2.1 The relationship between precipitation extremes and multifractals

The approach outlined below was used to identify EP events at the observation timescale. In the method of universal multifractals, two equivalent routes can be followed to investigate time series scaling: the probability distribution and the statistical moments. A fundamental property of multifractal fields related to the probability distribution is given by the following equation (Lovejoy and Schertzer, 2013; Schertzer and Lovejoy, 1987):

where λ represents the resolution of the measure (i.e., the ratio of the external scale L to the measurement scale l; $\mathit{\lambda }=L/l$), φ_λ is the intensity of the field (in this case accumulated precipitation) measured at resolution λ, γ is the order of singularity (maximum precipitation) corresponding to φ_λ, and the codimension function c(γ) characterizes the sparseness of the γ-order singularities (this function is a basic multifractal probability relation for cascades). Accordingly, the statistical moments are given by

where K(q) is the multiple scaling exponent function for moments; q is the order of the statistical moment, and 〈 〉 denotes the average of the field (averaged precipitation) at scale ratio λ. The two equivalent routes are related via a Legendre transform (Parisi and Frisch, 1985). The universal K(q) functions and the codimension function c(γ) are expressed as

where α is the multifractal index, which describes how rapidly the fractal dimensions vary as we leave the mean. For time series in this paper, $\mathrm{0}<\mathit{\alpha }<\mathrm{1}$, and α^′ is the auxiliary variable defined by $\mathrm{1}/{\mathit{\alpha }}^{\prime }+\mathrm{1}/\mathit{\alpha }=\mathrm{1}$ (Lovejoy and Schertzer, 2013). The term C₁, the codimension of the mean of the process, varies as $\mathrm{0}\le C_{\mathrm{1}}\le D$ (D is space dimension; D=1 for time series) and quantifies the sparseness of the mean. In this paper, the parameters α and C₁ of the multifractal model were estimated using the double trace moment (DTM) technique (Lavallée et al., 1993).

As noted by Gagnon et al. (2006) and Lovejoy and Schertzer (2007), the parameters C₁ and α characterize the mean of the field, whereas the extremes are expressed by the singularity, γ, and the codimension function c(γ) (Hubert et al., 1993). For $\mathrm{0}\le \mathit{\alpha }<\mathrm{1}$ and considering a time series of infinite length, a finite maximum order of singularity, γ₀, can be determined as follows:

However, in general, any time series of finite length will almost surely miss the presence of rare extremes in the field. In this case, the observed singularities will be bounded by an effective maximum singularity, γ_s:

where D is the embedding space dimension (D=1 for the time series). The total dimension of this problem is actually (D+D_s), where $D_{\mathrm{s}}=\log N_{\mathrm{s}}/\log \mathit{\lambda }$ is the sampling dimension, and N_s is the number of independent time series at each location. The parameter γ_s links the physical processes that generate precipitation events to the conceptual model of multiplicative cascades, and it allows the extremes to be cast in a probabilistic framework, γ_s>0. Thus, it was used to infer the extreme events of precipitation fields over well-defined scales.

For parameter estimation, the parameters α and C₁ of the multifractal model are estimated by the DTM technique (Lavallée et al., 1993). The q, η DTM at resolution λ and Λ is defined as

where the sum is obtained over all the disjoint D dimensional balls B_λ,i (with intervals of length $\mathit{\tau }=T/\mathit{\lambda }$) that are required to cover the time series. K(q, η) is the double trace scaling exponent, and $K(q,\mathrm{1})=K\left(q\right)$ is the scaling exponent. This relation can be expressed as

where the notation indicates that the multifractal φ at a (finest) resolution Λ is first raised to the power η then degraded to resolution λ, and the qth power of the result is averaged over the available data. The scaling exponent K(q, η) is related to $K(q,\mathrm{1})\equiv K\left(q\right)$ and is given by

In the case of universal multifractals, by plugging Eq. (3) into Eq. (9), K(q, η) has a particularly simple dependence on η:

where α can be estimated from a simple plot of K(q, η) versus log (η) for fixed q. Thus, based on the DTM technique, all parameters can be estimated.

2.2 Determination of the extreme precipitation threshold

The approach outlined below was used to estimate the EPTs and EP events for each station, using the singularity parameter (γ_s). Given that the multifractal parameters α and γ_s naturally characterize extremes, both of these parameters will change if we gradually remove extreme values from the data set. The singularity of the precipitation data series will be completely changed, and the two parameters will significantly change if all of the extreme values are deleted. To obtain a physically meaningful value for the EPT, we attempted to estimate the multifractal parameter series by applying the universal multifractal approach to our precipitation series and successively eliminating maximum values of precipitation. However, as shown by Fig. 1a and c, the degree of convergence of the original value is not unique, with the values fluctuating slightly around the original α and γ_s. Accordingly, the variance series of index series α_j and γ_s,j were computed to eliminate the fluctuation while identifying the point of convergence. The procedure is as follows:

1. eliminate the data point x_j, {x_j, $x_j\ge x_{\max }-d\times j\mathit{\}}$ from the precipitation time series {x_i, i=1, 2, …, n}, where x_max is the maximum, x_ave is the average, and d is the interval space (we set d to 1 mm in this case);

2. compute the selected parameters;

3. repeat (1) and (2) for j varying from 1 to int(($x_{\max }-x_{\mathrm{ave}})/d$).

Using the obtained parameter series, we applied the segmentation algorithm proposed by Bernaola Galván et al. (2001) to determine the point of abrupt change, which we define as the EPT. The segmentation algorithm is based on the calculation of the statistic t of each data point in a series or subseries:

where $s_{\mathrm{D}}=\left[\right(s_{\mathrm{L}}^{\mathrm{2}}+s_{\mathrm{R}}^{\mathrm{2}})/(N_{\mathrm{L}}+N_{\mathrm{R}}-\mathrm{2}\left){]}^{\mathrm{1}/\mathrm{2}}\right(\mathrm{1}/N_{\mathrm{L}}+\mathrm{1}/N_{\mathrm{R}}{)}^{\mathrm{1}/\mathrm{2}}$ is the pooled variance. The μ_L∕μ_R, s_L∕s_R, and N_L∕N_R are the mean, standard deviation, and number of points from the data to the left/right of the series, respectively. The significance level P(τ) of the largest value t_max obtained from Eq. (11) is defined as the probability of obtaining a value equal to or less than τ within a random sequence (Swendsen and Wang, 1987):

If the significance exceeds a selected threshold P₀ (usually taken to be 95 %), an abrupt point is selected and the series is cut into two subsets at this point.

Figure 1Procedure for the EPT determination of the Xingxian station. (a) The multifractal index α (black dots) and the alternative abrupt points (red dots). (b) Pooled variances (black dots) as calculated from the α series with the significant abrupt points (red dots) of the variances. (c) As in panel (a) but for the singularity γ_S. (d) As in panel (b) but for the variance calculated from the γ_S series. (e) The time variation of daily precipitation ranging from 1961 to 2015. The blue dotted line represents the determined EPT.

Download

The pooled variances and the abrupt points of the α and γ_s series are shown in Fig. 1b–d. The abrupt point, where s_D differs from its left-side points but is approximately equal to its right-side points, is selected to be the EPT. As shown for the determined EPT in Fig. 1, there is a steep slope of the pooled variance on the left and a gentle slope on the right of 37.1 mm daily precipitation. Thus, the EPT for Xingxian station is estimated to be 37.1 mm d⁻¹, and 90 EP events have occurred over the past 55 years (Fig. 1c).

2.3 EP indices

All of the variables characterizing spatiotemporal EP over the LP are shown in Table 1. For individual stations, the annual precipitation at each station was accumulated using daily data. Annual EP is accumulated using daily EP determined by the EPT, and EP frequency (EPF) is the number of daily EP events. Mean annual EP (MEP) is averaged from annual EP interpolated using ArcGIS 10.1. For a year with n EP events, the EP intensity (EPI) is calculated by the following equation:

where EP_i represents the magnitude of EP event i, respectively.

Table 1Variables (abbreviations) used in this study addressing precipitation variations.

Download Print Version | Download XLSX

As noted by IPCC (2007), neither the cases of high frequency with low intensity nor the cases of low frequency with high intensity can reflect the severity of EP events given a long-term time series for an area, whereas the severity of EP (EPS) events relies on both intensity and frequency. Consequently, we examine the extreme precipitation intensity (EPI), extreme precipitation frequency (EPF, defined below), and EPS to characterize the spatiotemporal nature of EP over the LP. To obtain the concordant EPS, each annual EPF/EPI series is standardized to the range from 0 to 1 using Eq. (10):

where X_min and X_max represent the lowest and highest annual EP frequency/intensity, respectively. The annual EPS for each station (0 ≤ EPS ≤ 1) is calculated from the standardized EPI and EPF by

where k₁ and k₂ are the weight coefficients of frequency and intensity influencing EP severity, respectively, and $k_{\mathrm{1}}+k_{\mathrm{2}}=\mathrm{1}$. In this paper, k₁ and k₂ are set to 0.5 according to IPCC (2007).

In addition, we compute the long-term mean daily precipitation (MDP) and the long-term accumulated daily EP events (ADEP) to help characterize the intra-annual precipitation and EP. As shown in Table 1, MDP is averaged from all 87 stations, and ADEP is accumulated from 87 stations over the past 55 years.

2.4 Spatiotemporal EP presentation

The spatial distributions of the EP indices were derived by interpolation via kriging (Oliver and Webster, 1990), using data observed at the gauging stations. All spatial analyses were carried out using the ArcGIS 10.1 software. Spatiotemporal trends for annual EP indices were computed for each pixel using the least squares method. Following Stow et al. (2003), the trend is defined as the slope of the least squares line that fits the inter-annual variability of individual EP indices during the study period and is given by

where m is the total of years, and P_i is value of the pixel in the j-th year.

3.1 Study area

The LP (640 000 km²) is a typical arid and semiarid area located in the middle reaches of the Yellow River (750 000 km²) and is characterized by a continental monsoon climate. Elevations range from 84 to 5207 m (Fig. 2). The desert steppe, typical steppe, and forest steppe (deciduous broadleaf forest) zones are distributed from northwest to southeast and correspond to mean annual isohyets of 250, 450, and 550 mm in the arid, semiarid, and semi-humid areas, respectively. The continuous loess covering ranges from 100 to 300 m in thickness on the mountains, hills, basins, and alluvial plains of different heights. The northwestern part of the region is dominated by flat sandy areas. The middle and southeastern parts are characterized by EP-induced water-erosion landforms (Zhang et al., 1997), with a rugged undulating ground surface that is broken, barren, and dissected by gullies and ravines (Cai, 2001). EP-induced flooding episodes occur occasionally in the summer, with sediment concentrations generally exceeding 300 kg m⁻³, although they have been observed to be as large as 1240 kg m⁻³. The hyperconcentrated flooding has historically resulted in numerous disasters, with severe consequences for people and livestock (Zhang et al., 2017). The amount of soil erosion has been estimated to be larger than 2×10⁹ to 3×10⁹ t yr⁻¹ (Tang, 1990). Soil erosion has resulted in the density of gullies and ravines in the LP being larger than 3–4 km km⁻², with the maximum exceeding 10 km km⁻².

Figure 2Location of the Loess Plateau in the middle reaches of the Yellow River, China (inset), and the distribution of the meteorological stations in and around the LP.

3.2 Database

To conduct the EP assessment, we used daily data that were available for 87 national meteorological stations in and around the LP (Fig. 2), consisting of continuous time series from 1961 to 2015. All of the precipitation data were obtained from the China Meteorological Data Sharing Service System (http://data.cma.cn/, last access: 19 February 2020). Missing data accounted for less than 0.1 % of the total sample, and they were replaced by a value of zero in this paper; this replacement of a few missing values does not influence the analysis. Data regarding the severity of soil erosion were provided by the LP Science Data Center of the data sharing infrastructure of the National Earth System Science Data Center of China (http://loess.geodata.cn, last access: 19 February 2020). These data were compiled during the Soil Erosion Census of the China Census for Water (Ministry of Water Resources and National Bureau of Statistics, 2013). Mean annual vegetation coverage at an 8 km spatial resolution and a 15 d temporal resolution were computed using data for the period from 1982 to 2006, which were produced by the Global Inventory Modeling and Mapping Studies (GIMMS) group from measurements of the Advanced Very-High-Resolution Radiometer (AVHRR) onboard the NOAA 7, NOAA 9, NOAA 11, and NOAA 14 satellites. Data from the National Center for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) Reanalysis Project (Kalnay et al., 1996) were also used in this study. The variables selected for analysis were the monthly mean geopotential height, monthly mean wind, daily mean sea level pressure, and daily mean wind from 1961 to 2013 on a 2.5^∘ × 2.5^∘ spatial grid (http://www.esrl.noaa.gov/psd/, last access: 19 February 2020).

4.1 Spatial characteristics of extreme precipitation

Figure 3a shows that the mean annual precipitation (for the period from 1961 to 2015) varied from 115 mm in the northwestern LP to the 845 mm in the southeastern LP. The associated EPTs ranged from 17 mm d⁻¹ in the northwest to 50 mm d⁻¹ in the southeast (Fig. 3b); these EPT isohyets are generally consistent with those of mean annual precipitation. Figure 3b indicates that the area around the Dongsheng station is a regional EP center: the EPTs around the station are higher than those of the surrounding areas, whereas the isohyets of mean annual precipitation are smooth. The spatial distribution of MEP is also similar to that of mean annual precipitation, increasing from northwest to southeast and ranging from 35 to 138 mm yr⁻¹ (as shown in Fig. 3c). MEP maximums occur in the southern and southeastern LP.

Figure 3Spatial distributions of (a) mean annual precipitation, (b) EPTs, (c) MEP, (d) total EPF, (e) mean EPI, and (f) mean annual EPS in the LP from 1961 to 2015.

Figure 3d indicates that EPFs have ranged from 54 to 115 d (i.e., mean annual EPF ranged from 1.0 to 2.1 d) over the region during the past 5 decades. Notable occurrences of high EPF can be seen in and around the Ziwuling Mountains in the mid-southern LP, whereas the highest frequency occurred to the east of the Fenhe Valley in the southeastern LP. Meanwhile, the lowest frequency occurred in the northwestern LP, the western Mu Us sandy land, and the western Liupan Mountains.

Figure 3e indicates that the averaged EPIs mainly ranged between 0.3 and 0.7. The spatial variations in EPI and EPF contrast with each other, with the highest EPIs centered in the mid-eastern LP, where EPFs were comparatively lower, and the lowest EPIs centered in the southeastern LP, which had the maximum mean annual precipitation and EPF. The highest values of EPI dominated the central LP area. The northern boundary of the area was positioned southeast of the Mu Us Desert (Dongsheng and Xingxian); the western boundary was positioned west of the Liupan Mountains (Guyuan County); the eastern boundary was positioned northeast of the Lüliang Mountains and the Fenhe Valley (Taiyuan and Linfen); the southern boundary was positioned to the north of the central Shaanxi Plain (Huashan, Xi'an and Changwu). The EPI presents the event EP power causing natural hazards. This high value of EPI partially explains why this area is characterized by very serious soil erosion that releases more than 2×10⁹ t of sediment into channels of the Yellow River annually.

Figure 3f indicates that the spatial distribution of EPS in the LP increased from the northwest to the southeast, ranging from 0.27 to 0.66, but the highest EPSs were centered in the southeast central LP. The areas with the highest EPSs covered the basins of the Jing, Luo, and Fen rivers. Although EP events always occurred over a small range, the spatial maps of EPI, EPF, and EPS indicate that the areas with serious EP events are regularly distributed.

4.2 Spatiotemporal variation of EP

Our results (Fig. 4a) indicate that 91.4 % of the LP was characterized by a negative annual precipitation trend over the study period, whereas only 8.6 % of the total area presented a positive trend. A total of 9 out of 87 stations showed a significant negative trend, whereas 2 stations showed positive trends (p<0.1). At the same time, the spatiotemporal trends in the annual EP ranged from −0.78 to +0.48 mm yr⁻¹ (Fig. 4b), with 23.8 % of the total area showing a positive trend and increased annual EP distributed mainly in the southwestern LP (west of Lanzhou) and the central southern LP (Beiluo and Jing river basins and an area around the Xingxian station). Meanwhile, the annual EPF has changed by −0.6 to +0.5 d over the past 55 years, with a change rate ranging from $-\mathrm{1.2}\times {\mathrm{10}}^{-\mathrm{2}}$ to $+\mathrm{0.95}\times {\mathrm{10}}^{-\mathrm{2}}$ d yr⁻¹ (as shown in Fig. 4c). Of the 87 stations, 4 stations showed a significant negative trend in EPF, whereas 3 stations showed positive trends (p<0.1). The areas with a negatively trending EPF covered 86.4 % of the total area, whereas the areas with positively trending EPF covered 13.6 %; the latter regions mainly occurred in the southwestern LP (around the Xining station) and in the areas around the Xi'an and Xingxian stations. The areas with notably decreasing trends mainly occurred in the central western and southeastern regions of the LP.

Figure 4The spatial distribution of the trends and the stations with significant trends (p<0.1) for (a) annual precipitation, (b) annual EP, (c) annual EPF, (d) annual EPI, and (e) annual EPS in the LP from 1961 to 2015.

Figure 4d indicates that the changes in annual EPI ranged from −0.18 to +0.27, with a changing rate ranging from $-\mathrm{0.34}\times {\mathrm{10}}^{-\mathrm{2}}$ to $\mathrm{0.52}\times {\mathrm{10}}^{-\mathrm{2}}$ yr⁻¹. We found that 34 of the 87 stations showed upward slopes (5 stations with a significance level of p<0.1), and 53 stations showed negative slopes (4 stations with a significance level of p<0.1). As shown in Fig. 4d, areas with positive trends in EPI accounted for 42.2 % of the total area, with the areas delineated by the Wulate, Yan'an, and Huashan stations and the Jingtai, Xiji, and Tianshui stations as well as the area west of the Minhe station. The areas with a negative slope covered 57.8 % of the total area.

Figure 4e indicates that the annual EPSs changed by −0.09 to +0.07 during the study period, with rates varying from $-\mathrm{0.34}\times {\mathrm{10}}^{-\mathrm{2}}$ to $\mathrm{0.52}\times {\mathrm{10}}^{-\mathrm{2}}$ yr⁻¹. Of the 87 stations, 39 stations showed a positive slope (3 stations with a significance level of p<0.05), while 54 stations exhibited a negative slope (2 station with a significance level of p<0.05). The areas with increased EPSs covered 25.4 % of the total area and were mainly found in an area delineated by the Wuqi, Tianshui, and Huashan stations and an area west of the Xiji station. The areas with negative trends accounted for 74.6 % of the total area.

Figure 5Intra-annual distribution of daily precipitation (MDP) and the number of daily EP events (ADEP) for the 87 stations from 1961 to 2015 and their fitting curves with respect to a Weibull function.

Download

The trend estimates computed for annual EP, EPF, and EPI are associated with strong uncertainty. For instance, the upward trend in annual EP in and around the Xingxian station relied heavily on the upward trend in the EPF and not the downward trend in the EPI. The EPF around the Changwu station decreased, but both the annual EP and EPS increased with the upward trend in the EPI. However, nearly all of the areas with positive trends for annual EP, EPI, EPF, and EPS had a negative trend in annual precipitation (Fig. 4). It should be noted that 62.1 % of the LP area with a negative trend in annual precipitation has more than one EP indices with positive trends, potentially indicating the risk of more serious hazardous situations.

4.3 Intra-annual EP characteristics and their relationship with large-scale atmospheric circulation

4.3.2 Atmospheric circulation factors for the spatial variation of extreme precipitation

Atmospheric circulation is the leading factor causing the abovementioned phenomena. The LP is located in the East Asian monsoon region. According to the average sea level pressure and winds in winter from 1961 to 2015 (see Fig. 6a), the dry winter in the region is influenced by the interactions between two high-pressure areas in southwestern China (the Tibet Plateau high-pressure system) and North China (the Mongolian high-pressure system). The prevailing East Asian winter monsoon (which has a north-northwest direction) circulates in East China and brings cold and dry airstreams. In contrast, the summer climate of the LP is affected by interactions between two high-pressure systems, the Pacific high-pressure system and the Tibet Plateau high-pressure system. Figure 6b shows that the prevailing East Asian summer monsoon (which has a south-southeast wind direction) brings warm and humid maritime airstreams that spread from the West Pacific to central China. However, the Tibet Plateau high-pressure system has a notable effect on the climate of the northwestern LP, and the airstream humidity decreases gradually as the distance from the Pacific increases. The resulting effect and decreased humidity form a vast arid region in northwestern China, including the northwestern LP, with a prevailing west-southwest wind direction. This explains why precipitation decreases from the southeast to the northwest, and precipitation is scarce in the northwestern LP.

Figure 6Average sea level pressure and winds: (a) the mean for all winters (from December to February) and (b) the mean for all summers (from June to August) from 1961 to 2015. Characteristics of the average 1000 hPa geopotential height and winds on (c) 1 August 1996 and (d) 3 August 1996. The data were derived from global NCEP/NCAR reanalysis average monthly and daily data.

Figure 7(a) Spatial distribution of mean vegetation coverage in summer (from June to August) on the Loess Plateau from 1982 to 2006 at a spatial resolution of 8 km. (b) Spatial distribution of the soil erosion intensity, which was resampled to a spatial resolution of 8 km.

Nevertheless, tropical cyclones occasionally enter the central LP, accompanied by EP events. For instance, in August 1996 a western Pacific cyclone landed in the southeastern coastal area of China and weakened gradually as it moved northwest, as shown by the 1000 hPa geopotential height and winds in Fig. 6c. Plenty of rainstorms or intense rainfall events that accompanied the cyclone occurred in its transit area. On 3 August 1996, the weakened cyclone reached the southeastern LP (as shown in Fig. 6d). However, due to the control of the Tibet Plateau high-pressure system, the central LP is generally the northwestern inland boundary with respect to the reach and impact of tropical cyclones. Thus, as shown in Fig. 6d, the cyclone was blocked from entering the northwestern LP, moved towards the northeast, and gradually dissipated. These phenomena illustrate why this region has limited precipitation but severe EP events.

5.1 Rationality of spatial EP characteristics

Natural hazards related to EP can be divided into two categories: (1) hazards accompanied by EP and (2) hazards that follow the occurrence of EP. For the former, one focus is the dependence between EP and storm surges in the coastal zone. Using such dependence structures, EP and storm surges can be quantified to provide information for successful hazard management (Svensson and Jones, 2004; Zheng et al., 2013). For the latter, the LP is such that the area suffers from EP-induced natural hazards that exceed the general tolerance of the natural environment, existing ecosystems, human life, and the social economy. In this case, the rational characteristics of EP responsible for spatial hazards can be studied.

Here, we use the widely distributed soil erosion to verify the rationality of our results. According to the universal soil loss equation (Wischmeier, 1976), the rational characteristics of EP should correlate well with soil erosion and vegetation coverage (Fig. 7). To examine this, partial correlation analyses were performed between soil erosion intensity and EPI/EPS as well as with the vegetation coverage. Our results indicate that water-based erosion intensity correlates significantly with vegetation coverage (negatively, Fig. 7a) and EPS (positively, Fig. 3f); the related coefficients are −0.61 (p<0.001) and 0.53 (p<0.001), respectively. For the correlations between water-erosion intensity and vegetation coverage as well as with EPI (Fig. 3e), the coefficients are −0.58 (p<0.001) and 0.76 (p<0.001), respectively. This finding demonstrates the rationality of our results. Note that, the higher correlation between EPI and soil erosion agrees with the results of plot experiments by Tang (1993), who noted that high-intensity precipitation is the primary driving force of erosion.

Figure 8EPTs determined by different methods and the corresponding EP frequencies for 87 stations over the Loess Plateau. The abscissa represents the stations with an increase in the mean annual precipitation from 104 to 918 mm.

Download

Zhou and Wang (1992) divided the LP into three zones of raindrop kinetic energy (<1000, 1000–1500, and 1500–2000 J m⁻² yr⁻¹, respectively), based on observations of the raindrop kinetic energies of rainstorms during 1980s. We found that the 30 and 35 mm d⁻¹ EPT contours closely overlap with the raindrop kinetic energy contours of 1000 and 1500 J m⁻² yr⁻¹. Further, soil erosion in the LP in recent decades has been found to be approximately 5000–10 000 t km⁻² yr⁻¹ (Ludwig and Probst, 1998; Shi and Shao, 2000). Such high rates of sediment erosion are generally induced by several rainstorm events during the year, with the top five daily sediment yields accounting for 70 %–90 % of the annual total soil loss (Rustomji et al., 2008; Zhang et al., 2017). For instance, a 200-year precipitation event in Wuqi on 30 August 1994 induced a flooding event with a daily sediment concentration of 1060 kg m⁻³. The streamflow was 2.41 times the mean annual streamflow from 2002 to 2011, and the sediment load was equivalent to 9.6 % of the total sediment yields from 1963 to 2011 (Zhang et al., 2016). Therefore, the EPF obtained in this study, about twice every year on average, is rational to explain such serious sediment erosion. In the LP, spring drought is the limiting factor with respect to vegetation (especially herbaceous vegetation) recovery from winter every year, and grass generally germinates on an extensive scale after the first effective rainfall event (>5 mm) in spring (Cai, 2001; Tang, 1993, 2004). However, as shown in Fig. 5, the highest EPF occurred 11 d earlier than the day of maximum daily precipitation in the LP. This means that the days on which the LP experiences most serious EP events tend to be days when precipitation is low. In other words, every year, the vegetation has not sufficiently recovered when frequent EP events occur in the LP. Such an intra-annual distribution of precipitation is one of the climatic reasons why there is serious soil erosion in the semiarid LP. Further, the sparse spatial nature of precipitation is insufficient for the growth of high-coverage vegetation, especially in the northwestern LP (Fig. 7a). However, the highest EPI provides the strongest erosion force, which contributes to the severe rates of erosion (Fig. 7b) in the central LP. Thus, these results regarding the EP responsible for hazardous situations (both spatially and temporally) are important for sustainable catchment management, ecosystem restoration, and water resource planning and management within the LP. Given that 62.1 % of the total LP with a negative trend in annual precipitation has one or more positive EP indices, the underlying upward trends in water erosion and sediment yield should be taken into account in catchment management efforts.

5.2 Uncertainty in EP identification

The uncertainties in the identification and assessment of EP events stem from two aspects: (1) the stochasticity in climate (Miao et al., 2019) and (2) the methodology (Papalexiou et al., 2013). For the former, significant spatiotemporal variations occur in EP events as a result of varying geographical and meteorological conditions (Pinya et al., 2015). Extreme precipitation events are relatively rare, poorly predictable, and often short-lived, resulting in uncertainty in EP event identification. The uncertainties regarding EPT determination from parametric and nonparametric methods has been discussed in Sect. 2.

Figure 8 shows the results of the EPF obtained using nonparametric methods for all 87 stations over the LP during the period from 1961 to 2015. Large variances among the results, calculated at different percentile levels, are shown in Fig. 8a–c. Trivially, the EPTs are smaller but with larger EPFs for lower percentiles. The “±3 SD (standard deviation)” method (Fig. 8d) provided results with a similar variance among stations in comparison with the universal multifractal method (Fig. 8g). The EPTs determined by individual methods generally increase as annual precipitation increases. As shown in Figs. 8e–f and 3a, no precipitation event exceeded 50 mm d⁻¹, as the mean annual precipitation is <200 mm in the northwestern LP. A 50 mm d⁻¹ threshold is probably suitable for the southeast LP, which has higher mean annual precipitation, whereas a 25 mm d⁻¹ threshold may be more suitable for some stations in the northwestern LP. Therefore, regardless of the varying geographical and meteorological conditions, the selection of these thresholds can be quite subjective and empirical. Note that, although a similar variance of EPTs among stations can be obtained using individual methods, the spatial causes for hazards situations cannot be theoretically explained by these methods in Fig. 8a–f.

Figure 9The passing rates of the goodness-of-fit test for individual distribution functions, with EP data series selected by different preset thresholds. (a1) The K–S test, (a2) the A–D test, and (a3) the C–S test for different distribution functions using preset percentile thresholds. (b1) The K–S test, (b2) the A–D test, and (b3) the C–S test for different distribution functions using thresholds of fixed values. The significance level is 0.05. The symbol lines of the passing rate of Gumbell functions in (a1)–(a3) were arbitrarily offset upward by −5 units, respectively, in order to separate them.

Download

Parametric methods require a predetermined threshold value, above which the data can be chosen as the EP series if the data series passed the goodness-of-fit test. As shown in Fig. 9, both fixed values and percentiles were adopted to preset EPT. The selected rainfall data series were fitted to the gamma, GPA (generalized Pareto distribution), Gumbel, Pearson type III, GEV (generalized extreme value distribution), and the GNO (generalized normal) distributions, whose parameters were estimated utilizing the L-moments method (Haddad et al., 2011) at a 0.05 significance level, using goodness-of-fit tests including the K–S (Kolmogorov–Smirnov test), A–D (Anderson–Darling K-Sample test) and C–S (Pearson's Chi-squared test) tests. As shown in Fig. 9a1–a3 and b1–b3, the results of the three tests are similar, although they differ with respect to the details for the preset fixed value and percentile thresholds.

Further, these results for different distribution functions are quite different from each other. As shown in Fig. 9a1 and a2 (by the K–S and A–D tests), the passing rates from the GEV, Gumbell, GNO, and Pearson type III distribution functions are high, whereas there are very low passing rates from the GPA and Gamma functions. In addition, the passing rates are different between the preset methods of percentiles and fixed values. As shown in Fig. 9, the GEV and Gumbell distribution function have high passing rates for EP series obtained by preset percentiles when the percentile is less than or equal to 99 % (Fig. 9a1, a2), whereas the passing rates for the series obtained using fixed values decrease with increasing values (Fig. 9b1, b2). We also found that these distribution functions are not sensitive to the percentile or fixed value changes. These findings indicates that the fitting accuracy can be greatly affected by the selection of the extreme value distribution functions, the goodness-of-fit tests, and the methods used for the EPT preset. Liu et al. (2013) noted that the fitting accuracy is also affected by the size of rainfall series. Thus, unavoidably, applications of parametric methods also depend on personal subjectivity and empiricism. We have attempted to explore EP using fixed values of spatiotemporal variation in precipitation analysis in the LP; however, we found that the results did not explain the rainfall-induced natural hazards well nor did they agree (spatially) with plot experiments by Wan et al. (2014) or results obtained using fixed values (Xin et al., 2009) and percentiles (Li et al., 2010a, b). These uncertainties may be the reason why prior EP studies over the LP tend to disagree with each other.

As noted by Pandey et al. (1998) and Douglas and Barros (2003), these methodological uncertainties arise due to the wide gap between mathematical modeling and the physical understanding of precipitation processes. As previously mentioned, the multifractal technique can be used to describe the statistical probability and physical processes associated with observed data (Lovejoy and Schertzer, 2013; Tessier et al., 1996), while the scale invariance of multifractals enables the multifractal technique to also overcome the influence of the sample size (Pandey et al., 1998; Tessier et al., 1996). Further, the segmentation algorithm helps to overcome the problem of uncertainty. In the present study, the general correspondence and the specific divergences between the EPT and precipitation isohyets (Fig. 3) further exhibits the varying meteorological and geographical influences. As shown in Table 2, by fitting EP series derived using universal multifractals to the six distribution functions, the 100 % passing rate of the goodness-fit test strongly supports that the universal multifractal approach is advanced in identifying EP events. Overall, the universal multifractal method provides a much superior approach to addressing uncertainties and providing a unique set of EPTs.

Table 2Passing rates of the goodness-of-fit test for EP events determined using the universal multifractals method.

Download Print Version | Download XLSX