## 2 Materials and Methods
### Methods
#### 2.3.1 Data Gap-Filling  
In the preprocessing phase, addressing the challenge of missing values is vital to ensure the integrity and precision of climate-data-driven predictions. The dataset undergoes optimization through the systematic removal or statistical imputation of these missing values, rendering it more suitable for subsequent predictive modeling or cluster analysis functions. Spatial-Temporal Ground-Observed Air Temperature Gap-Filling  
In the event of malfunctioning at the meteorological station, the \(T_{\text{air}}\) dataset encounters partial incompleteness. Recognizing the missing data adherence to a pattern of missing at random (MAR), ref. [72] suggests employing straightforward imputation methods to mitigate these gaps. When the extent of missing data remains below 5 percent, a mean substitution approach is advocated. This entails replacing missing values with the mean computed from observed values spanning multiple years. However, if the percentage of missing data exceeds 5% or spans over a month, a regression model is employed. Regression models, whether utilizing a single closest neighbor station [73] or multiple nearby stations [74; 75; 76], have proven effective in estimating daily weather observations [77; 78; 79]. The imputation processes, which are applied to both \(T_{\text{max}}\) and \(T_{\text{min}}\), are represented by  
\[E(Y|X)=\beta_{0}+\beta_{1}X, \tag{1}\]  
where \(Y\) is the predicted missing air temperature, \(X\) is the reference data from a neighboring station, \(E(Y|X)\) is the expected value of \(Y\) given the values of \(X\), \(\beta_{0}\) is the intercept, and \(\beta_{1}\) is the slope. Spatial-Temporal Satellite-Based Land Surface Temperature Gap-Filling  
Addressing gaps in daily LST data derived from MODIS platforms poses a significant challenge, necessitating innovative solutions for data reconstruction and prediction. Recent studies have explored various methods, including reconstructing LST from satellite datasets, forecasting daily LST using time series data, and estimating subpixel LST by fusing multisource data [80; 81; 82; 83]. Although machine learning techniques have been introduced to LST retrieval, their application remains limited [84]. To bridge this gap, we employ the RF machine learning algorithm, celebrated for its exceptional predictive ability and adaptability to nonlinear data [85]. The RF methodology of constructing multiple decision trees provides a robust framework for classification and regression tasks, making it particularly effective for filling gaps in LST data caused by cloud cover or other disruptions. The RF algorithm is adept at managing high-dimensional data and unraveling complex relationships, which is crucial for accurately predicting missing LST values. It is noted that the RF performance is enhanced when there is a strong correlation between the target and reference variables, a principle that holds true in the analysis of intricate environmental datasets [86]. By leveraging RF, we can address missing data without relying on standard scalar functions. Although these functions are useful for normalization and standardization, they can potentially introduce bias. Instead, our approach focuses on utilizing the Python-based scikit-learn module [87] for efficient gap-filling, ensuring that the integrity and natural variability of the LST dataset are maintained. This methodology not only facilitates the prediction of missing LST values with minimal bias but also underscores the potential of machine learning in advancing the understanding and monitoring of extreme climate events [88; 89; 90; 91]. In RF, the prediction is made by aggregating predictions from multiple decision trees. The prediction \(\hat{y}\) for a given sample \(x\) is calculated by  
\[\hat{y}=\frac{1}{N}\sum_{i=1}^{N}f_{i}(x), \tag{2}\]where \(N\) is the number of decision trees in the forest, and \(f_{i}(x)\) is the prediction of the \(i\)-th decision tree.