An analysis on temporal scaling behavior of extreme rainfall of Germany based on radar precipitation QPE data

We investigate the depth–duration relationship of maximum rainfall over the whole of Germany based on 16 yrs of radar derived Quantitative Precipitation Estimates (namely, RADKLIM–YW, German Meteorological Service) with a space–time resolution of 1 km and 5 min. Contrary to the long–term historic records that identified a smooth power law scaling behavior between the maximum rainfall depth and duration, our analysis revealed three distinct scaling regimes of which boundaries are approximately 1 h and 1 d. Few extraordinary events dominate a wide range of durations and deviate 5 to the usual power law. Furthermore, the shape of the depth–duration relationship varies with the sample size of randomly selected radar pixels. A smooth scaling behavior were identified when the sample size is small (e.g. 10 to 100), but the original three distinct scaling regimes became more apparent as the sample size increases (e.g. 1 000 to 10 000). Lastly, a pixel wise classification of the depth–duration relationship of the maximum rainfall at all individual pixels in Germany revealed three distinguishable types of scaling behavior, clearly determined by the temporal structure of the extreme rainfall events at a pixel. 10 Thus, the relationship might change with longer time series and can be improved once available.

. Extreme precipitation values for different durations: World extremes (World Meteorological Organization, 1994), Spanish extremes (Gonzalez and Bech, 2017), regional extremes for Germany (Dyck and Peschke, 1995) One of the main obstacles to identify the "true" scaling behavior of maximum rainfall is that the most rainfall is measured 55 from sparse ground gauge networks (Dyck and Peschke, 1995;Papalexiou et al., 2016). Remotely sensed precipitation products with high spatio-temporal resolution such as the ones provided by radar, satellite or microwave link networks may be applied to overcome this issue. Breña-naranjo et al. (2015) used a satellite based rainfall product to identify the scaling behavior of the maximum rainfall across the globe. They showed that the maximum of the areal rainfall averaged over the~20 km ×~20 km data grid has the scaling exponent of~0.43 which is similar to that of Jennings (1950) while the maximum rainfall values 60 were systematically underestimated. They attributed the main reason of the discrepancy to the coarse spatial resolution of the satellite data that easily misses the small scale rainfall variability that is closely associated with extreme values (Cristiano et al., 2017;Fabry, 1996;Gires et al., 2014;Kim et al., 2019;Peleg et al., 2013Peleg et al., , 2018. Taken these findings, we want to analyze rainfall depth-duration relationship for the whole Germany based on the 16 y of the Quantitative Precipitation Estimates (QPE) radar product with 1 km-5 min space-time resolution in order to answer Germany. Some time series are, for example, only available from 2014 onwards. 2) Some locations/raster cells have NA values potentially due to malfunction of the radar or general (radar) errors. Figure 2 (a) shows the proportion of the NA values of the time series developed for each of the pixels. The visible cones display the individual radar coverage, and the overlapping areas of the radar cones have a better data coverage than the areas without overlapping. Additionally, Fig. 2 (b) shows the maximum rainfall differences between right before and after a data gap, calculated for a time step of 5 min (Imputation bridge = Intensity 105 difference/gap length). The red spots could mean a difference of greater than 180 mm h −1 .
There are multiple options of handling missing values, but it is very hard to handle highly episodic geophysical events such as rainfall. Even harder is the imputation of NA values which might be potential extreme rainfall events. We chose not to do any correction, since the distribution of the imputation bridge is sporadic, which means that the maximum values that could have been missed at one pixel would be complemented by the observation at the adjacent pixel. Furthermore, the worst 110 consequence of this approach would mean the missing of an extreme value whereas the consequences of imputing potentially too high extreme values seemed more severe and uncertain. The radar data can be expressed as with N being the number of pixels for whole Germany and n p the number of observation in time series.
For one single time series/raster cell the values X 1 , ..., X np cell are aggregated for each τ according to Eq. (3), p being the index for each aggregated sample: The maxima for each τ are retrieved with Eq. (4).

Depth-Duration relationship for whole Germany
Finally, the overall maxima for each τ are taken from each time series' maxima according to Eq. (5).
The scaling relationship with scaling factor b and intercept B can be established as following. respectively 6 https://doi.org/10.5194/nhess-2020-192 Preprint. Discussion started: 20 July 2020 c Author(s) 2020. CC BY 4.0 License. algorithm (Scott and Knott, 1974). Erroneous pixels (due to too many NA values in the time series) were excluded from the cluster process in order to avoid disturbance. The data was rescaled to make the characteristics more comparable with each other. The elbow evaluation (compare Fig. 3 for estimating the best number of clusters gave no obvious suggestion (curve should bend like an elbow). We chose six clusters for a sufficiently detailed analysis.
3 Results and discussion

Scaling behavior of entire Germany for high-quantile rainfall
High rainfall values are associated with especially great uncertainty when obtained from radar data. Thus, we also investigated the scaling behavior of high-quantile rainfall values. Figure 5 shows the maximum depth-duration relationship of several quantiles: 0.99999 (fourth greatest cells), 0.9999 (39th greatest cells), 0.999 (392nd greatest cells), and 0.99 (3921st cells). The "three phase regime" from radar maximum values remains relatively stable, however, the "single event" effect between 50 min 160 and 1 d is smoothed out, because the degree of inflections in the curve becomes weaker. The lower the chosen quantile, the clearer the scaling regime appears. Figure 6 shows the location of the 0.9999, 0.9999, 0.999, and 0.99 quantile rainfall. The color of the circles represents the different rainfall durations. The lower the quantile, the sparser the location of the quantile rainfall occurrence, which suggests the reduction of the influence of one single rainfall event on the depth-duration relationship causing inflection in the curve.

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The map corresponding to the lower quantiles (e.g. Fig. 6 (c) and 6 (d)) reveals that the locations of maximum rainfall contributing to the development the rainfall-duration relationship are spread over the whole of Germany. Naturally, one would assume that this heterogeneity of the meteorological conditions and rainfall generating mechanisms will reflect rather regional characteristics and will exhibit some irregular scaling behavior. Contrary to this conjecture, the curve shows a very smooth scaling behavior, that suggests the extreme rainfall events at this degree of quantile (upper one percentile) share common 170 characteristics such as peak rainfall depth and correlation structures regardless of timescale that could have been derived from similar generation mechanisms. Figure 7 shows the spatial distribution of 5 min, 30 min, 1 h, 6 h, 1 d, and 3 d maximum rainfall over Germany. The red and yellow spots that are spatially distributed in Fig. 7 (a) suggest that 5 min extreme rainfall can happen at any place in Germany.

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Note that extreme rain occurred also outside the Alpine region at the southern edge of Germany, which suggests that fine-scale extreme rainfall is not necessarily governed by topography. The influence of fine-scale intense rainfall persists until hourly 8 https://doi.org/10.5194/nhess-2020-192 Preprint. Discussion started: 20 July 2020 c Author(s) 2020. CC BY 4.0 License.      Zhang et al. (2013). Both studies showed that the maximum rainfall-duration relationship at a given point location follow smooth and simple power law if the rainfall process can be 190 modeled with a set of simple stochastic processes.

Classification of maximum depth-duration relationship
The maximum dept-duration relationships in Fig. 8 were clustered since some show a similar shape with each other . The k-mean clustering algorithm successfully classified the depth-duration relationship into six categories revealing different curve characteristics regarding the curve shapes. Figure 9 shows a categorical map of Germany indicating each category with a certain 195 color and the depth-duration relationship at 100 grid cells randomly sampled from each category.
Cells belonging to Category 1 have the highest rainfall intensities over all scales until 1 d and show a strong inflection at around 1 h similar to the scaling curve for whole Germany (Fig. 4). The behavior of the curve between 5 min and 1 h is associated with strong convective rainfall events that pour for around 1 h and move on or weaken. Thus, these events are 13 https://doi.org/10.5194/nhess-2020-192 Preprint. Discussion started: 20 July 2020 c Author(s) 2020. CC BY 4.0 License.