# Integrating XAJ Model with GIUH Based on Nash Model for Rainfall-Runoff Modelling

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## Abstract

**:**

## 1. Introduction

## 2. Study Area and Data Collection

#### 2.1. Study Area

^{2}with abundant natural resources holding the average annual runoff volume at 1980 m

^{3}/s ranking at the seventh among the major rivers all around China, containing the percentage of land-covered forests as 59%, basically being classified as pinus massoniana forest, the evergreen broad-leaved forest, Chinese fir, and bamboo forest.

^{2}, at the land slope ranging from to 0.0015 to 1.04. Generally speaking, the higher the order of a river, the gentler the reflected channel slope. The total area of the Jianyang catchment is 3253 km

^{2}and the land slope varies from 0.0012 to 0.87. As for the Shuiji watershed, it is the largest one covering an area of 3470.5 km

^{2}, with the land slope from 0.0009 to 0.77. Figure 1 shows the delineation maps of the study areas.

#### 2.2. Hydrologic Data and Preprocessing

## 3. Methodology

#### 3.1. XAJ Model

- (1)
- when $\mathrm{WU}+\mathrm{P}\ge \mathrm{EP}$,$$\mathrm{EU}=\mathrm{EP},\text{}\mathrm{EL}=0,\text{}\mathrm{ED}=0$$
- (2)
- when $\mathrm{WU}+\mathrm{P}<\mathrm{EP}$ and $\mathrm{WL}\ge \mathrm{C}\times \mathrm{WLM},$$$\mathrm{EU}=\mathrm{WU}+\mathrm{P},\text{}\mathrm{EL}=(\mathrm{EP}-\mathrm{EU})\times \frac{\mathrm{WL}}{\mathrm{WLM}},\text{}\mathrm{ED}=0$$
- (3)
- when $\mathrm{WU}+\mathrm{P}<\mathrm{EP},\mathrm{C}\times (\mathrm{EP}-\mathrm{EU})\le \mathrm{WL}<\mathrm{C}\times \mathrm{WLM},$$$\mathrm{EU}=\mathrm{WU}+\mathrm{P},\text{}\mathrm{EL}=\mathrm{C}\times (\mathrm{EP}-\mathrm{EU}),\mathrm{ED}=0$$
- (4)
- when $\mathrm{WU}+\mathrm{P}<\mathrm{EP},\text{}\mathrm{WL}\mathrm{C}\times (\mathrm{EP}-\mathrm{EU}),$$$\mathrm{EU}=\mathrm{WU}+\mathrm{P},\mathrm{EL}=\mathrm{WL},\mathrm{ED}=\mathrm{C}\times (\mathrm{EP}-\mathrm{EU})-\mathrm{EP}$$

#### 3.2. GIUH based on Nash Model

#### 3.2.1. GIUH Derivation

_{p}u

_{p}and t

_{p}are expressed, as below:

_{Ω}(km) is a scale variable to describe the size of a watershed, which stands for the length of the highest order in the catchment. Thus, v (m/s) is the mean flow velocity. The product of Equations (9) and (10), which is defined as $H$, stated as:

#### 3.2.2. Estimation of the Average Flow Velocity

- ${S}_{m}$: the slope of the drainage area, m/m.
- $L$: the sum of the mean length of each order channel, approximate to the length of flow concentration, m, $L\approx {L}_{c}$.

#### 3.3. Calibration and Validation on Model Parameters

#### 3.4. Criteria on the Model Assessment

^{3}/s); ${Q}_{o}$: observed peak flow (m

^{3}/s); $R{D}_{c}$: calculated runoff depth (mm); $R{D}_{\mathrm{o}}$: observed runoff depth (mm); ${Q}_{oi}$ the observed discharge (m

^{3}/s); ${Q}_{ci}$: the calculated discharge (m

^{3}/s); $\overline{{Q}_{0}}$: is the mean of observed data ${Q}_{oi}$ (m

^{3}/s); and, $\mathrm{i}$: time interval. It has been proved that NSE is much more superior than the deterministic coefficient as a criterion to evaluate the model performance [37].

## 4. Results and Discussions

_{L}, R

_{B}, while $k$ is determined by, not only the topographic parameters, but the mean flow velocity in a watershed. There are several approaches in estimating the average flow velocity in previous research, two commonly used methods of which are chosen to generate the GIUH, meanwhile for selecting one of them to participate the final hydrologic response computation after comparison. The estimation of the average flow velocity $v$ the hydraulic factor, is operated in two methods, respectively, in virtue of the mean effective rainfall intensity ${i}_{e}$ (I), and the values of river channel characteristics (II). According to the method (I), the flow velocity can be easily obtained, which only depends on the rainfall intensity. However, it is unlikely to promptly obtain the average excess rainfall of a flood event in the real-time flood forecasting work. In contrast, method (II) is relatively an available approach that is only related to the topologic and fluvial characteristics, the flow length, and the average channel slope. Though method (I) considers different rainfall intensity rates to calculate the corresponding velocity, it neglects the impact from the important terrain and geomorphologic factors. In the meantime, the rainfall intensity variance leads to the instability of GIUH, causing the complexity of the runoff prediction based on the method (I). Hereafter, the latter one is selected to calculate the final GIUH of each basin and verify the modified GIUH model on flood simulation in this proposed study. The criteria, including the time to peak error (TPE), which is a kind of absolute error (AE), the relative peak error (RPE), and the relative runoff depth error (RRDE) in relative error (RE), and the Nash–Sutcliffe efficiency coefficient (NSE), for evaluating the performance are stated in Table 4. The summarized evaluation values of three catchments, respectively, in the calibration and validation phases are also listed in this table. The statistics that are written in bold are the outlier case showing that the flood event estimation is unsatisfied. At the calibration step, the value of NSE ranges from 0.812 to 0.941, with a mean value of 0.874 in Shaowu basin and two flood events (19920831, 19930615) are unqualified in the series, which illustrates that the result meets the second class; the average value of NSE in the Jianyang watershed reaches 0.882, 10 flood events are all qualified, satisfying the first level; as for Shuiji catchment, the NSE value ranges from 0.817 to 0.922 in this phase, with eight qualified flood events simulation. All of the XAJ parameters and the corresponding value in each catchment are listed in Table 2, and are respectively applied into the validation stage. At the validation stage, the value of criteria presents that the simulated runoff processes fit the observed well, with the average NSE value of 0.888 in three watersheds. In total, there are 11 parameters in Table 2. It illustrates that the modified model requires much less parameters than the traditional XAJ model, correspondingly greatly reducing the uncertainty of the rainfall-runoff modelling.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Location map of three watersheds selected in the study (the watershed delineation map indicates that, the selected basins (Shaowu, Jianyang, Shuiji) located in the north of Fujian Province in the southeastern of China, and elevation and terrain information of each). (

**a**) location of Fujian province in China, (

**b**) location of three studied watersheds in Fujian Province, (

**c**) watershed digital elevation model (DEM) map of Shaowu, Jianyang, and Shuiji watersheds.

**Figure 2.**The structure of modified Xin’anjiang (XAJ) model couple with geomorphologic instantaneous unit hydrograph (GIUH) in this study (GIUH is supposed to replace the former method to calculate the stream flow routing), and the explanation of all the items in the figure.

E: Evapotranspiration | S: Free water storage |

P: Precipitation | SM: Free water capacity |

EM: Evapotranspiration capability | EX: Free water capacity distribution exponent |

B: Exponent of tension water capacity distribution | KI: Outflow ratio of free water storage to interflow |

K: Ratio of potential evapotranspiration to the pan evapotranspiration | KG: Outflow ratio of free water storage to groundwater |

WM: Tension water capacity | RS: surface runoff |

FR: Ratio of the runoff generation area to the basin area | RI: Interflow |

UM: Upper layer tension water capacity | RG: Groundwater runoff |

LM: Lower layer tension water capacity | QS: Surface runoff inflow to river network |

C: Deep layer evapotranspiration coefficient | QI: Interflow to river network |

W: Tension water storage | QG: Groundwater inflow to river network |

WU: Upper soil moisture | QT: Total inflow to river network |

WL: Lower soil moisture | Q: Total outflow |

WD: Deep layer soil moisture |

**Figure 3.**A diagrammatic sketch of the structure of “Free water reservoir” (with such a nonfigurative structure, the runoff generation can be partition based on the runoff components.).

**Figure 4.**Various GIUH curves based on two different methods for Shaowu (

**a**); Jianyang (

**b**); and Shuiji (

**c**) watersheds, with the dotted line curves standing for method (I), and the solid lines for method (II).

**Figure 5.**The discharge simulating graphs of two flood events in each watershed: (

**a**,

**b**) belong to Shaowu basin; (

**c**,

**d**) are for Jianyang basin; (

**e**,

**f**) are for Shuiji catchment; the left graphs (

**a**,

**c**,

**e**) belong to the calibration stage and the right (

**b**,

**d**,

**f**) are in validation step.

**Figure 6.**The fitting lines between measured and calculated runoff value in validation stage via the method of least square. (

**a**–

**c**) are for Jiangyang basin, Shaowu basin and Shuiji basin respectively.

**Table 1.**The Horton geomorphologic indices and basic topographic values in Shaowu, Jianyang, and Shuiji basin.

Watershed | S_{0} (km^{2}) | ${\mathit{L}}_{\mathit{\Omega}}$ (km) | Stream Area $\mathbf{Ratio},\text{}{\mathit{R}}_{\mathit{A}}$ | Bifurcation $\mathbf{Ratio},\text{}{\mathit{R}}_{\mathit{B}}$ | Stream Length $\mathbf{Ratio},\text{}{\mathit{R}}_{\mathit{L}}$ |
---|---|---|---|---|---|

Shaowu | 2677 | 49.74 | 4.2 | 3.99 | 2.13 |

Jianyang | 3253 | 72.77 | 4.292 | 4.345 | 2.211 |

Shuiji | 3470.5 | 86.93 | 4.326 | 4.209 | 2.187 |

**Table 2.**Parameter of the XAJ model (in the sensitive degree column, S stands for sensitive and I for insensitive; the range of every parameter is for reference, and the corresponding values of Shaowu, Jianyang, Shuiji catchments are presented in the list are all under calibration).

Module | Parameter | Physical Significant (Unit) | Sensitive Degree | Range | Shaowu | Jianyang | Shuiji |
---|---|---|---|---|---|---|---|

Evaporation | KC | potential evaporation/pan evaporation | S | 0.8–1.2 | 0.8 | 0.9 | 1.35 |

UM | Volume of upper layer soil moisture storage capacity (mm) | I | 5–20 | 20 | 20 | 20 | |

LM | Volume of lower layer soil moisture storage capacity (mm) | I | 60–90 | 80 | 80 | 80 | |

C | Conversion coefficient of deep layer evaporation | I | 0.1–0.2 | 0.15 | 0.16 | 0.16 | |

Runoff generation | WM | Volume of average soil moisture storage capacity (mm) | I | 120–200 | 160 | 223 | 163 |

B | The power in the curve of soil moisture storage capacity | I | 0.1–0.4 | 0.3 | 0.3 | 0.79 | |

IM | A ratio impervious area/the area of saturated zone | I | 0.01–0.04 | 0.01 | 0.01 | 0.01 | |

Runoff partition | SM | Free water capacity in the soil surface (mm) | S | 18 | 40 | 20 | |

EX | The power in the curve of free water capacity in the soil surface | I | 1.0–1.5 | 0.9 | 0.9 | 1.5 | |

KG | Outflow coefficient of free water storage to ground water | S | 0.4 | 0.6 | 0.45 | ||

KI | Outflow coefficient of free water storage to subsurface runoff | S | 0.35 | 0.397 | 0.4 |

**Table 3.**Estimation of the average flow velocity in terms of two different methods (I) is calculated based on the Kirpich formula with the effective rainfall intensity in 3 flood events, and (II) is a path of computing the velocity merely dependent on the geomorphologic characteristics: the channel slope and the length of flow concentration.

Watershed | Method (I) | Method (II) | |||||
---|---|---|---|---|---|---|---|

Code of Flood Event | Average Effective Rainfall Intensity, ${\mathit{i}}_{\mathit{e}}$ (mm/h) | Mean Flow Velocity, $\mathit{v}$ (m/s) | The Mean Channel Slope of Whole Basin, ${\mathit{S}}_{\mathit{m}}$ | Length of Flow Concentration, ${\mathit{L}}_{\mathit{c}}$ (km) | Time of Concentration, ${\mathit{t}}_{\mathit{c}}$ (min) | Mean Flow Velocity, $\mathit{v}$ (m/s) | |

Shaowu | 19890522 | 2.94 | 1.20 | 0.035 | 104.895 | 520.3 | 3.36 |

19960530 | 3.05 | 0.77 | |||||

19980302 | 2.17 | 1.13 | |||||

Jianyang | 19880228 | 1.59 | 1.07 | 0.029 | 116.693 | 605.9 | 3.21 |

19950603 | 2.61 | 1.17 | |||||

19990715 | 2.65 | 1.17 | |||||

Shuiji | 19880520 | 2.43 | 1.16 | 0.032 | 130.692 | 635.0 | 3.43 |

19950425 | 2.36 | 1.15 | |||||

19930615 | 1.83 | 1.10 |

**Table 4.**Summarized values of evaluation criteria and performance of 10 calibrated and 6 validated flood events in Shaowu, Jianyang, and Shuiji watershed.

Watershed | Stage | Flood Code | TPE | RPE (%) | RRDE (%) | NSE |
---|---|---|---|---|---|---|

Shaowu | Calibration | 19880228 | 1 | 1 | 20 | 0.854 |

19880512 | 0 | −19 | 6.7 | 0.892 | ||

19890515 | 8 | −11 | −16.3 | 0.905 | ||

19890522 | 1 | 3 | 8 | 0.941 | ||

19890621 | 0 | −16 | 3.5 | 0.924 | ||

19890629 | 2 | 19 | 13.3 | 0.812 | ||

19920321 | 0 | 17 | 12.2 | 0.851 | ||

19920514 | 0 | −14 | −16 | 0.877 | ||

19920831 | 1 | −29 | −22.4 | 0.834 | ||

19930615 | 0 | −22 | −1.6 | 0.846 | ||

Validation | 19940501 | 0 | −5 | −9.9 | 0.832 | |

19960328 | −2 | −9 | 0 | 0.856 | ||

19960530 | 2 | 11 | −5.9 | 0.922 | ||

19980215 | 0 | 1 | −7.5 | 0.851 | ||

19980302 | 0 | −4 | −13.9 | 0.930 | ||

19980509 | 6 | −9 | −10.5 | 0.898 | ||

Jianyang | Calibration | 19880228 | −1 | −5 | 6.9 | 0.943 |

19880620 | 1 | −15 | 14.6 | 0.883 | ||

19890629 | −2 | −5 | −3.3 | 0.880 | ||

19920321 | −5 | −4 | 12 | 0.871 | ||

19920616 | −1 | −14 | 5.1 | 0.897 | ||

19920704 | −1 | −8 | 2.0 | 0.905 | ||

19920831 | 0 | −8 | 11.9 | 0.817 | ||

19930615 | −2 | 8 | 19.5 | 0.887 | ||

19930630 | −3 | −17 | −17.9 | 0.842 | ||

19950603 | 1 | −9 | 0 | 0.948 | ||

Validation | 19950622 | −2 | −5 | −5.8 | 0.844 | |

19950626 | 0 | 6 | 0.8 | 0.914 | ||

19970605 | 1 | 12 | 4.7 | 0.897 | ||

19980608 | −2 | 13 | 2.4 | 0.944 | ||

19990521 | 0 | −18 | −20 | 0.876 | ||

19990715 | 0 | −8 | −5.2 | 0.958 | ||

Shuiji | Calibration | 19880228 | 2 | −5 | 25.0 | 0.875 |

19880520 | 11 | −18 | −1.7 | 0.894 | ||

19880620 | 3 | −36 | 4.2 | 0.827 | ||

19890520 | 1 | −17 | 5.9 | 0.882 | ||

19890621 | −1 | −12 | −5.9 | 0.817 | ||

19900629 | 1 | 7 | 11.1 | 0.864 | ||

19920321 | 1 | −4 | 14.6 | 0.861 | ||

19920514 | −1 | −11 | 10.2 | 0.912 | ||

19930615 | −1 | −20 | 2.1 | 0.933 | ||

19930502 | −2 | −7 | −2.4 | 0.890 | ||

Validation | 19940521 | 4 | −28 | −4.7 | 0.883 | |

19940614 | −1 | −11 | 12 | 0.903 | ||

19950425 | 0 | −22 | −10.4 | 0.870 | ||

19950614 | −7 | −7 | −1.6 | 0.848 | ||

19980301 | 0 | −7 | −9.7 | 0.869 | ||

19990715 | 5 | −5 | −9.9 | 0.826 |

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**MDPI and ACS Style**

Chen, Y.; Shi, P.; Qu, S.; Ji, X.; Zhao, L.; Gou, J.; Mou, S.
Integrating XAJ Model with GIUH Based on Nash Model for Rainfall-Runoff Modelling. *Water* **2019**, *11*, 772.
https://doi.org/10.3390/w11040772

**AMA Style**

Chen Y, Shi P, Qu S, Ji X, Zhao L, Gou J, Mou S.
Integrating XAJ Model with GIUH Based on Nash Model for Rainfall-Runoff Modelling. *Water*. 2019; 11(4):772.
https://doi.org/10.3390/w11040772

**Chicago/Turabian Style**

Chen, Yingbing, Peng Shi, Simin Qu, Xiaomin Ji, Lanlan Zhao, Jianfeng Gou, and Shiyu Mou.
2019. "Integrating XAJ Model with GIUH Based on Nash Model for Rainfall-Runoff Modelling" *Water* 11, no. 4: 772.
https://doi.org/10.3390/w11040772