# Comparing Rainfall Erosivity Estimation Methods Using Weather Radar Data for the State of Hesse (Germany)

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## Abstract

**:**

^{2}and 5 min are used alongside rain gauge data to compare erosivity estimation methods used in erosion control practice. The aim is to assess the impacts of methodology, climate change and input data resolution, quality and spatial extent on the R-factor of the Universal Soil Loss Equation (USLE). Our results clearly show that R-factors have increased significantly due to climate change and that current R-factor maps need to be updated by using more recent and spatially distributed rainfall data. Radar climatology data show a high potential to improve rainfall erosivity estimations, but uncertainties regarding data quality and a need for further research on data correction approaches are becoming evident.

## 1. Introduction

^{2}grid [16]. There is evidence that rainfall distribution and intensity has changed since this time period [12,17], emphasising the need for updated precipitation datasets and methods that estimate rainfall erosivity.

^{2}) and temporal (up to 5 min) resolution [19]. The largely comprehensive nationwide detection of all precipitation events indicates a high potential for the derivation of spatial information to calculate the R-factor. The high temporal resolution of the data as well as recent advances in computer hardware enable the direct event-based calculation of the R-factor. However, the differences in measurement method and scale between radar and rain gauges, especially in detecting heavy rainfall, must be taken into account when interpreting the results. The precipitation totals in radar climatology tend to be slightly lower than the precipitation amounts measured by rain gauges and this underestimation by radar climatology is particularly pronounced for high precipitation intensities [20]. This is due to the averaging of precipitation over the area of the radar pixels and path-integrated rainfall-induced attenuation of the radar beam [21].

- The newly calculated R-factors from both datasets are higher than the R-factors from earlier calculations due to changes in climate, interannual rainfall distribution and rainfall intensity.
- Since radar data include small-scale convective cells without gaps, the R-factors derived from the radar climatology should be higher on average than those calculated from rain gauge measurements. At the same time, the radar measurements underestimate the maximum precipitation intensities. The latter can be compensated by the correction factors according to Fischer et al. [22].
- The spatial distribution of the R-factors derived from the radar climatology deviates from the patterns of the R-factors calculated and interpolated by means of the regression equation due to the comprehensive coverage of all heavy rainfall events.

## 2. Materials and Methods

#### 2.1. Study Area

^{2}. The area is characterised by a diverse topography with several low mountain ranges and highlands crossed by depressions and river valleys (see Figure 1). The highest elevation is 950 m.a.s.l., whereas the lowest elevation is about 73 m.a.s.l. A large portion of the intensively used agricultural areas in the lowlands are oriented in Rhenish direction (SSW-NNO) [24]. The study area is located in the humid midlatitudes in a transition zone between a maritime climate in north-western Germany and a more continental climate in the south and east of Germany. Westerly winds influence the distribution of precipitation and, thus, many of the intensively used agricultural areas are located in the rain shadow on the lee side east of the mountain ranges.

#### 2.2. Data Basis

#### 2.2.1. Radar Climatology Data

^{2}and a temporal resolution of up to 5 min starting from 2001. For this study, we used the YW product in 5-min resolution [18] and the RW product [25] in hourly resolution for the period 2001–2016. Their derivation procedure comprises various correction algorithms to compensate for typical radar-related errors and artefacts such as clutter, spokes, signal attenuation and bright band effects. Ground clutter can be caused by non-meteorological objects such as mountains, buildings, wind energy plants or trees that disturb the radar signal and cause non-precipitation echoes. If the radar beam is blocked in whole or in part by such objects, the sector behind these obstacles is shielded, which causes a linear artefact, the so-called negative spoke. Signal attenuation may cause significant underestimation of rainfall rates. It can be caused by a wet radome, by heavy precipitation events that shield the sector behind or by range degradation at far range from the radar. Bright Band effects occur in the melting layer where the comparatively large surface of melting snowflakes is covered by a film of water, which may cause very strong radar signals.

#### 2.2.2. Rain Gauge Data

#### 2.3. Methodology

#### 2.3.1. R-factor Calculation According to DIN 19708

_{30}. As defined by DIN 19708 [8], the total amount of precipitation is doubled and assigned to I

_{30}if an event lasts less than 30 min. Rainfall events are separated by a precipitation pause of at least 6 h.

_{30}[mm/h] and the kinetic energy E [kJ/m

^{2}] of the total rainfall during the event.

$i$ | 5-min interval of the rainfall event |

${E}_{i}$ | kinetic energy of the rainfall in period i [kJ/m^{2}] |

${N}_{i}$ | rainfall depth in period i, [mm] |

${I}_{i}$ | rainfall intensity in period i, [mm/h], that is ${I}_{i}={N}_{i}\text{}\xb7\frac{60\text{}\mathrm{Min}}{5\text{}\mathrm{Min}}$ |

_{event}products [kJ/m

^{2}mm/h = N/(ha a)] of all erosive rainfall events in a year. Due to the great interannual variability of erosivity, it is recommended to average the annual R-factors over a period of at least ten years [8]. For the calculations based on the radar climatology this criterion was fulfilled everywhere, whereas the time series of five rain gauges was limited to nine years.

#### 2.3.2. R-factor Calculation Using Regression

_{su}:

_{YW,DIN,Agri}.

- (a)
- all 1 km
^{2}pixels within Hesse (n = 23,320) - (b)
- all pixels containing at least ten hectares of cropland (n = 11,555)
- (c)
- all rain gauge stations (n = 110)

_{YW,DIN}and R

_{G,DIN}were used to determine two new regression equations. These serve to assess the following: the changes in the correlation between rainfall erosivity and precipitation sums, changes in comparison to the existing regression equation used for the erosion atlas, and the impact of sample size.

#### 2.3.3. Application of Scaling Factors

_{G,DIN}:

## 3. Results

#### 3.1. Statistical Comparison of the Calculated R-factors

_{YW,DIN}calculated from the original unscaled RADKLIM YW product according to DIN 19708 ranges between 28.8 and 173.2 kJ/m

^{2}mm/h with an average value of 58.0 kJ/m

^{2}mm/h (see Table 2 and Figure 2). It is thus 6.4% higher on average than the values of the erosion atlas R

_{EA}

_{,}whereas its range is 263.7% higher and its standard deviation is 122.7% higher. R

_{YW,DIN}shows thus a much higher variability than the strongly smoothed R

_{EA}which was derived from spatially interpolated rainfall data using a regression equation (Equation (3)).

_{G,DIN}has an average of 80.6 kJ/m

^{2}mm/h, which is 47.8% higher than the average value of R

_{EA}and 39% higher than the average of R

_{YW,DIN}. At 107 of 110 stations the rain gauges show higher R-factors than the corresponding pixels of the radar climatology. The average R-factor difference for all point-pixel pairs amounts to 20.5 kJ/m

^{2}mm/h between R

_{YWG,DIN}and R

_{G,DIN}. For the 72 stations operated by DWD, which were used for radar data adjustments, the average difference between R

_{YWG,DIN}and R

_{G,DIN}amounts to 19.1 kJ/m

^{2}mm/h, whereas the average difference at the 38 stations operated by HLNUG is slightly higher with 23.1 kJ/m

^{2}mm/h. Compared to the erosion atlas, all 110 rain gauge stations show higher R values with an average difference of 24.7 kJ/m

^{2}mm/h.

_{RW,Reg}yielded comparable values as R

_{EA}with a slightly lower mean and maximum, significantly lower minimum, but a slightly higher median and standard deviation. For R

_{G,Reg}, all statistical values were slightly higher than for R

_{EA}and R

_{RW,Reg}. Consequently, before scaling, the rain gauge dataset consistently produces the highest R-factors, but the magnitude of the differences is governed by the derivation method. The input dataset has little influence on the statistical characteristics of the outcome when using a regression equation and the major differences between these regression-based derivatives are the spatial resolutions and spatial distributions (see Section 3.2). When grouping all R-factor derivatives by the calculation method—irrespective of input data and spatial extent—the mean of those R-factors derived according to DIN 19708 (without scaling) is 9.1 kJ/m

^{2}mm/h higher than the mean of all R-factors derived using the regression equation. Furthermore, with 15.8 kJ/m

^{2}mm/h, the DIN method group showed on average a 122.2% higher standard deviation than the regression method group (7.1 kJ/m

^{2}mm/h), which underlines the smoothing effect that can be obtained by using a regression equation instead of the event-based method according to DIN 19708. The difference between both methods is particularly well illustrated by the very steep empirical cumulative distribution functions (ECDF) of all regression-based derivatives (see Figure 3).

_{YW,DIN}by 3.8 kJ/m

^{2}mm/h (−6.6%). The minimum did not change, while the maximum decreased by 27.1 to 146.1 kJ/m

^{2}mm/h (see Figure 2 and Figure 3). Taking into account only the pixels with cropland and rain gauges, the count was reduced to 54 (a total of 54 rain gauges are located in radar pixels with cropland), the average R-factor (R

_{YWG,DIN,Agri}) decreased also by 3.8 to 56.3 kJ/m

^{2}mm/h and the maximum decreased by 12.5 to 92.2 kJ/m

^{2}mm/h. For R

_{G,DIN}, the impact of the data selection on the statistical distribution is considerably higher due to the smaller sample size. Its average decreased by 6.1 to 74.5 kJ/m

^{2}mm/h, whereby the maximum decreased by 42.5 to 114 kJ/m

^{2}mm/h when selecting only pixels with cropland. Consequently, the removal of many high erosivity values in the mountainous regions (see Figure A1), for which the uncertainty and underestimation of the radar data is particularly high, leads to a slightly better agreement of the R-factors calculated according to DIN 19708 from RADKLIM and rain gauge data. Grouping the nine R-factor derivatives based on R

_{YW,DIN}, R

_{RW,Reg}and R

_{EA}by spatial extent resulted in a mean of 55.2 kJ/m

^{2}mm/h for all pixels of the study area, 52.9 kJ/m

^{2}mm/h for pixels with cropland and 56.3 kJ/m

^{2}mm/h for pixels with a rain gauge.

_{YW,F}is 8.8 kJ/m

^{2}mm/h higher than R

_{G,F}(see Table 2). In comparison to R

_{EA}, both R-factors were significantly higher after scaling. On average, R

_{YW,F}was 65.3% higher and R

_{G,F}was 58.5% higher than the R-factor R

_{EA}of the erosion atlas. Although the correction factor proposed by Panagos et al. [23] reduces the R-factor to a level close to R

_{YW,DIN}, R

_{G,P}still showed an 18.2% higher mean than R

_{EA}. Irrespective of the dataset used for derivation and the application of correction procedures, an increase of the R-factor compared to R

_{EA}can thus be determined without doubt.

#### 3.2. Spatial Distribution

_{YW,DIN}occur in the north of Hesse, around the West Hesse Depression, in an area for which no radar measurements were available during some months of the years 2007 and 2014 due to radar hardware upgrades. The average value of the annual R-factor without these two years shows that the minimum is nevertheless located in this area. This is therefore in accordance with the R-factor R

_{EA}(calculation based on regression), which also shows a minimum in this area (see Figure 4). The areas of relatively low R-factors northwest of Fulda and in the Upper Rhine Plain correspond well in both datasets, too. In the north-east of Hesse, however, the newly calculated R-factor R

_{YW,DIN}showed slightly lower erosivity over a large area with a similar spatial distribution. Both datasets showed an increase of the R-factor with increasing terrain height, whereby R

_{YW,DIN}showed significantly higher values over a large area, especially in the Odenwald, Taunus, Westerwald and at Vogelsberg. However, at Vogelsberg, a weakness of the radar climatology to correctly quantify precipitation at higher altitudes was evident as the increase of the R-factor in the lower slope areas was considerably higher than in the summit area. In the area of Wetterau, a negative spoke of the Frankfurt Radar was clearly visible in R

_{YW,DIN}and all other R-factors derived from RADKLIM. Still, in this area an increase of the R-factor compared to the erosion atlas can be seen in most of the grid cells, in some places even up to 45% (see Figure 5a).

_{G,DIN}in the entire study area, which has already been indicated in the previous section.

#### 3.3. Derivation of Updated Regression Equations

_{YW,DIN}, which are mainly located in the area of the radar gap in northern Hesse, are still below the regression line from the erosion atlas. For R

_{G,DIN}, however, all data points are above. Consequently, for the period 2001–2016, the regression equation used in the erosion atlas provides a value deviating from the R-factor according to DIN 19708 for all of the rain gauges.

_{YW,DIN}in the range between 400 and 500 mm summer precipitation. These are only included in the R-factor of the entire radar climatology dataset, but are not significantly reflected in the regression due to their relatively small number. Therefore, it can be assumed that extraordinarily intensive individual events have a strong impact due to the comparatively short time series. These events could only be detected by the high spatial resolution of the radar climatology and are not included in the rain gauge dataset.

_{Su}) with the summer precipitation sums of the RADKLIM RW product for the federal state of Hesse leads to a R-factor value range between 29.7 and 123.9 kJ/m

^{2}mm/h with an average of 73.2 kJ/m

^{2}mm/h. It has thus a significantly lower maximum than all event-based R-factor derivatives. Its mean value is slightly lower than that of R

_{G,DIN}(80.6 kJ/m

^{2}mm/h) due to the slight overall underestimation of precipitation by the radar climatology, and lies approximately in the centre between the averages of R

_{YW,DIN}(58 kJ/m

^{2}mm/h) and the corrected R

_{YW,F}(90.1 kJ/m

^{2}mm/h).

## 4. Discussion

_{EA}which is currently used in the technical information system erosion atlas Hesse [13] can be determined without doubt. This result highlights the need of updated R-factor methods for consultation and planning in Hesse.

_{G,F}is regarded as a correct reference for validation, the correction applied for R

_{YW,F}and R

_{YWG,F}appears somewhat too high, especially when looking at Figure 2. When considering the identical sample size and the largely consistent location of the point-pixel data pairs of R

_{YWG,F}, the advantage of the radar and the fact that more events tend to be recorded hardly matters. However, the median of R

_{YWG,F}almost corresponds to the third quartile of R

_{G,F}. Here, a direct transferability of the correction factors, which were derived from a four-year series of measurements of 12 rain gauges within one square kilometre in Bavaria [22], may be limited. Further research efforts and measurements to extend these time series and derive correction factors of higher spatial representativity from more than one single raster cell would have the potential to significantly reduce the uncertainty when using radar climatology data—not only for rainfall erosivity estimation but for applications related to heavy rainfall in general.

_{Su}) has a high transferability for most of Germany. However, for federal states in northern and eastern Germany which have a more maritime or continental climate, regional regression equations should be calculated from recent local rain gauge data.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**R-factor percentage change of R

_{YW,DIN,Agri}against R

_{EA,Agri}(

**a**), and percentage change of R

_{YW,F,Agri}against R

_{EA,Agri}(

**b**).

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**Figure 1.**(

**a**) Location, height above sea level [m] and selected landscape units of the federal state of Hesse, (

**b**) spatial distribution of cropland areas in the study area.

**Figure 2.**Boxplots of all R-factor derivatives grouped by spatial extent. In the lower subplots, the average of the rain gauges (R

_{G,DIN}) and the rain gauges in pixels with cropland (R

_{G,DIN,Agri}) have been added as a ground-truth reference. See Table 1 for explanation of the used abbreviations.

**Figure 3.**Empirical cumulative distribution functions (ECDF) for all spatially highly resolved R-factor derivatives. The ECDFs for the rain gauges (R

_{G,DIN}) and the rain gauges in pixels with cropland (R

_{G,DIN,Agri}) have been added as a ground-truth reference.

**Figure 4.**R-factor comparison between R

_{YW,DIN}, R

_{G,DIN}(

**a**), R

_{EA}(

**b**), R

_{RW,Reg}and R

_{G,Reg}(

**c**).

**Figure 5.**R-factor percentage change of R

_{YW,DIN}against R

_{EA}(

**a**), scaled R-factors R

_{YW,F}and R

_{G,F}(

**b**) and percentage change of R

_{YW,F}against R

_{EA}(

**c**).

**Figure 6.**Comparison of regression models between different R-factors and the respective mean summer precipitation sums.

Name | Derivation Method | Input Dataset | Spatial Extent | n |
---|---|---|---|---|

R_{YW,DIN} | DIN 19708 | RADKLIM YW (5 min) | All radar pixels in Hesse (1 × 1 km) | 23,320 |

R_{YW,DIN, Agri} | DIN 19708 | RADKLIM YW (5 min) | Radar pixels containing ≥ 10 ha of cropland | 11,555 |

R_{G,DIN} | DIN 19708 | Rain gauge data (5 min) | All rain gauges | 110 |

R_{YWG,DIN} | DIN 19708 | RADKLIM YW (5 min) | Pixels containing a rain gauge | 110 |

R_{EA} | $0.141\xb7\text{}{N}_{Su}-1.48$ | Interpolated rain gauge data (1971–2000) | 1 × 1 km grid for Hesse | 23,320 |

R_{EA,Agri} | $0.141\xb7{N}_{Su}-1.48$ | Interpolated rain gauge data (1971–2000) | Grid cells containing ≥ 10 ha of cropland | 11,555 |

R_{RW,Reg} | $0.141\xb7\text{}{N}_{Su}-1.48$ | RADKLIM RW (1 h) | All radar pixels in Hesse | 23,320 |

R_{RW,Reg,Agri} | $0.141\xb7{N}_{Su}-1.48$ | RADKLIM RW (1 h) | Radar pixels containing ≥ 10 ha of cropland | 11,555 |

R_{G,Reg} | $0.141\xb7\text{}{N}_{Su}-1.48$ | Rain gauge data | All rain gauges in Hesse | 110 |

R_{RWG,Reg} | $0.141\xb7\text{}{N}_{Su}-1.48$ | RADKLIM RW (1 h) | Pixels containing a rain gauge | 110 |

R_{EAG} | $0.141\xb7\text{}{N}_{Su}-1.48$ | Interpolated rain gauge data (1971–2000) | Grid cells containing a rain gauge | 110 |

R_{YW,F} | ${R}_{DIN}\xb7\left(\left(1.13+0.35\right)\xb71.05\right)$ | RADKLIM YW (5 min) | All radar pixels in Hesse | 23,320 |

R_{YW,F,Agri} | ${R}_{DIN}\xb7\left(\left(1.13+0.35\right)\xb71.05\right)$ | RADKLIM YW (5 min) | Radar pixels containing ≥ 10 ha of cropland | 11,555 |

R_{G,F} | ${R}_{DIN}\xb71.05$ | Rain gauge data | All rain gauges | 110 |

R_{YWG,F} | ${R}_{DIN}\xb7\left(\left(1.13+0.35\right)\xb71.05\right)$ | RADKLIM YW (5 min) | Pixels containing a rain gauge | 110 |

R_{G,P} | ${R}_{DIN}\xb70.7984$ | Rain gauge data | All rain gauges | 110 |

R-factor | n | Method | Data Source | Mean | Standard Deviation | Min | Median | Max |
---|---|---|---|---|---|---|---|---|

R_{YW,DIN} | 23,320 | DIN 19708 | RADKLIM | 58.0 | 14.7 | 28.8 | 54.6 | 173.2 |

R_{YW,DIN,Agri} | 11,555 | DIN 19708 | RADKLIM | 54.2 | 12.0 | 28.8 | 52.3 | 146.1 |

R_{G,DIN} | 110 | DIN 19708 | Gauges | 80.6 | 20.6 | 53.4 | 75.3 | 157.2 |

R_{YWG,DIN} | 110 | DIN 19708 | RADKLIM | 60.1 | 15.8 | 31.0 | 57.8 | 104.7 |

R_{EA} | 23,320 | Regression | Erosion atlas | 54.5 | 6.6 | 42.1 | 52.8 | 81.8 |

R_{EA,Agri} | 11,555 | Regression | Erosion atlas | 52.8 | 5.3 | 42.1 | 51.7 | 81.0 |

R_{RW,Reg} | 23,320 | Regression | RADKLIM | 53.2 | 6.8 | 32.8 | 53.0 | 77.0 |

R_{RW,Reg,Agri} | 11,555 | Regression | RADKLIM | 51.9 | 6.4 | 32.8 | 52.1 | 71.4 |

R_{G,Reg} | 110 | Regression | Gauges | 57.0 | 8.8 | 44.7 | 55.0 | 84.7 |

R_{RWG,Reg} | 110 | Regression | RADKLIM | 53.1 | 7.8 | 35.9 | 52.4 | 73.0 |

R_{EAG} | 110 | Regression | Erosion atlas | 55.9 | 8.1 | 45.2 | 53.7 | 81.8 |

R_{YW,F} | 23,320 | DIN scaled | RADKLIM | 90.1 | 22.8 | 44.5 | 84.8 | 269.1 |

R_{YW,F,Agri} | 11,555 | DIN scaled | RADKLIM | 84.2 | 18.6 | 44.5 | 81.3 | 227.0 |

R_{G,F} | 110 | DIN scaled | Gauges | 84.6 | 21.6 | 56.1 | 79.1 | 165.1 |

R_{YWG,F} | 110 | DIN scaled | RADKLIM | 93.4 | 24.6 | 48.0 | 89.8 | 162.7 |

R_{G,P} | 110 | DIN scaled | Gauges | 64.4 | 16.4 | 42.6 | 60.1 | 125.5 |

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**MDPI and ACS Style**

Kreklow, J.; Steinhoff-Knopp, B.; Friedrich, K.; Tetzlaff, B.
Comparing Rainfall Erosivity Estimation Methods Using Weather Radar Data for the State of Hesse (Germany). *Water* **2020**, *12*, 1424.
https://doi.org/10.3390/w12051424

**AMA Style**

Kreklow J, Steinhoff-Knopp B, Friedrich K, Tetzlaff B.
Comparing Rainfall Erosivity Estimation Methods Using Weather Radar Data for the State of Hesse (Germany). *Water*. 2020; 12(5):1424.
https://doi.org/10.3390/w12051424

**Chicago/Turabian Style**

Kreklow, Jennifer, Bastian Steinhoff-Knopp, Klaus Friedrich, and Björn Tetzlaff.
2020. "Comparing Rainfall Erosivity Estimation Methods Using Weather Radar Data for the State of Hesse (Germany)" *Water* 12, no. 5: 1424.
https://doi.org/10.3390/w12051424