the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Rain erosivity map for Germany derived from contiguous radar rain data

### Franziska K. Fischer

### Tanja Winterrath

### Robert Brandhuber

Erosive rainfall varies pronouncedly in time and space.
Severe events are often restricted to a few square kilometers. Radar rain
data with high spatiotemporal resolution enable this pattern of erosivity
to be portrayed with high detail. We used radar data with a spatial
resolution of 1 km^{2} over 452 503 km^{2} to derive
a new erosivity map for Germany and to analyze the seasonal distribution of
erosivity. The expected long-term regional pattern was extracted from the
scattered pattern of events by several steps of smoothing. This included
averaging erosivity from 2001 to 2017 and smoothing in time and space. The
pattern of the resulting map was predominantly shaped by orography. It
generally agrees well with the erosivity map currently used in Germany
(Sauerborn map), which is based on regressions using rain gauge data
(mainly from the 1960s to 1980s). In some regions the patterns of both maps
deviate because the regressions of the Sauerborn map were weak. Most
importantly, the new map shows that erosivity is about 66 % larger than
in the Sauerborn map. This increase in erosivity was confirmed by long-term
data from rain gauge stations that were used for the Sauerborn map and which
are still in operation. The change was thus not caused by using a different
methodology but by climate change since the 1970s. Furthermore, the seasonal
distribution of erosivity shows a slight shift towards the winter period
when soil cover by plants is usually poor. This shift in addition to the
increase in erosivity may have caused an increase in erosion for many crops.
For example, predicted soil erosion for winter wheat is now about 4 times
larger than in the 1970s. These highly resolved topical erosivity data will
thus have definite consequences for agricultural advisory services,
landscape planning and even political decisions.

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Soil erosion by heavy rain is regarded as the largest threat to the soil
resource. Rain erosivity is rain's ability to detach soil particles and
provide transport by runoff and thereby is one of the factors influencing
soil erosion. The most commonly used measure of rain erosivity is the
*R* factor of the Universal Soil Loss Equation (USLE; Wischmeier, 1959;
Wischmeier and Smith, 1958, 1978) or the Revised Universal Soil Loss
Equation (RUSLE; Renard et al., 1991), although other concepts also exist
(Morgan et al., 1999; Schmidt, 1991; Williams and Berndt, 1977). The *R* factor
is given as the product of a rain event's kinetic energy and its maximum
30 min intensity. Both components are usually derived from hyetographs recorded
by rain gauges. Such rain gauge data are spatially scarce. For instance, in
Germany only one rain gauge per 2571 km^{2} was available for
the currently used *R* map (Sauerborn, 1994; this map will be called
“Sauerborn map” in the following). Hence, point information has to be
spatially interpolated to derive an *R* map that enables us to estimate *R* for any
location. Different interpolation techniques have been applied. Most often
correlations (transfer functions) of *R* with other meteorological data
available at higher spatial density were used (for an overview see Nearing
et al., 2017). The Sauerborn map was based on correlations between *R* and
normal-period summer rain depth or normal-period annual rain depth, which
differ between federal states (Rogler and Schwertmann, 1981; Sauerborn, 1994,
and citations therein).

Recent research has shown that the erosivity of single events exhibits
strong spatial gradients (Fiener and Auerswald, 2009; Fischer et al., 2016,
2018b; Krajewski et al., 2003; Pedersen et al., 2010; Peleg
et al., 2016). This is due to the small spatial extent of convective rain
cells, which is typical for erosive rains. The resulting heterogeneity has
two consequences. First, interpolation of erosivity between two neighboring
rain stations will not be possible for individual rains because a rain cell
in between may be completely missed. Second, even long records of rain gauge
data may miss the largest events that occurred in close proximity to a rain
gauge and thus underestimate rain erosivity. This is illustrated nicely by
the data of Fischer et al. (2018b). They showed that the largest event
erosivity, which was recorded by contiguous measurements over only
2 months, was more than twice as large as the largest erosivity recorded by
115 rain gauges over 16 years and the same area. Furthermore, this single
event contributed about 20 times as much erosivity as the expected long-term
average. Even in a 100-year record this single event would thus still change
the long-term average erosivity. The large variability of erosivity in space
and time then directly translates to soil loss. This may be illustrated by
soil loss measurements in vineyards in Germany. Emde (1992) found a mean
soil loss of 151 t ha^{−1} yr^{−1} averaged over 10 plot years while
Richter (1991) only measured 0.2 t ha^{−1} yr^{−1}, averaged over
144 plot years. The difference is due to the largest event during the study by
Emde (1992), which obviously had too much influence on the mean compared to
the size of his data set. Such an event was missing entirely in Richter's (1991)
much larger data set. The inclusion of rare events when measured by
chance by a rain gauge leads to statistical problems due to their
extraordinary magnitude. They cause outliers in regression analysis and thus
strongly affect transfer functions. To avoid an effect by single events to
the transfer function, Rogler and Schwertmann (1981) excluded all events for
which the estimated return period was more than 30 years (assuming that event
erosivities followed a Gumbel distribution). In consequence, the largest
event was replaced by zero erosivity and, in turn, soil erosion was underestimated.

The demand for contiguous rain data to create *R* factor maps was only recently
met by satellite data (Vrieling et al., 2010, 2014) and by radar rain data
with considerably larger spatial (presently up to 9-fold) and temporal
resolution (presently up to 36-fold) (Fischer et al., 2016). Put simply, the
measurements are based on the principle that radar beams are reflected by
hydrometeors (Bringi and Chandrasekar, 2001; Meischner et al., 1997). The
intensity of the reflection depends on rain intensity and the travel time of
the reflected radar beam depends on the distance between the emitting and
receiving radar tower and the hydrometeors within the measurement volume.
Radars usually measure with a resolution of approx. 1^{∘} azimuth and
125 to 250 m in the direction of beam propagation. The data are then
typically transformed to grids of square pixels of 1 km^{2}
(Bartels et al., 2004; Fairman et al., 2015), 4 km^{2}
(Koistinen and Michelson, 2002; Michelson et al., 2010) or
16 km^{2} (Hardegree et al., 2008) after many refinement steps.

An *R* factor (map) can serve two purposes with contrasting requirements.
First, it can be used in combination with measured soil loss or reported
damages (e.g., Mutchler and Carter, 1983; Vaezi et al., 2017; Fischer et
al., 2018a). In this hindcast case, the highest possible spatial and temporal
resolution is recommended. The second application of the *R* factor is for
forecasting erosion, which is required, e.g., for field use planning
(Wischmeier and Smith, 1978). In this case, the long-term expectation is of
interest rather than the true *R* factor of the (near) past that was influenced
by the stochastic location of individual rain cells. Thus, for future
modeling, smoothing of the stochastic noise is necessary.

Existing *R* maps have also undergone a number of smoothing steps, although
this is not explicitly stated in the corresponding reports. Most *R* maps use
regressions between long-term averages of erosivity and long-term
meteorological parameters, e.g., annual rain depth. For long-term averages,
periods of more than 20 years are accepted (Chow, 1953; Wischmeier and
Smith, 1978) to remove the stochasticity of individual events and leave the
general pattern. In consequence, a temporal smoothing follows from using
long-term averages, and a spatial smoothing follows from the transfer
functions and their application to rainfall maps. These rainfall maps
include a third step of smoothing because the meteorological recommendation
is to use normal-period rainfall (30-year) data, and point data (meteorological
stations) have to be extended to create a map. For example for the *R* map in
Germany (Rogler and Schwertmann, 1981), the precipitation map of 1931
to 1960 was used, which was the last available normal period although rain
erosivities were derived mainly from measurements in the 1960s and 1970s.
This precipitation map was mainly based on educated guesses of the best
meteorologists at that time as geostatistical tools were only developed
later (Matheron, 1970). With the large increase in data availability by radar
measurements and the development of (geostatistical) smoothing tools, this
uncontrolled smoothing can be replaced by accepted statistical methods of
smoothing. The general recommendation is to apply smoothing until the
pattern that is intended to be shown can be seen (O'Haver, 2018; Simonoff,
1996; Quantitative Decisions, 2004) by using several smoothing steps in
sequence (O'Haver, 2018; Wedin et al., 2008).

In this study, we used the new RW product from the radar climatology RADKLIM
(RADar KLIMatologie) from the German Meteorological Service (Deutscher
Wetterdienst, DWD). RW data provide gauge-adjusted and further refined
precipitation for a pixel size of 1 km × 1 km (Winterrath et
al., 2017, 2018). RW data of 17 years (2001–2017) were available as a
contiguous source of rain information. Using these data to establish a new
*R* factor map for Germany should be a major step forward compared to the
Sauerborn map, which was derived from an inconsistent set of data compiled
by different researchers (e.g., some had winter precipitation data available
and used them while others did not; see Sauerborn, 1994) and with equations
developed independently for 16 federal states. Our data set is much larger
(by a factor of 2571 regarding locations) and, because of the contiguous
data source, it does not require interpolation with transfer functions. Our
first hypothesis was that there will be considerable changes in the pattern
of erosivity due to the removal of transfer-function weaknesses. Our second
hypothesis was that the *R* factor map will exhibit larger values than the
Sauerborn map, for two reasons. Very large and rare events will no longer be
missed, as occurred previously due to the large distances between
meteorological stations, and there is no longer any need to remove these
events to arrive at robust transfer functions. The second reason for larger
*R* factors is due to global climate change, as Rogler and Schwertmann (1981)
and Sauerborn (1994) mostly used data from the 1960s, 1970s and 1980s.
Global climate change is expected to increase rain erosivity (Burt et al., 2016).

## 2.1 Radar-derived precipitation data

DWD runs a Germany-wide network of presently 17 C-band Doppler radar systems (Fig. 1). This network underwent several upgrades during the analysis period. At the start of the time period considered, five single-polarization systems (DWSR-88C, AeroBase Group Inc., Manassas, USA) were operated without a Doppler filter, the latter being added between 2001 and 2004. Between 2009 and 2017, DWD replaced the network of C-band single-polarization systems of the types METEOR 360 AC (Gematronik, Neuss, Germany) and DWSR-2501 (Enterprise Electronics Corporation, Enterprise, USA) with modern dual-polarization C-band systems of the type DWSR-5001C/SDP-CE (Enterprise Electronics Corporation), all equipped with Doppler filters. During this period, a portable interim radar system of the type DWSR-5001C was installed at some sites.

The radar systems permanently scan the atmosphere to detect precipitation signals. Every 5 min, the radars perform a precipitation scan, each with terrain-following elevation angle to measure precipitation near the ground. The resulting local reflectivity information over a range of currently 150 km in real time and a constant 128 km in the climate approach is combined to form a Germany-wide mosaic of about 1100 km in the north–south direction and 900 km in the west–east direction. The reflectivity information is converted to precipitation rates applying a reflectivity–rain rate (ZR) relationship (Bartels et al., 2004). An operational quality control system screens the radar data. To further improve the quantitative precipitation estimates, the radar-derived precipitation rates are summed to hourly totals and immediately adjusted to gauge data from more than 1000 meteorological stations resulting in RADOLAN (RADar OnLine ANeichung, i.e., online-adjusted, radar-derived precipitation), which provides precipitation data in real time, mainly for applications in flood forecasting and flood protection (Bartels et al., 2004; Winterrath et al., 2012).

Based on RADOLAN, the climate version RADKLIM is derived. Compared to the
real-time approach, the data are additionally offline-adjusted to daily
gauge data, combining a total of more than 4400 rain gauges measuring hourly
and daily (equivalent to 1 rain gauge per 80 km^{2}). The data
are then reprocessed by new climatological correction methods, e.g., for
spokes, clutter or short data gaps. Spokes result from permanent obstacles
blocking the radar beam, while clutter is introduced by non-meteorological
targets like windmills or birds. The final product (called RW data) has a
temporal resolution of 1 h and a spatial resolution of 1 km × 1 km in
polar stereographic projection. For more detailed information on RADKLIM the
reader is referred to Winterrath et al. (2017). The RW data, restricted to
the German territory, are freely available (Winterrath et al., 2018). For
the first time, the RADKLIM data set provides contiguous precipitation data
with high temporal and spatial resolution. It includes local heavy or
violent precipitation events (for classification of heavy and violent see
UK Met Office, 2007) that are partly missed by point measurements alone. Thus,
it particularly improves the analysis of extreme precipitation events.

Two additional data sets were used to verify the validity of the approach and to examine effects of methodological details (see below). These data sets are erosivities derived from radar data at 5 min resolution taken from Fischer et al. (2016) and erosivities derived from rain gauge data of 115 stations in Germany from 2001 to 2016, which were taken from Fischer et al. (2018b).

## 2.2 Erosivity calculation procedure

According to Wischmeier (Wischmeier, 1959; Wischmeier and Smith, 1958,
1978), the erosivity of a single rain event (*R*_{e} in N h^{−1}) is the
product of the maximum 30 min rain intensity (*I*_{max30} in mm h^{−1}) and
the total kinetic energy per unit area (*E*_{kin} in kJ m^{−2}).

An erosive rain event is defined to have at least a total precipitation
amount (*P* in mm) of 12.7 mm or an *I*_{max30} of more than 12.7 mm h^{−1}
that is separated from the next rain by at least 6 h. In order to scan
and fulfil the 6 h criterion, we did not separate between days but used a
continuous 17-year data stream. Specific kinetic energy *e*_{kin,i} per millimeter
rain depth (in kJ m^{−2} mm^{−1}) is given for intervals *i* of constant
rain intensity *I* (in SI units according to Rogler and Schwertmann, 1981).

For 0.05 mm h${}^{-\mathrm{1}}\le I<\mathrm{76.2}$ mm h^{−1},

For *I*<0.05 mm h^{−1},

For *I*≥76.2 mm h^{−1},

We used the equation by Wischmeier and Smith (1978) to calculate specific kinetic energy although several others have also been proposed (van Dijk et al., 2002) with none being superior (Wilken et al., 2018). Our choice retained comparability with the Sauerborn map. Furthermore van Dijk et al. (2002) had shown that kinetic energy as obtained by the Wischmeier and Smith equation did not deviate from measured kinetic energy in Belgium neighboring Germany.

For all intervals *i*, *e*_{kin,i} is multiplied with the rain amount of
this interval and then summed to yield *E*_{kin} for the entire event. The
annual erosivity of a specific year is the sum of *R*_{e} of all erosive
events within this year. The average annual erosivity (*R*) is then the average
of all annual erosivities during the study period (17 years in this case).
While in the USA and other countries the unit MJ mm ha^{−1} h^{−1} is
often used for *R*_{e}, we use N h^{−1} because it is the unit most often
used in Europe and because of its simplicity. Both units can be easily
converted by multiplying the values in N h^{−1} with a factor of 10 to
yield MJ mm ha^{−1} h^{−1}. The unit for *R* is then N h^{−1} yr^{−1}.

Rain erosivity strongly depends on intensity peaks. Fischer et al. (2018b)
have shown that these peaks increasingly disappear the lower the spatial and
temporal resolution becomes. This can be accounted for by scaling factors
but these scaling factors can only adjust to an average behavior, while the
factors may either be too large or too small for a specific event. A high
spatiotemporal resolution should be used to determine *R*_{e} for individual
events. This is not required to determine the long-term average pattern like
an *R* factor map for planning and prediction purposes. In that case, data with
lower resolution and the application of appropriate scaling factors are
advantageous because this will reduce the noise introduced by large events
of small spatial extent that would not be leveled out by averaging alone. We
will use data in 1 h time increments as those are adjusted to rain gauge
measurements and the number of data is reduced by a factor of 12 compared to
5 min increments. This is especially important when all calculations,
including identification of rain breaks >6 h and periods of *I*_{max30},
have to be carried out for many years and many locations. In our
case, roughly 7×10^{10} 1 h increments had to be processed.

According to Fischer et al. (2018b), the following modifications in the
calculation of *R*_{e} had to be made to account for the temporal resolution
of 1 h, the spatial resolution of 1 km^{2} and the method of
measuring rain by radar: (i) *I*_{max30} was replaced by the maximum 1 h rain
intensity and the threshold for *I*_{max30} was lowered to 5.8 mm h^{−1},
while the total precipitation threshold remained at 12.7 mm. (ii) Five or
more subsequent 1 h intervals without rain separated events, which assumed
that rain events begin on average in the middle of the first nonzero rain
interval and end again in the middle of the last nonzero rain interval,
yielding a total rain break of at least 6 h. (iii) The temporal scaling
factor was 1.9 and the spatial scaling factor was 1.13, to which 0.35 had to
be added to account for the radar measurement instead of the rain gauge
measurement. The total scaling factor [$(\mathrm{1.13}+\mathrm{0.35})\times \mathrm{1.9}$] was then 2.81.

Gaps in the time series were considered when calculating annual sums of erosivity by scaling the total sum of erosivity over the whole time series to 365.25 days. If the effective number of missing values exceeded 2 months per year, the respective year was excluded from the calculation for that pixel. If the effective number of excluded years was larger than one, the respective pixel was excluded. This was the case for 0.6 % of all pixels.

## 2.3 Steps to generate an *R* factor map

The reduction of noise by using 1 h increments and a 17-year mean was still
not sufficient to level out the most extreme events. Two further smoothing
steps were therefore applied. The first step was to winsorize the annual
erosivities of the 17 years for each individual pixel by replacing the lowest
value with the second-lowest value and the highest value with the
second-highest value (Dixon and Yuen, 1974). Winsorizing is an appropriate
measure for calculating a robust estimator of the mean in symmetrically
distributed data, but it is biased for long-tailed variables like rain
erosivity. Thus, the country-wide mean of all winsorized data (94 N h^{−1} yr^{−1})
was smaller than the mean of the original data (96 N h^{−1} yr^{−1}).
In order to remove this bias, we binned all data in 26 bins of
20 N h^{−1} yr^{−1} width and calculated the mean *R* before and after
winsorizing. For bins with *R*<180 N h^{−1} yr^{−1}, comprising
95 % of all pixels, the bias increased linearly with *R* (*r*^{2}=0.92;
*n*=8) and amounted to 2.3 % of *R*. Above 180 N h^{−1} yr^{−1}
there was no further increase in the bias (*r*^{2}=0.01, *n*=18), which was,
on average, 3.4 N h^{−1} yr^{−1}. We removed
the bias by adding 2.3 % to all values <180 N h^{−1} yr^{−1} and 3.4 N h^{−1} yr^{−1}
to all values above.

The last smoothing step applied geostatistical methods. A semivariogram (over a range of 50 km) was calculated and ordinary kriging was applied. Geostatistical analysis was done using the program R (version 3.5.0; R Core Team, 2018) and gstat (Gräler et al., 2016). A block size of 10 km × 10 km was chosen to remove noise and to fill the pixels with data gaps, while the spatial resolution remained at 1 km. The missing information was obtained from neighboring pixels. The radar data outreached the German borders. In total, 452 503 pixels were used to ensure small kriging variances near borders or on islands, while the final map was restricted to the German land surface (357 779 pixels).

Using 1 h data instead of 5 min data reduced the effect of single extreme
events at certain locations. Winsorizing reduced the effect of extreme years
at a location, in addition to the effect of averaging 17 years. Finally,
kriging used the information from neighboring pixels to reduce the effect of
the extremes. This smooths among near neighbors (distance < 20 km)
but does not affect the regional pattern (>20 km). To evaluate
whether this was the case and to quantify the effect of all smoothing steps,
we used the data from Fischer et al. (2016). They had calculated rain
erosivity from 5 min resolution radar data for 2 years (2011 and 2012) and
an area of 14 358 km^{2} (yielding a total of 28 770 pixel
years), which is called “test region” in the following. Using these data
we calculated semivariograms from annual to biennial erosivities based on
5 min and 1 h resolution. These semivariograms were compared to those from
17-year average erosivities, 17-year winsorized average erosivities, and 17-year
winsorized and kriged erosivities for the test region and for the entire area
of Germany. Smoothing should reduce the influence of individual violent
thunderstorm cells and reveal the regional pattern. In geostatistical
analysis this decreases the sill of the semivariogram while the range
increases as it changes from being dominated by thunderstorm cells to being
dominated by the regional pattern. The regional trend was calculated as the
difference between the square root of semivariances at distances of 40 and
20 km divided by the difference in distance of 20 km to examine whether it
was influenced by the individual smoothing steps. The effect of violent rain
cells was calculated as the square root of the semivariance at a distance of
20 km divided by the difference in distance of 20 km minus the regional trend.

## 2.4 Annual erosivity return periods

Rain erosivity usually follows long-tailed distributions. This leads to the
question of how frequent years of extraordinarily large erosivity are. To
answer this question, the development of cumulative distribution curves (for
basic concepts see Stedinger et al., 1993) is required. A period of 17 years
is not sufficient to reliably estimate a cumulative distribution curve for
every pixel. Therefore, we combined all annual erosivities of the total data
set (452 503 pixels and 17 years) after expressing each of them relative to the
corresponding winsorized and bias-corrected mean of the pixel (in %).
This enabled the cumulative distribution curves to be calculated from a
large data set (*n*=7.7 million). The expected maximum relative annual
erosivity for a given return period could then be estimated from the
complementary cumulative distribution curve (exceedance). This was also done
for the relative annual erosivities of the test region, calculated from 1 h
rain data, to examine whether the general cumulative distribution curve also
applies to smaller regions.

The erosivities, when calculated from 1 h rain data, are already smoothed and do not adequately reflect the extremes that result from data that are better resolved, such as the 5 min rain data. The cumulative distribution curve for the test region was also calculated using the 5 min rain data. Given that the cumulative distribution curves of the entire study area and the test region agree for the relative erosivities calculated from 1 h data, we expect that the relative erosivities calculated from 5 min rain data of the test region can serve as a first estimate for the entire study region. The cumulative distribution curve for the test region calculated from 5 min data will then be a fair estimate of the return periods anywhere in the entire research area.

## 2.5 Seasonal distribution of erosivity

The seasonal distribution of erosivity, calculated as the relative
contribution of each day to total annual erosivity, is called the erosion index
distribution or EI distribution (Wischmeier and Smith, 1978). It is required
in erosion modeling to determine the influence of seasonally varying soil
cover due to crop development. The convolution of the seasonal effect of
soil cover with the seasonal EI distribution results in the so-called crop
and cover factor (*C* factor) of the USLE. The EI distribution was calculated
for each pixel and averaged over all 452 503 pixels. Seventeen years of data
still did not suffice to show similar amounts of erosivity on subsequent
days, despite the large number of pixels. There was still considerable
scatter that required smoothing to illustrate the seasonal distribution.
Smoothing between individual days during the year involved three steps (for
details of the methods see Tukey, 1977): first a 13-day centered median was
calculated for each day. A centered median smooths but preserves the common
trend signal (Gallagher and Wise, 1981), which is also true for the two
subsequent steps. A 3-day skip mean (leaving out the second day) was
calculated from the results, followed by a 25-day centered Hanning mean
(weighted mean with linearly decreasing weights). To account for the
periodic nature of the EI distribution and to allow the smoothing methods to
be applied at the start and the end of the year, the year was replicated and
shifted by plus or minus 1 year.

Radar measurements tend to have larger errors during wintertime with snowfall. The reduced reflectivity of snow particles may lead to an underestimation of the precipitation rate, while the increased reflectivity of melting particles in the bright band may cause an overestimation. Moreover, the lower boundary layer promotes a potential overshooting of the radar beam with regard to the precipitating cloud (Holleman et al., 2008; Wagner et al., 2012). Such measurement problems, if relevant, should especially influence the EI distribution during winter months and cause a deviation from measurements at meteorological stations. Therefore, we also calculated the EI distribution using data from 115 rain gauges distributed throughout Germany and covering 2001 to 2016. These data were taken from Fischer et al. (2018b). This data set will also be used in the discussion for comparison of recent radar-derived erosivities with recent rain-gauge-derived erosivities and with historic rain-gauge-derived erosivities taken from the literature.

## 3.1 The effects of smoothing

The effects of smoothing on the appearance of the maps were negligible
(compare Fig. 2 with Figs. S3 and S4) because smoothing had
only removed the extraordinarily large variability that exists on small
temporal and spatial scales. However, the high data density revealed that
even long-term averages were insufficient to remove all influence of erratic
cells of violent rain, and further attenuating steps had to follow. Annual
sums of rain erosivity from 5 min data for the test region varied most due
to the dominance of individual cells of violent rain that did not overlap or
fill the entire area (semivariogam I in Fig. 3a). This was indicated by the
short range (20 km) and high semivariance for that range
(2749 N^{2} h^{−2} yr^{−2}) (Table 1). The standard deviation (SD) of two pixels
separated by 20 km thus was 52 N h^{−1} yr^{−1} (square root of
2749 N^{2} h^{−2} yr^{−2}), which is more than half of the
average annual erosivity in Germany. After averaging both years (2011 and
2012), the semivariance for a distance of 20 km was only reduced to
1569 N^{2} h^{−2} yr^{−2} and the range stayed the same at
approximately 20 km (semivariogam III in Fig. 3a). Both findings indicated
that, even after averaging 2 years, the individual cells of violent rain
were still fully detectable and had not merged to form a larger pattern. In
consequence, the regional trend, albeit detectable, appeared minor (Table 1).

^{1} The regional trend was calculated as the difference between
the square roots of *γ* at distances of 40 and 20 km divided by the
difference in distance of 20 km. ^{2} The effect of violent rain cells was
calculated as the square root of *γ* at a distance of 20 km divided by
the difference in distance of 20 km minus the regional trend.

The effect when using data with a resolution of 1 h was almost as strong as
when 2 years were averaged. Semivariance at a distance of 20 km was only
1667 N^{2} h^{−2} yr^{−2} for annual values (semivariogam II in
Fig. 3a) and 953 N^{2} h^{−2} yr^{−2} for biennial averages
(semivariogam IV in Fig. 3a). Even more important, the regional trend became
more visible due to smoothing of the extreme events by using 1 h instead
of 5 min data. This regional trend is evident from the gradual increase in
semivariance over the entire distance of 50 km shown in Fig. 3. Importantly,
smoothing by using 1 h data did not change average erosivity because the
difference was adequately compensated for by the temporal scaling factor. The
biennial average for the test region was 115 N h^{−1} yr^{−1} when
calculated from 5 min data and 114 N h^{−1} yr^{−1} when
calculated from 1 h data.

Averaging annual erosivities of the test region over 17 years further reduced
variability (semivariogram V in Fig. 3a). Semivariance strongly decreased to
197 N^{2} h^{−2} yr^{−2} and the influence of individual cells
of violent rain became small relative to the regional trend. This led to an
almost linear increase in semivariance over distance. The influence of
extreme years in individual pixels was further reduced by winsorizing, which
slightly reduced semivariance at 20 km distance to 190 N^{2} h^{−2} yr^{−2}
(semivariogram VIII in Fig 3b). For all of Germany, winsorizing
reduced the standard deviation of a pixel over time from, on average,
49 to 39 N h^{−1} yr^{−1}, while bias
correction left the mean of erosivity over all pixels unchanged at
96 N h^{−1} yr^{−1}. The effect on the appearance of the map was small
(compare Figs. S3 and S4) because only small erratic patches
of extraordinarily high or low erosivity disappeared.

Finally, kriging reduced semivariance at 20 km distance to 121 N^{2} h^{−2} yr^{−2},
leaving mainly the regional trend (semivariogram VI in Fig. 3a). Thus, the step from 5 min to 1 h resolution reduced
semivariance at 20 km distance by a factor of 1.6 while averaging 17 years
reduced semivariance by a factor of 8.5. Winsorizing contributed a factor
of 1.04 and kriging a factor of 1.6. In total, semivariance was reduced by a
factor of 23, indicating a pronounced patchiness of erosive rains on the
annual scale that could not be leveled out by averaging 17 years alone. The
effect of each smoothing step decreased with increasing distance. For a
distance of 10 km, the combined factor was 32 while it was only 13 for a
distance of 30 km. This was due to the decreasing importance of thunderstorm
cells relative to the regional trend. Independent of the degree of
smoothing, the regional trend, extracted from the change in semivariance
between distances of 20 and 40 km, remained practically unchanged at
0.2 N h^{−1} yr^{−1} km^{−1} (Table 1). In contrast, the effect
of violent rain cells decreased greatly using the smoothing steps from
2.4 to 0.3 N h^{−1} yr^{−1} km^{−1}. The effect on the appearance of the map was again small
(compare Fig. S4 and Fig. 2) because only large contrasts
between close neighbors disappeared, which are hardly visible due to the small
pixel size. The main visible effect was the filling of the few gaps.

After winsorizing and kriging, the semivariances for the test region
followed a linear regression through the origin almost perfectly
(*r*^{2}=0.9889, *n*=50; line through semivariogram VI in
Fig. 3a). This indicated that the variation in erosivity over a distance of
50 km followed linear trends without any noise (nugget) or short-range
structures that could be attributed to individual cells of violent rain. The
semivariances, when calculated for the whole of Germany, were considerably
larger (twice as large at a distance of 50 km; Fig. 3b, semivariogam VII)
and close to a linear trend only for short distances (e.g., a linear
regression through the origin for the first 15 km yielded *r*^{2}=0.9905).
For longer distances, the semivariogram followed an exponential
model (nugget 4 N^{2} h^{−2} yr^{−2}, partial sill 970 N^{2} h^{−2} yr^{−2},
effective range 123 km). The larger semivariance
and the exponential model were both caused by the inclusion of mountain
areas with large erosivities and steep erosivity gradients that were missing
in the test region.

## 3.2 *R* factor map

Erosivity was on average 96 N h^{−1} yr^{−1} but varied between
46 and 454 N h^{−1} yr^{−1}. The regional pattern of
erosivity (Fig. 2) was mainly determined by orography (for a detailed
topographic map see Fig. S1 in the Supplement). The largest values (above 185 N h^{−1} yr^{−1})
were found in the very south where the northern chain
of the Alps reaches altitudes of almost 3000 m a.s.l. (above sea level).
Lower mountain ranges are also characterized by larger mean annual
erosivities than in their surrounding area (compare Fig. 1 or Fig. S1
with Fig. 2). For instance, the Bavarian Forest with elevations of
up to 1450 m a.s.l. exhibited annual erosivities of above
155 N h^{−1} yr^{−1}. The Ore Mountains with elevations of up to 1244 m a.s.l., had
erosivities mostly between 125 and 155 N h^{−1} yr^{−1}. Also mountain
ranges like the Black Forest or the Harz mountains clearly shape the
erosivity map. Additionally, upwind–downwind effects were detectable. For
example, the areas west-northwest (upwind) of the Harz mountains had
erosivities of between 70 and 80 N h^{−1} yr^{−1}, while the areas
east-southeast (downwind) received less than 65 N h^{−1} yr^{−1}.

## 3.3 Annual erosivity return periods

The cumulative distribution of the relative annual erosivities followed a
straight line in a probability plot fairly well when the logarithm was used
(Fig. 4). This indicated a log-normal distribution (log mean 1.96; log
SD 0.19). A very similar cumulative distribution was found for annual
erosivities derived from the 1 h data of the test region (log mean 1.97; log
SD 0.18). The distribution based on the less-smoothed 5 min data was
considerably wider (log mean 1.94; log SD 0.22). The annual expected
erosivity was 88 %, 216 % and 273 % of the respective long-term
mean for return periods of 2, 30 and 100 years when the 5 min data were
used (Fig. 4). It is important to note that these values apply for averages
of 1 km^{2} pixels and include the smoothing that results from
the radar measurement, the radar reprocessing and from using 5 min rain
increments. Even more extreme years are expected to occur in reality.

## 3.4 Seasonal distribution of erosivity

There was a pronounced peak in the seasonal distribution of relative
erosivity during summer months (Fig. 5). The daily erosion index increased
rapidly from mid-April to mid-May and was 0.61 % day^{−1} on average
in June, July and August. From mid-August to September the daily erosion
index declined rapidly. In winter months the daily erosion index was small
(mean of December, January, February and March: 0.08 % day^{−1}).
There was no detectable difference in the seasonal variation between
different regions in Germany (see Fig. S5). The cumulative
distribution functions of different regions correlated with at least
*r*^{2}=0.998 (*n*=365).

Even more striking was the fact that this pattern required considerable
smoothing to yield a continuous seasonal time course. The difference between
subsequent days in the unsmoothed data was enormous (e.g., 1.5 % day^{−1},
0.4 % day^{−1} and 0.4 % day^{−1} on 29, 30 and 31 July).
This was despite the large number of measurements (17 years and
455 309 pixels) that were averaged for each day. It highlights the exceptional
strength of some violent rains. Despite the rather small extent of
individual erosivity cells, many of them occurred in the same day, making a
large relative contribution to total erosivity for this day. While
particular days of the year were influenced by heavy precipitation, during
other days no erosive rainfall occurred anywhere within the research area.
A period of 17 years was not sufficient to level out the contrast between
subsequent days. The results of the smoothing procedure show that even
221 years (17 years multiplied by a moving-average window of 13 days) were not sufficient
to level out these differences. Two additional smoothing steps had to be
applied to arrive at a smooth time course. Despite the strong smoothing that
was necessary for the probability density function, the smoothing did not
change the cumulative distribution function (which is used for calculating
*C* factors). The cumulative distribution functions of the original data and of
the smoothed data correlated with *r*^{2}=0.9998 (*n*=365; both
functions are shown in Fig. S5).

The distribution of the daily erosion index calculated from rain gauge data (1840 station years) was very similar to the distribution calculated from the much larger radar data set (compare solid and dashed lines in Fig. 5). This was especially true during winter months, when values derived from both measurement methods were considerably larger than expected from previous analysis in the 1980s.

## 4.1 Increase in erosivity

The most striking difference between the Sauerborn map based on data from
the 1960s to 1980s and the radar-derived map is a pronounced increase in
erosivity. A German average of 58 N h^{−1} yr^{−1} was derived from
the Sauerborn map (Auerswald et al., 2009), while the radar-derived map
suggests an average of 96 N h^{−1} yr^{−1}. This increase will come
along with an equal increase in predicted soil losses by 69 %. An almost
identical increase resulted when the erosivity of meteorological stations,
as reported by Sauerborn (1994), was compared with the erosivity derived
from radar data at the same locations. This resulted in an increase of
63 % (open symbols in Fig. 6). Thus, the increase in erosivity is not an
effect of the regression approach that was previously used or due to better
capturing of extreme events by the contiguous radar data.

Fischer et al. (2018b) calculated erosivity for 33 of the Sauerborn stations from recent (2001 to 2016) rain gauge data. A comparison of these data with the Sauerborn data (1994) also showed a similar increase of 52 % (closed symbols in Fig. 6). The increase in erosivity between the Sauerborn map and the new radar-derived map is thus also not an artifact of using radar data but the result of a true change in erosivity over time. This is further corroborated by Fiener et al. (2013), who analyzed long-term records from 10 meteorological stations in western Germany. They found an increase in erosivity of 63 % between 1973 and 2007. Both independent findings leave little doubt that the pronouncedly higher values in the new erosivity map are a result of a change in weather properties and not a result of the difference in the applied methodologies, although we did expect the mean to increase due to the contiguous data set, which is better at recording rare extremes.

A time series of 17 years is regarded to be too short in meteorology for
calculating temporal trends. The data in Sauerborn (1994) were derived from
different periods for different states. If we calculate the statewide mean
*R* factors from her transfer functions relative to the statewide mean
*R* factors of the radar-derived map and plot this relative *R* factor against the
mean year from which the state-specific data originated, a 23-year-long period
can be covered by the means (Fig. 7; years < 1990; the total time
period of individual years covers an even wider range, mostly
about ±5 years around the mean year). During this period there was a slight but
insignificant increase in erosivity with time. This increase smoothly leads
over to the steeper increase in radar-derived Germany-wide annual *R* factors
if we express them again relative to the 17-year mean (Fig. 7; years > 2000).
Both data sets combined cover more than 60 years and
yield a very highly significant regression (*r*^{2}=0.7340,
*n*=27) that indicates an accelerating increase in erosivity likely due to
climate change. Furthermore, Fig. 7 indicates that at the end of the radar
time series (2017) the *R* factor likely is already 20 % higher than the
values depicted in Fig. 2.

## 4.2 Change in the regional pattern of erosivity

The regional patterns of the Sauerborn map and of the radar-derived map
generally agree well but with two exceptions. First, the radar-derived map
shows distinctly larger values southeast of the German Bight of the North
Sea where the air masses coming from the North Sea are channeled by the Elbe
river estuary and its Pleistocene meltwater valley and then hit the higher
areas of the north German moraines. A large frequency of large rains is not
unlikely in this situation. The reason that this was missed by Sauerborn (1994)
using the data obtained by Hirche (1990) for Lower Saxony might be
mainly due to the small data density and the regression with long-term
rainfall. Only 18 stations were available for the whole of Lower Saxony and
only five of them were in the area of large erosivity. Using the 18 stations
in the state of Lower Saxony only, and ignoring the difference between
landscapes, resulted in a rather poor regression with long-term annual
rainfall (*r*^{2} was only 0.32 for *n*=18), and therefore a large
prediction error and considerable smoothing of the true erosivity pattern
can be expected. For comparison, in Bavaria the regression with long-term
rainfall yielded *r*^{2} of 0.92 (for *n*=18; Rogler and Schwertmann, 1981).

The second difference in the pattern is that the radar-derived map reveals more detail than the regression-based map by Sauerborn (1994). This is especially evident in southern Germany where southwest–northeast-oriented structures seem to follow tracks of thunderstorm movement. In the northeast quarter of Germany, where the pattern is not shaped by mountain ranges, a rather patchy pattern resulted. Although Sauerborn (1994) had already found a patchy pattern in this area it appears to be patchier now. At present, it is difficult to decide whether this pattern is random due to large multicell clusters of rainstorms that will level out in the long term or whether landscape properties, e.g., the existence of large forests, cause a stable pattern in an area where other factors affecting the pattern are missing. More detailed variation may also be expected in mountainous areas but radar measurements cannot adequately show this variation. In the future, using data obtained by commercial microwave links as an additional source for retrieving precipitation (Chwala et al., 2012, 2016; Overeem et al., 2013) may improve high-resolution estimates, particularly in these areas.

## 4.3 Change in the seasonal distribution of erosivity

The third pronounced difference between past and recent erosivities was
found for the erosion index distribution. This distribution is needed for
*C* factor calculations (Wischmeier and Smith, 1978). A change in the
seasonality of erosivity was already suggested by Fiener et al. (2013)
analyzing an 80-year time series. However, Fiener et al. (2013) used data from
April to October only, and their results therefore cannot be compared
directly with our results that show the most pronounced changes for the
period from December to March.

At present, the *C* factors for all of Germany (DIN, 2017) are based on the
erosion index distribution developed for Bavaria by Rogler and Schwertmann (1981),
although unpublished erosion indices are also available for other
federal states (e.g., Hirche, 1990). The index distribution by Rogler and
Schwertmann (1981) is characterized by very low values during winter months,
which in turn causes a sharp increase during summer months. In contrast, the
radar-based index, although still having a pronounced summer maximum,
predicts a higher percentage of erosivity during winter. Rogler and
Schwertmann (1981) found that only 1.5 % of the annual erosivity fell
from January to March, while Fig. 5 indicates that these months contributed
6.9 % to annual erosivity. This deviation may be caused by a regional
variation in the erosion index because the unpublished indices for other
federal states also suggested a larger contribution by winter months (e.g.,
January to March contributed 7.5 % in Lower Saxony according to Hirche,
1990). However, restricting our data set to Bavaria led to a very similar
index during winter months (e.g., 6.2 % for January to March) to the
index for the whole of Germany, and the discrepancy with Rogler and
Schwertmann (1981) remained. Furthermore we could not find significant
differences when calculating the index distribution separately for different
regions (Fig. S5).

A second explanation might be that the Rogler and Schwertmann (1981) data were too limited to capture enough erosive rains during periods of infrequent erosive events. This explanation is corroborated by the large scatter between individual days that still existed in our data set (Fig. 5), although our data set was more than 50 000 times larger than the data set used by Rogler and Schwertmann (1981).

A third explanation could again be climate change. In Germany the number of extreme wet months increased in winter by 463 % from the first to the second half of the last century, while summer and autumn remained unchanged (Schönwiese et al., 2003).

The change in erosion index distribution may be regarded as being rather
unimportant at first glance because erosivity is still dominated by
precipitation in summer. This small increase in erosivity during the winter
months, however, could have important consequences for the *C* factor of crops
that provide only small soil coverage during winter. As there is practically
no growth during winter, these crops stay susceptible to erosion over a long
period. Thus they experience a considerable amount of erosivity, even though
erosivity per day is small. For example, the *C* factor for continuous winter
wheat increases from 0.04 to 0.10 when using the soil loss ratios taken from
Auerswald et al. (1986) that entered DIN (2017) and the new erosion indices
instead of those from Rogler and Schwertmann (1981).

## 4.4 Stochasticity

Soil erosion is characterized by a large temporal variability at a small spatial scale due to the stochastic character of erosive rains. About 20 years are necessary, according to Wischmeier and Smith (1978), until this variability levels out and average soil loss approaches values predicted with the (R)USLE. Our data set covered 17 years but significant additional smoothing was still necessary. One of the smoothing steps was to use hourly data, although 5 min data would have been available. In one or two decades the data series may be long enough to remove some of the smoothing steps. In particular, it would be desirable to use data of 30 min or even 5 min resolution.

This pronounced stochasticity is due to the small size of convective rain
cells. Just recently it has been shown by analyzing the radar-derived rain
pattern of the largest rainfall events that on average the rain amount is
halved within a distance of only 2 km around the central point of a rain
cell (Lochbihler et al., 2017). Given that rain amount is squared in the
calculation of rain erosivity, the *R* factor decreases to one fourth within
this distance. Larger areas are only covered if there is movement of the
rain cells. This small size of rain cells questions the use of sparsely
distributed rain gauges to derive rain erosivity. The inconsistent transfer
functions among German states to derive erosivity from rainfall maps likely
originated in the high stochasticity of rain gauge measurements under such
conditions. It was only the unintended but unavoidable smoothing that was
inherent in previous approaches that allowed deriving such maps. Radar
technology enables us to replace this unintended smoothing using clearly
defined statistical protocols and to quantify the effect of smoothing.

Another implication of this large variability is that 20 years will still not
be sufficient to level out extraordinary events. The largest event erosivity
that Fischer et al. (2016) found in 2 years on ∼15 000 km^{2} was
622 N h^{−1}. Even for a 20-year period, this event
will add 31 N h^{−1} yr^{−1} to the average annual erosivity at the
small location of only a few squared kilometers (km^{2}) where it occurred. Most
studies measuring soil erosion under natural rain use much shorter intervals
that usually cover only a few years and rarely exceed 10 years (see
Auerswald et al., 2009, for a meta-analysis of German studies and Cerdan et
al., 2010, for European studies). The interpretation of such short-term
studies and the applicability of the results are limited due to the
pronounced variability of natural rains.

In addition, the erosion index distribution required considerable smoothing
to improve representation of the seasonal variation. Without smoothing, the
shift in a certain crop stage by only 1 day can cause large discrepancies
in the resulting *C* factor, depending on whether a day of large erosivity in
the past is included or excluded at the bounds of the crop stage period.
Smoothing can prevent this. This is especially important for short crop
stage periods, while the effect becomes small for longer periods. For
instance, the monthly sums of the smoothed data correlated closely with the
sums of the unsmoothed data (coefficient of determination: 0.995;
Nash–Sutcliffe efficiency: 0.994).

Radar-derived rainfall data enable us to derive highly resolved and contiguous maps of erosivity with high spatial detail. This avoids errors in landscapes with insufficient rain gauge density. The analysis showed that present (2001 to 2017) rain erosivity is considerably higher than erosivity in the past (1960s to 1980s). Furthermore, the seasonal distribution of rain erosivity also deviates from that of the past period. Winter months contribute more to total erosivity than previously recorded. Considerably more erosion can be expected for crops that are at a highly susceptible stage of development in winter. In consequence, the predicted soil loss will change pronouncedly by using recent erosivity and the ranking of crops regarding their erosion potential will change. This will have definite consequences for agricultural extension and advisory services, landscape planning and even political decisions.

Data can be obtained from two sources:
https://doi.org/10.5676/DWD/RADKLIM_Rfct_V2017.002 (Fischer et al., 2019) and
https://opendata.dwd.de/climate_environment/CDC/grids_germany/annual/erosivity/precip_radklim/2017_002/ (Winterrath, 2019).
The first source provides a shape file containing *R* factors of the 16 German
states, the 401 German counties, and the 11 256 German communities as well as
the entire map as raster data with a resolution of 1 km^{2} in
GeoTIFF format. The second source provides a shape file containing the
*R* factors of the 16 German states, the 401 German counties, and the 11 256 German
communities based on the unsmoothed maps of all individual years
since 2001. Further information on the data is given in the corresponding
README files. Annual maps of future years will routinely be produced and published within
the framework of the annual RADKLIM update after the precipitation data have
undergone all steps of quality control and refinement. Be aware that the
annual maps based on 1 h rain data cannot be used to quantify
high-resolution site-specific erosion in a certain year because of the
potential smoothing of extreme rain intensities. These maps are only
designed for calculating long-term averages.

The supplement related to this article is available online at: https://doi.org/10.5194/hess-23-1819-2019-supplement.

KA, RB and TW were responsible for the concept of the study. KA designed the analysis, which was mainly carried out by FKF. TW provided most data and the knowledge about all steps involved in radar data creation. KA prepared and revised the manuscript with contributions by FKF and TW. RB secured funding for the project and contributed to the data analysis.

The authors declare that they have no conflict of interest.

This study was part of the project “Ermittlung des Raum- und
Jahreszeitmusters der Regenerosivität in Bayern aus radargestützten
Niederschlagsdaten zur Verbesserung der Erosionsprognose mit der Allgemeinen
Bodenabtragsgleichung” at the Bavarian State Research Center for Agriculture
funded by the Bayerisches Staatsministerium für Ernährung,
Landwirtschaft und Forsten (A/15/17). The RADKLIM data were provided by the
project “Erstellung einer dekadischen radargestützten
hochauflösenden Niederschlagsklimatologie für Deutschland zur
Auswertung der rezenten Änderung des Extremverhaltens von Niederschlag
(Kurztitel: Radarklimatologie)” financed by the Strategic Agencies' Alliance
(Strategische Behördenallianz) “Adaptation to Climate Change” consisting of the Federal Office of Civil
Protection and Disaster Assistance (BBK); the Federal Institute for Research
on Building, Urban Affairs and Spatial Development (BBSR); the Bundesanstalt
Technisches Hilfswerk (THW); the Umweltbundesamt (UBA); and the Deutscher
Wetterdienst (DWD). Melanie Treisch and Ewelina Walawender helped with ArcGIS
operations; Karin Levin provided language editing; Anton Vrieling provided
helpful comments during revision; and Helmut Rogler has, for many years,
requested that the German *R* factor map be updated.

This work was supported by the German Research Foundation (DFG) and the Technical
University of Munich (TUM) in the framework of the Open Access Publishing Program.

This paper was edited by Remko Uijlenhoet and reviewed by Anton Vrieling and one anonymous referee.

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