# Modeling River Runoff Temporal Behavior through a Hybrid Causal–Hydrological (HCH) Method

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Case Study

^{2}, which includes a significant extension of Los Alcornocales Natural Park and a small sector at the SW of Breña and Marismas in the Barbate Natural Park. The basin has a smooth orography, with more than the 70% of its area ranging between 0 and 100 m; however, in its northeast sector, there are mountainous reliefs that reach 1090 m above sea level (m.a.s.l.; Figure 1a–c) [55].

^{2}) and the Barbate aquifer (GWB 062.013, of 93 km

^{2}). Both systems, which are constituted by biocalcarenites and sands, are characterized by an intergranular porosity that is locally increased by dissolution processes [62].

^{3}) and its tributaries, the Álamo, Celemín, and Almodóvar rivers. The latter two are also regulated, in this case by the Celemín (45 Hm

^{3}) and Almodóvar (5.7 Hm

^{3}) dams, respectively (Figure 1c). The water resources for the Celemín and Barbate reservoirs are jointly used to irrigate an area measuring 122.3 Km

^{2}, whereas the Almodóvar reservoir is exploited for both irrigation (3.63 Km

^{2}) and for supply of the municipality of Tarifa. Finally, the basin’s water resources were estimated to be 247 Hm

^{3}/year for the period 1999–2016 [55], with an average annual demand of 102 Hm

^{3}[63].

#### 2.2. Dataset Description

#### 2.3. Justification of HCH Method Utility

#### 2.4. General Methodology

#### 2.4.1. Phase 1 Deterministic Module: Rainfall–Runoff Model Definition

_{i}) is a function of the precipitation (P

_{i}), soil moisture deficit (H

_{max}–H

_{i–1}), evapotranspiration (ET

_{i}), and the surplus parameter (C):

_{i}) and real evapotranspiration (ET

_{i}) can be obtained from the following expressions:

_{i}), is a function of the excess (T

_{i}) and maximum infiltration parameter (I

_{max}):

_{S}) that does not infiltrate to the aquifer becomes runoff and is obtained through the following expression:

^{−1}). The evolution of the volume stored in the aquifer and the discharge to the surface drainage network or to the sea can be obtained through the following expressions:

_{t}) corresponds to the excess water that does not infiltrate (T − I) plus the groundwater contribution:

_{C}is the unadjusted monthly evapotranspiration value (mm/month), d is the number of days in a month, h is the light hours depending on the latitude, t is the monthly average temperature (°C), I is the annual heat index (mm/year), i is the monthly heat index (mm/month), and a is the dimensionless parameter. For each sub-basin, monthly temperature series were obtained from the thermometric records belonging to nearby stations, following a similar procedure to that used for precipitation. In this case, however, it was necessary to apply a correction coefficient based on the altitude–temperature gradient in order to compensate for the discrepancies between a sub-basin’s average altitude and the average altitude of the thermometric stations considered.

_{max}

_{)}, coefficient of runoff (C), maximum infiltration (I

_{max}), and aquifer depletion curve (α). These parameters depend mainly on the land uses, geology, lithology, and morphology of the terrain [82,83].

#### 2.4.2. Phase 2 Stochastic Module

#### Subphase 2a ARMA Model Definition

_{t}is the value of the variable at a certain time step t; p is the number of autoregressive parameters; q is the number of moving average parameters; ${\varnothing}_{j}$ and ${\theta}_{j}$ are the coefficient of autoregressive and moving average model, respectively; and a

_{t}is a random variable that represents the historical residuals (error term).

#### Subphase 2b Bayesian Causal Modeling (BCM) Design

#### 2.4.3. Phase 3 Hybrid Causal–Hydrological (HCH) Modeling

#### 2.4.4. Phase 4 Assessment of the Basin’s Hydrological Runoff Memory

## 3. Results

#### 3.1. T-RRM Outputs

_{max}

_{)}, coefficient of runoff (C), maximum infiltration (I

_{max}), and aquifer depletion curve (α) parameters. The modeled runoff fits very well with the observed data for the validation period (Figure 5 and Figure 6).

^{3}/month was detected in the case of the Barbate reservoir, of 1 Hm

^{3}/month for Celemín, and of 0.2 Hm

^{3}/month for the Almodóvar reservoir. Considering the maximum contribution of each reservoir, the average deviation was 2.9% for Barbate, 3.9% for Celemín, and 3.5% for Almodóvar.

^{3}/month, implying annual values of between 6 and 130 Hm

^{3}. The greatest contributions are from the Barbate sub-basin, owing to its physiography and more abundant rainfall, which enable greater surface runoff. On the contrary, the Almodóvar basin gave the lowest contributions, mainly due to the reduced surface area of its basin (16.6 Km

^{2}, see Table 1).

^{2}) ranging between 0.8106 and 0.92514.

^{3}, for Celemín it is 45 Hm

^{3}, and for the Almodóvar reservoir it is 5.7 Hm

^{3}. Considering the average contributions, it can be deduced that the only infrastructure with multiannual regulation capacity within the basin (approximately 2 years of storage) is the Barbate reservoir. Moreover, for 11 of the 68 years of study (16%), the inputs exceeded the storage capacity. This situation accounts for 26% of the years in the case of Celemín and 46% for Almodóvar, showing the reduced capacity of the latter. In addition, the hydrographs also show pronounced dry periods. The first took place in the 1950s, while the two most extreme periods in terms of length and volume took place in the 1990s and in the 2000s. During these decades, there were also contribution peaks, which evidence greater irregularity in the precipitation events during the last decades.

#### 3.2. Stochastic Module: Statistical Parameters and Design of Bayesian Causal Modeling (BCM)

^{3}and 88.82 Hm

^{3}; standard deviation: 69.92 Hm

^{3}and 56.08 Hm

^{3}), which is in agreement with the results shown in the previous section.

^{©}software, which may also be seen as a result in itself. On the left side, the learning and preprocessing processes are shown. Here, the synthetic data were discretized into five intervals of the same length. On the other hand, the right side shows the developed hierarchical structure from top to bottom (initial to final year). Here, each decision variable is connected in such a way that it can influence the previous and following one in a natural way (trivial relationships). This defines the structure constraints process, in which the main “a priori” relationship among variables is considered as the natural behavior, i.e., between consecutive years. Subsequently, and by means of the analysis of the “a posteriori” probability distributions, non-trivial dependence relationships (time lag > 1) were extracted, owing to the power of analysis that CR supported by DMG offers. This information is implicitly present in hydrological data.

#### 3.3. Runoff Basin Memory Assessment through Hybrid Causal–Hydrological (HCH) Modeling

^{2}) of the resulting mathematical functions were almost 1.00 (0.99 in all cases), demonstrating the robustness of the adjustment process.

^{2}) for W-MAX and W-MIN mathematical functions (polynomial in all cases) show values close to 1.00, with 0.98 being the lowest value (Figure 10a), demonstrating the excellent fit of the mathematical functions.

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Situation map. (

**b**) Barbate River Basin orography map. (

**c**) Land occupation and hydrological elements. (

**d**) Rainfall and thermometric stations map [6].

**Figure 2.**Causal model. Conceptual scheme for the case of a runoff time series of 69 years (from the initial year, Y

_{1}, to the final year, Y

_{69}). The visible threshold of independence between variables is 5% (0.05), which means that up to 95% (0.95) of the dependence relationships between the variables are displayed. Each link between variables (consecutive or not, trivial or not) denotes a relationship.

**Figure 4.**(

**a**) The T-RRM conceptual scheme. (

**b**) Barbate River Basin. Sub-basin scheme associated with the availability of gauging data (Qi): Barbate reservoir (Q1, sub-basins 1, 2, and 3); Celemín reservoir (Q2, sub-basin 4); Almodóvar reservoir (Q3, sub-basin 5). (

**c**) Hypsometric curves of the 5 sub-basins.

**Figure 8.**Conceptual design of BCM. (

**Left**) Schemes of learning, preprocessing, and structure constraints processes. (

**Right**) Hierarchical structure from top to bottom (from initial year to final one). Note: This scheme was replicated in each of the 9 causal models. For this reason, just one causal model is shown (gauging records for Barbate Q1 from 2000 to 2015).

**Figure 9.**Dependence mitigation graphs (DMGs) for a short series (2000–2015). Comparison of causal model results from T-RRM versus gauging records: (

**a**) Barbate Q1 sub-basin; (

**b**) Celemín Q2 sub-basin; (

**c**) Almodóvar Q3 sub-basin).

**Figure 10.**Dependence mitigation graphs (DMGs) for a long series (1951–2017) using the T-RRM: (

**a**) Barbate Q1 sub-basin; (

**b**) Celemín Q2 sub-basin; (

**c**) Almodóvar Q3 sub-basin.

Sub-Basins/ | Q_{i} ^{1} | Geological Materials | Extension (Km^{2}) | Elevation (m) | ||||
---|---|---|---|---|---|---|---|---|

Code Number | Maximum | Minimum | Mean | Median | ||||

/1 | Q1 | ^{2} Flysch. Trias, Jurassic | 149.7 | 1087.4 | 11.3 | 209.1 | 181.2 | |

Barbate | /2 | Flysch | 59.7 | 1089.6 | 18.0 | 290.2 | 198.9 | |

/3 | Flysch | 141.8 | 911.8 | 18.0 | 202.2 | 180.9 | ||

Celemín | /4 | Q2 | Flysch | 94.1 | 630.3 | 19.4 | 196.7 | 185.1 |

Almodóvar | /5 | Q3 | Flysch | 16.6 | 763.4 | 98.0 | 283.2 | 248.4 |

^{1}Gauging control points (see Figure 4b).

^{2}Triassic-Jurassic period.

Sub-Basin ^{1}/Code Number | H_{max} | C | I_{max} | α (Month^{−1}) |
---|---|---|---|---|

/1 | 350 mm | 0.4 | 20 mm | 0.1 |

Barbate/2 Q1 | 280 mm | 0.3 | 3 mm | 0.1 |

/3 | 280 mm | 0.3 | 3 mm | 0.1 |

Celemín/4 Q2 | 280 mm | 0.3 | 3 mm | 0.1 |

Almodóvar/5 Q3 | 280 mm | 0.3 | 3 mm | 0.1 |

^{1}Gauging control points (see Figure 4b).

**Table 3.**Average monthly and annual runoff values for the Barbate, Celemín, and Almodóvar reservoirs (period: 1951–2017).

Sub-Basins | Month | Average (Hm^{3}) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

January | February | March | April | May | June | July | August | September | October | November | December | Monthly | Annual | |

Barbate | 25.0 | 24.5 | 18.8 | 10.5 | 5.5 | 1.2 | 1.0 | 0.9 | 0.9 | 4.2 | 11.3 | 25.2 | 10.8 | 129.1 |

Celemín | 6.7 | 6.3 | 5.0 | 2.3 | 1.1 | 0.2 | 0.1 | 0.1 | 0.1 | 0.8 | 2.8 | 6.3 | 2.7 | 31.8 |

Almodóvar | 1.1 | 1.2 | 0.9 | 0.5 | 0.2 | 0.0 | 0.0 | 0.0 | 0.0 | 0.2 | 0.6 | 1.3 | 0.5 | 6.2 |

Sub-Basins ^{1} | Parameters | T-RRM Results | Average SS ^{3} | |
---|---|---|---|---|

Long series: From 1951 to 2017 | Barbate/Q1 | Mean: | 129.10 Hm^{3} | 129.71 Hm^{3} |

Standard deviation: | 87.78 Hm^{3} | 93.87 Hm^{3} | ||

Skewness coefficient ^{2}: | 1.64 | 1.83 | ||

Variation coefficient: | 68% | 72% | ||

Hurst coefficient: | 0.66 | ---------- | ||

Celemín/Q2 | Mean: | 31.77 Hm^{3} | 31.89 Hm^{3} | |

Standard deviation: | 21.47 Hm^{3} | 22.39 Hm^{3} | ||

Skewness coefficient ^{2}: | 1.84 | 1.73 | ||

Variation coefficient: | 68% | 70% | ||

Hurst coefficient: | 0.66 | ---------- | ||

Almodóvar/Q3 | Mean: | 6.22 Hm^{3} | 6.16 Hm^{3} | |

Standard deviation: | 4.14 Hm^{3} | 3.92 Hm^{3} | ||

Skewness coefficient ^{2}: | 1.54 | 1.42 | ||

Variation coefficient: | 67% | 64% | ||

Hurst coefficient: | 0.66 | ---------- | ||

Sub-basins ^{1} | Parameters | T-RRM results | Average SS ^{3} | |

Short series: From 2000 to 2015 | Barbate/Q1 | Mean: | 110.98 Hm^{3} | 111.71 Hm^{3} |

Standard deviation: | 69.62 Hm^{3} | 62.46 Hm^{3} | ||

Skewness coefficient ^{2}: | 1.13 | 0.58 | ||

Variation coefficient: | 63% | 56% | ||

Hurst coefficient: | 0.66 | ---------- | ||

Celemín/Q2 | Mean: | 27.50 Hm^{3} | 27.76 Hm^{3} | |

Standard deviation: | 16.10 Hm^{3} | 15.62 Hm^{3} | ||

Skewness coefficient ^{2}: | 1.06 | 0.70 | ||

Variation coefficient: | 59% | 56% | ||

Hurst coefficient: | 0.61 | ---------- | ||

Almodóvar/Q3 | Mean: | 7.05 Hm^{3} | 7.25 Hm^{3} | |

Standard deviation: | 4.24 Hm^{3} | 4.23 Hm^{3} | ||

Skewness coefficient ^{2}: | 1.45 | 1.09 | ||

Variation coefficient: | 60% | 58% | ||

Hurst coefficient: | 0.69 | ---------- | ||

Sub-basins ^{1} | Parameters | Gauging data | Average SS ^{3} | |

Short series (Observed data): From 2000 to 2015 | Barbate/Q1 | Mean: | 88.82 Hm^{3} | 87.83 Hm^{3} |

Standard deviation: | 56.08 Hm^{3} | 52.16 Hm^{3} | ||

Skewness coefficient ^{2}: | 1.69 | 1.28 | ||

Variation coefficient: | 63% | 59% | ||

0.74 | ---------- | |||

Celemín/Q2 | Mean: | 26.58 Hm^{3} | 28.08 Hm^{3} | |

Standard deviation: | 18.54 Hm^{3} | 20.48 Hm^{3} | ||

Skewness coefficient ^{2}: | 1.89 | 1.54 | ||

Variation coefficient: | 70% | 73% | ||

0.77 | ---------- | |||

Almodóvar/Q3 | Mean: | 6.71 Hm^{3} | 6.53 Hm^{3} | |

Standard deviation: | 4.30 Hm^{3} | 4.13 Hm^{3} | ||

Skewness coefficient ^{2}: | 1.14 | 1.11 | ||

Variation coefficient: | 64% | 63% | ||

Hurst coefficient: | 0.57 | ---------- |

^{1}Gauging Control Point (see Figure 4b).

^{2}Classic skewness coefficient.

^{3}Synthetic series from ARMA (1,1).

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**MDPI and ACS Style**

Zazo, S.; Molina, J.-L.; Ruiz-Ortiz, V.; Vélez-Nicolás, M.; García-López, S.
Modeling River Runoff Temporal Behavior through a Hybrid Causal–Hydrological (HCH) Method. *Water* **2020**, *12*, 3137.
https://doi.org/10.3390/w12113137

**AMA Style**

Zazo S, Molina J-L, Ruiz-Ortiz V, Vélez-Nicolás M, García-López S.
Modeling River Runoff Temporal Behavior through a Hybrid Causal–Hydrological (HCH) Method. *Water*. 2020; 12(11):3137.
https://doi.org/10.3390/w12113137

**Chicago/Turabian Style**

Zazo, Santiago, José-Luis Molina, Verónica Ruiz-Ortiz, Mercedes Vélez-Nicolás, and Santiago García-López.
2020. "Modeling River Runoff Temporal Behavior through a Hybrid Causal–Hydrological (HCH) Method" *Water* 12, no. 11: 3137.
https://doi.org/10.3390/w12113137