2.2 Definition of dry spell and wet spell
According to the research conducted by Bichet et al. (Bichet and Diedhiou, 2018) when there are two or more consecutive dry days with precipitation less than 0.1 mm or 1 mm, the multi-year average days of the maximum continuous day without precipitation was defined as a dry spell. Similarly, when there are two or more consecutive wet days with precipitation greater than 0.1 mm or 1 mm, and the multi-year days of average maximum continuous precipitation day is defined as a wet spell. In addition, Pendergrass pointed out that different thresholds (0.1mm or 1mm) might lead to different conclusions during the analysis of changes in precipitation characteristics (Pendergrass, 2018). In this article, 1 mm is selected as the threshold for calculation.
2.3 Gamma distribution fitting of hydrological sequences with zero values
Generally speaking, if PDC is represented by an overly complex distribution with multiple parameters, a more accurate curve fitting may be obtained. However, some correlation between different parameters may exist, leading to uncertainty in parameter estimation, which may confuse the physical control on statistical parameters. Therefore, in order to achieve the goal of this paper, a simple statistical distribution (the gamma distribution) was selected. Considering the need for parameter simplicity and the need to link these parameters with climate attributes, a three-parameters continuous probability distribution is adopted, which is determined by the probability of occurrence of (zero value) , shape parameters α and scale parameters β . PDC and FDC are complementary cumulative distribution functions (CCDF) of daily precipitation and daily runoff respectively, which must adapt to the presence of flow (zero value condition), especially in arid areas. Therefore, in this study, we used the following gamma distribution to represent FDC, as shown in Eq. .
Where is the probability of occurrence of zero value (precipitation, flow, etc.); is the probability density function of gamma distribution, and is defined as Eq.
in which α and β are the shape and scale parameters respectively. The probability of exceedance distribution P can be calculated by Eq. , where G-1 is the inverse function of the CCDF.
According to the above formula, controls the zero part of the duration curve, while α and β control the shape of the non-zero part of the duration curve.
The scale parameter β has great impacts on the vertical offset of PDC/FDC. The larger the observed average flow rate, the higher the valueβ . In addition, the shape parameter α essentially controls the slope of FDC. The smaller the value α , the steeper the slope of FDC. The parameter was directly estimated from the observation results, which is the fraction of the number of days with zero precipitation (flow) in the data record, that is, the number of days with zero precipitation (flow) divided by the total number of days in the precipitation (flow) record. The parameters α and β of the function were estimated using the moment method (Saulo et al., 2018) through the relationship between the mean (µ ) and variance (v ) of the function distribution (Eqs. and ).
and are estimated from time series with q >0 (non-zero part of FDC). In addition, R-squared (R2 ) and Nash Sutcliffe efficiency coefficient (Nse ) (Nash and Sutcliffe, 1970) were selected to evaluate the performance of gamma distribution in providing good fitting for non-zero segments of different duration curves in each catchment becauseR2 (Eq. ) only measures the degree of linear correlation, while Nse (Eq. ) represents the matching degree between observations and estimates, and explains the bias at the same time.
where and are the predicted and the observed value, respectively;n represents the total number of days included in those years when representing the long-term time curve, and represents the number of days included in one year when representing the one-year time curve, respectively. and are the average values of observed and predicted values. When the value of R2 and Nse are close to 1, it represents that the gamma distribution fits well.