## 3 Uncertain Analysis Method of Thermodynamic Properties
### Uncertainty Characterization Method  
The principle behind quantifying soft clay rock parameter uncertainty in spatial random fields lies in understanding and modeling the inherent spatial variability of soft clay rock properties. Geotechnical parameters, such as permeability, porosity, and shear strength, exhibit spatial heterogeneity due to geological processes, depositional patterns, and environmental factors. Spatial random field theory provides a framework to represent this variability by characterizing the statistical properties of geotechnical parameters across a given area. This involves analyzing spatial autocorrelation structures using tools like variograms and covariance functions. Geostatistical techniques, including kriging and stochastic simulation, are then employed to interpolate or simulate geotechnical parameters, accounting for spatial dependency and uncertainty. The local averaging method in random fields, also known as the moving window technique, is a powerful approach used in geostatistics and spatial analysis to simulate spatially correlated data. This method involves partitioning a random field into a series of overlapping windows and computing the local average within each window. The process begins by defining a window size and shape, typically a square or circular area, and then moving this window across the field, calculating the average of the data points within each window. The center of the window is then assigned the average value obtained. This process is repeated for each window, resulting in a new simulated field with spatially correlated values. The local averaging method offers several advantages, including simplicity, computational efficiency, and the ability to preserve local spatial patterns [39; 40]. It is particularly useful for simulating spatially correlated phenomena with complex spatial structures, such as geotechnical properties, groundwater levels, or geological formations. Parameterization of the local averaging method involves determining the appropriate window size and shape based on the spatial characteristics of the data and the desired level of spatial correlation. The choice of parameters can significantly impact the simulated results, with larger windows capturing broader spatial trends and smaller windows preserving finer-scale variability. To optimize parameters, practitioners often employ cross-validation techniques or sensitivity analyses to assess the performance of different parameter combinations. By adjusting the parameters and comparing the simulated results to observed data or known spatial patterns, the most suitable parameter values can be identified for a given application. The three hydrothermal parameters of the freezing curtain are simulated as three two-dimensional random fields. It can be expressed as follows:  
\[X_{e}=\frac{1}{A_{e}}\int_{\Omega_{e}}X(x,y)dxdy \tag{16}\]  
The covariance can be expressed as  
\[Cov(X_{e},X_{e^{\prime}})=\frac{\sigma^{2}}{A_{e}A_{e^{\prime}}}\sum_{i=1}^{4} \sum_{j=1}^{4}\sum_{m=1}^{4}\sum_{l=1}^{4}H_{i}H_{j}H_{m}H_{l}\rho(\Delta x, \Delta y)|J||J^{\prime}| \tag{17}\]where \(A_{e}\) is the area of \(e\); \(A_{e^{\prime}}\) is the area of \(e^{\prime}\); \(H_{i}\), \(H_{j}\), \(H_{m}\), and \(H_{l}\) are the weighting coefficients; \(\rho\) (\(\Delta x\), \(\Delta y\)) is the correlation structure of original random field; \(\left|\left|\left|\right.\right.\right.\) and \(\left|\left|\left.\right.\right|^{\prime}\right|\) are the Jacobian determinants of coordinate transformation. It can be seen that the covariance can be calculated as long as \(\rho\) (\(\Delta x\), \(\Delta y\)) is given. The autocorrelation function in stochastic fields quantifies the degree of similarity between a random variable and a lagged version of itself across space or time. It is a fundamental tool in analyzing spatial or temporal dependence patterns within stochastic processes. The autocorrelation function describes how the correlation between points changes as the distance between them varies.