The topographical factors of RUSLE are the slope steepness factor (S) and a slope length factor (L). The S factor is generally computed by the continuous function of Nearing (1997): S=1.5+171+e(2.3-6.1∗sin⁡θ). And the L factor is computed according to Renard et al. (1997): L=l22.13m, where m=F1+F and F=(sin⁡θ/0.0896)3×sin⁡θ0.8+0.56, in which θ is the slope and l is the slope length in metres.

As seen in Eqs. (1)–(3), slope is a crucial parameter and thus an accurate estimation is essential in deriving accurate estimates of the L and S factors and soil erosion rates. For an accurate estimation of the slope, input elevation data from digital elevation models (DEMs) should capture the detailed spatial variability in elevation. However, global DEMs are often too coarse to capture the detailed topography because of the surface smoothening effect. To account for this problem it is assumed that topography is fractal. Following Klinkenberg and Goodchild (1992) and Zhang et al. (1999), slope can be expressed as a function of the spatial scale by applying the variogram equation. The variogram equation is used to approximate the fractal dimension of topography and is expressed as follows: Zp-Zq2=kdpq4-2D, so that |Zp-Zq|dpq=αdpq1-D, where Zp and Zq are the elevations at points p and q, dpq is the distance between p and q, k is a constant, α=k0.5, and D is the fractal dimension. Because the left side of Eq. (5) represents the slope, it can be assumed that the slope (θ) is related to the spatial scale or the grid size (d) in θ=αd1-D. This result implies that by calculating the fractal properties (D and α) Eq. (6) can be used to calculate slope at any specified d. The local fractal dimension (D) describes the roughness of the topography while the local value of α is related to the concept of lacunarity, which is a measure of the size of “gaps” (valleys and plains) in the topography (Zhang et al., 2002). To estimate the spatial variations of D and α, Zhang et al. (1999) proposed to relate these parameters to the standard deviation of elevation. Hereby it is assumed that the standard deviation of elevation does not change much with the DEM resolution. D is then calculated as a function of the standard deviation (σ) in a 3 pixel × 3 pixel moving window, as proposed by Zhang et al. (1999): D=1.13589+0.08452ln⁡σ. To estimate α we used the modified approach by Pradhan et al. (2006). They derived α directly from the steepest slope in a 3 pixel × 3 pixel moving window, called αsteepest in the following. Having obtained αsteepest and D from a grid at a given resolution, the scaled slope (θscaled) for a target grid resolution (dscaled) is obtained by θscaled=αsteepestdscaled1-D. Pradhan et al. (2006) also showed that in their case study the ideal target resolution for downscaling slope was 150 m. This is due to the breakdown of the unifractal concept at very fine scales, which was shown to happen at a scale of 50 m. Altogether, this fractal method shows that a high resolution slope can be obtained from a low resolution DEM as is needed by the RUSLE model.

In this study, we investigate the performance of the fractal method on a global scale using different global DEMs as a starting point. The target resolution of downscaling is put to 150 m (about 5 arcsec) according to Pradhan et al. (2006). It should be noted that the spatial scale on which the original RUSLE and USLE models are operating is usually between 10 and 100 m, which indicates that the 150 m target resolution may be still too coarse for a correct representation of slope. The DEMs that are used here are given in Table 1.

As reported in previous studies (Zhang et al., 1999; Chang and Tsai, 1991; Zhang and Montgomery, 1994), the average slope decreases with decreasing DEM resolution. This confirms the expectation of loss of detail in topography at lower DEM resolutions. A large difference is found between the unscaled global average slope from the 5 arcmin and the 30 arcsec DEMs, which is in the order of 0.017 m m-1 or 74 % (Table 2). After applying the fractal method, the scaled slopes at 150 m target resolution from all DEMs increased significantly compared to the unscaled slopes (Fig. 1). However, there is still a difference of about 0.05 m m-1 or 8.5 % between the scaled slopes from the 5 arcmin and the 30 arcsec DEMs (Table 2). This difference can be attributed to several factors. One factor could be the underlying assumption that the standard deviation of elevation (σ) is independent of the DEM resolution. Although σ does not change much when considering different resolutions, there is still a general decrease in mean global σ when going from the 5 arcmin to the 30 arcsec DEM (Table 2). Due to the dependence of the fractal dimension (D) on σ (Zhang et al., 1999), a decrease of σ leads to a decrease in D and therefore an increase in the scaled slope. Other factors that could play a role here are the dependence of αsteepest on the steepest slope, and the breakdown of the fractal method at certain scales and in certain environments. Zhang et al. (1999) mentioned that the scaling properties of slope are affected in very coarse resolution DEMs if σ changes considerably. On the other hand, Pradhan et al. (2006) mentioned the breakdown of the fractal method at very fine scales. This can indicate that the 150 m target resolution is not appropriate for some topographically complex regions in the world or, as addressed by Zhang et al. (1999), the DEMs used in this study are too coarse to scale down the slope to 150 m accurately for these regions.

After applying the fractal method on a 30 arcsec resolution DEM, the scaled slope shows a clear increase in detail, while the unscaled slope shows a strong smoothening effect (Fig. 2a, b). It is found that, after scaling, the slope values range from 0 to 85∘ and are less than 2∘ in 80 % of the area. In contrast, all slope values are less than 45∘ and range between 0 and 2∘ in 89 % of this area when slope is computed directly from the 30 arcsec DEM.

The scaled slope from the 30 arcsec DEM will be used in this study to estimate the global soil erosion rates by the RUSLE model.

Rainfall erosivity (R factor) is described by Hudson (1971) and Wischmeier and Smith (1978) as the result of the transfer of kinetic energy of raindrops to the soil surface. This causes a detachment of soil and the downslope transport of the soil particles, depending on the amount of energy, rainfall intensity, soil type and cover, topography and management (Da Silva, 2004). The original method of calculating erosivity is described by Wischmeier and Smith (1978) and Renard et al. (1997) as R=1n×∑j=1n∑k=1mjEI30k, where n is the number of years of records, mj is the number of storms of a given year j, and EI30 is the rainfall erosivity index of a storm k. The event's rainfall erosivity index EI30 (MJ mm ha-1 h-1) is defined as EI30=I30×∑r=1mervr, where er and vr are, respectively, the unit rainfall energy (MJ ha-1 mm-1) and the rainfall depth (mm) during a time period r, and I30 is the maximum rainfall intensity during a time period of 30 min (mm h-1). The unit rainfall energy, er, is calculated for each time period as er=0.29×1-0.72×e-0.05×ir, where ir is the rainfall intensity during the time period (mm h-1).

The information needed to calculate the R factor according to the method of Wischmeier and Smith (1978) is difficult to obtain on a large spatial scale or in remote areas. Therefore, different studies have been done on deriving regression equations for the R factor (Angulo-Martinez et al., 2009; Meusburger et al., 2012; Goovaerts, 1999; Diodato and Bellocchi, 2010). Most of these studies, however, concentrate on a specific area and can therefore not be implemented on the global scale. Studies on global soil erosion estimation by the RUSLE model or a modified version of it (Doetterl et al., 2012; Van Oost et al., 2007; Montgomery, 2007; Yang et al., 2003) have all used the method of Renard and Freimund (1994). Renard and Freimund (1994) related the R factor to the total annual precipitation based on erosivity data available for 155 stations in the USA, shown in the following equations: R=0.0483×P1.61,P≤850 mmR=587.8-1.219×P+0.004105×P2,P>850 mm. To test how this method performs globally, we calculated the R factor according to the method of Renard and Freimund (1994) (Eq. 12) first. Here we used the 0.25∘ resolution annual precipitation data from the Global Precipitation Climatology Centre (GPCC) product (Table 1). Then, we selected three regions to validate the resulting R values and their variability: the USA (EPA, 2001), Switzerland (Meusburger et al., 2012), and the Ebro Basin in Spain (Angulo-Martinez et al., 2009). For these regions, high resolution erosivity data are available from pluviographic data of local meteorological stations across the whole region.

Figure 3 shows that the R values computed with the Renard and Freimund (1994) method strongly overestimate R when compared to the high resolution R data of the selected regions. For the USA the R factor of Renard and Freimund (1994) shows an overall overestimation for the western USA and for a large part of the eastern USA when compared to the high resolution R factor (Table 7, Fig. 3a). In particular, a strong overestimation is seen for the north-west coast of the USA. This region is known to have complex rainfall patterns due to the presence of mountains and high local precipitation intensities with frequent snow fall (Cooper, 2011). It should be noted that the USA is not the best suited case study for testing the R values computed with the Renard and Freimund (1994) method, as this method is based on climate data from stations in the USA. The available high resolution or observed data on the R factor from Switzerland and the Ebro Basin are better suited for an independent validation.

For Switzerland, which has a complex precipitation variability influenced by the relief of the Alps (Meusburger et al., 2012), the R factor of Renard and Freimund (1994) shows a strong overall overestimation when compared to the high resolution R values (Table 7, Fig. 3b). For the Ebro Basin, located in Spain, the observed R data were available for the period 1997–2006 from Angulo-Martinez et al. (2009). Also here the method of Renard and Freimund (1994) overestimates the R factor and is not able to reproduce the high spatial variability of the R data (Table 7, Fig. 3c).

To better represent the R factor on a global scale, the R estimation was based on the updated Köppen–Geiger climate classification (Table 3, Fig. 4). The Köppen–Geiger climate classification is a global climate classification and is based on the vegetation distribution connected to annual cycles of precipitation and temperature (Lohmann et al., 1993). The reason for this approach is that this classification system includes annual cycles of precipitation and is thus indirectly related to precipitation intensity. Based on this, it is possible to derive regression equations for the R factor that are applicable for each individual climate zone of the classification. This provides a basis to calculate the R factor with coarse resolution data on a global scale.

As a basis for deriving the regression equations for the R factor we used high resolution R maps of the USA from the EPA (2001). The USA covers most of the world's climate zones and is also the largest region with available high resolution R data. Linear multiple regression was used to adjust R: log⁡(Ri)=β0+∑j=1nβij×log⁡Xij+εi,for i=1,2,…,n, where X is the independent explanatory variable, j is the number of explanatory variables, β is a constant and ε is the residual.

The regression operates on one or more of the following parameters (Xj): total annual precipitation (GPCC 0.25∘ product), mean elevation (ETOPO 5 DEM), and the simple precipitation intensity index, SDII. It should be mentioned that the SDII was only available on a very coarse resolution of 2.5∘ for certain regions on Earth, such as parts of Europe and the USA. The SDII is calculated as the daily precipitation amount on wet days (≥ 1 mm) in a certain time period divided by the number of wet days in that period. Previous studies that performed regression of R showed that precipitation and elevation were in most cases the only explanatory variables (Meusburger et al., 2012; Mikhailova et al., 1997; Goovaerts, 1999; Diodato and Bellocchi, 2010; Angulo-Martinez et al., 2009). Here, we added to the regression the SDII as it is a simple representation of precipitation intensity, which is an important explaining variable of the R factor. The precipitation and SDII data sets were rescaled to a 5 arcmin resolution (corresponding to 0.0833∘) to match the Köppen–Geiger climate classification data that was available at the resolution of 6 arcmin (corresponding to 0.1∘).

Furthermore, high resolution erosivity data from Switzerland (Meusburger et al., 2012) and annual precipitation from the GPCC 0.5∘ product were used to derive the regression equations for the R factor for the polar (E) climate zones. These climate zones are not present in the USA. For the rest of the climate zones that are not present in the USA it was difficult to obtain high resolution erosivity data. Therefore, we maintained the method of Renard and Freimund (1994) for those climate zones to calculate erosivity. Also, we kept the R factor of the Renard and Freimund (1994) method if no clear improvement of the R factor was found when using the new regression equations for a specific climate zone. Here, we mainly used the r2 combined with the residual standard error to evaluate if the new regression equations showed a clear improvement in the R factor. The Renard and Freimund (1994) R factors where kept for the hot arid climate zone (BWh) and the temperate climate zone with a hot summer (Csa) in the USA. These are just two climate zones out of the 17 evaluated ones, which show that the Renard and Freimund method performs as good as or slightly better than the regression method. All data sets for deriving the R factor are described in Table 1.

Tables 4 and 5 show the resulting regression equations for climate zones for which we found initially a low correlation between the R values calculated by the method of Renard and Freimund (1994) and the high resolution R values from the EPA (2001) and Meusburger et al. (2012). Figure 5 shows for each addressed climate zone how the method of Renard and Freimund (1994) and the new regression equations compare to the high resolution R of the USA. For the cold climate zones with a dry summer (Ds), the new regression equations show only a slight improvement as compared to the method of Renard and Freimund (1994). Also for the polar climate zones (E) the new regression equations still show a significant bias. However, they perform much better compared to the method of Renard and Freimund (1994). For most of the addressed climate zones the SDII explains a large part of the variability in the R factor. The elevation plays a smaller role here. Elevation can be an important explaining variable in regions with a high elevation variability, which then affects the precipitation intensity.

From Tables 4 and 6 it can be concluded that the R factor in climate zones without a dry season (f) can be easily explained by the total annual precipitation and the SDII. Dry climate zones, especially dry summer climate zones, showed a weaker correlation. This is most likely due to the fact that the SDII is too coarse to explain the variability in the low precipitation intensity in the summer. It is also interesting to see that even though the SDII was derived from a very coarse resolution data set, it turned out to be still important for deriving more accurate R values.

We also show for each addressed climate zone a comparison of the newly computed average R factor with the average high resolution R factor, and the uncertainty range (Table 6). The uncertainty range was computed by taking into account the standard deviation of each of the parameters in the regression equations. As mentioned before, the polar climate zones showed the largest uncertainty range. The new regression equations significantly improved the R values and spatial variability in the western USA and lead to an average R factor that was closer to the data mean (Table 7, Fig. 6a). Although the new regression equations show a bias for the polar climate zones (the minimum and maximum R values are not captured), the resulting mean R values for Switzerland show a strong improvement (Table 7, Fig. 6b).

Furthermore, the variability in the estimated R factor compares well with the variability of the high resolution R factor. It should be noted that Switzerland is not an independent case study for the polar climate zones, as the high resolution R values from this case study were used in our regression analysis. However, the Ebro Basin case study confirms the strong improvement for the polar climate zones (Fig. 6c). As the high resolution R values of the USA and Switzerland were used to derive the regression equations, the third case study, the Ebro Basin in Spain, provided an important independent validation. For the Ebro Basin, the new regression equations not only improve the overall mean but also capture the minimum R values better. This resulted in an improved representation of the R variability (Table 7, Fig. 6c). In Fig. 6c, however, there is a clear pattern separation in the newly computed R values, which is due to the fact that the SDII data are not available for part of the Ebro Basin. As mentioned before, SDII is an important explaining parameter in the regression equations for most of the addressed climate zones.

Figure 7a shows the global patterns of the estimated R factor from the method of Renard and Freimund (1994) and the new regression equations. Figure 7b shows a difference plot between the estimated R factor with the method of Renard and Freimund (1994) and the R factor estimated with the new regression equations. The new regression equations significantly reduced the R values in most regions. However, the tropical regions still show unrealistic high R values (maximum R values go up to 1×105 MJ mm ha-1 h-1 year-1). This is because the R factor was not adjusted for the tropical climate zones due to the lack of high resolution R data. Oliveira et al. (2013) found for the R factor in Brazil that the maximum R values for the tropical climate zones reach 22 452 MJ mm ha-1 h-1 year-1. We find R values in Brazil that exceed this maximum R value found by Oliveira et al. (2013).

Finally, it should be noted that the purpose of the adjusting methods for the S and R factors in this study is to capture more accurately the large-scale mean erosion rates rather than the extremes. Therefore, even though the new regression equations are still not accurate enough for certain climate zones, it is important that the average R factor is represented well. The approach for adjusting the R factor also showed that although there is no high temporal resolution precipitation intensity data available on a global scale, the R factor can still be represented well for most climate zones on a large spatial scale. This can be done by using other parameters, such as elevation, and especially one representative of precipitation intensity, such as the SDII. The SDII played an important role here as it improved the estimation of the R factor significantly, even though data was only available at a very low resolution as compared to the other data sets of precipitation, elevation and climate zone classification.

In the following we demonstrate the consequences of the new parameterizations of the S and R factors for global soil erosion rates. First, we compute the other individual RUSLE factors, soil erodibility (K) and crop cover (C). Estimations of the K factor were based on soil data from the gridded 30 arcsec Global Soil Data set for use in Earth System Models (GSCE). GSCE is based on the Harmonized World Soil database (HWSD) and various other regional and national soil databases (Shangguan et al., 2014). We used the method of Torri et al. (1997) to estimate the K factor, and gave volcanic soils a K factor of 0.08 t ha h ha-1 MJ-1 mm-1. This is because these soil types are usually very vulnerable to soil erosion, and the observed K values are beyond the range predicted by the method of Torri et al. (1997) (Van der Knijff et al., 1999). To account for the effect of stoniness on soil erosion we used a combination of the methods by Cerdan et al. (2010) and Doetterl et al. (2012), who based their methods on the original method of Poesen et al. (1994). For non-agricultural areas we used the method of Cerdan et al. (2010), where they reduced the total erosion by 30 % for areas with a gravel percentage larger or equal to 30 %. For agricultural and grassland areas we used the method of Doetterl et al. (2012), where erosion was reduced by 80 % in areas where the gravel percentage exceeded 12 %.

We calculated the C factor according to the method of De Jong et al. (1998), using 0.25∘ normalized difference vegetation index (NDVI) and land use data for the year 2002. An important limitation of this method is the fact that in winter the C factor is estimated too high (Van der Knijff et al., 1999). This is because the method does not include the effects of mulch, decaying biomass and other surface cover reducing soil erosion. To prevent the C factor from being too high, maximum C values for forest and grassland of 0.01 and 0.05 for pasture were used. Doetterl et al. (2012) showed that the slope length (L) and support practice (P) factors do not contribute significantly to the variation in soil erosion at the continental scale to global scale, when compared to the contribution of the other RUSLE factors (S, R and C). However, this does not mean that their influence on erosion should be ignored completely. They may play an important role in local variation of erosion rates. In our erosion calculations we do not include these factors because we have too little or no data of these factors on a global scale. Including them in the calculations would only add an additional large uncertainty to the erosion rates. This would make it more difficult to judge the improvements we made to the S and R factors.

We applied the RUSLE model with the settings mentioned in the previous paragraph at a 5 arcmin resolution on a global scale for the present time period (see time resolutions of data sets in Table 1). We calculated global soil erosion rates with four different versions of the RUSLE model: (a) the unadjusted RUSLE, (b) RUSLE with only an adjusted S factor, (c) RUSLE with only an adjusted R factor, and (d) the adjusted RUSLE (all adjustments included).

We found a global average soil erosion rate for the adjusted RUSLE of 6.5 t ha-1 year-1 (Fig. 8a). When including the uncertainty arising from applying the linear multiple regression method, the mean global soil erosion rate differs between 5.3 and 15 t ha-1 year-1. Furthermore, the RUSLE version with only an adjusted S factor shows the highest average global soil erosion rate, while the lowest rate is found for the RUSLE version with only the adjusted R factor (Table 8). Figure 8c shows the difference between the erosion rates of the S-adjusted RUSLE and the unadjusted RUSLE versions. The erosion rates are in general increased here and mostly pronounced in mountainous regions. This feature is “dampened” when adjusting the R factor. The difference between the R-adjusted RUSLE and unadjusted RUSLE versions (Fig. 8d) shows that the erosion rates are overall decreased in regions where the adjustments are made. When the erosion rates of the unadjusted RUSLE model are subtracted from the fully adjusted RUSLE model (Fig. 8b), we find that erosion rates are slightly decreased in areas where the R factor is adjusted. However, for the tropics an increase in erosion rates is found in the fully adjusted RUSLE due to the lack of adjusting the R factor there. This indicates that these two factors balance each other, and that it is important to have a correct representation of all the RUSLE factors on a global scale in order to predict reliable erosion rates.

In this study the K and C factors are not tested and adjusted for a coarse resolution at global scale and thus validation with existing empirical databases on soil erosion is not fully justified. However, to test if the global erosion rates are in an acceptable range, they are compared to erosion estimates from the NRI database for the USA and erosion estimates from the study of Cerdan et al. (2010) for Europe. These are to our knowledge the only large-scale high resolution empirical databases on soil erosion.

The NRI database contains USLE erosion estimates for the year 1997, which are available at the Hydrologic Unit Code 4 (HUC4) watershed level. We aggregated the resulting erosion rates from the adjusted and unadjusted RUSLE models to the HUC4 watershed level. The results show that the average erosion rates from the adjusted RUSLE model come closer to that of the NRI database (Table 9, Fig. 9a). However, the maximum average HUC4 soil erosion rate from the adjusted RUSLE is somewhat higher compared to the NRI database. From these results we can conclude that the erosion rates of the adjusted RUSLE fall in the range of observed values but that there are still some local overestimations. Some of these overestimations can be found in the south-west of the USA where the adjusted RUSLE shows a slightly worse performance compared to the unadjusted RUSLE. The R factor in this region was not changed as it was already estimated well by the method of Renard and Freimund (1994), however, the S factor increased due to the hilly terrain. Without adjusting the other RUSLE factors (K and C), this resulted in an overall increase in soil erosion rates. This indicates that the other RUSLE factors may play an important role in this region. Furthermore, we see that along the west coast of the USA the erosion values are not much improved with the adjusted RUSLE model. This is mainly because some climate zones such as the temperate climate zone with a dry and warm summer (Csb) prevail in this region, for which the R factor is still difficult to estimate in a correct way (Table 4).

For Europe, Cerdan et al. (2010) used an extensive database of measured erosion rates on plots under natural rainfall. They extrapolated measured erosion rates to all of Europe (European Union area) and adjusted them with a topographic correction. This correction was based on the L and S factors of the RUSLE model. They also applied a correction to account for soil stoniness. For comparison, the soil erosion rates from Cerdan et al. (2010) and the RUSLE estimates in our study are aggregated at country level. The performance of the adjusted RUSLE model was not as good for Europe as compared to the USA. This is not surprising as the RUSLE model is based on soil erosion data of the USA. However, also on the European scale the adjusted RUSLE model performed better than the unadjusted RUSLE model (Table 9, Fig. 9b). In particular, the large erosion rates in the south of Europe as observed in the results of the unadjusted RUSLE model are less extreme in the adjusted RUSLE model. Still, the overall average erosion rate for Europe is overestimated by approximately 2 times (Table 9).

The biases in erosion rates as seen for the south-west of the USA and southern Europe can be attributed to several factors. As mentioned before, the other RUSLE factors (K and C) and the way they interact with the R and S factors are not adjusted to the coarse resolution at global scale. We found no clear signal for the land cover types with which the adjusted RUSLE performs better or worse. In general, we can see that the adjusted RUSLE model still overestimates erosion rates for most land cover types. A short analysis for Europe showed that the largest biases are found for shrubs and the lowest for grassland. However, a more explicit analysis is needed to find out how we can improve the contribution of land cover and land use to erosion rates in the RUSLE model. Explicitly including the interaction between the C and R factor on a monthly timescale could be crucial. This is very important for example in areas with agriculture and areas with a strong seasonal character. Another aspect related to improving the C factor is looking at the location of land use in a certain grid cell. If the land use in a grid cell is located on steep slopes, the resulting erosion in that grid cell would be higher than when it was located in the flatter areas. In this study, however, only mean fractions of land cover and the NDVI are used for each grid cell. This can lead to possible biases in the resulting erosion rates.

Furthermore, land management is not accounted for in this study, which could introduce an important systematic bias in the soil erosion rates especially for agricultural areas. Land management is represented by the P factor in the original USLE; however, it is partly also incorporated in the C factor for agricultural land use through plant residues, cover crops and tillage. A limitation of the NDVI approach to estimate the C factor lies therefore in the inability to estimate this land management effect. Applying this method also limits the interaction between the R and C factors on a monthly to seasonal scale, because this interaction is partly based on land management.

Furthermore, uncertainties in the coarse resolution land cover/land use, soil and precipitation data sets that are not accounted for can lead to the model biases. Also, better adjustment of the R factor for climate zones such as the polar climates could help improve the overall results. Some biases in the erosion rates can also be attributed to the fact that stepped relief, where flat plateaus are separated by steep slopes, is not well captured by the 150 m target resolution used in the fractal method to scale slope. In this way erosion would be overestimated in these areas. Finally, errors and limitations in the observational data sets can also contribute to the differences between model and observations. The study of Cerdan et al. (2010) on Europe, for example, used extrapolation of local erosion data to larger areas, which could introduce some biases. Also, the underlying studies on measured erosion rates used different erosion measuring techniques that can be linked to different observational errors.