## Figures

## Abstract

Correlation analysis is popular in erosion- or earth-related studies, however, few studies compare correlations on a basis of statistical testing, which should be conducted to determine the statistical significance of the observed sample difference. This study aims to statistically determine the erosivity index of single storms, which requires comparison of a large number of dependent correlations between rainfall-runoff factors and soil loss, in the Chinese Loess Plateau. Data observed at four gauging stations and five runoff experimental plots were presented. Based on the Meng’s tests, which is widely used for comparing correlations between a dependent variable and a set of independent variables, two methods were proposed. The first method removes factors that are poorly correlated with soil loss from consideration in a stepwise way, while the second method performs pairwise comparisons that are adjusted using the Bonferroni correction. Among 12 rainfall factors, *I*_{30} (the maximum 30-minute rainfall intensity) has been suggested for use as the rainfall erosivity index, although *I*_{30} is equally correlated with soil loss as factors of *I*_{20}, *EI*_{10} (the product of the rainfall kinetic energy, *E*, and *I*_{10}), *EI*_{20} and *EI*_{30} are. Runoff depth (total runoff volume normalized to drainage area) is more correlated with soil loss than all other examined rainfall-runoff factors, including *I*_{30}, peak discharge and many combined factors. Moreover, sediment concentrations of major sediment-producing events are independent of all examined rainfall-runoff factors. As a result, introducing additional factors adds little to the prediction accuracy of the single factor of runoff depth. Hence, runoff depth should be the best erosivity index at scales from plots to watersheds. Our findings can facilitate predictions of soil erosion in the Loess Plateau. Our methods provide a valuable tool while determining the predictor among a number of variables in terms of correlations.

**Citation: **Zheng M, Chen X (2015) Statistical Determination of Rainfall-Runoff Erosivity Indices for Single Storms in the Chinese Loess Plateau. PLoS ONE 10(3):
e0117989.
https://doi.org/10.1371/journal.pone.0117989

**Academic Editor: **Vanesa Magar, Centro de Investigacion Cientifica y Educacion Superior de Ensenada, MEXICO

**Received: **June 16, 2014; **Accepted: **January 6, 2015; **Published: ** March 17, 2015

**Copyright: ** © 2015 Zheng, Chen. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited

**Data Availability: **All Data are deriven from the book “Yellow River Water Conservancy Commission, Ministry of Water Conservancy and Electric Power, PRC (1961–1971). Observed Data of Rainfall, Runoff and Sediment in the Zizhou Experimental Office over the Period 1959–1969.”, which is available on application at http://loess.geodata.cn/Portal/metadata/listMetadata.jsp?category=1160. Thanks to the Data Sharing Infrastructure of Earth System Science—Data Sharing Infrastructure of Loess Plateau for providing the data.

**Funding: **The research is funded by the National Natural Science Foundation of China (41271306;http://www.nsfc.gov.cn/) and the Non-profit Industry Financial Program of MWR (201201083; http://www.mwr.gov.cn). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

**Competing interests: ** The authors have declared that no competing interests exist.

## Introduction

Rainfall erosivity indicates the potential of a storm to erode soil. A single index of rainfall erosivity that can measure the composite effect of various rainstorm characteristics on soil erosion is highly desirable for predicting soil loss [1, 2]. It is well known that soil losses are frequently due to a few intense rainfall events [1, 3]. The most common erosivity index for single storms is the *EI*_{30} index (the product of the rainfall kinetic energy, *E*, and the maximum 30-min intensity, *I*_{30}), as is used in the Universal Soil Loss Equation (USLE) [4] and in the Revised Universal Soil Loss Equation (RUSLE) [5]. The calculation of *EI*_{30} is of high data requirements and labor intensive [2, 6]. For this reason, a large number of studies (e.g. [1, 2, 6, 7]) were devoted to developing a proxy, using more readily available data such as daily, monthly and annual precipitations, for *EI*_{30} and the *R* factor of the USLE (the mean annual total of *EI*_{30}).

Besides the *EI*_{30} index, other forms of erosivity index for storm events primarily include the *KE* > 25 index (the total kinetic energy for rainfall duration with intensity exceeding 25 mm h^{-1}) [8], the *PI*_{m} index (the product of the rainfall amount, *P*, and the peak rainfall intensity, *I*_{m}) [9], the *I*_{x}*E*_{A} index (the product of the excess rainfall rate, *I*_{x}, and the rainfall kinetic energy flux, *E*_{A}) [10], and the so-called *A* index [11]. Many local erosivity indices for single storms have also been used, such as *I*15 in Belgium [12], *EI*5 in NE Spain [13], *E* in Palestinian areas [14] and *EI*_{60} and *I*60 in Malaysia [15]. In the Chinese Loess Plateau, both *Wang* [16] and *Jia et al*. [17] suggested *EI*_{10} as the rainfall erosivity index; however, *EI*_{10} has been shown to be of similar effectiveness as *EI*_{30} [18]. Notably, *Chen et al*. [19] detected no significant difference among correlations between soil loss and a set of *EI*_{t} variables (*EI*_{10}, *EI*_{20}, *EI*_{30}, *EI*_{40}, *EI*_{50} and *EI*_{60}). Furthermore, it was found that *EI*_{t} did not greatly improve soil loss predictions compared with *PI*_{t} [19–21], which can thus serve as a surrogate for *EI*_{30}.

Both raindrops and runoff are drivers of soil erosion [22]. Runoff factors, mainly runoff volume and peak discharge, have frequently been included into an erosivity index [23–28]. The addition of runoff terms can improve the ability of models to predict soil loss or sediment yield especially for small to medium events [25, 27, 29]. Foster et al. [25] found that the lumped erosivity factors including rainfall amount, rainfall intensity and runoff amount performs better than *EI*_{30}, whereas erosivity factors with separate terms for rainfall and runoff erosivity performs best. However, Foster et al. [25] acknowledged their inability to determine whether the observed improvements were statistically significant or not. In the Modified Universal Soil Loss Equation (MUSLE), the *EI*_{30} index is replaced by a power of the product of runoff volume and peak discharge [24], as is also used in the Agricultural Policy/Environmental eXtender (APEX) model [28]. In another modified version of the USLE called the USLE-M [27], the erosivity index of single events is the product of *EI*_{30} and the runoff ratio. In the Loess Plateau, many studies included both terms of runoff volume and peak discharge, in a lumped or separated form, into models for predicting soil loss or sediment yield [30–32]. Nevertheless, it is well known that runoff volume alone can adequately predict sediment yield of the flood event in the Loess Plateau (*r*^{2} > 0.9) [33]. A proportional model of event runoff volume and sediment yield applies well over a wide range of spatial scales from hill slopes to large-sized watersheds [34–35].

To determine the erosivity index, a large number of correlations between rainfall-runoff factors and soil loss often need to be compared (e.g. [14–17, 23, 25, 26, 36–37]). For example, the *EI*_{30} index was established as the rainfall erosivity index of the USLE by comparing correlations of more than 40 factors with soil loss [38–40]. As a result of the indelible sampling error, sample correlation coefficients can never be identical to population ones. Because the sample difference does not fully represent the population difference, a statistical test is needed to determine the significance of the observed sample difference. To our knowledge, no studies have applied statistical tests while determining the erosivity index with the exceptions of [12] and [19], although a number of statistical tests for comparing dependent or independent correlations exist [41].

The object of this study is to determine erosivity indices for single storms on a statistical basis using data observed in the Chinese Loess Plateau. After describing the study area and data source, we present two methods that compare a large number of dependent correlations. The two methods build on the Meng’s tests [42], which has been widely used to compare correlations in psychological research. We then determine the rainfall-runoff erosivity indices among a large number of factors using the two methods. We finally made some discussions about rainfall and runoff factors with an emphasis on the best erosivity index in the Loess Plateau.

## Study Area and Data

The present study uses data observed at 5 runoff experimental plots and 4 gauging stations (Tables 1 and 2) within the Dalihe River watershed (See Fig. 1 in [35] for the location), a secondary-order river of the middle Yellow River. Typical of the Loess Plateau, the loess mantle of the Dalihe watershed is generally thicker than 100 m. The climate is typically semiarid with a mean annual precipitation of 440 mm (1960–2002). Soil erosion is primarily caused by localised short-duration, high-intensity convective rainstorms. A single storm can commonly cause a soil loss of greater than 10 000 t km^{-2}. Most of the lands were intensively cultivated with little soil conservation practices during the monitoring period (1959–1969). The terrain is very precipitous and deeply dissected.

All examined plots are located within the Tuanshangou subwatershed (latitude 37°41′N, longitude 109°58′E; See Fig. 1(c) in [35] for the location), a headwater basin of the Dalihe watershed. The plots were all under arable with crops varying between years, generally including millet, potato, mung bean, clover, sorghum and wheat. The vegetation cover rarely exceeded 25% in the plots. The recorded maximum *I*_{10} is 2.17 mm min^{-1} (1961–1969). Rill erosion is dominant on upland slopes. During the 1960s, the annual erosion intensity, averaging 41 000 t km^{-2} at Plots 4, 2 and 3 (Table 2), was maximized in 1966. Although rills only occurred on 5 out of 54 rainfall days, these five days contributed almost all of the annual soil loss (>96%) in 1966. Downslope, the valley side slope is generally very steep (>35°), allowing the emergence of permanent, incised gullies and mass wasting events.

Unless stated otherwise, all data used in this study were obtained from the Yellow River Water Conservancy Commission (YRWCC). The YRWCC stream-gauging crews conducted all measurements. Hyetograph data were obtained using a rainfall gauge near Plot 3 (See Fig. 1(c) in [35] for the location). The observation interval was generally smaller than 10 min, even 1 min in many cases of high rainfall intensity. The field monitoring programs of runoff and sediment have been described in detail in [43, 44].

Based on the instantaneous measurement of water discharge and sediment concentration, the runoff depth, *h* (mm: total runoff volume normalized to drainage area), and the specific sediment yield, *SSY* (t km^{−2}) of single events were calculated. Conventionally, the term “soil erosion” is used for hill slopes, and the term “sediment yield” is used for a river system or watershed. For simplicity, we use *SSY* to represent both cases hereafter. The event mean sediment concentration, *SC*_{e} (kg m^{−3}) was computed by dividing *SSY* by *h*. The maximum instantaneous sediment concentration (*SC*_{max}, kg m^{-3}) was also used to represent the level of the sediment concentration of a single event.

The data we used are on a single storm basis. The runoff factors we examined include 4 factors: *h*, *q*_{max} (peak flow discharge normalised for drainage areas, m^{3} s^{-1} km^{-2}), *hq*_{max} (the product of *h* and *q*_{max}) and h+*q*_{max} (the sum of *h* and *q*_{max})_{.} The use of h+*q*_{max} follows [23, 30]. The rainfall factors we examined include 12 factors: *P* (mm), *T* (rainfall duration, min), *I* (mean rainfall intensity, mm min^{-1}), *I*_{10} (mm min^{-1})_{,} *I*_{20} (mm min^{-1}), *I*_{30} (mm min^{-1}), *EI*_{10,} *EI*_{20}, *EI*_{30}, *PI*_{10,} *PI*_{20} and *PI*_{30}. The storm kinetic energy, *E* (J m^{-2}), was calculated as follows:
(1)
where *e*_{r} is the rainfall kinetic energy per unit depth of rainfall per unit area (J m^{-2} mm^{-1}), and *p*_{r} is the depth of rainfall (mm) for the *r*th interval among *m* intervals of the storm hyetograph. *e*_{r} is calculated by an empirical equation building on measurements of the drop size distribution of 195 storms in the Loess Plateau [45]:
(2)
where *i*_{r} (mm min^{-1}) represents the mean rainfall intensity for the *r*th interval. After unit conversion, Equation (3) is almost identical to the rainfall intensity-energy equation of the USLE [4]. The discrepancy between the two equations is less than 10% for rainfall intensities from 1 to 40 cm h^{-1}.

## Methodology

Assumed that *r*_{1} and *r*_{2} represent the correlation coefficients of any two rainfall factors with *SSY*. To compare *r*_{1} and *r*_{2}, *Sinzot et al*. [12] and *Chen et al*. [19] used the following statistic:
(3)
where and are the Fisher z-transformed values for *r*_{1} and *r*_{2}, *N is* the sample size. However, Equation (4) is applicable only to independent correlations [46] and cannot be used to compare *r*_{1} and *r*_{2} because they have a common dependent variable, *SSY*.

Meng’s tests [42] are widely applied for comparing correlations between a dependent variable and a set of independent variables. This study used these tests because they take a rather simple and thus easy-to-use form and perform as well as other statistical tests in terms of controlling the Type I error and power [41]. To compare *r*_{1} and *r*_{2}, a *Z* (standard normal) test (termed “Meng’s Z_{1} test” in the following section for simplicity) is used [42]:
(4)
where *r*_{x} is the correlation between the two rainfall factors under examination,
(5)
(6)
where , and *f* should be set to 1 if the right term of Equation (7) is larger than 1.

If the comparison involves *k* rainfall factors (*k* > 2), the statistic (termed “Meng’s χ^{2} test” hereafter) used to test the heterogeneity of the correlations of the *k* factors with *SSY* is as follows [42]:
(7)
where *z*_{i} is the Fisher z-transformed correlation coefficient for the *i*th rainfall factor (*i* ≤ *k*), and is the mean of the *z*_{i} values. In the definition of *H* given by Equation (5), becomes the mean of the , and *r*_{x} becomes the median intercorrelation among the factors under testing. The resulting χ^{2} statistic is χ^{2} distributed on *k*-1 degrees of freedom.

By comparing a correlation with the average of the *k*-1 other correlations, *Meng et al*. [42] also designed a standard *Z* test (termed “Meng’s Z_{2} test” hereafter) to determine whether a contrast exists among the *k* factors under examination:
(8)
where r_{λx} represents the correlation coefficient between *z*_{i} and λ_{i}. The values of λ_{i} are the contrast weights assigned to each *z*_{i}. The sum of the λ_{i}s must be zero. If we wish to determine whether the first factor differs among four factors in terms of their correlations with *SSY*, for instance, λ_{i}s should be-3, 1, 1 and 1, respectively.

Based on Meng’s tests, we would use two methods to determine the rainfall-runoff erosivity indices. Method one repeatedly uses Meng’s Z_{2} test (Equation (8)) to remove factors that are poorly correlated with *SSY* in a stepwise way. Method two performs all paired comparisons using Meng’s Z_{1} test (Equation (5)). The Type I error, however, would increase when multiple comparisons are conducted simultaneously. We used Hochberg’s Sharpened Bonferroni correction to counteract the problem of multiple comparisons [47, 48]. Given a *p* value resulting from Meng’s Z_{1} test, the corrected value according to the Hochberg approach is *p*’ = *Rp*, where *R* is the rank value in descending order of the given *p* value among all obtained *p* values. The *p* values given below are one-tailed for the Meng’s χ^{2} test, and two-tailed for all other tests.

Using the two methods above, we would determine the rainfall erosivity index among the 12 rainfall factors and the runoff erosivity index among the 4 runoff factors. We limited our analyses of rainfall factors to the six experimental sites in the Tuanshangou subwatershed (#3 in Table 1 and the five plots in Table 2) due to the lack of reliable rainfall data at larger scale. A total of 222 events were used. Events without detailed hyetograph data were excluded. The analyses of runoff factors involve 379 events observed at all nine experimental sites listed in Tables 1 and 2.

## Results

### Method One—a stepwise procedure using Meng’s Z_{2} test

Table 3 presents the correlation coefficients between *SSY* and the 12 rainfall factors we examined. Factors other than *T* are generally well correlated with *SSY* (*p* < 0.01). The obtained correlation coefficients for *T*, *I* and *P* are much smaller than those for *I*_{t} (*I*_{10,} *I*_{20} and *I*_{30}), *EI*_{t} (*EI*_{10,} *EI*_{20} and *EI*_{30}) and *PI*_{t} (*PI*_{10,} *PI*_{20} and *PI*_{30}). The relationship between *I*_{30} and *SSY* for four of the sites was plotted in Fig. 1.

To determine whether a rainfall factor represents a contrast, we performed 12 Meng’s Z_{2} tests at each of the six sites in the Tuanshangou subwatershed. The result (Test 1 in Table 4) shows that the correlation coefficients of *T*, *I* and *P* with *SSY* are significantly smaller (*p* < 0.005) than the average of factors other than itself at every site. Factors of *PI*_{20} and *PI*_{30} are not contrasts (*p* > 0.05) at any site. Hence, these five factors (*T*, *I*, *P*, *PI*_{20} and *PI*_{30}) were excluded as candidates for the erosivity index. Meng’s Z_{2} tests of the seven remaining factors shows that *PI*_{10} at Plot 4 and *I*_{10} at Plot 7, Plot 9 and #3 are less correlated with *SSY* (Test 2 in Table 4; *p* < 0.02). When factors of *I*_{10} and *PI*_{10} were removed (Test 3 in Table 4), no contrast was detected among the five remaining factors of *I*_{20}, *I*_{30}, *EI*_{10,} *EI*_{20} and *EI*_{30} at all sites (*p* > 0.08). The same result holds when using Meng’s χ² test, which returns a quite high *p* value at the six sites (0.96, 0.92, 0.43, 0.90, 0.31 and 0.39, respectively), to examine the heterogeneity of the correlations of the five factors with *SSY*.

The four runoff factors of *h*, *q*_{max}, *h+q*_{max} and *hq*_{max} are all highly correlated with *SSY* (*p* < 0.01) at all nine sites, from plots to watersheds (Table 5). The derived correlation coefficients between *h* and *SSY* are either at the maximum or simply slightly smaller than the maximum (generally < 0.01) at each site. Meng’s Z_{2} tests (Table 6) show that the correlation between *h* and *SSY* is significantly higher than the average of the three remaining factors at seven of nine examined sites (*p* < 0.0006). In contrast, the correlation between *q*_{max} and *SSY* is significantly lower than the average of the three remaining factors at eight sites (*p* < 0.031). Hence, *h* should be preferred to *q*_{max} as the predictive factor of *SSY*. The factor of *h*+*q*_{max} is better correlated with *SSY* than the average of the three remaining factors at four sites (*p* < 0.04), whereas the factor of *hq*_{max} is less correlated with *SSY* than the average of the three remaining factors at six sites (*p* < 0.02). This demonstrates that the combination of *q*_{max} with *h* would impair rather than improve the ability of *h* to predict *SSY*.

### Method Two—multiple Meng’s Z_{1} tests adjusted using the Hochberg approach

Meng’s Z_{2} tests used above have clearly demonstrated that *T*, *I* and *P* are inferior to other factors for application as the erosivity index. To reduce complexity, these factors are not considered in this section.

To test the significance of the difference between correlations of the nine rainfall factors of *I*_{t}, *EI*_{t} and *PI*_{t} with *SSY*, we performed 36 pairwise comparisons using Meng’s Z_{1} test at each of six sites within the Tuanshangou subwatershed. The resultant *p* values, together with the *p*’ values after the Bonferroni correction, are presented in Table 7. Fifty-one among the 216 comparisons produced significant differences (*p* < 0.05) in the absence of the Bonferroni correction, with most (33) involving comparisons between *PI*_{t} and *EI*_{t}. When the Bonferroni correction was performed, significant differences remained for only 9 comparisons (*p*’ < 0.04), all of which involved the comparison between *PI*_{t} and *EI*_{t}. The results adjusted using the Bonferroni correction demonstrate that *EI*_{t} is better correlated with *SSY* than *PI*_{t} in some cases, and the correlations with *SSY* was not significantly different among six factors of *EI*_{t} and *I*_{t} (*p*’ > 0.11).

To compare the strength of correlations between *SSY* and the four runoff factors, we performed six Meng’s Z_{1} tests at each of nine sites within the Dalihe watershed. At seven sites, *h* is more correlated with *SSY* than *q*_{max} (*p* < 0.001, *p*’ < 0.01; Table 8), regardless of whether the Bonferroni correction was applied. Only at Plot 4 was the obtained correlation coefficient between *h* and *SSY* smaller than that between *q*_{max} and *SSY* (Table 5). This difference, however, was not statistically significant (*p* = 0.26, *p*’ = 0.52). A total of 18 comparisons at the nine sites were made between the correlations of *h* and the combined factors of *h* and *q*_{max} (i.e. *h+q*_{max} and *hq*_{max}) with *SSY*. The Meng’s Z_{1} test suggested a significant difference for 11 among the 18 comparisons, and almost all (10) remain significant after the Bonferroni correction (Table 8). Among the ten comparisons, *h* is more correlated with *SSY* for nine (*p* < 0.007, *p*’ < 0.007) and less correlated for only one (*p* < 0.01, *p*’ < 0.01; #12). This again indicates that both *q*_{max} and its combination with *h* are inferior to the single factor of *h* for the *SSY* predictions.

## Discussion

### Rainfall factors

For rainfall factors, the results of two methods slightly differ: Method one suggested that *I*_{20}, *I*_{30}, *EI*_{10,} *EI*_{20} and *EI*_{30} are superior to other factors as a predictor of *SSY*; Method two excessively accepted *I*_{10} as an optimal predictor. This result may be related to the Bonferroni correction, which increases the likelihood of accepting the null hypothesis of identical correlations thereby increasing the risk of committing the type II errors [49].

The Loess Plateau is typically dominated by infiltration excess overland flows, and the runoff yield is determined by rainfall intensity rather than rainfall amount. In the Tuanshangou subwatershed, the median *T* is about 170 min. In contrast, the runoff duration at Plots 4, 2 and 3, with a median value of approximately 16 min, hardly exceeded 40 min. Rainfall during the low-intensity period is thus of little consequence to runoff yield and thus, to soil erosion. As a result, *P* is a poor indicator of *SSY*, as was also reported in [25, 40].

The single factor of *I*_{30}, although equally as effective in predicting *SSY* as factors of *I*_{20}, *EI*_{10,} *EI*_{20} and *EI*_{30}, can be preferentially used as the rainfall erosivity index in practices because *I*_{30} is in form simpler than *EI*_{t} and can be measured somewhat more accurately than *I*_{20}. *Wang* [16] also noted that the predictive ability of *EI*_{t} is only marginally higher than that of *I*_{t} in the Loess Plateau. The calculation of *E* involves data which are rarely available. Our finding shows that *E* is not necessarily included into the rainfall erosivity index thereby facilitating the obtainment of rainfall erosivity in the Loess Plateau.

Our calculations at six sites within the Tuanshangou subwatershed indicate that *I*_{30} summed over a year can explain 71 to 89% of the variation in yearly soil loss, an accuracy that is comparable to that of the use of the *EI*_{30} index in the USA [40]. Considering the fact that our data are from cropped plots, *I*_{30} can be directly applied to predicting soil loss in cropped areas, as opposed to the USLE, which first predicts erosion for the unit plot (bare fallow areas 22.1 m long on a 9% slope) and then predicts for the area of interest by introducing the topographic factors and the cover and management factors. Nevertheless, the mean of the annual cumulative *I*_{30} promises to act as the *R* factor of the USLE due to the sparse vegetation cover in the plots.

### Runoff factors

For runoff factors, the two methods present the same result: *h* is not only superior to *q*_{max} but is also superior to the combined factors of *h* and *q*_{max} as a predictor of *SSY*.

Similar to the rainfall case, flow discharge during flood events is primarily concentrated during the high-flow period, especially during the peak-flow period. Consequently, *q*_{max} correlates well with *h* (*r* > 0.77) and in turn, with *SSY* at every site (*r* > 0.83, Table 5). However, sediment concentrations at moderate discharges are more or less the same as those at high discharges in the Loess Plateau. Extremely high concentrations even primarily occur at low discharges from plots to watersheds (see Fig. 6 in [44] and Fig. 2 in [34]). This mismatch between sediment concentration and water discharge can be related to hyperconcentrated flows, which are well developed from upland slopes to river channels in the Loess Plateau [50, 51]. It is known that no direct relationship exists between sediment concentration and water discharge for hyperconcentrated flows [52]. In stream channels of the Chabagou watershed (#9), *SC*_{max} generally occur at flow discharges that are approximately 30–50% lower than *q*_{max} [53]. Hence, contrary to the rainfall case, *h* is more correlated with *SSY* than *q*_{max}.

Runoff factors can provide better *SSY* predictions than rainfall factors in the Loess Plateau in terms of correlations (See Tables 3 and 5). The obtained correlation coefficients between *h* and *SSY* is larger than those between *I*_{30} and *SSY* at all six sites within the Tuanshangou subwatershed, and five of them being statistically valid (*p* < 0.02). Interrill erosion is closely related to rainfall factors, whereas rill erosion is primarily dependent on runoff factors [25, 54]. The higher correlation of runoff factors with *SSY* relative to rainfall factors can thus be linked to the dominance of rill erosion and the mass wasting over the interrill erosion in our study area.

### The best erosivity index

The rainfall-runoff factors we examined are generally inter-correlated. As a result, it can hardly be expected to improve the prediction accuracy by introducing more factors. *SSY* equals the product of *h* and *SC*_{e}. There is no need to include factors that do not affect *SC*_{e} into the erosivity index if *h* has been included. We hereafter examine the correlations between the rainfall-runoff factors and *SC*_{e}.

Except for *T*, *I* and *P*, nine other rainfall factors correlate well with *SC*_{e} at all sites (*p* < 0.01). However, almost all of these correlations become insignificant with only two exceptions when only major sediment-producing events are considered (see Table 3). As in [35], we defined major sediment-producing events as high-concentrated events that accumulatively contribute 90% to the total sediment yield of all examined events. Scatter plots of *SC*_{e} and *I*_{30} at all sites are generally parallel to the x-axis for major sediment-producing events (Fig. 2), a result contrary to that observed by *Kinnel* [29] and *Chaplot et al*. [55]. The same observation holds for scatter plots of *SC*_{max} and *I*_{30} (Fig. 3).

For major sediment-producing events, *SC*_{e} and *SC*_{max} also remain independent of the four runoff factors although there are five exceptions among the 36 derived correlations (Table 5). Fig. 4 and Fig. 5 depict the relationships between *SC*_{e} and *q*_{max} and between *SC*_{max} and *q*_{max}, respectively.

In terms of the correlation with *SSY*, the single factor of *h* is the best erosivity index among factors we examined at scales from plots to watersheds. It is not expected that the prediction accuracy would be further improved by introducing additional rainfall-runoff factors because these factors are ineffective in altering *SC*_{e} for major sediment-producing events. As was the case of *q*_{max}, little was added to the prediction accuracy by combining *h* with *I*_{30}. Among the six sites within the Tuanshangou subwatershed, *h*+*I*_{30} performs better than *h* only at one site (*p* = 0.03; Table 9), and *hI*_{30} is not as good as *h* at three sits (*p* < 0.046). When the runoff coefficient (*a*, given by *h* divided by *P*) was introduced, as did in the USLE-M [27], neither *aI*_{30} nor *aEI*_{30} provide better predictions of *SSY* than *h* at any site in terms of correlations. Moreover, *aI*_{30} and *aEI*_{30} are less correlated with *SSY* than *h* at three and two sites, respectively (*p* < 0.006; Table 9).

## Conclusions

Based on Meng’s tests, this study presents two methods to determine the erosivity index among a number of rainfall-runoff factors by comparing their correlations with *SSY*. The first method involves a stepwise procedure to remove factors that are poorly correlated with *SSY*. The second method involves multiple comparisons that are adjusted using a Bonferroni correction. It appears that few studies have compared correlations on a statistical basis, not only within the soil erosion community but also within the entire geoscience community. Our methods therefore have wide significance, not only for determining the best predictor, but also in other respects, such as comparing model performance, which is often indexed by the correlation between observed and modeled values.

Using the methods described above, we determined the erosivity indices of rainfall and runoff in a typical Chinese Loess Plateau watershed. Among 12 rainfall factors under examination, *I*_{20}, *I*_{30}, *EI*_{10,} *EI*_{20} and *EI*_{30} were found to be the most correlated with *SSY* at scales from plots to subwatersheds (< 1 km^{2}). We suggested the use of *I*_{30} as the rainfall erosivity index, although it is equally effective as the three remaining factors. The value of *I*_{30} summed over one year is also a good predictor of annual soil loss (*r*^{2} > 0.7).

Runoff factors are more correlated with SSY than rainfall factors almost at all examined sites. Among the four studied runoff factors, h is correlated best with SSY at scales from plots to watersheds. Moreover, the combination of h with other rainfall-runoff factors, including rainfall intensity and peak discharge (as used in the MUSLE [24]), does not show enhanced ability to predict SSY compared with the single factor of h because these factors are of little importance in determining sediment concentration for major sediment-producing events. Introducing the runoff coefficient (as used in the USLE-M [27]) also added little to the prediction accuracy. Hence, we considered the single factor of h as the best erosivity index, although I30 would be useful in many cases considering the difficulty of measuring runoff.

## Acknowledgments

We appreciate the suggestions of the anonymous reviewer and the editor. Thanks to the Data Sharing Infrastructure of Earth System Science-Data Sharing Infrastructure of Loess Plateau for providing our data.

## Author Contributions

Conceived and designed the experiments: MGZ. Performed the experiments: MGZ XAC. Analyzed the data: MGZ. Contributed reagents/materials/analysis tools: MGZ XAC. Wrote the paper: MGZ.

## References

- 1. Angulo-Martínez M, Beguería S (2009) Estimating rainfall erosivity from daily precipitation records: A comparison among methods using data from the Ebro Basin (NE Spain). J Hydrol 379:111–121.
- 2. Lee J-H, Heo J-H (2011) Evaluation of estimation methods for rainfall erosivity based on annual precipitation in Korea. J Hydrol 409: 30–48.
- 3. Shi ZH, Cai CF, Ding SW, Wang TW, Chow TL (2004) Soil conservation planning at the small watershed level using RUSLE with GIS: a case study in the Three Gorge Area of China. Catena 55: 33–48.
- 4.
Wischmeier WH, Smith DD (1978) Predicting rainfall erosion losses: a guide to conservation planning. USDA Handbook 537, Washington, DC.
- 5.
Renard KG, Foster GR, Weesies GA, McCool DK, Yoder DC (1997) Predicting soil erosion by water: a guide to conservation planning with the revised universal soil loss equation (RUSLE). USDA Handbook 703, Washington, DC.
- 6. Yue BJ, Shi ZH, Fang NF (2014) Evaluation of rainfall erosivity and its temporal variation in the Yanhe River Catchment of the Chinese Loess Plateau. Natural Hazards 74:585–602.
- 7. Diodato N, Bellocchi G (2007) Estimating monthly (R)USLE climate input in a Mediterranean region using limited data. J Hydrol 345:224–236.
- 8.
Hudson N (1971) Soil Conservation. Ithaca: Cornell University Press.
- 9. Lal R (1976) Soil erosion on Alfisols in Western Nigeria, III-Effects of rainfall characteristics. Geoderma 16: 389–401.
- 10.
Kinnell PIA (1995) The I
_{x}E_{A}erosivity index: An index with the capacity to give more direct consideration to hydrology in predicting short-term erosion in the USLE modeling environment. J Soil Water Conserv 50: 507–512. - 11. Sukhanovski YP, Ollesch G, Khan KY, Meißner R (2001) A new index for rainfall erosivity on a physical basis. Journal of Plant Nutrition and Soil Science 65 (1), 51–57.
- 12. Sinzot A, Bollinne A, Laurant A, Erpicum M, Pissart A (1989) A contribution to the development of an erosivity index adapted to the prediction of erosion in Belgium. Earth Surf Process Landf 14: 509–515.
- 13. Usón A, Ramos MC (2001) An improved rainfall erosivity index obtained from experimental interrill soil losses in soils with a Mediterranean climate. Catena 43: 293–305.
- 14. Abu Hammad AH, Borresen T, Haugen LE (2005) Effects of rain characteristics and terracing on runoff and erosion under the Mediterranean. Soil Tillage Res 87 39–47.
- 15. Sharifah Mastura SA, Al-Toum S, Jaafar O (2003) Rainsplash erosion: a case study in Tekala River catchment, East Selangor, Malaysia. Geografia 4: 44–59.
- 16. Wang WZ (1983) Relationship between soil loss and rainfall characteristics for the Chinese loess areas. Bull Soil Water Conserv (5): 62–64. (in Chinese. )
- 17. Jia ZJ, Wang XP, Li JY (1987) Determination of the rainfall erosivity index for hilly loess areas in the western Shanxi. Soil Water Conserv Chin (6):18–20. (in Chinese. )
- 18. Wang WZ (1987) Determination of the rainfall erosivity index for Chinese loess areas. Soil Water Conserv Chin (12): 34–38. (in Chinese. )
- 19. Chen XA, Cai QG, Zheng MG, Nie BB, Cui PW (2010) Study on rainfall erosivity of Chabagou watershed in a hilly loess region on the Loess Plateau. J Sediment Res (1): 5–10. (in Chinese. )
- 20. Jia Z, Jiang ZS, Liu Z (1990) Studies of the relationship between rainfall characteristics and soil loss. Memoir of NISWC, Academia Sinica & Ministry of Water Conservancy 12: 9–15. (in Chinese).
- 21. Wang WZ, Jiao JY (1996a) A quantitive study of soil erosion factors in China. Bull Soil Water Conserv 16(5): 1–20. (in Chinese. )
- 22. Shi ZH, Yue BJ, Wang L, Fang NF, Wang D, Wu FZ (2013) Effects of mulch cover rate on interrill erosion processes and the size selectivity of eroded sediment on steep slopes. Soil Sci Soc Am J 77: 257–267.
- 23. Dragoun FJ (1962) Rainfall energy as related to sediment yield. J Geophys. Res 67:1495–1501.
- 24.
Williams JR (1975) Sediment-yield prediction with universal equation using runoff energy factor. In: Present and Prospective Technology for Predicting Sediment Yield and Sources, Publ. ARS-S-40. US Dept. Agric., Washington, DC, pp. 244–252.
- 25. Foster GR, Lambaradi F, Moldenhauer WC (1982) Evaluation of rainfall-runoff erosivity factors for individual storms. Trans Am Soc Agric Eng 25:124–129.
- 26. Hussein MH, Awad MM, Abdul-Jabbar AS (1994) Predicting rainfall-runoff erosivity for single storms in northern Iraq. Hydrol Sci J 39:535–547.
- 27. Kinnell PIA, Risse LM (1998) USLE-M: empirical modelling rainfall erosion through runoff and sediment concentration. Soil Sci Soc Am J 62: 1667–1672.
- 28.
Williams JW, Izaurralde RC, Steglich EM (2008) Agricultural Policy/Environmental Extender Model Theoretical Documentation. BRC Report # 2008–17. Blackland Research and Extension Center, Temple, Texas.
- 29. Kinnell PIA (2010) Event soil loss, runoff and the Universal Soil Loss Equation family of models: A review. J Hydrol 385: 384–397.
- 30.
Mou JZ, Xiong GS (1980) Prediction of sediment yield and calculation of trapped sediment by soil conservation measures from small catchments in Northern Shaanxi, China. In: Proceedings of the First International Symposium on River Sedimentation. Beijing: Guanghua Press, pp. 73–82 (in Chinese).
- 31. Cai QG, Liu JG, Liu QJ (2004) Research of sediment yield statistical model for single rainstorm in Chabagou drainage basin. Geog Res 23(4): 433–439. (in Chinese. )
- 32.
Yu GQ, Zhang MS, Li ZB, Li P, Zhang X, Cheng SD (2013) Piecewise prediction model for watershed‐scale erosion and sediment yield of individual rainfall events on the Loess Plateau, China. Hydrol Process. https://doi.org/10.1002/hyp.10020
- 33. Wang ML, Zhang R (1990) Study on sediment yield model under single storm in Chabagou watershed. J Soil Water Conserv 4 (1): 11–18. (in Chinese. )
- 34. Zheng MG, Cai QG, Cheng QJ (2008) Modelling the runoff-sediment yield relationship using a proportional function in hilly areas of the Loess Plateau, North China. Geomorphology 93: 288–301.
- 35. Zheng MG, Yang JS, Qi DL, Sun LY, Cai QG (2012) Flow-sediment relationship as functions of spatial and temporal scales in hilly areas of the Chinese Loess Plateau. Catena 98: 29–40.
- 36. Salako F, Obi ME, Lal R (1991) Comparative assessment of several rainfall erosivity indices in Southern Nigeria. Soil Technol 4,: 93–97.
- 37. Kiassari EM, Nikkami D, Mahdian MH, Pazira E (2012) Investigating rainfall erosivity indices in arid and semiarid climates of Iran. Turk J Agric For 36: 365–378.
- 38. Wischmeier WH, Smith DD (1958) Rainfall energy and its relationship to soil loss. Trans. Am. Geophys. Union, 392, 285–291.
- 39. Wischmeier WH, Smith DD, Uhland RE (1958) Evaluation of factors in the soil loss equation. Agric Eng 39: 458–462.
- 40. Wischmeier WH (1959) A rainfall erosion index for a universal soil-loss equation. Soil Sci Soc Am J 23: 246–249.
- 41. Silver NC, Hittner JB, May K (2006) A FORTRAN 77 program for comparing dependent correlations. Appl Psychol Meas 30(2): 152–153.
- 42. Meng XL, Rosenthal R, Rubin DB (1992) Comparing correlated correlation coefficients. Psychol Bull 111: 172–175.
- 43. Zheng MG, Qin F, Sun LY, Qi DL, Cai QG (2011) Spatial scale effects on sediment concentration in runoff during flood events for hilly areas of the Loess Plateau, China. Earth Surf Processes Landforms 36: 1499–1509.
- 44. Zheng MG, Qin F, Yang JS, Cai QG (2013) The spatio-temporal invariability of sediment concentration and the flow-sediment relationship for hilly areas of the Chinese Loess Plateau. Catena 109: 164–176.
- 45. Jiang ZS, Song WJ, Li XY (1983) Studies of the raindrop characteristics for Chinese loess area. Soil Water Conserv Chin (3): 32–36. (in Chinese. )
- 46.
Goyal JK, Sharma JN (2009) Mathematical Statics. Meerut: Krishna Prakashan Media (P) Ltd., 464pp.
- 47. Hochberg Y (1988) A sharper Bonferroni procedure for multiple tests of significance. Biometrika 75: 800–802.
- 48. Wright SP (1992) Adjusted P-Values for Simultaneous Inference. Biometrics, 48: 1005–1013.
- 49. Rice TK, Schork NJ, Rao DC (2008) Methods for handling multiple testing. Adv Genet 60: 293–308. pmid:18358325
- 50. Xu JX (1999) Erosion caused by hyperconcentrated flow on the Loess Plateau. Catena 36: 1–19.
- 51. Liu QJ, Shi ZH, Fang NF, Zhu HD, Ai L (2013) Modeling the daily suspended sediment concentration in a hyperconcentrated river on the Loess Plateau, China, using the Wavelet-ANN approach. Geomorphology 186: 181–190.
- 52.
Pierson TC (2005) Hyperconcentrated flow-transitional process between water flow and debris flow. In: Jakob M, Hungr O, editors. Debris-Flow Hazards and Related Phenomena. Berlin Heidelberg: Springer, pp. 159–202.
- 53. Wang WZ, Jiao JY (1996b) Statistic analysis on process of gully runoff and sediment yield under different rain pattern in loess plateau region. Bull. Soil Water Conserv 16 (6): 12–18 (in Chinese).
- 54. Shi ZH, Fang NF, Wu FZ, Wang L, Yue BJ, Wu GL (2012) Soil erosion processes and sediment sorting associated with transport mechanisms on steep slopes. J Hydrol 454: 123–130.
- 55. Chaplot VAM, Le Bissonnais Y (2003) Runoff features for interrill erosion at different rainfall intensities, slope lengths, and gradients in an agricultural loessial hillslope. Soil Sci Soc Am J 67: 844–851.