Multiple-point statistics (MPS) has shown promise in representing complicated subsurface structures. For a practical three-dimensional (3-D) application, however, one of the critical issues is the difficulty in obtaining a credible 3-D training image. However, bidimensional (2-D) training images are often available because established workflows exist to derive 2-D sections from scattered boreholes and/or other samples. In this work, we propose a locality-based MPS approach to reconstruct 3-D geological models on the basis of such 2-D cross sections (3DRCS), making 3-D training images unnecessary. Only several local training subsections closer to the central uninformed node are used in the MPS simulation. The main advantages of this partitioned search strategy are the high computational efficiency and a relaxation of the stationarity assumption. We embed this strategy into a standard MPS framework. Two probability aggregation formulas and their combinations are used to assemble the probability density functions (PDFs) from different subsections. Moreover, a novel strategy is adopted to capture more stable PDFs, where the distances between patterns and flexible neighborhoods are integrated on multiple grids. A series of sensitivity analyses demonstrate the stability of the proposed approach. Several hydrogeological 3-D application examples illustrate the applicability of the 3DRCS approach in reproducing complex geological features. The results, in comparison with previous MPS methods, show better performance in portraying anisotropy characteristics and in CPU cost.

3-D characterization of geological architectures plays a crucial role in the quantification of subsurface water, oil and ore resources (Chen et al., 2017, 2018; Foged et al., 2014; Hoffman and Caers, 2007; Jackson et al., 2015; Kessler et al., 2013; Raiber et al., 2012; Wambeke and Benndorf, 2016). Heterogeneity and connectivity of sedimentary reservoirs exert controls on underground fluid transport (Gaud et al., 2004; Renard and Allard, 2013; Weissmann et al., 1999) which is vital to quantify and forecast the formation and distribution of subsurface resources. For a practical 3-D application, however, these attributes are notoriously difficult to characterize and model since the informed data we can acquire are very sparse. Two-point geostatistics (Pyrcz and Deutsch, 2014; Ritzi, 2000) and object-based methods (Deutsch and Tran, 2002; Maharaja, 2008; Pyrcz et al., 2009) are not effective at reproducing anisotropic features and connectivity patterns properly (Heinz et al., 2003; Klise et al., 2009; Knudby and Carrera, 2005; Vassena et al., 2010) due to the lack of high-order statistics and the difficulty in parameterization. To overcome the abovementioned limitations, multiple-point statistics (MPS) was developed over recent years and has shown prospects in modeling subsurface anisotropic structures, such as porous media, hydrofacies, reservoirs and other sedimentary structures (Dell Arciprete et al., 2012; Hajizadeh et al., 2011; Hu and Chugunova, 2008; Oriani et al., 2014; Pirot et al., 2015; Wu et al., 2006).

The first MPS approach was suggested by Guardiano and Srivastava (1993) and is designed to reproduce heterogeneous geometries by extracting spatial patterns from training images directly rather than through variograms. A training image is a conceptual model derived from observations, and it bears a crucial role in MPS-based stochastic simulation. The first efficient implementation of MPS was developed by Strebelle (2002) on the basis of a tree structure. Several attempts have thereafter focused on improving MPS algorithms (Arpat and Caers, 2007; Caers, 2001; Mariethoz et al., 2010; Straubhaar et al., 2011; Tahmasebi et al., 2012; Wu et al., 2008; Yang et al., 2016; Zhang et al., 2006). With these methods, training images are scanned with a fixed search template and the MPS patterns are stored in a tree or a list data structure. An important difficulty lies in choosing the size of data template to best reproduce large-scale structures (Strebelle, 2002). The larger the size of the data event, the fewer replicates of this data event will be found over the training images for inferring the corresponding conditional probability density function (CPDF). However, when the size of data template is too small, large-scale structures of the training image cannot be reproduced (Mariethoz et al., 2010). In addition, a search template including too many nodes can lead to storing a large number of patterns, increasing CPU cost and memory consumption. The multiple grids concept (Tran, 1994; Strebelle, 2002) mitigates the abovementioned limitations, but they still present due to the rigidity of data templates and multiple grids. A more straightforward MPS method, direct sampling (DS), was proposed in a study by Mariethoz et al. (2010), in which the high order statistics are sampled directly from the training image without storing patterns and without the need for multiple-point grids. One of the main advantages of this approach is that several types of distances between patterns can be considered, making it possible to simulate continuous variables, or even multivariate simulation. As an approximation, pattern distance was used to express the matching degree between the neighborhood of a node and a data event in the training image (Chugunova and Hu, 2008; Mariethoz et al., 2010, 2015).

No matter which MPS algorithm is used, a suitable training image is the fundamental requirement. Although such algorithms are gaining popularity in hydrogeological applications (Hermans et al., 2015; He et al., 2014; Høyer et al., 2017; Hu and Chugunova, 2008; Huysmans et al., 2014; Jha et al., 2014; Mahmud et al., 2015), they still suffer from one vital limitation: the lack of training images, especially for 3-D situations. Object-based or process-based techniques are one possibility to build 3-D training images (de Marsily et al., 2005; de Vries et al., 2009; Feyen and Caers, 2004; Maharaja, 2008; Pyrcz et al., 2009). Besides inherent limitations in the parameterization of these algorithms, it is also challenging to reproduce the various aspects of geological geometries from a high-resolution outcrop map, or even from insufficient borehole data (Comunian et al., 2014; Pirot et al., 2015). To overcome this difficulty of obtaining 3-D training images, scholars have attempted to use low-dimensional data (e.g., boreholes, cross sections, outcrops and remote sensing and geophysical images) to reconstruct 3-D models directly instead of a training image in the entire 3-D domain (Bayer et al., 2011; Comunian et al., 2011; Hu et al., 2011; Weissmann et al., 2015). In particular, a reconstruction method of partial datasets was proposed by Mariethoz and Renard (2010) by using and adapting the DS algorithm. However, large-scale 3-D models contain millions of nodes, and thus a very large number of scan attempts will be carried out for each simulated node by using this method, especially in the early stages of a simulation due to the sparse known data. Therefore, this method still suffers from a severe computational burden for fine 3-D applications. Moreover, it assumes stationarity of the modeled domain, which is not often the case in practice. Comunian et al. (2012) proposed an approach to tackle the lack of a full 3-D training image using sequential 2-D simulations with conditioning data (s2Dcd): a 3-D domain is filled by preserving an overall coherence due to a series of 2-D simulations performed using 2-D training images along orthogonal directions. However, this strategy is not effective at characterizing the connectivity of structures in all directions of a 3-D domain, because each 2-D simulation only considers the high-order statistics in this direction. Moreover, it also suffers from the limitation of nonstationarity of geological phenomena due to the global search in a 2-D plane. To integrate the benefits of both approaches, Gueting et al. (2017) proposed a new combination of the two existing approaches. The combination is achieved by starting with the sequential two-dimensional approach (Comunian et al., 2012), and then switching to the three-dimensional reconstruction approach (Mariethoz and Renard, 2010). However, the abovementioned limitations of the two approaches still remain because this combination is an optimization of the workflow and does not substantially improve the methods. To combine the CPDFs from different directions, several probability aggregation methods were tested and discussed (Allard et al., 2012; Bordley, 1982; Genest and Zidek, 1986; Journel, 2002; Krishnan, 2008; Mariethoz et al., 2009; Stone, 1961). Other 3-D applications to represent geological structures using MPS and partial data include filling in the shadow zone of a 3-D seismic cube (Wu et al., 2008), generating small-scale 3-D models of porous media (Okabe and Blunt, 2007) and building a 3-D training image with digital outcrop data (Pickel et al., 2015).

From another perspective, using very common workflows, geologists can obtain 2-D geological maps or sections from scattered boreholes and/or other samples by studying analogs (Caumon et al., 2009). With increasingly sophisticated data acquisition methods, 2-D high-resolution images can be acquired. For example, large-scale outcrop maps can be captured by using terrestrial lidar (Dai et al., 2005; Heinz et al., 2003; Nichols et al., 2011; Pickel et al., 2015; Zappa et al., 2006), and fine-scale pore images can be derived from progressive imaging techniques (Zhang et al., 2010). Therefore, there are many ways to acquire low-dimensional data for reconstructing a full 3-D model. In practice, however, these methods using real geological analogs or sections as training images still face significant nonstationarity due to the heterogeneity of geological phenomena and processes (Comunian et al., 2011; de Vries et al., 2009).

To address the insufficient access to a 3-D training image and the challenge
of nonstationarity, we present a new strategy to reconstruct 3-D geological
heterogeneities using 2-D cross sections (3DRCS) instead of an entire
training image. Compared to previous MPS implementations relying on partial
data, our proposal is to use only several local subsections closer to the
simulated node as training images, rather than full planes perpendicular to
the

The remainder of this paper is organized as follows. Section 2 gives background information used in the following sections. Section 3 presents the main concepts of the locality-based 3-D MPS reconstruction using 2-D cross sections and the detailed steps of the proposed approach. Section 4 shows a parameter sensitivity analysis and the performance comparison with other MPS algorithms. Section 5 gives a synthetic example in hydrogeology to illustrate the effectiveness of the 3DRCS approach when facing the real geological field data. The final section contains some concluding remarks and ideas for future work.

A pattern distance

Allard et al. (2012) presented a comprehensive literature review for aggregating probability distributions. These can be divided into additive methods and multiplicative methods according to their mathematical properties. The linear pooling formula (Stone, 1961) is a widely used method (for example, it was used by Okabe and Blunt, 2007) based on the addition of probabilities. It is appealing because of its flexibility and simplicity. Multiplicative methods include Bordley and Tau models and log-linear pooling (based on odd ratios) (Bordley, 1982; Journel, 2002; Genest and Zidek, 1986).

The linear pooling formula, proposed by Stone (1961), is probably the most
intuitive way of aggregating the probabilities

Local subsections divided by their spatial relationships and the
corresponding training images.

The log-linear pooling formula is a linear operator of the logarithms of the
probabilities (Genest and Zidek, 1986). If a prior probability

Tan et al. (2014) proposed a distance-based approach to evaluate the quality of multiple-point simulation outcomes where the Jensen–Shannon (JS) divergence is used to depict the dissimilarity of MP histograms as a quantitative metric. The information in the dissimilarity of MP histograms can be visualized using multidimensional scaling (MDS) (Caers, 2011). MDS approximates these distances by a lower-dimensional Euclidean distance in Cartesian space, which facilitates the visualization of the dissimilarity of MP histograms.

Hermans et al. (2015) used an adaptive kernel smoothing (see Park et al., 2013) to estimate
the probability density of the data variable for each kind of realizations

In the abovementioned MPS methods, when using partial data, whether searching an entire 3-D domain or complete sections, any locations of the training images are scanned even if they are far away from the simulated node, so that one spatial pattern will be carried to a distant position. Therefore, the use of these methods is restricted to stationary training images, which are in practice seldom available. In this work, we propose a local search strategy that allows this problem to be palliated, by taking into account the spatial relationships of the real geological cross sections in a given 3-D domain.

As illustrated in Fig. 1, a 3-D domain is segmented into nine small blocks
by six cross sections from three orthogonal directions where there are two
sections in each direction. Every local block is surrounded by

Another important point is related to handling of the search window when scanning a subsection. Here, we allow all locations of a subsection to be visited by the central node of a data event. The neighbor nodes of the data event can be placed in other adjacent subsections when matching with the training images. As shown in Fig. 2, the area inside the blue line is the search window. If only the nodes of the data event are from the subsection itself (case 1 on the figure), the training patterns are seriously reduced. We adopt a search strategy where neighbor nodes can be searched in the neighboring subsections (case 2 on the figure). Its main advantages are the coherence of the spatial patterns in a realization and the larger number of training patterns available. In addition, the size of the data events is constrained by the boundary of the global section, as illustrated in Mariethoz et al. (2010).

If more cross sections are available, a finer spatial subdivision can be used. In this case, the size of each subsection is smaller and the computational cost is reduced significantly. However, extremely small training images cannot offer enough spatial patterns, thus a minimal subsection size has to be considered. In practice, if there are many sections in each direction, a feasible solution is to select several ones as the references and use others as conditioning data only.

As an additive aggregation method, the linear pooling formula corresponds to
a mixture model, which is related to the union of events and to the logical
operator OR (Allard et al., 2012). This method is thus used to unite several independent
probabilities into a global term

Search window in subsections.

In this study,

Aggregate the PDFs collected along the same direction for parallel subsections using the linear pooling formula described in Sect. 2.2.1.

Aggregate the orthogonal PDFs from the above step by using the log-linear pooling formula described in Sect. 2.2.2.

For step 2, an influence of the prior probability is desired to tune the
other orthogonal PDFs. Thus, we usually use

An example of multiple grids and the corresponding neighborhoods,
search radius

When large neighborhoods are considered, it is more difficult to find
matching data events in the training image and thus a larger distance
threshold

As illustrated in Fig. 3, an example of multiple grids with three levels
is used to show the relationship between neighborhoods, search radius

Based on the strategies proposed in the above sections, the detailed steps of our simulation algorithm proceed as illustrated in Algorithm 1.

As mentioned above, we capture the MP statistics from several subsections of a local domain. Thus, the corresponding prior proportion should also be computed on the basis of these surrounding subsections (step 2). Comparing to s2Dcd, we use a fully random path on each multiple grid in the 3-D space and not within a specific section. For the current node, however, the MP statistics are only captured from several subsections in three orthogonal directions, because we only have 2-D cross sections to scan and not a 3-D training image. Obviously, step 8 is the most important procedure in our simulation algorithm, and the idea is inspired by ENESIM (Guardiano and Srivastava, 1993) and DS (Mariethoz et al., 2010). The main procedure is demonstrated in Algorithm 2.

The fraction of the scanned training image

In this section, we apply 3DRCS on several synthetic cases where the
cross sections are extracted from existing 3-D references. Using these
examples, we perform a parameter sensitivity analysis and compare it with
two widely used methods, DS-based 3-D reconstruction (Mariethoz and Renard, 2010) and s2Dcd
(Comunian et al., 2012). The workflows and algorithms proposed in this work are developed
in the C

A sample of Berea sandstone from Okabe and Blunt (2007) is used as a
3-D reference (

The majority of parameters of 3DRCS are similar to DS. Therefore, only the sensitivity of three parameters specific to 3DRCS are tested against the 3-D reference shown in Fig. 4, considering CPU cost and statistical and geometrical features of the realizations obtained. All cross sections used in the following tests in Sect. 4.1 are extracted from this 3-D model.

Reconstructions and their statistical properties when increasing the
number of sections in each direction. The first section along

The number of cross sections

Comparison of the performance of the tests in Fig. 5. All the statistics are the averages of 20 realizations.

Comparison of the performance for 20 realizations with three sections
in each direction, varying the maximum of matched patterns from each
training image

Table 2 shows the statistics of 20 realizations obtained by varying the
maximum of matched patterns from each training image

Reconstructions and their statistical properties with

Three realizations obtained by varying the weights of the probability aggregation formulas. Three sections in each direction are used and other parameters are same with the test of Fig. 5.

In this work, the strategy for aggregating the PDFs from local subsections
includes two steps. In the first step the weights of the linear pooling formula
for two parallel subsections are selected depending on the distances
between the current location and the two subsections in the first step.
Therefore, the weights are automatically set and do not need to be set. In
the second step, the appropriate weights for the prior probability
distribution and three orthogonal CPDFs are to be selected by the user.
Figure 7 shows different realizations obtained by varying the four weights

Interaction between

For the other parameters involved in our algorithm, most of them are similar to the parameterization of DS, which have been tested thoroughly in Meerschman et al. (2013). However, 3DRCS allows larger initial values for the neighborhood size and the distance threshold because multiple grids are used so that these initial values are decreased when increasing the level of multiple grids.

In this section, we compare the interaction between two important parameters
of DS (distance threshold

Realizations of three different MPS reconstruction methods.

Proportions of the facies for 20 reconstructions by using three MPS methods. The black and red horizontal lines represent the proportions of facies in the 3-D reference and the cross sections used as training images respectively.

To verify the validity of the 3DRCS approach for reproducing heterogeneous
structures, we compare it with two MPS implementations that use partial
data: DS (Mariethoz and Renard, 2010) and s2Dcd (Comunian et al., 2012). As shown in Fig. 9, six
cross sections extracted from a 3-D model of folds (

Comparison of the variograms between DS, s2Dcd and 3DRCS.

Comparison of the connectivity functions in three directions with three MPS methods.

To further compare the models obtained using the three different MPS approaches, MDS plots are constructed by calculating the distance of MP histograms between all the realizations of the three approaches and a 3-D reference. The resulting MDS map is shown in Fig. 13 and it can be observed that the realizations of 3DRCS are closer to the reference in the MDS map than the results obtained by the other two approaches. In addition, kernel smoothing is used to estimate the density distribution of the realizations of three different MPS approaches around the reference. The probabilities of the realizations are calculated from kernel density estimation by using Eq. (4) described in Sect. 2.3. According to the reference model, the three different approaches have quite similar probabilities with 29 %, 33 %, and 38 % for DS, s2Dcd and 3DRCS, respectively. However, the 3DRCS approach still gains the highest probability.

MDS representation for 20 realizations of each MPS method.

In practice, there is no fully informed 3-D reference and we only have several informed cross sections. Thus, the statistical features of the reconstructions (e.g., variograms, connectivity functions and MDS plots) are close to the reference but no one can surpass it in the above test. However, these comparisons are still able to validate the reproduction of spatial patterns for the different MPS approaches.

Comparison of computational performance between DS, s2Dcd and 3DRCS
when increasing the size of output grid:

Section 4.1.1 and 4.1.2 have already analyzed the influence of the number of
cross sections

A comparison of computational performance between DS, s2Dcd and 3DRCS is presented in Fig. 14. Because 3DRCS is sensitive to the number of input cross sections, we offer two and four sections in each direction respectively, and the computational efficiencies when increasing the total number of grid cells are shown in Fig. 14a and b. Other parameters are the same as the test in Sect. 3.2. Note that a different time axis is used for DS-based reconstruction because it uses much more CPU time than the other two methods, even though four processors are used for DS-based reconstruction. As shown in Fig. 14a, the 3DRCS approach presents better computational performance than DS-based 3-D reconstruction since the MP statistics are captured from a smaller domain composed of several 2-D sections in s2Dcd and 3DRCS. Because four processors are used in DS and s2Dcd, 3DRCS presents a speedup of about 4 compared to s2Dcd and about 120 compared to DS in this test (Fig. 14a). When increasing the number of cross sections, the search space is divided into more subdomains in 3DRCS so as to achieve a much better performance than s2Dcd and DS (see Fig. 14b).

Descalvado aquifer analog dataset (Bayer et al., 2015).

To further demonstrate the applicability of our algorithm, an
example from a real geological application is presented in this section.
The Descalvado aquifer analog dataset (Fig. 15) depicts the complex
hydrofacies of a small area (28 m

Three realizations using three different MPS approaches:

Figure 16 shows realizations obtained by using three different MPS
approaches on the basis of the abovementioned dataset. The white lines
indicate the locations of informed sections in each realization. Note that
an auxiliary variable along the

MDS maps of sections extracted from realizations using three different
MPS approach.

Comparison of reproduction of nonstationary patterns.

3DRCS is able to reduce the nonstationarity effect of real geological data to a certain extent due to the local search strategy. As shown in the above analysis, the patterns in the informed cross sections are very complicated and the distribution of hydrofacies is anisotropic and nonstationary, especially for the facies with a lower proportion. As illustrated in Fig. 18a, a local domain is surrounded by four segments from the informed cross sections. It should be noted that there is no facies 2 in any of the four segments. We extract the local parts from three realizations by using different MPS approaches. Then we check all the segments of the three local models, and we find that facies 2 is reproduced in this local area in the realizations of DS and s2Dcd. Three segments are randomly selected from the three local models, and they are shown in Fig. 18b where the boundaries of facies 2 are marked by red lines. Figure 18c shows the histograms of the four informed segments and the local models of 10 realizations for each MPS method. It can be observed that, although there is no facies 2 in the closest four segments, it is reproduced in this local area by DS and s2Dcd. Conversely, 3DRCS can maintain the distribution of facies well since all the MP statistics are captured from the surrounded subsections. If the surrounding subsections of a local area do not contain an attribute but it exists in other locations, the patterns with this attribute will not be moved to this local area in the 3DRCS approach. This indicates that 3DRCS allows the involvement of the nonstationary geological analogs in the 3-D real applications, and spatial patterns are restricted to a local domain so that they are not carried to faraway locations.

In the real-world applications, the geological sections or other analogs are not always straight or orthogonal. Therefore we need to project them in orthogonal directions. Figure 19 illustrates the process of projecting the tortuous sections to the parallel planes along a given direction. The same strategy can be used to address the issues in other directions. After that the original sections will be used as hard data and the projected sections will only be used as training images. Thus other scattered samples (e.g., boreholes, outcrops) also can be involved as hard data.

Process of projecting real-world sections to parallel planes along a
given direction: the process in

In this paper, we presented a novel method (3DRCS) for reconstructing 3-D complex heterogeneous structures by using partial lower dimensional data. Indeed, this is a very general issue since inferring high-dimensional patterns from low-dimensional data (e.g., boreholes, outcrops and other analogs) is a very common workflow for geologists. In practice, reliable 3-D models of complex geological structures are still difficult to construct due to the heterogeneity of geological phenomena and processes, even though there are many real geological analogs or sections that can be used. 3DRCS makes it possible to reconstruct 3-D structures with MPS when no 3-D training image is available. The synthetic experiments and practical applications presented in this paper demonstrate the capacity to reconstruct such heterogeneous structures.

As compared to the previous MPS implementations that use partial data, the proposed method requires several local training subsections surrounding a simulated node, rather than a full section (Comunian et al., 2012) or points in a 3-D domain (Mariethoz and Renard, 2010). The local search strategy proposed in this paper allows more reliable MPS to be computed because it avoids spatial patterns from faraway locations being considered in the simulation of the current node. In this strategy, the original cross sections are divided into many subsections according to their spatial relationships. Therefore, the nonstationarity of real geological analogs is reduced to a certain extent because the training patterns cannot be borrowed from further than a local subdomain. Of course, besides cross sections, other scattered samples also can be included as hard data.

Moreover, 3DRCS increases the computational efficiency compared with existing MPS methods. The local search strategy allows MP statistics from the local subsections to be acquired so that the searches are significantly reduced. Its good computational performance makes it potentially applicable to real 3-D modeling problems such as porous media, hydrofacies, reservoirs and other complex sedimentary structures. In addition, a new parameter, the maximum of matched patterns from each training image, is adopted to avoid the unnecessary searches. The experimental results demonstrated that a reasonable choice for this parameter can not only ensure to capture a stable CPDF, but also gain a further performance speed-up.

The method presented here retains many advantages of DS (Mariethoz et al., 2010), such as
unnecessary storing for MP statistics, pattern distances and a flexible
neighborhood. Nevertheless, we propose an adaptive and flexible
implementation of the search template on multiple grids where the radius of
the neighborhood, the distance threshold and the size of data events
decrease linearly with the rising of levels of multiple grids. As a result,
a big data event is divided into several small parts placed on the different
grids, which results in a smaller neighborhood on each grid. An acceptable
distance threshold is assigned to the first grid to make it easier to obtain
a stable CPDF and to capture the large-scale features from the original
sparse samples. For the last grid, the radius of neighborhood is reduced to
one and the highest criterion is carried out for the threshold (i.e.,

Another important advantage of 3DRCS is the probability aggregation strategy in which the combinations of two different formulas are used to combine the CPDFs from different subsections. First, an additive aggregation method (linear pooling formula) is used to combine two disjunctive probability distributions from each pair of parallel subsections to obtain a more stable PDF. The weights of this step are related to the distances between the current location and the two parallel subsections. Such parameterization is able to ensure the pattern trend changing from one subsection to another one. And then, we aggregate the orthogonal PDFs and prior probability distribution by using a multiplicative method, the log-linear pooling formula. This step can enhance the capability for reconstructing connectivity of spatial patterns in comparison with the method using a series of 2-D MPS simulations to fill a 3-D domain along given orthogonal directions (Comunian et al., 2012).

The limitations of the 3DRCS method come from the fact that it is not always possible to obtain abundant sections in each direction, and extremely small local blocks cannot offer enough spatial patterns; thus, a minimal subsection size has to be considered. In addition, 3DRCS is not able to perform the simulation of continuous variables. The proposed method can be further improved to overcome these limitations. Another possible direction is to parallelize the proposed MPS implementation and further enhance its computational performance.

An executable program of the proposed algorithm is available
on the website of the first author (

The test data used in this paper are available on the website of the first
author (

QC, GM and GL designed this study. QC implemented the
C

The authors declare that they have no conflict of interest.

We are grateful to Thomas Hermans, Kashif Mahmud and one anonymous referee for their insightful comments and suggestions towards improving the research enclosed in this paper. This work was supported in part by the National Natural Science Foundation of China (U1711267, 41172300) and the Ministry of Education Key Laboratory of Geological Survey and Evaluation (CUG2019ZR03). The authors wish to thank Philippe Renard and Julien Straubhaar for providing the MPS algorithm DeeSse as well as Moctar Dembele, Min Zeng and Luiz Gustavo Rasera for the fruitful discussions. Edited by: Philippe Ackerer Reviewed by: Kashif Mahmud, Thomas Hermans, and one anonymous referee