This study focuses on mainland China as the study region (18–53° N, 73–135° E; Fig. 1). The region exhibits pronounced topographic heterogeneity characterized by a stepwise terrain pattern, including the Tibetan Plateau, inland basins, and eastern plains. Such complex terrain leads to strong regional contrasts in climatic background and water vapor transport and convergence mechanisms, resulting in pronounced spatial heterogeneity in PWV.

To systematically evaluate the influence of training sample size and spatial configuration on the generalization capability of all-weather PWV models, a total of 244 GNSS stations were selected as modeling samples. In addition, 16 GNSS stations were randomly excluded from the training process and reserved as an independent validation set. Furthermore, 80 radiosonde stations from the Integrated Global Radiosonde Archive (IGRA) were selected as an external independent validation dataset. The spatial distribution of all stations is shown in Fig. 1, where Fig. 1b highlights the randomly selected 16 independent GNSS validation stations.

The reconstruction aims to restore PWV continuity under cloudy or missing-data conditions by establishing a nonlinear mapping between PWV and multiple auxiliary variables, including meteorological parameters, topography, time, and spatial information. To ensure spatiotemporal consistency across all input data, a rigorous pre-processing procedure was applied. Temporally, the ERA5 data were aligned to match the exact observation hours of the full-disk PWV product. Spatially, the PWV data and the DEM over the study domain were resampled onto the ERA5's 0.25°×0.25° grid using a bilinear interpolation method, thereby unifying the spatial resolution and grid alignment for all datasets prior to model training.

A random forest (RF) model was employed to describe the nonlinear relationship between PWV and auxiliary variables: 5PWVFY-4A=fRF(Lat,Lon,Dem,Time,T2m,TCW,SP) where T2m, SP, and PWV denote near-surface temperature, surface pressure, and water vapor-related parameters, respectively; Dem represents elevation; Lat and Lon denote spatial location; and TCW is total column water. After training, the model was applied to generate spatiotemporally continuous reconstructed PWV fields.

The model performance was evaluated using the test set, and the results show that the reconstructed model achieved an RMSE of 1.12 mm and a Bias of 0.51 mm. The reconstructed PWV has been verified to effectively retrieve the spatial distribution characteristics of water vapor over the entire region, and its distribution is consistent with that of ERA5 PWV. Detailed validation of the reconstruction has been reported in previous studies (Wang et al., 2026) and is not repeated here.

Although reconstruction improves data completeness, regional systematic biases remain in PWV. Therefore, GNSS PWV was used as a reference to calibrate reconstructed PWV. To address the spatial mismatch between site-based GNSS observational data and gridded satellite PWV, this study employs a bilinear interpolation method to interpolate gridded data to the GNSS site locations. Temporal mismatches were handled by averaging GNSS PWV at the two nearest epochs before and after satellite overpass times.

An RF-based calibration model was constructed to map reconstructed PWV, spatiotemporal information, NDVI, and land cover characteristics to GNSS PWV: 6PWVGNSS=fRF(Lat,Lon,Elv,Time,NDVI,PLCTi,PWVFY-4A) where PLCTi denotes the area fraction of the ith land cover type within a 2 km buffer around each GNSS station.

To quantitatively assess the effects of training station density and spatial structure on model generalization, two groups of comparative experiments were designed, as shown in Fig. 2b. 1.

Training station number experiment. Different numbers of GNSS stations were randomly selected from the available 228 stations to construct training datasets. Seven training sample sizes were considered: No. = 30, 40, 80, 120, 160, 200, and 244. Among them, Experiment No. 244 served as the reference. The remaining GNSS stations were used as independent validation samples, and IGRA radiosonde PWV data were further introduced for external validation.

Through this experimental design, the relationships among sample size, spatial coverage, and error response were systematically examined, enabling the identification of major limiting factors governing model performance improvement and the corresponding saturation thresholds.

Key RF parameters were optimized using out-of-bag (OOB) error as the evaluation criterion. Bayesian optimization was applied to automatically search the parameter space. This process was repeated for more than 5 cycles to minimize random variability in the optimal results. The range of tree numbers was 50–250 with an interval of 10. The final optimal parameter settings are summarized in Table 1.

Model performance was evaluated at three levels, including training GNSS stations, independent GNSS validation stations, and IGRA radiosonde stations, to distinguish fitting capability from spatial extrapolation performance. The coefficient of determination, RMSE, and mean bias were adopted as the primary evaluation metrics: 7Bias=1m∑i=1mDatamodel-Dataref8RMSE=1m∑i=1mDatamodel-Dataref29R2=1-∑Dataref-Datamodel2∑Dataref-Datamodel‾2 where Data^model denotes model-derived PWV and Data^ref represents reference PWV observations.

In addition to overall statistical metrics, model robustness was further assessed in terms of spatial error patterns, PWV time series at representative stations, and monthly aggregated statistics. These analyses were conducted to evaluate model stability, seasonal dependence, and spatial consistency across different observational networks.

A central finding of our experiments is that in-sample fitting accuracy is remarkably insensitive to the number of training stations, whereas spatial generalization (validation at independent sites) critically depends on it. As shown in Fig. 3, for training stations, the accuracy of all-weather PWV model remains consistently high as the number of training stations increases from 30–244. The coefficient of determination R2 remains stable within the range of 0.988–0.990, while RMSE varies only slightly between 1.45 and 1.66 mm. Mean bias is close to zero throughout all configurations (|Bias|≤0.02 mm). These results suggest that the random forest model is capable of sufficiently learning the regional-scale mapping relationship between GNSS PWV and PWV, and that further increases in training samples yield only marginal improvements in fitting accuracy.

In contrast, the number of training stations has a pronounced effect on model performance at independent GNSS validation stations. As training station numbers increase, retrieval accuracy at independent stations improves steadily, with R2 increasing from approximately 0.96–0.97 and RMSE decreasing markedly from 3.24–2.28 mm. Meanwhile, Bias gradually converges from 0.42 mm toward near-zero values. These results demonstrate that enlarging the training dataset enhances the model's ability to represent regional PWV heterogeneity and effectively suppresses systematic errors during spatial extrapolation. Notably, when the number of training stations exceeds approximately 160, improvements in independent-station accuracy become marginal, indicating a saturation behavior in model performance and suggesting that the dominant statistical characteristics of PWV over the study region have been adequately captured.

This saturation identifies a cost-effective range for station numbers. Beyond this range, merely adding stations yields diminishing returns. To dissect the drivers of this saturation and explore how to optimize performance within a fixed station budget, we now turn to a spatial analysis of errors.

Figure 4 presents the inversion performance of models trained with different numbers of stations at 16 GNSS stations that were not involved in the training process. Uncalibrated PWV exhibits substantial systematic underestimation (Bias: -2.17 mm) accompanied by large random errors (RMSE: 3.93 mm). With increasing training station numbers, calibration errors are significantly reduced and spatial performance becomes more stable, with particularly pronounced improvements observed in low-latitude regions. When the training station number reaches 200, the mean Bias decreases to 0.07 mm, while RMSE shows a monotonic decline from 3.93–1.90 mm.

The consistency of these improvements across independent stations indicates that increasing training sample size effectively mitigates systematic bias and enhances model generalization. From a mechanistic perspective, when training data are limited, the model cannot adequately represent the spatial heterogeneity of regional PWV, leading to error accumulation and amplified random errors in regions with distinct climatic backgrounds, surface conditions, and moisture variability. As training station numbers increase, the diversity of PWV scenarios represented in the training data expands, enabling the model to learn more representative regional statistical features and thereby improving both accuracy and stability at independent locations.

Independent validation using IGRA radiosonde PWV is summarized in Table 2. When the number of training stations increases from 30–120, model errors decrease substantially, with RMSE reduced from 2.68–2.36 mm and Bias converging to near-zero values (0.02 mm). This indicates that a moderate training sample size is enough to achieve a favorable balance between accuracy and robustness. Further increasing the number of training stations to 160–244 results in only minor fluctuations in RMSE (approximately 2.37–2.39 mm) and Bias, confirming that model performance gradually approaches saturation. Moreover, it was found that the model trained on 120 stations achieved higher accuracy than those trained on a larger number of stations. This phenomenon can be attributed to two main reasons. First, as the number of training stations increases, the model is compelled to learn more generalizable physical patterns, which may slightly reduce its fitting accuracy to the training data (i.e., a shift from “overfitting” toward “generalization”). Second, redundant stations fail to provide additional useful information; instead, they introduce a smoothing effect that dilutes the critical spatial features present in the original data.

Overall, the consistent patterns revealed by Figs. 3 and 4 and Table 2 demonstrate that increasing training station density is essential for improving the spatial generalization capability of GNSS PWV calibration models. However, larger sample sizes do not necessarily translate into proportional performance gains. From the perspective of station number and random spatial distribution alone, a training station scale of approximately 120–200 provides an effective compromise between accuracy improvement and computational efficiency for regional-scale PWV calibration over China.

While the analysis above confirms the role of station number and its saturation, a more pressing practical question emerges under resource constraints: Can the spatial layout of a fixed number of stations serve as a lever to enhance performance beyond what the number alone dictates? To answer this, we analyze the spatial patterns of model errors.

Figure 5 presents the spatial distributions of RMSE and Bias over China based on IGRA radiosonde validation under different training station numbers, together with the spatial distribution of training GNSS stations. Overall, both RMSE and Bias exhibit pronounced spatial heterogeneity, reflecting the complex spatial variability of atmospheric water vapor and its controlling factors.

When the number of training stations is limited (e.g., No. =30 and No. =40), spatial discrepancies in model errors are particularly pronounced. RMSE is generally higher in regions with more active moisture variability, especially in eastern and southern China, where RMSE at some stations exceeds 3.0 mm. At the same time, Bias shows a clear pattern of alternating positive and negative values, indicating strong regional dependence during spatial extrapolation. These results suggest that insufficient training samples with limited spatial coverage hinder the model's ability to capture regional-scale PWV heterogeneity, leading to increased systematic errors in specific regions.

As the number of training stations increases to 80 and 120, the spatial error structure improves substantially. Overall RMSE levels decrease, the extent of high-error regions shrinks markedly, and RMSE at most stations concentrates within approximately 2.0–2.5 mm. Meanwhile, the spatial distribution of Bias becomes smoother, with extreme positive and negative deviations significantly reduced. This improvement indicates a substantially enhanced capacity of the model to adapt to regional differences in water vapor variability. When the number of training stations further increases to 160, the improvement in RMSE and Bias becomes relatively limited, suggesting that the dominant spatial variability patterns of PWV have largely been captured under the current data and regional setting, and that further increases in training samples yield diminishing returns in terms of spatial error reduction.

Overall, these spatial patterns confirm that small training datasets with insufficient spatial coverage are unable to represent regional PWV heterogeneity, resulting in localized systematic biases. In contrast, increasing training station numbers enhances the model's adaptability to diverse climatic backgrounds and moisture regimes, thereby effectively reducing spatially non-uniform errors. Once the number of training stations reaches approximately 120–160, model performance gradually approaches saturation, consistent with the statistical results discussed previously. This consistency further emphasizes the importance of jointly considering training sample size and spatial coverage in regional-scale PWV calibration.

To further examine the role of spatial density, Table 3 summarizes the relationship between calibration accuracy and station spatial resolution, quantified by the mean inter-station distance derived from a Delaunay triangulation. RMSE shows an overall positive correlation with mean station spacing, indicating that denser station networks provide stronger spatial constraints on the model. When the number of training stations exceeds approximately 120, improvements in accuracy become notably weaker, again exhibiting a diminishing marginal benefit. This behavior can be attributed to the effective reduction in inter-station distance with increasing station numbers, which enhances the spatial representativeness of training data and improves the model's ability to characterize PWV spatial variability. These results demonstrate that, beyond sample size, the spatial configuration of training stations is a critical factor limiting further improvements in calibration performance.

Due to the combined influence of land–sea contrast, temperature fields, and large-scale circulation, southern China (black-box region) exhibits stronger spatiotemporal variability in water vapor and more pronounced short-term fluctuations than northern China. Under limited station coverage, the model struggles to capture fine-scale variability in this region, resulting in generally lower accuracy compared with northern China. For example, for the RF model, increasing the training configuration from RF_N30 to RF_N160 reduces mean station spacing by 46.40 % in southern China and leads to a 13.04 % reduction in RMSE, whereas in northern China, mean spacing is reduced by approximately 55 % and RMSE decreases by 17.10 %. These comparisons further demonstrate that reducing inter-station distance and improving spatial coverage density can significantly enhance calibration accuracy, although the magnitude of improvement is constrained in regions with more intense moisture variability.

Figure 6 further compares the spatial distribution of PWV retrieved by different models under varying training station numbers. A comparison of Fig. 6a–c indicates that the all-weather model can effectively fill in regional missing values. At large spatial scales, the RF_N244 retrieval exhibits strong consistency with the GNSS-derived PWV field, characterized by a gradual increase from northwestern to southeastern China. To better illustrate the influence of training sample size, RF_N244 is used as a reference benchmark, and spatial differences between other models and RF_N244 are analyzed, as shown Fig. 6d–i. When training station numbers are small (e.g., RF_N30 and RF_N40), deviations relative to RF_N244 are large and spatially heterogeneous, with local differences exceeding 6 mm in some regions. These discrepancies are particularly pronounced in areas with complex terrain or strong moisture gradients, indicating elevated uncertainty in representing local PWV structures under sparse training conditions.

As training station numbers increase, spatial differences relative to RF_N244 decrease markedly, with patterns transitioning from scattered to more continuous and smoother distributions. Extreme difference regions shrink substantially, and when training station numbers reach 160–200, only minor small-scale fluctuations remain. Notably, although station numbers increase beyond 120, the overall spatial coverage pattern does not improve substantially, reinforcing the conclusion that station layout and coverage uniformity are as important as sample size in constraining model performance.

The persistent spatial error patterns under sparse conditions suggest that the spatial representativeness and coverage uniformity of stations may be a more critical performance controller than sheer quantity. To test this hypothesis directly, we designed a controlled experiment isolating the effect of spatial configuration while holding the station number constant.

To explicitly isolate the effect of spatial configuration, calibration models were trained using GNSS stations arranged under three different spatial distribution structures, and their performance was independently evaluated using the 16 GNSS stations that were not involved in the model training process. The validation results are summarized in Table 4.

As shown in Table 4, the randomly distributed configuration consistently achieves the best calibration performance. Compared with the clustered and surrounding configurations, the random distribution reduces RMSE by 27.08 % and 26.59 %, respectively, and reduces Bias by 65.96 % and 15.69 %, respectively. These results indicate that a spatially uniform and representative training set is more effective in constraining regional PWV variability and suppressing systematic errors.

In contrast, the surrounding configuration exhibits the poorest performance among the three layouts. This degradation is likely attributable to the spatial mismatch between training and validation stations: most of the remaining validation stations are concentrated in the mid- and low-latitude regions, where water vapor variability is stronger and more dynamic. The lack of representative training information for these regions limits the model's ability to characterize intense moisture variability, leading to increased random errors and systematic bias during spatial extrapolation.

Overall, Table 4 show that, for regional-scale PWV calibration, spatial representativeness and coverage uniformity of training stations are more critical than merely increasing station numbers. Local clustering or surrounding layouts do not guarantee improved accuracy and may even degrade spatial extrapolation capability, whereas uniformly distributed random configurations offer the most robust and transferable solution for GNSS-constrained satellite PWV calibration.

Figure 7 compares model performance under different station layouts while keeping the training station number fixed (No. =200). Figure 7d–f visualizes the spatial difference patterns of RF_N244 retrievals under distinct station layouts, while Fig. 7g–i depicts the actual geographical arrangements corresponding to clustered, surrounding, and randomly distributed stations, respectively. Even with identical sample sizes, significant performance differences emerge among spatial configurations: random distributions perform best (RMSE: 1.44 mm, Bias: -0.34 mm), followed by surrounding configurations (RMSE: 1.75 mm, Bias: -0.68 mm), while clustered configurations perform worst (RMSE: 2.39 mm, Bias: -0.36 mm). This finding highlights the decisive role of spatial coverage uniformity and representativeness in determining model generalization capability.

Further inspection reveals that performance differences among configurations are relatively small over low-elevation plains, whereas the largest discrepancies occur in regions characterized by strong topographic and climatic gradients, such as plateaus, basins, and transition zones. A more balanced station layout provides more comprehensive spatial constraints, thereby improving the representation of water vapor variability in complex terrain. For instance, compared with a surrounding configuration, a more uniformly distributed configuration yields markedly improved performance over the Tibetan Plateau and the Yunnan–Guizhou Plateau, indicating reduced systematic bias at high elevations.

It is noteworthy that although clustered configurations increase station density locally, they result in the poorest overall performance, as shown Fig. 7d. In regions with strong spatial heterogeneity in water vapor and surface conditions, locally clustered observations can introduce spatial bias in the training dataset, making it difficult for a single global mapping function to simultaneously represent diverse environments such as plateaus, mountains, plains, lakes, and urban areas. Consequently, fitting errors accumulate and spatial extrapolation capability deteriorates. These results demonstrate that blindly increasing station density through localized clustering may be counterproductive, and that optimized spatial distribution is essential for robust regional calibration.

Taken together, the spatial analysis confirms that while training station number is important, optimizing spatial layout and coverage uniformity under a given sample size is even more critical for improving calibration accuracy and robustness in practical applications.

Finally, we evaluate temporal stability to assess whether optimizations in station number and layout yield robust performance under dynamic atmospheric conditions. Figure 8 presents PWV time series comparisons at three GNSS stations located at different latitudes. All three stations exhibit pronounced seasonal cycles, characterized by high PWV in summer and low PWV in winter. Across different training configurations, RF-based models generally capture the seasonal evolution of GNSS PWV well; however, differences remain in amplitude representation and short-term variability.

When training station numbers are small (e.g., RF_No. =30), the model tends to underestimate peak PWV during high-moisture periods, with evident smoothing effects in some intervals. As training station numbers increase to 80 and 200, agreement with GNSS observations improves substantially, particularly at mid- and low-latitude stations. This improvement indicates that enlarging the training dataset enables the model to better learn both the amplitude and temporal structure of PWV variability.

Analysis of error time series further reveals that, although errors generally fluctuate around zero, their magnitude and dispersion strongly depend on training station number. Sparse training configurations exhibit larger positive and negative oscillations and higher error dispersion, whereas increasing training station numbers leads to more concentrated error distributions and a marked reduction in extreme errors. These results demonstrate that increasing training station numbers enhances not only mean accuracy but also temporal stability and robustness. Nevertheless, occasional synchronized biases across different training configurations persist during some high-PWV periods, suggesting that retrieval errors are not solely controlled by training sample size, but may also be influenced by rapid moisture transport, strong convective activity, and observation or matching uncertainties.

Monthly-scale variations in Bias and RMSE under different training station numbers are shown in Fig. 9, with the original PWV product included for comparison. Overall, calibrated models exhibit substantially lower Bias and RMSE than the original product throughout the year, while also displaying a clear seasonal dependence, with larger errors in summer and smaller errors in winter.

In terms of Bias, monthly mean values fluctuate around zero for all calibrated models, but their amplitudes depend on training station number. Models trained with fewer stations show more pronounced positive and negative deviations in certain months, particularly during seasonal transitions or under high-PWV conditions. As training station numbers increase to 80–120 and beyond, monthly Bias variations converge markedly toward zero, indicating effective suppression of systematic errors and improved monthly stability.

RMSE exhibits a consistent seasonal pattern across all models, with maximum values occurring during June–August and minimum values in winter months. Under sparse training conditions, RMSE during high-PWV months can reach 4–5 mm. As training station numbers increase, overall RMSE decreases significantly and monthly variability becomes smoother. When training station numbers reach approximately 120–200, RMSE differences among months are substantially reduced, demonstrating enhanced robustness under varying seasonal moisture regimes.

It should be noted that even with increased training station numbers, RMSE remains relatively high during summer months characterized by intense moisture variability. This suggests that monthly-scale errors are not solely determined by training sample size, but are also influenced by enhanced convective activity, rapid moisture transport, and increased observational uncertainties. Nevertheless, the convergence of Bias and RMSE with increasing training station numbers provides a solid basis for further investigations into seasonal dependence and extreme moisture processes.