is too simplified to quantify the uncertainty of geological model. Conditional entropy [25] is more suitable for quantifying the uncertainty of geological model than information entropy. As shown in Figure 8, the geological laws of the simple model are as follows:
With the condition of without outcrop data, the subsurfaces h1 and h2 must be deeper than ground surface. Let the depth of ground surface equal to 0. The uncertainty of geological model must be constrained by P{B > 0, C > 0}.
The depth values of h1 and h2 cannot be equal to infinity. We can limit the maximum depth of h1 and h2. The uncertainty of geological model must be constrained by P{B < maxd, C < maxd}, where maxd is the maximum depth of the model.
We may know that h1 is earlier than h2 according to horizon calibration. So the uncertainty of geological model must be constrained by P{B < C}.
Integrating the above geological laws, the probabilities p _{1}, p _{2}, and p _{3} must be constrained by the following condition.
where X = P{0 < B < maxd, 0 < C < maxd, B < C}.
The approach of uncertainty quantification using conditional entropy integrates the geological laws. It effectively avoids the difficulties caused by the rationality of the geological model of the Monte Carlo simulation.
3 Results
We apply our methods to assess the uncertainty of geological model of a survey, which is located north of China.
As shown in Figure 9, the nappe is poorly stratified (or complicated by faults). The carboniferous system is repeatedly pushed over. Due to the complexity of geological structure and lithology, the quality of seismic imaging is very poor. The interpretation data have high uncertainty. In addition, the reliability of velocity model is very low.
The bottom map is shown as Figure 10. The boreholes are concentrated on a small region southern of survey, and relatively shallow. So the borehole data cannot control the whole geological model. the geological model has high uncertainty.
3.1 Preprocessing of interpretation data
The geologist interpreted six horizons and nine faults in the survey. As shown in Figure 9, the survey is split into 16 geological blocks by the 15 geological subsurfaces. The interpretation data are shown in Figure 11.
The task of preprocessing of interpretation data includes two aspects:
Checking the rationality and conflict of interpretation data. The interpretation data has a great impact on the mean model. We try to improve the rationality of interpretation data. Especially, we must handle the conflict of interpretation.
Evaluating the uncertainty of interpretation data. The interpretation data are the uncertainty source. We must initialize the variance of each interpretation data based on the skill and cognition of geologists. In order to simplify the process, we hypothesize that the variance of interpretation data is equal.
3.2 Mean model and variance model
The process of creating mean model is to reconstruct all the subsurfaces. As shown in Figure 12, we reconstructed 15 geological subsurfaces with the approach of Kriging [26]. Because of the lack of certain cross lines, we reconstructed each geological surface only with the constraints of range of survey. So, each surface covers the whole survey. Similarly, the variance model can be created based on the uncertainty diffusion model and the uncertainty merging model. Here we set the initial variance of each interpretation data δ ^{2} = 20, the scale factor d = 3,500, and the merging factor p = 4. The variance model is shown in Figure 13.
3.3 Geological laws and conditional information entropy
According to the formula (9), the information entropy model must be satisfied with the constraints of geological structural rationality. So, the uncertainty information entropy is a conditional information entropy. It is difficult to calculate the conditional probability from the aspect of mathematics [27,28]. From Figure 14, we can see six horizons (named K, J, T, p2w, p2x, and p1f, respectively) and nine faults (named F2, F3, F4, F5, F6, F3_{2}, F4_{1}, F6_{1}, and F6_{2}, respectively). With the geological processes of fault, erosion, nappe, etc., we can distinguish 16 geological blocks (named B1, B2,…, B16). So, each cell of geological model may belong to each of the blocks. The probabilities are marked as p _{1}, p _{2},…, p _{16}, respectively.
According to the process of sedimentary and geological action, the horizon K splits the model space into two parts: B_{1} ^{(K)} and B_{2} ^{(K)}, the former is B_{1} and the latter is split by other subsurfaces. Then, horizon J splits B_{2} ^{(K)} into two parts: B_{1} ^{(J)} and B_{2} ^{(J)}, the former is B_{2}. So, we can achieve the result as shown in Figure 15. We can calculate the probabilities p _{1}, p _{2},…, p _{16} and then we can get the conditional information entropy.
3.4 Visual analysis of uncertainty
The uncertainty of geological model can be quantified by means of conditional information entropy. The conditional information entropy is a 3D data field. Each data point represents a cell. The value of each point is the conditional information entropy. The larger the value, the higher the uncertainty. The software Voreen is a smart visualization platform [29]. Figure 16 is the result of visualization.
We can discover the high uncertainty region with the help of adjusting the visual parameters through human–machine interaction.
4 Conclusion
This work proposes an improved method and framework to quantify the uncertainty of geological model based on limited data. Based on the results obtained, the following conclusions can be drawn:
The diffusion model and the merging model of uncertainty abiding by the cognition of uncertainty (two axioms) are presented in this work. And then the rationality of the two models is analyzed. Compared with the existing methods (such as Monte Carlo simulation, Bayesian fusion, the coupled Markov chain, etc.), the two models are more satisfied with the uncertainty cognition of geological model under limited data.
Information entropy is a smart tool to quantify the uncertainty of geological model. But the geological model must be satisfied with the constraints of geological laws. This study proposes an uncertainty quantification approach of geological models based on conditional information entropy. Conditional information entropy takes full account of sedimentary relations and structural constraints. So, the method of uncertainty quantification is more reasonable than existing technology.
According to the uncertainty models and quantification methods of uncertainty, a framework of uncertainty assessment in geological model is presented. Unlike other methods [30–32] (such as Monte Carlo simulation, etc.), it is not necessary to create multiple geological models which is a timeconsuming and laborious task. Especially, it is difficult to take into account the geological laws. Finally, this study carried out an application of Hashan survey located north of China. The results further show high accuracy and stability of this method.
It should be noted that the research of geological uncertainty is inadequate. This study just discusses the distribution and diffusion model of geological modeling uncertainty. Some special complex uncertainty in geological models still depends on the experience of geologists. In future, we will continue to study the uncertainty of geological modeling, especially on uncertain inversion of geological parameter, uncertain reservoir prediction. etc.

Funding information: This study is financially supported by the National Science Technology Project (No. 41974147) and Guoteng Venture Capital Fund (GTKY05) of CDUESTC.

Author contributions: All authors contributed to the study’s conception and design. material preparation and data collection: Kai Luo; original draft and analysis: Cai Lu. The manuscript was written and reviewed by Xiaoyan Nie. All authors read and approved the final manuscript.

Conflict of interest: Authors state no conflict of interest in this article.
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