Fractional Linear Reservoir Model as Elementary Hydrologic Response Function

Abstract

This paper presents a fractional linear reservoir model as the elementary response function of hydrologic systems corresponding to the classical linear reservoir model and tests its applicability to rainfall–runoff modeling. To this end, we formulate a fractional linear reservoir model in terms of fractional calculus following the same procedure as the classical linear reservoir model and, at the simplest level, compare its performance of rainfall–runoff modeling with the linear and nonlinear reservoir models. The impulse response function of a fractional linear reservoir model, a probability density function (PDF) following the Mittag–Leffler distribution, shows nonlinearity due to its time-variant behavior compared to that of a linear reservoir model. In traditional linear hydrologic system theory, the lag and route version of a fractional linear reservoir model produces the fast-rising and slow-recession of runoff hydrographs, implying the mixed response of linear and nonlinear reservoir models to rainfall. So, a fractional linear reservoir model could be considered a fundamental tool to effectively reflect the nonlinearity of rainfall–runoff phenomena within the framework of the linear hydrologic system theory. In this respect, the fractional order of the storage relationship specifying a fractional linear reservoir model can be viewed as a kind of parameter to quantify the heterogeneity of runoff generation within a river basin.

1. Introduction

The nonlinear properties of rainfall–runoff phenomena are one of the intrinsic attributes of natural systems [1]. It is well known that analytical exact solutions might be hard to obtain from mathematical models to describe water particles’ movement within a river basin via classical calculus due to their inherent nonlinearity. To cope with this, several alternatives have been developed in hydrology; for instance, a linearization of the rainfall–runoff problems through the concept of unit hydrographs [2] or a linear reservoir model [3], as well as a numerical approach to such nonlinear models as the storage function method [4]. However, it could be difficult to figure out the fundamental explanations of rainfall–runoff phenomena only through these circumvent ways due to their own nonlinearity, as mentioned before. Furthermore, since the drainage structure of river basins could be addressed in non-integer spatial dimensions through the fractal property of channel networks [5,6,7], it is unlikely that conducting a systematic approach to rainfall–runoff phenomena will be easy within the framework of classical calculus.

Fractional calculus is a branch of mathematics that deals with the integration and differentiation of the non-integer order, which has recently been widely used in various science and engineering fields [8,9,10,11]. There can also be found recent research on fractional calculus relevant to hydrology, such as [12,13,14,15,16,17,18]. Among those, Borthwick [19] suggests the fractional order derivative of storage relationships with respect to time and, thereby, the potential to extend rainfall–runoff models rooted in linear hydrologic system theory [20,21,22,23]. However, he mainly concentrates on the fractional reformulation of the existing rainfall–runoff models based on classical calculus without inspecting its hydrologic implications. This motivates us to test the hydrologic applicability of fractional calculus to rainfall–runoff modeling. In fact, fractional calculus could be expected to provide a new approach to the existing various mathematical models that have been developed based on classical calculus; therefore, it could be a valuable tool in terms of bridging the theoretical gaps and relaxing the practical restrictions relating to rainfall–runoff modeling.

In this study, we propose a fractional linear reservoir model as the elementary response function of hydrologic systems in correspondence to the classical linear reservoir model. We formulate the response functions of a fractional linear reservoir model in terms of fractional calculus following the same procedure as a linear reservoir model. Then, a comparison was performed with the models based on linear and nonlinear reservoirs in order to test the applicability of a fractional linear reservoir model to rainfall–runoff modeling at the simplest level. It should be noted that the main purpose of this study is not to provide a new rainfall–runoff model in the complete form but to provide an elementary hydrologic response function for river basins, a fractional linear reservoir concept. As commonly observed in many classical rainfall–runoff models, such as [20,21,22,23], the concept of a linear reservoir model has been frequently used as a basic element that consists of more complicated rainfall–runoff models. Therefore, this study tries to suggest a fractional linear reservoir model as an alternative to the classical linear reservoir model, especially focusing on the distinct behavior of the former compared to the one of the latter, in order to provide a new essential tool for the reasonable and systematic construction of rainfall–runoff models.

2. Theoretical Background

2.1. Linear Reservoir Model

A linear reservoir model [3] consists of a continuity equation and a storage relationship.

$$i\left(t\right)-q\left(t\right)=\frac{dS\left(t\right)}{dt}$$

(1)

$$S\left(t\right)=Kq\left(t\right)$$

(2)

where $i\left(t\right)$, $q\left(t\right)$, and $S\left(t\right)$ represent the inflow, outflow, and storage of a linear reservoir, while $K$ is a proportional constant in the temporal dimension that specifies a linear relationship between $q\left(t\right)$ and $S\left(t\right)$. Assuming $i\left(t\right)$ to be Dirac’s delta function $\delta \left(t\right)$, $q\left(t\right)$ amounts to an impulse response function $h\left(t\right)$ of a linear reservoir model. Thereby, substituting Equation (2) into Equation (1) results in a nonhomogeneous linear ordinary differential equation of the first-order for $h\left(t\right)$ with respect to time

$$\frac{dh\left(t\right)}{dt}+\frac{1}{K}h\left(t\right)=\frac{1}{K}\delta \left(t\right)$$

(3)

Taking Laplace transforms of Equation (3) yields

$$H\left(s\right)=\frac{1}{K}\left(\frac{1}{s+\frac{1}{K}}\right)+h(0)\left(\frac{1}{s+\frac{1}{K}}\right)$$

(4)

where $H\left(s\right)$ is the counterpart of $h\left(t\right)$ in the Laplace transform domain. Considering a linear reservoir in the empty state at the initial ($S\left(0\right)=0$, $h\left(0\right)=0$) and, then, omitting the second term on the right-hand side of Equation (4), $H\left(s\right)$ can be inversely transformed into

$$h\left(t\right)=\frac{1}{K}{e}^{-\frac{t}{K}}$$

(5)

Equation (5) is the elementary response function of the linear hydrologic system to rainfall, the so-called instantaneous unit hydrograph (IUH) of a linear reservoir model, along with a linear channel model.

2.2. Fractional Calculus

Fractional calculus has been built on the formula of Cauchy’s repeated integral for an arbitrary function $f\left(t\right)$ [24]

$${}_{a}I_t^{n}f\left(t\right)=\frac{1}{\left(n-1\right)!}{\int }_a^t{\left(t-\tau \right)}^{n-1}f\left(\tau \right)d\tau =\frac{1}{\Gamma \left(n\right)}{\int }_a^t{\left(t-\tau \right)}^{n-1}f\left(\tau \right)d\tau$$

(6)

where $I$ denotes an integration operator, with $\Gamma \left(·\right)$ being the gamma function. Equation (6) indicates the n-fold integration of $f\left(t\right)$ on $a\le \tau \le t$. However, it is noted that the first equality in Equation (6) implies $n$ in integer numbers to the second equality in Equation (6), which holds for $n$ in real numbers with the help of $\Gamma \left(·\right)$. Therefore, the $\alpha$th-order fractional derivative of $f\left(t\right)$ can be defined as follows.

$${}_{a}D_t^{\alpha }f\left(t\right)=\frac{1}{\Gamma \left(n-\alpha \right)}{\int }_a^t{\left(t-\tau \right)}^{n-\alpha -1}{f}^{\left(n\right)}\left(\tau \right)d\tau$$

(7)

where $D$ is a differentiation operator with $\alpha$ being real numbers. It can be seen that the subscripts in front and back of $I$ in Equation (6) and $D$ in Equation (7) specify the interval of time of interest according to the standard notation of fractional calculus. Fractional integration is based on the mathematical structure shown in Equation (6), whereas fractional differentiation undergoes various development stages originating from the classical definition of Riemann–Liouville [25,26,27]. Equation (7) is taken from Caputo’s definition of fractional derivatives, which has recently been widely used in diverse fields owing to the better availability of the observable initial conditions in the real world than the others. Ref. [28] points out that fractional derivatives can provide the description of memory and hereditary properties of various materials and processes as recognized in the integral on the right-hand side of Equation (7), and this is the main advantage of fractional derivatives in comparison with classical integer-order models. Therefore, we focus on the applicability of fractional calculus to rainfall–runoff modeling with Equation (7); therefore, this allows for considering the global variation of hydrologic variables over time, whereas the traditional approach in terms of classical calculus only the local variation of hydrologic variables with respect to time.

2.3. Mittag–Leffler Function

The exponential function in Equation (5) can be expressed as the sum of the infinite series:

$$e^z={\sum }_{k=0}^{\mathrm{\infty }}\frac{z^k}{k!}={\sum }_{k=0}^{\mathrm{\infty }}\frac{z^k}{\Gamma \left(k+1\right)}$$

(8)

The Mittag–Leffler function $E_{\alpha ,\beta }\left(z\right)$ is a generalization of Equation (8) [29,30]

$$E_{\alpha ,\beta }\left(z\right)={\sum }_{k=0}^{\mathrm{\infty }}\frac{z^k}{\Gamma \left(\alpha k+\beta \right)}$$

(9)

where $\alpha$ and $\beta$ are two parameters of $E_{\alpha ,\beta }\left(z\right)$, which could be real or complex numbers. Letting $\alpha =\beta =1$, Equation (9) reduces to Equation (8) ($E_{\mathrm{1,1}}\left(z\right)=e^z$).

The Laplace transform relevant to Equation (9) can be defined as follows [28,29,30]

$$L\left[t^{\beta -1}{E}_{\alpha ,\beta }\left(-\lambda t^{\alpha }\right)\right]=\frac{s^{\alpha -\beta }}{s^{\alpha }+\lambda }$$

(10)

where $L\left[·\right]$ denotes a Laplace transform operator.

3. Fractional Linear Reservoir Model

3.1. Impulse Response Function

Following [19], let the first-order derivative of $S\left(t\right)$ with respect to time in Equation (1) be rewritten in terms of the $\alpha$th-order fractional derivative of $S\left(t\right)$

$$\frac{dS\left(t\right)}{dt}=K\frac{dq\left(t\right)}{dt}\to {}_{0}D_t^{\alpha }S\left(t\right)=K^{\alpha }{}_{0}D_t^{\alpha }q\left(t\right)$$

(11)

where the lower limit of fractional derivatives ($a$ in Equation (7)) goes to 0 for convenience, which is denoted according to the standard notation of fractional calculus similar to Equation (7). It can be noticed that $K$ is raised by the power of $\alpha$ in order to preserve dimensionality. Thereby, corresponding to Equation (3), we can have a fractional nonhomogeneous linear ordinary differential equation of the $\alpha$th-order for $h\left(t\right)$ with respect to time

$${}_{0}D_t^{\alpha }h\left(t\right)+\frac{1}{K^{\alpha }}h\left(t\right)=\frac{1}{K^{\alpha }}\delta \left(t\right)$$

(12)

Similar to Equation (4), Equation (12) can be rearranged for $H\left(s\right)$ in the Laplace transform domain

$$H\left(s\right)=\frac{1}{K^{\alpha }}\left(\frac{1}{s^{\alpha }+\frac{1}{K^{\alpha }}}\right)+{\sum }_{i=0}^{m-1}{s}^{\alpha -i-1}{h}^{\left(i\right)}(0)\left(\frac{1}{s^{\alpha }+\frac{1}{K^{\alpha }}}\right);m-1<\alpha <m$$

(13)

where $m$ is integer numbers dependent on $\alpha$. Taking inverse transforms of Equation (13) with the zero initial conditions ($h^{\left(i\right)}\left(0\right)=0$, $i=\mathrm{0,1},\cdots m-1$), we can get the solution to Equation (12) with the help of Equation (10)

$$h\left(t\right)=\frac{1}{K^{\alpha }}{t}^{\alpha -1}{E}_{\alpha ,\alpha }\left[{-\left(\frac{t}{K}\right)}^{\alpha }\right]$$

(14)

From the viewpoint of linear hydrologic system theory [31,32], Equation (14) could be viewed as the IUH of a fractional linear reservoir model. In addition, when $\alpha =1$, Equation (14) reduces to Equation (5); therefore, Equation (14) could be a generalized expression for the impulse response function of a linear reservoir model within the framework of fractional calculus. Figure 1 demonstrates $h\left(t\right)$ in Equation (14) with $K=10 \mathrm{h}$, where Figure 1a depicts $h\left(t\right)$ for $0<\alpha \le 1$ while Figure 1b $h\left(t\right)$ for $1<\alpha \le 2$. We calculate the ordinates of $h\left(t\right)$ with the help of the Matlab code written by [33] in order to estimate $E_{\alpha ,\beta }\left(z\right)$.

One can readily see that Figure 1b, $h\left(t\right)$ for $1<\alpha \le 2$, would be unsuitable for an impulse response function of a hydrologic system due to the presence of oscillations, even including the negative values of $h\left(t\right)$. In fact, this kind of unrealistic behavior of $h\left(t\right)$ has already been reported in previous research, and it is mainly associated with the determination of an optimal IUH in the framework of a linear time-invariant hydrologic system [34,35]. Traditionally, it has been treated as an undesirable byproduct attributed to observational errors in hydrologic practice. In this regard, Figure 1b could provide new insight into the long-standing hydrologic problems related to the emergence of IUH with negative values.

In contrast to Figure 1b, ref. [36] proves Figure 1a, $h\left(t\right)$ for $0<\alpha \le 1$, to be a PDF following the so-called Mittag–Leffler distribution [37,38]. In hydrology, an IUH is regarded as a PDF of water particles’ travel time to the outlet of a river basin [39]. So, Figure 1a can be thought of as a variety of IUHs emerging from the single linear reservoir model with constant $K$ but varying $\alpha$. In this figure, the black bold curve depicts $h\left(t\right)$ for $\alpha =1$, which is equivalent to an impulse response function of a linear reservoir model in Equation (5). It is clearly seen that the remaining curves of $h\left(t\right)$ for various $\alpha$ vary faster with time in the vicinity of the origin but slower in the tail part, in comparison with the black bold one. So, we can expect that a fractional linear reservoir model could produce the impulse response functions in a nonlinear nature, which is characterized by a different rate of variation with time subject to $\alpha$ compared to a linear reservoir model.

3.2. Unit Pulse Response Function

Letting $i\left(t\right)$ in Equation (1) be the unit step function $u\left(t\right)$, $q\left(t\right)$ in Equation (1) corresponds to an S hydrograph $\mathrm{s}h\left(t\right)$, which is generally used in hydrology [40]. Therefore, $\mathrm{s}h\left(t\right)$ of a linear reservoir model can come from integrating Equation (5)

$$\mathrm{s}h\left(t\right)=1-e^{-\frac{t}{K}}$$

(15)

Referencing the first term on the right-hand side of Equation (13), the $\mathrm{s}h\left(t\right)$ of a fractional linear reservoir model can be expressed in the form of Laplace transforms

$$SH\left(s\right)=\frac{1}{K^{\alpha }}\left(\frac{1}{s}\frac{1}{s^{\alpha }+\frac{1}{K^{\alpha }}}\right)$$

(16)

where $SH\left(s\right)$ denotes the Laplace transform of $\mathrm{s}h\left(t\right)$, which can be inversely transformed with the help of Equation (10)

$$\mathrm{s}h\left(t\right)=\frac{1}{K^{\alpha }}{t}^{\alpha }{E}_{\alpha ,\alpha +1}\left[{-\left(\frac{t}{K}\right)}^{\alpha }\right]$$

(17)

when $\alpha =1$ Equation (17) is converted into Equation (15) [28]. So, the unit pulse response function of both linear reservoir models, $U\left(t\right)$, can be formulated by using Equations (15) and (17)

$$U\left(t\right)=sh\left(t\right)-sh\left(t-∆t\right)$$

(18)

Equation (18) can be seen equivalent to the duration unit hydrograph with the finite rainfall duration of $∆t$ [40]. So, we can expect that given an effective rainfall hyetograph $R_m$, a direct runoff hydrograph $Q\left(t\right)$ can be estimated by using a discrete convolution of $U\left(t\right)$ with $R_m$ in the exact same way as unit hydrographs

$$Q_n={\sum }_{m=1}^{n\le M}{R}_m{U}_{n-m-1}$$

(19)

where $t=n∆t$, so that $n$ and $m$ are indices for discrete time steps with $M$ being the total rainfall duration.

Figure 2 illustrates various $\mathrm{s}h\left(t\right)$ and $U\left(t\right)$ for $0<\alpha \le 1$, which are relevant to the PDF of the Mittag–Leffler distribution depicted in Figure 1a. Despite the traditional assumption of a finite base time of unit hydrographs [2], the base time of $h\left(t\right)$ is theoretically infinite. So, it has been practically approximated as the time in which the value of $\mathrm{s}h\left(t\right)$ closely approaches unity within the allowable tolerance (for instance, $\mathrm{s}h\left(t\right)=0.99$ in HEC-1). In this respect, Figure 2a implies that the base time of $h\left(t\right)$ could grow extremely long depending on $\alpha .$ In this figure, $\mathrm{s}h\left(t\right)$ for $\alpha =1$ (the black bold curve) already arrives nearby the value of unity at the lapsed time of about $50 \mathrm{h}$, whereas the remaining ones do not reach the unity value within the time span of $100 \mathrm{h}$. The long span of the base time for direct runoff hydrographs often happens in the separation procedure of total runoff hydrographs when using a straight line. To avoid this, several methods have been recommended for base flow separation, such as the fixed base method and variable slope method in the hydrologic literature [40]. However, they are known to be empirical ones and, furthermore, there is no evidence to guarantee that they are free of subjectivity in determining the cease point of direct runoff on recession curves. Rather, we can say that the straight line method could yield the most objective estimate of base flow, although it is the simplest among the others. So, if allowing the validity of the straight line method, Figure 2a implies that it could be possible for the span of base time to be much longer than the commonly acknowledged one in hydrologic practice. This might also be reminiscent of the old water paradox [41] in which the extremely long travel time of water particles to the outlet is reported within the framework of isotope hydrology.

4. Results and Discussions

4.1. Methodology

In order to test the applicability of a fractional linear reservoir model to rainfall–runoff modeling at the simplest level, we picked an illustrative example problem concerned with the storage function method from the textbook of hydrology in Korea [42]. The storage function method is the most frequently used rainfall–runoff model in the hydrologic practice of Korean water management, which can be written by a continuity equation and a nonlinear storage relationship

$$i\left(t-T_l\right)-q\left(t\right)=\frac{dS\left(t\right)}{dt}$$

(20)

$$S\left(t\right)=K{q}^p\left(t\right)$$

(21)

where $T_l$ is a basin lag time, with $p$ being an exponent that specifies a nonlinear relation of $q\left(t\right)$ to $S\left(t\right)$. Therefore, this model could be viewed as a kind of lag and route model [43]. However, unlike the traditional lag and route model, the storage function method does not lend itself to an explicit form of a response function to rainfall due to Equation (21), so a finite difference scheme has been used for routing of a nonlinear reservoir

$$\frac{K{q}_n^p}{∆t}+\frac{q_n}{2}=\frac{S_{n-1}}{∆t}-\frac{q_{n-1}}{2}+i_{n,l}$$

(22)

where the subscript $n$ stands for an index of discrete time steps ($t=n∆t$) with $l$ denoting $T_l$. To take account into the initial loss of rainfall, the storage function method is equipped with the saturation rainfall $R_{sa}$ and runoff ratio $f$, by which total drainage area of $A$ is divided into the runoff area ($=fA$) and the percolation area ($=\left(1-f\right)A$). Through Equation (22), $Q\left(t\right)$ is estimated only over the runoff area until total rainfall arrives at $R_{sa}$, while for both areas after total rainfall arrives at $R_{sa}$. Additionally, then, two runoff components are summed to each other. So, a total of five parameters is required to perform the storage function method.

Similar to the storage function method, we extend a linear reservoir model in Equation (5) as well as a fractional linear reservoir model in Equation (14) to lag and route versions in

Table 1. Lag and route versions of linear reservoir models.

	Linear Reservoir	Fractional Linear Reservoir
$h\left(t\right)$	$\frac{1}{K}u(t-T_l)e^{-\frac{t-T_l}{K}}$	$\frac{1}{K^{\alpha }}{u(t-T_l)(t-T_l)}^{\alpha -1}{E}_{\alpha ,\alpha }\left[{-\left(\frac{t-T_l}{K}\right)}^{\alpha }\right]$
$sh\left(t\right)$	$(t-T_l)\left(1-e^{-\frac{t-T_l}{K}}\right)$	$\frac{1}{K^{\alpha }}{u(t-T_l)(t-T_l)}^{\alpha }{E}_{\alpha ,\alpha +1}\left[{-\left(\frac{t-T_l}{K}\right)}^{\alpha }\right]$

. It is noted that the former, in the same form as the traditional lag and route model, has two parameters of $K$ and $T_l$, whereas the latter three parameters of $K$, $\alpha$, and $T_l$. For the sake of simplicity, we adopt the same $T_l$ as the one used in the storage function method to both of models in lag and route versions. Then, we estimated the remaining parameters based on the golden section search method to minimize the sum of the squared errors by using the rainfall–runoff data in the example problem.

4.2. Results

Figure 3 depicts an overview of the rainfall–runoff event in the example problem considered in this study, which has been originally designed to demonstrate the performance of the storage function method concerning the river basin with a drainage area of $373.6 {\mathrm{k}\mathrm{m}}^2$ [42]. In this figure, the red curve with circles represents the observation of total runoff hydrographs $Q_{total}\left(t\right)$ while the black one is coming from the storage function method with the prescribed parameters for the demonstration; $K=19.1$, $p=0.825$, $T_l=3 \mathrm{h}$, $f=0.7$, and $R_{sa}=160 \mathrm{m}\mathrm{m}$ [42]. It is noted that the base flow, the deep blue curve with circles, is separated from the observation by means of a straight line.

Table 2. Results of parameter estimation for lag and route models.

	Linear Reservoir	Fractional Linear Reservoir
$K \left(\mathrm{h}\right)$	10.669011	10.669011 (10.901208)
$\alpha$		0.947744 (0.945768)

lists the estimated parameters of two lag and route models presented in Table 1, in which the values in parenthesis of the third column come from the multi-dimensional golden section search method recently suggested by [44]. In fact, the golden section search method has been originally developed to solve the one-dimensional search problems; therefore, we estimate $K$ (the value in the second column of Table 2) for the lag and route version of a linear reservoir model with the same $T_l$, such as the one used in the storage function method in the first step. Then, we successively estimated $\alpha$ (the value outside of parenthesis in the third column of Table 2) for the lag and route version of a fractional linear reservoir model with the same $K$ and $T_l$ as the previous step through the golden section search method again in the second step. We use $U_i$ in a dimension of $N-M+1$ following Equation (19) in the first step, in which $U_i$ denotes the discrete counterpart of $U\left(t\right)$ with $N$ being the base time of direct runoff. However, in the second step, we use $U_i$ in a dimension of 30,000. This is intended to guarantee that every $\mathrm{s}h\left(t\right)$ for any value of $\alpha$ within the search interval approaches at least 0.99 at each iteration procedure. Then, we applied the multi-dimensional golden section search method [44] to the lag and route version of a fractional linear reservoir model with $U_i$ in a dimension of 30,000 in order to validate the estimates from the previous successive search procedures. Comparing two parameter sets for a fractional linear reservoir model in the third column of Table 2, a significant difference between them cannot be found. Therefore, we choose the parameter set coming from the successive estimation processes with the one-dimensional golden section search method mentioned above.

It can be seen in Figure 3a that all three models tend to underestimate peak discharges relative to the observed one. Generally, the convolution operation in Equation (19) is well known to unduly attenuate input signals, so the lower peak discharges of the computed hydrographs in Figure 3 might be due to the routing procedure operated in the lag and route models used in this study. However, it is noted that two lag and route models based on linear and fractional linear reservoirs seem to be more suitable in terms of rainfall than the storage function method in that the formers produce faster-rising limb and closer peak discharge to the observation than the latter. So, we could expect that the models based on linear and fractional linear reservoirs could better reflect rainfall variations in the rising limb of runoff hydrographs than those based on a nonlinear reservoir.

In contrast, rather different features can be found in terms of the recession limb in Figure 3b, which is depicted on semi-logarithmic paper. In this figure, the storage function method produces recession curve that is more similar to the observation than the remaining two models. Nevertheless, it is also noted that the recession curve from a linear reservoir model reduces the base flow swiftly, whereas a fractional linear reservoir model makes a slower drawdown of the recession curve along with the trajectory of the red curve with circles despite the lesser magnitude of discharge than the observation.

In summary, a linear reservoir model would quickly respond to rainfall in both rising and recession periods of runoff processes while more slowly to a nonlinear reservoir in both periods. However, a fractional linear reservoir model shows a mixed behavior with fast-rising and slow-recession, so it might be located somewhere between a purely linear reservoir and purely nonlinear reservoir models. It is interesting that though a fractional linear reservoir model is operated in the same framework of the linear hydrologic system through Equation (19), it can effectively produce the nonlinear property of runoff processes.

Table 3. MBE and RMSE for lag and route models.

	Storage Function Method	Linear Reservoir	Fractional Linear Reservoir
MBE (${\mathrm{m}}^3/\mathrm{s}$)	6	15	12
RMSE (${\mathrm{m}}^3/\mathrm{s}$)	72.9	83.4	71.7

details the mean biased error (MBE) and the root mean squared error (RMSE) of the three calculated hydrographs compared to the observation, as shown in Figure 3. It is noted that the fractional linear reservoir model could produce a comparable result to that of the storage function method, also showing a slightly better performance than the linear reservoir model.

4.3. Discussions

Figure 4 shows $h\left(t\right)$, which was removed from the translation effect due to $T_l$ for the basin of interest, which is considered in the example problem. It is superposed by $h\left(t\right)$ for $\alpha =1$ and $\alpha =0.9$. Figure 4a is described on semi-logarithmic paper, while Figure 4b is described on double-logarithmic paper. In Figure 4a, the curve in black represents $h\left(t\right)$ for $\alpha =1$, the analytical solution to the linear reservoir model in Equation (5) for the basin of interest. This curve seems to be in a linear form on semi-logarithmic paper due to the exponential function in Equation (5). Therefore, we can think that the curvilinear behavior of the remaining two curves would indicate the nonlinear property of a fractional linear reservoir model, as mentioned in Section 3.1. It can be clearly observed that these two curves increasingly deviate from the black one as time proceeds, so the nonlinearity of those tends to be more profound with time. In [45], the equivalence of the fractional relaxation equations to differential equations with time-varying coefficients is shown. In this context, the deviations of two curves from the black one in Figure 4a could be attributed to the time-varying feature of the proportional constant $K$ of a linear reservoir model, which is usually interpreted as the mean residence time. In this regard, ref. [46] estimates $K$ as follows by applying geomorphologic IUH theory based on a linear reservoir model

$$K=\frac{L}{V}$$

(23)

where $L$ is the geomorphologic characteristic path length of a river basin, with $V$ being the average flow velocity. Equation (23) implies that the time-variant property of $K$ might come from the variation of $V$. According to the saturation excess overland flow mechanism [47], it could be reasonable to assume overland flow with relatively faster $V$, which occurs within the lower region around the channel networks in a river basin, can likely govern the whole runoff process in the vicinity of the origin of Figure 4a. However, as time goes on, the water particles undergoing the subsurface runoff, characterized by a relatively slower $V$, could have a greater contribution to the runoff process in the tail part of Figure 4a. Therefore, the different origins of runoff generation could result in the time-variant property of $h\left(t\right)$ characterized by time-varying $V$, as depicted in Figure 4a. Based on the results above, we could say that a fractional linear reservoir model has the potential to reflect the nonlinear property of rainfall–runoff phenomena through its time-variant characteristics owing to the heterogeneous movement of water particles. In this respect, $\alpha$ can be viewed as a kind of parameter to quantify the heterogeneity of runoff generation within a whole river basin.

Figure 4b clearly depicts the time-variant property of $h\left(t\right)$ with two distinct distributional features. In this figure, the black curve represents $h\left(t\right)$ for $\alpha =1$, which is the same as that shown in Figure 4a; the remaining two curves show a similar shape to the exponential distribution (the black curve) in the vicinity of the origin, whereas a similar shape to the power law distribution is evident in the tail part. Therefore, we can think that, in a fractional linear reservoir model, the travel time distribution of water particles to the outlet tends to follow an exponential distribution initially but follows a power law distribution later. Generally, the power law distribution implies scale invariance within the extremely long range of random variables; therefore, their statistical moments cannot be defined [48]. This is parallel with the infinite characteristic time of transfer functions, as shown in [13]. So, by allowing for the possibility of the long span of base times depicted in Figure 2a, a much wider time window would be required to separate the rainfall–runoff event from the continuous observation records in the context of a fractional linear reservoir model.

5. Conclusions

The following are the noteworthy results of this study.

(1)

The impulse response function of a fractional linear reservoir model has a nonlinear nature, which can be characterized by different rates of variation in terms of time.

(2)

The lag and route versions of fractional linear reservoir models produce the fast-rising and slow-recession of runoff hydrographs, that is, the mixed response of linear and nonlinear reservoir models to rainfall. So, a fractional linear reservoir model could be considered to be an effective tool in terms of reflecting the nonlinearity of rainfall–runoff phenomena within the framework of linear hydrologic system theory.

(3)

The fractional order, specifying a fractional linear reservoir model, can be viewed as a kind of parameter to quantify the heterogeneity of runoff generation within a river basin.

(4)

It could be possible for the span of base time to be much longer than that commonly acknowledged in hydrological practice, so a much wider time window would be required to separate the rainfall–runoff event from the continuous observation records in the context of a fractional linear reservoir model.