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.