# Simple and Low-Cost Procedure for Monthly and Yearly Streamflow Forecasts during the Current Hydrological Year

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{2}; the Root-Mean-Square Error RMSE; the Mean Absolute Error MAE; the Index of Agreement IOA; the Mean Absolute Percent Error MAPE; the Coefficient of Nash-Sutcliffe Efficiency NSE; and the Inclusion Coefficient IC) were used and the results showed good levels of accuracy (improving as the number of observed months increases). The model forecast outputs are the mean monthly and yearly streamflows along with the 10th and 90th percentiles. The methodology has been successfully applied to two headwater reservoirs within the Guadalquivir River Basin in southern Spain, achieving an accuracy of 92% and 80% in March 2017. These risk-based predictions are of great value, especially before the intensive irrigation campaign starts in the middle of the hydrological year, when Water Authorities have to ensure that the right decision is made on how to best allocate the available water volume between the different water users and environmental needs.

## 1. Introduction

- Use free hydrological data sources available in the public domain (to be downloaded in an easy, free, and quick manner from a reliable, official online resource where possible);
- minimise the number of hydrological variables;
- integrate easily and regularly new observed hydrological data in the model (an automatic process where possible);
- minimise computational and simulation running times (instantaneously where possible); and
- offer a user-friendly interface.

## 2. Materials and Methods

#### 2.1. Overview

#### 2.2. Model Time Step, Data Sources, and Treatment

#### 2.3. Simple Regression Analysis and Best Fit Predictor

_{annual}) from relevant hydrological descriptors. We selected the cumulative monthly precipitation (hereinafter referred to as P

_{cum}) and the cumulative monthly streamflow (hereinafter referred to as A

_{cum}) as the best predictors.

^{2}). This allowed us to select the best fit regression model and best fit hydrological predictor (cumulative rainfall or cumulative streamflow) for each specific month of study. The estimated total annual streamflow from the regression models is an intermediary output (to be used as an input to obtain the conditioned gamma, as shown in Figure 1).

#### 2.4. Two-Parameter Gamma Cumulative Probability Distribution Function

^{2}/β

#### 2.5. Conditioned Two-Parameter Gamma Cumulative Probability Distribution Function (Using the Output from the Regression Models)

_{cond}(Conditioned Gamma) = Mean (Conditioned Gamma)/β =

= Estimated total annual streamflow (from the regression analysis)/β

_{cond},β).

_{cond},β) allowed us to assign a probability to each year of the historical series based on their total annual streamflow. Therefore, we obtained greater probabilities for those observed years whose annual streamflow is similar to the estimated annual streamflow (‘${\mathrm{A}}_{\mathrm{annual}}$’) and lower probabilities for those observed years whose annual streamflow differs from the estimated annual streamflow. For each year of the historical series, we have its probability (p), given by:

^{3}), α

_{cond}is the shape parameter of the conditioned gamma, and $\mathsf{\beta}$ is the scale parameter of the gamma distribution (the same as per the non-conditioned gamma distribution).

#### 2.6. Monte Carlo Method

#### 2.7. Model Running Time, Test, and Validation

^{2}), were used (as shown in Section 3).

#### 2.8. Application and Limitations

## 3. Application to Canales and Quéntar Reservoirs (Upper Guadalquivir River Basin, Spain)

#### 3.1. Study Area Description

^{2}and is delimited by Sierra Morena to the north, the Betic mountain to the south, and the Atlantic Ocean. The altitude at the mountainous borders varies between 1000 m Above Ordenance Datum (AOD) and 3480 m AOD, which contrasts with the lower altitudes of the Guadalquivir River valley. The climate is Mediterranean, which is defined by the warm temperatures (16.8 °C annual average) and the irregularity of precipitations (550 mm annual average). The rains are frequently torrential and occur after long periods of drought and high temperatures, with a distinct susceptibility to erosion [31]. Figure 3 shows the mean annual precipitation and temperature, altitude, and land uses in the Guadalquivir River Basin.

^{2}. The dam was built in 1989 with a total capacity of 70 hm

^{3}and average streamflow of 80.42 hm

^{3}/year. This catchment area is affected by snow storage and melting processes in the Sierra Nevada. The snow appears between the months of November and May, reaching the maximum accumulation of snow in March with an average water-equivalent volume of 20 hm

^{3}[33].

^{2}. This dam was built in 1975 with a total volume capacity of 13.5 hm

^{3}. The average streamflow is 28.84 hm

^{3}/year. This catchment area is affected by subterranean inflows due to the aquifer and lithology present in the region.

#### 3.2. Data Sources and Treatment

#### 3.3. Regression Analysis, Correlation, and Selection of the Best Fit Estimator and Regression Model

^{2}values shown in Table 2.

^{2}) allowed us to select the best fit regression model and best fit hydrological predictor (cumulative rainfall or cumulative streamflow) for each reservoir and specific month of study. The estimated total annual streamflow from the regression models is an intermediary output (to be used as an input to obtain the conditioned gamma).

^{2}values (varying from 0.7253 to 0.9083) for the first three, four, and five observed months whilst the cumulative streamflow achieves the best R

^{2}values (varying from 0.9429 to 0.9839) for the six, seven, and eight observed months. This fact is considered to be due to the influence of snow storage and melting processes distinctive of this basin. The Canales catchment river hydrograph shows a pluvio-nival pattern with two flow peaks. The first one was in January and February due to the surface water runoff from rainfall events. The second one was around April and May due to snow-melt water flows from the Sierra Nevada.

_{annual}is the annual streamflow (hm

^{3}), x

_{cum}. is the cumulative monthly precipitation (for the first three, four, and five observed months in mm) and cumulative monthly streamflow (for the first six, seven, and eight months in hm

^{3}), and a and b are the lineal regression model coefficients.

^{2}values (varying from 0.5442 to 0.9894) for all the months of study. The most important hydrological process given within the Quéntar catchment is due to the underground aquifer flows. The snow storage and melting processes are not relevant if compared with the Canales reservoir. Therefore, the base flow contribution into the total annual reservoir inflow is important. This might be the reason why the strongest correlation is found between the cumulative monthly streamflow and the annual streamflow.

_{cum}. is the cumulative monthly streamflow (hm

^{3}), and c and d are the regression model coefficients.

#### 3.4. Two-Parameter Gamma Cumulative Probability Distribution Function (Observed Data) and Conditioned Gamma Cumulative Probability Distribution Function (Using the Output from the Regression Models)

#### 3.5. Assigning Probabilities and Application of Monte Carlo Method

#### 3.6. Model Validation

- (a)
- Coefficient of determination (R
^{2}); - (b)
- root-mean-square error (RMSE);
- (c)
- Mean Absolute Error (MAE);
- (d)
- Index of Agreement (IOA);
- (e)
- Mean Absolute Percent Error (MAPE);
- (f)
- Coefficient of Nash-Sutcliffe efficiency (NSE); and
- (g)
- Inclusion Coefficient (IC) defined as the percentage of times that the observed streamflow falls within the 10th and 90th percentiles (our pessimistic and optimistic forecast values).

^{2}values vary from 0.53 to 0.83 and average of 0.69) reservoir are better than those found for the Quéntar reservoir (R

^{2}values vary from 0.23 to 0.51 and average of 0.41). The optimal results in our study are given for the Canales reservoir for the seven observed months that achieved the highest R

^{2}= 0.83 and lowest MAPE = 32.45.

^{2}, IOA, and NSE increase while the values of RMSE, MAE, and MAPE decrease). The higher values of error correspond to those observed hydrological years where hydrographs differ considerably from the mean historical observed year. Particularly, the worst MAPE value for the Canales and Quéntar reservoirs occurs in the driest years of 1994–1995 and 2004–2005.

^{2}values varying from 0.10 to 0.63 (average of 0.37) and MAPE values varying from 25.02 to 1112.38. Myronidis et al. acknowledged similar results achieved by other authors also using ARIMA models. Their optimal results obtained in this study were found for catchment number 6, which achieved the highest R

^{2}= 0.63 and lowest MAPE = 25.02.

#### 3.7. Model Outputs

^{®}spreadsheet.

#### 3.8. First Operational Year 2016–2017: Results

^{3}whilst the observed value was 38.34 hm

^{3}(compared with the observed historical mean annual streamflow of 66.71 hm

^{3}). Logically, the prediction results improved as the number of observed months increased.

^{3}whilst the observed value was 10.68 hm

^{3}(compared with the observed historical mean annual streamflow of 20.81 hm

^{3}).

## 4. Conclusions and Future Research Directions

^{2}, and NSE) were used during the validation process, which showed good levels of accuracy and increased with the number of months observed. The model was first put into operation during the last hydrological year (2016–2017), obtaining accuracy levels of 92% and 80% respectively, for Canales and Quéntar, for the annual streamflow forecast given in March 2017.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Location of the Guadalquivir River Basin [31] in Spain and the area of study (Latitude: 37.15855 Longitude: −3.470605).

**Figure 3.**Guadalquivir River Basin (

**a**) altitude, (

**b**) mean annual temperature, (

**c**) mean annual precipitation, and (

**d**) land uses.

**Figure 4.**Location of Canales reservoir sub-basin (Latitude: 37.15855 Longitude: −3.470605) and the Quéntar reservoir sub-basin (Latitude: 37.206527 Longitude: −3.433458). Source: own elaboration.

**Figure 5.**Canales reservoir: Regression analysis for the first three, four, five, six, seven, and eight months using the cumulative monthly reservoir inflow and precipitation.

**Figure 6.**Quéntar reservoir: Regression analysis for the first three, four, five, six, seven, and eight months using the cumulative monthly reservoir inflow and precipitation.

**Figure 7.**Two-parameter gamma CDF and conditioned gamma distribution—1st April 2017: (

**a**) Canales reservoir (observed data October 1988–September 2016); and (

**b**) Quéntar reservoir: (observed data October 1977–September 2016).

**Figure 8.**Example of typical model forecast outputs results compared with the historical observed streamflow data.

**Figure 9.**Observed monthly streamflows (hm

^{3}) and predictions using the model with the first six months of observed data of the hydrological year and 10th and 90th percentiles (lower and upper limits) (hm

^{3}) of the (

**a**) Canales reservoir and (

**b**) Quéntar reservoir.

**Figure 10.**Observed annual streamflows (hm

^{3}) and predictions using the model with the first six months of observed data of the hydrological year and 10th and 90th percentiles (lower and upper limits) (hm

^{3}) of the (

**a**) Canales reservoir and (

**b**) Quéntar reservoir

**Figure 12.**Predictions of annual and monthly cumulative streamflow (hm

^{3}) and observed annual and monthly cumulative streamflow (hm

^{3}) of the (

**a**) Canales reservoir and (

**b**) Quéntar reservoir.

Number of Observed Months | Observed Period | Number of Forecasted Months | Forecasted Period |
---|---|---|---|

3 | Oct–Dec | 9 | Jan–Sept |

4 | Oct–Jan | 8 | Feb–Sept |

5 | Oct–Feb | 7 | Mar–Sept |

6 | Oct–Mar | 6 | Apr–Sept |

7 | Oct–Apr | 5 | May–Sept |

8 | Oct–May | 4 | Jun–Sept |

Period | Canales Reservoir (R^{2}) | Quéntar Reservoir (R^{2}) | ||||||
---|---|---|---|---|---|---|---|---|

No. Observed Months | y = ax + b | y = cx^{d} | y = ax + b | y = cx^{d} | ||||

P cum | A cum | P cum | A cum | P cum | A cum | P cum | A cum | |

3 (Oct–Dec) | 0.7253 | 0.5939 | 0.6353 | 0.6314 | 0.3370 | 0.3801 | 0.4100 | 0.5442 |

4 (Oct–Jan) | 0.8257 | 0.8098 | 0.7453 | 0.7836 | 0.4675 | 0.5772 | 0.5653 | 0.7594 |

5 (Oct–Feb) | 0.9083 | 0.8664 | 0.8567 | 0.8449 | 0.6428 | 0.8069 | 0.6714 | 0.8909 |

6 (Oct–Mar) | 0.9154 | 0.9429 | 0.8366 | 0.8872 | 0.7187 | 0.9190 | 0.7341 | 0.9483 |

7 (Oct–Apr) | 0.8996 | 0.9658 | 0.8656 | 0.9374 | 0.6767 | 0.9567 | 0.6724 | 0.9710 |

8 (Oct–May) | 0.8736 | 0.9839 | 0.8487 | 0.9714 | 0.6479 | 0.9844 | 0.6539 | 0.9894 |

Forecast Period | No. Observed Months | Canales Reservoir–Model Fit Statistics | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Monthly Series | Yearly Series | ||||||||||||||

R^{2} | RMSE (hm^{3}) | MAE (hm^{3}) | IOA | MAPE (%) | NSE | IC (%) | R^{2} | RMSE (hm^{3}) | MAE (hm^{3}) | IOA | MAPE (%) | NSE | IC (%) | ||

Jan–Sept | 3 (Oct–Dec) | 0.53 | 3.75 | 2.26 | 0.83 | 46.72 | 0.53 | 79.01 | 0.70 | 21.56 | 14.82 | 0.89 | 25.32 | 0.69 | 85.19 |

Feb–Sept | 4 (Oct–Jan) | 0.59 | 3.51 | 2.08 | 0.86 | 42.46 | 0.59 | 82.87 | 0.82 | 16.69 | 11.53 | 0.94 | 19.83 | 0.82 | 88.89 |

Mar–Sept | 5 (Oct–Feb) | 0.64 | 3.37 | 1.91 | 0.88 | 37.34 | 0.64 | 83.07 | 0.88 | 13.87 | 9.21 | 0.96 | 15.42 | 0.87 | 88.89 |

Apr–Sept | 6 (Oct–Mar) | 0.75 | 2.69 | 1.63 | 0.92 | 34.18 | 0.75 | 82.72 | 0.95 | 9.26 | 6.74 | 0.98 | 13.23 | 0.94 | 88.89 |

May–Sept | 7 (Oct–Apr) | 0.83 | 2.16 | 1.36 | 0.95 | 32.45 | 0.83 | 81.48 | 0.98 | 6.67 | 5.40 | 0.99 | 10.25 | 0.97 | 88.89 |

Jun–Sept | 8 (Oct–May) | 0.79 | 2.01 | 1.19 | 0.93 | 34.24 | 0.79 | 81.48 | 0.99 | 5.08 | 4.08 | 1.00 | 7.49 | 0.98 | 88.89 |

Forecast Period | No. Observed Months | Quéntar Reservoir–Model Fit Statistics | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Monthly Series | Yearly Series | ||||||||||||||

R^{2} (−) | RMSE (hm^{3}) | MAE (hm^{3}) | IOA (−) | MAPE (%) | NSE (−) | IC (%) | R^{2} (−) | RMSE (hm^{3}) | MAE (hm^{3}) | IOA (−) | MAPE (%) | NSE (−) | IC (%) | ||

Jan–Sept | 3 (Oct–Dec) | 0.23 | 1.95 | 1.09 | 0.63 | 69.17 | 0.20 | 82.75 | 0.39 | 10.72 | 7.60 | 0.75 | 39.23 | 0.37 | 84.21 |

Feb–Sept | 4 (Oct–Jan) | 0.33 | 1.82 | 0.99 | 0.72 | 62.28 | 0.28 | 78.62 | 0.59 | 8.82 | 5.82 | 0.87 | 27.84 | 0.57 | 86.84 |

Mar–Sept | 5 (Oct–Feb) | 0.50 | 1.39 | 0.76 | 0.83 | 53.61 | 0.45 | 79.32 | 0.81 | 6.03 | 4.04 | 0.95 | 18.52 | 0.80 | 89.47 |

Apr–Sept | 6 (Oct–Mar) | 0.51 | 0.93 | 0.55 | 0.85 | 46.26 | 0.53 | 78.95 | 0.92 | 3.85 | 2.55 | 0.98 | 11.19 | 0.92 | 84.21 |

May–Sept | 7 (Oct–Apr) | 0.48 | 0.71 | 0.46 | 0.88 | 45.21 | 0.62 | 78.95 | 0.96 | 2.78 | 1.73 | 0.99 | 8.03 | 0.96 | 84.21 |

Jun–Sept | 8 (Oct–May) | 0.41 | 0.52 | 0.36 | 0.93 | 44.45 | 0.75 | 77.63 | 0.98 | 1.72 | 1.19 | 1.00 | 5.77 | 0.98 | 81.58 |

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## Share and Cite

**MDPI and ACS Style**

Delgado-Ramos, F.; Hervás-Gámez, C.
Simple and Low-Cost Procedure for Monthly and Yearly Streamflow Forecasts during the Current Hydrological Year. *Water* **2018**, *10*, 1038.
https://doi.org/10.3390/w10081038

**AMA Style**

Delgado-Ramos F, Hervás-Gámez C.
Simple and Low-Cost Procedure for Monthly and Yearly Streamflow Forecasts during the Current Hydrological Year. *Water*. 2018; 10(8):1038.
https://doi.org/10.3390/w10081038

**Chicago/Turabian Style**

Delgado-Ramos, Fernando, and Carmen Hervás-Gámez.
2018. "Simple and Low-Cost Procedure for Monthly and Yearly Streamflow Forecasts during the Current Hydrological Year" *Water* 10, no. 8: 1038.
https://doi.org/10.3390/w10081038