# Subdaily Rainfall Estimation through Daily Rainfall Downscaling Using Random Forests in Spain

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Study Area and Information Sources

## 3. Methodology

#### 3.1. Beuchat’s Model

#### 3.2. RFB Model

#### 3.3. Model Evaluation

#### 3.4. Synthetic Rainfall Generation

## 4. Results

#### 4.1. Model Comparison

#### 4.2. Performance Analysis of RFB

#### 4.3. Performance of Simulated Rainfall

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(Left) Location of the hourly (735, black stars) rainfall gauges used in the present study with rainfall records in the period 1990–2015. (Right) Spatial distribution of Köppen–Geiger climate groups developed by AEMET [32]. The acronyms shown in the legend correspond to the following climatic subgroups: BWh (hot desert), BWk (cold desert), Bsh (hot semi-arid), Bsk (cold semi-arid), Csa (temperate hot-summer), Csb (temperate warm-summer), Csc (temperate cold-summer), Csa (temperate humid subtropical), Cfb (temperate oceanic climate), Dsb (Mediterranean-influenced hot-summer humid continental), Dsc and Dfc (Subarctic climates), Dfb (Warm-summer humid continental climate), and ET (tundra climate)

**Figure 2.**Scheme of the methodology. First, hourly rainfall data are aggregated into daily data. Second, supradaily and subdaily rainfall statistics are estimated. Third, monthly atmospheric series from reanalysis data over the period of observation of the corresponding hourly rainfall series are extracted in each gauge location. Then, monthly statistics (mean and variance) are estimated from atmospheric predictors. Daily and supradaily rainfall statistics together with monthly atmospheric statistics and elevation are used as a predictors. The regressor is trained to fit the observed statistics of hourly rainfall. SLP, sea-level pressure; MARS, multivariate adaptive regression splines.

**Figure 3.**Performance evaluation of the random forests-based model (RFB) for variance, proportion of dry intervals (Pdry), skewness coefficient, and autocorrelation lag-one (ACF-lag1) for 1-h intervals. The first and third columns show the scatter plots of the observed versus downscaled (predicted) values; colors indicate climate type; dashed black line correspond to the linear regression. The second and fourth columns show the relative error (%) distributions through a kernel density estimator representation ($err=(Ob{s}^{2}-Pre{d}^{2})/Ob{s}^{2}$).

**Figure 4.**Performance evaluation of the random forests-based model (RFB) for the transition probabilities dry-dry and wet-wet for 1-h intervals. Results are shown in the same way as defined in Figure 3.

**Figure 5.**Location of the hourly rainfall gauges used to perform the simulation of the rainfall by the Neyman–Scott rectangular pulse model (NSRPM).

**Figure 6.**One-hour rainfall performance of the NSRPM simulation of variance, Pdry, and skewness. Black color corresponds to the “exact scenario”; the line and squares represent, respectively, the observed and simulated 1-h statistics. Red color corresponds to the “target scenario”; the dashed-line and squares represent, respectively, the RFB-predicted and the simulated 1-h statistics. The blue-hatched area shows the range of statistics for the calibration in the “simple scenario”. Each row corresponds to a case study.

**Figure 7.**One-hour rainfall performance of the NSRPM simulation of ACF-lag1, ${\varphi}^{DD}$ and ${\varphi}^{WW}$. Black color corresponds to the “exact scenario”; the line and squares represent, respectively, the observed and simulated 1-h statistics. Red color corresponds to the “target scenario”; the dashed-line and squares represent, respectively, the RFB-predicted and the simulated 1h-statistics. The blue-hatched area shows the range of statistics for the calibration in the “simple scenario”. Each row corresponds to a case study.

**Figure 8.**Empirical intensity-frequency curves derived from observed and simulated rainfall at 1-h aggregation. Black dots represent exceedance probability values of the observed rainfall series. Black and red lines correspond to the exceedance probability of “exact scenario” and “target scenario”, respectively. The blue-colored area represents the range of exceedance probability values between the 10 calibrations in the “simple scenario”. Each panel corresponds to a case study.

Organization | Number of Gauges |
---|---|

AEMET [38] | 38 |

Cuenca Mediterránea Andaluza [39] | 109 |

C.H.Segura [40] | 114 |

C.H. Miño-Sil [41] | 89 |

C.H. Cantábrico [42] | 56 |

C.H. Jucar [43] | 185 |

C.H. Ebro [44] | 69 |

Organismo Autónomo Parques Nacionales [45] | 16 |

Sistema de Información Agroclimática para el Regadio [46] | 237 |

Servei Meteorològic de Catalunya [47] | 43 |

**Table 2.**Coefficient of determination (${R}^{2}$) computed through 10-fold cross-validation for the 1-h and 12-h variance, probability of a dry interval (Pdry), skewness, and lag-1 autocorrelation coefficient for Beuchat’s model (B) and for the random forests-based model (RFB) proposed in the present paper. Bold letters highlight the best performing model for every climate and statistic.

Variance | Pdry | Skewness | ACF-lag1 | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 h | 12 h | 1 h | 12 h | 1 h | 12 h | 1 h | 12 h | |||||||||

B | RFB | B | RFB | B | RFB | B | RFB | B | RFB | B | RFB | B | RFB | B | RFB | |

BWh | 0.79 | 0.65 | 0.96 | 0.96 | 0.84 | 0.89 | 0.99 | 0.99 | 0.59 | 0.62 | 0.91 | 0.91 | −0.13 | 0.40 | −12.16 | 0.04 |

BWk | 0.65 | 0.72 | 0.98 | 0.98 | 0.53 | 0.75 | 0.98 | 0.98 | 0.62 | 0.68 | 0.94 | 0.93 | −0.54 | 0.00 | −52.59 | 0.32 |

BSh | 0.73 | 0.63 | 0.96 | 0.96 | 0.92 | 0.94 | 0.99 | 0.99 | 0.69 | 0.65 | 0.91 | 0.92 | −0.68 | 0.33 | −7.89 | 0.34 |

BSk | 0.72 | 0.75 | 0.97 | 0.97 | 0.92 | 0.93 | 0.99 | 0.99 | 0.65 | 0.65 | 0.93 | 0.93 | 0.14 | 0.49 | −13.83 | 0.38 |

Csa | 0.80 | 0.81 | 0.98 | 0.98 | 0.92 | 0.94 | 0.98 | 0.99 | 0.73 | 0.74 | 0.92 | 0.92 | 0.26 | 0.57 | −11.51 | 0.53 |

Csb | 0.84 | 0.85 | 0.99 | 0.99 | 0.93 | 0.97 | 0.98 | 0.99 | 0.65 | 0.65 | 0.92 | 0.91 | 0.14 | 0.67 | −8.11 | 0.65 |

Cfa | 0.61 | 0.70 | 0.95 | 0.94 | 0.79 | 0.84 | 0.97 | 0.97 | 0.37 | 0.42 | 0.87 | 0.87 | 0.27 | 0.63 | −11.45 | 0.44 |

Cfb | 0.69 | 0.72 | 0.97 | 0.97 | 0.93 | 0.95 | 0.99 | 0.99 | 0.59 | 0.62 | 0.93 | 0.93 | 0.29 | 0.65 | −7.16 | 0.57 |

D | 0.66 | 0.77 | 0.94 | 0.97 | 0.73 | 0.69 | 0.92 | 0.88 | −0.56 | 0.21 | 0.78 | 0.79 | 0.26 | 0.43 | −13.77 | 0.03 |

Total | 0.78 | 0.83 | 0.98 | 0.98 | 0.94 | 0.96 | 0.99 | 0.99 | 0.71 | 0.73 | 0.93 | 0.93 | 0.24 | 0.61 | −9.74 | 0.56 |

**Table 3.**Predictors used (first column of the table) by models B and RFB (heading row) for the prediction of each statistic (subheading row). Predictors are: average daily precipitation (${\mu}_{24}$), daily precipitation variance (${\sigma}_{24}^{2}$), probability of a dry day (${\varphi}_{24}$), probability of two consecutive dry days (${\varphi}_{48}$), daily rainfall skewness (${\gamma}_{24}$), lag-1 daily autocorrelation coefficient (${\rho}_{24}^{1}$), probability of two consecutive wet days (${\varphi}_{24}^{WW}$), probability of two consecutive dry days (${\varphi}_{24}^{DD}$), average surface air temperature (TAS), surface air temperature variance (${\sigma}_{TAS}^{2}$), relative air humidity (HUR), and elevation of the station. Predictands are: rainfall variance (${\sigma}_{T}$), probability of a dry interval (${\varphi}_{T}$), rainfall skewness (${\gamma}_{T}$), rainfall lag-1 autocorrelation coefficient (${\rho}_{T}^{1}$), probability of two adjacent wet intervals (${\varphi}_{T}^{WW}$), and probability of two adjacent dry intervals (${\varphi}_{T}^{DD}$) at the ${T}^{should}$ it be italics? (in hours) aggregation scale. A colored cell indicates that a predictor has been used to predict a given predictand for the specific model (B: Beuchat’s model, in red; RFB: random forests-based model, in green).

Predictors | Predictands ($\mathit{T}\in \left\{1,2,3,6,12\right\}$) | ||||||||
---|---|---|---|---|---|---|---|---|---|

B | RFB | ||||||||

${\sigma}_{T}$ | ${\varphi}_{T}$ | ${\gamma}_{T}$ | ${\sigma}_{T}$ | ${\varphi}_{T}$ | ${\gamma}_{T}$ | ${\rho}_{T}^{1}$ | ${\varphi}_{T}^{WW}$ | ${\varphi}_{T}^{DD}$ | |

${\mu}_{24}$ | |||||||||

${\sigma}_{24}$ | |||||||||

${\varphi}_{24}$ | |||||||||

${\varphi}_{48}$ | |||||||||

${\gamma}_{24}$ | |||||||||

${\rho}_{24}^{1}$ | |||||||||

${\varphi}_{24}^{WW}$ | |||||||||

${\varphi}_{24}^{DD}$ | |||||||||

TAS | |||||||||

${\sigma}_{TAS}^{2}$ | |||||||||

HUR | |||||||||

Elevation |

**Table 4.**Coefficient of determination (${R}^{2}$) computed through 10-fold cross-validation for the 1-h and 12-h variance for the random forests-based model (RFB) proposed in the present paper.

${\mathit{\varphi}}^{\mathit{DD}}$ | ${\mathit{\varphi}}^{\mathit{WW}}$ | |||
---|---|---|---|---|

1 h | 12 h | 1 h | 12 h | |

BWh | 0.97 | 0.99 | 0.59 | 0.72 |

BWk | 0.97 | 0.97 | 0.12 | 0.83 |

BSh | 0.96 | 0.98 | 0.59 | 0.85 |

BSk | 0.98 | 0.99 | 0.67 | 0.87 |

Csa | 0.97 | 0.99 | 0.75 | 0.86 |

Csb | 0.96 | 0.99 | 0.80 | 0.90 |

Cfa | 0.87 | 0.97 | 0.73 | 0.79 |

Cfb | 0.95 | 0.98 | 0.8 | 0.90 |

D | 0.61 | 0.93 | 0.04 | 0.20 |

Total | 0.97 | 0.98 | 0.77 | 0.89 |

**Table 5.**Set of statistics used to fit NSRPM and associated weights. d and h indicate daily and hourly level of aggregation, respectively.

Mean | Variance | Skewness | Proportion of Dry Intervals | Lag-1 Correlation | ${\mathit{\varphi}}^{\mathit{DD}}$ | ${\mathit{\varphi}}^{\mathit{WW}}$ | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Time scale | 1 d | 1 h | 1 d | 1 h | 1 d | 1 h | 1 d | 1 h | 1 d | 1 h | 1 d | 1 h | 1 d |

Weights | 5 | 4 | 2 | 3 | 2 | 3 | 2 | 3 | 2 | 3 | 2 | 4 | 2 |

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**MDPI and ACS Style**

Diez-Sierra, J.; del Jesus, M.
Subdaily Rainfall Estimation through Daily Rainfall Downscaling Using Random Forests in Spain. *Water* **2019**, *11*, 125.
https://doi.org/10.3390/w11010125

**AMA Style**

Diez-Sierra J, del Jesus M.
Subdaily Rainfall Estimation through Daily Rainfall Downscaling Using Random Forests in Spain. *Water*. 2019; 11(1):125.
https://doi.org/10.3390/w11010125

**Chicago/Turabian Style**

Diez-Sierra, Javier, and Manuel del Jesus.
2019. "Subdaily Rainfall Estimation through Daily Rainfall Downscaling Using Random Forests in Spain" *Water* 11, no. 1: 125.
https://doi.org/10.3390/w11010125