The study area is the Feldbach region located in the southeast of the federal state of Styria, Austria (Fig. ). The eastern Styrian area is characterized by diverse meteorological and climatological conditions, as well as a varied orographic structure. The Feldbach is geographically part of the southeastern Alpine foothills. The landscape is composed partly of the East Styrian hills and the valley floors of the foothill region. The hills are characterized by elongated, often asymmetrical ridges with steep western slopes and more gentle eastern slopes. The elevation range in the study area spans from 250 to 598 m a.s.l. The valley floor of the foothill region consists of the broad alluvial valley of the Raab river, which stretches approximately 23 km within the study area and has a relatively uniform elevation ranging from 250 to 300 m a.s.l., resulting in a rather low gradient. The alluvial valley in the study area varies in width from 800 to 2500 m. The ridges are predominantly forested, while the slopes are mainly used for orchards and other agricultural activities. The valley landscape is also primarily used for agriculture, except for the town of Feldbach . The climate of the study area is shaped, on the one hand, by the region's location in the southeastern Alpine foothills and its proximity to the Alpine arc itself and on the other hand, by factors affecting the overall Alpine climate. On a large scale, the Alpine climate is primarily influenced by the weather systems of the North Atlantic, the effects of the Mediterranean Sea, and the Eurasian landmass . Based on the Köppen-Geiger climate classification, the Feldbach region can be categorized as cold climate without dry season but warm summers .

On a fine scale, the Feldbach region is dominated in summer by heavy rainfall from thunderstorms . This is endorsed by lightning information accessible from the Austrian Lightning Detection and Information System (ALDIS; ), indicating the federal states Styria (including the Feldbach region) and Carinthia as regions with highest frequency of thunderstorms (see ). Furthermore, large hail events and severe wind gusts are typical for the Feldbach region according to the European Severe Weather Database (ESWD; ). The mean wet spell duration in summer is approx. 3 h, the overall mean dry spell duration ranges between 30 and 32 h . points out historical region-dependent differences in precipitation-generating processes on a monthly basis for Austria, but concludes that precipitation events in winter result from large-scale advective processes. identified the strongest precipitation events on a daily basis in the period July–October. Although and use data sets with different temporal resolution and station density, their findings were combined by with their own findings, classifying the Feldbach region as Eastern mixed region and thus highlighting its transition characteristic.

The WegenerNet climate station network is characterized by its exceptionally high spatial density of climate stations. It is designed for long-term studies of small-scale weather and climate developments, providing measurements at high spatial and temporal resolutions. The network consists of 168 climate stations located in Austria, 155 of which have operated in the Feldbach region since 2007. Due to relocation of 12 stations during the study period these stations had to be removed from the data set and the remaining 143 stations are used for the analyses of this study. The studied time series spans from 2007 to 2022, 15 years with a temporal resolution of 5 min. Additionally, a regionalized raster product from the WegenerNet is used, which covers a total area of 330 km² with 75 × 110 grid cells. A grid of 200 m × 200 m in the Universal Transverse Mercator projection is available for each time step. The data collected by the climate stations undergoes an automated Quality Control System, followed by a Data Product Generator.

The Data Product Generator regionalizes the gauge-based time series with an inverse distance-squared method (IDSW; ). Inverse distance-based methods lead to similar results as Ordinary Kriging for a high station network density . Also, additional improvements from e.g., external drifts, can be questioned for such a dense network. IDSW is able to represent the spatial variance of the precipitation field, by simultaneously representing the extreme values well due to favouring the immediate stations by squaring the distance. Using the regionalized data product also enables comparisons with other studies.

For a more detailed description of the study area and the WegenerNet data, the authors refer to and . For the validation of the estimated point precipitation extreme values the official Austrian design precipitation values from eHYD (Electronical HYdrographic Data) are applied.

The ARF is determined as the ratio of areal precipitation Pa(D,T) to point precipitation Pp(D,T) for each duration D and return period T of interest, as displayed in Eq. () : 2ARF(D,T)=Pa(D,T)Pp(D,T) where the Pa(D,T) and Pp(D,T) correspond to areal and point precipitation for given duration and return periods extracted from the time series using the method of Extreme Value Analysis (EVA) . The EVA method is elaborately described in the following subsection, followed by the description of how areal precipitation extreme values were derived.

The estimated precipitation point extreme values are depicted in Fig. . As expected, for both return periods, the precipitation depth P increases with duration. Similarly, with increasing return periods, higher precipitation depths are observed. For a return period of one year, the mean values span from approx. 9 mm for D = 5 min to about 51 mm for D = 24 h, while higher values are observed for T = 10 years, ranging from approx. 15 to 86 mm. The quantiles show that 10 % of the values are responsible for a significant portion of the variability. The estimated precipitation extreme values are compared with the official Austrian design values (ÖKOSTRA values from the eHYD catalogue) for the Feldbach region (entry 5433) in the Appendix. The mean over all raster cells is very similar to the official data, but for higher return periods, subtle deviations are identified.

Figure  the two bottom rows show the mean values of seasonal point extreme precipitation for the return periods of 1 and 10 years. For all seasons, the expected pattern of increasing precipitation with increasing duration is evident. The mean precipitation depth also increases with higher return periods. Additionally, the range of values (distance between minimum and maximum of precipitation amount for each duration) increases with the return period across all seasons. The progression of the mean values differs between seasons. In spring, autumn and winter, the slope decreases for all return periods for durations longer than 720 min compared to durations shorter than 720 min. However, it increases slightly in summer due to higher precipitation amounts, associated with longer durations and higher return periods in comparison to other seasons. For T = 10 years and D = 24 h, values of 85 mm are reached in summer, compared to 34 mm in winter. The lowest precipitation depths occur in winter, where the mean precipitation for T = 1 year and D = 24 h is only 19 mm. The range indicates that during summer, this combination of return period and duration occasionally results in maximum values exceeding 100 mm. Additionally, the range of precipitation depth is considerably smaller in winter.

The top row panels in Fig.  show the mean precipitation values for two areas of 1 and 25 km² for the whole year. As the area increases, the precipitation amounts decrease for the same return period and duration. For example, for T = 1 year and D = 24 h, the precipitation amounts are 34.2 mm for 1 km², but only 31.3 mm for 25 km². This behaviour confirms the decrease of precipitation amounts with increasing area size for the same return periods and durations as identified by e.g., . The range increases with return period for areal extreme values, while the range decreases as the area increases. The decrease in range with area is primarily due to the decreasing number of time series available for analysis as the area grows. The increase of extreme precipitation depth with duration and return period is evident here as well. Additionally, the range of values increases with higher return periods.

Figure  presents an example of the yearly ARF values for D = 60 min and T = 10 years. The ARF increases with increasing area, and so does its range. It is observed that the mean and the median of the ARF values are almost identically, with values below 1, and both decrease with increasing area. For areas larger than 200 km², the mean and median values show a slight increase, which can be attributed to the small sample size, since as the area size increases, the number of areal time series decreases, e.g. only two time series are available for areas of 225 km². Additionally, the ARF range increases up to an area size of approximately 15 km² and stagnates or decreases for larger areas. Therefore, the largest ARFs are associated with an area size of roughly 15 km² and are very likely attributable to isolated outliers, as the 95th quantile does not show a significant increase at this area. The inter-quantile-range (IQR) as distance between q = 0.95 and q = 0.05 increases with increasing area, but only slightly compared to the total range, and starts to decrease for areas larger than 90 km². Overall, the deviation of outliers from the mean appears to be more significant in the upper bound than the lower bound in all cases, as indicated by the range and the IQR depicted in Fig. . As mentioned above, the sample size for areas larger than 81 km² is notably small and leads to high uncertainty in the results. Therefore, from this point on, the analysis will be limited to area sizes up to 81 km².

Figure  presents the seasonal ARFs for a duration of 60 min at a return period of 10 years. The smallest ARF values observed belong to spring, while the largest values occur in winter. The slopes of the mean and median of the winter ARF are both noticeably constant over area sizes. For all seasons, the range of ARF increases with area size, while in autumn, the range shows a more irregular pattern of increase and decrease. Cases of ARF>1 are identified in all seasons, where in spring, summer and autumn such values belong almost exclusively to the upper 5 % of the sample, probably indicating outliers. However, in winter, a noticeable part of the ARF values which fall into the IQR are higher than 1.

Figure  depicts ARFs for the entire year, different duration and different return periods. For ease of communication, only 10 of the 16 durations are displayed in these plots. For all return periods, the ARF increases with increasing durations and decreases with increasing area. An exception is the duration D = 24 h, which occasionally exhibits lower ARFs compared to the D = 6 h and D = 12 h (observed across all return periods). The 5 min duration has the lowest ARFs, with the smallest value 0.595 occurring at a return period of 1 year. The generally higher values of ARF for the return period of 10 years indicates the increase of ARF with increasing return period. This behavior is clearly demonstrated for durations between 5 to 20 min. In contrast, the ARF decreases with return period at durations between 60 min to 12 h for areas exceeding 9 km². The durations in between 30 and 45 min show no clear behaviour in this regard. Overall, the ARF curves appear to converge as the area decreases, meaning the ARF increases with area and approximates the value of 1. The exact ARF values for the entire year analysis are provided in the Table  for the return periods of 1, 2, 5 and 10 years.

The difference between the ARFs between 1 and 10 years return periods is the largest at 5 min duration and 81 km². This indicates that the greatest discrepancy in ARFs is between different return periods. Since all other differences are smaller, it is concluded that the changes in ARF are more pronounced for smaller durations and larger areas.

Figure  presents ARF values separately for each season for T = 1 year. The same pattern of increase with duration and decrease with area size is detectable in all seasons. Winter exhibits the highest ARF values across most durations. In contrast, spring shows the lowest ARF values, particularly for shorter durations, specifically up to 6 h. The minimum ARF value is 0.497 for the duration of 5 min and an area size of 81 km². Summer and autumn show intermediate values which fall between those of spring and winter. In summary, seasonal variations are most prominent in the shorter durations (up to 6 h) and smaller areas.

To analyze the relationship of the ARF with the topography, the Spearman correlation coefficient (rs) is calculated between the ARF values and the elevation of the stations for different area sizes, durations and return periods. Figure  shows the correlation between ARF and elevation for the area sizes of 1 to 25 km². The value of the correlation coefficients is also presented in the two plots. It is clearly noticeable that there is a low correlation between the ARF and the elevations. For the duration of 60 min and the return period of 1 year rs spans between 0 and 0.19, increasing with area. For the same duration, as the return period increases to 10 years, the rs values decrease and show a negative correlation between ARF and elevation, ranging from -0.14 for 1 km² and -0.02 for 25 km². It is noteworthy that the correlation decreases with increasing area for T = 10 years.

Evidently, in all cases, across all durations and seasons, the ARF reduces with area, which complies with the definitions and findings among the related literature . ARF decreases more rapidly with area for shorter durations, as in . This behavior can be attributed to the different types of precipitation events associated with various durations. Short-duration events generally exhibit higher spatial variability compared to long-duration events . As the mean ARF is in most of the cases below 1, this study underlines the necessity for reducing point design values for determining the areal design storm. The largest reduction for a 1 km² area occurs in spring at a return period of 1 year, with an ARF of 0.928, corresponding to a 7.2 % reduction in areal precipitation. However, it is noteworthy that across seasons and return periods, values above 1 appear, which are in the upper 5 % of the extreme and contradicts the base assumption of the reduction of precipitation in ARF. discuss this vastly in their study, where they attribute an increase in areal precipitation with increasing area (meaning ARF > 1), to the shortcomings of the sparse measurement networks and the conventional fixed-area spatial sampling methods. It is important to acknowledge that even with a dense network like WegenerNet, there are still events whose center of highest precipitation depth cannot be detected with the conventional sampling approaches, as discuss. Furthermore, our results show that the assumption of approximation of the point extreme precipitation value, even with the areal extreme of 1 km² is not accurate and will lead to the overestimation in infrastructure planning. Widely used guidelines like suggest that point precipitation can represent the areal precipitation of catchments up to 25 km², with an intentional overestimation for planning purposes. However, applying this assumption for an area of 25 km² at a duration of D = 5 min would result in a 26.5 % overestimation of areal precipitation, which seems excessive as a safety margin. Especially, when summed up with other safety margins resulting from conservative estimations, as, e.g., using the same return period for all durations within one design event . Without the application of ARF for small areas, overestimation of the urban drainage infrastructure dimensions is a significant risk.

As widely documented in the literature the ARF has a direct relationship with duration in general. However, it is observed in this study that the ARF of the two longest durations, 18 and 24 h, does not follow the expectations. This irregularity is shown in Figs.  and for the duration of 24 h. For the whole year results, for T = 1 year this behavior is observable for almost all area sizes, whereas for T = 10 years this happens only for areas larger than 25 km². The seasonal analysis show this pattern only in summer and winter for T = 10 years. Precipitation events with extensive spatial coverage may influence the statistics at these durations, extending well beyond the study area. Such regional-scale events may not be fully captured in the data, leading to these deviations.

The four return periods analyzed exhibit varying effects on ARF, with the strongest impact observed in winter, where the largest differences between ARFs at T = 1 and 10 years occur. For the whole year, the smallest ARF was found at the smallest return period T = 1 year, contradicting literature findings , which generally report smaller ARF values for larger return periods. This deviation can be explained by the distinct behavior of short-duration events, where areal precipitation increases more significantly with increasing return periods than point precipitation. The spatial variability of the short-duration precipitation events decreases with increasing return periods. An example of similar behavior of ARF for short durations across different return periods could not be found in the literature. This can be caused by the rarity of such a data set: the spatial and temporal high-resolution station-based data set from WegenerNet appears to be one of only a few data sets existing globally, and it is the first data set of this kind used to estimate ARFs. Again, due to the limited study area it remains unclear, if this finding represents rather a general behaviour or an observational-scale effect, or if it is related to the regional climate. determined ARF using an almost identical method and concluded the influence of return periods to be minimal, and suggested that the return period can be disregarded in the ARF calculation. However, based on the differing ARFs observed in this study as a function of return periods, as well as the distinct behavior of short durations with increasing return periods, the influence of return periods on ARF cannot be neglected.

The dependence of ARF on seasons, as identified by , was partially confirmed in this study. found that the ARF in warm seasons (April–September) has lower values compared to those of the cold seasons (October–March). This pattern is confirmed in this study, particularly for the shorter return periods. The smallest ARFs are observed in spring and summer, which can be attributed to the predominant convective precipitation events. These events exhibit high spatial variability , which leads to smaller ARF values as the area size increases. The autumn ARF values show a very similar mean and median to those of summer. However, the largest ARF values belong to autumn when looking at the range of the values (Fig. ). In contrast, winter shows higher values on average. While the dependence on seasons can be identified, the combined effects of return periods and duration complicate definitive conclusions, such as those made by . A deeper insight could be achieved by multifractal analysis of the scaling behavior as done by , but is beyond the scope of this study.

As the correlations showed in Fig.  in this study, an insignificant relationship between the ARF and the elevation is observed. Although the correlation increases with area for shorter return periods, even the strongest correlations do not reach the value of 0.2, which indicates a weak relationship with elevation. found no significant relationship between extreme precipitation and elevation for WegenerNet stations either. It is interesting to note that ARF show no correlation with elevation, since extreme values themselves show clear dependencies (see e.g. and references within). However, it should also be noted that the spatial interpolation of precipitation for the WegenerNet was conducted without accounting for elevation, which was justified by the relatively small elevation variation among the climate stations . As a result, the areal precipitation between stations may not have been accurately interpolated, which must be considered when interpreting the results of this analysis.