E-proceedings of the 38th IAHR World CongressSeptember 1-6, 2019, Panama City, Panama

doi:10.3850/38WC092019-0161

3750

RELATIONSHIP BETWEEN DROUGHT OCCURRENCE AND ENSO IN SOUTHERN PERU: A COPULAS ANALYSIS

JUAN WALTER CABRERA CABRERA (1) & JESÚS ABEL MEJÍA MARCACUZCO(2)

(1,2) Programa de Doctorado en Recursos Hídricos, Universidad Nacional Agraria La Molina, Lima, Peru.

20180857@lamolina.edu.pe

ABSTRACT

Southern Peru is a semiarid area with frequent occurrence of drought events. Causes are often attributed to the

occurrence of El Niño Phenomenon however; this relationship has not been fully confirmed. In this article, the

possible relationship between the occurrences of drought with ENSO in southern Peru is analyzed under a

copula’s analysis. For this purpose, the Standardized Precipitation Index is used for a period of three months

(SPI3) as an indicator of drought and NINO34, NINO 1+2, and ICEN indices as indicators of El Niño

Phenomenon occurrence. First, SPI should be estimated for every gage station in Candarave and Cairani

irrigation district and the marginals distributions to be fitted to every group of data. Also, marginals to ENSO

indexes should be evaluated. Finally, copulas are built in base to marginals and looking for the best correlation.

Copulas analysis is a statistical technic which will establish whether there are relationships between these

variables and can be used as a basis for the development of mitigation plans against the occurrence of droughts.

The results show significant relationship between the occurrence of droughts and NINO34 index in Cairani but

not relationship in Candarave.

Keywords: Drought, ENSO, Copulas.

1 INTRODUCTION

The study of droughts is not a new issue in the field of water resources, nor is it a problem restricted only to the

national level. Its consequences on the population include economic losses and affect the normal development

of socio-economic activities (Santos Pereira et al., 2002, Salvadori et al., 2005, Knutson 2008, Wilhite and

Glantz 1985), and may cause massive waves of migration, such as those that occurred in the first half of the

eighties in Peru (INEI, 2009).

Due to the high complexity of these phenomena, and in order to make predictions of occurrence, in the last fifty

years a strong tendency towards the use of stochastic models has developed. The works of Salas (Cancelliere

and Salas 2004, Tables and Salas 1985, Chung and Salas 2000), Cancellieri (Cancelliere et al., 2007, Serinaldi

et al., 2009) analyzing the application of techniques such as AR models, ARMA, Markov chains, among others,

and including non-stationary series, represent an attempt to reproduce and predict the occurrence of the

phenomenon.

In the last fifteen years, the International Association of Hydrological Sciences (IAHS) has begun to test the use of copulas for the study of droughts based on Sklar's theorem (Rüschendorf 2013). The studies carried out by (Madadgar and Moradkhani 2013, Shiau 2006, Dupuis 2007, Kole et al 2007, Salvadori et al 2005, Serinaldi et al 2009) show that the method is efficient to perform multivariate frequency analysis on different descriptors of droughts. These experiences suggest that the technique allows to sufficiently represent complex relationships between hydrometeorological variables (AghaKouchak 2014) similar to those that occur in our country.

2 DEFINITIONS

2.1 Drought Droughts are considered like climatic deviations with respect to normal or desirable conditions in a region

(Wilhite and Glantz, 1985). Usually, they can be described using 3 parameters: magnitude, length and intensity however, they could be studied by using indices. These indices define the level of severity according to predefined boundaries. The more widely used indices are the Palmer Drought Severity Index (PDSI) and the Standardized Precipitation Index (SPI).

The PDSI is an algorithm of soil moisture for relatively homogeneous areas based on the precipitation,

temperature and volume of water available locally. It uses a model of monthly water balance with two layers of

soil and expresses the humidity conditions in a standardized way with a scale that goes from approx. -6 to 6

E-proceedings of the 38th IAHR World CongressSeptember 1-6, 2019, Panama City, Panama

3750

(Cancelliere, 2010). Due to the high information requirement, McKee (1993) proposes a new index based on

standardized precipitation, defined as a standard normal variable:

𝑆𝑃𝐼 =y−μ

σ [1]

where,

y it is the monthly aggregate precipitation;

is the average of the monthly aggregate precipitation variable; and

is the standard deviation of the variable monthly aggregate precipitation.

Precipitation can be accumulated on a monthly, trimester or semiannual basis, defining in this way the so-called SPI1, SPI3 and SPI6. The scale of aggregation depends on the type of drought to be evaluated and its manifestation in the short or medium term. In the case of agricultural drought, its occurrence is associated with the SPI3 index.

2.2 ENSO El Niño-Southern Oscillation (ENSO) is a cycle of warming and cooling events of the equatorial Pacific

Ocean and its atmosphere that occurs with interannual frequency, which oscillate between 2 to 7 years. These temperatures anomalies could show in different regions of Pacific Ocean and their impact will affect different regions on the world

Figure 1 : “El Niño” regions. SST temperatures anomalies in 3-4 and 1+2 have more influence in south America.

Source: https://climatedataguide.ucar.edu

ENSO include two stages: warming (called like “El Niño” phenomenon), and cooling (called “La Niña” phenomenon). To detect the presence of the El Niño and La Niña phenomenon, various indices have been designed to represent anomalies of related variables related such pressure, sea temperature, etc. For this study, we will focus on El Niño Coastal Index (ICEN), and El Niño 1+2 and 3-4 SST anomalies indices.

El Niño Coastal Index (ICEN), is based on sea surface temperature for zone 1 + 2. For their calculation the following protocol has been established: "It consists of the three-month average run of the monthly anomalies of sea surface temperature (SST) in the Niño 1 + 2 region. These anomalies will be calculated using the monthly climatology calculated for the base period 1981-2010. The data source for this index are the absolute TSMs of the ERSST v3b product from NOAA (USA) for the Niño 1 + 2 region "(ENFEN, 2012). This index has shown good performance to describe El Niño phenomenon occurrence at northern Peru.

2.3 Copulas Copula is a function that unites or couples multivariate distribution functions to its one-dimensional marginal

distribution functions (Rayens and Nelsen 2000). The concept was introduced by Sklar (Rüschendorf 2013), who proposed that every function of n-dimensional distribution F can be decomposed into two parts, a function of marginal distribution Fi and a copula C, which represents the dependence of the distribution.

Copulas could be subdivided in three categories: fundamental copulas, which represent number of important special dependence structures; implicit copulas, which are extracted from known multivariate distributions using Sklar’s Theorem but without simple closed-form expressions, usually; and, explicit copulas, which have simple closed-form expressions. Probably, gumbel, normal (or Gaussian), frank, and clayton copula are the most usual used in hydrology and climatology; the first two are implicit copula meanwhile the two last are explicit copulas. These four will be used in this research.

Table 1 summarize definition for these copulas and Figure 1 shows characteristic plots.

E-proceedings of the 38th IAHR World CongressSeptember 1-6, 2019, Panama City, Panama

3751

Table 1. Definition of most important copulas.

Copula Bivariate Copula C(u,v) Parameter

Normal --

Clayton 𝜃𝜖[−1,∞)\{0}

Frank 𝜃𝜖𝑅\{0}

Gumbel 𝜃𝜖[1,∞)

Source: https://www.senamhi.gob.pe

Figure 2 : Characteristic distribution to some copulas used in this research. Source: By Avraham - Own work, CC BY 4.0, https://commons.wikimedia.org/w/index.php?curid=38108215

3 MATERIALS AND METHODS

3.1 Study zone and data Candarave is located in Tacna, in southern Peru. Geographically, this area is defined by UTM Coordinates

353000 - 384000East and 8087000 – 8134000South, with altitudes between 3200m to 4800m above sea level. Its main economic activity is agriculture, with two main irrigation districts: Candarave and Cairani (see Figure 3). This region has a moderately cold weather, with temperatures varying between 16.8º C (maximal mean temperature in November) to 1.9 ºC (minimum mean temperature in July). Total monthly precipitation in Candarave varies between 0 mm and 55 mm, with December, January, February and March as wetted month and the driest months are May, Jun and July, defining a permanent water scarcity.

To evaluate SPI, total monthly precipitation corresponding to Candarave and Cairani gauge stations for the period 1964 – 2009 were used (See Table 2).

Table 1.Meteorological Stations used for this research.

Station Operated by

Geographic Coordinates – ZONE 19 Datum WGS’ 84

Length (m)

Latitude (m)

Altitude (m.a.s.l.)

CANDARAVE SENAMHI 70°15’ O 17°16’ S 3415

CAIRANI SENAMHI 70º 22' O 17º 17' S 3205

Source: https://www.senamhi.gob.pe

To evaluate ENSO occurrence, NINO3-4, NINO 1+2, and ICEN indices were download from Geophysical Institute of Peru (IGP, by the acronyms in Spanish) in http://www.met.igp.gob.pe/ and the webpage from National Weather Services from NOAA (https://origin.cpc.ncep.noaa.gov).

E-proceedings of the 38th IAHR World CongressSeptember 1-6, 2019, Panama City, Panama

3752

Figure 3 : Study area. In red circles, Cairani and Candarave irrigation districts. Source: Modified from UNI(2010)

3.2 Methods For compare drought an ENSO occurrence, the SPI to Cairani and Candarave will be estimated and after

that the empirical cumulative distribution function will be estimated for every index: SPI, ICEN, ENSO34 and ENSO1+2. This analysis was developed by using the package “copula” in R.

In a second step, pseudo observations will be estimated, and cross correlation will be evaluated to recognize any kind of relationship. For this goal, Spearman’s rho and Kendall’s tau parameters will be used like simple scalar measures of dependence.

Finally, if it’s possible, the copula model will be built. For this goal, the Multivariate Copula Analysis Toolbox

(MvCAT) was employed. This software was developed by University of California Irvine (UCI) and employs

Markov Chain Monte Carlo (MCMC) simulation within a Bayesian framework to estimate copula parameters and

the underlying uncertainties (Sadegh et al, 2017) and ranks the performance of copulas based on maximum

likelihood, Akaike Information Criterion (AIC), and Bayesian Information (BIC).

4 RESULTS Figure 4 show SPI3 for Candarave and Cairani irrigation districts. According with this, Candarave suffered

three extreme droughts in 1988, 1991 and 1992 however, in none of them ENSO don’t reach the category of “strong”. This is a first sign of not relationship. On the other hand, Cairani show extreme droughts in 1982-1983 (were an strong ENSO was registered) and 1992 (without ENSO occurrence); this discordance suggest the necessity of more evidences for a final conclusions. The most important El Niño phenomenon registered in the period analysis occurred in 1997-1998 however, these year didn’t show unusual severe or extreme drought.

E-proceedings of the 38th IAHR World CongressSeptember 1-6, 2019, Panama City, Panama

3753

Figure 4 : SPI for Candarave and Cairani irrigation district. Period 1964-2009.

Empirical cumulative distribution was estimated for every index considered, getting curves with a good distribution: SPIs shown a normal distribution because their definition but, ICEN and ENSO1+2 shown asymmetry (see Figure 5).

Figure 5 : Empirical cumulative distribution function (ecdf) to analyzed data. Up left, SPI in Candarave gauge station. Up middle, SPI in Cairani gauge station. Up right, ICEN. Down left, ENSO1+2. Down

right, ENSO34. Period 1964-2009. Like complement, a cross analysis for data pairs was realized. (see Figures 6 and 7). This analysis shown

no relationship between SPI and ENSO indices in Candarave nor in Cairani however, there is high correlation

-4

-3

-2

-1

0

1

2

3

4

Mar

-64

Jun

-65

Sep

-66

Dec

-67

Mar

-69

Jun

-70

Sep

-71

Dec

-72

Mar

-74

Jun

-75

Sep

-76

Dec

-77

Mar

-79

Jun

-80

Sep

-81

Dec

-82

Mar

-84

Jun

-85

Sep

-86

Dec

-87

Mar

-89

Jun

-90

Sep

-91

Dec

-92

Mar

-94

Jun-

95

Sep

-96

Dec

-97

Mar

-99

Jun

-00

Sep

-01

Dec

-02

Mar

-04

Jun

-05

Sep

-06

Dec

-07

Mar

-09

SPI - Candarave

-4

-3

-2

-1

0

1

2

3

4

Mar

-64

Jun-

65

Sep

-66

Dec

-67

Mar

-69

Jun

-70

Sep

-71

Dec

-72

Mar

-74

Jun

-75

Sep

-76

Dec

-77

Mar

-79

Jun

-80

Sep

-81

Dec

-82

Mar

-84

Jun-

85

Sep

-86

Dec

-87

Mar

-89

Jun

-90

Sep

-91

Dec

-92

Mar

-94

Jun

-95

Sep

-96

Dec

-97

Mar

-99

Jun

-00

Sep

-01

Dec

-02

Mar

-04

Jun

-05

Sep

-06

Dec

-07

Mar

-09

SPI - Cairani

E-proceedings of the 38th IAHR World CongressSeptember 1-6, 2019, Panama City, Panama

3754

between ICEN and ENSO1+2. This relationship is evident because ICEN is defined in base to anomalies of SST in region El Niño 1+2. Also, there is a notorious correlation between ICEN and ENSO34. This correlation could mean a joint effect of El Niño phenomenon in southern Peruvian coast but not on study region.

Figure 6 : Observed data matrix. Candarave gauge station. SPI don’t show relationship with ENSO indices.

Figure 7 : Observed data matrix. Cairani gauge station. SPI don’t show relationship with ENSO indices. Finally, dependence between two input variables was evaluated by using Kendall’s tau and Spearman’s

rho ranks and all every analysis was found non-significant, like is shown in Table 2; only ENSO3-4 shown significant relationship. For this pair, maxima likelihood, Akaiko and Bayesian Information Criteria were

E-proceedings of the 38th IAHR World CongressSeptember 1-6, 2019, Panama City, Panama

3755

evaluated to select copula with the best performance. All of them suggest that the best performance correspond to Gumbel copula (see Table 3).

Table 2. Kendall rank and Spearman rank for SPI and ENSO indices. Region ENDO index Kendall rank Spearman's rank-order Significant at 5%?

Candarave ICEN -0.0360 -0.0558 NO ENSO1+2 -0.0365 -0.572 NO ENSO34 -0.0435 -0.0630 NO

Cairani ICEN -0.0266 -0.0421 NO ENSO1+2 -0.0310 -0.0493 NO ENSO34 -0.0976 -0.1423 YES

Table 3. Sort copulas based on different criteria. Rank Max-Likelihood AIC BIC

1 Gumbel Gumbel Gumbel 2 Clayton Clayton Clayton 3 Gaussian (normal) Gaussian (normal) Gaussian (normal) 4 Frank Frank Frank

Parameters for these four copulas were evaluated and RMSE and NSE estimated. Gumbel copula shown an RMSE near to zero and NSE near to one, confirming that have the best performance (see Table 4) however, Clayton copula shown similar values to any parameter. Figure 8 show copulas probability for both and similar performance is clear.

Table 4.Estimated copula parameters.

Copula RMSE NSE Parameter ()

Gaussian 0.5679 0.988 -0.1364

Clayton 0.4686 0.9918 0

Frank 0.5894 0.987 -0.8759

Gumbel 0.467 0.9919 1.0036

Figure 8 : Copulas probability for SPI vs ENSO3-4.

5 CONCLUSIONS According with results, there was not found a relationship between agricultural drought occurrence and El

Niño phenomenon in Candarave but Cairani. The first analysis, in base a SPI, didn’t find relationship between extreme droughts and strong ENSO. The second cross analysis didn’t find relationships neither. Finally, Kendall rank and Spearman rank shown significance between SPI and ENSO34 in Cairani. For this last condition, Gumbel and Clayton copulas shown best performance and copulas parameters were estimated. Results suggest necessity of more research because Cairani is not far from Candarave and the main geographic difference is related to altitude; this could mean effects of El Niño phenomenon restricted to some level above

E-proceedings of the 38th IAHR World CongressSeptember 1-6, 2019, Panama City, Panama

3756

sea surface. Also, is necessary to evaluate performance of copular to replicate events according with historical data.

REFERENCES

AghaKouchak, A. (2014). Entropy–Copula in Hydrology and Climatology. Journal of Hydrometeorology. Cancelliere, A; Di Mauro, G; Bonaccorso, · B; Rossi, G. (2007). Drought forecasting using the Standardized

Precipitation Index. Water Resources Management 21(5):801-819. Cancelliere, A; Salas, JD. (2004). Drought length properties for periodic-stochastic hydrologic data. Water

Resources Research 40(2). INEI. (2009). Perú: Migraciones Internas. Lima, INEI. 166 p. Knutson, C. (2008). Methods and Tools for Drought Analysis and Management. Eos, Transactions American

Geophysical Union . Mckee, TB; Doesken, NJ; Kleist, J. (1993). The relationship of drought frequency and duration to time scales.

In Applied Climatology. s.l., s.e. p. 17-22 Nkemdirim, L. (2015). Palmer Drought Severity Index (online). s.l., s.e., vol.3. 224-231 p. Rayens, B; Nelsen, RB. 2000. An Introduction to Copulas. Technometrics. DOI:

https://doi.org/10.2307/1271100. Rüschendorf, L. (2013). Copulas, Sklar’s Theorem, and Distributional Transform. s.l., s.e. DOI:

https://doi.org/10.1007/978-3-642-33590-7. Sadegh, M., Ragno, E., and AghaKouchak, A. (2017), Multivariate Copula Analysis Toolbox (MvCAT):

Describing dependence and underlying uncertainty using a Bayesian framework, Water Resources. Res., 53, doi:10.1002/2016WR020242.

Salvadori, G; De Michele, C; Kottegoda, NT; Rosso, R. (2005). Extremes in Nature: An approach using Copulas. s.l., s.e.

Wilhite, D. A. and Glantz, M. (1985) “Understanding the drought phenomenon: The role of definitions”. Water International, 10 (3), 111–120.

Yevjevich, V. (1967). August 1967. Hydrology papers 146(August):1967.