Introduction
Prediction of rain-induced soil erosion using models like the Universal Soil
Loss Equation (USLE) requires quantification of the potential of rain to
cause soil detachment and transport. This potential is called rainfall
erosivity and is typically obtained from point rainfall measurements using
rain gauges. For the conversion of erosivities from point to spatial
information, isolines, interpolation techniques and relations to parameters
such as the mean summer rainfall depth have been used (Rogler and Schwertmann,
1981; Wischmeier, 1959; Wischmeier and Smith, 1958, 1978). The
characteristic relation between erosivity and rain depth of the same period
was termed erosivity density and used in RUSLE2 (Dabney et al., 2012; USDA,
2013). It is recommended for areas with poor data availability (Nearing et
al., 2017).
Rainfall is now able to be measured contiguously by radars and adjusted by
rain gauges so that information about the spatio-temporal distribution of
rain is combined with hyetographs measured at ground level. Several
countries provide rain-gauge-adjusted radar data products with spatial
resolutions of, for example, 1×1 km2 (Bartels et al., 2004;
Fairman et al., 2015), 2×2 km2 (Koistinen and Michelson,
2002; Michelson et al., 2010) or 4×4 km2 (Hardegree et
al., 2008). Contiguous data of even coarser scale may result from other
sources such as satellite data (Vrieling et al., 2010, 2014) or the output
of regional climate models (e.g. Christensen et al., 2007; Flato et al.,
2013).
Despite the important advantage that radar rain data are contiguous and
temporally resolved, they cannot easily be used in place of rain gauge data
for erosivity estimations because the scales of measurement differ a lot
between both techniques. While rain gauges measure the rain near ground
level at point scale (in Germany the collection area is 200 cm2), radars usually deliver rain measurements with an
azimuthal resolution of approx. 1∘ and a range of 125 to 1000 m.
The data are then typically aggregated in grids of square pixels 1 to 16 km2 in size. Rain intensity may differ greatly between point
and grid measurements due to reduction in peak intensities with decreasing
temporal and spatial resolution. Furthermore, sources of error differ
between both measurement techniques. For radar measurements, errors may
result from shading of rain cells by objects such as buildings, orographic
elevations or hydrometeors and from the influence of the melting layer
causing bright-band effects (Wagner et al., 2012). Major limitations of rain
gauges are caused by adhesion, evaporation, wind drift and splashing (Habib
et al., 2001). Finally, strong gradients can, in particular, be expected for
thunderstorm cells of limited spatial extent. Thus, heterogeneity within
pixels will be especially pronounced for erosive rains (Fiener and
Auerswald, 2009; Fischer et al., 2016; Krajewski et al., 2003; Pedersen et
al., 2010; Peleg et al., 2016). This heterogeneity cannot be resolved but
needs to be quantified because it is the uncertainty that can be expected
for predictions at a resolution higher than the pixel size. This uncertainty
also applies in cases where a point measurement of rain erosivity is within
a certain distance (e.g. 1 km) from the target area for which erosion is to
be calculated. The resulting deviation between point measurement and grid
pixel average will be called “positional effect” in the following. The
positional effect also determines the uncertainty, caused by the spatial
variability of rain, of soil loss predications in the proximity of a point
rain measuring location. This positional effect should level out in
long-term measurements as long as grid pixels are small enough not to
include a consistent orographic pattern.
By definition in the USLE, erosivity is the product of a rain event's maximum
30 min intensity and its total kinetic energy (Wischmeier and Smith, 1958).
Both factors depend on rain intensity; thus, intensity is squared in
erosivity. Consequently, a difference in rain intensity of just 10 % would
result in difference in erosivity of 21 %. Therefore, larger
effects of variation in rain intensity can be expected for erosivity than
for rainfall. In particular, an average of squares, as obtained from several
point measurements within an area of non-uniform rainfall, will always be
higher than the square of the average calculated from the same measurements.
This difference between both squares caused by the difference in spatial
scale of the measurements is expected to be a robust factor in the long run.
We will call this the “spatial-scale effect”. A spatial-scale effect for
erosivity, to the best of our knowledge, has not been studied. This is
probably due to the novelty of operational radar measurements and the lack
of long-term data sets required for erosivity estimations. Long-term and
revised radar rain data now exist and can help to improve contiguous
erosivity and soil loss estimations. Therefore, it is crucial to know to
what extent erosivity, and subsequently also soil loss, is underestimated
due to the spatial-scale effect by gridded rain data as provided by radar
measurements and also by climate models or satellites that employ an even
coarser spatial resolution than typical radars (Chen and Knutson, 2008;
Vrieling et al., 2014). Rain intensities from radar may additionally be
smoothed by measuring and subsequent processing procedures. The contribution
of erosivity underestimation due to these procedures is called the “method
effect” in the following. Thus, the difference in erosivity from rain gauge
data and from radar data is caused by spatial-scale and method effects.
Another effect is induced by the temporal scale of the data used for
erosivity calculations. With decreasing temporal resolution, maximum
30 min intensity and hence erosivity are increasingly underestimated.
Therefore, temporal scaling factors are required to compensate for this
underestimation (e.g. Auerswald et al., 2015; Agnese et al., 2006; Istok et
al., 1986; Williams and Sheridan, 1991; Weiss, 1964; Yin et al., 2007).
These are especially important for contiguous data, for which temporal
resolution of rain data is decreased, often to 60 min, as a requirement for
the adjustment to rain gauge data and to reduce the enormous amount of data
caused by the high spatial resolution and wide spatial and temporal
coverage.
We therefore hypothesize that (1) with decreasing temporal and spatial
resolution of rain data, calculated erosivities decrease due to a smoothing
of intensities; (2) radar measurements cause an additional underestimation of
erosivities due to the measuring principle and the required calculation and
correction steps; and (3) large uncertainty of erosivity within 1 km2 is due to strong gradients of erosive rains as determined
by the positional effect. The effects of hypotheses (1) and (2) have to be
compensated for by changes in the calculation of erosivity, while the effect of
hypothesis (3) quantifies uncertainty of erosivity of individual events at
any location within an area of 1 km2 around a rain gauge. We
will quantify these effects and discuss their implications.
Material and methods
Data sets
To cover a wide range of spatial and temporal resolutions, several large and
overlapping data sets had to be combined (for an overview see Table 1). The
spatial resolution from point scale to 1 km pixel width (with an
intermediate pixel width of 0.5 km) was covered by a high-density network of
12 rain gauges which operated over 4 years within an area of 1 km2
(taken from Fiener and Auerswald, 2009; for location of
the measuring site see Fig. 1a; for the spatial distribution of rain gauges
see Fig. 1c). The data of the network comprised 542 events at point scale.
The spatially integrated hyetographs at 0.5 or 1 km pixel width generated
by the Thiessen polygon method (see Fig. 1c) will be referred to as
“pseudo-radar” data.
(a) Locations of the 115 rain gauges (dots), the coverage (circles)
of the 17 weather radars (crosses) and the location of the 12 rain gauges
used for the pseudo-radar data (square; size exaggerated) in Germany. (b) One
rain gauge (dot) within one 1×1 km2 pixel (bounding box)
and isolines of rain depth (taken from Fiener and Auerswald, 2009)
illustrating the variability of a single erosive rain event at 1×1 km2
grid scale causing positional effects. (c) Distribution of
the 12 rain gauges (dots) within an area of 1×1 km2
(bounding box) and their corresponding Thiessen polygons. Dashed lines
separate the area to a spatial scale of 0.5×0.5 km2.
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Point scale and 1 km pixel width were also compared for a much wider data
set covering 16 years and the whole of Germany. Erosivities at 115 rain
gauges were compared to erosivities obtained from radar data with 1 km
resolution (for location of the rain gauges and the coverage of weather
radars see Fig. 1a). Rain gauge data were taken from the Climate Data Center
of the German Weather Service (Deutscher Wetterdienst: DWD; ftp://ftp-cdc.dwd.de/pub/CDC/, last access: 11 December 2018).
DWD also provided the radar data, which were a revised version of the
RADar OnLine ANeichung (RADOLAN) radar rain data product (Winterrath et al., 2012, 2017). This resulted in point–pixel pairs
for >20 000 erosive rain events. For this data set the
effect of temporal resolution was also evaluated. For spatial resolutions lower
than 1 km pixel width (up to 18 km pixel width), a third data set was used.
It comprised 1.9×106 events at 1 km pixel width determined by radar
measurements within an area of 800×600 km2 (Table 1).
Overview of the data used to determine the positional effect, the
spatial-scale effect, the temporal-scale effect and the method effect.
Purpose
Measurement
Spatial
Temporal
Number of
Period
Event
scale
scale
stations/pixels
number
Positional and spatial-scale effect
Rain gauge
Point
60 min
115
16 yr
29 610
Radar
1 km2
60 min
115
16 yr
25 884
Spatial-scale and method effect
Rain gauge
Point
1 min
12
4 yr, Apr–Oct
542
Radar
1 km2
60 min
480×103
2 months
1.9×106
Temporal-scale effect
Rain gauge
Point
1 min
17
16 yr
4599
Radar
1 km2
5 min
17
16 yr
3924
Precipitation measurements of the DWD station network were conducted with
OTT Pluvio weighing rain gauges (OTT Hydromet GmbH, Kempten, Germany) with a
collector area of 200 cm2, a measurement range of 0–1800 mm h-1
and a
1 min resolution of 0.1 mm h-1. The precipitation data passed a quality
control system testing for completeness, carrying out climatological tests,
and checking consistency over time as well as internal and spatial consistency
(Spengler, 2002; Kaspar, 2013). The data were neither corrected for wind
drift effects nor homogenized. A thorough overview of the precision of rain
gauge measurements is given in Vuerich et al. (2009). Information on the
stations' metadata can be found in the Climate Data Center
(ftp://ftp-cdc.dwd.de/pub/CDC/observations_germany/climate/hourly/precipitation/historical/; last access: 11 December 2018) of DWD.
The DWD weather radar network underwent several upgrades during the analysis
period. In the beginning of the considered time period five
single-polarization systems (DWSR-88C, AeroBase Group Inc., Manassas, USA)
operated without a Doppler filter, the latter being added between 2001 and
2004. Between 2009 and today, DWD has exchanged the network of C-band
single-polarization systems of the next generation of type METEOR 360 AC
(Gematronik, Neuss, Germany) and DWSR-2501 (Enterprise Electronics
Corporation, Enterprise, USA) by modern dual-polarization C-band systems of
type DWSR-5001C/SDP-CE (Enterprise Electronics Corporation), all equipped
with a Doppler filter. During the time of exchange, a portable interim radar
system of type DWSR-5001C was installed at some of the sites. Radar data
underwent an operational quality control system. They were adjusted to gauge
data within a reprocessing suite applying a consistent software version
(version 2017.002) and optimized quality control algorithms with 5 min
resolution (Winterrath et al., 2018a) and 60 min resolution (Winterrath et
al., 2018b).
Erosivity calculation procedures
Following Wischmeier (1959) and Wischmeier and Smith (1978) erosivity of a
single rain event (Re) was calculated as the product of the maximum
30 min rain intensity (Imax30) and the kinetic energy (Ekin) (Eq. 1). A rain event is erosive by definition if it has a total precipitation
(P) of at least 12.7 mm or a minimum Imax30 of 12.7 mm h-1
(min(Imax30)).
Re=Imax30⋅Ekin
The Ekin,i per millimetre rain depth (in kJ m-2 mm-1) was
calculated for intervals i of constant rain intensity I following Eq. (2a)–(2c).
For all intervals i, Ekin,i was multiplied by the rain
amount of this interval and then summed up to yield Ekin for the entire
event.
Ekin,i=11.89+8.73×log10I×10-3for0.05mmh-1≤I<76.2mmh-1Ekin,i=0forI<0.05mmh-1Ekin,i=28.33×10-3forI≥76.2mmh-1
When Imax30 was derived from data with intervals longer than 30 min,
Imax30 was determined as the maximum rain intensity of the event.
Erosive events are separated from each other by rain breaks of at least 6 h
(Wischmeier and Smith, 1958, 1978). For example, using 60 min rain
data, we defined events as being separate when five subsequent 60 min
intervals without rain occurred. This assumes that rain events stop and
start on average in the middle of the first and the last non-zero rain
interval. The same concept was used for all data sets with temporal
resolutions >60 min.
The annual erosivity of a specific year (Ry) is the sum of Re of all
n erosive events within this year. The long-term average annual erosivity
(R) is then calculated as
R=1k∑jk(∑inRe,i)j=1k∑jkRy,j,
which is the average of Ry for a number of k years, in the case of this study
16 years.
While in the USA and other countries often the unit MJ mm ha-1 h-1
is used, we use N h-1 for Re, because it is the unit most often
used in Europe and because of its simplicity. The units can be easily
converted by multiplying the values in N h-1 by a factor of 10 to
yield MJ mm ha-1 h-1.
Determination of scale effects
The smoothing caused by decreasing resolution in time and space mainly
decreases intensity, while the total amount of rainfall should, in
principle, be unaffected. This decrease in intensity has two consequences.
First, the intensity threshold min(Imax30) that defines an erosive event
is less often met and thus has to be adjusted to arrive at the same number
of erosive rains irrespective of resolution. Second, scaling factors for
Re are required. A temporal scaling factor tτ,σ scales
from temporal resolution τ to 1 min resolution at a certain spatial
scale with pixel width σ. A spatial scaling factor sσ
scales from spatial resolution σ to point resolution (rain gauge). A
method effect m may additionally occur, which quantifies the difference
between erosivities obtained from rain gauges and from radar measurements at
identical spatial and temporal scales. It is caused by the additional
smoothing resulting from the radar technique and the adjustment and
correction steps subsequently required. It may also include the errors of
rain measurement that differ between the rain gauge and radar methods. The
positional effect pRe describes the average relative deviation of
erosivity of single events derived at 1 km resolution and at point scale
from rain gauges located within the respective 1 km pixel including the
spatial-scale and method effects. The positional effect cannot be used for
correction, but it is a measure of variability within a certain pixel.
Adjusting the intensity threshold to account for smoothing at low resolution
is appropriate only for the temporal resolution. With decreasing spatial
resolution some areas will be included within a pixel that actually received
erosive rain, while other areas within the pixel did not. Without adjustment
of the intensity threshold the entire pixel may be classified as
non-erosive, while adjustment of the threshold would then indicate an
erosive event also in those areas within a pixel where no erosive rain had
occurred. Adjusting the intensity threshold with decreasing spatial
resolution could not correct both errors simultaneously. Even more
important, the criterion of breaks that separate between events is biased
for large areas. Any rain at some place within a large pixel abrogates an
existing break even if it does not fall at a site that experienced an
erosive rain event. The loss of a break with increasing pixel size decreases the
number of events even when all events are considered. Adjusting the number
of events in this case would be a wrong correction. Hence for the spatial
resolution the threshold effect was included in sσ, while for the
temporal-scale effect the intensity threshold could be adjusted. As a result
the number of erosive events can correctly be estimated at low temporal
resolution with this adjustment at point scale, while for a spatial
resolution lower than point scale the number of erosive events will be wrong
compared to point scale. Only the sum of erosivities over a longer period of
time (months, years or longer) can then be corrected with the spatial
scaling factor.
The hyetographs of the high-density network of 12 rain gauges were spatially
integrated to yield hyetographs at 0.5 or 1 km pixel width. The average
deviation of annual erosivities calculated from hyetographs at point scale
and from spatially integrated hyetographs at 0.5 or 1 km pixel width
yielded the spatial scaling factors sσ=0.5 and sσ=1. The
individual deviation of event erosivities at point scale from the average
was due to the positional effect pRe (for an example see Fig. 1b). The
average positional effect pRe was calculated as the geometric mean
of the k ratios of Re derived from rain gauge (σ=0) and 1 km2 pixel data (σ=1), for which neither rain
gauge Re nor pixel Re was zero:
pRe=10(∑i=1klog10(Re,σ=0/Re,σ=1)i/k).
The positional effects were determined separately for events with
Re,σ=1 larger and Re,σ=1 lower than Re,σ=0.
Rains that were erosive at only one of both spatial scales were excluded
from the calculation of the geometric mean, and the percentages of these
events were determined for both cases.
Erosivity at point scale and at 1 km2 pixel scale were also
compared based on >20 000 erosive rain events at 115 locations
distributed over Germany, where a rain gauge was situated within a radar
pixel. The long-term (16 years) average deviation of R between point and
pixel scale was due to the smoothing effects of the spatial-scale effect and
the radar technique (method effect). The method effect was quantified by
subtracting the spatial-scale effect, as obtained from the dense rain gauge
network, from the combined effect, as obtained by comparing erosivities from
rain gauges with radar-derived erosivities. The combined effects of spatial
scale and method were also tested for seasonal variation.
For spatial resolution lower than 1 km pixel width, radar data were
aggregated to yield pixel widths of up to 18 km. Erosivities were calculated
from the aggregated rain data and compared to the erosivities at 1 km pixel
width, which were averaged for the pixel width being examined. This
comparison was carried out for radar data covering an area of 800×600 km2 over 2 months (1.9×106 events at 1 km pixel width;
Table 1).
The temporal resolutions of the rain gauge data and the radar data differed
(1, 5, 60 min). Erosivities derived from these data were adjusted to
1 min resolution with the appropriate temporal scaling factor. The temporal
scaling factors were determined on two spatial scales, at point scale and at
1 km pixel width. To this end, 17 out of the 115 point–pixel pairs were
selected randomly, and rain data for the period 2001 to 2016 (16 years) with
1 min resolution from rain gauges and 5 min resolution from radar
measurements were used. The rain gauge data yielded a total of 4599 erosive
events, for which rain data were aggregated to 2, 5, 10, 15,
30, 45, 60, 80, 100 and 120 min intervals, and Re
was determined as described in Sect. 2.1. The intensity threshold
min(Imax30)τ was adjusted until the annual number of erosive
rain events at the respective temporal resolution τ was equal to that
at τ=1 min. The temporal scaling factor (tτ=x,σ=y)
for Re was then obtained at point scale (σ=0) from
tτ=x,σ=0=∑i=1N(Re,τ=1,σ=0)i/∑i=1N(Re,τ=x,σ=0)i,
which is the ratio of the sums of Re derived from 1 min data and
Re derived from data with τ>1 min at point scale.
Additionally, for 1 km pixel width tτ=x,σ=1 was estimated by
using an intermediate radar product of RADOLAN with a temporal resolution of
5 min that was recursively adjusted corresponding to the 60 min RADOLAN data
(analogously to Fischer et al., 2016). This was done for the 17 grid pixels
where the 17 rain gauges were located. The temporal scaling factors were derived from RADOLAN data as described above (Eq. 5) but relative
to τ=5 min. The resulting factors were then multiplied by the
scaling factor for τ=5 min obtained from the rain gauge data to
yield scaling factors relative to a temporal resolution τ=1 min.
The temporal scaling factors tτ=x,σ=0 were additionally
determined for each month (January–December) and separately for rain gauges
located in the northern and southern halves of Germany (7 and 10 rain
gauges, respectively) to test for any seasonal or regional dependence of the
factors.
Finally, the combined procedure of an adjusted intensity threshold and a
temporal scaling factor was validated by comparing annual Ry obtained
from 60 min RADOLAN data to Ry derived from RADOLAN data with 5 min
resolution. This was done for the remaining 98 (115–17) grid pixels and
16 years, yielding a total of 1568 Ry.
(a) Time periods influencing the underestimation of Imax30 when
temporal resolution is 30 min (or higher) or when temporal resolution is 60 min
(or any resolution >30 min). (b) Minimum threshold for Imax30
(min(Imax30)τ) derived from rain gauge (solid circles) and radar data
(open squares) required to obtain the same number of erosive events as with
a temporal resolution of 1 min; lines show Eq. (6a) and (6b) (RMSE is
0.10 and 0.39). (c) Scaling factor tτ,σ to scale Re or R for
temporal resolution τ when spatial resolution σ is either rain
gauge scale (solid circles) or 1×1 km2 (open squares),
respectively; lines show Eq. (7a), (7b) and (7c) (for all
RMSE ≤0.04). The x axes in (b) and (c) are square-root-scaled.
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Statistics
We mainly used arithmetic means even though most distributions were strongly
skewed. Arithmetic means are less robust than other measures like geometric
means, but our huge sample size compensated for this. Using arithmetic means
instead of robust measures is a requirement of the USLE, which sums up
erosivities over 1 year or longer. The arithmetic mean provides an
unbiased estimator of event erosivity that allows sums to be calculated over
longer periods of time (e.g. 1 year). Otherwise different scaling factors
would become necessary for individual events and for temporal sums depending
on their temporal length.
Statistical spread is quantified by the standard deviation (SD) or the root
mean squared error (RMSE), and the uncertainty of the scaling factors is
quantified by their 95 % interval of confidence (CI). Validation included
the calculation of the Nash–Sutcliffe efficiency (Nash and Sutcliffe,
1970).
Results
Temporal-scale effect
With 17 rain gauges operating at 1 min resolution, 4599 erosive events were
determined in 16 years. Re ranged from 0.1 to 178.4 N h-1 with an average of 5.8 N h-1. The number of events with P≥12.7 mm or Imax30>12.7 mm h-1 decreased
pronouncedly when resolution decreased from 1 min down to 120 min (by 1,
14 and 16 % at a resolution of 2, 60 and 120 min,
respectively). To avoid this loss of events, min(Imax30)τ was
decreased continuously with decreasing temporal resolution (Fig. 2b). The
decrease was less steep below a temporal resolution of 30 min than above:
min(Imax30)=-0.59τ0.5+13.23forτ≤30min,min(Imax30)=147τ-0.79forτ>30min.
This change at a resolution of 30 min is because 30 min is the time interval
in which the maximum is searched for. For resolutions higher than 30 min,
there is a discrepancy between the true period of Imax30 and the period
of Imax30 that is coerced by the temporal resolution (see grey bars in
Fig. 2a). The error caused by this discrepancy only results from the
difference in intensity immediately before and after true Imax30. When
the temporal resolution becomes less than 30 min, attenuation caused by the
period exceeding the 30 min interval additionally decreases in intensity (see
60 min resolution in Fig. 2a). This attenuation increases the lower the
temporal resolution becomes, and it caused Eq. (6b) to be much steeper than
Eq. (6a).
The decrease in min(Imax30)τ was identical for both the rain
gauge scale and the 1 km2 scale (slope between both scales:
1.0067, r2=0.9858, n=9). For both scales combined,
RMSE was only 0.10 and 0.39 for Eq. (6a) and (6b), respectively. Thus,
both equations were valid for point scale and for a grid width of 1 km.
Rain erosivity also decreased with decreasing temporal resolution; in
turn, the scaling factor tτ,σ
increased (Fig. 2c; Eq. 7a–7c). For intervals τ≤30 min, the increase was
identical for rain gauge scale and for radar pixels of 1 km pixel width. The
increase of tτ,σ was much steeper when τ became longer
than 30 min. This increase then depended on the spatial scale and was larger
for rain gauge scale than for radar pixels of 1 km pixel width (Fig. 2c).
The behaviour of tτ,σ was caused by underestimating
Ekin and underestimating Imax30. The underestimation of
Imax30 was the stronger effect (data not shown). It prevailed for time
intervals greater than 30 min and caused the break at a temporal resolution
of 30 min, as already shown for min(Imax30)τ. The identical
behaviour of intensity with decreasing temporal resolution at rain gauge
scale and at 1 km2 radar pixel scale that was already evident
for min(Imax30)τ thus also led to identical tτ,σ
for both spatial scales as long as τ was less than 30 min. For τ>30 min the attenuation of intensity peaks came into play. This
attenuation was less for the 1 km radar data than for the rain gauge data
because the time a moving intensity peak remains in a 1 km2
grid pixel is longer than the time it requires to pass a rain gauge. In
consequence, three equations for tτ,σ (Eq. 7a–7c)
were necessary to adjust Re, Ry or R to 1 min resolution at the
respective spatial scale.
Forτ≤30minand point or1×1km2grid scale:tτ,σ=τ100+1Forτ≥30minand point scale or:tτ,σ=0=τ40+0.55Forτ≥30minand1×1km2grid scale:tτ,σ=1=τ50+0.70
The RMSE of all three equations was less than 0.04. The validity of
combining the effects of min(Imax30)τ=60 and tτ=60,σ=1 was supported by the close correlation of temporally scaled Ry
derived from 5 and 60 min RADOLAN data, for which the Nash–Sutcliffe
efficiency was 0.9483 (n=1568) while RMSE was 8.8 N h-1 yr-1.
Variation among monthly tτ,σ=0 was small, especially for τ≤60 min. The coefficient of variation among monthly tτ,σ=0 was ≤6 % for τ≤60 min and
11 % to 14 % for τ>60 min. It was not clear if there
was seasonality in this variation because for some temporal resolutions tτ,σ=0 was higher for summer than for winter months, while for other
resolutions the opposite was the case.
There was also a negligible regional variation for τ>30 min, while no difference could be found for τ≤30 min. For
intervals longer than 30 min the scaling factor tτ,σ=0
increased slightly more in northern Germany (+4 %) than in southern
Germany (-2 %), compared to the whole of Germany. This small difference
will only become relevant if data of very low temporal resolution are used.
Spatial-scale effects
Erosivities from all data of rain gauge–radar pixel pairs were calculated by
application of appropriate min(Imax30)τ and temporal scaling
factors to enable comparison. Annual erosivity Ry for the 0.5×0.5 km2 pseudo-radar data set was 7.3 % lower than the average
Ry of the rain gauges. This resulted in a factor sσ=0.5 of
1.08 (CI: 1.00–1.16). This factor increased to sσ=1=1.15
(CI: 1.04–1.26) when Ry was calculated from 1×1 km2
pseudo-radar data (Fig. 3).
Spatial scaling factors for long-term average annual R. Open circles result from rain
gauges aggregated to pseudo-radar pixels. Open squares result from radar and
aggregation of radar data. Error bars represent the 95 % confidence
interval. Lines denote a multiple regression (see text). The x axis is
square-root-scaled to improve visibility at low pixel width.
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For the rain gauges of the 115 rain gauge–radar pixel pairs, long-term
average annual R varied between 42 and 223 N h-1 yr-1 over 16 years and was on average 90.2 N h-1 yr-1. For the radar pixels,
R varied between 26 and 146 N h-1 yr-1 but was on average only 62 N h-1 yr-1
(Fig. 4). In this case the deviation was equal to a
factor of 1.48 (CI: 1.43–1.52), which was considerably larger than
sσ=1 obtained from pseudo-radar data, for which no difference in
measurement method occurred between point scale and pixel scale. This
difference was hence assigned to a method effect (Fig. 3).
Percentage of cases that were erosive at point (115 rain gauges) or
pixel scale (115 radar pixels) relative to a total of 35 124 point–pixel
pairs of rain events that were erosive on at least one of both scales.
Point scale
Pixel scale
Percentage
Erosive
Not erosive
27 %
Not erosive
Erosive
16 %
Erosive
Erosive
57 %
The monthly comparison of the 115 rain gauge–radar pixel pairs over 16 years
did not yield significant differences between months due to the large CI of
the combined scale and method effects (CI between ±4 % and ±9 % for the individual months), but on average this combined effect was
lower during the hydrological winter months (1.16; CI: 1.12–1.21) than
during the hydrological summer months (1.42; CI: 1.30–1.53). This
difference, despite being significant (p<0.001), was unimportant
because of the small contribution of winter months to annual erosivity.
For the large and contiguous radar data set of 800×600 pixels, 1.9×106 events were recorded at 1×1 km2 scale. For these
events, Re was on average 5.1 N h-1 and ranged from 0.5 to 1270 N h-1.
Aggregating these pixels to larger square pixels decreased
Re. At 18×18 km2, Re was on average 4.4 N h-1
and ranged from 0.2 to 221.6 N h-1. In consequence, the spatial scaling
factor sσ increased further (Fig. 3). The increase in scaling
factors over the entire range from point scale to 18 km grid width could be
described by a multiple regression (r2=0.9995, n=21)
accounting for pixel width σ (in km) and the method effect m
depending on the method μ (which is 0 for rain gauges and 1 for radar
data):
m+sσ=1+0.35μ+0.092σ3/4.
The CI was ±0.004 for the slope of σ and ±0.02 for the
method effect.
Annual erosivity Ry (grey points) and multi-annual mean erosivity
R (black circles) derived from radar pixel and rain gauge data for 115
point–pixel pairs and 16 years. The difference in slope between the solid
line and unity (dashed line) is due to the spatial scale and method
effects.
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On average for the pseudo-radar pixel, rain was erosive for only 10 out of
12 rain gauges. Hence only 83 % of the 1 km2 pixel was
covered by an erosive event. The fraction covered by the erosive event
decreased further the larger the pixel size became (fraction =83 % -10.3×ln(pixel size (km2)), r2=0.9974, n=18). On average only about 50 % of a 5×5 km2 pixel and 25 % of a 17×17 km2
pixel received an erosive rain event. This makes it increasingly difficult to
detect erosive rains the larger pixel size becomes.This caused the strong
increase in the spatial scaling factor and indicated a strong positional
effect.
Positional effects
The positional effect as defined here describes the variability of Re
within 1×1 km2. Using the pairs with the true radar data,
29 610 erosive rain events were recorded during 16 years at the 115 rain
gauges. On average, Re was 5.6 N h-1 and ranged from 0.1 to 547.2 N h-1.
For the corresponding 115 radar pixels, 25 884 erosive events were
recorded during the 16 years. Mean Re was 4.4 N h-1 and ranged from
0.2 to 318.9 N h-1.
Combining all events of the 115 rain gauge–radar pixel pairs during 16 years
that were at least erosive at rain gauge scale or at radar pixel scale
resulted in 35 124 events. Only 57 % of them were erosive at both scales,
while the criteria for an erosive event were met exclusively at pixel scale
for 16 % of all events and exclusively at rain gauge scale for 27 % of all events
(Table 2). The gradients of erosivity within 1 km2 were huge. The largest event that was recorded at a rain
gauge while the radar pixel indicated no erosive event was 156 N h-1.
The largest event for the opposite case, i.e. that radar recorded an erosive
event while the rain gauge recorded no erosive event, was similarly high
(180 N h-1). The mean Re of erosive events which were recorded for
the radar pixel while Re at the corresponding rain gauge was zero was
2.9 N h-1 (SD: ±4.9 N h-1). The mean Re of events
which were erosive at a rain gauge but not for the corresponding radar
pixel was also 2.9 N h-1 (SD: ±5.6 N h-1).
The percentage of unpaired events was not significantly related to the
geographical location, neither longitude (r=-0.02, p=0.23) nor
latitude (r=-0.01, p=0.83). It was also independent of the
distance to the adjacent radar station (r=-0.02, p=0.79), which might
be used as a proxy for increasing noise in the radar data. The percentage was
higher in winter (October–March) with 34 % (SD: ±2.4 %) than in
summer (April–September) with 25 % (SD: ±2.4 %). The probability of
remaining just below the threshold of an erosive event on one of both scales
was higher in winter than in summer as in general winter events are less
intensive than summer events. Mean Re in winter was only 35 % of mean
Re in summer.
Rain gauge Re was larger than radar Re for 74 % of those
point–pixel pairs (points above the line of unity in Fig. 5) which were
erosive on both scales (19 944 events). Mean pRe was 1.54 (CI:
±0.01) for these events. This value quantifies the mean deviation of
all locations within a 1 km2 pixel that experience a higher
erosivity than the mean. For individual locations, the deviation can be much
larger, which was already evident from the magnitude of the largest events
that were recorded only on one of both scales. For individual locations with
an erosive event on both scales, pRe could be considerably higher
than 10 (see “outliers” in Fig. 5). Rain gauge Re was lower than radar
Re for only 26 % of all events (points below the line of unity in
Fig. 5), and pRe was 0.72 (CI: ±0.01). Again, the deviation of
individual locations within 1 km2 could be much larger.
Comparison of event erosivity Re calculated from radar data and
Re
from rain gauge data for 115 radar pixels that enclose a rain gauge. Only
events that were erosive at both scales (19 944 events) during the 16-year
period are shown. The dashed line represents unity. Axes are log-scaled.
Note that no spatial scaling factor or method factor was applied because these
factors also included the effect of incomplete coverage of the pixel by an
erosive rain cell.
![]()
For the dense rain gauge field used to create pseudo-radar data, 579
point–pixel pairs of events were at least erosive at rain gauge scale or at
pseudo-radar pixel scale. For these 579 events, Re derived from rain
gauge data ranged from 0 to 45.5 N h-1 (mean: 3.9 N h-1), and
Re derived from pseudo-radar data ranged from 0 to 28.1 N h-1
(mean: 3.4 N h-1) (Fig. 6). For 9 % of these events, the event was not
erosive with pseudo-radar but at the rain gauge, and for 6 % the opposite
was true (Table 3).
Event erosivity Re at 12 rain gauges located within a 1 km2
pixel vs. Re based on pseudo-radar data calculated
from the hyetographs of the 12 rain gauges (open grey circles). Filled black
circles show the average Re of all 12 rain gauges vs. the Re from
pseudo-radar rainfall. Note that the average Re can be considerably larger
than zero while the averaged rainfall of the pseudo-radar remains below the
thresholds of erosivity (black circles along the y axis). Rectangular frame
shows variation of Re for a single day. Axes are square-root-scaled to
improve resolution at low Re.
![]()
Percentage of cases that were erosive at point (rain gauge) or
pixel scale, using the pseudo-radar data; in total 579 point–pixel pairs of
rain events were erosive on at least one of both scales.
Point scale
Pixel scale
Percentage
Erosive
Not erosive
9 %
Not erosive
Erosive
6 %
Erosive
Erosive
85 %
For 67 % of those events which were erosive at both scales, rain gauge
Re was larger than pseudo-radar Re and pRe was 1.28 (CI: 1.25–1.30).
For 33 % of these events, rain gauge Re was lower than
pseudo-radar Re and pRe was 0.81 (CI: 0.77–0.85). Also in this
case, where measurement errors could be excluded because rain gauge
Re and pseudo-radar Re were calculated from the same data, the
variation within 1 km2 was again huge. For the single days
with erosive events, Re varied greatly between rain gauges. For an
example see height of the rectangle in Fig. 6. Although this was the largest
event in this data set, one rain gauge remained below the threshold and
hence recorded no erosive event. This large variation was also reflected by
the large coefficient of variation between rain gauge Re for the same
day (mean: 68 %).
Discussion
Our analysis showed pronounced effects of temporal scale, spatial scale,
position and measuring method. These effects were all caused by the
sensitivity of erosivity calculation to intensity peaks and because
thresholds were used for the definition of erosivity. These strong effects
call for using temporally and spatially highly resolved rain gauge
measurements, like those used in the development of the USLE and most
subsequent studies. Our study, however, also showed strong gradients in
erosivity that were also caused by the sensitivity to intensity peaks and by
the thresholds which earlier studies also showed (Fiener and Auerswald,
2009; Fischer et al., 2016; Krajewski et al., 2003; Pedersen et al., 2010;
Peleg et al., 2016). Erosivity can thus reliably be recorded at the position
of a rain gauge, but this information cannot even be extrapolated over a
distance of only 500 m (half of our radar pixel widths). This was
illustrated by the fact that, within this distance, Re could be zero or
>150 N h-1, which is more than twice the annual erosivity
in Germany (Auerswald, 2006; Sauerborn, 1994). It is also illustrated by the
fact that the largest Re that was recorded within only 2 months was
1270 N h-1 when contiguous measurements were used, while the largest
Re that occurred during 16 years when the same region was covered by 115 rain
gauges was only 547 N h-1. Hence rain gauge measurements fail to
record many erosive events that occur in their close vicinity (even
<500 m). Erosivity determined by a rain gauge cannot be
extrapolated to a small watershed, to farms or even to fields. Discrepancies
between model predictions and measurements of erosion that can be found in
many studies (Govers, 1991; Liu et al., 1997; Risse et al., 1993;
Rüttimann et al., 1995; Zhang et al., 1996) probably originate in part
from this strong positional effect. Such strong discrepancies during
individual events even exist between replicates of bare plots (Nearing et
al., 1999) or between replicated vegetated plots and cannot be explained by
plot characteristics. They do not appear in subsequent runoff and soil loss
observations (Wendt et al., 1986). Erosion prediction and model development
are thus strongly limited by the unexplained variability caused by
short-range erosivity gradients. Hence, there is no alternative to using
contiguous rain measurements. Radar technology provides, for the first time,
measurements that fulfil this need.
Contiguous measurements, on the other hand, suffer from the fact that they
cannot be carried out at the same temporal and spatial scale as rain gauge
measurements, and the method of measurement differs. Here we provide scaling
factors that help to partly overcome this problem and that allow radar
measurements to be used for erosivity calculations. These factors, however,
do not solve the problem that contiguous measurements integrate over a
certain space and time and thus that the information about the variation within
these domains is lost. In particular, the positional effect can only be used
to quantify uncertainty within a radar pixel, but it cannot be used to
predict erosivity at specific locations within a pixel. This large
uncertainty is probably also one of the main reasons for the discrepancy
between observed soil loss and predicted soil loss based on radar rain data
for individual fields, whereas this discrepancy disappeared as soon as many
fields were grouped, irrespective of how this grouping was done (Fischer et
al., 2018a; Auerswald et al., 2018). With future improvements in technology
it may become possible to further improve temporal and spatial resolution of
contiguous rain data and, thus, to reduce the uncertainty of event
erosivities.
Temporal scaling factors had already been developed (Auerswald et al., 2015;
Agnese et al., 2006; Istok et al., 1986; Williams and Sheridan, 1991; Weiss,
1964; Yin et al., 2007) because they are also required for rain gauge
measurements of low temporal resolution (in data storage). Our temporal
scaling factors were of a similar order of magnitude to those in other
studies. However, our data showed that using a scaling factor is not
sufficient because the intensity threshold also has to be adjusted in order
to identify the correct number of erosive events. The existence of an
erosive event and long-term sums of erosivity will otherwise be incorrect,
even with a temporal scaling factor. To our knowledge our study provides,
for the first time, a function that enables the intensity threshold to be
adjusted according to the temporal resolution of the rain data. Adjustment
of the total rain depth threshold is not necessary because total rain depth
should be independent of the temporal resolution, as long as it is still
short enough to identify the rain breaks that separate individual events.
Despite providing intensity thresholds and scaling factors for Re,
Ry and R for different temporal resolutions, we advocate using a high
resolution in order to not lose information. All scaling factors can only
represent average behaviour and cannot reflect the characteristic of an
individual event. A high resolution is easier to achieve in the time domain
than in the spatial domain. In particular, it is advantageous to have a
temporal resolution that is higher than 30 min because scaling factors
increased strongly for less resolved data. For shorter time increments, only
compensation for the error that resulted from an imperfect identification of
the period of Imax30 was necessary. Longer time increments than 30 min
additionally attenuated Imax30 and thus blurred this information.
The spatial scale was more difficult to consider than the temporal scale due
to the large positional effect. In particular, large parts of a pixel
remained below the thresholds of an erosive event even when measurement
errors could be excluded, like in the case of the pseudo-radar pixel that
used rain gauge measurements. On average, 17 % of the rain gauges within a
1 km2 pixel remained below the erosivity threshold while the
other rain gauges recorded an erosive event. This percentage increased
strongly with increasing pixel size. In consequence, the spatial-scale
effect cannot be corrected for individual events but only for the averages
of many events.
The spatial scaling factor is conceptually the inverse of the so-called
areal reduction factors, which are used to reduce rain intensity from rain
gauge measurements when scaled to catchment areas depending on the duration
and return period of the rain event (Allen and DeGaetano, 2005; De Michele
et al., 2001; Stewart, 1989). This conceptual difference is due to the
difference in the intended purpose of contiguous rain data. While in
catchment hydrology the average and the relative distribution of rain depth
within a watershed is of interest (Asquith and Famiglietti, 2000), for
erosion analysis rain intensities are important at point and field scale,
where erosion occurs.
The method effect combines all differences in measurement and measuring
errors (e.g. the wind effect in the case of rain gauges). It is thus highly
dependent on the specific configuration of rain gauge measurements and radar
measurements, including all subsequent data manipulation steps. These
configurations are usually fairly standardized within a country (e.g. rain
gauge height and diameter are usually defined) but differ from country to
country. Our method effect may thus only be valid for Germany, whereas
application to other countries, even if they use similar rain gauge and
radar protocols (e.g. Goudenhoofdt and Delobbe, 2016; Koistinen and
Michelson, 2002), should be done with care. The same is true for using
satellite data or data of commercial microwave links, which recently have
been identified as additional source for retrieving precipitation (Chwala et
al., 2012; Overeem et al., 2013) and which will require the method effect to
be adapted for this particular approach. The approach is based on analysing
the signal attenuation that depends on rain intensity. These data are
especially valuable in regions with sparse coverage by conventional
measurement devices, e.g. in parts of the African continent, but may
also improve high-resolution precipitation estimates and forecasts in
hydrometeorological applications (Chwala et al., 2016).
As an example, for the new German RADOLAN product that recently became
publicly available (spatial resolution: 1 km2; temporal
resolution: 60 min) the Imax30 threshold has to be lowered to
5.79 mm h-1, while the total precipitation threshold remains at 12.7 mm. The
temporal scaling factor becomes t=1.9, and the spatial scaling factor becomes
s=1.13, to which the method effect of m=0.35 has to be added. In total,
the correction factor is 2.81((1.13+0.35)×1.9). Hence the
change of the Imax30 threshold and the combined scaling factor are
large, and ignoring both would considerably underestimate erosivity. The large
change of the Imax30 threshold and the large temporal scaling factor
also show that much information is lost when using data of 60 min
resolution.
This loss of information can be either an advantage or a disadvantage. It
would be a disadvantage in hindcasting, wherein usually the true pattern of
erosivity is wanted. In this case a better-resolved product like 5 min data
should be used. The Imax30 threshold would then be 11.9 mm h-1, and
the temporal scaling factor would only be t=1.05, indicating a minor
loss of information. The spatial scaling factor is already rather low, and
the method effect cannot be avoided.
On the other hand the loss of information would be an advantage in
forecasting, which aims at the likely regional pattern of erosivity. The
loss of information removes the influence of randomly occurring local events
of extraordinarily high magnitude that add noise to the regional pattern of
erosivity. The finding that the largest Re within only 2 months was
1270 N h-1 while the expected long-term average R was only about 70 N h-1 yr-1 (Sauerborn, 1994) shows that this single event would add
64 N h-1 yr-1 to a 20-year record of radar data. Even in a 100-year
record this single event would still be detectable. Using data of 60 min
resolution thus reduces the need for smoothing the map statistically to
remove the influence of such local events.