# A Case Study Comparing Methods for Coal Thickness Identification in Complex Geological Conditions
## 3 Methods
### Seismic Multi-Attribute Method
#### 3.2.1 Attribute System Optimization  
Seismic attributes refer to the geometrical, kinematic, dynamic, or statistical characteristics of seismic waves derived from pre-stack or post-stack seismic data through mathematical transformations [33]. The characteristics of seismic signals are directly influenced by the physical properties and variations of rock layers. Seismic information reflecting anomalies related to coal thickness mainly includes amplitude attributes, frequency attributes, velocity attributes, absorption attenuation information, and other attributes related to coal thickness [34]. Since many attribute anomalies have certain ambiguities, it is crucial to extract sensitive parameters of seismic attributes that are related to coal thickness [35]. By selecting the most sensitive seismic attributes or combinations of attributes with the fewest numbers, the prediction accuracy of seismic attributes can be improved. This selection is integrated with geological data for comprehensive analysis and judgment, facilitating the prediction of target coal seam thickness. By calculating correlation coefficients and cross-correlation, the accuracy of seismic attribute extraction for coal thickness is ensured. This process optimizes the combination and selection of attributes to minimize prediction errors with multi-parameter, selected, effective attribute combinations. (1) Seismic Layer Attribute Extraction and Normalization  
Extracting seismic layer (interlayer or along-layer) attributes provides information on the lateral variation of various attributes along the target layer interface. For the No. 2 coal seam, 20 types of seismic attributes (including amplitude attributes, trajectory attributes, and frequency (energy) spectrum statistical attributes) were extracted along the layer (with an extraction window length of 20 ms). The extracted attributes were then normalized using the following formula:  
\[y_{(i)}=(x_{(i)}-x_{\min})/(x_{\max}-x_{\min}) \tag{5}\]  
where \(x_{(i)}\) is the value at the \(i\)th point before normalization, \(y_{(i)}\) is the value at the \(i\)th point after normalization, and \(x_{\min}\) and \(x_{\max}\) are the minimum and maximum values for normalization, respectively. (2) Attribute Optimization  
Attribute optimization is mainly divided into the following two steps: first, correlation analysis to preliminarily screen for attributes sensitive to coal thickness changes; second, cross-correlation analysis to independently screen the initially selected parameters and eliminate redundant attributes with strong correlations [36], as follows:  
\[R_{xy}=\frac{\sum{(X_{i}-\overline{X})(Y_{i}-\overline{Y})}}{\sqrt{\sum{(X_{i }-\overline{X})}^{2}(Y_{i}-\overline{Y})^{2}}} \tag{6}\]  
The correlation coefficient between \(X\) and \(Y\), denoted as \(R_{xy}\), is calculated as follows: \(\overline{X}\) and \(\overline{Y}\) represent the \(i\)th observations of variables \(X\) and \(Y\), respectively. \(X_{i}\) and \(Y_{i}\) are the mean values of \(X\) and \(Y\). The value of the correlation coefficient ranges between \(-1\) and \(1\), where \(1\) indicates a perfect positive correlation, \(-1\) indicates a perfect negative correlation, and \(0\) indicates no linear correlation. The expression for cross-correlation is similar to that of the correlation coefficient, but with different parameter values. (3) Optimal Attribute Combination  
Stepwise regression is used for system attribute selection to screen for the best attribute combination that minimizes prediction errors [37]. Assuming that the optimal combinationof M attributes is known, the optimal combination of M + 1 attributes includes the previous M attributes. The steps are as follows:  
a. Perform an exhaustive search to identify the optimal attribute, attribute 1; b. Incrementally add the next best attribute to the system, targeting the smallest prediction error to determine attribute 2; c. Combine the attributes according to their correlation and iteratively seek the optimal combination, determining attribute 3, and so on.