## 2 Principles and Algorithms
### 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  
Figure 1: BME workflow. 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:  
\[f_{G}(x_{map})=C^{-1}\exp\left[\sum_{\kappa=1}^{Ne}\mu_{a}g_{\alpha}(x_{map})\right] \tag{13}\]  
where \(x_{map}\) is a random variable, \(f_{G}=(x_{map})\) is a prior probability density function of \(x_{map}=[x_{1},\ldots,x_{m}]^{T}\), \(g_{\alpha}\) is a known function of the relationship between \(x_{i}\), \(C\) is a constant regularization constraint, and \(\mu_{a}\) 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.