# Fractional Linear Reservoir Model as Elementary Hydrologic Response Function

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Linear Reservoir Model

#### 2.2. Fractional Calculus

#### 2.3. Mittag–Leffler Function

## 3. Fractional Linear Reservoir Model

#### 3.1. Impulse Response Function

#### 3.2. Unit Pulse Response Function

## 4. Results and Discussions

#### 4.1. Methodology

#### 4.2. Results

#### 4.3. Discussions

## 5. Conclusions

- (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.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Impulse response function of fractional linear reservoir model: (

**a**) $0<\alpha \le 1$; (

**b**) $1<\alpha \le 2$.

**Figure 2.**Hydrologic response function of the fractional linear reservoir model: (

**a**) unit step re-sponse function, (

**b**) unit pulse response function.

**Figure 3.**Overview of rainfall–runoff event: (

**a**) ${Q}_{total}\left(t\right)$ in normal scale, (

**b**) ${Q}_{total}\left(t\right)$ in logarithmic scale.

**Figure 4.**IUH of fractional linear reservoir model for the basin of interest: (

**a**) semi-logarithmic scale; (

**b**) double-logarithmic scale.

Linear Reservoir | Fractional Linear Reservoir | |
---|---|---|

$h\left(t\right)$ | $\frac{1}{K}u(t-{T}_{l}){e}^{-\frac{t-{T}_{l}}{K}}$ | $\frac{1}{{K}^{\alpha}}{u(t-{T}_{l})(t-{T}_{l})}^{\alpha -1}{E}_{\alpha ,\alpha}\left[{-\left(\frac{t-{T}_{l}}{K}\right)}^{\alpha}\right]$ |

$sh\left(t\right)$ | $\text{}(t-{T}_{l})\left(1-{e}^{-\frac{t-{T}_{l}}{K}}\right)$ | $\frac{1}{{K}^{\alpha}}{u(t-{T}_{l})(t-{T}_{l})}^{\alpha}{E}_{\alpha ,\alpha +1}\left[{-\left(\frac{t-{T}_{l}}{K}\right)}^{\alpha}\right]$ |

Linear Reservoir | Fractional Linear Reservoir | |
---|---|---|

$K\left(\mathrm{h}\right)$ | 10.669011 | 10.669011 (10.901208) |

$\alpha $ | 0.947744 (0.945768) |

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 |

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**MDPI and ACS Style**

Yoon, Y.-J.; Kim, J.-C.
Fractional Linear Reservoir Model as Elementary Hydrologic Response Function. *Water* **2023**, *15*, 4254.
https://doi.org/10.3390/w15244254

**AMA Style**

Yoon Y-J, Kim J-C.
Fractional Linear Reservoir Model as Elementary Hydrologic Response Function. *Water*. 2023; 15(24):4254.
https://doi.org/10.3390/w15244254

**Chicago/Turabian Style**

Yoon, Yeo-Jin, and Joo-Cheol Kim.
2023. "Fractional Linear Reservoir Model as Elementary Hydrologic Response Function" *Water* 15, no. 24: 4254.
https://doi.org/10.3390/w15244254