The different climate regimes between coastal and inland regions over Norway compel the analysis of hydro-meteorology based on divided regions. Based on similar seasonal cycle characteristics, Michel et al. (2021) and Konstali and Sorteberg (2022) divided the Norwegian continent into eight regions. Taking into account the spatial distribution of rain gauges and ensuring that each region has at least one hourly rain gauge, we combined the south and southwest into the south and the middle inland and north into the north. Therefore, in this study, mainland Norway is divided into five regions: east (E), south (S), west (W), middle (M) and north (N) (as shown in Fig. 1).

The study areas cover mainland Norway, which has unique climate characteristics within different regions. Precipitation in Norway primarily occurs along the coast in late autumn and winter, while inland areas receive more precipitation in summer. The eastern region, with stratiform precipitation originating from the south, is dominated by continental climate, with convective precipitation in summer. The western coast of Norway is strongly affected by the North Atlantic storm track, where precipitation from frontal systems and land-falling storms is enhanced due to the orographic uplift over Scandinavia (Poujol et al., 2021). Most extreme events occurring in the western region, with abrupt topography, are mainly related to atmospheric rivers (ARs), which are generally linked to extratropical cyclones during cooler seasons (Whan et al., 2020). Additionally, in the summer, ARs coincide with more frequent convective activities (Poujol et al., 2021). The southern region lies at the end of the climatological jet and is regularly affected by the ARs, especially during the zonal and Atlantic trough weather regimes (Michel et al., 2021), while convective activities play a crucial role in the southern regions in summer (Li et al., 2020b). For the middle and northern regions, 59 % of extremes are associated with ARs, and the precipitation rate decreases moving inland (Konstali and Sorteberg, 2022).

We utilize the outputs of double-nested model simulations from the HCLIM38 model, which includes different configuration settings for each spatial resolution: HCLIM3 and HCLIM12. HCLIM12 covers most of Europe with 313 × 349 grid points using the ERA-Interim reanalysis (∼ 80 km) as the boundary condition, and HCLIM3 spans the Fenno-Scandinavian region with 637 × 853 grid points using the output of the HCLIM12 as the boundary condition for every 3 h. Importantly, the convection parameterization scheme was switched off in HCLIM3, allowing for an explicit representation of convection processes. The present-day simulations from HCLIM3 and HCLIM12 span the years 1997–2018. For more comprehensive information, refer to the work of Lind et al. (2020) and Médus et al. (2022).

This study primarily focuses on assessing the performance of HCLIM3 and HCLIM12 in simulating hourly and daily extreme-precipitation events in mainland Norway for the present-day period (1997–2018, with the first 2 years excluded). The model outputs from HCLIM3 and HCLIM12 are specifically extracted for mainland Norway. Before analysis, HCLIM3 data were remapped to the HCLIM12 grid (12 km) using a bilinear interpolation method.

Precipitation from the SeNorge_2018 (SeNorge) gridded dataset, covering Norway with 1 d temporal resolution and 1 km spatial resolution since 1957 (Lussana et al., 2019), is used as the observation dataset to evaluate the performance of HCLIM3 and HCLIM12 during 1999–2018. Precipitation from the SeNorge2 gridded dataset, with 1 h temporal resolution and 1 km spatial resolution, is also applied to evaluate the hourly result during 2010–2018 (Lussana et al., 2018). In addition, in situ precipitation observations, including both 1 h and 1 d resolutions, are downloaded from the Norwegian Meteorological Institute Frost API (https://met.no, last access: 24 April 2025).

To evaluate the characteristics of precipitation extremes between HCLIM3 and HCLIM12, we compared the historical simulations with the daily SeNorge gridded dataset, the hourly SeNorge2 gridded dataset and in situ observations. We only keep the stations that have less than 10 % of the data missing during 1999–2018 and consider station distribution uniformity, giving a total of 192 daily stations and 10 hourly stations, over Norway (Fig. 1, Tables S1 and S2 in the Supplement). In this study, the evaluation based on in situ observations and gridded datasets (SeNorge and SeNorge2) was defined at the local scale and regional scale, respectively.

Remapping finer-resolution data to a coarser resolution reduces the influence of such artifacts by averaging out the variability. This approach is consistent with the methodology used by Lind et al. (2020) and Médus et al. (2022), who also remapped all data to a coarser grid when comparing the performance of HCLIM3 and HCLIM12. Lind et al. (2020) observed that the differences between HCLIM3 data remapped to the coarser native grid of HCLIM3 and the HCLIM12 grids were minimal. Importantly, they found that the improvements in HCLIM3 persisted even after spatial aggregation, indicating that the enhanced resolution of the model offered benefits that were preserved when viewed on a coarser grid. Therefore, HCLIM3, SeNorge and SeNorge2 were remapped to the HCLIM12 grid ∼ 12 km for the evaluation at the regional scale. For the SeNorge- and SeNorge2-based assessments, the extreme indices are first calculated at the grid point level, and then the regional averages are computed. For the evaluation based on in situ observations, HCLIM3 and HCLIM12 were interpolated to the 192 daily rain gauges and 10 hourly rain gauges to calculate the indices using a bilinear interpolation method.

For the evaluation of extreme precipitation, we examined the maximum 1 d precipitation (Rx1 d); maximum 1 h precipitation (Rx1 h); return-period-based precipitation amounts at 5-, 10-, 20- and 50-year return periods; and seasonality of frequency and/or intensity from the regional to local scale. The calculation of seasonal Rx1d or Rx1h was based on the maximum value within one season per year.

The generalized extreme value (GEV) distribution was used to estimate precipitation intensity for specific return periods (e.g., 5, 10, 20 and 50 years). The return levels were calculated by fitting the annual maximum discharge derived from observed and simulated daily data (both gridded and rain gauges) and hourly data (only 10 rain gauges) to the GEV distribution. Then the quantile Zp of the GEV distribution with a return period of 1p can be obtained. The GEV distribution has been widely used to model extreme events in meteorology (Coles et al., 2003). The cumulative distribution function Fx and probability density function fx of the GEV were used as follows to calculate the return level Zp: 1Fx=exp⁡-1-kx-ξα1/kk≠02fx=1α1-kx-ξα1/k-1×exp⁡-1-kx-ξα1/k,3Zp=ξ-αk1--log⁡1-p-k, where α, ξ and k indicate the scale, location and shape parameters, respectively. Kolmogorov–Smirnov, Anderson–Darling and chi-square tests were performed to determine if the GEV was accepted to fit the maxima series.

The orographic effect on Rx1 h and Rx1 d precipitation was explored by looking at the relationship of the annual and seasonal maxima with elevation from regional to local scales. A linear regression model (Di Piazza et al., 2011) was utilized to approximate the relations. The relationship of elevation with observations (Rx1 h (SeNorge2), Rx1 d (SeNorge) and daily in situ observation) and simulations (HCLIM3 and HCLIM12) was fitted to compute the linear regression slope. To eliminate the impact of the unit (Rx1 h and Rx1 d), the slope is converted to a relative slope with respect to the average value of extreme precipitation, expressed as percentage precipitation (%) per kilometer of elevation. This is done by dividing the mean extreme-precipitation value for the entire study region computed separately for daily and hourly extremes. The orographic effect at the local scale was only based on daily in situ observations due to the limited hourly in situ observations. At the local scale, the elevation for each rain gauge was extracted according to the digital elevation model. At the regional scale, the grid of the digital elevation model and HCLIM3 was resample to the same grid resolution as HCLIM12 (12 km) before calculation. Only the grids and stations above sea level (0 m) are included to quantify the orographic effect.

If the precipitation increases with elevation, there is an orographic effect on extreme precipitation; if the precipitation decreases with elevation, this results in a reverse orographic effect on extreme precipitation.

Figures 9 and 10 show the comparison of seasonality (indicated by monthly distribution) of annual Rx1 d (e.g., frequency and magnitude) from both HCLIM3 and HCLIM12 compared to SeNorge and in situ observations, respectively. From the seasonality of observed daily extreme precipitation in Fig. 9, we can see that winter–autumn precipitation dominates in the eastern, southern and western regions, while, in the middle and northern regions, spring–summer precipitation is more prevalent. HCLIM3 captures the seasonality of Rx1 d frequency at the regional scale in most regions, except in middle region, where winter–autumn precipitation dominates. In contrast, HCLIM12 performs poorly in the eastern and middle regions. It is particularly noteworthy that HCLIM3 has an enhanced ability to capture the seasonality of extreme-precipitation frequency over the western region compared to HCLIM12. Heavy precipitation over 50 mm d⁻¹ mainly occurs in the south and west, which is also simulated by HCLIM3 and HCLIM12. In general, both HCLIM3 and HCLIM12 demonstrate competence in capturing the magnitude of extreme daily precipitation seasonally at the regional scale in most regions, except in the middle region. The seasonal performance of Rx1 d from HCLIM3 and HCLIM12 at the local scale is also confirmed by the in situ observations, as shown in Fig. 10. A larger magnitude of annual Rx1 d across five regions at the local scale than at the regional scale is shown for the observations, HCLIM3 and HCLIM12. Generally, both HCLIM3 and HCLIM12 capture the seasonality of daily extreme precipitation well; HCLIM3 does not consistently show added value in simulating them.

For the seasonality of annual Rx1 d at the local scale, as shown in Fig. 11, HCLIM3 more accurately represents the seasonality of Rx1 h compared to HCLIM12, which tends to underestimate the frequency of hourly extremes at most sites. Compared with RCMs, CPRCMs demonstrate better potential performance in simulating the seasonality of extreme precipitation, with particularly improved accuracy for the hourly extremes at the local scale.

To further explore the uncertainty of different observation datasets in local-scale model evaluation, we investigate the bias of SeNorge's annual and seasonal Rx1 d based on daily in situ observations (see Fig. 15). Our analysis in Fig. 15a shows that SeNorge mostly underestimates the annual Rx1 d compared to in situ observations at 192 stations, with an average bias of -5.8 % and a range between -28 % and 25 %. Although SeNorge data are designed to improve hydrological simulations (Lussana et al., 2019), their dry biases still persist in most seasons and regions, especially in summer. It is noteworthy that SeNorge slightly overestimates the winter Rx1 d in the eastern, middle and northern regions. Moreover, SeNorge underestimates the return levels of Rx1 d for different return periods (e.g., 5, 10, 20 and 50 years) in all regions (Fig. S3).

The larger differences between SeNorge and in situ observations in simulating Rx1 d are manifested annually in the summer and autumn in the south and west and in the summer in the east, where SeNorge tends to underestimate Rx1 d more than the in situ observations. This discrepancy helps explain the differences between HCLIM3's performance in simulating Rx1 d in summer in the east and in autumn in the south and west at a regional scale compared to at the local scale, as shown in Figs. 2c and 5c. Generally, the difference between SeNorge and in situ observations at a daily scale is not very large, which is why, in most regions, the added value of HCLIM3 in Rx1 d at the regional scale is similar to that at the local scale. However, it should be noted that the interpolated precipitation from SeNorge may introduce uncertainties in assessing the performance of CPRCMs at the local scale due to the sparse distribution of daily and hourly rain gauges at high altitudes. In particular, for the hourly extremes at the local and regional scales, larger uncertainties should be considered due to the limited data from only 10 rain gauges at the local scale and from 9 years of data series at the regional scale. The impact of station density on the errors of gridded datasets was also highlighted by Gervais et al. (2014b), who suggested that low station density is an important source of errors in such datasets. To address these challenges and to enhance the accuracy of extreme-precipitation assessments, future studies should prioritize expanding in situ datasets and improving the spatial coverage of observational networks, especially at the 1 h timescale.

HCLIM3 demonstrates clear advantages over HCLIM12 in capturing the annual Rx1 d in most regions. In terms of regional averages, HCLIM12 underestimates Rx1 d in most regions, except in the east, and is biased towards wetter conditions, while HCLIM3 shows relatively smaller biases in most regions, except in the east, due to improvements in microphysics and convection schemes (Lind et al., 2020).

Despite the overall better performance of HCLIM3, slightly larger biases in summer over the east, south and west may result from the model's sensitivity to convective processes and limitations in accurately resolving localized dynamics under moisture-rich and unstable atmospheric conditions. These challenges are particularly pronounced during summer, when the intensity of convective activity increases, leading to rapid atmospheric feedback and localized extremes (Poujol et al., 2021). In contrast, in winter, the southern region is mainly affected by atmospheric rivers (ARs) associated with extratropical cyclones, and HCLIM3 can better capture this feature due to its finer resolution.

In terms of annual Rx1 h, HCLIM3 outperforms HCLIM12, although it exhibits a wet bias compared to SeNorge2. HCLIM12 underestimates Rx1 h in most grids, likely due to its reliance on parameterization schemes that fail to capture extremes (Médus et al., 2022). However, HCLIM3 shows larger biases in seasonal Rx1 h in the west in all seasons, except in spring; in the eastern and middle regions, in summer, the overestimation of HCLIM3 over Norway may be attributed to the underestimation in the hourly SeNorge2 (Lussana et al., 2018). Compared with daily extremes, both HCLIM3 and HCLIM12 exhibit larger biases in simulating hourly extremes compared to daily extremes at both the annual scale and seasonal scale. It is important to note the limitations of the SeNorge2 dataset, which only spans 8 years and is interpolated from sparse hourly rain gauges.

In summary, HCLIM3 demonstrates better agreement with observations across most regions of Norway and across most seasons at the regional scale, with the exception of the east and summer. This is consistent with previous studies highlighting the advantage of convection-permitting models, especially in capturing extreme-precipitation events over complex terrain (Kendon et al., 2023; Médus et al., 2022; Lucas-Picher et al., 2021).

The analysis of local-scale convection-permitting climate models (CPRCMs) highlights their better performance in capturing precipitation extremes. HCLIM3 demonstrates notable advantages over HCLIM12, especially in terms of hourly precipitation extremes (Rx1 h). For example, HCLIM3 achieves near-zero bias for the annual Rx1 d in Norway (Fig. 5) and a relatively smaller bias for hourly extremes (Figs. 7 and 8) at all stations, while HCLIM12 consistently underestimates the return levels for hourly extremes at most stations (Fig. 8) and for daily extremes in all regions. Médus et al. (2022) also pointed out that RCMs underestimate the return levels of Rx1 h in Norway. Thomassen et al. (2023) compared the performance of HCLIM3 and HCLIM12 based on local rain gauge data in Denmark and found that HCLIM12 indeed underestimates the hourly extreme events and that HCLIM3 agrees well with observations. Despite these benefits, the added value of HCLIM3 is not uniform across all stations and seasons, struggling to capture summer daily extremes in the south and west and the return level in the east and west. However, it should also be noted that the analysis is based on data from only 10 sites, which limits the generalizability of the findings to local hourly extreme events. Further studies of hourly extreme events at more stations are needed to validate these results and to provide a more comprehensive understanding. Additionally, the uncertainties in the extreme-precipitation analysis based on the stationary GEV method with a 20-year data series should also be noted.

The added value of CPRCMs in simulating hourly precipitation extremes is more obvious at the local scale than at the regional scale. The damped extremes caused by grid-scale averaging may explain the smaller return level observed for HCLIM3 and HCLIM12 compared to station-level observations. As discussed in Sect. 5.1, this discrepancy between regional and local scales may be partly due to the inadequate density of in situ observations.

Few studies have systematically compared hourly and daily rainfall in RCMs due to the challenges in reliably simulating hourly extremes. In line with Ban et al. (2014), we find that RCMs such as HCLIM12 demonstrate reasonably good performance for daily extremes, with biases of less than 50 %. However, CPRCMs such as HCLIM3 perform better for hourly extremes. This is consistent with previous studies (Jiang et al., 2013; Thomassen et al., 2023) which showed that it is challenging to capture sub-daily extreme rainfall using RCMs with a resolution of 10 km in the southwestern United States. The better performance of CPRCMs compared to RCMs at an hourly scale is consistent with the findings by Médus et al. (2022) and Ban et al. (2014), emphasizing that the CPRCMs have significantly better sub-daily precipitation characteristics, including spatial distributions and duration–intensity characteristics. Nonetheless, further improvements in the observation networks and longer observational datasets are necessary to fully verify and realize the benefits of CPRCMs at finer spatial and temporal scales.

Comparison of regional and local extreme-precipitation seasonality confirms that HCLIM3 is able to represent the seasonality of daily extremes, although both HCLIM3 and HCLIM12 fail to capture the spring–summer events in the middle region. Moustakis et al. (2021) also highlighted the adequacy of CPRCMs (∼ 4 km) in capturing seasonality observed over the United States. In particular, we observe that HCLIM3 better represents the seasonality of hourly precipitation at the local scale. The persistent underestimation of hourly extremes by HCLIM12 may be attributed to higher uncertainty in its convective parameterization scheme or numerical uncertainties at the local scale.

An unclear relation of daily extreme precipitation with elevation was also seen from the study of Dallan et al. (2023), in which they analyzed annual daily return levels based on CPRCMs and in situ observations over an Alpine region. By comparing the relationship between the elevation and seasonal variation of extreme precipitation, HCLIM3 represents the reverse orographic effect well at regional and local scales, although there is a weak relationship between extreme precipitation and elevation at the regional scale. The reverse orographic effects on hourly and daily extremes vary with the season, indicating the influence of topography on extreme precipitation at different timescales and emphasizing the reliability of the simulation of extreme precipitation over complex terrain. Unlike Rx1 d at the regional scale, which is less affected by topography, the slope of the reverse orographic effect of daily extreme precipitation at the local scale is more clear. From a seasonal perspective, the reverse orographic effect of extreme precipitation in summer is not well captured in HCLIM3 and HCLIM12, which may be related to the intense orographically sustained convection affected by the atmospheric, aerosol conditions, local terrain slope and shadowing effects, which RCMs and CPRCMs fail to capture (Dallan et al., 2023; Poujol et al., 2021).

For hourly extremes, the reverse orography effect of the seasonal Rx1 h in this study is consistent with the reverse orographic effect of the hourly return level, as found by Dallan et al. (2023) over the Alpine region. HCLIM3 and HCLIM12 capture the reverse orography effect on seasonal Rx1 h well, especially in HCLIM3, although a stronger decrease in Rx1 h with elevation is observed from SeNorge2, except in spring. In comparison, lower Rx1 h and a weak reverse orography effect is found in HCLIM12 in all seasons. Our findings confirm the reverse orographic effect on Rx1 h, as demonstrated by Marra et al. (2021) for hourly precipitation and by Formetta et al. (2022) for the sub-hourly scale.

Furthermore, we demonstrate the reverse orographic effect for both seasonal Rx1 h and Rx1 d, which contrasts with the findings of Formetta et al. (2022), who identified an orographic enhancement for durations of approximately 8 h or longer, although a reverse orographic effect for hourly and sub-hourly durations was shown. These differences may be attributed to the combined effects of latitude, climate, altitude zones, static atmospheric or aerosol conditions, and shadowing effects (Amponsah et al., 2022; Napoli et al., 2019).

It should be noted that a simple relationship between extreme precipitation and elevation is difficult to build due to the fact that several land surface characteristics could influence the precipitation; a complex regression model should be considered to more realistically quantify the reverse orographic effect (Zhang et al., 2018) in the future. The interpolated gridded dataset and limited rain gauges over the complex orography, along with the decreasing station density at higher elevations, may also limit the reliable analysis of the reverse orographic effect. The sparsity of rain gauges and undercatch problems could also lead to the underestimation of precipitation, especially in the case of complex orography (Lussana et al., 2018, 2019; Gervais et al., 2014b).