# Improving Stochastic Modelling of Daily Rainfall Using the ENSO Index: Model Development and Application in Chile

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Meteorological Stations and Climatic Data

#### 2.2. Precipitation Occurrence Submodel

_{t}with value 1 or 0, in case of non-zero or no precipitation, respectively. The first-order Markov chain model is characterized by the transition probabilities ${p}_{01}$ (i.e., probability of having a rainy day after a dry day), and ${p}_{11}$ (i.e., probability of having a rainy day following a rainy day):

_{t}

_{−1}, while the ENSO index is the second covariable. In this way, the conditional probabilities of the first-order Markov chain are being considered. Additional covariables, such as a seasonal cycle or other climatic indices, could be included if needed. The following expression was used to obtain ${p}_{1}\left(t\right)$.

_{1}) is generated. If ${p}_{1}\left(t\right)$ ≤ υ

_{1}a rainfall event is simulated, otherwise the day is considered to be dry (Figure 3).

#### 2.3. Precipitation Intensity Submodel

_{2}between 0 and 1, then the precipitation follows an exponential distribution with mean M = ${\mathrm{\xdf}}_{1}$, otherwise M = ${\mathrm{\xdf}}_{2}$. Note that ${\mathrm{\xdf}}_{1}$ and ${\mathrm{\xdf}}_{2}$ correspond to magnitudes thar are low and high respectively. Finally, the precipitation of day t (P

_{t}) is obtained as:

## 3. Results

#### 3.1. Precipitation Occurrence

_{0}, C

_{1}, and C

_{2}of the GLM model for precipitation occurrence including ENSO (values of the coefficients for the model that do not incorporate this covariable are not shown), and the basic goodness of fit statistics (i.e., coefficient of determination (R

^{2}), Akaike information criterion (AIC) [52], and the root mean square error (RMSE)) for both models. This comparison is shown for August, although similar results were obtained for the autumn and winter seasons. Incorporating the ENSO index improves the estimation of the precipitation occurrence during these months, mostly in arid and Mediterranean areas. For example, the La Serena rain gauge R

^{2}increased from 0.5 to 0.7, and the AIC decreased from 966.9 to 890.9 in August, once the ENSO index was added to the GLM model only based on K

_{t}

_{-1.}

#### 3.2. Precipitation Intensity

#### 3.2.1. Annual Precipitation

#### 3.2.2. Monthly Precipitation

_{3}, of the synthetically generated values is the same as the maximum historical precipitation (Figure 7a). This difference is explained by the fact that a few precipitation outliers affected the distribution of synthetically generated data in that month. On the other hand, the model is able to generate monthly precipitation values similar to the observed ones for autumn and winter months, as shown in Figure 8b,d. In Santiago, the minimum and maximum observed daily precipitation values in July were 0.2 mm and 27.1 mm, while the simulated ones were 0.5 mm and 27 mm. Similarly, the observed 75th, 50th, and 25th percentiles (i.e., quartiles Q

_{1}, Q

_{2}, and Q

_{3}) are 12.0 mm, 2.9 mm, and 0.9 mm, values that compare very well with the simulated ones, which are 11.9 mm, 3.2 mm, and 0.7, mm (Figure 8b).

#### 3.2.3. Mixed Exponential Distribution Parameters

#### 3.2.4. Interannual Variability

_{t}

_{−1}) underestimated the interannual variability, particularly during those months with less precipitation in arid and Mediterrenean climates. In contrast, the standard deviation values were better represented when incorporating the ENSO index, which implies an improvement in the simulation of the interannual variability. Standard deviations of the number of wet days from the observations, and those from simulations conducted with and without the ENSO index models, are given in Figure 11. Using the ENSO index as a covariate improved the representation of the interannual variability of the frequency of wet days. This indicates that the overdispersion problem in the model for monthly precipitation totals is the result of its inability to produce a sufficiently large variance in the number of wet days.

#### 3.2.5. Wet and Dry Spell Durations

## 4. Discussion and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

WGs | Weather Generators |

WGEN | Weather Generate by Richardson and Wright |

GLM | Generalized Lineal Model |

ENSO index | Monthly Sea Surface Temperature Anomalies of the Region 3.4 of El Niño-Southern Oscillation |

CLIGEN | CLImate GENerator |

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**Figure 3.**Flowcharts for daily precipitation generation using the proposed model. The numbers in square brackets refer to equations in the text.

**Figure 4.**Conditional probabilities obtained with the parameters of a GLM model for different months as a function of the ENSO index. Panels above (

**a**,

**b**) show conditional probabilities of having a rainy day following dry day for La Serena and Santiago, respectively. Panels below (

**c**,

**d**) show the conditional probabilities of having wet days following a wet day for La Serena and Santiago, respectively. Lines in yellow and red are the observed probabilities, whereas black lines are the simulated values.

**Figure 5.**Differences between historic and synthetically generated average annual precipitation. Values were generated with 500 simulations of 64 years.

**Figure 6.**Plots of the simulated versus observed 25-year mean of annual maximum daily precipitation (

**a**) and standard deviation of the annual precipitation (

**b**).

**Figure 7.**Box plot of rainy days calculated from observed and simulated data, grouped in spring–summer months (

**Left**) and autumn-winter months (

**Right**) for La Serena (

**a**,

**b**) and Balmaceda (

**c**,

**d**). In each box plot, the upper triangle, rhombus and lower triangle denote quartile one (Q

_{1}), quartile two (Q

_{2}) and quartile three (Q

_{3}). Five hundred simulations of 64 years were produced to generate the box plots for the simulated data.

**Figure 8.**Box plot of rainy days calculated from observed and simulated data, grouped in spring–summer months (

**Left**) and autumn–winter months (

**Right**) for Santiago (

**a**,

**b**) and Puerto Aysén (

**c**,

**d**). In each box plot the upper triangle, rhombus and lower triangle denote quartile one (Q

_{1}), quartile two (Q

_{2}) and quartile three (Q

_{3}). Five hundred simulations of 64 years were produced to generate the box plots for the simulated data.

**Figure 9.**Seasonal variation of the parameters of the mixed exponential distribution from synthetically generated (dotted lines) and historical data (lines); (

**a**–

**c**) Rancagua gauge (Mediterranean climate); (

**d**–

**f**) Puerto Aysén gauge (temperate climate).

**Figure 10.**Simulated versus observed standard deviation of monthly total precipitation, from simulations carried out with GLM (

**a**) with the ENSO index as a covariable and (

**b**) not incorporating the ENSO index as a covariable.

**Figure 11.**Simulated versus observed standard deviation of monthly number of wet days from simulations carried out with GLM (

**a**) incorporating the ENSO index as a covariable and (

**b**) not incorporating the ENSO index as a covariable.

**Figure 12.**Plots of simulated vs. observed lengths of wet or dry spells for all seasons: (

**a**) average length of wet spells; (

**b**) average length of dry spells.

Station Name | Climate | Latitude (° South) | Longitude (° West) | Elevation (m a.s.l.) | Begginning | End | % Missing Data | Average Annual Precipitation (mm) |
---|---|---|---|---|---|---|---|---|

La Serena | Arid | 29°55′ | 71°12′ | 142 | 01/01/1950 | 31/12/2013 | 3.6 | 84.3 |

Embalse La Laguna | Arid | 30°12′ | 70°02′ | 3160 | 01/01/1964 | 31/12/2013 | 0.7 | 159.7 |

Santiago | Mediterrean | 33°27′ | 70°40′ | 527 | 01/01/1950 | 31/12/2013 | 0 | 314.1 |

Rancagua | Mediterrean | 34°46′ | 71°07′ | 239 | 01/09/1971 | 31/12/2013 | 1.5 | 691.7 |

Curicó | Mediterrean | 34°58′ | 71°13′ | 225 | 01/01/1950 | 31/12/2013 | 10.4 | 675,5 |

Parral | Temperate | 36°11′ | 71°49′ | 175 | 01/02/1993 | 31/12/2013 | 0 | 952.2 |

Diguillin | Temperate | 36°52′ | 71°38′ | 670 | 01/05/1959 | 31/12/2013 | 0.8 | 2091.5 |

Chillán | Temperate | 36°35′ | 72°02′ | 151 | 01/01/1950 | 31/12/2013 | 12.6 | 1044.0 |

Concepción | Temperate | 36°46′ | 73°03′ | 12 | 01/01/1950 | 31/12/2013 | 0 | 1119.5 |

Temuco | Temperate | 38°46′ | 72°38′ | 92 | 01/01/1950 | 31/12/2013 | 13.0 | 1163.4 |

Futaleufu | Temperate | 43°11′ | 71°51′ | 350 | 01/01/1960 | 31/12/2013 | 2.7 | 2133.8 |

Osorno | Temperate | 40°36′ | 73°03′ | 61 | 01/01/1950 | 31/12/2013 | 10.8 | 1358.1 |

Balmaceda | Temperate | 45°54′ | 71°41′ | 520 | 01/01/1958 | 31/12/2013 | 0 | 576.3 |

Puerto Aysén | Temperate | 45°24′ | 72°42′ | 10 | 01/01/1950 | 31/12/2013 | 7.9 | 2661.6 |

Punta Arenas | Temperate | 53°00′ | 70°50′ | 39 | 01/01/1950 | 31/12/2013 | 3.2 | 399.7 |

Puerto Williams | Temperate | 54°55′ | 67°37′ | 30 | 01/01/1950 | 31/12/2013 | 19.9 | 480.8 |

Valdivia | Temperate | 39°39′ | 73°04′ | 18 | 01/01/1950 | 31/12/2013 | 3.7 | 1836.2 |

**Table 2.**Parameters of the generalized lineal model (GLM) (Equation (3)) for August, considering precipitation occurrence in the previous day (K

_{t}

_{−1}) and the El Niño-Southern Oscillation (ENSO) index as a covariable. Bold values denote statistical significance (p value < 0.05) in the parameters.

Station Name | GLM Parameters | Goodness of Fit with ENSO Index | Goodness of Fit without ENSO Index | ||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{C}}_{0}$ | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | R^{2} | RMSE | AIC | R^{2} | RMSE | AIC | |

La Serena | −2.80 | 1.81 | 0.40 | 0.70 | 0.10 | 890.90 | 0.51 | 0.70 | 966.91 |

Embalse La Laguna | −3.11 | 2.45 | 0.40 | 0.62 | 0.61 | 645.22 | 0.43 | 2.31 | 670.46 |

Santiago | −1.83 | 1.64 | 0.10 | 0.56 | 0.94 | 1800.51 | 0.33 | 3.10 | 2100.67 |

Rancagua | −1.71 | 1.72 | 0.10 | 0.76 | 0.91 | 1277.10 | 0.52 | 4.58 | 1877.21 |

Curicó | −1.43 | 1.43 | 0.20 | 0.48 | 0.70 | 1744.11 | 0.34 | 0.91 | 1750.84 |

Parral | −1.15 | 1.32 | 0.05 | 0.67 | 0.97 | 1863.85 | 0.53 | 1.19 | 1975.91 |

Diguillin | −1.03 | 1.51 | 0.08 | 0.42 | 1.15 | 2044.91 | 0.47 | 1 | 2043.22 |

Chillán | −1.10 | 1.42 | 0.10 | 0.53 | 1 | 2060.73 | 0.53 | 1 | 2061.11 |

Concepción | −0.92 | 1.51 | 0.01 | 0.33 | 1.10 | 2434.81 | 0.30 | 1.16 | 2436.80 |

Temuco | −0.53 | 1.41 | −0.07 | 0.52 | 1.12 | 2119.91 | 0.41 | 1.17 | 2125.91 |

Futaleufu | −0.45 | 1.43 | −0.09 | 0.22 | 1.18 | 1934.75 | 0.20 | 1.19 | 1934.69 |

Osorno | −0.40 | 1.51 | −0.01 | 0.43 | 1.09 | 2112.81 | 0.43 | 1.11 | 2114.88 |

Balmaceda | −0.81 | 1.15 | −0.01 | 0.52 | 1.32 | 2226.31 | 0.50 | 1.21 | 2228.39 |

Puerto Aysén | 0.10 | 1.51 | −0.10 | 0.64 | 1.05 | 2058.81 | 0.62 | 1.06 | 2057.61 |

Punta Arenas | −0.91 | 0.83 | 0.09 | 0.42 | 1.05 | 2315.61 | 0.40 | 1.10 | 2416.10 |

Puerto Williams | −1.01 | 0.92 | −0.12 | 0.33 | 1 | 1900.11 | 0.21 | 1.12 | 1955.41 |

Valdivia | −0.33 | 1.71 | −0.01 | 0.34 | 1.06 | 2146.54 | 0.32 | 1 | 2100.62 |

**Table 3.**Contingency table of daily frequencies of dry and wet days from observations and simulations obtained from 10 random years.

Climate Type | Station Name | Observed | GLM with ENSO Index | HSS | GLM without ENSO Index | HSS | ||
---|---|---|---|---|---|---|---|---|

(0) | (1) | (0) | (1) | |||||

Arid | La Serena | (0) | 3340 | 95 | 0.52 | 3303 | 132 | 0.01 |

(1) | 100 | 118 | 208 | 10 | ||||

Arid | Embalse La Laguna | (0) | 3320 | 119 | 0.46 | 3318 | 121 | 0.03 |

(1) | 105 | 109 | 201 | 13 | ||||

Mediterrean | Santiago | (0) | 3000 | 272 | 0.16 | 2956 | 316 | 0.04 |

(1) | 290 | 90 | 330 | 50 | ||||

Mediterrean | Rancagua | (0) | 2756 | 350 | 0.12 | 2744 | 367 | 0.10 |

(1) | 421 | 120 | 423 | 118 | ||||

Mediterrean | Curicó | (0) | 2621 | 430 | 0.13 | 2631 | 420 | 0.11 |

(1) | 441 | 161 | 452 | 150 | ||||

Temperate | Parral | (0) | 2306 | 573 | 0.12 | 2365 | 514 | 0.11 |

(1) | 523 | 250 | 551 | 222 | ||||

Temperate | Diguillin | (0) | 2212 | 586 | 0.12 | 2130 | 668 | 0.09 |

(1) | 574 | 281 | 573 | 282 | ||||

Temperate | Chillán | (0) | 2194 | 556 | 0.15 | 2183 | 567 | 0.13 |

(1) | 590 | 312 | 599 | 303 | ||||

Temperate | Concepción | (0) | 2054 | 638 | 0.13 | 1995 | 661 | 0.10 |

(1) | 632 | 364 | 649 | 347 | ||||

Temperate | Temuco | (0) | 1411 | 756 | 0.10 | 1403 | 773 | 0.09 |

(1) | 821 | 655 | 818 | 658 | ||||

Temperate | Futaleufu | (0) | 1100 | 849 | 0.09 | 1100 | 849 | 0.04 |

(1) | 900 | 804 | 900 | 804 | ||||

Temperate | Osorno | (0) | 1096 | 806 | 0.09 | 1133 | 769 | 0.08 |

(1) | 859 | 891 | 902 | 848 | ||||

Temperate | Balmaceda | (0) | 1640 | 777 | 0.05 | 1663 | 754 | 0.01 |

(1) | 780 | 456 | 844 | 392 | ||||

Temperate | Puerto Aysén | (0) | 1008 | 766 | 0.04 | 1025 | 709 | 0.02 |

(1) | 1000 | 878 | 1090 | 828 | ||||

Temperate | Punta Arenas | (0) | 1973 | 618 | 0.22 | 2004 | 73 | 0 |

(1) | 1500 | 61 | 1516 | 59 | ||||

Temperate | Puerto Williams | (0) | 1008 | 726 | 0.04 | 1025 | 709 | 0.02 |

(1) | 1040 | 878 | 1090 | 828 | ||||

Temperate | Valdivia | (0) | 898 | 629 | 0.09 | 898 | 629 | 0.07 |

(1) | 1043 | 1082 | 1087 | 1038 |

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**MDPI and ACS Style**

Urdiales, D.; Meza, F.; Gironás, J.; Gilabert, H.
Improving Stochastic Modelling of Daily Rainfall Using the ENSO Index: Model Development and Application in Chile. *Water* **2018**, *10*, 145.
https://doi.org/10.3390/w10020145

**AMA Style**

Urdiales D, Meza F, Gironás J, Gilabert H.
Improving Stochastic Modelling of Daily Rainfall Using the ENSO Index: Model Development and Application in Chile. *Water*. 2018; 10(2):145.
https://doi.org/10.3390/w10020145

**Chicago/Turabian Style**

Urdiales, Diego, Francisco Meza, Jorge Gironás, and Horacio Gilabert.
2018. "Improving Stochastic Modelling of Daily Rainfall Using the ENSO Index: Model Development and Application in Chile" *Water* 10, no. 2: 145.
https://doi.org/10.3390/w10020145