# Multi-Step Ahead Probabilistic Forecasting of Daily Streamflow Using Bayesian Deep Learning: A Multiple Case Study

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Bayesian Long Short-Term Memory (BLSTM)

#### 2.1.1. Epistemic Uncertainty

#### 2.1.2. Aleatoric Uncertainty

#### 2.1.3. Combining Aleatoric and Epistemic Uncertainty

#### 2.2. Performance Evaluation Metrics

- 1.
- Metric for Deterministic Forecasting

- 2.
- Metrics for Probabilistic Forecasting

## 3. Case Study

#### 3.1. Study Area

^{2}.

#### 3.2. Experiment Setup

^{®}GeForce

^{®}RTX 2070 SUPER and an Intel

^{®}Core i9-10920X central processing unit at 3.5 GHz utilizing 128 GB random access memory. For a fair comparison among the prediction models, a grid search for hyperparameter tuning was used to ensure identical evaluation.

## 4. Result and Discussion

#### 4.1. Probabilistic Prediction Performance Assessment

#### 4.2. Impact of Forecast Horizon in Probabilistic Prediction Performance

^{3}/s and Std. of 90 m

^{3}/s, achieved the worst prediction results for all scenarios. From Scenarios I to II in case study I for BLSTM, LSTM-BNN, BNN, and LSTM-MC, PICP decreased by approximately 25, 38, 48, and 53%, respectively, indicating the best performance of BLSTM in maintaining its predictability in case study I, with the highest peak and Std in the extended horizon prediction. Moreover, by increasing the horizon, prediction performance for case study I dramatically decreased, whereas, in terms of PICP for BLSTM, case studies II and III from Scenarios I to III decreased by approximately 1–2% and 2–4%, respectively. Furthermore, LSTM-MC and BNN achieved the worst overall prediction performance for all the scenarios. The catchment area of case study I was relatively large, and heavy rain was the primary source of streamflow. These two characteristics cause the seasonal and annual variations in streamflow to be greater than those in the other two case studies. In this case study, the streamflow was very stable and small during the dry season, whereas in the rainy season, the streamflow increased steeply and then decreased. This made forecasting challenging and resulted in a higher uncertainty than that in the other case studies. Therefore, for this type of catchment, using more in-situ meteorological predictors, such as precipitation and temperature, along with available high-resolution large-scale hydroclimate data, can improve forecasting accuracy.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Example of the proposed BLSTM network with a zoomed-in plot of the forget gate at time step t in the first layer.

**Figure 7.**Probabilistic streamflow forecasting results obtained by BLSTM for case study II for (

**a**) forecast horizon 1, (

**b**) forecast horizon 7, and (

**c**) forecast horizon 30.

**Figure 8.**Probabilistic streamflow forecasting results obtained by LSTM-BNN for case study II for (

**a**) forecast horizon 1, (

**b**) forecast horizon 7, and (

**c**) forecast horizon 30.

**Figure 9.**Probabilistic streamflow forecasting results obtained by BNN for case study II for (

**a**) forecast horizon 1, (

**b**) forecast horizon 7, and (

**c**) forecast horizon 30.

**Figure 10.**Probabilistic streamflow forecasting results obtained by LSTM-MC for case study II (

**a**) forecast horizon 1, (

**b**) forecast horizon 7, and (

**c**) forecast horizon 30.

**Figure 11.**Change in probabilistic streamflow forecasting results by increasing horizon for (

**a**) case study I, (

**b**) case study II, and (

**c**) case study III.

**Figure 12.**Prediction results of all models with 1, 7, and 30 days ahead forecasting for (

**a**) case study I, (

**b**) case study II, and (

**c**) case study III.

**Figure 13.**Kernel density estimation plots of daily streamflow prediction of all models in case study II for (

**a**) forecast horizon 1, (

**b**) forecast horizon 7, and (

**c**) forecast horizon 30.

Field | Probabilistic Method | Base Models | Posterior Approximation * | Evaluation Metrics | Ref. | ||
---|---|---|---|---|---|---|---|

VI | MCM | Deterministic | Probabilistic | ||||

Streamflow | LSTM-HetGP | ANN, HetGP, GLM, LSTM | - | - | NSE, RMSE, MRE, MSLE | percentage of coverage (POC) and the average interval width (AIW) | [39] |

Flood | LSTM | ARIMA | - | - | RMSE, MAE | [40] | |

Streamflow | LSTM with multiparameter ensemble and dropout ensemble | _ | - | ✓ | PBIAS, MARE, RMSE, NSE, KGE | POC, average width (AW), average interval score (AIS) | [41] |

Streamflow | Variational Bayesian Long Short-Term Memory network (VB-LSTM) | Bayesian model Averaging (BMA) | ✓ | - | MAE | CRPS | [42] |

Runoff | XGBoost (XGB) and Gaussian process regression (GPR) with Bayesian optimization algorithm (BOA) | GBR, LGB, CNN, LSTM, ANN, SVR, QR, GPR—combined with GPR | - | - | RMSE, MAPE, R^{2} | Coverage probability, Mean width percentage, Suitability metric, CRPS | [38] |

Runoff | B-spline quantile regression model combined with kernel density estimation | QR, QRNN | - | - | RMSE, R^{2}, Q_{r} | PICP, PINAW, CRPS | [43] |

Streamflow | Bayesian LSTM model | physics-based hydrologic model (Precipitation-Runoff Modeling System) | - | ✓ | NSE, RMSE-observations standard deviation ratio (RSR) | [44] |

Criteria | Case Study 1 | Case Study 2 | Case Study 3 |
---|---|---|---|

No. Samples | 27,146 | 34,205 | 22,645 |

Mean (m^{3}/s) | 58 | 30 | 31 |

Std (m^{3}/s) | 90 | 51 | 42 |

Min (m^{3}/s) | 2.5 | 0.6 | 3 |

25% (m^{3}/s) | 13 | 4 | 9 |

50% (m^{3}/s) | 31 | 11 | 20 |

75% (m^{3}/s) | 64 | 32 | 36 |

Max (m^{3}/s) | 1654 | 702 | 1274 |

Case Study. No. | Station ID | G-Name | Elev. (m) | Drainage Area (km^{2}) | Lon. (°) | Lat. (°) | Period |
---|---|---|---|---|---|---|---|

1 | 03364000 | EAST FORK WHITE RIVER AT COLUMBUS, IN | 183.8 | 4421 | 85°55′32″ | 39°12′00″ | 1948–2022 |

2 | 05131500 | LITTLE FORK RIVER AT LITTLEFORK, MN | 330.3 | 4403 | 93°32′57″ | 48°23′45″ | 1928–2022 |

3 | 11368000 | MCCLOUD R AB SHASTA LK CA | 335.3 | 1564 | 122°13′07″ | 40°57′30″ | 1945–2007 |

Forecast Horizon = 1 | Forecast Horizon = 7 | Forecast Horizon = 30 | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Model | Metric | Case I | Case II | Case III | Case I | Case II | Case III | Case I | Case II | Case III | |||||||||

BLSTM | PICP | 0.950 | 0.964 | 0.956 | 0.709 | 0.958 | 0.941 | 0.477 | 0.943 | 0.921 | |||||||||

MPIW | 0.021 | 0.006 | 0.016 | 0.023 | 0.024 | 0.023 | 0.032 | 0.042 | 0.028 | ||||||||||

CRPS | 0.087 | 0.035 | 0.066 | 0.375 | 0.212 | 0.214 | 0.576 | 0.437 | 0.337 | ||||||||||

LSTM-BNN | PICP | 0.957 | 0.967 | 0.971 | 0.591 | 0.962 | 0.956 | 0.371 | 0.953 | 0.942 | |||||||||

MPIW | 0.023 | 0.008 | 0.018 | 0.035 | 0.037 | 0.032 | 0.039 | 0.076 | 0.040 | ||||||||||

CRPS | 0.086 | 0.034 | 0.070 | 0.400 | 0.226 | 0.237 | 0.615 | 0.457 | 0.367 | ||||||||||

BNN | PICP | 0.955 | 0.953 | 0.961 | 0.496 | 0.779 | 0.870 | 0.281 | 0.630 | 0.865 | |||||||||

MPIW | 0.027 | 0.009 | 0.022 | 0.045 | 0.050 | 0.040 | 0.053 | 0.141 | 0.047 | ||||||||||

CRPS | 0.101 | 0.041 | 0.083 | 0.412 | 0.240 | 0.262 | 0.634 | 0.461 | 0.371 | ||||||||||

LSTM-MC | PICP | 0.973 | 0.994 | 0.981 | 0.454 | 0.948 | 0.913 | 0.268 | 0.945 | 0.909 | |||||||||

MPIW | 0.046 | 0.040 | 0.027 | 0.049 | 0.050 | 0.032 | 0.057 | 0.076 | 0.039 | ||||||||||

CRPS | 0.109 | 0.071 | 0.106 | 0.414 | 0.251 | 0.248 | 0.636 | 0.545 | 0.376 |

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**MDPI and ACS Style**

Ghobadi, F.; Kang, D.
Multi-Step Ahead Probabilistic Forecasting of Daily Streamflow Using Bayesian Deep Learning: A Multiple Case Study. *Water* **2022**, *14*, 3672.
https://doi.org/10.3390/w14223672

**AMA Style**

Ghobadi F, Kang D.
Multi-Step Ahead Probabilistic Forecasting of Daily Streamflow Using Bayesian Deep Learning: A Multiple Case Study. *Water*. 2022; 14(22):3672.
https://doi.org/10.3390/w14223672

**Chicago/Turabian Style**

Ghobadi, Fatemeh, and Doosun Kang.
2022. "Multi-Step Ahead Probabilistic Forecasting of Daily Streamflow Using Bayesian Deep Learning: A Multiple Case Study" *Water* 14, no. 22: 3672.
https://doi.org/10.3390/w14223672