## 3 Materials and Methods
### ASTER LST Data Product and D-S Evidence Theory Fusion Method
Thus, the final threshold is defined as the mean temperature of each sub-region plus 1.282 times its standard deviation. 4. Fault Structure Buffer Zone Method: Geothermal drilling results indicate a strong correlation between fault structures and geothermal anomalies [48]. Regional geothermal temperature measurements show that the heat diffusion halo of deep fault zones extends approximately 1000 m [47]. Considering the fault characteristics in the study area, a 1 km buffer zone was defined to identify geothermal anomalies, forming evidence layer A4. 5. D-S Evidence Theory Fusion Method: D-S evidence theory provides a robust framework for integrating uncertain information, making it especially effective for combining data from diverse sources to improve anomaly detection [53]. In this study, D-S evidence theory was applied to integrate geothermal anomaly evidence extracted from the four criteria (A1-A4) mentioned above [39]. D-S evidence theory defines a frame of discernment \(\Theta\), representing a finite set of mutually exclusive and collectively exhaustive hypotheses. In this study, \(\Theta=\) [anomaly, non-anomaly], representing the two possible states of each pixel in geothermal anomaly detection. The set of all subsets of \(\Theta\) is denoted as \(2^{\Theta}\). The mass function m(A) defines the basic probability assignment (BPA) for each subset A\(\in 2^{\Theta}\), and satisfies the following conditions:  
\[\text{m(A)}\in[0,1],\ \text{m(\varnothing)}=0,\ \bigg{(}\sum\limits_{A\subseteq 2 ^{\Theta}}\bigg{)}\text{m(A)}=1 \tag{9}\]  
Here, \(\emptyset\) represents the empty set, and m(A) is the BPA assigned to set A. In this study, BPA values were assigned based on local temperature gradients. A local window of size n \(\times\) n was defined, with the central pixel temperature denoted as t(k). The temperature gradient \(g_{i}\) is calculated as  
\[g_{i}=t_{i}(k)-\overline{t(k)},i=1,2,\ldots,n \tag{10}\]  
Here, \(\overline{t(k)}\) denotes the mean temperature of all pixels within the window, \(n\) represents the window size, and \(i\) indicates the position index of the temperature value. The probability of a geothermal anomaly at the center pixel is then calculated as  
\[p_{k} =\frac{g(k)}{\sum\limits_{i=1}^{n}g_{i}},\ i=1,2,\ldots,n \tag{11}\]  
A larger temperature gradient corresponds to a higher probability of geothermal anomaly, whereas a smaller gradient suggests a lower probability. The D-S combination rule integrates multiple sources of evidence to derive a comprehensive probability distribution of geothermal anomalies. For four evidence sources \(m_{1}\), \(m_{2}\), \(m_{3}\) and \(m_{4}\), the combination rule is expressed as  
\[(m_{1}\oplus m_{2}\oplus m_{3}\oplus m_{4})(\text{A})=\frac{1}{k}\bigg{(} \sum\limits_{A_{1}\cap A_{2}\cap A_{3}\cap A_{4}}=A\bigg{)}m_{1}(A_{1})\cdot m _{2}(A_{2})\cdot m_{3}(A_{3})\cdot m_{4}(A_{4}) \tag{12}\]\[\text{K}=1-\left(\sum\limits_{A_{1}\cap A_{2}\cap A_{3}\cap A_{4}=0}\right)m_{1}( A_{1})\cdot m_{2}(A_{2})\cdot m_{3}(A_{3})\cdot m_{4}(A_{4}) \tag{13}\]  
Here, K is the normalization constant that ensures the combined probability distribution sums to 1. To enhance the reliability of anomaly detection, geothermal anomaly probabilities were directly assigned a value of 1 within the fault buffer zone and 0 outside it [30]. This strategy takes advantage of the strong spatial correlation between fault structures and geothermal anomalies. Finally, an empirical threshold was applied to extract the final geothermal anomaly distribution [54].