1. INTRODUCTION
Flow duration curves (FDC) is a statistical expression of the flow changes observed during the period of record (long-term flow duration curves or annual flow duration curves) based on the daily runoff series, which explains the relationship between the flow size and the probability of occurrence (Ghotbi et al., 2020; Leong and Yokoo, 2021). The probability value of a specified flow equal to or exceeding a certain flow size can be read from FDC, so it is essentially the cumulative distribution function of daily flow (Yokoo and Sivapalan, 2011). There are two options about the data recording time of FDC: the total duration method (Cheng et al., 2012) and the multi-year average method (Cheng et al., 2012; Karst et al., 2019; Liang, 2019). Cheng et al. used the three-parameter mixed gamma distribution to characterize the shape of the normalized FDC using the total data record, and observed the change of FDC year by year in eight representative catchments. The total duration method more closely represents the rainfall and runoff of a specific basin, while the fitting curve calculated by the multi-year average method can fully capture the changes of FDC between years (Cheng et al., 2012; Costa and Fernandes, 2021; Liang, 2019). FDC can be used to diagnose the rainfall-runoff response of a watershed to help develop or validate rainfall-runoff models through the transition from precipitation change to runoff (Wang et al., 2023). It can be also used to analyze the similarity and difference between watersheds, establish a model using precipitation and watershed characteristics as input, and predict the hydrological elements of ungauged sub-watersheds from a limited number of stations observed in the past (Burgan and Aksoy, 2022a; Wolff and Duarte, 2021).
In the past, the physical control of FDC involved graphics (non-parametric method), statistics (parametric method) and process-based methods (Ghotbi et al., 2020; Ghotbi et al., 2020; Karst et al., 2019; Müller and Thompson, 2016). The graphic method is based on exploring the relationship between the shape characteristics of FDC and the climate and geomorphologic characteristics of the catchment, such as the steepness or quantile of the curve, to estimate the shape of the FDC which represents the flow condition of unmeasured catchments (MOHAMOUD, 2008). The statistical method aims to fit the empirical FDC through appropriate distribution functions (such as gamma distribution, lognormal distribution, generalized Pareto distribution, kappa distribution, etc.) (Almeida et al., 2021; Burgan and Aksoy, 2022b; Cheng et al., 2012; Ghotbi et al., 2020; Yire Shin, 2022), and then find the quantitative and qualitative relationship between the estimated parameters of the fitting function and the characteristics of the local climate and geographical environment of the measured catchment. Villalobos and Neelin explained why the gamma distribution can well fit the daily precipitation distribution. The control of climate and watershed geographical environment characteristics on FDC always tends to be empirical (Ridolfi et al., 2020). Based on these empirical properties and the diversity of climate and geographical impact factors in different watersheds, the main hydrological processes are not explicitly included. The main hydrological processes are complex and specific in different watersheds and climatic regions, so the follow-up research is mostly the process-based method. The essence of the process-based method is to combine graphic with statistical methods, take into account the dominant process of time flow change, take rainfall series as random input, take seasonal and other meteorological factors as random influence, establish a deterministic model of runoff involving precipitation, and derive the explicit statistical distribution form of FDC (Chouaib et al., 2018; Leong and Yokoo, 2021).
Among these three methods, the parametric method shows that the geographical characteristics of the catchment will affect the shape of FDC. For example, topographic features, including the area, average elevation and gradient of the catchment area (Luan et al., 2021; Yang et al., 2023), have a great impact on the shape of FDC. Stephen Oppong Kwakye and András confirmed the control effect of rainfall events and dry conditions on the FDC of most rivers in West Africa (Kwakye and Bárdossy, 2022). Cheng et al. found that in the control of 197 watersheds in the United States, daily precipitation, probability of non-rainy days and Aridity index (AI ) has a great impact on the parameter α estimated by gamma distribution. Although there may be significant differences in the key control factors of FDCs at different geographical locations and times, the shape of FDCs represents the comprehensive impact of rainfall, topography, soil, geology and other climatic and geomorphologic characteristics.
The main problem of the three methods is that the process of precipitation input and runoff generation involves different time scales (Huang et al., 2020). For example, the runoff generation process usually includes fast land flow and groundwater flow. The fast flow is the hydrological response to the change of precipitation and the result of land flow generated by the mechanism of excess infiltration, which is related to many factors such as rainfall intensity, soil infiltration capacity and early soil moisture content, etc. While the slow flow is related to climate seasonality and is controlled by soil permeability and terrain slope. Therefore, when establishing hydrological models and performing numerical simulation, the total flow was always divided into two parts: fast flow and slow flow, so as to have a deeper understanding of the shape characteristics and process control of FDC. Based on the work of Yokoo and Sivapalan (2011) and Cheng et al. Ghotbi et al. (2020) proposed a new stochastic framework to construct total flow, involving three parts of partition (total, fast and slow flow), fitted their distributions and then combined them together as an FDC model of total flow. In this way, the FDC is usually divided into total flow duration curve (TFDC), fast flow duration curve (FFDC) and slow flow duration curve (SFDC). However, the fast flow is usually highly intermittent, while the slow flow is usually more continuous, which poses a challenge to explain the statistical dependence between slow flow and fast flow. In addition, in previous work, Ghotbi et al. (2020) did not further implement it to explore the climate and landscape controls of FDC. At present, most research about FDC is distributed in many countries such as the United States, Africa and Italy (Chouaib et al., 2019; Cislaghi et al., 2020; Kwakye and Bárdossy, 2022; Over T M, 2018; SMAKHTIN et al., 1997; Ye et al., 2012). Due to China’s vast area, complex terrain, and many unmeasured catchment areas, it is very meaningful to study the FDC of unmeasured catchments across China, and provide a good theoretical and practical basis for its research.
In this paper, based on the continuous daily precipitation data (56yr) of 698 weather stations in China and streamflow data (30yr) of 244 gauging stations in watersheds in the middle and lower Yangtze River basin, we carried out the gamma distribution fitting and parameter estimation of PDC and FDCs, and explored the internal and external interfering factors of each fitting parameter. Firstly, data sources were described for the analyses and a brief overview of methodology adopted were gave. Secondly, the fitting images of PDC and FDCs were shown and the values of R2 and Nse were calculated to evaluate the fitting effect. Then the spatial distribution pattern of fitting parameters which controls the shape of PDC and FDCs were presented. Finally we analyzed the estimated parameters with their correlation with regional meteorological physical characteristics, took the middle and lower Yangtze River basin as an example with base flow segmentation, and studied the relationship, similarity, regional patterns and response mechanism of parameters between PDC and FDCs.