Study on Spatial Interpolation Methods for High Precision 3D Geological Modeling of Coal Mining Faces

Abstract

High-precision three-dimensional geological modeling of mining faces is crucial for intelligent coal mining and disaster prevention. Accurate spatial interpolation is essential for building high-quality models. This study focuses on the 25214 workface of the Hongliulin coal mine, addressing challenges in interpolating terrain elevation, stratum thickness, and coal seam thickness data. We evaluate eight interpolation methods (four kriging methods, an inverse distance weighting method, and three radial basis function methods) for terrain and stratum thickness, and nine methods (including the Bayesian Maximum Entropy method) for coal seam thickness, using cross-validation to assess their accuracy. Research results indicate that for terrain elevation data with dense and evenly distributed sampling points, linear kriging achieves the highest accuracy (MAE = 1.01 m, RMSE = 1.20 m). For the optimal interpolation methods of five layers of thickness data with sparse sampling points, the results are as follows: Q₄, spherical kriging (MAE = 2.13 m, RMSE = 2.83 m); N₂b, IDW (p = 2), MAE = 2.08 m, RMSE = 2.44 m; J₂y³, RS-RBF (MAE = 0.89 m, RMSE = 1.05 m); J₂y², TPS-RBF (MAE = 1.96 m, RMSE = 2.25 m); J₂y¹, HS-RBF (MAE = 2.36 m, RMSE = 2.71 m). A method for accurately delineating the zero line of strata thickness by assigning negative values to virtual thickness in areas of missing strata has been proposed. For coal seam thickness data with uncertain data (from channel wave exploration), a soft-hard data fusion interpolation method based on Bayesian Maximum Entropy has been introduced, and its interpolation results (MAE = 0.64 m, RMSE = 0.66 m) significantly outperform those of eight other interpolation algorithms. Using the optimal interpolation methods for terrain, strata, and coal seams, we construct a high-precision three-dimensional geological model of the workface, which provides reliable support for intelligent coal mining.

1. Introduction

Spatial interpolation is a computational technique used to predict attribute values at any location in a region. It involves filling the gaps between sampled points and converting discrete sample data into a continuous surface [1,2,3]. This method is based on Tobler’s [4] first law of geography and the principle of distance decay. Currently, spatial interpolation is widely used in various fields such as spatial analysis, 3D geological modeling, management of geographic data, smart city geological management, and spatial assessment [5,6,7,8,9,10]. Recently, as China’s coal mining industry moves towards intelligent mining practices [11,12,13], constructing high-precision 3D geological models has become a critical technical necessity. The quality and reliability of these models directly hinge upon the selection of a scientifically sound and appropriate spatial interpolation method [14,15].

Spatial interpolation methods are broadly classified into three categories: deterministic interpolation, geostatistical interpolation, and machine learning interpolation. Deterministic interpolation relies on the similarity of sample points or surface smoothness to develop a predictive model. Common deterministic methods include inverse distance weighting (IDW) and radial basis function (RBF) [16]. Nistor et al. [17] employed IDW for 3D geological modeling of groundwater levels. Liu et al. [18,19] conducted a systematic analysis of IDW parameters and introduced the concept of “Relative distance” for 3D geological modeling. Addressing the issue of RBF stability, Iske [20] proposed a method for 3D geological modeling utilizing scattered data. Cuomo et al. [21] explored a method for reconstructing implicit curves and surfaces through RBF interpolation. Deterministic interpolation algorithms offer advantages in terms of simplicity, flexibility, and speed. However, they require a high density of spatially distributed sample points. Therefore, interpolation precision reduces when sample points are sparsely distributed.

Geostatistical interpolation analyzes natural characteristics by referencing spatial or spatiotemporal data. Central to this field are kriging [22] and its variations, including ordinary kriging, simple kriging, probabilistic kriging, and directed kriging, along with their optimized algorithms [23,24]. These methods utilize the semivariogram to characterize spatial autocorrelation and achieve optimal, linear, and unbiased estimations. Adhikary et al. [25] developed a spatially interpolated precipitation prediction employing collaborative kriging. Cheng et al. [26] generated a multi-attribute 3D geological model of a high-geostress tunnel utilizing indicator kriging. An et al. [27] evaluated the effectiveness of discrete smooth interpolation (DSI), kriging, and function interpolation for transparent surfaces by analyzing the resultant errors. Che et al. [28] built a 3D fault model utilizing weighted kriging. However, kriging methods necessitate stable, consistently distributed data, and when significant spatial variability exists between sampling points, the resulting interpolation is excessively smooth, failing to capture localized fluctuations. With advances in geostatistics, Bayesian theory and its derived methods are widely applied in fields such as engineering geology, spatiotemporal interpolation, and public health. Yang HQ et al. [10,29,30,31] precisely identified the soil–rock boundary using Bayesian evidence theory. They inverted issues such as rainfall-induced slope instability based on Bayesian theory. In the field of spatiotemporal interpolation, Bayesian Maximum Entropy (BME) interpolation [32,33] has become prominent in the field. In contrast to kriging, BME is knowledge-driven, capable of integrating diverse data sources with varying levels of precision under a rigorous and comprehensive theoretical framework. This enhances predictive precision and offers adaptability [34,35]. Reyes et al. [36] applied BME to predict PM2.5 concentration distributions, achieving a 21.89% error reduction compared to ordinary kriging.

Machine learning also has certain applications in the field of spatial interpolation. Wang et al. [37] introduced a leapfrog algorithm to optimize the variation function of kriging experimentation, proposing the Shuffled Frog Leaping Algorithm (SFLA)–kriging algorithm, integrating borehole data from an open-pit mine. Zhang et al. [38] employed a Particle Swarm Optimization (PSO) algorithm for variogram parameter estimation, while Ding et al. [39] developed a PSO–kriging 3D geological modeling method based on a PSO-optimized kriging algorithm. Liu and Zhang [40] advanced a k-nearest neighbor (KNN)–kriging hybrid method for 3D geological modeling. In contrast to geostatistical interpolation, machine learning operates without reliance on data trends or assumptions; however, overfitting remains a central challenge affecting the predictive precision of machine learning. Currently, its primary application in 3D geological modeling is reflected by optimizing the parameters of geostatistical interpolation algorithms.

For high-precision 3D geological modeling of a coal face, the data sources for terrain elevation, strata, and coal seam thickness vary. Terrain elevation data sources are typically dense and widespread, whereas strata thickness data sources are sparse, exhibit inconsistent variations between strata, and demonstrate significant shifts in spatial data characteristics. Coal seam thickness data sources comprise not only “hard” data, such as those derived from drilling and tunnel realities, but also “soft” data, such as coal thickness estimations from channel wave exploration or other geophysical prospecting methods, which contain uncertainties. Effectively integrating these “soft” and “hard” datasets for interpolation presents a challenge. Currently, research on selecting scientifically robust interpolation methods for high-precision 3D geological modeling of a workface, especially specific data sources, is limited. Similarly, studies focusing on 3D geological modeling of coal faces, addressing issues of missing strata and the integrated use of “soft” and “hard” data, are scarce. Focusing on the 25214 workface in the Hongliulin coal mine, this study systematically evaluates the interpolation data sources for terrain elevation, strata, and coal seam thickness. The precision of nine interpolation methods, including kriging, IDW, RBF, and BME, was evaluated through cross-validation. By combining stratigraphic sequence analysis with the optimal interpolation method, a high-precision 3D geological model of the 25214 workface was constructed. The objective of this research is to develop a high-precision three-dimensional geological model of the coal face, reflecting the actual geological characteristics, and to offer scientific support for intelligent coal mining operations.

2. Principles and Algorithms

2.1. Deterministic Interpolation Method

Inverse distance weighting (IDW) [41] interpolates values based on spatial proximity. It operates on the principle that closer distances exert greater influence on the interpolated result. The interpolation formula is:

$$Z(x)={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^N\frac{Z(X_i)}{d{(x,x_i)}^p}}}{{\displaystyle {\sum }_{i=1}^N\frac{1}{d{(x,x_i)}^p}}}}$$

(1)

where $Z(x)$ represents the predicted value for the insertion point, $Z(x_i)$ denotes the known value of point $x_i$, $d(x,x_i)$ expresses the Euclidean distance between the insertion point $x$ and the known point $x_i$, $p$ indicates a power parameter of distance, usually 2, and the degree of attenuation of the control distance, and N refers to the number of known points involved in the interpolation.

RBF (radial basis function) interpolation is an interpolation method based on radial basis function, which produces smooth surface by fitting the known point position and attribute value. The basic formula for RBF is:

$$Z(x)={\displaystyle \sum _{i=1}^N{\lambda }_i·\phi (d(x,x_i))}$$

(2)

where $Z(x)$ represents the predicted value of $x$ to be interpolated, ${\lambda }_i$ denotes the weight coefficient, determined by the location and value of known points, $\phi$ expresses the radial basis function, the common radial basis functions include thin plate spline, regular spline, higher-order spline et al., and $d(x,x_i)$ refers to the distance between the interpolation point $x$ and the known points $x_i$.

Common radial basis functions are:

(1) Thin plate spline function (TPS):

$$\phi (r)=r^2\ln (r)$$

(3)

(2) Regular spline function (RS):

$$\phi (r)=r^p$$

(4)

(3) Higher-order spline function (HS):

$$\phi (r)=r^{2k-1},k\in {\rm N},k\ge 2$$

(5)

where $r$ represents the Euclidean distance between a point to be inserted and a known point, $\ln (r)$ denotes the natural logarithm; $p>0$ expresses the parameter that controls the order of polynomials, and $k$ indicates a non-negative integer that determines the order of functions.

2.2. Kriging Interpolation

Kriging interpolation [22] leverages spatial autocorrelation statistics. Its core principle is minimizing the variance of estimation error. The fundamental kriging algorithm, often termed ordinary kriging (OK) interpolation, assumes a spatially constant but unknown mean (i.e., stationary data). The interpolation formula is:

$$Z(x)={\displaystyle \sum _{i=1}^N{\lambda }_{i}Z(x_i)}$$

(6)

where $Z(x)$ represents the predicted value of the point $x$ to be inserted, ${\lambda }_i$ denotes the weight coefficient determined by minimizing the variance of the prediction error, and $Z(x_i)$ expresses the value of the known point $X_i$.

The weight ${\lambda }_i$ coefficients of ordinary kriging satisfy the following constraints:

$${\displaystyle \sum _{i=1}^N{\lambda }_i}=1$$

(7)

The weights are calculated utilizing a semivariance function $\gamma (h)$ that describes the variation in similarity between known points as distance $h$ increases:

$$\gamma (h)={\displaystyle \frac{1}{2N(h)}}{\displaystyle \sum _{i=1}^{N(h)}{(Z(x_i)-Z(x_i+h))}^2}$$

(8)

where $h$ represents the distance between two points, and $N(h)$ denotes the logarithm of points whose distance is $h$.

The semivariogram model is fitted by the semivariogram theory $\gamma (h)$. The commonly utilized semivariogram models are:

(1)

Spherical:

$$\gamma (h)=\left\{\begin{array}{l}0,h=0\\ C_0+C({\displaystyle \frac{3h}{2a}}-{\displaystyle \frac{h^3}{2a^3}}),0\le h\le a\\ C_0+C,h>a\end{array}\right.$$

(9)

(2)

Exponential:

$$\gamma (h)=C_0+C(1-e^{-{\displaystyle \frac{h}{a}}})$$

(10)

(3)

Gaussian:

$$\gamma (h)=C_0+C(1-e^{{(-{\displaystyle \frac{h}{a}})}^2})$$

(11)

(4)

Linear:

$$\gamma (h)=C_0+kh$$

(12)

$\gamma (h)$ represents a semi-variogram model, $C_0$ denotes the nugget effect, $C$ expresses the base value, $a$ depicts the variation.

2.3. Bayesian Maximum Entropy (BME)

The Bayesian Maximum Entropy (BME) interpolation method is a geostatistical approach that integrates Bayesian statistical theory with the principle of maximum entropy. This method effectively combines high-precision measurement data (hard data) with estimation data containing uncertainties (soft data), accounting for data errors and uncertainties throughout the spatial interpolation process to achieve more accurate variable estimates.

According to data precision, the BME method classifies interpolation data into two categories. Hard data is high-precision measurement data with negligible error, such as coal seam thickness measurements from boreholes and roadways; soft data is estimated data with inherent uncertainty, such as coal seam thickness inferred from channel wave exploration inversion. Soft data can be further classified into interval-type, probability-type, and function-type data.

Within the theoretical framework of BME (Figure 1), all data and information are collectively referred to as the knowledge base (K). Based on the nature of the information, the knowledge base can be further divided into:

General knowledge (G) describes the global statistical characteristics of the variable, such as the mean, covariance, etc.; specific knowledge (S) refers to site-specific observational information within the study area, including both hard and soft data.

The BME method involves three computational stages: prior, intermediate, and posterior. The spatial variable to be estimated is treated as a random variable x, which is assumed to follow a certain probability distribution at each location within the study area. The goal of spatial interpolation is to estimate the random variable x at any unsampled location. To obtain the final estimate, x_map is introduced to represent the optimal estimate of the random variable x at a given location.

In this study, x_map represents the coal seam thickness. In other contexts, it can represent elevation, stratigraphic thickness, or other spatial variables. Essentially, BME is applicable to the interpolation of any spatial variable, provided that spatial correlation exists. This assumption is widely applicable in geosciences. For example, elevation data typically exhibit smooth and continuous spatial variations, whereas coal seam thickness data may present zero-thickness or abrupt changes. However, within a certain region, coal seam thickness still exhibits spatial correlation. Therefore, the BME method is suitable for interpolating various geoscientific variables with spatial correlation, including but not limited to elevation, stratigraphic thickness, and coal seam thickness.

Prior stage: Based on general knowledge G, the prior probability density function of the random variable x is established as follows:

$$f_G(x_{\mathit{map}})=C^{-1}\exp \left[{\displaystyle \sum _{\alpha =1}^{Ne}{\mu }_{\alpha }{g}_{\alpha }(x_{\mathit{map}})}\right]$$

(13)

where $x_{\mathit{map}}$ is a random variable, $f_G=(x_{\mathit{map}})$ is a prior probability density function of $x_{\mathit{map}}={\left[x_1,\dots ,x_m\right]}^T$, $g_{\alpha }$ is a known function of the relationship between $x_i$, $C$ is a constant regularization constraint, and ${\mu }_{\alpha }$ is a Lagrange multiplier.

Intermediate stage: All identified observational data (hard and soft data) are integrated to construct the specific knowledge base S, which is incorporated into the BME model.

Posterior Stage: By integrating specific knowledge S (i.e., hard data and soft data), the posterior probability density function of the random variable is derived using Bayesian inference. In this stage, the final estimate is typically obtained as the expectation of the posterior probability density function.

$$f_k(x_k\left|x_{\mathit{hard}},x_{\mathit{soft}}\right.)={\displaystyle \frac{{\displaystyle \int f_G(x_{\mathit{map}})d{x}_{\mathit{soft}}}}{{\displaystyle \int f_G(x_{\mathit{hard}},x_{\mathit{soft}})d{x}_{\mathit{soft}}}}}$$

(14)

where $f_k(x_k\left|x_{\mathit{hard}},x_{\mathit{soft}}\right.)$ is a posteriori probability density distribution function, $f_G(x_{\mathit{hard}},x_{\mathit{soft}})$ is a joint probability density distribution function for “hard data” and “soft data”, and finally, ${\overline{x}}_{k/K}$ is obtained based on a posteriori probability density function and utilized as an estimate of the variable:

$${\overline{x}}_{k/K}={\displaystyle \int x_k{f}_K(x_k)d{x}_k}$$

(15)

To quantitatively evaluate the performance of the aforementioned nine interpolation methods, mean absolute error (MAE) and root mean square error (RMSE) were calculated as evaluation indexes utilizing cross-validation. Considering that the present dataset is not extensive, leave-one-out cross-validation was employed. Each data point was iteratively excluded, and the mean MAE and RMSE across all data points were finally computed.

$$MAE(X,h)={\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m\left|h(x_i)-y_i\right|}$$

(16)

$$RMSE(X,h)=\sqrt{{\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m{(h(x_i)-y_i)}^2}}$$

(17)

where $m$ represents the number of sampling points, $h(x_i)$ denotes the true value of sampling point $x_i$, and $y_i$ expresses the predicted value in $x_i$.

3. Study Area and Data Source Analysis

3.1. Study Area

This study focuses on the 25214 workface at the Hongliulin coal mine, north of Yulin City, Shaanxi Province. The region presents a complex geomorphology, represented by rolling sand dunes to the west and Loess Hills and gullies in the northeast. The overall topography slopes downwards from northwest to southeast (Figure 2).

Borehole data, collected over a 300 m near the 25214 workface (Figure 3), indicates the presence of Quaternary (Q₄), Neogene Baode Formation (N₂b), and Jurassic Yan’an Formation (J₂y) strata. The Quaternary system consists primarily of Holocene eolian sand deposits and Lishi Formation loess. The red soil of the Baode Formation constitutes the principal aquiclude in the workface roof. Near the 25214 workface, erosion has removed J₂y⁵ and J₂y⁴ strata, leaving only J₂y³, J₂y², and J₂y¹. The primary coal seams cap each of these sections: 3⁻¹ Coal sits atop J₂y³, 4⁻² Coal above J₂y², and 5⁻² Coal crowns J₂y¹. The main mining face 25214 is extracting from the 5⁻² coal, therefore no interpolation has been conducted for the 3⁻¹ and 4⁻² coal. This study primarily evaluates the topography Q₄, N₂b, J₂y³, J₂y², 5⁻² Coal, and J₂y¹.

3.2. Data Source and Feature Analysis

An analysis was conducted of the spatial interpolation data sources—terrain elevation, stratum, and coal seam thickness—within 300 m near the 25214 workface.

(1)

Terrain data

The data sources for terrain interpolation consist of elevation measurement points obtained from aerial photogrammetry using an unmanned aerial vehicle (UAV) equipped with a high-resolution camera, and from Real-Time Kinematic (RTK) positioning surveys. Both datasets cover the entire mining area; however, this study extracted a total of 1700 elevation points within the 25214 workface and its 300 m extended boundary (Figure 4). Within this range, the UAV photogrammetric data points are densely distributed. These points were obtained through the deployment of ground control points and aerial triangulation densification processing, ensuring high positional accuracy. In contrast, the RTK survey data points are more sparsely distributed, as they were collected using conventional ground surveying methods. Although the two datasets differ in acquisition techniques and individual point accuracies, both are considered high-precision measurements. Therefore, they are treated as “hard data” in the interpolation process, providing a reliable basis for terrain modeling.

(2)

Stratigraphic data

Formation thickness data for the study area were obtained from 27 boreholes, the spatial distribution of which is shown in Figure 5. These boreholes provide thickness measurements for five primary stratigraphic units: the Quaternary (Q₄), Neogene Baode Formation (N₂b), and Jurassic Yan’an Formation members (J₂y³, J₂y², J₂y¹). Due to variations in borehole depths, data for the J₂y² and J₂y¹ units were available from 20 boreholes, whereas the remaining 7 boreholes did not reach these deeper strata.

The thickness ranges of the five stratigraphic units are summarized in

Table 1. Formation thickness data (m).

	BK1-1	2-HB1	HS-5	2-1	HS-4	HS-3	HS-2	HS-1	3-2	326	BK5	3-3	4-5	4-4
Q4	36.25	7.50	27.40	54.18	25.00	46.00	26.00	45.80	46.42	76.91	39.44	57.00	26.30	51.60
N2b	0.00	56.30	55.80	29.62	45.04	41.08	64.49	36.80	43.98	0.00	75.91	31.50	59.43	44.25
J2y3	48.15	34.28	35.89	42.25	35.81	29.42	34.49	21.90	22.10	25.47	27.29	19.00	21.88	25.55
J2y2	76.13	77.50	NaN	79.05	NaN	NaN	NaN	NaN	81.40	80.17	NaN	79.35	69.86	80.85
J2y1	19.75	14.43	NaN	10.60	NaN	NaN	NaN	NaN	10.45	30.15	NaN	12.76	14.41	14.69
	4-6	5-2	332	5-3	BK3-2	6-6	6-5	B6	B7	B8	B9	B10	B12
Q4	42.95	24.19	90.29	30.60	0.00	22.45	42.60	60.40	63.80	53.90	46.50	88.25	40.55
N2b	39.10	59.04	0.00	38.30	92.00	63.24	47.10	17.80	27.00	26.40	21.20	14.45	53.00
J2y3	20.23	32.49	23.01	15.40	11.30	24.05	21.00	27.07	28.27	20.32	25.80	32.97	30.15
J2y2	56.52	79.79	64.42	68.25	58.70	51.06	71.00	68.49	65.53	78.90	65.40	68.05	NaN
J2y1	9.50	9.88	10.17	8.85	10.78	11.16	6.85	17.40	23.48	11.97	25.12	22.87	NaN

as follows: Q₄ (0–90.29 m), N₂b (0–92 m), J₂y³ (11.3–48.15 m), J₂y² (51.06–81.4 m), and J₂y¹ (6.85–30.15 m). The Q₄ and N₂b formations exhibit substantial thickness variability across the workface, reflecting notable stratigraphic undulation and local erosion. In contrast, the J₂y³, J₂y², and J₂y¹ units generally show more uniform thicknesses, though localized variations are still present.

Furthermore, local absences of certain strata, particularly Q₄ and N₂b, were observed in some boreholes, indicating erosional truncation or non-deposition in specific areas. These spatial discontinuities further highlight the heterogeneity in the overlying strata. Collectively, these stratigraphic data provide the fundamental basis for subsequent geological modeling and interpolation analyses.

(3)

Coal seam data

The interpolation data for coal seam thickness in the 25214 workface were obtained from multiple sources, including borehole measurements, roadway exposure measurements during tunnel excavation (Figure 6), and channel wave exploration inversion results. During tunnel excavation, coal seam thickness was measured along the roadway with an average spacing of approximately 50 m between measurement points. These measurements were used to assist in roadway positioning and adjust the excavation direction. Notably, seam thinning was observed approximately 500 m from the retreat support roadway, where a sandstone scour zone had eroded the seam. To further delineate this thinning anomaly, channel wave exploration was conducted within the range of 310–690 m from the retreat roadway, and the seam thickness was inferred by inverting the channel wave group velocity.

The layout for the channel wave exploration at 25214 workface is depicted in Figure 7. Twenty shot points and forty-two receiver points were planned for each of the two survey lines. Shot points were spaced 20 m apart, and receiver points 10 m apart. In practice, shots S21 and S25 were not fired, resulting in data acquisition from 38 shots and 1596 data points (

Table 2. Channel wave construction data point.

Workface	Name of Roadway	Stimulate Point Number	Number of Stimulate Points	Distance Between Firing Points (m)	Recieve Point Number	Recieve the Number of Points	Recieve Point Spacing (m)
25214	Return wind	S1–S20	20	20	R43–R84	42	10
	Shipping	S2–S24 S26–S40	18	20	R1–R42	42	10
	Subtotal		38			84

).

The channel wave exploration results indicate that the coal seam thinning zone correlates with a high-velocity anomaly. Figure 8 presents the seismic record for shot S11 in the 25214 workface. The magenta line represents an auxiliary line at 925 m/s, while the green line demonstrates the actual arrival time of each channel wave. From this figure, the channel wave velocity is approximately 925 m/s for receivers 1 through 29. However, arrivals at receivers 30 through 42 exhibit an earlier arrival time, indicating a velocity increase and, therefore, coal seam thinning. Picking different channel waves produces the velocity distribution across the workface (Figure 9). While the channel wave velocity is generally stable, the data define an anomalous high-velocity range, referred to as SYC1.

High-velocity anomalies typically indicate local coal seam thinning or the presence of a sandstone scoured zone, as channel waves travel faster in sandstone than in coal. In the SYC1 region, on-site observations confirmed that the coal seam thinning was caused by a localized sandstone scour body. This scour body replaced part of the coal seam with harder sandstone, resulting in a velocity increase in the channel wave propagation.

Notwithstanding their dense and uniform distribution across the workface, the inverted coal thickness data from channel waves are considered “soft data”, indicating lower precision and uncertainty. In subsequent interpolation, a comparison was made between the measured data from the tunnel and the inversion coal thickness data of the nearby trough waves. It was found that the error range was within ±20% of the measured data. Therefore, in the BME interpolation, the trough wave exploration inversion coal thickness data was used as interval-type [lb, ub] soft data for the calculations.

4. Analysis and Visualization of Interpolation Results

4.1. Terrain Modeling

To develop an accurate model for estimating the spatial distribution of topographic elevation data in the study area, spherical kriging, exponential kriging, Gaussian kriging, linear kriging, IDW (p = 2), TPS-RBF, RS-RBF, and HS-RBF, were employed to create topographic contour maps (Figure 10). In the interpolation, the neighbor points are selected as 15. Visually, the eight contour maps exhibited minimal differences and effectively represented the undulating nature of the terrain. To quantitatively assess each interpolation algorithm, cross-validation was performed utilizing the mean absolute error (MAE) and root mean square error (RMSE) as evaluation criteria (

Table 3. Terrain interpolation result error.

		Spherical	Exponential	Gaussian	Linear	IDW	TPS	RS	HS
Terrian	MAE (m)	2.77	2.49	2.88	1.01	2.95	1.86	1.64	1.70
	RMSE (m)	3.39	3.61	4.17	1.20	3.87	2.41	2.67	2.84
	Calculate time (s)	51.87	63.10	55.63	52.26	42.05	42.99	45.94	49.36
	Consume memory (KB)	184.00	164.00	136.00	124.00	172.00	212.00	128.00	144.00

). In terms of computation time and memory consumption, there is little difference among the eight interpolation methods. The kriging methods take 51.87–63.10 s, and the IDW and RBF methods take 42.05–49.36 s, which is related to their weighting methods. Kriging-type methods require a relatively long time to fit the variogram, but overall the time is acceptable, with a memory consumption of between 124–212 KB, which is negligible on modern computer hardware.

The results indicate a high degree of precision for both geostatistical and deterministic interpolation methods, fulfilling the requirement for high-precision modeling. The linear kriging interpolation method demonstrated the highest precision, with MAE = 1.01 m and RMSE = 1.20 m. The linear variogram proved most suitable for fitting the variogram of the topographic elevation points. This suggests that the linear kriging method effectively captures the spatial distribution characteristics of topographic elevation points and represents the optimal interpolation method for this topographic model. A 3D terrain model was generated based on linear kriging interpolation (Figure 11), visualizing the relief characteristics of the surface elevation in the study area. The 3D terrain model and contour map were translated along the Z-axis, and the 3D terrain model was stretched by a factor of two along the Z-axis to enhance visual clarity. The details and overall trends of the terrain in the study area were clearly reflected. After selecting the linear kriging method, the sensitivity experiment of its neighbor points parameters is carried out. Figure 12 represents the MAE and RMSE change diagrams of the interpolation results under the condition of different number of neighbor points. It can be seen that when the number of interpolation points is 0–9, the MAE and RMSE values of the model show a rapid downward trend, which is due to the elevation of the terrain interpolation data; the value is large, the terrain points are numerous, the neighbor range of 0–9 points cannot well capture the local differences of the terrain; the neighbor range tends to be stable at 10–25 points of MAE and RMSE, proving that the neighbor range of 10–25 points is able to capture the local variation and undulation characteristics of the terrain. The parameter sensitivity experiments show that the model is stable when the neighborhood points are more than 10.

4.2. Formation Modeling

For the strata (Q₄, N₂b, J₂y³, J₂y², and J₂y¹), the optimal interpolation method was selected (

Table 4. Strata interpolation result error.

		Spherical	Exponential	Gaussian	Linear	IDW	TPS	RS	HS
Q4	MAE (m)	2.13	2.88	2.82	2.86	2.45	3.73	3.41	2.67
	RMSE (m)	2.83	3.51	3.34	3.65	3.24	4.43	3.98	3.28
N2b	MAE (m)	3.71	3.68	3.33	3.77	2.08	5.47	4.56	4.53
	RMSE (m)	4.23	4.13	3.79	4.24	2.44	6.19	5.34	5.80
J2y3	MAE (m)	1.39	1.63	1.78	1.65	1.55	1.05	0.89	1.06
	RMSE (m)	1.64	1.83	1.95	1.80	1.67	1.2	1.05	1.19
J2y2	MAE (m)	2.74	2.72	2.8	2.77	2.38	1.96	2.17	2.42
	RMSE (m)	3.28	3.19	3.25	3.15	2.79	2.25	2.45	2.83
J2y1	MAE (m)	2.72	2.96	2.72	3.78	2.55	2.60	2.43	2.36
	RMSE (m)	3.01	3.22	2.80	4.32	2.79	2.74	2.81	2.71

) using the same eight interpolation methods as for terrain interpolation and evaluated through cross-validation (MAE and RMSE). Taking the Q₄ stratum as an example, its best result was spherical kriging (MAE = 2.13 m, RMSE = 2.83 m). From the eight interpolation results in Figure 13, the spatial distribution of the spherical kriging image is also more reasonable, without abnormal breakpoints or irregular polylines. The same indicators and criteria were used to determine the optimal interpolation methods for the other four strata, N₂b was IDW (p = 2), J₂y³ was RS-RBF, J₂y² was TPS-RBF, and J₂y¹ was HS-RBF.

The data source of stratum interpolation is borehole data, which is less distributed. Figure 14 shows that the change of MAE and RMSE is very small when the number of neighbor points is 3–10, which is between 0.05 and 0.1 m. It proves that the number of neighbor points has little effect on the interpolation of strata and the model is stable. Because the number of formation interpolation boreholes is small, the calculation time and memory consumption are very small, so the statistical analysis is not carried out separately.

For certain formations (such as Q₄ and N₂b), a virtual thickness assignment method addressed missing borehole data. Specifically, the missing zone is determined by the contour map generated by the optimal interpolation method, and the thickness of the missing zone is assigned to a negative value, then the interpolated surface is regenerated, while the missing line (0 line) is drawn automatically. During 3D layer modeling, grid intersection eliminated these areas of missing stratum data, allowing for accurate definition of these zones (Figure 15). Employing the optimal interpolation method in conjunction with this stratum deletion method produced the layer models for the Q₄, N₂b, and J₂y³ strata (Figure 16), verifying the effectiveness and scientific basis of the missing data handling procedure.

4.3. Coal Modeling

Considering that the interpolation results inform the three-dimensional geological model and recognizing that the channel wave exploration does not consist of the entire workface, the spatial distribution of 5⁻² coal thickness was analyzed across the entire 25214 workface, as well as in the channel wave exploration area, to ensure accurate representation.

Assessment of coal thickness distribution across the entire workface employed eight interpolation methods. Low MAE and RMSE values indicate the reliability of these methods (

Table 5. 5⁻² coal interpolation result error (hard data).

	Hard Data	Spherical	Exponential	Gaussian	Linear	IDW	TPS	RS	HS
5−2 Coal	MAE (m)	0.85	0.85	0.89	0.86	1.06	1.03	1.01	0.91
	RMSE (m)	1.06	1.06	1.13	1.06	1.37	1.51	1.30	1.27

). Among these methods, spherical, exponential, and linear kriging demonstrated comparably high precision. Visual inspection of the interpolated distributions (Figure 17) indicates a more uniform color distribution and aesthetically preferable representation from spherical kriging in the 25214 workface. Therefore, spherical kriging was selected as the optimal interpolation method for 5⁻² coal thickness in this workface.

In addition to the eight methods described above, Bayesian Maximum Entropy (BME) interpolation was employed to estimate the spatial distribution of coal thickness specifically in the channel wave exploration area. BME interpolation was implemented utilizing Seksgui V1.0.9, an open-source MATLAB R2020a-based software package available at Spatiotemporal Epistematics Knowledge Synthesis Graphical User Interface. Construction of “soft data” is central to BME interpolation. In this study, interval “soft data” were constructed based on the error analysis of channel wave exploration. Specifically, considering a 20% error in the retrieved coal thickness near roadway exposures, interval “soft data” [lb, up] were generated. For instance, a measured coal thickness of 4.6 m yields corresponding “soft data” of [3.68 m, 5.52 m], thus offering comprehensive input for the BME interpolation.

Eight interpolation methods, including kriging, IDW, and RBF, cannot accommodate data uncertainty during the interpolation process. Therefore, when interpolating coal thickness, data from channel wave inversion, borehole data, and mine roadway data were incorporated as input sources. The MAE and RMSE values for the nine interpolation methods are lower (

Table 6. 5⁻² Coal interpolation result error (hard and soft data).

	Data	BME	Spherical	Exponential	Gaussian	Linear	IDW	TPS	RS	HS
5−2 Coal	MAE (m)	0.64	0.99	1.01	0.98	1.00	1.20	1.96	2.00	1.58
	RMSE (m)	0.66	1.18	1.18	1.12	1.13	1.27	2.05	2.11	1.63
	Calculate time (s)	83.57	75.45	69.03	61.70	70.13	45.21	72.07	64.02	67.28
	Consume memory (KB)	387.00	207.00	191.00	157.00	163.00	183.00	218.00	154.00	201.00

) accordingly with the uniform distribution and gridded output of the geophysically inverted coal thickness data. The BME method has the highest accuracy but consumes more time and memory than the other eight algorithms, which is determined by the computational complexity of the BME algorithm itself. The time consumption is 83.57 s and the memory consumption is 387 KB. Figure 18 illustrates that BME interpolation offers greater precision in areas where the coal seam thins and in the vicinity of the maximum coal thickness of 7.04 m, effectively capturing localized differences in the coal seam. This enhanced performance is attributed to the inability of the other eight methods to utilize coal thickness data uncertainty; thus, these data are treated as “Hard Data” during interpolation, leading to greater discrepancies between interpolated and observed values. BME interpolation, in contrast, incorporates both “Hard Data” and “Soft Data”, utilizing uncertainty intervals and analyzing interpolation weights at various locations to achieve an optimal solution. Accordingly, BME interpolation was chosen for accurate coal thickness predictions in the channel wave exploration area, while spherical kriging was applied to predict coal thickness in other sections of the 25214 workface.

According to the results of BME parameter sensitivity experiments (Figure 19), the accuracy of the model (MAE and RMSE) tends to be stable when the number of neighbor points exceeds 5, and it performs best at 10 neighbor points. This result shows that there is a sensitivity between the interpolation accuracy of the BME method and the number of neighbor points, and in this experiment, 10 neighbor points were selected as the best balance points.

The BME method considers both soft data and hard data in the process of interpolation, and uses these data to model uncertainty. Due to the large number of points in the in-seam wave inversion data, selecting 10 neighbor points can effectively capture the spatial variation of the data and provide enough local information for accurate interpolation. After this point, adding more neighbor points has little effect on the accuracy improvement, indicating that the BME method has a limited demand for the number of neighbor points, and 10 neighbor points are sufficient to ensure the stability and accuracy of the model.

In the actual recovery process of the trough wave exploration area, three measurement lines were conducted at distances of 2409 m, 2515 m, and 2632 m from the starting recovery direction to measure the coal thickness, totaling 39 actual coal thickness measurement points to validate the interpolation results (Figure 20).

Table 7. Comparison of interpolation results of 5⁻² coal thickness with measured values.

Error Between the Interpolation Results and Real Measurement Points	BME	Spherical	Exponential	Gaussian	Linear	IDW	TPS	RS	HS
Min (m)	0.01	0.02	0.01	0.00	0.00	0.02	0.01	0.01	0.01
Max (m)	0.54	0.90	0.88	0.91	0.81	1.23	1.86	1.46	1.32
Average (m)	0.20	0.29	0.27	0.25	0.30	0.28	0.35	0.26	0.27

provides a quantitative error analysis comparing BME with other interpolation methods (such as spherical, exponential, Gaussian, linear, IDW, TPS, RS, HS) against the actual coal seam thickness measurement points. Among the minimum errors, Gaussian kriging and linear kriging performed the best with 0.00 m, followed by BME, exponential kriging, TPS-RBF, RS-RBF, and HS-RBF methods with 0.01 m. In terms of maximum error, the BME method performed the best with 0.57 m, while the other methods ranged between 0.81 m and 1.86 m. For average error, the BME method also performed the best with 0.20 m, while the other methods slightly lagged behind BME, ranging from 0.25 m to 0.35 m. These results indicate that BME has high accuracy in predicting coal seam thickness and can effectively integrate soft and hard data information for predictions

The BME and spherical kriging interpolation results were combined to create a 3D geological model of the 5⁻² coal seam (Figure 21). The yellow area represents the BME interpolation domain, while the gray surface depicts the spherical kriging results. The BME results lie beneath the gray surface, offering a more accurate representation in the coal seam thinning zone.

5. 3D Geological Modeling of 25214 Workface

Three-dimensional geological modeling hinges on the accurate mapping of stratum boundaries. Following the experimental results detailed in Chapter 3, a sequence analysis of the 25214 workface terrain, strata, and coal seam was conducted, resulting in a high-precision 3D geological model. Leveraging optimal interpolation, the layer model was developed according to the sequence presented in Figure 22. The layer was discretized into a 10 × 10 m mesh, triangulated utilizing the shortest distance method. This approach facilitated a smooth and accurate representation of missing strata areas. In Figure 23, the triangular mesh between the upper and lower boundaries of the 5⁻² coal seam appears denser and more constricted. This is attributed to the grid splicing employed for the 5⁻² coal seam, where the higher grid density in the trough wave exploration zone created a mismatch of grid nodes between upper and lower levels. This mismatch resulted in a denser triangular network upon connection of the upper and lower levels. While in practical application, separate fine models for coal seams and strata may be generated, or the stratum grid resolution could match the coal seam’s, the focus of this study is on applying high-precision interpolation methods to 3D geological modeling, and therefore this aspect is not further optimized here.

6. Discussion

In this study, nine interpolation methods were applied to the spatial interpolation of surface elevation, topography, and coal seam thickness data, with cross-validation to select the best method. A novel approach for redrawing zero lines based on negative virtual thickness values was introduced, addressing the challenge of defining missing stratigraphic regions. For coal seam thickness, a BME interpolation method was proposed, integrating “soft data” and “hard data” from channel wave exploration, boreholes, and mine roadway data, enabling more accurate identification of abnormal coal thickness variations, which were effectively visualized in the 3D model.

The advantage of this study lies in the selection of diverse interpolation methods, including kriging, IDW, RBF, and BME. Compared to other studies, such as Ding et al. [39] and Che et al. [28], which focused on kriging methods for coal seam interpolation, the methods in this paper demonstrate higher accuracy. While these studies focused on entire mine coal seams, the present study is limited to the mining face, allowing for more detailed results. Additionally, the integration of soft and hard data through BME offers improved results compared to traditional methods like kriging and Bayesian kriging (BK), particularly in handling data gaps and inconsistencies [42].

However, this study also has limitations. The in-seam exploration area does not cover the entire workface, and the method performs better with evenly distributed and large volumes of data. BME has also been shown to perform well with only soft data [43]. Future research should explore enhanced soft data construction methods, similar to probabilistic soft datasets, and apply machine learning to improve interpolation accuracy, especially as data sources increase. At present, small datasets in mining environments may cause machine learning models to overfit, but this approach remains worth exploring [44,45].

In geologically complex regions, such as fault zones and fractured coal seams, traditional interpolation methods (e.g., kriging, IDW, RBF) often fail to capture local characteristics due to spatial heterogeneity and geological discontinuities. The BME method mitigates this by integrating soft and hard data, allowing for more accurate interval estimates for missing data and reducing errors caused by geological discontinuities. However, BME may face challenges in highly discontinuous environments, particularly in fault zones, where interpolation accuracy may decline. Future improvements could involve increasing auxiliary data (e.g., geophysical exploration) and optimizing search radii to limit data influence from across faults, ensuring more accurate results in fault and fracture zones.

In three-dimensional geological modeling, real-time updates are crucial for mining accuracy, particularly for coal seam models. While terrain and strata data are updated less frequently, coal seam models must be continuously updated with real-time data from mining operations. Despite its high accuracy, BME’s computational complexity can cause delays in real-time updates. However, coal seam changes are generally stable, except in areas with abnormal geological structures (such as fault zones). In stable conditions, the model updated during mining progress can provide valuable guidance for subsequent mining cuts, especially as mining typically advances 8–12 cuts per day.

To mitigate these delays, future research could focus on accelerating the BME calculation process via parallel computing or GPU acceleration. Additionally, machine learning could be applied to predict coal thickness trends, enabling faster and more precise real-time updates.

Although this article studies the mining operations that are currently being extracted, the proposed interpolation methods are also highly applicable to closed coal mines, such as those in Europe. In such areas, geological data remain critical for monitoring geological hazards (e.g., subsidence, surface water leakage) and environmental concerns (e.g., groundwater contamination, wastewater treatment, ecological restoration). Three-dimensional geological models, reconstructed from historical data using BME, can aid in assessing disaster risks and guiding environmental remediation efforts. For example, accurate coal seam data can support groundwater flow and subsidence monitoring, providing essential data for long-term environmental management after mine closure. Furthermore, integrating soft and hard data through BME can help evaluate environmental risks and support mine reclamation and pollution remediation.

Thus, while real-time updates may not be as crucial in closed mines, updating three-dimensional geological models based on historical data still provides valuable insights for disaster monitoring, environmental protection, and reclamation efforts. In particular, regions like Europe, with closed coal mines, will benefit from these advanced modeling techniques in post-mining environmental restoration and management.

7. Conclusions

Based on the statistical characteristics and spatial distribution features of the topographic elevation, strata, and coal seam thickness data within a 300 m range near the 25214 workface of the Hongliulin coal mine, the accuracies of nine interpolation methods were evaluated through cross-validation. Among these, the topographic and strata data were subjected to eight interpolation methods, while the coal seam thickness interpolation included an additional BME method. A method for accurately delineating areas with missing strata was proposed, and a fine three-dimensional geological model of the 25214 workface was established using the optimal interpolation methods for each layer.

Interpolation Results for Terrain Elevation Points: For the terrain elevation data, which is evenly distributed and dense, the performance of eight interpolation methods, including kriging, IDW, and RBF, was generally satisfactory. Among these methods, linear kriging exhibited the highest accuracy (MAE = 1.01 m, RMSE = 1.20 m), capturing the local variations in terrain elevation most effectively.

Interpolation Results for Stratigraphic Thickness Data: For stratigraphic thickness data with sparse sampling points, the optimal interpolation methods for five strata are as follows: For Q₄ stratum, spherical kriging (MAE = 2.13 m, RMSE = 2.83 m); for N₂b stratum, IDW (p = 2) (MAE = 2.08, RMSE = 2.44); for J₂y³ stratum, RS-RBF (MAE = 0.89 m, RMSE = 1.05 m); for J₂y² stratum, TPS-RBF (MAE = 1.96 m, RMSE = 2.25 m); and for J₂y¹ stratum, HS-RBF (MAE = 2.36 m, RMSE = 2.71 m).

Method for Handling Missing Stratigraphic Areas: A novel method for addressing missing stratigraphic regions was proposed. By virtually redrawing the zero line of stratum thickness based on negative virtual thickness values, a 3D layer model was constructed. Grid intersection techniques were then employed to eliminate gaps within the model, enabling the accurate definition of missing stratigraphic regions in the 3D geological model.

Precise Prediction Interpolation Method for Coal Seam Thickness: A precise interpolation method for predicting coal seam thickness based on Bayesian Maximum Entropy (BME) was proposed. This method integrates hard data from boreholes and mine roadway measurements with soft data from channel wave exploration. The interpolation accuracy after integration (MAE = 0.64 m, RMSE = 0.66 m) is significantly superior to that of the other eight methods. Compared to the real 5⁻² coal thickness measurement points, the BME method shows an error range between 0.01 and 0.57 m, with an average error of 0.20 m, demonstrating the highest accuracy.

Construction of a High-Precision 3D Geological Model: By employing the optimal interpolation methods identified for terrain, individual strata, and coal seams, a high-precision 3D geological model of the coal mining face was constructed using TIN grid modeling. This model provides reliable 3D geological support for intelligent coal mining operations.