# An Ensemble Flow Forecast Method Based on Autoregressive Model and Hydrological Uncertainty Processer

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods and Data

#### 2.1. Postprocessing of Ensemble Weather Forecasts

#### 2.2. Establishment of the Ensemble Flow Forecast Model

#### 2.2.1. The Hydrological Uncertainty Processor Methodology

_{n}and S

_{n}are, respectively, the implementation values of the random variables. The prior probability density function (PDF) under different prediction periods can be expressed as the following equation.

_{n}, and c is the Pearson correlation coefficient obtained from the normal quantile transform.

_{n}are the parameters calculated in the normal quantile conversion space and the expression are as follows.

_{n}and W

_{n–}

_{1}is governed by a normal-linear equation.

_{n}is Pearson’s correlation coefficient, $\mathrm{\Xi}$ is a stochastically independent and normally distributed variable of W

_{n}with a mean of zero and a variance of ${\tau}_{n}^{2}=1-{c}_{n}^{2}$.

_{n}and W

_{n}, W

_{n−}

_{1}, W

_{0}is governed by a normal-linear equation.

_{n}, b

_{n}, and d

_{n}are the regression coefficients, and ${\mathrm{\Theta}}_{n}$ is a stochastically independent and normally distributed variable of (W

_{n}, W

_{0}) with a mean of zero and a variance of ${\sigma}_{n}^{2}$.

#### 2.2.2. Precipitation-Dependent Hydrological Uncertainty Processer Based on Gaussian Mixture Model

_{n}, where ${\theta}_{\mathrm{\Gamma}}$ and ${\theta}_{{\mathrm{\Lambda}}_{n}}$ are the distribution parameters to be estimated. More details about the GMM can be found in Feng et al. [28]. Then, on the conditional of observed inflow at time t

_{0}, the prior distribution of H

_{n}can be expressed as:

_{0}is h

_{0}and the simulated flow is s

_{n}, the posterior probability density function of H

_{n}is the following.

_{0}under the condition V = v where $v=P(V=1)$ represents the probability of effective rainfall in the lead time period. Then, the marginal PDF of H

_{n}is as follows.

_{0}, the probability of effective and negligible rainfall is as follows.

_{0}, the PDF of the S

_{n}is as follows.

_{0}and simulated inflow is s

_{n}, the probability of effective and negligible rainfall is as follows.

#### 2.2.3. Generation of Ensemble Flow Forecasts

_{n,t}represents the simulated value of the nth flood ensemble forecast member at time t, and E(H

_{n,t}) is the expected value of H

_{n,t}. Then, the ensemble flow forecast result can be expressed as follows.

#### 2.3. Data

^{6}km

^{2}, the surface area of the reservoir is about 1080 km

^{2}, and the average width is about 1100 m [29].

## 3. Results and Discussion

#### 3.1. Correction of Ensemble Weather Forecasts

_{t}represents the forecast precipitation, and O

_{t}represents the observed precipitation. The MAE and RMSE value of each GEFS member before and after correction are shown in Table 1.

^{3}/s, while RMSE decreased by an average of 16.05 m

^{3}/s. It shows that the AR model can effectively reduce the precipitation deviation of GEFS. However, there are still some bias and uncertainties in the corrected GEFS precipitation data. Therefore, the PD-HUP model was adopted in this paper to further deal with the uncertainty of hydrological forecast caused by rainfall forecast bias.

#### 3.2. Estimated Parameters of Ensemble Flow Forecast Model

#### 3.2.1. Generation of Ensemble Flow Forecasts

#### 3.2.2. Parameters of PD-HUP

#### 3.3. Ensemble Forecast Analysis

#### 3.3.1. Comparison of Forecast Accuracy

#### 3.3.2. Uncertainty Evaluation of Ensemble Prediction Results

^{7}, indicating that the simulation results are highly uncertain. However, by using the adjusted GEFS rainfall data to drive the hydrological model, the variance of the ensemble members of the inflow prediction in different periods will significantly decrease, and the uncertainty of the prediction results will significantly decrease. When the Bayesian ensemble forecast method proposed by this paper is adopted, the uncertainty of hydrological prediction results is further reduced.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Flow of methods. The PD-HUP-GMM represents the precipitation-dependent hydrological uncertainty processor (PD-HUP) based on a Gaussian mixture model (GMM).

**Figure 4.**Fitting of the marginal CDF of observed value and simulated flood value. (

**a**) Marginal CDF of observed value. (

**b**) Marginal CDF of simulated value. GMM (v = 1) and GMM (v = 0), respectively, represent the fitting of the distribution function in the case of effective rainfall and negligible rainfall.

**Figure 5.**Simulated result of the four scenarios: (

**a**) Observed data + Xin’an Jiang model, (

**b**) GEFS data + Xin’an Jiang model, (

**c**) Corrected GEFS data + Xin’an Jiang model, and (

**d**) Bayesian ensemble forecast.

**Figure 6.**The variance of ensemble flow forecast members at different moments of the 2017 flood season under different scenarios.

GEFS Member | MAE | RMSE | ||
---|---|---|---|---|

Before Correction | After Correction | Before Correction | After Correction | |

1 | 30.46 | 11.25 | 32.42 | 15.98 |

2 | 29.95 | 11.49 | 31.89 | 16.36 |

3 | 29.45 | 12.09 | 31.54 | 16.30 |

4 | 28.87 | 11.49 | 31.03 | 15.82 |

5 | 28.13 | 11.75 | 30.29 | 15.14 |

6 | 28.54 | 10.44 | 30.86 | 14.17 |

7 | 29.47 | 10.73 | 31.83 | 14.94 |

8 | 29.74 | 11.61 | 31.72 | 15.36 |

9 | 28.92 | 10.55 | 30.71 | 14.18 |

10 | 28.76 | 10.83 | 30.97 | 14.91 |

11 | 29.98 | 10.91 | 32.19 | 15.77 |

Dataset | Components | Weight | Mean | Variance |
---|---|---|---|---|

Observed value | 1 | 0.49 | 1.59 × 10^{4} | 1.16 × 10^{7} |

2 | 0.17 | 3.61 × 10^{4} | 9.36 × 10^{7} | |

3 | 0.34 | 2.49 × 10^{4} | 2.20 × 10^{7} | |

Simulated value | 1 | 0.51 | 1.75 × 10^{4} | 1.26 × 10^{7} |

2 | 0.12 | 3.96 × 10^{4} | 9.19 × 10^{7} | |

3 | 0.36 | 2.77 × 10^{4} | 2.18 × 10^{7} |

Dataset | Components | Weight | Mean | Variance |
---|---|---|---|---|

Observed value | 1 | 0.46 | 1.63 × 10^{4} | 1.12 × 10^{7} |

2 | 0.16 | 2.59 × 10^{4} | 6.03 × 10^{7} | |

3 | 0.38 | 1.06 × 10^{4} | 3.29 × 10^{7} | |

Simulated value | 1 | 0.57 | 1.25 × 10^{4} | 4.55 × 10^{7} |

2 | 0.12 | 2.82 × 10^{4} | 4.79 × 10^{7} | |

3 | 0.31 | 1.92 × 10^{4} | 1.20 × 10^{7} |

V | c | a_{n} | d_{n} | b_{n} | ${\mathit{\sigma}}_{\mathit{n}}$ | t_{n} |
---|---|---|---|---|---|---|

1 | 0.931 | 1.058 | −0.150 | −0.080 | 0.420 | 0.365 |

0 | 0.974 | 0.868 | 0.715 | 0.043 | 0.384 | 0.225 |

V | A_{n} | B_{n} | D_{n} | T_{n} |
---|---|---|---|---|

1 | 0.434 | 0.065 | 0.538 | 0.269 |

0 | 0.236 | −0.169 | 0.765 | 0.200 |

Scenarios | Description |
---|---|

Observed data + Xin’an Jiang model | Use observed data only to run Xin’an Jiang model and get deterministic forecast |

GEFS data + Xin’an Jiang model | Use GEFS to run the Xin’an Jiang model and get ensemble forecast |

Corrected GEFS data + Xin’an Jiang model | Corrected GEFS data to run the Xin’an Jiang model and get ensemble forecast |

Bayesian ensemble forecast | Use HUP to postprocess the result of corrected GEFS data with the Xin’an Jiang model and get the ensemble result |

Scenarios | NSE | RMSE |
---|---|---|

Observed data + Xin’an Jiang model | 0.92 | 1364.02 |

GEFS data + Xin’an Jiang model | −7.14 | 13613.12 |

Corrected GEFS data + Xin’an Jiang model | 0.56 | 3167.20 |

Bayesian ensemble forecast | 0.91 | 1501.23 |

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**MDPI and ACS Style**

Yang, X.; Zhou, J.; Fang, W.; Wang, Y.
An Ensemble Flow Forecast Method Based on Autoregressive Model and Hydrological Uncertainty Processer. *Water* **2020**, *12*, 3138.
https://doi.org/10.3390/w12113138

**AMA Style**

Yang X, Zhou J, Fang W, Wang Y.
An Ensemble Flow Forecast Method Based on Autoregressive Model and Hydrological Uncertainty Processer. *Water*. 2020; 12(11):3138.
https://doi.org/10.3390/w12113138

**Chicago/Turabian Style**

Yang, Xin, Jianzhong Zhou, Wei Fang, and Yurong Wang.
2020. "An Ensemble Flow Forecast Method Based on Autoregressive Model and Hydrological Uncertainty Processer" *Water* 12, no. 11: 3138.
https://doi.org/10.3390/w12113138