Anomaly Detection and Correction for High-Spatiotemporal-Resolution Land Surface Temperature Data: Integrating Spatiotemporal Physical Constraints and Consistency Verification

Highlights

What are the main findings?

• A two-stage anomaly detection framework following “temporal physical pre-screening first, spatial statistical verification later” is proposed, effectively integrating diurnal variation constraints and spatial consistency checks.
• The method significantly outperforms STL and double standardization, achieving average improvement rates of 23.61%, 18.79%, 16.46%, and 61.33% in RMSE, MAE, R, and R², respectively, on hourly LST data from the Heihe River Basin.

What are the implications of the main findings?

• It provides a physically interpretable and operationally robust solution for quality control of high-temporal-resolution LST data, enhancing reliability in complex land surface thermal infrared scenarios.
• By prioritizing the reduction of “mis-correction” risk, the method is particularly suitable for applications demanding high physical consistency, such as energy balance analysis and trend studies.

Abstract

High-spatiotemporal-resolution land surface temperature (LST) data are crucial for analyzing surface energy balance, modeling temperature-related processes, and monitoring thermal environments. However, despite advancements in multi-source fusion and reconstruction techniques, high-frequency LST data remain susceptible to anomalies such as abrupt changes and outliers due to retrieval uncertainties and varying observation conditions. Conventional statistical outlier detection methods risk misidentifying physically plausible rapid weather changes as data errors, introducing systematic biases. To address this, we propose a two-stage anomaly detection framework that follows a “temporal physical pre-screening first, spatial statistical verification later” logic. First, a piecewise empirical model, based on typical diurnal LST variation characteristics, is constructed to identify points violating physical patterns. Subsequently, a spatial consistency test using median absolute deviation (MAD) is introduced to distinguish real weather-driven fluctuations from genuine data anomalies from a spatial synergy perspective. This sequential design effectively reduces the risk of mis-correcting physically reasonable temperature variations. Validated using hourly seamless LST data (2016–2021) and ground observations in the Heihe River Basin, our method outperformed Seasonal-Trend decomposition using Loess (STL), double standardization methods, and robust Holt–Winters. For over 87% of the detected anomalies, the proposed method demonstrated positive improvement rates in RMSE, MAE, R, and R². The overall average improvement rates reached 23.61%, 18.79%, 16.46%, and 61.33%, respectively, indicating robust performance. The results underscore that explicitly incorporating physical constraints enhances the reliability and interpretability of quality control for high-temporal-resolution remote sensing LST data.

1. Introduction

With the increasing demand for precision in early forest fire detection, urban heat environment assessment, and permafrost monitoring, spaceborne thermal infrared remote sensing technology is rapidly evolving towards high spatiotemporal resolution, all-weather capability, and intelligence [1,2,3,4]. Breakthroughs in the spatiotemporal resolution and quantitative capabilities of new-generation Earth observation systems have made continuous and dynamic monitoring of surface thermal processes possible [5,6]. In this context, enhancing the quantitative modeling capability for complex land surface thermal infrared scenes is of great significance for linking sensor design, physical retrieval, and scientific applications [7].

Land surface temperature (LST) is a key physical parameter characterizing land surface thermal infrared scenes. It is closely coupled with processes such as evapotranspiration, soil moisture, and vegetation status and is widely used in studies on urban heat islands, drought monitoring, ecosystem modeling, and climate change [8,9,10,11,12,13,14]. In recent years, developments in multi-source data fusion and reconstruction techniques have facilitated various high-continuity LST products, significantly enhancing the capacity for regional thermal process simulation and long-term trend analysis [15,16,17]. By integrating passive microwave data, reanalysis data, and physical models, some products have improved all-weather LST product quality while maintaining high temporal coverage [18,19,20,21].

However, influenced by multiple factors such as the inherent limitations of model parameterization [22], uncertainties in bias correction methods, theoretical assumption constraints of multi-source data fusion algorithms [23], “phase lags” between multi-source data [24], and physical definition differences between observations and simulations, high-spatiotemporal-resolution LST products inevitably still contain anomalies like abrupt changes, outliers, or trend breaks. If not identified and processed, these anomalies can disrupt the physical consistency of time series, leading to issues such as biases in energy balance estimation, misjudgment of thermal anomalies, and instability in trend analysis. Therefore, developing highly robust anomaly detection methods for high-spatiotemporal-resolution LST data has become a critical step in enhancing data quality.

Currently, anomaly detection methods for meteorological and remote sensing time series mainly fall into three categories: statistical methods, machine learning, and deep learning. Statistical methods (e.g., Seasonal-Trend decomposition using Loess (STL) [25,26] and double standardization methods [27]) are structurally simple and computationally efficient, having been widely applied in the quality control of ground weather stations and historical temperature series. Various machine learning methods can, to some extent, capture the complex spatial distribution and nonlinear relationships of multi-station temperature fields, demonstrating performance superior to traditional spatial regression and Inverse Distance Weighting (IDW) in spatial quality control and interpolation for station networks [28]. Deep learning models, by virtue of their theoretical strength in nonlinear fitting and automatic feature extraction, are particularly suited for large-scale, multivariate time series data analysis [29]. Nevertheless, current methods face notable challenges when dealing with high-temporal-frequency LST data characterized by strong non-stationarity, significant feature uncertainty, and difficulty in sample selection: statistical methods are sensitive to parameter settings and prone to false deletions or missed detections; machine learning methods rely on feature engineering and regional parameter tuning, which limits their generalizability; and deep learning methods require large sample sizes and suffer from weak interpretability. For hourly high-temporal-frequency LST data series, there is currently a lack of a systematic anomaly identification framework that balances physical rationality and detection robustness, which constrains the in-depth application of such data in refined thermal environment analysis.

Many studies have primarily focused on improving the accuracy of single detection algorithms applied solely to either the temporal or spatial dimension, with comparatively less attention paid to the detection structure and process that integrates spatiotemporal correlations [27,30]. Most existing methods formally adopt a single-stage, symmetric detection logic. In this approach, anomalies are directly determined within a single step based on the degree of statistical deviation in both time and space. This does not differentiate between the levels of preliminary screening and final confirmation, nor does it prioritize the physical laws of the temporal dimension [27,30,31]. When processing high-frequency hourly land surface temperature data, this approach has significant limitations. On one hand, the diurnal variation in LST is radiation-dominated and exhibits strong periodicity. On the other hand, rapid weather processes can lead to fluctuations that are physically reasonable but statistically anomalous [16,32]. This makes it difficult for traditional methods based on a single statistical anomaly criterion to effectively distinguish between real weather changes and data errors [30]. Therefore, this paper proposes a sequential two-stage detection framework. Its core contribution is not an improvement to a single detection method, but a reconstruction of the anomaly detection process. The fundamental difference between this work and existing spatiotemporal anomaly detection methods lies in the following: instead of attempting to compromise between missed detections and false alarms using a single threshold or a unified discriminant function, it decouples these two types of risk through an asymmetric, sequential decision-making process. These risks are then independently managed by physical constraints and spatial constraints, respectively.

To tackle these challenges, this paper proposes a two-stage anomaly detection framework following the principle of “temporal physical pre-screening first, spatial statistical verification later”. First, an empirical model based on the physical characteristics of LST diurnal variation is constructed for preliminary temporal screening (Section 3), prioritizing the retention of temporal variations conforming to physical laws. Subsequently, a spatial consistency test is introduced to filter and finally confirm the genuine anomalies from the preliminary candidates. This sequential design aims to reduce the risk of mis-deleting real weather changes due to statistical outliers from the outset, thereby enhancing the physical rationality and robustness of anomaly identification. Utilizing multi-year, multi-site observed land surface temperature series data from the Heihe River Basin (Section 2), we systematically validate the proposed method’s effectiveness in anomaly identification and correction. In addition, we conduct a comparative analysis with the STL and double standardization methods (Section 4) to evaluate their performance in practical applications. This method provides a physically interpretable and operationally robust solution for the quality control of high-temporal-frequency LST data, contributing to improved reliability and applicability of the data in complex land surface thermal infrared scenarios.

2. Selection and Analysis of Research Data

2.1. Hourly LST Dataset

The hourly LST data used in this study were sourced from the 0.02° seamless hourly land surface temperature dataset over East Asia (2016–2021) [23,33]. This product utilizes Himawari-8/Advanced Himawari Imager (AHI) thermal infrared observations as the core data source and combines the iTES retrieval algorithm, bias correction techniques, and the Multiresolution Kalman Filter (MKF) method to achieve dynamic optimal fusion of multi-source data. The product features a high temporal resolution of 1 h and a spatial resolution of approximately 2 km. It maintains good data continuity under both clear-sky and partially cloudy conditions, making it suitable for hourly monitoring and modeling analysis of land surface thermal processes. The main attributes of the dataset are summarized in Table 1.

Table 1. Attribute summary of the 0.02° seamless hourly land surface temperature dataset over East Asia (2016–2021).

	Attribute	Description
	Spatiotemporal Coverage	East Asia, 2016–2021
	Spatiotemporal Resolution	1 h, 0.02°
	Data Sources	Himawari-8 AHI data, CLDAS reanalysis data
	Core Methods	iTES retrieval, bias correction, MKF fusion
	Accuracy (RMSE)	Clear-sky: 2.59 K; Cloudy-sky: 3.38 K
	Data Archive Source	National Tibetan Plateau Data Center [34]

2.2. Ground Observation Data

The Heihe River Basin Integrated Observatory Network covers various typical underlying surface types, including alpine meadows, oasis farmlands, desert, wetlands, and shrub forests. It exhibits significant water–ecology–climate interactions, making it an ideal test area for validating remote sensing LST products [35]. This study selected seven long-term, continuously operational ground observation stations within the Heihe River Basin (http://data.tpdc.ac.cn) [36,37], as shown in Table 2. The geographical distribution of these stations is shown in Figure 1. The land surface temperature ($T_{\mathrm{g}}$) observed at the stations was calculated from upwelling and downwelling longwave radiation data using the following formula:

$$T_{\mathrm{g}}=\sqrt[4]{{\displaystyle \frac{L^{\uparrow }-\left(1-{\epsilon }_{\mathrm{BBE}}\right){\times L}^{\downarrow }}{{\epsilon }_{\mathrm{BBE}}\times \sigma }}}$$

(1)

where $T_{\mathrm{g}}$ is the station-measured land surface temperature in Kelvin (K), $L^{\uparrow }$ and $L^{\downarrow }$ are the measured upwelling and downwelling longwave radiation at the station in $\mathrm{W}/{\mathrm{m}}^2$, ${\epsilon }_{\mathrm{BBE}}$ is broadband emissivity, obtained in this study from the Global Land Surface Satellite (GLASS) Broadband Emissivity product (spatial resolution: 1 km, temporal resolution: 8 days) [38,39,40], and σ is the Stefan–Boltzmann constant (5.67 × 10⁻⁸ W/(m²·K⁴)).

Table 2. Overview of the ground validation sites.

Site Name (Code)	Underlying Surface Type	Longitude	Latitude	Elevation (m)
A’rou Superstation (S1)	subalpine meadow	100.46°E	38.05°N	3033
Daman Superstation (S2)	maize	100.37°E	38.86°N	1556
Dashalong Station (S3)	marsh alpine meadow	98.94°E	38.84°N	3739
Heihe Remote Sensing Station (S4)	grassland	100.48°E	38.83°N	1560
Ejin Desert Station (S5)	Reaumuria desert	100.99°E	42.11°N	1054
Sidaoqiao Superstation (S6)	Tamarix	101.14°E	42.00°N	873
Zhangye Wetland Station (S7)	reed	100.45°E	38.98°N	1460

Figure 1. Distribution of ground validation sites in the Heihe River Basin. The numbers 1–7 correspond to the sites listed in Table 2.

To test the applicability of the proposed method in other regions, this study selected Huailai Station in the Haihe River Basin, located in the transitional zone between the North China Plain and mountainous areas, as an independent validation site. The underlying surface of this station is a single-cropping corn field, and its climate is classified as temperate sub-humid continental, which differs significantly from the arid/semi-arid conditions of the Heihe River Basin sites. The geographical location of the station is shown in Figure 2 [41,42,43].

Figure 2. Geographical location of Huailai Station in the Haihe River Basin.

Due to systematic differences in physical meaning, sensor characteristics, and spatial representativeness between remotely sensed LST and the infrared radiometric temperature obtained from ground stations, previous work has commonly reported a stable absolute bias between the two time series, while their diurnal variation phase and relative patterns remain highly consistent [44,45,46,47,48,49,50]. Under strict quality control conditions, the station radiometric temperature and remotely sensed LST show high synchrony at the hourly scale. Therefore, these sites provide a reliable physical baseline for evaluating LST anomalies.

To assess the consistency in temporal patterns between the original remote sensing LST data and the station-observed land surface temperature, we calculated the correlation coefficient (R) for 14,589 valid days across the seven sites in the Heihe River Basin from 2016 to 2021. Statistical analysis (Figure 3) indicates that the remote sensing LST and station observation sequences exhibit highly synchronous diurnal variation patterns on the vast majority of days, with an overall average correlation coefficient as high as 0.95 and with over 90.20% of days having an R value above 0.90.

Figure 3. (a) Histogram of correlation coefficient (R) distribution; (b) average correlation coefficient for different sites.

The distribution of R values shows excellent overall data quality, with approximately 74.32% of days falling within the extremely high correlation range (R ≥ 0.95). However, the analysis also identified a very small number of anomalies: a total of 7 days (0.048%) showed a negative correlation (R < 0), and a few other days exhibited a weak correlation.

3. Methodology

3.1. Typical Anomaly Types

Pixel-level time series analysis of the hourly LST sequences from 2016 to 2021 identified two representative types of anomalous features, as shown in Figure 4. Both types can cause significant interference to time series modeling and spatial analysis and, therefore, require effective identification and correction.

Figure 4. Examples of anomaly types in remote sensing LST and ground station temperature time series: (a) single-point spike anomaly on 29 April 2017 (site S1); (b) single-point spike anomaly on 4 October 2016 (site S7); (c) continuous segment anomaly on 27 November 2016 (site S5); (d) mixed anomalies on 21 January 2018 (site S2). The red star denotes the identified spatiotemporal anomaly point, and the red dashed line indicates the deviation between the continuous anomalous LST curve and the station temperature curve.

Single-point Spike Anomaly: This is manifested as a sudden, sharp peak or drop in temperature at a specific moment, exhibiting an extreme difference from preceding and following time points.

Continuous Segment Anomaly: This is manifested as a systematic deviation of LST from the ground observation trend over a period of time.

3.2. General Idea and Framework

Anomaly detection in time series is of great importance in the quality control of remote sensing data. Owing to their intuitive methodologies and high computational efficiency, statistical approaches are extensively employed for this purpose. Classical methods such as Seasonal-Trend decomposition using Loess (STL) and the double standardization method have demonstrated good robustness in various time series scenarios: STL identifies anomalies by decomposing a time series into trend, seasonal, and residual components and detecting outliers within the residuals; the double standardization method performs standardization in both the temporal and spatial dimensions, respectively, and identifies anomalies based on set thresholds. In addition, the robust Holt–Winters method, a classic robust filtering technique, effectively mitigates the impact of outliers on trend and seasonal component estimation by recursively updating and cleaning observations in real time. However, STL is sensitive to seasonal period length and smoothing parameters, making it difficult to accurately characterize diurnal variation features at an hourly scale. Furthermore, although the double standardization method considers both temporal and spatial information, it is essentially a global outlier detection method and struggles to distinguish between different anomaly types. Although the robust Holt–Winters method can mitigate the impact of anomalies, it is still a purely temporal statistical method that lacks spatial information constraints, making it difficult to distinguish between real weather fluctuations and data errors.

To address the aforementioned shortcomings, this paper proposes a two-stage anomaly detection method following a “temporal first, spatial later” sequence, based on the physical mechanisms of temperature change. This method first conducts preliminary screening of anomalies based on physical constraints within the time series, followed by further verification through spatial statistical testing. Compared to the traditional approach that directly identifies anomalies based on outlier characteristics, this design can effectively distinguish between genuine weather events that are “physically plausible but statistically outliers” and data errors that are “physically implausible and spatially incoherent,” thereby systematically reducing the risk of misjudgment. This study follows a technical framework of “data input → anomaly point detection → anomaly point correction → method evaluation.” The overall research framework is illustrated in Figure 5.

Figure 5. Overall research approach and framework. The dashed line demarcates the two core stages of the anomaly detection module: the upper temporal dimension pre-screening and the lower spatial consistency verification.

3.3. Two-Stage Anomaly Detection Method

3.3.1. Temporal Anomaly Pre-Screening: Empirical Model Based on Diurnal Variation Physical Characteristics

The diurnal variation in LST is controlled by the periodic change in solar radiation, typically manifesting as a continuous temperature change process of cooling at night and warming during the day. As shown in Figure 6, a single day is divided into three stages: early morning/nighttime, daytime, and evening/nighttime. Exponential decay models and a sinusoidal function are employed to fit the hourly LST for each stage, respectively [51].

Figure 6. Simulation results of the piecewise fitting model for the diurnal cycle (site S1, 18 September 2020).

Pre-dawn and nighttime segment ($0\le t<t_{\mathrm{sr}}$):

$$T\left(t\right)=A_1\exp \left(B_{1}t\right)+C_1$$

(2)

Daytime segment ($t_{\mathrm{sr}}\le t\le t_{\mathrm{ss}}$):

$$T\left(t\right)=A_2\sin \left[\omega \left(t-\varphi \right)\right]+B_2$$

(3)

Evening and nighttime segment ($t_{\mathrm{ss}}<t<24$):

$$T\left(t\right)=A_3\mathit{exp}\left(B_{3}t\right)+C_3$$

(4)

where $t_{\mathrm{sr}}$ and $t_{\mathrm{ss}}$ are the sunrise and sunset times, respectively, calculated using a standard sunrise equation and rounded to the nearest hour. A, B, C, $\omega$, $\varphi$ are model parameters for each segment.

By analyzing the residual distribution $r_i=T_i-\widehat{T}\left(t_i\right)$ between the remote sensing temperature value $T_i$ and the fitted value $\widehat{T}\left(t_i\right)$, a dynamic threshold $\theta =k\cdot {\sigma }_r$ is constructed based on the statistical characteristics of the residuals, where ${\mathrm{\sigma }}_{\mathrm{r}}$ is the standard deviation of the annual residual sequence, used to identify moments that significantly deviate from the theoretical trajectory. When the absolute value of the residual exceeds the threshold, the corresponding time point is flagged as a temporal suspect anomaly.

Through a full-dataset parameter sensitivity analysis (see Section 4.1 for details), this study comprehensively evaluated the impact of the temporal threshold coefficient k on detection performance and ultimately determined k = 2. This parameter achieves an optimal balance between correction magnitude and correction reliability, while providing a moderate-sized candidate set for the subsequent spatial consistency check.

3.3.2. Spatial Consistency Test: Robust Statistical Method Based on MAD

Mutations in the time series may originate from genuine weather events (e.g., cold front passage). To distinguish such physical anomalies from data errors, a spatial consistency test is necessary. Its core logic is that genuine weather changes exhibit spatial synergy, whereas data anomalies manifest as local incoherence.

For each temporal suspect anomaly point, a spatial neighborhood window is constructed centered on it, and a robust Z-score is calculated for that point:

$$Z_{\omega }={\displaystyle \frac{T_c-M_{\omega }}{1.4826\times {\mathrm{M}\mathrm{A}\mathrm{D}}_{\omega }}}$$

(5)

where $T_c$ is the temperature at the center point, and $M_{\omega }$ and ${MAD}_{\omega }$ are the median and median absolute deviation of the temperatures within the neighborhood, respectively (the constant 1.4826 is used for consistency with the standard deviation under a normal distribution). If $\left|Z_{\omega }\right|>Z_t$, the pixel is identified as a final anomaly, where $Z_t$ is the preset spatial anomaly threshold. This study comprehensively evaluated the impact of the spatial anomaly threshold $Z_t$ on detection performance through a full-dataset parameter sensitivity analysis (see Section 4.1 for details), and ultimately determined $Z_t=3$. The selection of the window size was also based on systematic sensitivity experiments (see Section 4.1 for details), with the 7 × 7 window ultimately determined as the globally optimal parameter.

Figure 7 comprehensively illustrates the “temporal identification–spatial verification” process using data from site S1 in 2020 as an example. Figure 7a shows the residual distribution from the temporal anomaly detection, where two temporal suspect anomaly points (Point 1 and Point 2) are annotated, both exhibiting significant local mutations in the time series. The subsequent spatial test intuitively demonstrates the difference in spatial synergy through temperature heatmaps of a 7 × 7 window (Figure 7d,e)—the heatmap corresponding to the normal fluctuation point (Figure 7d) shows a continuous and coherent temperature distribution, whereas the heatmap corresponding to the suspect anomaly point (Figure 7e) shows evident temperature incoherence between the central pixel and its surroundings. Finally, Figure 7f presents the residual distribution corresponding to the anomaly points retained after the spatial consistency test.

Figure 7. (a) Residual distribution of temporal anomaly detection for the 2020 LST data at site S1; (b) local temperature anomaly in the time series corresponding to Point 1 in (a) (12:00, 20 July 2020, site S1); (c) local temperature anomaly in the time series corresponding to Point 2 in (a) (16:00, 28 July 2020, site S1). (d) The thermal map of a 7 × 7 pixel window around Point 1 during the local temperature anomaly shown in (b) demonstrates good spatial consistency. (e) The thermal map of a 7 × 7 pixel window around Point 2 during the local temperature anomaly shown in (c) shows significant spatial inconsistency; (f) residual distribution of spatial anomaly detection for site S1 in 2020.

3.4. Anomaly Value Correction Method

For anomaly points confirmed through two-stage detection, interpolation methods are required for correction to restore reasonable temperature values. This study evaluated the correction performance of two methods: Inverse Distance Weighting (IDW) interpolation and bilinear interpolation. IDW interpolation is a local deterministic method whose weights are solely distance-dependent, allowing it to flexibly utilize the spatial correlation of valid pixels in the neighborhood. Bilinear interpolation, on the other hand, constructs an interpolation surface based on piecewise linear functions. The comparative results indicate that in the specific scenario of correcting localized spatial anomaly pixels, the IDW method exhibits stronger adaptability and robustness, with significantly better correction accuracy. Therefore, considering both the correction effectiveness and computational efficiency, the IDW interpolation method was ultimately selected. This study employs IDW interpolation using a 7 × 7 window as the neighborhood range. The specific formula is as follows [52]:

$$T_{\mathrm{cor}}={\displaystyle \frac{{\sum }_{i=1}^n{\omega }_i{T}_i}{{\sum }_{i=1}^n{\omega }_i}},\hspace{1em}{ \omega }_i={\displaystyle \frac{1}{d_i^2}}$$

(6)

where $T_i$ is the temperature of the $i$-th neighboring pixel, $d_i$ is its distance to the central anomaly point, and $n$ is the number of valid pixels within the neighborhood. A detailed comparative analysis of the two interpolation methods, an explanation of the principles of bilinear interpolation, and the complete correction performance data are provided in Appendix A.

3.5. Evaluation Metrics

To objectively evaluate the effects of anomaly point screening and correction, we use ground observation station data as the reference benchmark. Due to differences in spatial representativeness, observation mechanisms, and physical definitions between satellite-retrieved LST and station-observed land surface temperature, their absolute values are generally not directly comparable. However, their diurnal variation patterns and relative fluctuations are highly synchronous. Therefore, in this study, we primarily evaluate the consistency in variation trends and relative patterns between the remote sensing LST sequence and the station observation sequence. To eliminate potential systematic biases and focus the evaluation on sequence patterns, we adopt a dynamic normalization strategy based on a local temporal window.

For each point identified as a spatiotemporal anomaly, a local temporal window (n = 11) is constructed by extracting remote sensing LST and station-observed land surface temperature data from 5 h before to 5 h after its corresponding time point (totaling 11 h). To ensure that statistical calculations are not affected by the anomaly point itself, an “exclude the center point” normalization method is used. First, the mean (μ) and standard deviation (σ) are calculated using the other 10 data points in the window, excluding the center point. Then, all data within the window (including the center point) undergo Z-score normalization. Based on this, to quantitatively assess the agreement between the original remote sensing LST sequence (and the corrected LST sequence) and the station-observed temperature sequence, the following metrics are calculated:

Root Mean Square Error (RMSE): This metric measures the overall level of difference between the remote sensing LST sequence and the station-observed land surface temperature sequence.

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\left[T_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}\left(i\right)-T_{\mathrm{r}\mathrm{e}\mathrm{m}\mathrm{o}\mathrm{t}\mathrm{e}}\left(i\right)\right]}^2}$$

(7)

Mean Absolute Error (MAE): This metric measures the average absolute magnitude of the difference between the remote sensing LST sequence and the station-observed land surface temperature sequence.

$$MAE={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n\left|T_{station}\left(i\right)-T_{remote}\left(i\right)\right|$$

(8)

Pearson Correlation Coefficient (R): This metric measures the linear correlation in the direction and phase of change between the remote sensing LST sequence and the station-observed land surface temperature sequence.

$$R={\displaystyle \frac{{\sum }_{i=1}^n\left[T_{\mathrm{station}}\left(i\right)-\overline{T_{\mathrm{station}}}\right]\left[T_{\mathrm{remote}}\left(i\right)-\overline{T_{\mathrm{remote}}}\right]}{\sqrt{{\sum }_{i=1}^n{\left[T_{\mathrm{station}}\left(i\right)-\overline{T_{\mathrm{station}}}\right]}^2}\cdot \sqrt{{\sum }_{i=1}^n{\left[T_{\mathrm{remote}}\left(i\right)-\overline{T_{\mathrm{remote}}}\right]}^2}}}$$

(9)

Coefficient of Determination (R²): This metric measures the consistency in the magnitude of variation between the remote sensing LST sequence and the station-observed land surface temperature sequence.

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^n{\left[T_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}\left(i\right)-T_{\mathrm{r}\mathrm{e}\mathrm{m}\mathrm{o}\mathrm{t}\mathrm{e}}\left(i\right)\right]}^2}{{\sum }_{i=1}^n{\left[T_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}\left(i\right)-\overline{T_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}}\right]}^2}}$$

(10)

Let ${\mathrm{RMSE}}_{\mathrm{raw}}$, ${\mathrm{MAE}}_{\mathrm{raw}}$, $R_{\mathrm{raw}}$, and ${R^2}_{\mathrm{raw}}$ denote the respective metric values calculated based on the original remote sensing LST sequence and the station observation sequence, and let ${\mathrm{RMSE}}_{\mathrm{cor}}$, ${\mathrm{MAE}}_{\mathrm{cor}}$, $R_{\mathrm{cor}}$, and ${R^2}_{\mathrm{cor}}$ denote the metric values calculated based on the corrected remote sensing LST sequence and the station observation sequence. For error metrics (RMSE, MAE), lower values are better. The relative improvement rate for RMSE, ${∆}_{\mathrm{RMSE}}$, is defined as

$${∆}_{\mathrm{RMSE}}={\displaystyle \frac{{\mathrm{RMSE}}_{\mathrm{raw}}-{\mathrm{RMSE}}_{\mathrm{cor}}}{{\mathrm{RMSE}}_{\mathrm{raw}}}}\times 100\%$$

(11)

The relative improvement rate for MAE, ${∆}_{\mathrm{MAE}}$, is calculated as

$${∆}_{\mathrm{MAE}}={\displaystyle \frac{{\mathrm{MAE}}_{\mathrm{raw}}-{\mathrm{MAE}}_{\mathrm{cor}}}{{\mathrm{MAE}}_{\mathrm{raw}}}}\times 100\%$$

(12)

Here, ${∆}_{\mathrm{RMSE}}>0$ or ${∆}_{\mathrm{MAE}}>0$ indicates reduced error after correction, i.e., performance improvement.

For correlation metrics (R, R²), higher values are better. The relative improvement rate for R, ${∆}_R$, is defined as

$${∆}_R={\displaystyle \frac{R_{\mathrm{cor}}-R_{\mathrm{raw}}}{\left|R_{\mathrm{raw}}\right|}}\times 100\%$$

(13)

The relative improvement rate for R², ${∆}_{R^2}$, is calculated as

$${∆}_{R^2}={\displaystyle \frac{{R^2}_{\mathrm{cor}}-{R^2}_{\mathrm{raw}}}{\left|{R^2}_{\mathrm{raw}}\right|}}\times 100\%$$

(14)

Here, ${∆}_R>0$ or ${∆}_{R^2}>0$ indicates enhanced correlation after correction, i.e., performance improvement.

Based on the relative improvement rate for individual anomaly points, the following statistics are further defined to comprehensively evaluate method performance in terms of different dimensions:

Improvement Proportion: This refers to the percentage of points among all detected anomaly points whose relative improvement rate is greater than 0.

Overall Average Improvement Rate: This refers to the arithmetic mean of the relative improvement rates of all anomaly points.

The formulas for calculating the overall average of the original standardized metric values (${\overline{X}}_{\mathrm{raw},\mathrm{all}}$) and the overall average of the corrected standardized metric values (${\overline{X}}_{\mathrm{cor},\mathrm{all}}$) are as follows:

$${\overline{X}}_{\mathrm{raw},\mathrm{all}}={\displaystyle \frac{1}{N_{\mathrm{total}}}}{\sum }_{k=1}^{N_{\mathrm{total}}}{X}_{\mathrm{raw},k}$$

(15)

$${\overline{X}}_{\mathrm{cor},\mathrm{all}}={\displaystyle \frac{1}{N_{\mathrm{total}}}}{\sum }_{k=1}^{N_{\mathrm{total}}}{X}_{\mathrm{cor},k}$$

(16)

where $X$ represents any evaluation metric (RMSE, MAE, R, or R²), $N_{\mathrm{total}}$ is the total number of 7 × 7 spatial anomaly points across all monitoring sites, ${\mathrm{X}}_{\mathrm{raw},\mathrm{k}}$ and ${\mathrm{X}}_{\mathrm{cor},\mathrm{k}}$ are the original standardized metric value and the corrected standardized metric value for the $k$-th anomaly point, respectively.

4. Results and Analysis

4.1. Parameter Sensitivity Analysis

The two-stage anomaly detection framework proposed in this paper involves three key parameters: the temporal dimension threshold coefficient k, the spatial consistency check window size, and the spatial anomaly threshold $Z_{\mathrm{t}}$. To scientifically determine these parameters and evaluate their stability under different conditions, a layer-by-layer parameter sensitivity analysis was conducted based on hourly LST data from seven stations in the Heihe River Basin from 2016 to 2021. This analysis progressed from preliminary single-station experiments to systematic evaluation on the full dataset, and further examined the performance of the parameters under different seasons and land cover types. The detailed analysis process and complete data are provided in Appendix B.

First, the temporal dimension threshold coefficient k was examined. Figure 8 shows the number of detected anomaly points, overall average improvement rates, and improvement proportions for different k values. Considering both the number of detections and correction performance, k = 2 achieves a relatively high improvement magnitude while ensuring good correction reliability (RMSE improvement of 23.61%, improvement proportion of 91.24%). Across different seasons and land cover types, the improvement rates and improvement proportions for k = 2 exhibit minimal fluctuation (standard deviation of 1.0% for improvement rates and 2.8% for improvement proportions), demonstrating good stability.

Figure 8. Sensitivity analysis of the temporal threshold coefficient k: (a) number of detected anomalies; (b) overall average improvement rates; (c) improvement proportions.

The selection of the spatial consistency check window size was also based on systematic evaluation. Figure 9 presents the sensitivity analysis results for different window sizes. The 7 × 7 window achieves improvement rates and improvement proportions that are at or near optimal levels, with minimal differences compared to larger windows (9 × 9, 11 × 11). Its performance remains stable across different seasons and land cover types (standard deviation of 0.9% for improvement rates and 2.8% for improvement proportions). Further increasing the window size yields limited gains in correction performance, whereas the 7 × 7 window achieves a favorable balance between computational efficiency and detection recall.

Figure 9. Sensitivity analysis of window size: (a) number of detected anomalies; (b) overall average improvement rates; (c) improvement proportions.

Regarding the spatial anomaly threshold $Z_t$, the analysis indicates that $Z_t=3$ achieves a relatively good balance between detection recall and correction reliability. Figure 10 shows the sensitivity analysis results for different $Z_t$ thresholds. The improvement rates and improvement proportions for the $Z_t=3$ threshold remain stable across different seasons and land cover types (standard deviation of 1.7% for improvement rates and 1.9% for improvement proportions). Its improvement magnitude is essentially comparable to that of $Z_t=3.5$, but with a more reasonable number of detections.

Figure 10. Sensitivity analysis of the spatial anomaly threshold ${\mathit{Z}}_{\mathbf{t}}$: (a) number of detected anomalies; (b) overall average improvement rates; (c) improvement proportions.

Based on the comprehensive analysis above, this study selected k = 2, a 7 × 7 window, and $Z_t=3$ as the final parameter combination. This combination consistently achieves a favorable balance between correction magnitude and correction reliability across different seasons and land cover types, demonstrating the robustness of the parameter selection.

4.2. Processing Results and Analysis of the Proposed Method

To systematically evaluate the performance of the proposed two-stage anomaly detection framework—“temporal physical pre-screening first, spatial statistical verification later”—we first analyze the detection results of the method itself. All analyses are based on hourly remote sensing LST data from 2016 to 2021 and concurrent ground-observed land surface temperature data from the seven sites in the Heihe River Basin.

As shown in Table 3, during the study period from 2016 to 2021, the proposed method detected a total of 1175 spatiotemporal anomaly data points from the hourly LST sequences of all sites, with an annual average of approximately 195.83.

Table 3. Annual statistics of anomaly detection counts and improvement proportions of the proposed method (2016–2021).

Year	Total Anomalies	RMSE Improvement Proportion (%)	MAE Improvement Proportion (%)	R Improvement Proportion (%)	R2 Improvement Proportion (%)
2016	215	90.23	90.23	88.84	90.23
2017	188	92.55	92.55	91.49	92.55
2018	200	95.50	95.00	92.50	95.50
2019	184	90.76	90.76	82.07	90.76
2020	213	90.14	90.14	86.38	90.14
2021	175	88.00	88.57	85.14	88.00
all/Avg.	1175	91.23	91.23	87.83	91.23

Using the ground-observed land surface temperature data as the benchmark, a quantitative evaluation of the performance of all detected anomaly points before and after correction was conducted. The results indicate that our method exhibits high accuracy in identifying highly suspected data anomalies. As shown in Table 3, among all 1175 anomaly points, the improvement proportions for RMSE and MAE are as high as 91.23% and 91.23%, respectively, meaning the absolute error was effectively reduced for the vast majority of anomaly points. Similarly, the respective improvement proportions for R and R² reach 87.83% and 91.23%. As shown in Table 4 and Table 5, the proposed method demonstrates stable positive improvement across all four evaluation metrics. The six-year average overall improvement rates for RMSE and MAE are 23.61% and 18.79%, respectively, indicating that the correction effectively reduced the absolute deviation between remote sensing LST and ground-observed land surface temperature. Meanwhile, the average overall improvement rates for R² and R are as high as 61.33% and 16.46%, indicating the method’s effectiveness in restoring time series pattern and phase consistency. The standard deviations of the improvement rates for each metric are generally low (e.g., 2.12% and 1.65% for RMSE and MAE improvement rates), indicating that the performance of the proposed method is less affected by interannual variations and maintains robust correction effectiveness across different years, demonstrating good robustness and adaptability.

Table 4. Standardized error metrics before and after correction and overall average improvement rates of the proposed method (2016–2021).

Year	RMSE (Original) (K)	RMSE (Corrected) (K)	RMSE Overall Avg. Improvement Rate (%)	MAE (Original) (K)	MAE (Corrected) (K)	MAE Overall Avg. Improvement Rate (%)
2016	0.50	0.375	24.44	0.37	0.301	19.26
2017	0.42	0.31	24.43	0.31	0.249	19.24
2018	0.61	0.399	27.18	0.41	0.311	21.67
2019	0.58	0.446	21.55	0.43	0.355	17.07
2020	0.60	0.453	22.82	0.45	0.365	18.48
2021	0.57	0.426	20.76	0.42	0.342	16.65
Avg.	0.55	0.40	23.61	0.40	0.32	18.79

Table 5. Standardized correlation metrics before and after correction and overall average improvement rates of the proposed method (2016–2021).

Year	R (Original)	R (Corrected)	R Overall Avg. Improvement Rate (%)	R2 (Original)	R2 (Corrected)	R2 Overall Avg. Improvement Rate (%)
2016	0.84	0.87	35.65	0.59	0.75	41.47
2017	0.90	0.93	10.46	0.75	0.86	80.06
2018	0.81	0.87	15.14	0.19	0.73	52.01
2019	0.82	0.85	10.17	0.46	0.69	73.75
2020	0.80	0.85	14.15	0.41	0.69	68.84
2021	0.80	0.85	10.23	0.40	0.69	54.05
Avg.	0.83	0.87	16.46	0.47	0.73	61.33

To further investigate the performance differences in the method across various sites, the seven stations were classified into four categories based on their underlying surface attributes: alpine meadow (S1, S3), farmland/artificial vegetation (S2, S4), desert shrubland (S5, S6), and wetland (S7). The correction performance for each station category was statistically analyzed (see the k = 2 columns in Table A11, Table A12, Table A13 and Table A14 of Appendix B for details). The results indicate that the proposed method exhibits stable correction effectiveness across different land surface types.

Taking the overall average improvement rate of RMSE as an example (see the k = 2 column in Table A11 of Appendix B), the values for alpine meadow, farmland/artificial vegetation, desert shrubland, and wetland are 23.99%, 23.46%, 24.21%, and 22.29%, respectively. The maximum difference among categories is 1.92 percentage points, and these values are relatively close to the overall average level of the Heihe River Basin (23.61%). Regarding the RMSE improvement proportion, the values are 91.47% for alpine meadow, 88.07% for farmland/artificial vegetation, 94.86% for desert shrubland, and 91.10% for wetland. Except for farmland/artificial vegetation, which is slightly lower, the other three categories all exceed 91%, indicating minimal fluctuation.

For MAE, the overall average improvement rates (see the k = 2 column in Table A12 of Appendix B) are 19.24%, 18.46%, 19.23%, and 17.95% for alpine meadow, farmland/artificial vegetation, desert shrubland, and wetland, respectively, with differences among categories within 1.29 percentage points. The MAE improvement proportions are 91.18%, 88.07%, 94.86%, and 91.62%, respectively. Except for farmland/artificial vegetation, which is slightly lower, the other three categories all exceed 91%, a trend generally consistent with the RMSE improvement proportion.

The overall average improvement rate for R (see the k = 2 column in Table A13 of Appendix B) shows that farmland/artificial vegetation achieves the highest improvement (23.36%), followed by alpine meadow (21.49%), while desert shrubland and wetland have relatively lower values of 4.12% and 13.62%, respectively. This may be related to the spatial heterogeneity characteristics of different land surface types: desert shrubland and wetland areas have relatively complex land cover and stronger spatial heterogeneity, whereas the land cover in farmland/artificial vegetation and alpine meadow is relatively homogeneous. Notably, however, the R improvement proportion exceeds 84% for all categories, with desert shrubland and wetland reaching 91.78% and 90.58%, respectively. This indicates that the correlation between the corrected sequence and the observed sequence is enhanced for the vast majority of anomaly points.

The overall average improvement rate for R² (see the k = 2 column in Table A14 of Appendix B) is 67.57%, 59.04%, 25.24%, and 109.60% for alpine meadow, farmland/artificial vegetation, desert shrubland, and wetland, respectively. The R² improvement proportions are 91.47%, 88.07%, 94.86%, and 91.10%, respectively, which are highly consistent with the RMSE improvement proportions, further supporting the reliability of the correction effectiveness.

As shown in Figure 11, after detection by the proposed method and correction by IDW interpolation, the anomalous time points are reasonably replaced. The diurnal variation curve is restored to a smooth shape and closely aligns with the ground-observed LST time series. These examples collectively demonstrate that the proposed method possesses effective detection and correction capabilities for various typical anomalies, enhancing the physical plausibility and visual coherence of the time series.

Figure 11. Comparison of time series before and after screened anomaly correction: (a) S1 site, 12 October 2016; (b) S4 site, 7 December 2016; (c) S7 site, 22 April 2021; (d) S5 site, 27 November 2016. The red star denotes the identified spatiotemporal anomaly point.

To further evaluate the method’s effectiveness from the perspective of the entire anomaly detection and correction process, we compare and analyze the results at each stage from the viewpoint of residual distribution. Figure 12 systematically illustrates the changes in residual distribution throughout the entire process of “temporal detection → anomaly identification → correction recovery” for the 2020 data of site S1.

Figure 12. Residual distributions at different stages of anomaly detection and correction for site S1 in 2020: (a) residual distribution from temporal anomaly detection; (b) residual distribution of original anomaly points; (c) residual distribution after IDW correction.

Figure 12a shows the residual distribution calculated based on the piecewise fitting model during the temporal anomaly detection stage, where points significantly deviating from the theoretical trajectory are identified as suspected anomalies. The original residual distribution of pixels finally confirmed as anomalies through two-stage detection is shown in Figure 12b, exhibiting discrete, high-amplitude “single-point spike” characteristics, indicating the presence of localized anomalies that severely deviate from physical laws in the original data.

After IDW interpolation correction, the residual distribution of the corresponding anomaly points (Figure 12c) significantly converges: the range of residual distribution narrows, high-value areas become more concentrated, and the residual amplitudes of different anomaly points tend to be uniform. This indicates that the correction effectively restored the temperature plausibility of the anomaly points, making the LST sequence more aligned with the theoretical diurnal variation pattern in the temporal dimension, thereby indicating the effectiveness of the proposed method in correcting local anomalies and enhancing the physical consistency of the sequence.

4.3. Ablation Experiment

To systematically verify the necessity of each component in the two-stage framework of “temporal physical preliminary screening first, spatial statistical verification second” proposed in this paper, a comprehensive ablation experiment was designed. The performance of the following three methods was compared and analyzed: (A) temporal physical pre-screening only; (B) spatial statistical verification only; (C) the complete sequential two-stage framework (i.e., the proposed method).

Table 6 summarizes the total number of detected anomalies for the three methods during 2016–2021. Method A detected the largest total (22,175), which is about 19 times that of Method C. Method B also detected a relatively large total (11,583), about 10 times that of Method C. Method C, through the dual constraints of physics and space, detected only 1175 anomalies over the six years, indicating that the two-stage screening effectively filters out the numerous suspected points generated by a single step.

Table 6. Comparison of anomaly detection counts and improvement rates for three methods (2016–2021 Overall average).

Method	Total Anomaly Detections	RMSE Improvement Proportion (%)	MAE Improvement Proportion (%)	R Improvement Proportion (%)	R2 Improvement Proportion (%)
A	22,175	70.47	70.11	68.84	69.92
B	11,583	40.04	40.11	39.26	38.38
C	1175	91.23	91.23	87.83	91.23

In terms of improvement proportion (Table 6), Method C exceeds 87% across all four metrics, while Methods A and B only achieve between 38% and 71%. Regarding error metrics (Table 7), Method C shows improvements of 23.61% and 18.79% in RMSE and MAE, respectively, whereas Methods A and B both yield negative values, indicating that the error actually increases after correction. For the correlation metrics (Table 8), Method C achieves improvements of 16.46% in R and 61.33% in R², outperforming Methods A and B.

Table 7. Comparison of RMSE and MAE correction performance for three methods (2016–2021 Overall average).

Method	RMSE (Original) (K)	RMSE (Corrected) (K)	RMSE Overall Avg. Improvement Rate (%)	MAE (Original) (K)	MAE (Corrected) (K)	MAE Overall Avg. Improvement Rate (%)
A	0.40	0.36	−9.79	0.31	0.29	−8.11
B	0.34	0.37	−24.93	0.27	0.30	−20.88
C	0.55	0.40	23.61	0.40	0.32	18.79

Table 8. Comparison of R and R² correction performance for three methods (2016–2021 Overall average).

Method	R (Original)	R (Corrected)	R Overall Avg. Improvement Rate (%)	R2 (Original)	R2 (Corrected)	R2 Overall Avg. Improvement Rate (%)
A	0.89	0.91	1.77	0.75	0.80	6.85
B	0.91	0.89	−1.02	0.79	0.77	−2.24
C	0.83	0.87	16.46	0.46	0.73	61.33

As shown in Table 7 and Table 8, Method C exhibited higher error metrics (RMSE 0.55 K, MAE 0.40 K) and lower correlation metrics (R 0.83, R² 0.46) than Methods A and B before correction. This stems from the fundamental difference in their screening strategies. Methods A and B employ single-dimensional screening, resulting in candidate sets that contain numerous physically reasonable temperature fluctuations (e.g., normal variations caused by weather processes). These points have relatively small initial errors and higher correlation with the observed sequence, leading to seemingly “better” overall pre-correction metrics. In contrast, Method C, through its dual constraints of “temporal pre-screening + spatial verification,” retains only those data points that violate both temporal physical laws and exhibit spatial isolation—points that typically have large initial errors and are the primary targets for correction. Therefore, the relative disadvantage of Method C in pre-correction metrics corroborates the precision of its screening strategy.

The results of the ablation experiment indicate that single-dimension testing has significant limitations. Although the temporal physical model is based on diurnal variation patterns, it fits a typical trend and is sensitive to rapid weather changes such as short-term cloud cover and local convection, making it prone to misclassifying such physically reasonable changes as anomalies. Spatial dimension testing, on the other hand, relies on neighborhood statistics. Real weather processes (e.g., cold front passages) often exhibit gradual spatial variations, and surface heterogeneity can form inherent thermal patches. When the scale of these reasonable spatial variations is comparable to the testing window, the central pixel can easily be misjudged as an anomaly.

The complete two-stage framework, through its “temporal first, spatial second” sequential design, first uses the temporal physical model to filter out the vast majority of physically reasonable temperature fluctuations. This ensures that the candidate points entering the spatial test are themselves highly suspicious in the temporal dimension. Subsequently, spatial statistics are applied for the final discrimination on this significantly reduced candidate set, fundamentally lowering the probability of misidentifying real weather changes as data errors. The results of the ablation experiment indicate that this sequential logic is not a simple additive process but rather an effective architecture for achieving high-reliability quality control.

4.4. Cross-Site Validation

To test the applicability of the proposed method in different regions, an independent validation was conducted at Huailai Station. Hourly observed temperature data for this station from 2016 to 2021, along with LST product data corresponding to the same spatiotemporal range, were obtained and processed according to the two-stage detection and IDW correction procedure described in Section 3.

Applying the proposed method, a total of 321 spatiotemporal anomaly points were detected at Huailai Station between 2016 and 2021. Based on the evaluation index system outlined in Section 3.5, the correction effectiveness for the detected anomaly points was assessed. The number of anomaly points per year and the performance metrics before and after correction are shown in Table 9 and Table 10, respectively, while the improvement proportions are summarized in Table 11.

Table 9. Comparison of error metrics and overall average improvement rates before and after anomaly correction at Huailai Station (2016–2021).

Year	Number of Anomaly Points	RMSE (Original) (K)	RMSE (Corrected) (K)	RMSE Overall Avg. Improvement Rate (%)	MAE (Original) (K)	MAE (Corrected) (K)	MAE Overall Avg. Improvement Rate (%)
2016	62	0.50	0.37	22.83	0.36	0.29	18.39
2017	47	0.60	0.41	23.54	0.40	0.32	18.23
2018	68	0.59	0.44	23.44	0.45	0.35	21.54
2019	23	0.44	0.37	16.19	0.32	0.28	13.62
2020	44	0.64	0.47	21.68	0.50	0.39	16.84
2021	77	0.57	0.42	20.01	0.42	0.34	17.33
Total/Average	321	0.56	0.42	21.75	0.41	0.33	18.23

Table 10. Comparison of correlation metrics and overall average improvement rates before and after anomaly correction at Huailai Station (2016–2021).

Year	Number of Anomaly Points	R (Original)	R (Corrected)	R Overall Avg. Improvement Rate (%)	R2 (Original)	R2 (Corrected)	R2 Overall Avg. Improvement Rate (%)
2016	62	0.85	0.89	6.57	0.61	0.78	55.37
2017	47	0.82	0.88	11.55	0.38	0.76	51.76
2018	68	0.80	0.85	6.38	0.38	0.69	79.65
2019	23	0.89	0.90	0.87	0.75	0.81	79.41
2020	44	0.78	0.86	27.58	0.47	0.72	63.60
2021	77	0.81	0.86	61.93	0.39	0.71	30.42
Total/Average	321	0.82	0.87	23.01	0.46	0.73	56.85

Table 11. Annual statistics of improvement proportion for anomaly correction at Huailai Station (2016–2021).

Year	RMSE Improvement Proportion (%)	MAE Improvement Proportion (%)	R Improvement Proportion (%)	R2 Improvement Proportion (%)
2016	80.65	80.65	80.65	80.65
2017	85.11	85.11	85.11	85.11
2018	92.65	92.65	89.71	92.65
2019	82.61	82.61	82.61	82.61
2020	79.55	81.82	75.00	79.55
2021	87.01	85.71	80.52	87.01

From the perspective of error metrics (Table 9), the overall average improvement after correction is 21.75% for RMSE and 18.23% for MAE. Regarding the correlation metrics (Table 10), the overall average improvement is 23.01% for R and 56.85% for R². In terms of improvement proportion (Table 11), the values for all four metrics exceed 80% in the vast majority of years.

The validation results at Huailai Station demonstrate that the proposed method is equally effective in regions with different underlying surfaces and climatic conditions. Its overall average improvement rates (RMSE 21.75%, MAE 18.23%, R 23.01%, R² 56.85%) are essentially comparable to those obtained in the Heihe River Basin (RMSE 23.61%, MAE 18.79%, R 16.46%, R² 61.33%), and the improvement proportions are also at similar levels. The comparable performance at Huailai Station and the Heihe River Basin indicates that the method has good generalization potential within the studied regions.

4.5. Comparative Evaluation and Analysis Among Different Methods

To systematically evaluate the performance of the proposed two-stage anomaly detection framework, this section presents a comparative analysis and discussion with the STL method, the dual standardization method, and the robust Holt–Winters method described in Section 3.1.

4.5.1. Settings of Comparative Methods

• STL Anomaly Detection Method

The seasonal period parameter ${\mathrm{n}}_{\mathrm{p}}$ was set to 24 h to reflect diurnal periodicity. Other parameters were primarily set according to recommendations by Cleveland et al. (1990) [53] and Kwok et al. (2023) [25]: The seasonal smoothing parameter ${\mathrm{n}}_{\mathrm{s}}$ was calculated via the empirical formula $n_s=\mathrm{odd}\left(\alpha \times L/n_p\right)$, where $L$ is the total data length, $\alpha =0.5$ is the smoothing coefficient, and $\mathrm{odd}\left(\cdot \right)$ denotes taking the nearest odd number while ensuring $n_s\ge 7$. The trend smoothing parameter $n_t$ was automatically calculated using the formula $n_t=\left[1.5n_p/\left(1-1.5n_s^{-1}\right)\right]\mathrm{odd}$, where $\left[\cdot \right]\mathrm{odd}$ represents rounding up to the nearest odd number. Anomaly identification was based on the residual sequence ${\mathrm{r}}_{\mathrm{i}}$, using the median absolute deviation to set a dynamic threshold. The decision criterion is

$$\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}=\mathrm{median}\left(r_i\right)+k·\mathrm{R}\mathrm{S}\mathrm{D}$$

(17)

where RSD is the robust standard deviation calculated based on the Interquartile Range (IQR), and the coefficient k is set to 4 [26] to identify significant outliers.

2.

Double Standardization Method [27]:

First, global standardization is performed on the time series of each pixel itself, using the median and median absolute deviation (MAD).

$$z_{it}^{\ast }={\displaystyle \frac{x_{it}-M_i}{{\mathrm{MAD}}_i}}$$

(18)

where

$$M_i=\mathrm{median}\left(x_{it}\right)$$

(19)

$${\mathrm{M}\mathrm{A}\mathrm{D}}_{\mathrm{i}}=\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{n}\left(\left|x_{it}-M_i\right|\right)$$

(20)

Subsequently, at each common observation time, spatial standardization is performed on the results $z_{it}^{\ast }$ from the previous step for all pixels, calculating the double-standardized value $z_{it}^{\ast \ast }$ for each pixel.

$$z_{it}^{\ast \ast }={\displaystyle \frac{z_{it}^{\ast }-M_t^{\ast }}{1.4826\cdot {MAD}_t^{\ast }}}$$

(21)

where

$$M_t^{\ast }=\mathrm{median}\left(z_{it}^{\ast }\right)$$

(22)

$${\mathrm{M}\mathrm{A}\mathrm{D}}_{\mathrm{t}}^{\ast }=\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{n}\left(\left|z_{it}^{\ast }-M_t^{\ast }\right|\right)$$

(23)

If $z_{it}^{\ast \ast }$ exceeds the preset threshold of 4, the pixel is marked as an anomaly.

3.

Robust Holt–Winters Method [54]

To further compare with robust filtering methods, this paper introduces the robust Holt–Winters filtering approach proposed by Gelper et al. This method is a classic online robust filtering technique that effectively mitigates the impact of anomalies on trend and seasonal components by recursively updating and cleaning observations in real time. Its specific parameter settings and detailed results are provided in Appendix C.

4.5.2. Comparison of Improvement Proportions

Table 12, Table 13, Table 14 and Table 15 present the annual improvement proportions for the RMSE, MAE, R, and R² metrics, respectively, from 2016 to 2021 for the STL method, the double standardization method, Robust Holt–Winters, and the proposed method. An illustrative comparison of these improvement proportions across methods and metrics is provided in Figure 13.

Table 12. RMSE improvement proportion for the four methods (%, 2016–2021).

Year	STL Method	Double Standardization Method	Robust Holt–Winters	Proposed Method
2016	66.92	71.56	63.41	90.20
2017	61.99	67.39	63.21	92.55
2018	65.85	62.58	58.09	95.50
2019	65.09	58.80	58.94	90.76
2020	68.42	60.65	59.07	90.14
2021	56.73	55.57	58.18	88.00
Average	64.16	61.93	60.15	91.23

Table 13. MAE improvement proportion for the four methods (%, 2016–2021).

Year	STL Method	Double Standardization Method	Robust Holt–Winters	Proposed Method
2016	65.41	71.26	61.98	90.23
2017	64.21	67.75	64.01	92.55
2018	60.98	63.25	59.40	95.00
2019	63.31	57.87	58.69	90.76
2020	67.46	61.47	59.64	90.14
2021	56.73	55.87	57.94	88.57
Average	63.18	61.83	60.28	91.23

Table 14. R improvement proportion for the four methods (%, 2016–2021).

Year	STL Method	Double Standardization Method	Robust Holt–Winters	Proposed Method
2016	66.92	70.51	62.88	88.84
2017	62.36	65.40	62.32	91.49
2018	64.39	61.42	57.78	92.50
2019	65.68	56.48	58.15	82.07
2020	68.42	59.70	58.42	86.38
2021	57.21	55.13	58.06	85.14
Average	64.16	60.44	59.60	87.83

Table 15. R² improvement proportion for the four methods (%, 2016–2021).

Year	STL Method	Double Standardization Method	Robust Holt–Winters	Proposed Method
2016	66.92	71.56	62.51	90.23
2017	62.36	67.39	61.97	92.55
2018	64.39	62.42	57.46	95.50
2019	65.09	58.64	58.01	90.76
2020	68.90	60.52	57.69	90.14
2021	57.21	55.57	57.69	88.00
Average	64.16	61.85	59.22	91.23

Figure 13. Comparison of improvement proportions for the four methods on different evaluation metrics: (a) RMSE; (b) MAE; (c) R; (d) R².

The proposed method consistently outperforms the three comparative methods in terms of improvement proportions across all metrics. The average annual improvement proportions for the STL, double standardization method, and Robust Holt–Winters method generally range between 60% and 64% (STL and double standardization methods around 61–64%, and Robust Holt–Winters around 60%). This indicates that, among the anomaly points flagged by the three comparative methods, over 35% did not improve or even worsened in performance after correction, suggesting a relatively high risk of misjudgment.

This performance difference primarily stems from the methodological design philosophy. The STL and double standardization methods primarily determine anomalies based on statistical outlier characteristics, and although the Robust Holt–Winters method can mitigate the impact of anomalies, it remains a purely temporal statistical method. In contrast, the proposed method employs a two-step process of “temporal physical pre-screening first, spatial statistical verification later.” This process aims to identify anomalies that are both inconsistent with the physical laws of diurnal temperature variation and spatially incoherent with surrounding pixels. In the temporal step, the method first retains temperature fluctuations conforming to typical diurnal patterns, subjecting only the remaining suspected anomalies to spatial consistency testing. Consequently, although the proposed method detects fewer anomaly points annually (195.83) compared to the STL method (214.17) and the double standardization method (647), and Robust Holt–Winters (2258), its detection results are more targeted, effectively reducing the risk of mis-correction.

The lower detection count is not due to insufficient sensitivity but is determined by our method’s screening criteria. Through the dual judgment of physical regularity and spatial consistency, the proposed method can effectively distinguish between real weather changes (e.g., rapid temperature fluctuations caused by cold front passage) and data anomalies. Therefore, the flagged points are more likely to be genuine data errors, which also explains why it has the lowest detection count but the highest post-correction improvement proportion.

For high-temporal-frequency LST data, “mis-correction” (i.e., erroneously replacing real weather changes) may introduce systematic bias and compromise the physical authenticity of the sequence. Its harm is often greater than “missed detection” (i.e., a small number of anomalies not being identified). The proposed method prioritizes controlling the “risk of mis-correction,” ensuring the reliability of correction operations and making it more suitable for application scenarios with high demands for physical data consistency (e.g., energy balance analysis, trend studies).

4.5.3. Comparison of RMSE and MAE

Table 16 and Table 17, respectively, present the overall average improvement rates for RMSE and MAE values of the four methods from 2016 to 2021. A visual comparison of these overall average improvement rates is provided in Figure 14.

Table 16. Overall average improvement rate for RMSE of the four methods (%, 2016–2021).

Year	STL Method	Double Standardization Method	Robust Holt–Winters	Proposed Method
2016	6.26	7.46	1.71	24.44
2017	5.08	2.70	1.66	24.43
2018	6.29	−2.64	2.04	27.18
2019	3.01	−4.20	0.97	21.55
2020	6.52	−7.80	2.33	22.82
2021	4.70	−13.89	1.45	20.76
Average	5.41	−3.67	1.69	23.61

Table 17. Overall average improvement rate for MAE of the four methods (%, 2016–2021).

Year	STL Method	Double Standardization Method	Robust Holt–Winters	Proposed Method
2016	5.44	5.24	0.91	19.26
2017	4.56	1.02	1.01	19.24
2018	5.71	−5.56	1.07	21.67
2019	2.55	−6.75	0.63	17.07
2020	5.82	−10.82	1.28	18.48
2021	3.99	−16.78	0.82	16.65
Average	4.77	−6.21	0.95	18.79

Figure 14. Comparison of overall average improvement rates for the four methods: (a) RMSE; (b) MAE.

Regarding correction effectiveness, the proposed method demonstrates stable and effective performance in reducing absolute error. The STL method also achieves positive error improvement in most years, but its improvement magnitude is noticeably smaller than our method, with annual average overall improvement rates for RMSE and MAE being only 5.41% and 4.77%, respectively.

The performance of the double standardization method is unstable and overall poor. Since 2018, its overall improvement rates for RMSE and MAE have fallen, and the negative magnitude shows an increasing trend year by year. By 2021, the deterioration rates for RMSE and MAE had reached −13.89% and −16.78%, respectively. The improvement magnitude of the Robust Holt–Winters method is very limited, with six-year average improvement rates of only 1.69% for RMSE and 0.95% for MAE. This indicates that while the STL and double standardization methods have varying degrees of mis-correction risk, and the Robust Holt–Winters method can identify some anomaly points, its correction magnitude is far lower than that of the proposed method due to its reliance on temporal information alone and lack of spatial constraints.

4.5.4. Comparison of R and R²

Table 18 and Table 19, respectively, present the overall average improvement rates for R and R² of the four methods from 2016 to 2021. A visual comparison of these overall average improvement rates is provided in Figure 15.

Table 18. Overall average improvement rate for R of the four methods (%, 2016–2021).

Year	STL Method	Double Standardization Method	Robust Holt–Winters	Proposed Method
2016	8.37	9.39	0.49	35.65
2017	6.34	4.44	0.56	10.46
2018	6.52	4.84	0.58	15.14
2019	3.52	2.77	0.30	10.17
2020	8.17	3.14	0.83	14.15
2021	2.31	5.72	0.46	10.23
Average	6.07	4.59	0.54	16.46

Table 19. Overall average improvement rate for R² of the four methods (%, 2016–2021).

Year	STL Method	Double Standardization Method	Robust Holt–Winters	Proposed Method
2016	19.15	16.18	0.86	41.47
2017	7.68	27.96	1.11	80.06
2018	11.10	16.64	1.02	52.01
2019	1.44	26.55	0.50	73.75
2020	20.94	19.69	1.40	68.84
2021	6.16	17.62	0.83	54.05
Average	11.56	21.87	0.95	61.33

Figure 15. Comparison of overall average improvement rates for the four methods: (a) R; (b) R².

In restoring sequence pattern consistency, the proposed method exhibits significant advantages, leading the three comparative methods in all years. In contrast, the improvement effects of the STL and double standardization methods are limited: the STL method’s R² improvement rate fluctuates considerably (1.44%~20.94%, average 11.56%), and its R improvement rate averages only 6.07%; the double standardization method achieves some improvement in R² (average 21.87%) but provides very limited enhancement in phase synchrony (R), averaging only 4.59%. The Robust Holt–Winters method achieves improvement rates of less than 1% for both R and R², indicating that it has limited capability the capability to restore sequence morphology. The improvement magnitudes of all three comparative methods on the R and R² metrics are lower than those of the proposed method, further indicating the better comprehensive performance of our method in comprehensively enhancing the correlation between the sequence and the real physical process.

5. Discussion

The two-stage anomaly detection framework proposed in this paper, with its core sequential logic of “temporal physical preliminary screening first, spatial statistical verification second,” aims to address a key challenge in anomaly identification for high-frequency LST data: effectively distinguishing between physically reasonable weather fluctuations and data errors.

In this study, an “anomaly” is defined as a data point within a cloud-corrected continuous LST sequence that both significantly deviates from typical diurnal variation patterns in the temporal dimension and manifests as a locally isolated anomaly in the spatial dimension. This definition, based on dual constraints, establishes a logical foundation for distinguishing genuine thermal anomalies from data errors. Phenomena such as cloud cover, fires, or other events that may cause instantaneous temperature changes in a localized area can disrupt the temporal coherence of temperature variation. Although the authenticity of temperature data anomalies can often be distinguished through spatial correlation, a risk of misjudgment exists when the spatial coverage of such events is smaller than the pixel scale.

While the method has been validated using data from the Heihe River Basin and an additional site (Huailai), its applicability to other regions with different climatic and surface conditions requires further investigation. Its overall performance in the Heihe River Basin suggests its simplicity and robustness in this region. However, over surfaces with significant heterogeneity, an interpolation strategy based on homogeneous pixels is theoretically more physically meaningful. Future research will integrate high-resolution land cover products from the same period to develop an adaptive interpolation algorithm constrained by land cover types. By prioritizing homogeneous pixels for interpolation, this approach aims to further improve correction accuracy while preserving the heterogeneity of surface thermal characteristics.

At the level of anomaly detection logic, most existing methods adopt a single-stage, symmetric detection approach. This approach incorporates temporal information, spatial information, and physical constraints into a single decision function for a one-time judgment, making it difficult to reflect the priority of physical laws within the process. In contrast, the proposed asymmetric, sequential detection framework uses the physical laws of diurnal variation as the primary criterion for judgment, conducting spatial consistency checks only on points preliminarily identified as suspicious in the temporal dimension. This design is not aimed at optimizing algorithmic precision but is grounded in the physical nature of LST: the diurnal cycle is the most stable constraint in the time series. Therefore, prioritizing physical laws over statistical rules for discrimination helps to avoid misclassifying real weather fluctuations as data errors. The experimental results show that the overall average improvement rates of the proposed method for the R and R² metrics are better than those of the comparison methods, supporting the effectiveness of this detection logic for quality control of high-frequency LST data under the conditions tested.

The proposed method demonstrates good robustness and effectiveness under the current data conditions, but it still has the following limitations: (1) the validation scope is limited to the Heihe River Basin and one additional site (Huailai); (2) its ability to distinguish genuine thermal anomalies or weather events with spatial scales smaller than the testing window is limited by the spatial resolution of current remote sensing data; (3) the interpolation method does not explicitly account for land cover heterogeneity, which may introduce uncertainty in areas with strong spatial heterogeneity; (4) the piecewise empirical model does not incorporate complex physical factors such as cloud cover and terrain shadowing, and its adaptability in complex terrain and during periods of rapid weather change needs improvement. Future research will address these directions, including utilizing higher-resolution remote sensing data, introducing dynamic land cover information, integrating multi-source auxiliary data (e.g., topographic indices, wind speed), and exploring hybrid “physical constraint + data-driven” models, with the aim of constructing a more universally applicable and accurate solution for improving LST data quality.

6. Conclusions

This paper proposes a two-stage anomaly detection method that integrates empirical constraints derived from the physical characteristics of diurnal variation and spatial consistency verification for anomaly identification in high-temporal-frequency LST data. Analysis based on hourly LST data from the Heihe River Basin indicates the method’s advantages in anomaly identification, rationality, and multi-year stability. In most cases, correcting the detection results using spatially correlated IDW interpolation improves the consistency between the LST sequence and ground station-observed land surface temperature. These findings provide a reference for the quality control of high-temporal-frequency remote sensing LST data.

The fundamental essence of the proposed method lies in the sequential design of its detection logic: by following the sequence of “temporal physical pre-screening first, spatial statistical verification later,” this method effectively retains genuine weather events that are “physically plausible but statistically outliers” while only classifying points that are “physically implausible and spatially incoherent” as anomalies. This sequential design has demonstrated strong robustness in the quality control of high-temporal-frequency LST data within the studied areas.

Although the method performs well under the current data conditions, it still has limitations in terms of validation scope, applicability in highly heterogeneous areas, and performance under complex weather conditions. Future research will focus on integrating multi-source auxiliary data, introducing dynamic land cover information, and exploring hybrid “physical constraint + data-driven” models to further enhance the method’s universality and accuracy.

Appendix B.1. Sensitivity of the Temporal Dimension Threshold Coefficient k

Appendix B.1.1. Preliminary Single-Station Experiment (Temporal Anomaly Pre-Screening)

Taking the hourly LST data from the Arou Superstation (S1) in the Heihe River Basin for the year 2020 as an example, the temporal residual detection method (Section 3.3.1 of the main text) was applied alone to perform preliminary temporal anomaly screening under the settings of k = 1, 2, and 3. The number of suspected anomaly points detected in the temporal dimension under each threshold was statistically analyzed. The results indicate that: with k = 1, 1595 suspected anomaly points were detected, resulting in an excessively large candidate set prone to incorporating numerous normal weather fluctuations; with k = 3, only 206 points were detected, leading to an overly small candidate set with a high risk of missed detection; with k = 2, 514 points were detected (accounting for approximately 6.01% of the annual temporal phases), representing a moderate quantity that can broadly capture potential anomalies while providing a manageable candidate set for the subsequent spatial consistency check (Section 3.3.2 of the main text).

Preliminary experiments demonstrate that k = 2 achieves a favorable balance between detection recall rate and candidate set size. It should be noted that this experiment only completed the temporal dimension preliminary screening step, and the detected “suspected anomaly points” are not the final spatiotemporal anomaly points; the complete two-stage framework requires further confirmation through the spatial consistency check, and its final detection quantity and correction performance will be presented in the full dataset analysis below.

Appendix B.1.2. Overall Performance Comparison on the Full Dataset (Complete Two-Stage Framework)

Extending the sensitivity analysis to all data from the seven stations over 2016–2021, with the window size fixed at 7 × 7 and the spatial anomaly threshold $Z_t=3$ fixed, the two-stage anomaly detection framework proposed in this paper (temporal screening + spatial verification) was fully implemented. The total number of final spatiotemporal anomaly points output under the settings of k = 1, 2, and 3 was statistically analyzed, and the overall average improvement rates and improvement ratios for each parameter were calculated as defined in Section 3.5 of the main text.

Table A5 summarizes the overall average improvement rates under different k values. It can be observed that: k = 3 achieves the highest overall average improvement rates across all four error/correlation metrics (RMSE 28.50%, MAE 22.26%, R 20.41%, R² 87.02%); however, the total number of spatiotemporal anomaly points detected is only 652, less than 56% of that under k = 2, indicating a significant risk of missed detection. k = 1 detects the largest number of spatiotemporal anomaly points (2086), but its four overall average improvement rates are the lowest among the three settings, suggesting that numerous normal fluctuations are misjudged as anomalies, posing a considerable risk of incorrect correction. k = 2 exhibits a limited gap compared to k = 3 in terms of overall average improvement rates, yet its detected number of spatiotemporal anomaly points is 1.8 times that of k = 3.

Table A5. Overall average improvement rates and anomaly detection counts under different k values (7 × 7 Window, $Z_t=3$, 2016–2021).

K Value	Number of Anomalies	Overall Avg. Improvement Rate of RMSE (%)	Overall Avg. Improvement Rate of MAE (%)	Overall Avg. Improvement Rate of R (%)	Overall Avg. Improvement Rate of R2 (%)
1	2086	16.60	13.45	11.59	44.59
2	1175	23.61	18.79	16.46	61.33
3	652	28.50	22.26	20.41	87.02

Table A5. Overall average improvement rates and anomaly detection counts under different k values (7 × 7 Window, $Z_t=3$, 2016–2021).

K Value	Number of Anomalies	Overall Avg. Improvement Rate of RMSE (%)	Overall Avg. Improvement Rate of MAE (%)	Overall Avg. Improvement Rate of R (%)	Overall Avg. Improvement Rate of R2 (%)
1	2086	16.60	13.45	11.59	44.59
2	1175	23.61	18.79	16.46	61.33
3	652	28.50	22.26	20.41	87.02

Table A6 presents the improvement ratios under different k values. For k = 2, the improvement ratios across all four metrics exceed 91%, significantly outperforming k = 1 and slightly surpassing k = 3. Combined with Table A5, k = 2 achieves the optimal correction reliability while maintaining a relatively high improvement magnitude.

Table A6. Improvement proportions under different k values (%, 7 × 7 Window,${ Z}_t=3$ , 2016–2021).

K Value	RMSE Improvement Ratio (%)	MAE Improvement Ratio (%)	R Improvement Ratio (%)	R2 Improvement Ratio (%)
1	82.41	82.26	79.53	82.41
2	91.23	91.23	87.83	91.23
3	95.09	94.94	92.02	95.09

Table A6. Improvement proportions under different k values (%, 7 × 7 Window,${ Z}_t=3$ , 2016–2021).

K Value	RMSE Improvement Ratio (%)	MAE Improvement Ratio (%)	R Improvement Ratio (%)	R2 Improvement Ratio (%)
1	82.41	82.26	79.53	82.41
2	91.23	91.23	87.83	91.23
3	95.09	94.94	92.02	95.09

Appendix B.1.3. Stability Analysis Across Different Seasons

To examine the performance differences in various k values under different seasonal climatic conditions and to evaluate the stability of the selected parameter k = 2 across the four seasons, this study further divided the data into four groups by month: spring (March–May), summer (June–August), autumn (September–November), and winter (December–February). The complete two-stage detection framework was applied to calculate the overall average improvement rates and improvement ratios for k = 1, 2, and 3 under each parameter across seasons (Table A7, Table A8, Table A9 and Table A10).

In terms of overall average improvement rates, k = 3 achieves the optimal performance across all seasons for RMSE, MAE, and R². For the R metric, k = 3 also performs best in all seasons except spring, where k = 2 is slightly higher than k = 3. Regarding improvement ratios, k = 3 attains the highest values across all metrics in each season, followed by k = 2. For the selected parameter k = 2, the overall average improvement rates for RMSE in spring, summer, autumn, and winter are 22.85%, 25.58%, 25.62%, and 22.36%, respectively; for MAE, they are 18.61%, 20.25%, 19.87%, and 17.68%; for R, they are 13.21%, 8.11%, 10.10%, and 28.42%; and for R², they are 51.67%, 33.38%, 64.86%, and 82.31%. The improvement ratios for all four metrics remain stable between 85% and 94% across seasons, with minimal fluctuation. The above data demonstrate that k = 2 achieves relatively high improvement magnitudes across all metrics, with a negligible gap in correction reliability (improvement ratio) compared to k = 3, while detecting a greater number of anomaly points and exhibiting good seasonal stability.

Table A7. Seasonal RMSE improvement rates and improvement proportions under different k values (%, 7 × 7 Window, $Z_t=3$ , 2016–2021).

Season	k = 1	k = 2	k = 3
Overall Avg. Improvement Rate of RMSE (%)
Spring	17.53	22.85	26.67
Summer	21.50	25.58	32.42
Autumn	18.45	25.62	31.40
Winter	12.48	22.36	28.00
RMSE Improvement Ratio (%)
Spring	85.41	93.42	96.53
Summer	88.09	92.14	97.06
Autumn	81.09	89.43	92.56
Winter	78.07	89.37	93.71

Table A7. Seasonal RMSE improvement rates and improvement proportions under different k values (%, 7 × 7 Window, $Z_t=3$ , 2016–2021).

Season	k = 1	k = 2	k = 3
Overall Avg. Improvement Rate of RMSE (%)
Spring	17.53	22.85	26.67
Summer	21.50	25.58	32.42
Autumn	18.45	25.62	31.40
Winter	12.48	22.36	28.00
RMSE Improvement Ratio (%)
Spring	85.41	93.42	96.53
Summer	88.09	92.14	97.06
Autumn	81.09	89.43	92.56
Winter	78.07	89.37	93.71

Table A8. Seasonal MAE improvement rates and improvement proportions under different k values (%, 7 × 7 Window, $Z_t=3$ , 2016–2021).

Season	k = 1	k = 2	k = 3
Overall Avg. Improvement Rate of MAE (%)
Spring	14.14	18.61	21.39
Summer	17.57	20.25	25.43
Autumn	14.41	19.87	23.86
Winter	10.46	17.68	21.35
MAE Improvement Ratio (%)
Spring	85.11	93.42	96.53
Summer	88.09	92.14	97.06
Autumn	81.30	90.24	92.56
Winter	77.78	88.79	93.14

Table A8. Seasonal MAE improvement rates and improvement proportions under different k values (%, 7 × 7 Window, $Z_t=3$ , 2016–2021).

Season	k = 1	k = 2	k = 3
Overall Avg. Improvement Rate of MAE (%)
Spring	14.14	18.61	21.39
Summer	17.57	20.25	25.43
Autumn	14.41	19.87	23.86
Winter	10.46	17.68	21.35
MAE Improvement Ratio (%)
Spring	85.11	93.42	96.53
Summer	88.09	92.14	97.06
Autumn	81.30	90.24	92.56
Winter	77.78	88.79	93.14

Table A9. Seasonal R improvement rates and improvement proportions under different k values (%, 7 × 7 Window, $Z_t=3$ , 2016–2021).

Season	k = 1	k = 2	k = 3
Overall Avg. Improvement Rate of R (%)
Spring	9.10	13.21	12.56
Summer	5.89	8.11	13.86
Autumn	7.87	10.10	15.51
Winter	18.82	28.42	39.25
R Improvement Ratio (%)
Spring	82.56	90.25	92.71
Summer	85.56	89.29	95.59
Autumn	78.26	85.77	90.91
Winter	75.00	85.63	90.29

Table A9. Seasonal R improvement rates and improvement proportions under different k values (%, 7 × 7 Window, $Z_t=3$ , 2016–2021).

Season	k = 1	k = 2	k = 3
Overall Avg. Improvement Rate of R (%)
Spring	9.10	13.21	12.56
Summer	5.89	8.11	13.86
Autumn	7.87	10.10	15.51
Winter	18.82	28.42	39.25
R Improvement Ratio (%)
Spring	82.56	90.25	92.71
Summer	85.56	89.29	95.59
Autumn	78.26	85.77	90.91
Winter	75.00	85.63	90.29

Table A10. Seasonal R² improvement rates and improvement proportions under different k values (%, 7 × 7 Window, $Z_t=3$ , 2016–2021).

Season	k = 1	k = 2	k = 3
Overall Avg. Improvement Rate of R2 (%)
Spring	38.01	51.67	70.81
Summer	31.00	33.38	53.40
Autumn	45.00	64.86	84.46
Winter	56.22	82.31	128.54
R2 Improvement Ratio (%)
Spring	85.41	93.42	96.53
Summer	88.09	92.14	97.06
Autumn	81.09	89.43	92.56
Winter	78.07	89.37	93.71

Table A10. Seasonal R² improvement rates and improvement proportions under different k values (%, 7 × 7 Window, $Z_t=3$ , 2016–2021).

Season	k = 1	k = 2	k = 3
Overall Avg. Improvement Rate of R2 (%)
Spring	38.01	51.67	70.81
Summer	31.00	33.38	53.40
Autumn	45.00	64.86	84.46
Winter	56.22	82.31	128.54
R2 Improvement Ratio (%)
Spring	85.41	93.42	96.53
Summer	88.09	92.14	97.06
Autumn	81.09	89.43	92.56
Winter	78.07	89.37	93.71

Appendix B.1.4. Stability Analysis Across Different Land Cover Types

To evaluate the stability of the selected parameter combination (k = 2, 7 × 7 window, and spatial anomaly threshold) under different land surface types, this study classified the seven ground stations in the Heihe River Basin into four categories based on the underlying surface attributes in Table 2: alpine meadow (S1, S3), farmland/artificial vegetation (S2, S4), desert shrubland (S5, S6), and wetland (S7). The hourly LST data from all stations within each category for 2016–2021 were aggregated, and the complete two-stage anomaly detection and IDW correction process was executed with the fixed parameter combination.

Table A11, Table A12, Table A13 and Table A14 present the overall average improvement rates and improvement ratios for various metrics under k = 1, 2, and 3 across different land cover types. Regarding overall average improvement rates, RMSE and MAE exhibit a consistent pattern: k = 3 achieves the highest values across all land cover types, followed by k = 2, with k = 1 being the lowest. For the R² metric, except in desert shrubland, k=3 also attains the highest values, followed by k = 2, with k = 1 being the lowest across other land cover types; however, in desert shrubland, k = 1 (27.86%) is slightly higher than k = 2 (25.24%). For the R metric (Table A13), k = 3 performs best in alpine meadow, farmland/artificial vegetation, and desert shrubland, while in wetland, k = 2 (13.62%) exceeds k = 3 (9.84%). Although k = 3 achieves slightly higher improvement magnitudes for most metrics, the total number of spatiotemporal anomaly points detected by k = 2 (1175) is 1.8 times that of k = 3 (652), indicating a significantly lower risk of missed detection.

In terms of improvement ratios, all metrics consistently indicate (Table A11, Table A12, Table A13 and Table A14): k = 3 achieves the optimal performance across all land cover types, followed by k = 2, with k = 1 being the lowest. The gap between k = 2 and k = 3 is relatively small, while k = 2 significantly outperforms k = 1.

From the perspective of cross-land cover stability, both the overall average improvement rates and improvement ratios of k = 2 exhibit minimal fluctuation across different land cover types, closely aligning with the overall average levels of the Heihe River Basin. These results indicate that k = 2 consistently achieves a favorable balance between correction magnitude and correction reliability under various land surface types, suggesting the robustness of the parameter selection in this study.

Table A11. Seasonal RMSE improvement rates and improvement proportions under different land cover types and k values (%, 7 × 7 Window, $Z_t=3$ , 2016–2021).

Land Cover Type	k = 1	k = 2	k = 3
Overall Avg. Improvement Rate of RMSE (%)
Alpine Meadow	18.35	23.99	28.92
Farmland/Artificial Veg.	20.69	23.46	28.39
Desert Shrubland	19.34	24.21	28.88
Wetland	5.58	22.29	27.49
RMSE Improvement Ratio (%)
Alpine Meadow	83.54	91.47	95.60
Farmland/Artificial Veg.	85.53	88.07	91.85
Desert Shrubland	90.37	94.86	98.22
Wetland	66.82	91.10	94.87

Table A11. Seasonal RMSE improvement rates and improvement proportions under different land cover types and k values (%, 7 × 7 Window, $Z_t=3$ , 2016–2021).

Land Cover Type	k = 1	k = 2	k = 3
Overall Avg. Improvement Rate of RMSE (%)
Alpine Meadow	18.35	23.99	28.92
Farmland/Artificial Veg.	20.69	23.46	28.39
Desert Shrubland	19.34	24.21	28.88
Wetland	5.58	22.29	27.49
RMSE Improvement Ratio (%)
Alpine Meadow	83.54	91.47	95.60
Farmland/Artificial Veg.	85.53	88.07	91.85
Desert Shrubland	90.37	94.86	98.22
Wetland	66.82	91.10	94.87

Table A12. Seasonal MAE improvement rates and improvement proportions under different land cover types and k values (%, 7 × 7 Window, $Z_t=3$ , 2016–2021).

Land Cover Type	k = 1	k = 2	k = 3
Overall Avg. Improvement Rate of MAE (%)
Alpine Meadow	15.12	19.24	22.99
Farmland/Artificial Veg.	16.69	18.46	21.68
Desert Shrubland	16.02	19.23	22.44
Wetland	3.82	17.95	21.77
MAE Improvement Ratio (%)
Alpine Meadow	84.07	91.18	95.60
Farmland/Artificial Veg.	85.53	88.07	91.30
Desert Shrubland	90.37	94.86	98.22
Wetland	65.42	91.62	94.87

Table A12. Seasonal MAE improvement rates and improvement proportions under different land cover types and k values (%, 7 × 7 Window, $Z_t=3$ , 2016–2021).

Land Cover Type	k = 1	k = 2	k = 3
Overall Avg. Improvement Rate of MAE (%)
Alpine Meadow	15.12	19.24	22.99
Farmland/Artificial Veg.	16.69	18.46	21.68
Desert Shrubland	16.02	19.23	22.44
Wetland	3.82	17.95	21.77
MAE Improvement Ratio (%)
Alpine Meadow	84.07	91.18	95.60
Farmland/Artificial Veg.	85.53	88.07	91.30
Desert Shrubland	90.37	94.86	98.22
Wetland	65.42	91.62	94.87

Table A13. Seasonal R improvement rates and improvement proportions under different land cover types and k values (%, 7 × 7 Window, $Z_t=3$ , 2016–2021).

Land Cover Type	k = 1	k = 2	k = 3
Overall Avg. Improvement Rate of R (%)
Alpine Meadow	17.01	21.49	24.90
Farmland/Artificial Veg.	17.10	23.36	37.12
Desert Shrubland	3.71	4.12	4.69
Wetland	7.25	13.62	9.84
R Improvement Ratio (%)
Alpine Meadow	79.12	86.47	89.01
Farmland/Artificial Veg.	82.28	84.38	90.22
Desert Shrubland	86.85	91.78	96.45
Wetland	67.29	90.58	93.16

Table A13. Seasonal R improvement rates and improvement proportions under different land cover types and k values (%, 7 × 7 Window, $Z_t=3$ , 2016–2021).

Land Cover Type	k = 1	k = 2	k = 3
Overall Avg. Improvement Rate of R (%)
Alpine Meadow	17.01	21.49	24.90
Farmland/Artificial Veg.	17.10	23.36	37.12
Desert Shrubland	3.71	4.12	4.69
Wetland	7.25	13.62	9.84
R Improvement Ratio (%)
Alpine Meadow	79.12	86.47	89.01
Farmland/Artificial Veg.	82.28	84.38	90.22
Desert Shrubland	86.85	91.78	96.45
Wetland	67.29	90.58	93.16

Table A14. Seasonal R² improvement rates and improvement proportions under different land cover types and k values (%,7 × 7 Window, $Z_t=3$ , 2016–2021).

Land Cover Type	k = 1	k = 2	k = 3
Overall Avg. Improvement Rate of R2 (%)
Alpine Meadow	52.08	67.57	80.75
Farmland/Artificial Veg.	48.46	59.04	96.15
Desert Shrubland	27.86	25.24	30.42
Wetland	50.83	109.60	164.20
R2 Improvement Ratio (%)
Alpine Meadow	83.54	91.47	95.60
Farmland/Artificial Veg.	85.53	88.07	91.85
Desert Shrubland	90.37	94.86	98.22
Wetland	66.82	91.10	94.87

Table A14. Seasonal R² improvement rates and improvement proportions under different land cover types and k values (%,7 × 7 Window, $Z_t=3$ , 2016–2021).

Land Cover Type	k = 1	k = 2	k = 3
Overall Avg. Improvement Rate of R2 (%)
Alpine Meadow	52.08	67.57	80.75
Farmland/Artificial Veg.	48.46	59.04	96.15
Desert Shrubland	27.86	25.24	30.42
Wetland	50.83	109.60	164.20
R2 Improvement Ratio (%)
Alpine Meadow	83.54	91.47	95.60
Farmland/Artificial Veg.	85.53	88.07	91.85
Desert Shrubland	90.37	94.86	98.22
Wetland	66.82	91.10	94.87

Appendix B.2. Sensitivity Analysis of Window Size

Appendix B.2.1. Preliminary Single-Station Experiment (Spatial Dimension Consistency Check)

Taking the 2020 data from the Arou Superstation (S1) in the Heihe River Basin as an example, with k = 2 and $Z_t=3$ the spatial anomaly threshold fixed, the spatial consistency check was performed using five window sizes: 3 × 3, 5 × 5, 7 × 7, 9 × 9, and 11 × 11. The number of final spatiotemporal anomaly point output under each window size and the detection overlap rate with adjacent windows were statistically analyzed. The results (Figure A1) indicate that: as the window size increases, the number of screened anomaly points gradually stabilizes, with the trend becoming particularly gentle after the 7 × 7 window. Further calculations show that the detection overlap rate between the 7 × 7 window and the 9 × 9 window reaches as high as 91.7%, and the 7 × 7 window has the fewest unique anomaly points (only 2). It should be noted that this experiment only demonstrates the impact of window size on detection results at a single station; the full dataset analysis below will provide a systematic performance comparison.

Figure A1. (a) Number of anomaly points screened by different window sizes; (b) number of unique anomaly points for different window sizes.

Figure A1. (a) Number of anomaly points screened by different window sizes; (b) number of unique anomaly points for different window sizes.

Appendix B.2.2. Overall Performance Comparison on the Full Dataset (Complete Two-Stage Framework)

Extending the window size sensitivity analysis to all data from the seven stations over 2016–2021, with k = 2 and $Z_t=3$ the spatial anomaly threshold fixed, the complete two-stage detection framework was implemented. The overall average improvement rates and improvement ratios for each window size were calculated, and the total number of spatiotemporal anomaly points for each window was statistically analyzed.

Table A15 and Table A16 summarize the overall correction performance under different window sizes. It can be observed that: the 7 × 7 window achieves either the highest overall average improvement rates or values very close to the optimum for RMSE, MAE, and R (compared to the 9 × 9 window, the overall average improvement rate for RMSE is only 0.65 percentage points lower, and for MAE it is 0.37 percentage points lower). The overall average improvement rate for R² is slightly lower than that of larger windows, yet its improvement ratio is on par with the 9 × 9 and 11 × 11 windows. The improvement ratios of the 7 × 7 window (RMSE 91.23%, MAE 91.23%, R 87.83%, R² 91.23%) are significantly higher than those of the 3 × 3 and 5 × 5 windows, while being comparable to those of the 9 × 9 and 11 × 11 windows. In terms of the number of detected anomaly points, the 7 × 7 window (1175) falls between the 5 × 5 (1367) and 9 × 9 (1128) windows, achieving the favorable balance between detection recall rate and correction reliability.

Table A15. Overall average improvement rates and anomaly detection counts under different window sizes (k = 2, $Z_t=3$ , 2016–2021).

Window Size	Number of Anomalies	Overall Avg. Improvement Rate of RMSE (%)	Overall Avg. Improvement Rate of MAE (%)	Overall Avg. Improvement Rate of R (%)	Overall Avg. Improvement Rate of R2 (%)
3 × 3	1598	17.03	13.57	9.66	44.56
5 × 5	1367	22.17	17.84	14.33	68.67
7 × 7	1175	23.61	18.79	16.46	61.33
9 × 9	1128	24.26	19.16	13.85	68.45
11 × 11	1068	24.68	19.55	13.46	69.21

Table A15. Overall average improvement rates and anomaly detection counts under different window sizes (k = 2, $Z_t=3$ , 2016–2021).

Window Size	Number of Anomalies	Overall Avg. Improvement Rate of RMSE (%)	Overall Avg. Improvement Rate of MAE (%)	Overall Avg. Improvement Rate of R (%)	Overall Avg. Improvement Rate of R2 (%)
3 × 3	1598	17.03	13.57	9.66	44.56
5 × 5	1367	22.17	17.84	14.33	68.67
7 × 7	1175	23.61	18.79	16.46	61.33
9 × 9	1128	24.26	19.16	13.85	68.45
11 × 11	1068	24.68	19.55	13.46	69.21

Table A16. Improvement proportions under different window sizes (%, 2016–2021).

Window Size	RMSE Improvement Ratio (%)	MAE Improvement Ratio (%)	R Improvement Ratio (%)	R2 Improvement Ratio (%)
3 × 3	81.79	81.50	78.56	81.28
5 × 5	90.38	90.55	86.47	90.21
7 × 7	91.23	91.23	87.83	91.23
9 × 9	91.68	91.59	88.44	91.68
11 × 11	91.43	91.43	88.30	91.43

Table A16. Improvement proportions under different window sizes (%, 2016–2021).

Window Size	RMSE Improvement Ratio (%)	MAE Improvement Ratio (%)	R Improvement Ratio (%)	R2 Improvement Ratio (%)
3 × 3	81.79	81.50	78.56	81.28
5 × 5	90.38	90.55	86.47	90.21
7 × 7	91.23	91.23	87.83	91.23
9 × 9	91.68	91.59	88.44	91.68
11 × 11	91.43	91.43	88.30	91.43

Appendix B.2.3. Stability Analysis Across Different Seasons

To examine the performance differences in various window sizes under different seasonal climatic conditions and to evaluate the stability of the selected 7 × 7 window across the four seasons, this study further stratified the data by season and calculated the overall average improvement rates and improvement ratios for each window size in each season (Table A17, Table A18, Table A19 and Table A20).

In terms of the overall average improvement rate for RMSE (Table A17), the 7 × 7 window remains stable between 22% and 26% across all four seasons, with relatively small gaps compared to the 9 × 9 and 11 × 11 windows. Regarding the overall average improvement rate for MAE (Table A18), the performance difference between the 7 × 7 window and the 9 × 9 window in each season is within one percentage point, and it is also very close to that of the 11 × 11 window. For the overall average improvement rate of R (Table A19), except for winter, where the 7 × 7 window significantly outperforms larger windows, its performance in the remaining seasons is generally comparable to that of the 9 × 9 and 11 × 11 windows. The overall average improvement rate for R² (Table A20) is on par with the 9 × 9 window in autumn but slightly lower than larger windows in the other seasons.

In terms of improvement ratios, the 7 × 7 window remains stable across the four seasons within the ranges of 89–94% (RMSE), 88–94% (MAE), 85–91% (R), and 89–94% (R²). These values are at the same level as those of the 9 × 9 and 11 × 11 windows, significantly higher than those of the 3 × 3 window, and show mixed results compared to the 5 × 5 window—either at a comparable level or slightly better.

The above results indicate that there is a clear “performance saturation zone” around the window size of 7 × 7: further increasing the window size yields very limited improvement in correction performance. Meanwhile, the 7 × 7 window achieves an optimal balance among computational efficiency, detection recall rate, and correction reliability, and remains unaffected by seasonal climatic conditions, demonstrating good cross-season stability.

Table A17. Seasonal RMSE improvement rates and improvement proportions under different window sizes (%, k = 2, $Z_t=3$ , 2016–2021).

Season	3 × 3	5 × 5	7 × 7	9 × 9	11 × 11
Overall Avg. Improvement Rate of RMSE (%)
Spring	16.74	22.85	22.90	24.04	24.48
Summer	14.06	25.58	25.33	25.61	26.93
Autumn	19.02	25.62	25.87	24.93	24.94
Winter	18.44	22.36	22.23	23.49	23.93
RMSE Improvement Ratio (%)
Spring	83.33	93.42	93.56	94.54	94.02
Summer	76.53	92.14	91.52	91.54	93.16
Autumn	82.26	89.43	90.02	88.85	88.24
Winter	83.54	89.37	89.04	90.33	90.00

Table A17. Seasonal RMSE improvement rates and improvement proportions under different window sizes (%, k = 2, $Z_t=3$ , 2016–2021).

Season	3 × 3	5 × 5	7 × 7	9 × 9	11 × 11
Overall Avg. Improvement Rate of RMSE (%)
Spring	16.74	22.85	22.90	24.04	24.48
Summer	14.06	25.58	25.33	25.61	26.93
Autumn	19.02	25.62	25.87	24.93	24.94
Winter	18.44	22.36	22.23	23.49	23.93
RMSE Improvement Ratio (%)
Spring	83.33	93.42	93.56	94.54	94.02
Summer	76.53	92.14	91.52	91.54	93.16
Autumn	82.26	89.43	90.02	88.85	88.24
Winter	83.54	89.37	89.04	90.33	90.00

Table A18. Seasonal MAE improvement rates and improvement proportions under different window sizes (%, k = 2, $Z_t=3$ , 2016–2021).

Season	3 × 3	5 × 5	7 × 7	9 × 9	11 × 11
Overall Avg. Improvement Rate of MAE (%)
Spring	13.85	17.75	18.61	19.45	19.93
Summer	10.92	18.36	20.25	20.58	21.71
Autumn	14.73	18.42	19.87	19.23	19.25
Winter	14.48	17.24	17.68	18.18	18.55
MAE Improvement Ratio (%)
Spring	82.74	91.10	93.42	94.54	94.02
Summer	76.90	89.63	92.14	91.54	93.16
Autumn	82.26	91.90	90.24	89.23	88.24
Winter	82.91	89.26	88.79	89.73	90.00

Table A18. Seasonal MAE improvement rates and improvement proportions under different window sizes (%, k = 2, $Z_t=3$ , 2016–2021).

Season	3 × 3	5 × 5	7 × 7	9 × 9	11 × 11
Overall Avg. Improvement Rate of MAE (%)
Spring	13.85	17.75	18.61	19.45	19.93
Summer	10.92	18.36	20.25	20.58	21.71
Autumn	14.73	18.42	19.87	19.23	19.25
Winter	14.48	17.24	17.68	18.18	18.55
MAE Improvement Ratio (%)
Spring	82.74	91.10	93.42	94.54	94.02
Summer	76.90	89.63	92.14	91.54	93.16
Autumn	82.26	91.90	90.24	89.23	88.24
Winter	82.91	89.26	88.79	89.73	90.00

Table A19. Seasonal R improvement rates and improvement proportions under different window sizes (%, k = 2, $Z_t=3$ , 2016–2021).

Season	3 × 3	5 × 5	7 × 7	9 × 9	11 × 11
Overall Avg. Improvement Rate of R (%)
Spring	9.86	11.15	13.21	13.41	14.05
Summer	3.68	6.71	8.11	8.77	9.55
Autumn	7.23	8.69	10.10	9.86	9.94
Winter	16.62	26.70	28.42	19.54	16.81
R Improvement Ratio (%)
Spring	79.76	87.21	90.25	91.45	91.87
Summer	73.65	86.59	89.29	87.69	89.74
Autumn	79.25	87.04	85.77	85.38	84.31
Winter	80.38	84.97	85.63	87.31	86.36

Table A19. Seasonal R improvement rates and improvement proportions under different window sizes (%, k = 2, $Z_t=3$ , 2016–2021).

Season	3 × 3	5 × 5	7 × 7	9 × 9	11 × 11
Overall Avg. Improvement Rate of R (%)
Spring	9.86	11.15	13.21	13.41	14.05
Summer	3.68	6.71	8.11	8.77	9.55
Autumn	7.23	8.69	10.10	9.86	9.94
Winter	16.62	26.70	28.42	19.54	16.81
R Improvement Ratio (%)
Spring	79.76	87.21	90.25	91.45	91.87
Summer	73.65	86.59	89.29	87.69	89.74
Autumn	79.25	87.04	85.77	85.38	84.31
Winter	80.38	84.97	85.63	87.31	86.36

Table A20. Seasonal R² improvement rates and improvement proportions under different window sizes (%, k = 2, $Z_t=3$ , 2016–2021).

Season	3 × 3	5 × 5	7 × 7	9 × 9	11 × 11
Overall Avg. Improvement Rate of R2 (%)
Spring	32.74	81.49	51.67	67.13	64.74
Summer	15.84	25.47	33.38	35.28	39.89
Autumn	39.03	48.64	64.86	64.25	68.39
Winter	93.21	88.36	82.31	86.45	85.89
R2 Improvement Ratio (%)
Spring	82.74	90.87	93.42	94.54	94.02
Summer	75.45	89.63	92.14	91.54	93.16
Autumn	82.26	90.69	89.43	88.85	88.24
Winter	83.23	89.26	89.37	90.33	90.00

Table A20. Seasonal R² improvement rates and improvement proportions under different window sizes (%, k = 2, $Z_t=3$ , 2016–2021).

Season	3 × 3	5 × 5	7 × 7	9 × 9	11 × 11
Overall Avg. Improvement Rate of R2 (%)
Spring	32.74	81.49	51.67	67.13	64.74
Summer	15.84	25.47	33.38	35.28	39.89
Autumn	39.03	48.64	64.86	64.25	68.39
Winter	93.21	88.36	82.31	86.45	85.89
R2 Improvement Ratio (%)
Spring	82.74	90.87	93.42	94.54	94.02
Summer	75.45	89.63	92.14	91.54	93.16
Autumn	82.26	90.69	89.43	88.85	88.24
Winter	83.23	89.26	89.37	90.33	90.00

Appendix B.2.4. Stability Analysis Across Different Land Cover Types

To evaluate the stability of the spatial consistency check window size under different land surface types, this study classified the seven ground stations in the Heihe River Basin into four categories based on the underlying surface attributes in Table 2: alpine meadow (S1, S3), farmland/artificial vegetation (S2, S4), desert shrubland (S5, S6), and wetland (S7). With the temporal dimension threshold coefficient fixed at k = 2 and the spatial anomaly threshold $Z_t=3$ fixed, the complete two-stage anomaly detection and IDW correction process was implemented using five window sizes: 3 × 3, 5 × 5, 7 × 7, 9 × 9, and 11 × 11. The overall average improvement rates and improvement ratios for each window size were calculated.

In terms of the overall average improvement rate for RMSE (Table A21), the 7 × 7 window achieves relatively high improvement magnitudes across all land cover types: alpine meadow (23.99%), farmland/artificial vegetation (23.46%), desert shrubland (24.21%), and wetland (22.29%). The differences between the 7 × 7 window and the 9 × 9 and 11 × 11 windows range from 0.21 to 1.64 percentage points, while it significantly outperforms the 3 × 3 and 5 × 5 windows. Regarding the improvement ratio for RMSE, the 7 × 7 window exhibits excellent performance across all land cover types, reaching its highest value in desert shrubland (94.86%), demonstrating good correction reliability.

In terms of MAE, R, and R² metrics (Table A22, Table A23 and Table A24), the 7 × 7 window performs well across all land cover types. For the overall average improvement rate of MAE, the differences between the 7 × 7 window and the 9 × 9 and 11 × 11 windows are relatively small, and it significantly outperforms the 3 × 3 and 5 × 5 windows. Regarding the overall average improvement rate of R, except for farmland/artificial vegetation, where the 7 × 7 window (23.36%) is significantly higher than the 9 × 9 (13.17%) and 11 × 11 (9.59%) windows, the 7 × 7 window is generally comparable to larger windows in other land cover types. For the overall average improvement rate of R², the 7 × 7 window is lower than the 9 × 9 and 11 × 11 windows in most land cover types but outperforms the 3 × 3 and 5 × 5 windows; in farmland/artificial vegetation, although the 7 × 7 window (59.04%) is lower than the 5 × 5 window (99.53%), it remains significantly better than the 3 × 3 window (39.56%). In terms of improvement ratios, the 7 × 7 window is close to the 9 × 9 and 11 × 11 windows across all metrics and land cover types, and significantly outperforms the 3 × 3 and 5 × 5 windows.

From the perspective of cross-land cover stability, the performance of the 7 × 7 window exhibits minimal fluctuation across different land cover types, closely aligning with the overall average levels of the Heihe River Basin. The above results indicate that the 7 × 7 window consistently achieves a favorable balance between correction magnitude and correction reliability under different surface thermal backgrounds. Further increasing the window size yields very limited performance improvement, while the 7 × 7 window achieves a favorable trade-off among computational efficiency, correction magnitude, and correction reliability.

Table A21. RMSE improvement rates and improvement proportions under different window sizes and land cover types (%, k = 2, $Z_t=3$ , 2016–2021).

Land Cover Type	3 × 3	5 × 5	7 × 7	9 × 9	11 × 11
Overall Avg. Improvement Rate of RMSE (%)
Alpine Meadow	18.76	23.92	23.99	23.82	24.20
Farmland/Artificial Veg.	15.45	20.13	23.46	24.96	25.10
Desert Shrubland	16.90	22.82	24.21	24.80	25.38
Wetland	16.90	21.42	22.29	22.81	23.36
RMSE Improvement Ratio (%)
Alpine Meadow	86.22	91.17	91.47	92.79	91.56
Farmland/Artificial Veg.	74.09	85.50	88.07	88.17	89.15
Desert Shrubland	82.27	94.37	94.86	94.31	94.40
Wetland	86.57	91.10	91.10	92.49	91.72

Table A21. RMSE improvement rates and improvement proportions under different window sizes and land cover types (%, k = 2, $Z_t=3$ , 2016–2021).

Land Cover Type	3 × 3	5 × 5	7 × 7	9 × 9	11 × 11
Overall Avg. Improvement Rate of RMSE (%)
Alpine Meadow	18.76	23.92	23.99	23.82	24.20
Farmland/Artificial Veg.	15.45	20.13	23.46	24.96	25.10
Desert Shrubland	16.90	22.82	24.21	24.80	25.38
Wetland	16.90	21.42	22.29	22.81	23.36
RMSE Improvement Ratio (%)
Alpine Meadow	86.22	91.17	91.47	92.79	91.56
Farmland/Artificial Veg.	74.09	85.50	88.07	88.17	89.15
Desert Shrubland	82.27	94.37	94.86	94.31	94.40
Wetland	86.57	91.10	91.10	92.49	91.72

Table A22. MAE improvement rates and improvement proportions under different window sizes and land cover types (%, k = 2, $Z_t=3$ , 2016–2021).

Land Cover Type	3 × 3	5 × 5	7 × 7	9 × 9	11 × 11
Overall Avg. Improvement Rate of MAE (%)
Alpine Meadow	15.00	19.17	19.24	19.28	19.58
Farmland/Artificial Veg.	12.12	16.22	18.46	19.32	19.52
Desert Shrubland	13.49	18.49	19.23	19.71	20.43
Wetland	13.66	17.14	17.95	17.72	18.08
MAE Improvement Ratio (%)
Alpine Meadow	85.71	91.17	91.18	92.49	91.56
Farmland/Artificial Veg.	74.09	85.80	88.07	88.17	88.89
Desert Shrubland	81.99	94.37	94.86	94.31	94.40
Wetland	86.11	91.62	91.62	92.49	92.36

Table A22. MAE improvement rates and improvement proportions under different window sizes and land cover types (%, k = 2, $Z_t=3$ , 2016–2021).

Land Cover Type	3 × 3	5 × 5	7 × 7	9 × 9	11 × 11
Overall Avg. Improvement Rate of MAE (%)
Alpine Meadow	15.00	19.17	19.24	19.28	19.58
Farmland/Artificial Veg.	12.12	16.22	18.46	19.32	19.52
Desert Shrubland	13.49	18.49	19.23	19.71	20.43
Wetland	13.66	17.14	17.95	17.72	18.08
MAE Improvement Ratio (%)
Alpine Meadow	85.71	91.17	91.18	92.49	91.56
Farmland/Artificial Veg.	74.09	85.80	88.07	88.17	88.89
Desert Shrubland	81.99	94.37	94.86	94.31	94.40
Wetland	86.11	91.62	91.62	92.49	92.36

Table A23. R improvement rates and improvement proportions under different window sizes and land cover types (%, k = 2, $Z_t=3$ , 2016–2021).

Land Cover Type	3 × 3	5 × 5	7 × 7	9 × 9	11 × 11
Overall Avg. Improvement Rate of R (%)
Alpine Meadow	15.12	18.96	21.49	22.94	25.30
Farmland/Artificial Veg.	8.86	20.41	23.36	13.17	9.59
Desert Shrubland	3.15	3.58	4.12	4.18	5.13
Wetland	11.89	12.26	13.62	13.47	13.99
R Improvement Ratio (%)
Alpine Meadow	81.95	85.47	86.47	87.39	87.34
Farmland/Artificial Veg.	70.98	80.97	84.38	85.35	85.53
Desert Shrubland	80.33	91.72	91.78	91.46	91.79
Wetland	82.87	89.53	90.58	91.91	91.08

Table A23. R improvement rates and improvement proportions under different window sizes and land cover types (%, k = 2, $Z_t=3$ , 2016–2021).

Land Cover Type	3 × 3	5 × 5	7 × 7	9 × 9	11 × 11
Overall Avg. Improvement Rate of R (%)
Alpine Meadow	15.12	18.96	21.49	22.94	25.30
Farmland/Artificial Veg.	8.86	20.41	23.36	13.17	9.59
Desert Shrubland	3.15	3.58	4.12	4.18	5.13
Wetland	11.89	12.26	13.62	13.47	13.99
R Improvement Ratio (%)
Alpine Meadow	81.95	85.47	86.47	87.39	87.34
Farmland/Artificial Veg.	70.98	80.97	84.38	85.35	85.53
Desert Shrubland	80.33	91.72	91.78	91.46	91.79
Wetland	82.87	89.53	90.58	91.91	91.08

Table A24. R² improvement rates and improvement proportions under different window sizes and land cover types (%, k = 2, $Z_t=3$ , 2016–2021).

Land Cover Type	3 × 3	5 × 5	7 × 7	9 × 9	11 × 11
Overall Avg. Improvement Rate of R2 (%)
Alpine Meadow	57.58	64.97	67.57	85.37	102.63
Farmland/Artificial Veg.	39.56	99.53	59.04	62.99	49.85
Desert Shrubland	12.93	17.86	25.24	24.26	29.49
Wetland	82.29	102.36	109.60	118.86	119.15
R2 Improvement Ratio (%)
Alpine Meadow	85.71	90.88	91.47	92.79	91.56
Farmland/Artificial Veg.	73.83	85.20	88.07	88.17	89.15
Desert Shrubland	82.27	94.37	94.86	94.31	94.40
Wetland	84.72	91.10	91.10	92.49	91.72

Table A24. R² improvement rates and improvement proportions under different window sizes and land cover types (%, k = 2, $Z_t=3$ , 2016–2021).

Land Cover Type	3 × 3	5 × 5	7 × 7	9 × 9	11 × 11
Overall Avg. Improvement Rate of R2 (%)
Alpine Meadow	57.58	64.97	67.57	85.37	102.63
Farmland/Artificial Veg.	39.56	99.53	59.04	62.99	49.85
Desert Shrubland	12.93	17.86	25.24	24.26	29.49
Wetland	82.29	102.36	109.60	118.86	119.15
R2 Improvement Ratio (%)
Alpine Meadow	85.71	90.88	91.47	92.79	91.56
Farmland/Artificial Veg.	73.83	85.20	88.07	88.17	89.15
Desert Shrubland	82.27	94.37	94.86	94.31	94.40
Wetland	84.72	91.10	91.10	92.49	91.72

Appendix B.3. Sensitivity Analysis of the Spatial Anomaly Threshold $Z_t$

Appendix B.3.1. Overall Performance Comparison on the Full Dataset (Complete Two-Stage Framework)

Extending the spatial anomaly threshold sensitivity analysis to all data from the seven stations over 2016–2021, with k = 2 and the 7 × 7 window fixed, the complete two-stage detection framework was implemented. The overall average improvement rates and improvement ratios were calculated for threshold $Z_t$ settings of 2.0, 2.5, 3.0, and 3.5, and the total number of spatiotemporal anomaly points under each threshold was statistically analyzed.

Table A25 and Table A26 summarize the overall correction performance under different thresholds. It can be observed that: a threshold $Z_t=3$ achieves the highest overall average improvement rate for R (16.46%), while its overall average improvement rates for RMSE and MAE are slightly lower than those of $Z_t=3.5$, with gaps of 0.95 and 0.66 percentage points, respectively. The overall average improvement rate for R² under 3.0 is 4.15 percentage points lower than that under $Z_t=3.5$. Its improvement ratios are slightly lower than those of $Z_t=3.5$, with all gaps within 1.3 percentage points (RMSE difference 1.26, MAE difference 1.05, R difference 0.65, R² difference 1.26 percentage points), while being significantly higher than those under $Z_t=2.5$ and $Z_t=2$.

In terms of the number of detected anomaly points, the threshold $Z_t=2$ detects the largest number (1858), but its overall average improvement rates and improvement ratios are the lowest, indicating that numerous spatially consistent normal fluctuations are misjudged as anomalies. The threshold $Z_t=3.5$ detects the fewest anomaly points (1046), with slightly higher overall average improvement rates; however, its improvement ratios are generally comparable to those of $Z_t=3$, and the risk of missed detection increases. The threshold $Z_t=3$ achieves a good balance between detection recall rate (1175 points) and correction reliability, indicating good comprehensive performance.

Table A25. Overall average improvement rates and anomaly detection counts under different $Z_t$ thresholds (k = 2, 7 × 7 window, 2016–2021).

${\mathit{Z}}_{\mathit{t}}$ Threshold	Number of Anomalies	Overall Avg. Improvement Rate of RMSE (%)	Overall Avg. Improvement Rate of MAE (%)	Overall Avg. Improvement Rate of R (%)	Overall Avg. Improvement Rate of R2 (%)
2.0	1858	20.58	16.41	11.82	48.65
2.5	1511	22.42	17.89	13.96	52.87
3.0	1175	23.61	18.79	16.46	61.33
3.5	1046	24.56	19.45	15.35	65.48

Table A25. Overall average improvement rates and anomaly detection counts under different $Z_t$ thresholds (k = 2, 7 × 7 window, 2016–2021).

${\mathit{Z}}_{\mathit{t}}$ Threshold	Number of Anomalies	Overall Avg. Improvement Rate of RMSE (%)	Overall Avg. Improvement Rate of MAE (%)	Overall Avg. Improvement Rate of R (%)	Overall Avg. Improvement Rate of R2 (%)
2.0	1858	20.58	16.41	11.82	48.65
2.5	1511	22.42	17.89	13.96	52.87
3.0	1175	23.61	18.79	16.46	61.33
3.5	1046	24.56	19.45	15.35	65.48

Table A26. Improvement proportions under different $Z_t$ thresholds (%, k = 2, 7 × 7 window, 2016–2021).

${\mathit{Z}}_{\mathit{t}}$ Threshold	RMSE Improvement Ratio	MAE Improvement Ratio	R Improvement Ratio	R2 Improvement Ratio
2.0	88.43	88.16	85.74	88.32
2.5	90.25	90.04	87.29	90.18
3.0	91.23	91.23	87.83	91.23
3.5	92.49	92.28	88.48	92.49

Table A26. Improvement proportions under different $Z_t$ thresholds (%, k = 2, 7 × 7 window, 2016–2021).

${\mathit{Z}}_{\mathit{t}}$ Threshold	RMSE Improvement Ratio	MAE Improvement Ratio	R Improvement Ratio	R2 Improvement Ratio
2.0	88.43	88.16	85.74	88.32
2.5	90.25	90.04	87.29	90.18
3.0	91.23	91.23	87.83	91.23
3.5	92.49	92.28	88.48	92.49

Appendix B.3.2. Stability Analysis Across Different Seasons

To examine the performance differences in various thresholds $Z_t$ under different seasonal climatic conditions and to evaluate the stability of the selected threshold $Z_t=3$ across the four seasons, this study further stratified the data by season. With k = 2 and the 7 × 7 window fixed, the overall average improvement rates and improvement ratios for each metric under each threshold were calculated for each season (Table A27, Table A28, Table A29 and Table A30).

Table A27, Table A28, Table A29 and Table A30 systematically present the seasonal decomposition comparison of the four core evaluation metrics under different thresholds $Z_t$. In terms of overall average improvement rates, the performance of $Z_t=3$ across seasons is generally consistent with the overall results of $Z_t=3.5$. For RMSE and MAE, $Z_t=3.5$ achieves slightly higher overall average improvement rates than $Z_t=3$, with differences within 2 percentage points. For the overall average improvement rate of R, $Z_t=3$ significantly outperforms $Z_t=3.5$ in winter (28.42% vs. 20.08%), demonstrating better effectiveness. Regarding the overall average improvement rate of R², $Z_t=3.5$ is slightly higher than $Z_t=3$ in all seasons, with relatively larger gaps in autumn and winter (5.61 and 6.18 percentage points, respectively), while $Z_t=3$, still maintaining a relatively high improvement magnitude. Both $Z_t=3$ and $Z_t=3.5$ are significantly better than $Z_t=2$ and generally outperforms $Z_t=2.5$ overall.

In terms of improvement ratios, the fluctuation ranges of $Z_t=3$ across seasons are 89–94% for RMSE, 88–94% for MAE, 85–91% for R, and 89–94% for R², which are at the same level as $Z_t=3.5$. Compared to $Z_t=2.5$, $Z_t=3$ is slightly better or comparable in spring, summer, and winter, while $Z_t=2.5$ is slightly lower in autumn, though the differences are minimal. The improvement ratios of both $Z_t=3$ and $Z_t=3.5$ are significantly higher than those of $Z_t=2$.

From the perspective of cross-season stability, the performance of $Z_t=3$ exhibits minimal fluctuation across seasons, closely aligning with the overall average levels. Considering the number of detected anomaly points ($Z_t=3$ detects 1175 points, falling between the 1858 points of $Z_t=2$ and the 1,046 points of $Z_t=3.5$), $Z_t=3$ achieves a favorable balance between correction magnitude and correction reliability, remains unaffected by seasonal climatic conditions, and demonstrates good cross-scenario stability.

Table A27. Seasonal RMSE improvement rates and improvement proportions under different $Z_t$ thresholds (k = 2, 7 × 7 window, 2016–2021).

Season	${\mathit{Z}}_{\mathit{t}}=2$	${\mathit{Z}}_{\mathit{t}}=2.5$	${\mathit{Z}}_{\mathit{t}}=3$	${\mathit{Z}}_{\mathit{t}}=3.5$
Overall Avg. Improvement Rate of RMSE (%)
Spring	20.63	21.91	22.85	23.36
Summer	20.31	22.50	25.58	26.62
Autumn	22.05	24.63	25.62	27.60
Winter	19.45	21.42	22.36	23.22
RMSE Improvement Ratio (%)
Spring	89.43	91.33	93.42	94.41
Summer	89.34	89.80	92.14	94.34
Autumn	88.70	90.46	89.43	91.18
Winter	86.35	88.89	89.37	90.46

Table A27. Seasonal RMSE improvement rates and improvement proportions under different $Z_t$ thresholds (k = 2, 7 × 7 window, 2016–2021).

Season	${\mathit{Z}}_{\mathit{t}}=2$	${\mathit{Z}}_{\mathit{t}}=2.5$	${\mathit{Z}}_{\mathit{t}}=3$	${\mathit{Z}}_{\mathit{t}}=3.5$
Overall Avg. Improvement Rate of RMSE (%)
Spring	20.63	21.91	22.85	23.36
Summer	20.31	22.50	25.58	26.62
Autumn	22.05	24.63	25.62	27.60
Winter	19.45	21.42	22.36	23.22
RMSE Improvement Ratio (%)
Spring	89.43	91.33	93.42	94.41
Summer	89.34	89.80	92.14	94.34
Autumn	88.70	90.46	89.43	91.18
Winter	86.35	88.89	89.37	90.46

Table A28. Seasonal MAE improvement rates and improvement proportions under different $Z_t$ thresholds (k = 2, 7 × 7 window, 2016–2021).

Season	${\mathit{Z}}_{\mathit{t}}=2$	${\mathit{Z}}_{\mathit{t}}=2.5$	${\mathit{Z}}_{\mathit{t}}=3$	${\mathit{Z}}_{\mathit{t}}=3.5$
Overall Avg. Improvement Rate of MAE (%)
Spring	16.74	17.80	18.61	19.11
Summer	16.20	17.96	20.25	20.96
Autumn	17.46	19.26	19.87	21.20
Winter	15.23	16.96	17.68	18.15
MAE Improvement Ratio (%)
Spring	89.28	91.14	93.42	94.41
Summer	88.60	89.80	92.14	94.34
Autumn	89.19	91.12	90.24	91.18
Winter	85.54	87.92	88.79	89.80

Table A28. Seasonal MAE improvement rates and improvement proportions under different $Z_t$ thresholds (k = 2, 7 × 7 window, 2016–2021).

Season	${\mathit{Z}}_{\mathit{t}}=2$	${\mathit{Z}}_{\mathit{t}}=2.5$	${\mathit{Z}}_{\mathit{t}}=3$	${\mathit{Z}}_{\mathit{t}}=3.5$
Overall Avg. Improvement Rate of MAE (%)
Spring	16.74	17.80	18.61	19.11
Summer	16.20	17.96	20.25	20.96
Autumn	17.46	19.26	19.87	21.20
Winter	15.23	16.96	17.68	18.15
MAE Improvement Ratio (%)
Spring	89.28	91.14	93.42	94.41
Summer	88.60	89.80	92.14	94.34
Autumn	89.19	91.12	90.24	91.18
Winter	85.54	87.92	88.79	89.80

Table A29. Seasonal R improvement rates and improvement proportions under different $Z_t$ thresholds (k = 2, 7 × 7 window, 2016–2021).

Season	${\mathit{Z}}_{\mathit{t}}=2$	${\mathit{Z}}_{\mathit{t}}=2.5$	${\mathit{Z}}_{\mathit{t}}=3$	${\mathit{Z}}_{\mathit{t}}=3.5$
Overall Avg. Improvement Rate of R (%)
Spring	10.51	11.22	13.21	15.41
Summer	5.26	6.42	8.11	9.07
Autumn	7.94	9.61	10.10	11.43
Winter	20.38	24.31	28.42	20.08
R Improvement Ratio (%)
Spring	87.08	88.56	90.25	90.78
Summer	86.40	87.24	89.29	90.57
Autumn	86.49	87.83	85.77	86.76
Winter	82.93	85.27	85.63	86.18

Table A29. Seasonal R improvement rates and improvement proportions under different $Z_t$ thresholds (k = 2, 7 × 7 window, 2016–2021).

Season	${\mathit{Z}}_{\mathit{t}}=2$	${\mathit{Z}}_{\mathit{t}}=2.5$	${\mathit{Z}}_{\mathit{t}}=3$	${\mathit{Z}}_{\mathit{t}}=3.5$
Overall Avg. Improvement Rate of R (%)
Spring	10.51	11.22	13.21	15.41
Summer	5.26	6.42	8.11	9.07
Autumn	7.94	9.61	10.10	11.43
Winter	20.38	24.31	28.42	20.08
R Improvement Ratio (%)
Spring	87.08	88.56	90.25	90.78
Summer	86.40	87.24	89.29	90.57
Autumn	86.49	87.83	85.77	86.76
Winter	82.93	85.27	85.63	86.18

Table A30. Seasonal R² improvement rates and improvement proportions under different $Z_t$ thresholds (k = 2, 7 × 7 window, 2016–2021).

Season	${\mathit{Z}}_{\mathit{t}}=2$	${\mathit{Z}}_{\mathit{t}}=2.5$	${\mathit{Z}}_{\mathit{t}}=3$	${\mathit{Z}}_{\mathit{t}}=3.5$
Overall Avg. Improvement Rate of R2 (%)
Spring	45.39	46.06	51.67	52.19
Summer	21.79	25.94	33.38	34.79
Autumn	56.42	56.89	64.86	70.47
Winter	61.43	71.60	82.31	88.49
R2 Improvement Ratio (%)
Spring	89.43	91.33	93.42	94.41
Summer	89.34	89.80	92.14	94.34
Autumn	88.70	90.46	89.43	91.18
Winter	85.94	88.65	89.37	90.46

Table A30. Seasonal R² improvement rates and improvement proportions under different $Z_t$ thresholds (k = 2, 7 × 7 window, 2016–2021).

Season	${\mathit{Z}}_{\mathit{t}}=2$	${\mathit{Z}}_{\mathit{t}}=2.5$	${\mathit{Z}}_{\mathit{t}}=3$	${\mathit{Z}}_{\mathit{t}}=3.5$
Overall Avg. Improvement Rate of R2 (%)
Spring	45.39	46.06	51.67	52.19
Summer	21.79	25.94	33.38	34.79
Autumn	56.42	56.89	64.86	70.47
Winter	61.43	71.60	82.31	88.49
R2 Improvement Ratio (%)
Spring	89.43	91.33	93.42	94.41
Summer	89.34	89.80	92.14	94.34
Autumn	88.70	90.46	89.43	91.18
Winter	85.94	88.65	89.37	90.46

Appendix B.3.3. Stability Analysis Across Different Land Cover Types

To evaluate the stability of the selected spatial anomaly threshold $Z_t=3$ under different land surface types, this study classified the seven ground stations in the Heihe River Basin into four categories based on underlying surface attributes: alpine meadow (S1, S3), farmland/artificial vegetation (S2, S4), desert shrubland (S5, S6), and wetland (S7). The hourly LST data from all stations within each category for 2016–2021 were aggregated. With the temporal dimension threshold coefficient fixed at k = 2 and the spatial consistency check window size fixed at 7 × 7, the complete two-stage anomaly detection and IDW correction process was implemented under four threshold $Z_t$ settings (2.0, 2.5, 3.0, and 3.5).

Table A31, Table A32, Table A33 and Table A34 present the overall average improvement rates and improvement ratios for various metrics across the four land cover types under different thresholds $Z_t$. In terms of overall average improvement rates, $Z_t=3.5$ is slightly higher than $Z_t=3$ in most cases, but the differences are generally small. Specifically, for RMSE, $Z_t=3.5$ is slightly higher across all land cover types, with gaps of 1.12, 1.12, 0.02, and 1.69 percentage points in alpine meadow, farmland/artificial vegetation, desert shrubland, and wetland, respectively, with the desert shrubland gap being nearly negligible. For MAE, except in desert shrubland, where $Z_t=3$ slightly surpasses by 0.05 percentage points, $Z_t=3.5$ outperforms by 0.63 to 1.17 percentage points in the other land cover types. For the R metric, $Z_t=3$ (23.36%) is significantly higher than $Z_t=3.5$ (15.36%) in farmland/artificial vegetation, while in other land cover types, $Z_t=3.5$ is slightly higher with gaps all below 2.4 percentage points. For R², $Z_t=3$ exceeds $Z_t=3.5$ by approximately 1.48 and 1.93 percentage points in farmland/artificial vegetation and desert shrubland, respectively, with even larger gaps of $Z_t=3.5$ in wetland and alpine meadow—15.05 and 5.11 percentage points, respectively. Overall, the performance of $Z_t=3$ and $Z_t=3.5$ is very close, and both significantly outperform $Z_t=2$ and $Z_t=2.5$, whose improvement magnitudes and ratios are notably lower.

In terms of improvement ratios, $Z_t=3$ remains highly consistent with $Z_t=3.5$. For example, the difference in RMSE improvement ratios between the two thresholds ranges from 0.27 to 2.15 percentage points, and the difference in MAE improvement ratios ranges from 0.27 to 2.11 percentage points. For the R metric, the improvement ratio of $Z_t=3$ in desert shrubland (91.78%) is even higher than that of $Z_t=3.5$ (91.15%), while differences in other land cover types are all within 1.79 percentage points. The differences in R² improvement ratios also do not exceed 2.15 percentage points. This indicates that $Z_t=3$ is essentially comparable to $Z_t=3.5$ in terms of correction reliability.

From the perspective of cross-land cover stability, the performance of $Z_t=3$ exhibits minimal fluctuation across different land cover types, closely aligning with the overall average levels of the Heihe River Basin. Considering the number of detected anomaly points ($Z_t=3$ detects 1,175 points, falling between the 1858 points of $Z_t=2$ and the 1046 points of $Z_t=3.5$), while maintaining correction performance similar to $Z_t=3.5$, $Z_t=3$ detects more anomaly points, has a significantly lower risk of missed detection than $Z_t=3.5$, and a much lower risk of false detection than $Z_t=2$. Therefore, $Z_t=3$ achieves a favorable balance between correction magnitude and correction reliability across different land surface types, indicating good robustness.

Table A31. RMSE improvement rates and improvement proportions under different |${\mathrm{Z}}_{\mathrm{t}}$| thresholds and land cover types (k = 2, 7 × 7 window, 2016–2021).

Land Cover Type	${\mathit{Z}}_{\mathit{t}}=2$	${\mathit{Z}}_{\mathit{t}}=2.5$	${\mathit{Z}}_{\mathit{t}}=3$	${\mathit{Z}}_{\mathit{t}}=3.5$
Overall Avg. Improvement Rate of RMSE (%)
Alpine Meadow	20.41	22.64	23.99	25.11
Farmland/Artificial Veg.	20.99	22.53	23.46	24.58
Desert Shrubland	21.75	23.22	24.21	24.23
Wetland	18.25	20.49	22.29	23.98
RMSE Improvement Ratio (%)
Alpine Meadow	87.77	90.62	91.47	93.62
Farmland/Artificial Veg.	87.03	87.90	88.07	88.77
Desert Shrubland	92.00	93.46	94.86	95.13
Wetland	86.17	88.74	91.10	93.25

Table A31. RMSE improvement rates and improvement proportions under different |${\mathrm{Z}}_{\mathrm{t}}$| thresholds and land cover types (k = 2, 7 × 7 window, 2016–2021).

Land Cover Type	${\mathit{Z}}_{\mathit{t}}=2$	${\mathit{Z}}_{\mathit{t}}=2.5$	${\mathit{Z}}_{\mathit{t}}=3$	${\mathit{Z}}_{\mathit{t}}=3.5$
Overall Avg. Improvement Rate of RMSE (%)
Alpine Meadow	20.41	22.64	23.99	25.11
Farmland/Artificial Veg.	20.99	22.53	23.46	24.58
Desert Shrubland	21.75	23.22	24.21	24.23
Wetland	18.25	20.49	22.29	23.98
RMSE Improvement Ratio (%)
Alpine Meadow	87.77	90.62	91.47	93.62
Farmland/Artificial Veg.	87.03	87.90	88.07	88.77
Desert Shrubland	92.00	93.46	94.86	95.13
Wetland	86.17	88.74	91.10	93.25

Table A32. MAE improvement rates and improvement proportions under different |${\mathrm{Z}}_{\mathrm{t}}$| thresholds and land cover types (k = 2, 7 × 7 window, 2016–2021).

Land Cover Type	${\mathit{Z}}_{\mathit{t}}=2$	${\mathit{Z}}_{\mathit{t}}=2.5$	${\mathit{Z}}_{\mathit{t}}=3$	${\mathit{Z}}_{\mathit{t}}=3.5$
Overall Avg. Improvement Rate of MAE (%)
Alpine Meadow	16.53	18.32	19.24	20.17
Farmland/Artificial Veg.	16.41	17.62	18.46	19.09
Desert Shrubland	17.34	18.62	19.23	19.18
Wetland	14.75	16.43	17.95	19.12
MAE Improvement Ratio (%)
Alpine Meadow	87.18	90.12	91.18	93.29
Farmland/Artificial Veg.	87.03	87.90	88.07	88.42
Desert Shrubland	91.80	93.46	94.86	95.13
Wetland	85.85	88.31	91.62	93.25

Table A32. MAE improvement rates and improvement proportions under different |${\mathrm{Z}}_{\mathrm{t}}$| thresholds and land cover types (k = 2, 7 × 7 window, 2016–2021).

Land Cover Type	${\mathit{Z}}_{\mathit{t}}=2$	${\mathit{Z}}_{\mathit{t}}=2.5$	${\mathit{Z}}_{\mathit{t}}=3$	${\mathit{Z}}_{\mathit{t}}=3.5$
Overall Avg. Improvement Rate of MAE (%)
Alpine Meadow	16.53	18.32	19.24	20.17
Farmland/Artificial Veg.	16.41	17.62	18.46	19.09
Desert Shrubland	17.34	18.62	19.23	19.18
Wetland	14.75	16.43	17.95	19.12
MAE Improvement Ratio (%)
Alpine Meadow	87.18	90.12	91.18	93.29
Farmland/Artificial Veg.	87.03	87.90	88.07	88.42
Desert Shrubland	91.80	93.46	94.86	95.13
Wetland	85.85	88.31	91.62	93.25

Table A33. R improvement rates and improvement proportions under different |${\mathrm{Z}}_{\mathrm{t}}$| thresholds and land cover types (k = 2, 7 × 7 window, 2016–2021).

Land Cover Type	${\mathit{Z}}_{\mathit{t}}=2$	${\mathit{Z}}_{\mathit{t}}=2.5$	${\mathit{Z}}_{\mathit{t}}=3$	${\mathit{Z}}_{\mathit{t}}=3.5$
Overall Avg. Improvement Rate of R (%)
Alpine Meadow	15.49	18.70	21.49	23.83
Farmland/Artificial Veg.	16.33	19.34	23.36	15.36
Desert Shrubland	3.73	4.15	4.12	4.28
Wetland	11.06	11.67	13.62	15.15
R Improvement Ratio (%)
Alpine Meadow	83.69	85.68	86.47	88.26
Farmland/Artificial Veg.	84.02	84.70	84.38	84.91
Desert Shrubland	89.20	91.36	91.78	91.15
Wetland	86.50	88.31	90.58	91.41

Table A33. R improvement rates and improvement proportions under different |${\mathrm{Z}}_{\mathrm{t}}$| thresholds and land cover types (k = 2, 7 × 7 window, 2016–2021).

Land Cover Type	${\mathit{Z}}_{\mathit{t}}=2$	${\mathit{Z}}_{\mathit{t}}=2.5$	${\mathit{Z}}_{\mathit{t}}=3$	${\mathit{Z}}_{\mathit{t}}=3.5$
Overall Avg. Improvement Rate of R (%)
Alpine Meadow	15.49	18.70	21.49	23.83
Farmland/Artificial Veg.	16.33	19.34	23.36	15.36
Desert Shrubland	3.73	4.15	4.12	4.28
Wetland	11.06	11.67	13.62	15.15
R Improvement Ratio (%)
Alpine Meadow	83.69	85.68	86.47	88.26
Farmland/Artificial Veg.	84.02	84.70	84.38	84.91
Desert Shrubland	89.20	91.36	91.78	91.15
Wetland	86.50	88.31	90.58	91.41

Table A34. R² improvement rates and improvement proportions under different |${\mathrm{Z}}_{\mathrm{t}}$| thresholds and land cover types (k = 2, 7 × 7 window, 2016–2021).

Land Cover Type	${\mathit{Z}}_{\mathit{t}}=2$	${\mathit{Z}}_{\mathit{t}}=2.5$	${\mathit{Z}}_{\mathit{t}}=3$	${\mathit{Z}}_{\mathit{t}}=3.5$
Overall Avg. Improvement Rate of R2 (%)
Alpine Meadow	58.96	59.28	67.57	72.68
Farmland/Artificial Veg.	47.36	51.69	59.04	57.56
Desert Shrubland	20.43	23.13	25.24	23.31
Wetland	79.13	93.07	109.60	124.65
R2 Improvement Ratio (%)
Alpine Meadow	87.77	90.62	91.47	93.62
Farmland/Artificial Veg.	87.03	87.90	88.07	88.77
Desert Shrubland	91.80	93.46	94.86	95.13
Wetland	85.85	88.31	91.10	93.25

Table A34. R² improvement rates and improvement proportions under different |${\mathrm{Z}}_{\mathrm{t}}$| thresholds and land cover types (k = 2, 7 × 7 window, 2016–2021).

Land Cover Type	${\mathit{Z}}_{\mathit{t}}=2$	${\mathit{Z}}_{\mathit{t}}=2.5$	${\mathit{Z}}_{\mathit{t}}=3$	${\mathit{Z}}_{\mathit{t}}=3.5$
Overall Avg. Improvement Rate of R2 (%)
Alpine Meadow	58.96	59.28	67.57	72.68
Farmland/Artificial Veg.	47.36	51.69	59.04	57.56
Desert Shrubland	20.43	23.13	25.24	23.31
Wetland	79.13	93.07	109.60	124.65
R2 Improvement Ratio (%)
Alpine Meadow	87.77	90.62	91.47	93.62
Farmland/Artificial Veg.	87.03	87.90	88.07	88.77
Desert Shrubland	91.80	93.46	94.86	95.13
Wetland	85.85	88.31	91.10	93.25