The present study focuses on the Cevennes region in the southern Central Massif (see Fig. ). This mountainous study area is characterized by rugged terrain, including high plateaus and forested hills, with elevations reaching 1700 m. Annual precipitation experiences significant spatial variability. The higher elevations can collect more than 2000 mm while the lower foothills present a Mediterranean climate with annual precipitation ranging from 600–1200 mm . Winter precipitation can fall as snow at high elevations, even if snow cover is intermittent. Autumn is the wettest season. The moisture-laden air from the warm Mediterranean Sea faces colder air from the north, leading to frequent extreme rainfall events of 300–500 mm and generating heavy floods in the valleys .

To capture the spatial variability of precipitation induced by the complex topography, a dense network of 973 rain gauges is available, including 197 stations in the study domain and 776 additional stations near the borders of the study domain. Rain gauges are up to 1500 m, encompassing the full elevation distribution. The number of available rain gauges on a daily timescale varies over the years, ranging from a minimum of 625 to a maximum of 836. The observed data were not quality-controlled in this study. Because we do not work with climatological statistics, temporal discontinuity is not a major issue. However, outliers may still occur and could affect the interpolated fields. Such outliers may influence local results, but we do not expect them to alter the conclusions of this study. Before applying the method in an operational or real-time context, a quality control procedure is needed, including a spatial anomaly analysis to identify outliers that exceed physically plausible differences from nearby stations under similar terrain characteristics, followed by a homogeneity test such as the Standard Normal Homogeneity Test SNHT,.

The study period ranges from 1982–2018, which corresponds to the availability of AROME simulations. 786 precipitation events (nearly 20 events per year), defined as the days with at least 50 mm recorded at a minimum of five rain gauges, are selected. Most of the 786 precipitation events arise from southerly atmospheric flow and central depression patterns, where strong orographic intensification is expected. Some winter events also include snowfall at high elevations.

This study develops a probabilistic geostatistical framework for daily precipitation interpolation, using four variograms derived from rain gauges and CP-RCM simulations. Unlike traditional approaches that rely solely on rain gauge data, our method exploits CP-RCM fields to estimate daily isotropic and anisotropic covariance structures. In the following, we detail the stochastic modeling and evaluation procedures.

In our analysis, the ensemble mean of the conditional simulations is equivalent to the result obtained using deterministic kriging with external drift (KED). Later in the article, we will use the terms “kriging” and “ensemble mean” interchangeably. While kriging captures the most likely precipitation value, the conditional simulations capture the entire ensemble of possible realizations, providing spatial uncertainty. To assess the performance of the spatial interpolation models, we evaluate both deterministic and probabilistic interpolated precipitation fields. The evaluation focuses on three key aspects:

point-scale accuracy using cross-validation on rain gauge stations;

Leave-one-cluster-out cross-validation evaluates the accuracy of ensemble means for all rain gauge stations in the Cévennes region. We compute the mean error and mean absolute error on intense precipitation (≥50mm) for the 786 events.

SPAZM has lower spatial interpolation skills than rgISO, rgANISO, arISO, and arANISO (Fig. ). Notably, SPAZM systematically underestimates intense precipitation in the northernmost high-altitude areas, with mean errors ranging from -30% to -50%, whereas other models (rgISO, rgANISO, arISO, arANISO) show less severe underestimation (-20% to -30%). SPAZM simultaneously underestimates and overestimates precipitation at neighboring stations, raising concerns about its robustness. rgISO, rgANISO, arISO, and arANISO all underestimate intense precipitation on average. The anisotropy contribution is only visible in the mean absolute error for the most northeasterly rain gauge stations.

To complement cross-validation, the TWS investigates the image gradient similarities, taking COMEPHORE fields as references. Lower TWS values indicate a better resemblance to COMEPHORE. Figure  illustrates the distribution of TWS values for three event classes: (1) Low anisotropy events (0.75≤η≤1), (2) Medium anisotropy events (0.5≤η≤0.75), (3) Strong anisotropy events (0≤η≤0.5). η results from the anisotropy ratio estimation with the daily AROME fields. TWS scores show that SPAZM exhibits lower image gradient similarity to COMEPHORE than all geostatistical approaches, regardless of the event type, indicating a poorer representation of spatial gradients. For both isotropic and anisotropic configurations, AROME-based methods (arISO and arANISO) consistently outperform their counterparts that use only rain gage data (rgISO and rgANISO). This systematic improvement suggests that the use of AROME fields allows for a more accurate estimation of the precipitation covariance structure and, consequently, a better reproduction of spatial gradients. Using only rain gauges, rgANISO outperforms rgISO for strongly anisotropic events, whereas the opposite behavior is observed for weakly anisotropic events. This indicates that accounting for anisotropy can improve gradient similarity when anisotropy is pronounced, but the limited robustness of anisotropic variogram estimation from rain gauges alone prevents drawing firm conclusions for weakly anisotropic cases. In contrast, arANISO consistently outperforms arISO, with a marked improvement for strongly anisotropic events and a slight but systematic improvement for weakly anisotropic ones. As no robustness issues arise in the estimation of anisotropic variograms when using AROME fields, these results provide stronger evidence of the added value of explicitly accounting for anisotropy in the representation of the spatial variability of precipitation. COMEPHORE relies on kriging for stratiform precipitation and radar observations for convective precipitation. While COMEPHORE uses isotropic covariance in the kriging step, anisotropy is introduced through the radar component. This suggests that anisotropic variograms, particularly when informed by AROME fields, help better reproduce the spatial structure of radar-derived precipitation patterns.

Figure  presents the spectral slopes of the reference and predicted precipitation fields. SPAZM systematically underestimates the highest slopes, indicating an overestimation of fine-scale spatial variability. Compared to rgISO, rgANISO degrades the representation of spatial variability for weakly anisotropic events and provides only limited improvement for strongly anisotropic events. In contrast, arISO and arANISO improve the representation of spatial variability, particularly for strongly anisotropic events. Among them, arANISO better reproduces spatial variability, as the highest spectral slopes are no longer underestimated.

To assess the reliability of ensemble simulations, we apply the same cross-validation approach used for evaluating ensemble means (see Sect. ). The Continuous Ranked Probability Score (CRPS) is computed to quantify probabilistic prediction accuracy, where lower values indicate better performance. Distribution of CRPS scores reveals similar performances for rgISO, rgANISO, and arISO but a notable improvement with arANISO (Fig. ). The error reduction is substantial for strong anisotropy events. This suggests that integrating anisotropy into the covariance structure using AROME enhances the ability of the ensemble to capture observed daily precipitation. Figure S1 in the Supplement demonstrates that this result holds across all weather regimes. Figure S2 in the Supplement further shows that the result also applies to lower precipitation intensity ranges (1–20 and 20–50 mm). For this reason, we later quantify precipitation uncertainty using conditional simulations from anisotropic covariance, derived from AROME simulations.

For the 20 most intense events, the mean catchment precipitations derived from the ensembles of 100-member arANISO from conditional simulations closely agree with COMEPHORE estimates (Fig. ) with a high rain gauge density. The interquartile range of the simulations frequently contains the mean COMEPHORE precipitation for the Chassezac at Sainte-Marguerite and the Thines at Pont de Gournier catchments. For the Chassezac at Pont du Mas catchment, the interquartile range does not necessarily encompass COMEPHORE precipitation, but the full range of the ensemble does encompass it. This suggests the reliability of the uncertainty estimates provided by the ensemble. arANISO is robust, as conditional simulations nearly agree with COMEPHORE estimates in an ungauged scenario. The artificial decrease of rain gauge density causes a slight negative bias in the conditional simulations. The latter frequently contain COMEPHORE precipitations through wider confidence intervals.

As expected, uncertainty increases with precipitation intensity, decreases with catchment size and rain gauge density, and provides asymmetrical distributions on mean precipitations. For instance, on 18 September 2014, the mean rainfall on the small Thines at Pont de Gournier catchment ranges from 175 mm to more than 450 mm with the complete set of rain gauges, representing a factor higher than 2.5. In contrast, the uncertainty is lower (factor of 1.33) for larger catchments such as the Chassezac at Sainte-Marguerite.

To further illustrate model performance, we analyze precipitation fields for two selected events, using COMEPHORE as the reference. Section  describes the COMEPHORE data. Figure  presents daily precipitation maps for four catchments. While all models broadly reproduce the spatial structure of COMEPHORE fields, important differences emerge. SPAZM exhibits both local overestimation and underestimation, producing overly broad precipitation cells. rgISO also yields overly smooth fields, leading to high-intensity underestimation. arISO slightly mitigates this issue, as seen in the 18 September 2014 case, where it captures more intense precipitation than rgISO. rgANISO captures the radar anisotropy for the 26 May 2008 events but produces an inaccurate spatial structure for the 18 September 2014 event. arANISO provides the most realistic interpolation, reducing both excessive smoothing and underestimation of high intensities. It is the only stochastic model able to capture the highest precipitation values (388–444 mm) in the 18 September 2014 event, demonstrating the advantage of incorporating directional effects.

A single precipitation field does not reflect the interpolation uncertainty, leading to unwanted smoothing spatial patterns. Conditional simulations provide an attractive alternative to derive a set of equiprobably plausible fields. Figure  illustrates five conditional simulations with anisotropic covariance (arANISO) for the 26 May 2008 and 18 September 2014 days, revealing different spatial patterns and precipitation intensities. The Thines at Pont de Gournier catchment, located near the center of the strong cell intensities, has a substantial mean precipitation uncertainty on 18 September 2014. Simulation 55 gives rise to nearly 200 mm mean catchment precipitation, compared to more than 500 mm for simulation 95.

Figure  displays the AROME fields, as well as the isotropic and anisotropic fitted variograms using rain gauges or the corresponding AROME fields, for the two case studies. Although the AROME fields do not show the same spatial variability and precipitation intensity as COMEPHORE (Fig. ), they include nearly the same anisotropy. The 26 May 2008 event shows a higher spatial correlation in the north–south direction. The 18 September 2014 one is characteristic of a Cevenol episode with a higher spatial correlation in the southwest-to-northeast direction. Figure  summarizes the anisotropy parameters η (anisotropy ratio) and θ (anisotropy angle) estimated from AROME and COMEPHORE. Both datasets show similar preferred anisotropy directions, predominantly oriented south–north (S–N) and southwest–northeast (SW–NE), with AROME exhibiting a slightly stronger S–N component. In both cases, the anisotropy is generally more pronounced along the SW–NE direction.

This study highlights that Trans-Gaussian Random Fields (kriging with external drift) outperform local regression models in robustness and spatial variability. SPAZM, a local regression interpolator, excessively spreads precipitation cells and overcorrelates precipitation with altitude. This overcorrelation results in poor performance for convective events, leading to regional inconsistencies where SPAZM significantly underestimates or overestimates intense precipitation in localized areas. In contrast, the ensemble mean from Trans-Gaussian Random Fields shows better regional coherence, despite the global underestimation of intense precipitation. This bias is well-known in kriging studies . Conditional simulations from Trans-Gaussian Random Fields provide an alternative to generate more realistic precipitation. This approach accurately captures large-scale convective events. Further evaluation is needed in ungauged areas with localized convection cells and complex topography, where orographic effects drive uncertainty.

Estimating anisotropic parameters typically requires a high density of rain gauge stations, a condition rarely met in practice. Our results show that CP-RCM simulations, despite their imperfections, effectively capture anisotropic precipitation structures, making them valuable for informing variogram parameters in regions with limited observational data.

In this study, we estimate four covariance models using weighted least squares: (1) isotropic covariance from rain gauge stations (rgISO), (2) anisotropic covariance from rain gauge stations (rgANISO), (3) isotropic covariance from CP-RCM simulations (arISO), and (4) anisotropic covariance from CP-RCM simulations (arANISO). Considering anisotropy in the covariance structure using rain gauges (rgANISO) does not necessarily improve interpolation performances compared to isotropic covariance (rgISO). More specifically, for both point cross-validation and spatial variability evaluation, rgANISO outperforms rgISO during strongly anisotropic events, whereas the opposite behavior is observed for weakly anisotropic events. This indicates that anisotropy can be significant but remains difficult to robustly estimate using a rain-gauge network alone, even when the network is dense. This result agrees with the few studies on covariance comparisons . Using a limited number of rain gauges generates a loss of robustness in estimating the covariance parameters despite the actual anisotropy presented in the daily precipitation fields. arISO shows similar performances to rgISO. This suggests that the rain gauge density is high enough to capture spatial autocorrelation in the study region. This also implies that CP-RCM simulations could serve as a viable alternative for estimating spatial autocorrelation in areas with sparse rain gauge station coverage. Estimating anisotropic parameters usually requires a large number of rain gauge stations, which is rarely met in practice. arANISO outperforms the other stochastic models. This approach aligns with previous studies that have used radar data to infer anisotropy through nonparametric estimation . High-quality CP-RCM simulations, increasingly available worldwide, might correctly carry anisotropic information.

The covariance modeling in this study has two main limitations. Firstly, CP-RCM simulations may miss or inaccurately generate precipitation cells, leading to biases in the estimation of covariance parameters such as the range, the anisotropy ratio, and the anisotropy angle. Secondly, the assumption of second-order stationarity may not hold on large spatial domains. Topographical effects might induce spatial non-stationarity in the covariance structure. The windward mountainsides might encounter more spatial variability than the leeward ones. The study region has daily oceanic or Mediterranean influences, but not both simultaneously, limiting spatial non-stationarity. Several authors implement spatial non-stationarity of the covariance (; ; ). For computational reasons, we decided not to include non-stationarity. Extending the method to larger and topographically complex domains would require non-stationary covariance. A practical option is to partition the region into climatologically homogeneous sub-regions, ideally preserving major watershed boundaries to maintain hydrological consistency. Alternatively, a solution is to incorporate non-stationary covariance structures, for example, through geographical coordinate deformation or locally stationary covariance models , which would allow spatial dependence to evolve smoothly across the domain. These approaches would make the method suitable for operational applications over larger domains.

Despite those limitations, we are confident in the study's transposability to other regions of similar size with high-quality CP-RCM simulations. Furthermore, the scientific community should explore whether the same methodology can be applied using RCM simulations in regions where CP-RCM simulations are unavailable. The methodology could also be extended to real-time, sub-daily interpolation. CP-RCM simulations are not continuously updated, so their replacement by hourly numerical weather forecasts (NWP) should be investigated. NWP assimilate past radar reflectivity and should therefore display a higher correlation with observations than CP-RCM simulations. As a result, NWP could allow us to extract precipitation intensity, spatial patterns, and spatial variability, while quantifying interpolation uncertainty through conditional simulations and the use of ensemble NWP forecasts. A natural follow-up would be to use NWP forecasts as both drift and covariance structures within a kriging-with-external-drift framework . At the daily timescale, timesteps are typically considered independent, but this assumption no longer holds at the hourly scale. To address this, temporal dependence should be incorporated into the model, as done in . A major limitation is the limited availability of sub-daily rain gauges. One potential solution to bypass this shortcoming is to disaggregate daily interpolated fields using radar data or numerical weather forecasts.

We assess precipitation interpolation uncertainty at the catchment scale, which is relevant for hydrological applications. For very small catchments (17 km2), the simulated mean catchment precipitation can vary widely by a factor of 2.5, highlighting substantial uncertainty despite high rain gauge density. Using a gamma distribution to normalize daily observations creates an asymmetrical distribution of mean catchment precipitation, which appears to be physically reasonable. Other authors frequently used the Box–Cox transformation to normalize data, causing uncertainties in the back-transformation step. Box–Cox transformations can lead to over- or under-normalization, artificially inflating or deflating the uncertainty in spatial interpolation. Standard square root transformation, a special case of Box–Cox (parameter equal to 0.5), often leads to flattening high precipitation values , causing underestimation of spatial interpolation uncertainty.

Uncertainty in precipitation analysis mainly arises from precipitation undercatch and spatial interpolation methods.

In this study, rainfall prevails over snowfall, and precipitation undercatch might be considered negligible. However, in snow-dominated regions, precipitation undercatch is significant under windy conditions. We propose fitting an asymmetrical gamma distribution to the catch ratio to handle this issue. The parameters of this distribution would depend on several factors, including the precipitation phase (rain or snow), precipitation intensity, rain gauge type, its exposure, and wind speed.

While we modeled the primary sources of uncertainty related to spatial interpolation, additional uncertainty remains unaddressed:

We assume fixed parameters for the gamma distribution used to normalize the data. A natural extension would be to generate multiple sets of gamma parameters and run conditional simulations for each.