## 3 Observational Data Sources and Processing  
### Source and Processing Method of Rock Geothermal Observation Data  
From 2016 to 2019, the Institute of Geology of the China Earthquake Administration and the Earthquake Administration of the Xinjiang Uygur Autonomous Region jointly established a number of bedrock temperature monitoring stations along the Keping fault (F1 in Figure 1) to detect regional fault stress changes and earthquake development processes. The bedrock temperature is monitored using wireless geothermal telemetry equipment developed by the Institute of Geology, China Earthquake Administration, with a sampling period of 10 min and an accuracy of 0.00003 \({}^{\circ}\)C. Holes are drilled in the bedrock, and a temperature sensor is installed in each hole using high-grade cement, which is closely fitted to the bedrock (Figure 2). The sensors are vertically distributed at multiple levels between the surface and 50 m underground. The observation stations are selected along the fault, with relatively complete bedrock and far away from water sources. The topographic environment is similar, without special terrain. The observed bedrock temperature data are stable and smooth, and the layer below 10m is not directly affected by the air temperature. The borehole is sealed by a cement tank, which is hard to be affected by external interference factors. The observed data reflect the seasonal variation of shallow ground temperature, but the deep ground temperature is constant. In this study, the bedrock temperature observation data were processed by general polynomial piecewise fitting. To ensure the continuous smoothness of the fitting curves of adjacent segments, the following constraints were introduced: the \(r\)-order derivatives of the fitting curves in adjacent data segments at the common points are equal, \(r\) = 0, 1, 2..., and \(r\) is less than the order of the fitting polynomial. The data are divided into \(p\) data segments, and each segment is fitted by \(m\) polynomials. The total number of parameters to be calculated is thus \(p\)\(\times\)\((m+1)\); the constraint condition is \((p-1)\)\(\times\)\(m\), and the independent parameters are  
\[r=p\cdot(m+1)-(p-1)\cdot m=p+m \tag{1}\]  
Figure 2: Schematic diagram of the installation of bedrock temperature observation equipment. The brown column is a borehole. The grey area is rock mass. The blue rectangle is the temperature sensor. The arrow direction indicates below the surface. The required solution is obtained by polynomial fitting of the \(r\) parameters. Because the continuous smoothness of the curve at the point should be considered, the required curve is written as  
\[y(x)=\sum_{i=0}^{m}a_{i}x^{i}+\sum_{j=1}^{p-1}a_{m+j}(x-\xi_{j})^{m}_{(+)} \tag{2}\]  
\[(x-\xi_{j})^{m}_{(+)}=\left\{ \begin{matrix}(x-\xi_{j})^{m}&(x\geq\xi_{j})\\ 0&(x<\xi_{j})\end{matrix} \right. \tag{3}\]  
where \(\xi_{j}\) is the \(x\) value at the \(j\)-th knot point. From formula (2), we obtain  
\[\left. \begin{array}{l}f_{j+1}=f_{j}(x)+a_{m+j}(x-\xi_{j})^{m}\\ f^{\prime}_{j+1}(x)=f^{\prime}_{j}(x)+ma_{m+j}(x-\xi_{j})^{m-1}\\ f^{\prime\prime}_{j+1}(x)=f^{\prime\prime}(x)+m(m-1)\cdot a_{m+j}(x-\xi_{j})^{m -2}\end{array} \right\} \tag{4}\]  
When \(x=\xi_{j}\), the second term of the above equations is 0, and the function value of the fitting curve at the point is equal to the first and second derivatives. That is, the fitting curve is continuous and smooth at each point \((x=\xi_{j})\). We treat \(x-\xi_{1},x-\xi_{2},\cdots,x-\xi_{p-1}\) as a new observed quantity and take \(a_{m+1},a_{m+2},\cdots,a_{m+p-1}\) as the new undetermined coefficient. Least-squares processing of \(p+m\) undetermined coefficients is performed to solve for all fitted observed data samples \(n\). The residual error equation is  
\[v_{k}=y_{k}-\sum_{i=0}^{m}a_{i}x_{k}^{i}-\sum_{i=1}^{p-1}a_{m+j}(x_{k}-\xi_{j}) ^{m} \tag{5}\]  
The least-squares method is used to obtain all the fitting coefficients according to the least-square sum of errors.