The construction of the flow duration curve and the regionalization parameters analysis in the northwest of China

Water resources are one of the most important natural resources. Daily runoff series is the basic data to estimate the water resources in a river basin. However, in many catchments, the daily discharge is not well observed. Therefore, predicting daily runoff time series in ungauged catchments is both important and challenging. The flow duration curve is a characteristic curve that illustrates the relationship between frequency and magnitude of streamflow (Ganora et al. 2009; Cheng et al. 2012; Mendicino & Senatore 2013). Using the flow duration curve, it is possible to estimate the percentage of time that a specified streamflow is equaled or exceeded. The flow duration curve is an important method in hydrological calculations for the estimation of water resources and engineering designs, such as hydropower, shipping and others, because the designs of these projects depend not only on the timing of the flow, but also on the duration of the flow (Liang et al. 2019). There are two commonly used methods to construct flow duration curves: the total duration method and the multi-year average method (Liang et al. 2019). The total duration method more closely represents the change of the runoff within a given period of time (Cheng et al. 2012), while the multi-year average method calculates virtual curves that demonstrate typical rainfall runoffs in a certain basin based on the long-term average (Liang et al. 2019).

In many areas of the world there is not enough long-term observed runoff data. These areas are referred to as ungauged areas (Hrachowitz et al. 2013). Accurate runoff prediction in these areas is almost impossible due to the lack of data (Atieh et al. 2015) because hydrological models need long-term observed runoff to calibrate (Hrachowitz et al. 2013). The flow duration curves of gauged sub-regions can be obtained based on measured runoff data, and are used to obtain the regional flow duration curve-based parameter regionalization, which can be transformed to ungauged sub-regions to meet the needs of water resources engineering design in those areas. There are usually two ways to establish a flow duration curve in an ungauged area: a process-based modeling method and a statistical method (Blum et al. 2017). Although the process-based hydrological models are applied in areas with sufficient streamflow data to investigate the physical features of the basin (Liu et al. 2019), the application of process-based models in ungauged areas is limited due to the lack of necessary data and the uncertainty of basic runoff and climate mechanisms (Yokoo & Sivapalan 2011; Schaefli et al. 2013; Basso et al. 2015; Mueller & Thompson 2016; Reichl & Hack 2017). On the other hand, the statistical model is simple and relatively easy to implement. Many researchers have used statistical models for the development of flow duration curves in ungauged areas. Based on the assumption of logarithmic normal distribution, Li et al. (2010) estimated the distribution parameters of flow duration curves in southeastern Australia, and an exponential model was proposed to fit the geomorphologic and climatic features of the sub basin. Viola et al. (2011) distinguish dry periods and wet periods according to flows, using a three parameters power law to describe the frequency distribution of flows in wet periods in Sicily, Italy. Cheng et al. (2012) found that the three-parameter mixed gamma distribution can fully capture the shape of the flow duration curve and its changes between catchments and years. The possible relationships between the fitted gamma distribution parameters and the climate and physical characteristics of the basin are explored to explain and point out potential physical controls. Zhang et al. (2015) compared the application of the rainfall-runoff model and the flow duration curve method to predict the daily runoff in ungauged catchments of southeastern Australia, and results indicate that both methods can be further improved to simulate daily hydrographs describing the range of flow metrics in ungauged catchments. Blum et al. (2017) researched nearly 400 unregulated, perennial streams across the United States and found that the four-parameter kappa distribution provides a very good representation of daily streamflow across the majority of physiographic regions. Further, for some regions of the US, the three-parameter generalized Pareto and log normal distributions also provide a good approximation to flow duration curves. Mendicino & Senatore (2013) used five-parameter models and two statistical models to develop flow duration curves of 19 sub-basins in southern Italy (Calabria). Regression analysis of the model parameters as functions of the geomorphic and climatic features of the sub basin demonstrated that the statistical model was more reliable. Longobardi & Villani (2013) introduced the base flow index and estimated the distribution parameters of the flow duration curve in the Mediterranean region with the consideration of the influence of basin lithology and climate. Booker & Snelder (2012) studied the basins of the entire New Zealand. Polynomial and probability distribution functions were used to fit the flow duration curves. Stepwise regression analysis and random forest analysis were used to calculate the distribution parameters based on the characteristics of the sub-basins. Atieh et al. (2015) introduced an entropy-based neural network model assuming a logarithm normal distribution for the flow duration curve and predicted the distribution parameters of the flow duration curve based on the geomorphic and climatic characteristics of the basin. It was found that, compared to ordinary neural network models, the accuracy of the location and scale parameters were increased by 7 and 21%, respectively.

The above studies have provided a good theoretical and practical basis for flow duration curve research, but most of the research on flow duration curve is currently located in several countries (Fennessey & Vogel 1990; Castellarin et al. 2004; Booker & Snelder 2012; Boscarello et al. 2016; Cislaghi et al. 2020), such as the United States, Australia and Italy. Due to China's vast area and complex topography, there are also many ungauged catchments. As a result, water conservancy projects in these areas have no runoff data as a reference. These methods provided a new guideline for developing small hydropower in rural areas without runoff data. Northern Shaanxi in China is the main sediment source region of the Yellow River and about 20,000 check dams have been built in this region to reduce the sediment load (Fu et al. 2011; Wang et al. 2015; Fu et al. 2017; Li et al. 2017). New check dams and other projects will be built in the future. Because the catchment areas of the check dams are very small and usually are several km², the virgin discharge of these small catchments were not usually observed in the past.

The motivation of this study is to select an optimal distribution function to describe the flow duration curve in the study area and construct the regional parameter model to calculate the parameter of flow duration curve from geomorphologic and climatic characteristics, and the spatial and temporal characteristics of the parameters will be revealed. The method in this study will provide guidance for construction and planning of soil and water conservancy projects in areas without measured flow data.

The following parts of the paper are organized as follows. The study area, the data, the distribution functions, evaluation indices and regional regression model are described in the following section. In the next section, the distribution function and their parameters are evaluated, and the multivariate linear regression model of distribution parameters is constructed. Then, the spatial distribution and temporal change of the parameters are analyzed followed by the conclusions.

The study area is mainly located in Yulin, Shaanxi, in the northwest of China, and covers a small part of Ningxia and Inner Mongolia as well (Figure 1(a)). The study area is monitored by 19 hydrological stations, which are shown in Table 1. The study area mainly includes the Wuding River Basin (including the Hailiutou River, Heimudou River, Yuxi River, Xiaoli River and Dali River), the Jialu River Basin, the Tuwei River Basin, the Kuye River Basin, the Gushan River Basin, and the Huangfu River Basin (including Shilichang River). The 19 hydrological stations and river basins are shown in Figure 1(b) and Table 1. Table 1 shows the geomorphologic and climatic parameters of each hydrological station, including sub-basin area A (km²) covered by hydrological stations, elevation difference ΔH (m) of sub-basins, mean elevation H (m) of sub-basins, length of main channel L, perennial flow index IPF (–) (Mendicino et al. 2008), average annual precipitation P (mm), average annual precipitation of dry season P_sum (mm), month of minimum flow, mean monthly precipitation in minimum flow month P_mmin (mm), and monthly precipitation of maximum flow month P_n (mm).

In this study, taking Northern Shaanxi in China as the study area, polynomial and probability distribution functions are used to construct the flow duration curve in the study area. Multiple regression analysis of the characteristic parameters of the curve as functions of the corresponding geomorphology and climatic characteristics of the sub-basin is conducted to analyze the relationship between the parameters and the spatial features. Finally, the temporal changes of the model parameters at certain sites are studied to evaluate the impacts of land use change over time.

The daily average flow data of 19 stations is derived from the Yellow River hydrological data (the upper reaches of the Yellow River middle reaches) and are assumed to be virgin flows. The longest data sequence is from 1955 to 2012, a total of 58 years. The shortest data sequence is from 1975 to 2012, a total of 38 years. The length of the data series is sufficient to meet the research requirements. Figure 2 shows the temporal availability of the data series of 19 hydrological stations. The x-axis is arranged according to the number in Table 1. The DEM data used in this study is from the official website of the US Geological Survey, which were extracted from NASA Advanced Spaceborne Thermal Emission and Reflection Radiometer Global Digital Elevation Model (ASTER GDEM) with a spatial resolution of 30 m. The precipitation data (1961–2015) is from China's ground precipitation monthly value 0.5 × 0.5° grid data set (V2.0), on China Meteorological Data Sharing Service Network (http://data.cma.cn).

There are various functions available to simulate a flow duration curve, and the suitable functions are region-specific (Franchini & Suppo 1996; Ganora et al. 2009; Viola et al. 2011). The distribution functions used in this study to fit the flow duration curve are summarized in Table 2. This study uses the total duration method to construct the flow duration curve.

Flow data collected at the 19 hydrological stations were used to select the best distribution function for the Northern Shaanxi Region from the 11 distribution functions listed in Table 2. Tables 3–5 show the values of NSE, RMSRE and R² at each station, respectively. Based on these values, the F03, i.e. the log normal function, is the best distribution function for this region, because the F03 has high NSE, low RMSRE and high R² for most stations. When the log normal function is used, 11 out of the 19 stations have NSE values higher than 0.9, and the NSE values at 14 out of the stations are higher than 0.8. Among the 19 stations, only two hydrological stations have NSE values lower than 0.7, which are Huangfu station (H10) and Wenjiachuan station (H14). At the same time, compared to the other 10 distribution functions, the log normal model has the lowest RMSRE values. For the log normal model, the maximum RMSRE is 0.0118 at Station H11, and the lowest RMSRE is 0.0003 at Station H15.

Table 6 shows the best-fitted parameters of the log normal distributions function (F03) for the 19 hydrological stations.

Figure 3 shows the observed flow and simulated flow duration curves using the log normal model at four typical sites, Baijiachuan station and Dingjiagou station in Wuding River Basin, Gaojiachuan station in Tuwei River Basin and Wenjiachuan station in Kuye River Basin. These four stations control the largest areas in their rivers. For all of the sites, the log normal function fits the observed flow frequency well while this model fits poorly at some stations. This ‘log normal function’ is listed as F03 in Table 2 and it is not the log normal distribution function as we usually know it, which is H08 in Table 2. The LN function in Blum et al. (2017) is the log normal distribution function. So the fitted curves are different from that in Blum et al. (2017).

For this fitting, R² = 0.9351, F = 37.4535, RMSRE = 0.0140.

The regression values of a and b parameters all pass the F_0.05 confidence test (Steiger 2004), which demonstrates that the prediction model proposed in this study is reasonable for the Northern Shaanxi Region of China. The a and b values calculated according to Equations (5) and (6) are compared with the values in Table 6 (as shown in Figure 4), and it is found that the parameter prediction model is relative reasonable.

The spatial distributions of the parameters a and b of 19 hydrological stations in Table 6 are shown in Figures 5 and 6, respectively. As shown in Figure 5, the value of parameter a is high in the east (downstream) and low in the west (upstream). The hydrological stations in the lower reaches of the rivers (into the Yellow River) have higher a value, and the hydrological stations in the middle and western rivers show lower a value. Figure 6 shows that parameter b and parameter a basically show the opposite spatial distribution, which is, the b value is relatively large in the west and relatively small in the east. The changes of parameters a and b are caused by changes in regional climate, geomorphology and sub-catchment area. This can be explained by Equations (5) and (6). Based on the spatial distribution of the parameters a and b, the values of parameters a and b in different sub-basin ranges can be roughly estimated, so that the flow duration curve of any river section in the region can be effectively estimated.

Due to the impacts of human activities on the underlying surface and climate, the geomorphologic and climatic conditions in the area have changed over time. These changes also affect the distribution parameters of the flow duration curves. This section reveals the temporal changes of distribution parameters in the Northern Shaanxi Region. Baijiachuan hydrological station, which covers the entire Wuding River Basin, is used as an example to present the temporal changes of the parameters. The flow duration curves of different periods are analyzed and the values of the parameters are shown in Table 7. Before 1978, the land use was relatively unchanged; after 1978, agricultural area increased quickly. Therefore, the parameters of the flow duration curve are evaluated in different periods as shown in Table 7.

Table 7 demonstrates decreasing trends for both parameters a and b over time and the flow duration curves are shown in Figure 7. These changes may be caused by the change of land use and vegetation status, and the interaction between humans and water (Liu et al. 2015) in the basin, especially reservoir operation (Zhang et al. 2021). Based on the Shaanxi Provincial Statistical Yearbook (1984–2012), it was found that the irrigation water consumption and the construction of reservoirs in the Wuding River Basin showed a gradually increasing and then decreasing trend, peaking in the 1990s. This also led to the longest duration of the low-value flow in Figure 7 from 1995 to 2004. In this process, the hydrological conditions are also changing (Huang et al. 2015).

The simulated flow duration curves by log normal function (F03) at Baijiachuan are plotted in Figure 7. At the medium flow part and high flow part, simulated flow duration curves fit well in the periods of 1995–2004 and 2005–2012, but the low flow part is not fitted satisfactorily. At Baijiachuan (H11 in Figure 3(a)), the best function is F11 (Generalized Pareto distribution function). At Baijiachuan (H11, upper-left figure in Figure 3), the best function is F11 (Generalized Pareto distribution function). The simulated flow duration curves by F11 at Baijiachuan are plotted in Figure 8. At the medium flow part and low flow part, simulated flow duration curves fit well, but the high flow part is not fitted satisfactorily. So perhaps the performance of the current functions for the flow duration curve is not good enough in this region. New functions for the flow duration curve should be tested to find a better function to simulate the flow duration curve in the region.

In this study, the flow duration curve of 19 hydrological stations in Northern Shaanxi in China was studied, and it was found that the log normal function is the optimal model among a total of 11 function models to simulate the flow duration curves in Northern Shaanxi. Then the two parameters of the log normal function are studied in parameter regionalization, and it is found that the two parameters have a strong regression relationship with regional climate and geomorphologic characteristics. The spatial distribution of parameter a and b of the log normal model is also evaluated for the entire region, which shows high values of parameter a in the east and low values of parameter a in the west. Parameter b and parameter a showed the opposite spatial distribution, which is that b value is relatively large in the west and relatively small in the east.

Taking Baijiachuan hydrological station as an example, the temporal changes of distribution parameters are analyzed. The study shows that, for the Wuding River Basin, which is covered by the Baijiachuan station, both parameters a and b show a decreasing trend over the study period. This change may be related to the land use, vegetation status, reservoir operation and water abstractions for irrigation.

This study presents an innovative approach to evaluate regionalized parameters of flow duration curves based on geomorphologic and climatic characteristics, which can be used to estimate the water resources and provide guidance for construction and planning of water conservancy projects in the catchments without measured runoff data in China. New functions for the flow duration curve should be tested to find a better function to simulate the flow duration curve in the region, and attention should be paid to the fact that due to the change of land use/land cover, and reservoir operation, the parameters of the flow duration curve will change. Therefore, in the water resources management of the river basin, the construction of the flow duration curve is an on-going task as the evolution of the underlying conditions in a river basin.

This work was partially supported by the National Natural Science Foundation of China (NSFC) (Grant No. 51779203, 51609270, 51939009), Basic Research Plan of Natural Science of Shaanxi Province (2016JQ5105) and Planning project of science and technology of water resources of Shaanxi (2014slkj-04). The authors would like to thank the editor and the anonymous reviewers for their constructive comments.

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