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

Daily runoff is the data to estimate the water resources in a river. In many catchments, the daily discharge is not well observed. Flow duration curve is an important characteristic of daily runoff, and important for the design of water conservancy projects. In the ungauged catchments, the evaluation of distribution functions and the parameters of ﬂ ow duration curve is a helpful method to understand the characteristics of the ﬂ ow. This study uses data from 19 hydrological stations to evaluate the applicability of 11 distribution functions to simulate ﬂ ow duration curves in the northwest of China. The ﬁ tted ﬂ ow duration curves are evaluated by Nash-Sutcliffe ef ﬁ ciency, the root mean square relative error and the coef ﬁ cient of determination. The evaluation shows that, among the 11 distribution functions, the log normal model is the most suitable model to construct ﬂ ow duration curves of 19 hydrological stations. Based on a multivariate linear regression model, a regional model of distribution parameters is constructed, including functions of watershed geomorphologic and climatic characteristics. The analysis of Baijiachuan hydrological station shows that the parameters a and b showed a decreasing trend. This study presents an innovative approach to evaluate regionalized parameters of ﬂ ow duration curves considering the impacts of geomorphologic and climatic characteristics.


INTRODUCTION
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 stream- 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. ).
In many areas of the world there is not enough longterm observed runoff data. These areas are referred to as ungauged areas (Hrachowitz et al. ). Accurate runoff prediction in these areas is almost impossible due to the lack of data (Atieh et al. ) because hydrological models need long-term observed runoff to calibrate (Hrachowitz et al. ). 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. ). 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. ), 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 &  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.

Study area
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    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.

Materials
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 where O i is the observed data on day i, P i is the predicted value for day i, and O is the mean of the observed data for the entire period (day i to N), N is the total number of daily observations.
NSE is unity minus the ratio of the mean square error to the variance in the observed data, and ranges from -∞ to 1.0.
High NSE and low RMSRE indicate a suitable fit of the distribution function to the observed flow duration curve. If NSE is larger than 0.7, the models are considered as suitable (Nash & Sutcliffe ; Gupta et al. ).
The root mean square relative error, RMSRE is calculated as: The coefficient of determination R 2 is calculated as: where, P is the mean of the predicted data for the entire period (day i to N).

Regional regression model
Multiple linear regression models are usually used to study the relationship between a dependent variable and multiple independent variables. If the dependence of the two can be described in a linear form, a multivariate linear model can be established for analysis. This study applies multivariate linear regression analysis to evaluate the impacts of geomorphic and climatic factors of sub-basins on the parameters of polynomial model and probability distribution function. The regression model is expressed as: where θ are the estimated parameters, such as a, b, μ, σ, k, β in Table 2; x i , i ¼ 1, 2, . . . , n are the variables representing the impacts of geomorphologic and climatic characteristics in the regional model, A 0 , A 1 , . . . , A n are the equation coefficients of the regional predictive models, ξ is the error term of the model. In this study, 10 variables are used to represent the geomorphologic and climatic characteristic as shown in Table 1, namely, n is equal to 10.

Determination of parameters for flow duration curve and evaluation of distribution function
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      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

Impacts of geomorphological and climatic factors
Based on the best-fitted values of the model parameters (Table 6), multivariate linear regression of distribution   (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.

Spatial distribution of the parameters
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    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.    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.

CONCLUSIONS
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