# A Continuous Multisite Multivariate Generator for Daily Temperature Conditioned by Precipitation Occurrence

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Multivariate Precipitation Occurrence (Dry–Wet)

#### 2.2. Multivariate Maximum and Minimum Temperature

#### 2.3. Evidence for the Goodness of Fit

#### 2.4. Generation of Multivariate Synthetic Series

#### 2.5. Study Area

^{2}. Information regarding the zone was obtained from the official website of the confederation (www.chj.es). The most relevant surface runoff is the Jucar River, which captures the surface runoff of all sub-basins [66]. The most significant reservoirs are Alarcon (1088 hm

^{3}) and Contreras (852 hm

^{3}). The river rises from the Tragacete (1600 ms.n.m) and subsequently arrives at reservoirs Toba, Alarcon, Molinar, and Tous. The study area’s limit ends where the Mediterranean Sea is reached (Figure 2). Rainfall in the Jucar River Basin has decreased since 1980 [67,68]. Temporal and spatial variation characteristics of meteorological elements in the Jucar River Basin are presented in Appendix A (Figure A1, Figure A2, Figure A3 and Figure A4).

## 3. Results

#### 3.1. Multivariate Occurrence Synthetic Series

#### 3.2. Stochastic Multisite Multivariate Temperature Series

#### 3.3. Generation of Multivariate Synthetic Temperature Series

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Spatial distribution of annual (

**a**) maximum temperature (°C) and (

**b**) temperature range (°C).

**Figure A2.**Spatial distribution of annual (

**a**) precipitation occurrence (wet days/year) and (

**b**) rainfall (mm/year).

**Figure A3.**Interannual variation trends of the average (

**a**) maximum temperature (°C) and (

**b**) temperature range (°C).

**Figure A4.**Interannual variation trends of the average (

**a**) precipitation occurrence (wet days year

^{−1}) and (

**b**) rainfall (mm year

^{−1}).

**Figure A5.**Daily correlation function for residual series considering the ten lag days: (

**a**) Model 1 (M1) and (

**b**) Model 2 (M2).

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**Figure 1.**Proposed methodology for multisite multivariate precipitation occurrence, daily and annual temperatures.

**Figure 3.**Fourier series and confidence limits for Alarcon: (

**a**) ${p}_{01}$ (

**b**) ${p}_{11}$ with four parameters. Confidence limits at 95% (lower and upper limits).

**Figure 4.**Skewness coefficient (daily average) of normalized series (66 years): (

**a**) maximum temperature and (

**b**) temperature range. Anderson confidence limits (95%).

**Figure 5.**Fourier series (daily average) of maximum temperature series (66 years): (

**a**) mean wet days, (

**b**) mean dry days, (

**c**) standard deviation wet days, and (

**d**) standard deviation dry days.

**Figure 6.**Fourier series (daily average) of temperature range series (66 years): (

**a**) mean wet days, (

**b**) mean dry days, (

**c**) standard deviation wet days, and (

**d**) standard deviation dry days.

**Figure 7.**Theoretical normal distribution (blue) and histogram for residual series for all sub-basins: (

**a**) Model 1 (maximum temperature) and (

**b**) Model 2 (temperature range).

**Figure 8.**Scatter plots for rainfall occurrence (mean 66 years observed and 1000 simulated series) for the five sub-basins: (

**a**) daily mean for each calendar day (green) and (

**b**) monthly mean for each calendar moth (green) for M1 and M2.

**Figure 9.**Scatter plots for observed mean versus generated temperature (mean 66 years observed and 1000 simulated series) for two models: (

**a**) maximum temperature for each calendar day (blue) and (

**b**) temperature range for each calendar day (red).

**Figure 10.**Scatter plots for observed standard deviations versus generated (mean 66 years observed and 1000 simulated series) for two models: (

**a**) maximum temperature for each calendar day (blue) and (

**b**) temperature range for each calendar day (red).

**Figure 11.**Scatter plots for observed skewness coefficient versus generated (mean 66 years observed and 1000 simulated series) for two models: (

**a**) maximum temperature for each calendar day (blue) and (

**b**) temperature range for each calendar day (red).

**Figure 12.**Monthly temperature for all sub-basins (mean 66 years observed and 1000 simulated series) observed and simulated for (

**a**) maximum temperature for each month (blue) and (

**b**) temperature range for each month (red), for 66 years, 5 sub-basins, and all months.

**Figure 13.**Yearly temperature for all sub-basins observed (obs) and simulated (sim) for (

**a**) maximum temperature and (

**b**) temperature range. (1) Alarcon, (2) Contreras, (3) Molinar, (4) Tous, and (5) Huerto Mulet. The outliers are plotted individually using the ‘+’ marker symbol.

Sub-Basin | Wet Day Threshold (mm) | |||
---|---|---|---|---|

0.001 * | 0.01 | 0.10 | 0.25 | |

Alarcon | −590.2 | −515.3 | −362.2 | −310.9 |

Contreras | −681.5 | −551.2 | −459.5 | −421.3 |

Molinar | −562.3 | −420.7 | −261.2 | −215.1 |

Tous | −587.4 | −463.6 | −380.5 | −340.0 |

Huerto Mulet | −610.5 | −554.7 | −467.3 | −427.3 |

Model | Statistical/Sub-Basin | Alarcon | Contreras | Molinar | Tous | Huerto Mulet |
---|---|---|---|---|---|---|

1 * | Mean | −6.7 × 10^{−5} | −2.0 × 10^{−4} | 7.5 × 10^{−5} | −2.6 × 10^{−4} | 5.6 × 10^{−5} |

Deviation | 0.842 | 0.821 | 0.834 | 0.812 | 0.850 | |

Skewness coefficient | −0.185 | −0.242 | −0.245 | −0.089 | 0.026 | |

Lag-one autocorrelation | 0.005 | 0.003 | 0.009 | −0.041 | −0.042 | |

AIC | −8369 | −9508 | −8814 | −10,152 | −7958 | |

2 ** | Mean | −3.09 × 10^{−4} | −5.98 × 10^{−4} | −6.83 × 10^{−5} | −2.70 × 10^{−4} | −2.65 × 10^{−4} |

Deviation | 0.937 | 0.920 | 0.942 | 0.900 | 0.942 | |

Skewness coefficient | −0.009 | −0.038 | −0.044 | 0.112 | −0.028 | |

Lag-one autocorrelation | 0.036 | 0.043 | −0.030 | −0.082 | −0.039 | |

AIC | −6471 | −8233 | −5957 | −10,294 | −5896 |

Parameter | Model 1 (M1) | Model 2 (M2) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 1 | 2 | 3 | 4 | 5 | |

RMSE (°C/day) | 1.881 | 1.107 | 1.392 | 1.156 | 0.767 | 1.786 | 1.019 | 1.290 | 1.078 | 0.780 |

MAE (°C/day) | 1.503 | 0.820 | 1.092 | 0.896 | 0.572 | 1.455 | 0.773 | 1.021 | 0.852 | 0.582 |

PE (%) | 0.043 | 0.027 | 0.017 | 0.031 | 0.021 | 0.035 | 0.049 | 0.040 | 0.025 | 0.012 |

Maximum Temperature Cross-Correlation | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Sub-Basin | Alarcon | Contreras | Molinar | Tous | Huerto M | * Alarcon | * Contreras | * Molinar | * Tous | * Huerto M |

Alarcon | 1.000 | 1.000 | ||||||||

Contreras | 0.833 | 1.000 | 0.834 | 1.000 | ||||||

Molinar | 0.760 | 0.797 | 1.000 | 0.765 | 0.819 | 1.000 | ||||

Tous | 0.239 | 0.492 | 0.598 | 1.000 | 0.243 | 0.489 | 0.610 | 1.000 | ||

Huerto M | 0.223 | 0.371 | 0.390 | 0.802 | 1.000 | 0.230 | 0.377 | 0.398 | 0.800 | 1.000 |

Temperature Range Cross-Correlation | ||||||||||

Sub-Basin | Alarcon | Contreras | Molinar | Tous | Huerto M | * Alarcon | * Contreras | * Molinar | * Tous | * Huerto M |

Alarcon | 1.000 | 1.000 | ||||||||

Contreras | 0.719 | 1.000 | 0.718 | 1.000 | ||||||

Molinar | 0.891 | 0.592 | 1.000 | 0.895 | 0.590 | 1.000 | ||||

Tous | 0.563 | 0.494 | 0.754 | 1.000 | 0.565 | 0.499 | 0.759 | 1.000 | ||

Huerto M | 0.504 | 0.331 | 0.710 | 0.918 | 1.000 | 0.502 | 0.333 | 0.708 | 0.917 | 1.000 |

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**MDPI and ACS Style**

Hernández-Bedolla, J.; Solera, A.; Paredes-Arquiola, J.; Sanchez-Quispe, S.T.; Domínguez-Sánchez, C. A Continuous Multisite Multivariate Generator for Daily Temperature Conditioned by Precipitation Occurrence. *Water* **2022**, *14*, 3494.
https://doi.org/10.3390/w14213494

**AMA Style**

Hernández-Bedolla J, Solera A, Paredes-Arquiola J, Sanchez-Quispe ST, Domínguez-Sánchez C. A Continuous Multisite Multivariate Generator for Daily Temperature Conditioned by Precipitation Occurrence. *Water*. 2022; 14(21):3494.
https://doi.org/10.3390/w14213494

**Chicago/Turabian Style**

Hernández-Bedolla, Joel, Abel Solera, Javier Paredes-Arquiola, Sonia Tatiana Sanchez-Quispe, and Constantino Domínguez-Sánchez. 2022. "A Continuous Multisite Multivariate Generator for Daily Temperature Conditioned by Precipitation Occurrence" *Water* 14, no. 21: 3494.
https://doi.org/10.3390/w14213494