# A Hybrid Forecasting Model to Simulate the Runoff of the Upper Heihe River

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Data

#### 2.1.1. Study Area

^{2}[36]. The upper reaches of the Heihe River, located in the Qilian Mountains at the northern foot of the Qinghai–Tibet Plateau, China, are situated in the Qilian Mountains–Qinghai Lake climatic zone, which is characterized by high precipitation, low evaporation, and low temperatures [37]. Affected by the climate and the flood season, with annual variations in flow, runoff is mainly concentrated from April to September, accounting for 80% of the annual flow; the runoff from October to March of the next year is relatively limited [38]. Notably, the runoff increases due to the rising temperatures from April to May, which result in snow and glacial melting [39]. Then, in the rainy season, with supplementary alpine ice and snow melt water, mountain runoff reaches a peak between July and August; runoff then decreases continuously after October, with decreasing temperatures and the weakening of the influence of warm and wet air flows [40]. The study area is shown in Figure 1.

#### 2.1.2. Data

#### 2.2. Methodology

#### 2.2.1. Variational Mode Decomposition

^{2}= −1; $\delta $ is the Dirac function; and $\ast $ is a convolution operation.

#### 2.2.2. Mutual Information

#### 2.2.3. LSTM

#### 2.2.4. Nonparametric Kernel Density Estimation

#### 2.2.5. Evaluation of the Model’s Performance

#### 2.3. Model Implementation

#### 2.3.1. Determining Network Parameters

#### 2.3.2. Process of Training the VMD-LSTM Model

- (1)
- The required monthly runoff sample data are selected, and a model training set and a test set are established.
- (2)
- The VMD method is used to decompose the original runoff sequence to obtain several components, and each component and the original runoff sequence are normalized.
- (3)
- The MI method is used to determine the model input delay (in this paper, the time step), and each component is input into the LSTM model for prediction.
- (4)
- After the forecast is completed, the data are denormalized, and the prediction results for each mode component are combined to obtain the final runoff forecast sequence.
- (5)
- The runoff series forecasting error is calculated, the nonparametric KDE method is applied to estimate the runoff series interval, and the accuracy of prediction results is evaluated.

## 3. Results

#### 3.1. Determination of the Number of VMD Components

#### 3.2. Runoff Estimation

#### 3.3. Comparing Single and Hybrid Forecasting Models

#### 3.4. Runoff Interval Simulation

## 4. Discussion

## 5. Conclusions

- (1)
- The VMD method can effectively reduce the non-stationarity of hydrological time series, extract important hydrological feature information, and significantly improve the accuracy of runoff predictions. Compared with EMD, VMD can better control center frequency aliasing and noise levels.
- (2)
- Based on the MI method, the constructed VMD-LSTM model can effectively determine the input characteristics for deep learning. Overall, the proposed model performs well in runoff predictions. However, it should be noted that the accumulation of forecast errors for each subsequence affects the forecast result.
- (3)
- In interval prediction, the proposed model also yields satisfactory results. The prediction interval coverage and the simulation accuracy are high, and the average width is small. Interval prediction can be used to quantify the uncertainty of runoff predictions, estimate reasonable fluctuation ranges, and provide a certain reference for the establishment of hydrological prediction models and water resource management plans.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Upper reaches of the Heihe River Basin. The red triangles correspond to three hydrologic stations. The blue circles represent the meteorological stations in the upper Heihe River Basin.

**Figure 2.**The basic structure of an LSTM cell. The black arrows indicate the directions of the data flows. The gray rectangles indicate activation functions. The orange symbols represent calculation steps.

**Figure 3.**Monthly runoff forecasting process based on VMD-LSTM. The dashed black box denotes the runoff simulation steps in the LSTM model.

**Figure 4.**The center frequency of the final modes after decomposition when the number of components is 5 and 6.

**Figure 5.**Decomposition results for all modes. The red, orange, yellow, green, and blue line segments represent the five modes after the completion of VMD.

**Figure 6.**Runoff predictions of the VMD-LSTM model. In the (

**left**) half, the black and red lines represent the observed and estimated runoff values, respectively. On the (

**right**) side, the red line indicates the linear regression trendline for the observed and simulated runoff values. The black dotted line is a 1:1 line.

**Figure 9.**Error distributions of different models; (

**left**) is the Gaussian frequency histogram, and (

**right**) shows the fitting distribution curves.

**Figure 10.**Error distribution for runoff estimation; (

**left**) is the probability density function (PDF), and (

**right**) is the cumulative distribution function (CDF).

**Figure 11.**Estimation of runoff intervals at different confidence levels: (

**a**) 95% confidence interval; (

**b**) 90% confidence interval; and (

**c**) 80% confidence interval.

Category | Sub-Category | Advantages | Limitations |
---|---|---|---|

Process-driven models | Conceptual models (tank model, storage function) | Relatively easy to calculate; can express various runoff patterns | Parameters lack physical meaning |

Physical models (distributed models) | Runoff process is expressed in detail; reflects topography and rainfall distribution | Required data are difficult to obtain; model building is time consuming | |

Data-driven models | Time-series models (AR, ARMA, ARIMA, etc.) | Models are easily constructed | Cannot simulate complex and nonlinear runoff |

Machine learning (linear regression, SVM, ANN, RNN, etc.) | Strong ability to deal with nonlinear problems | Calculation process is “black box”; requires a considerable amount of data |

Runoff Sample | Length/Months | Mean/10^{8} m^{3} | Standard Deviation/10^{8} m^{3} | Coefficient of Variation | Skewness |
---|---|---|---|---|---|

Total | 480 | 1.49 | 1.256 | 0.842 | 1.31 |

Training period | 360 | 1.417 | 1.216 | 0.858 | 0.92 |

Verification period | 120 | 1.715 | 1.351 | 0.788 | 1.2 |

Model | R | RMSE | NSE |
---|---|---|---|

XGBoost | 0.879 | 0.65 | 0.766 |

LSTM | 0.951 | 0.522 | 0.849 |

EMD-LSTM | 0.95 | 0.427 | 0.899 |

VMD-XGBoost | 0.979 | 0.406 | 0.909 |

VMD-LSTM | 0.988 | 0.24 | 0.968 |

Confidence Interval/% | Estimation Error Interval |
---|---|

95 | [−0.4252, 0.3983] |

90 | [−0.3863, 0.2189] |

80 | [−0.2645, 0.1148] |

Model | Confidence Interval/% | Number of Measured Values within the Interval | PICP | MPIW |
---|---|---|---|---|

XGBoost | 95 | 98 | 0.8167 | 1.5615 |

90 | 86 | 0.7167 | 1.3678 | |

80 | 72 | 0.6 | 1.0125 | |

LSTM | 95 | 96 | 0.8 | 1.4012 |

90 | 89 | 0.7417 | 1.2924 | |

80 | 74 | 0.6167 | 0.9085 | |

EMD-LSTM | 95 | 106 | 0.8833 | 1.1137 |

90 | 97 | 0.8083 | 0.9124 | |

80 | 86 | 0.7166 | 0.6114 | |

VMD-XGBoost | 95 | 106 | 0.8833 | 1.0685 |

90 | 99 | 0.825 | 0.8861 | |

80 | 87 | 0.725 | 0.5124 | |

VMD-LSTM | 95 | 116 | 0.9667 | 0.8235 |

90 | 109 | 0.9083 | 0.6052 | |

80 | 92 | 0.7667 | 0.3793 |

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## Share and Cite

**MDPI and ACS Style**

Xue, H.; Wu, H.; Dong, G.; Gao, J.
A Hybrid Forecasting Model to Simulate the Runoff of the Upper Heihe River. *Sustainability* **2023**, *15*, 7819.
https://doi.org/10.3390/su15107819

**AMA Style**

Xue H, Wu H, Dong G, Gao J.
A Hybrid Forecasting Model to Simulate the Runoff of the Upper Heihe River. *Sustainability*. 2023; 15(10):7819.
https://doi.org/10.3390/su15107819

**Chicago/Turabian Style**

Xue, Huazhu, Hui Wu, Guotao Dong, and Jianjun Gao.
2023. "A Hybrid Forecasting Model to Simulate the Runoff of the Upper Heihe River" *Sustainability* 15, no. 10: 7819.
https://doi.org/10.3390/su15107819