## 2 Materials and Methods
### Fractional Instantaneous Unit Hydrogen
#### 2.2.2 Derivation of Fractional Instantaneous Unit Hydrograph  
According to the Nash confluence theory, the influence of a watershed on net rainfall storage can be viewed as the regulating effect of a series of linear reservoirs connected in series. For the first reservoir, the net rainfall \(h\) in the basin is the inflow and \(Q_{1}\) is the outflow. The continuity equation for the first reservoir can be expressed as [67]  
\[h(t)-Q_{1}(t)=\frac{\mathrm{d}W_{1}}{\mathrm{d}t} \tag{2}\]  
Figure 1: Location of the study area and stations.  
where \(W_{1}\) represents the storage capacity of the first reservoir, and the storage and discharge equation of the linear reservoir is [67]  
\[W_{1}=K_{1}Q_{1}(t) \tag{3}\]  
where \(K_{1}\) is the storage and discharge coefficient of the first reservoir.  
The storage equation of the first reservoir can be obtained by combining the above Equations (2) and (3) [67]:  
\[K_{1}\frac{\mathrm{d}Q_{1}}{\mathrm{d}t}=h(t)-Q_{1}(t) \tag{4}\]  
This equation is a conventional integer order differential equation, and if Caputo fractional order derivatives express the integer order derivatives, then Equation (4) will become the following fractional order differential equation [88]:  
\[K_{1}^{\alpha C}D_{1}^{\alpha}Q_{1}(t)=h(t)-Q_{1}(t) \tag{5}\]  
where \(0<\alpha<1\). After further analysis of Equation (5), the following equation is obtained [88]:  
\[Q_{1}(t)=\frac{1}{1+K_{1}^{\alpha C}D_{1}^{\alpha}}h(t) \tag{6}\]  
Similarly, the outflow of the first reservoir is used as the inlet of the second reservoir. After \(n\) linear reservoirs with the same storage and discharge coefficients, the basin outlet process is obtained [88]:  
\[Q_{n}(t)=\frac{1}{\left(1+K_{0}^{\alpha C}D_{t}^{\alpha}\right)^{n}}h(t) \tag{7}\]  
According to the definition of instantaneous unit hydrograph, when the period tends to infinity, \(h(t)\) is the instantaneous unit net rainfall and \(Q_{n}(t)\) is the instantaneous unit hydrograph \(u(t)\); applying the principle of impulse induction and replacing \(h(t)\) with the impulse response function \(\delta(t)\), Equation (7) becomes the following equation [88]:  
\[u(t)=\frac{1}{\left(1+K_{0}^{\alpha C}D_{t}^{\alpha}\right)^{n}}\delta(t) \tag{8}\]  
Equation (8) is the Laplace transformation. The initial condition is set as \(u(0)=u^{\prime}(0)=\cdots=u^{(n-1)}(0)=0\). Since \(L[\delta(t)]=1\), the following equation is obtained [88]:  
\[L[u(t)]=\left(1+K^{\alpha}s^{\alpha}\right)^{-n}=K^{-n\alpha}s^{-n\alpha} \big{(}1+K^{-\alpha}s^{-\alpha}\big{)}^{-n} \tag{9}\]  
The Laplace inversion of the above equation is further performed to obtain the following fractional instantaneous unit hydrograph [88]:  
\[u(t)=\frac{t^{n\alpha-1}}{K^{n\alpha}}E_{\alpha,n\alpha}^{n}[-[\frac{t}{K}]^{ \alpha}] \tag{10}\]  
where \(E_{\alpha,\beta}^{\gamma}()\) is the generalized Mittag-Leffler function, which is defined as follows [88]:  
\[E_{\alpha,\beta}^{\gamma}(t)=\sum_{\tau=0}^{\infty}\frac{\Gamma(\tau+\gamma)t ^{\tau}}{\Gamma(\gamma)\Gamma(\alpha\tau+\beta)\tau!} \tag{11}\]  
where \(\alpha\), \(\beta\), and \(\gamma\) are parameters greater than \(0\). The derivation of Equation (9) to (10) utilizes the following properties of the Laplace transform of the generalized Mittag-Leffler function [88]:  
\[L\big{[}t^{\beta-1}E_{\alpha,\beta}^{\gamma}(\lambda t^{\alpha})\big{]}=s^{- \beta}\big{(}1-\lambda s^{-\alpha}\big{)}^{-\gamma} \tag{12}\]The Nash instantaneous unit hydrograph is a particular fractional order differential equation case. Accordingly, the Nash instantaneous unit hydrograph should be a specific case of the fractional instantaneous unit hydrograph. When the order \(\alpha=1\) of the fractional instantaneous unit hydrograph, Equation (10) becomes the following [88]:  
\[u(t)=\frac{1}{KT(n)}(\frac{t}{K})^{n-1}\sum_{r=0}^{\infty}\frac{1}{r!}(-\frac{t }{K})^{r}=\frac{1}{KT(n)}(\frac{t}{K})^{n-1}e^{\frac{1}{K}} \tag{13}\]