# Real Time Flow Forecasting in a Mountain River Catchment Using Conceptual Models with Simple Error Correction Scheme

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Case Study

#### 2.1. Catchment Description

^{2}(Figure 1).

#### 2.2. Data Set

_{e}= 0.48), make the routing process along the river reach a dominant driver explaining the outflow values at the downstream end. Consequently, the downstream peak flows tend to be attenuated, as shown in Figure 3 and Table 1.

## 3. Real-Time Forecast Modeling

#### 3.1. Model-1 Basic Component

_{0}should be a non-negative value.

_{0}in Equation (4) would result in a zero value, and Equation (3) can be then rewritten as follows (Equation (5)):

_{t}and O

_{t}are known values at time t, and the forecasted discharge at the downstream end of the reach would be ${O}_{t+\mathsf{\Delta}t}$.

#### 3.2. Model-2 Basic Component

^{3}/s) at the downstream gauge station, and I the inflow rate (m

^{3}/s) at the upstream gauge station. σ is a non-dimensionless attenuation parameter, associated with the ratio between outflow and inflow discharge rates. Its value should be expected to be less than 1 during the rising limb of the flood hydrograph and over 1 during the hydrograph recession. Concerning H, estimates will be taken as the time lag associated with flood hydrograph routing process along the river reach. Although estimated values will be equivalent to those of Section 3.1, it is convenient to use a different notation, as the parameter itself plays a different role in this second formulation.

_{k}is the last observed flow value at Campo station, and I

_{k−1}is the penultimate one. A vector of n predictions is also generated, for all intermediate intervals ($\mathsf{\Delta}t=15$ min) between k and (k + H), Equation (8).

#### 3.3. Error Updating Scheme

#### 3.4. Model Evaluation

_{obs}(t) is the outflow discharge at Campo station and O

_{pred}(t) is the predicted discharge at Graus station. ${\mu}_{obs}$ and ${\mu}_{pred}$ refer to the respective average values of the series.

_{obs}(t) is the measured flow discharge at time t in the downstream gauge station. O

_{pred}(t) is the predicted outflow value for that given time t, and ${\mu}_{obs}$ is the average value of the flow discharge at the downstream gauge station. Both summations were extended over all predictions provided by the model and compared to the corresponding observed values at the Graus station.

_{pred}is the forecasted flow value, and O

_{obs}is the observed one. Therefore, $\eta <1$ implies flow underestimation, while $\eta >1$ indicates overestimation.

^{3}/s). Finally, the goodness of flood peak forecasts was evaluated in terms of the average errors (absolute and relative) and maximum errors, computed only over the hydrograph peaks for both samples (calibration and validation).

## 4. Results and Discussion

#### 4.1. Results for Model-1

#### 4.2. Results for Model-2

_{2}in model-2, the same formal functional relationship was assumed (Equations (16) and (17)):

_{i}and b

_{i}(i = 1,2) directly estimated from the calibration hydrograph data. For the case study considered herein, the estimated values that best represented the observed empirical relationships where ${\mathrm{a}}_{1}=447.890,{b}_{1}=0.210,{a}_{2}=2.597,{b}_{2}=0.236$ (Figure 4 and Figure 9).

#### 4.3. Comparison of Results and Discussion

^{3}/s). The main peak of the flood event (250 m

^{3}/s), though, was correctly anticipated by both modeling schemes. With regards to the family of validation hydrographs, model-1 showed a better ability to predict more accurately the flow peaks, with flow forecasting errors under 4% (20% for model-2), for a 2 h lead-time. Model-2 tended to overestimate the hydrographs peaks ($\eta =1.059$), that is, an average approximate overestimation of around 6%.

^{3}/s, versus 9 m

^{3}/s for model-2. On the other hand, model-2 tended to overestimate outflows (average $\eta $ values over 1, for all the lead-times considered in the case study), while $\eta $ values remained very close to 1 for the model-1 flow real-time forecasts.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**(

**a**) Flow forecasts vs measured flows at the Graus station. (

**b**) Empirical frequency distribution of the errors (calibration data set).

**Figure 6.**(

**a**) Flow forecasts vs measured flows at the Graus station. (

**b**) Empirical frequency distribution of the errors (validation data set).

**Figure 7.**Comparison of flow forecasts and observed outflows at the Graus station (calibration event C-2).

**Figure 8.**Comparison of 2 h lead time flow forecasts and observed outflows at the Graus station (validation event V3).

**Figure 10.**(

**a**) Flow forecasts vs measured flows at the Graus station (left). (

**b**) Empirical frequency distribution of the errors (calibration data set) (right).

**Figure 11.**(

**a**) Flow forecasts vs measured flows at the Graus station. (

**b**) Empirical frequency distribution of the errors (validation data set).

**Figure 12.**Comparison of flow forecasts and observed outflows at the Graus station (calibration event C2).

**Figure 13.**Comparison of 2 h lead time flow forecasts and observed outflows at the Graus station (validation event V3).

Event | Initial Date | Q_{max}Campo (m ^{3}/s) | Q_{max}Graus (m ^{3}/s) | Volume Campo (10 ^{6} m^{3}) | Volume Graus (10 ^{6} m^{3}) | Time lag (hour) |
---|---|---|---|---|---|---|

Calibration Events | ||||||

C1 | 19 September 1999 | 154.8 | 127.8 | 18.2 | 17.7 | 1.75 |

C2 | 18 October 1999 | 169.4 | 145.0 | 49.1 | 46.8 | 2.00 |

C3 | 22 November 2000 | 103.7 | 95.7 | 11.0 | 11.1 | 2.50 |

C4 | 20 October 2004 | 64.8 | 56.8 | 7.7 | 7.0 | 3.25 |

Validation Events | ||||||

V1 | 22 September 2006 | 117.5 | 109.7 | 21.1 | 22.8 | 2.25 |

V2 | 16 October 2006 | 210.5 | 178.7 | 65.4 | 56.6 | 3.25 |

V3 | 08 June 2010 | 229.1 | 219.8 | 34.7 | 39.4 | 2.75 |

V4 | 1 November 2011 | 151.7 | 158.1 | 19.8 | 28.3 | 2.50 |

**Table 2.**Model-1 performance evaluation coefficients and forecasting error statistics (calibration data set).

Lead Time (min) | Calibration | Error Distribution | 5% and 95% Percentiles | |||||
---|---|---|---|---|---|---|---|---|

r | NS | PC | $\mathit{\eta}$ | Mean (m^{3}/s) | Std. dev. (m^{3}/s) | Err. (m^{3}/s) | ||

60 | 0.986 | 0.971 | 0.244 | 1.026 | −0.038 | 5.077 | −7.28 | 7.75 |

75 | 0.984 | 0.968 | 0.401 | 1.030 | −0.029 | 5.268 | −7.93 | 7.96 |

90 | 0.982 | 0.963 | 0.489 | 1.034 | −0.015 | 5.331 | −7.88 | 8.24 |

105 | 0.980 | 0.959 | 0.536 | 1.038 | −0.037 | 5.342 | −8.01 | 8.00 |

120 | 0.978 | 0.954 | 0.574 | 1.045 | −0.060 | 5.360 | −8.01 | 7.97 |

**Table 3.**Model-1 performance evaluation coefficients and forecasting error statistics (validation data set).

Lead Time (min) | VALIDATION | Error Distribution | 5% and 95% Percentiles | |||||
---|---|---|---|---|---|---|---|---|

r | NS | PC | $\mathit{\eta}$ | Mean (m^{3}/s) | Std. dev. (m^{3}/s) | Err. (m^{3}/s) | ||

60 | 0.987 | 0.974 | 0.205 | 0.996 | −0.110 | 6.777 | −10.29 | 9.29 |

75 | 0.987 | 0.974 | 0.387 | 0.997 | −0.104 | 6.911 | −10.40 | 9.45 |

90 | 0.986 | 0.973 | 0.491 | 0.998 | −0.108 | 6.963 | −10.46 | 9.41 |

105 | 0.986 | 0.972 | 0.557 | 0.999 | −0.110 | 6.986 | −10.58 | 9.37 |

120 | 0.985 | 0.970 | 0.601 | 1.001 | −0.121 | 6.996 | −10.59 | 9.37 |

Average | Maximum | |||||||
---|---|---|---|---|---|---|---|---|

Lead Time (min) | h | Absolute Error m ^{3}/s | Relative. Error% | Min h _{sub} | Max h _{over} | Absolute Error m ^{3}/s | Relative. Error % | |

Calibration | 120 | 0.972 | −3.581 | −2.822 | 0.955 | 0.986 | −6.77 | −4.54 |

Validation | 120 | 1.059 | 9.712 | 5.945 | 0.997 | 1.193 | 31.38 | 19.33 |

**Table 5.**Model-2 performance evaluation coefficients and forecasting error statistics (calibration data set).

Lead Time (min) | Calibration | Error Distribution | 5% and 95% Percentiles | |||||
---|---|---|---|---|---|---|---|---|

r | NS | PC | $\mathit{\eta}$ | Mean(m^{3}/s) | Std. dev(m^{3}/s) | Err.(m^{3}/s) | ||

60 | 0.984 | 0.965 | 0.468 | 0.818 | −0.728 | 6.186 | −10.80 | 5.99 |

75 | 0.983 | 0.963 | 0.570 | 1.008 | −0.751 | 6.487 | −11.45 | 6.90 |

90 | 0.982 | 0.960 | 0.631 | 1.014 | −0.745 | 6.790 | −11.44 | 7.09 |

105 | 0.981 | 0.958 | 0.676 | 1.024 | −0.727 | 7.129 | −11.56 | 7.36 |

120 | 0.979 | 0.955 | 0.712 | 0.890 | −0.723 | 7.460 | −12.07 | 7.66 |

**Table 6.**Model-2 performance evaluation coefficients and forecasting error statistics (validation data set).

Lead Time (min) | Validation | Error Distribution | 5% and 95% Percentiles | |||||
---|---|---|---|---|---|---|---|---|

r | NS | PC | $\mathit{\eta}$ | mean(m^{3}/s) | Std. dev(m^{3}/s) | Err.(m^{3}/s) | ||

60 | 0.988 | 0.977 | 0.318 | 1.006 | −0.560 | 8.486 | −10.25 | 9.09 |

75 | 0.988 | 0.975 | 0.465 | 1.006 | −0.549 | 8.695 | −9.84 | 9.47 |

90 | 0.987 | 0.974 | 0.545 | 1.006 | −0.567 | 8.869 | −11.01 | 9.98 |

105 | 0.986 | 0.972 | 0.601 | 1.007 | −0.546 | 8.997 | −11.42 | 10.63 |

120 | 0.985 | 0.970 | 0.648 | 1.008 | −0.483 | 9.096 | −12.79 | 11.19 |

Average | Maximum | |||||||
---|---|---|---|---|---|---|---|---|

Lead Time (min) | h | Absolute Error m ^{3}/s | Relative Error % | Min h _{sub} | Max h _{over} | Absolute Error m ^{3}/s | Relative. Error % | |

Calibration | 120 | 1.031 | 3.630 | 3.060 | 0.958 | 1.114 | 12.68 | 11.44 |

Validation | 120 | 1.059 | 9.712 | 5.945 | 0.997 | 1.193 | 31.38 | 19.33 |

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**MDPI and ACS Style**

Montes, N.; Aranda, J.Á.; García-Bartual, R.
Real Time Flow Forecasting in a Mountain River Catchment Using Conceptual Models with Simple Error Correction Scheme. *Water* **2020**, *12*, 1484.
https://doi.org/10.3390/w12051484

**AMA Style**

Montes N, Aranda JÁ, García-Bartual R.
Real Time Flow Forecasting in a Mountain River Catchment Using Conceptual Models with Simple Error Correction Scheme. *Water*. 2020; 12(5):1484.
https://doi.org/10.3390/w12051484

**Chicago/Turabian Style**

Montes, Nicolás, José Ángel Aranda, and Rafael García-Bartual.
2020. "Real Time Flow Forecasting in a Mountain River Catchment Using Conceptual Models with Simple Error Correction Scheme" *Water* 12, no. 5: 1484.
https://doi.org/10.3390/w12051484