SPEI calculation

The SPI has been applied using various durations k of droughts, which varied from three to 72 months (^{Vicente-Serrano et al., 2010}). The monthly precipitation record is processed under the moving sum scheme, so that for the duration of 3 months, the first sequence is obtained by adding the historical data one, two and three, the second sequence is the sum of the data two, three and four and so on until the last sequence, which is made up of the antepenultimate, penultimate and last data. Then the number of ns sequences that can be formed depends on k and is defined by expression:

in which, NA (for its Spanish initials) is the number of completed years of the processed record (> 30 years). The fundamental difference between the calculation of SPI and SPEI is that the second uses as historical data the differences (D _j,i ) between the monthly precipitation (P _j,i ) and the monthly potential evapotranspiration (ETP _j,i , for its Spanish initials), that is:

Logically, j varies from one to 12 and i from one to NA. In order to proceed with the calculation of the SPEI, firstly, the sequences of differences designated by Dlk are formed in which t varies from one to ns. When taking into account that the differences Dlk are most of them negative, there is a need to use a probabilistic model of three fit parameters whose location parameter u is less than the smallest of the sequences to be processed. ^{Vicente-Serrano et al. (2010)} and ^{Beguería et al. (2014)} contrasted four distributions: Log-Normal, Pearson type III, General of Extreme values and Log-Logistics; recommend the latter, fitted by the method of biased weighted probability moments (βs), whose equations are (^{Stedinger, Vogel, & Foufoula-Georgiou, 1993}; ^{Vicente-Serrano et al., 2010}):

being:

In equation 3 the sequences are used ordered in increasing form D1k≤D2k≤Λ≤Dnsk. ^{Beguería et al. (2014)} have proposed to use the unbiased βs, when biased estimators do not lead to a numerical solution. ^{Stagge, Tallaksen, Gudmundsson, Van Loon, & Stahl (2015)} have suggested using the General Distribution of Extreme Values, fitted with the maximum-likelihood method. The equation of the cumulative probability distribution function [F (x)] of the Log-Logistic distribution is (^{Haktanir, 1991}):

being, γ > 0, α > 0 and u < x _mo the parameters of form, scale and location. x _mo is the minimum sequence observed. The values of the fitting parameters are estimated with the following expressions (^{Haktanir, 1991}):

in which Γ (·) is the Gamma factorial function, it was estimated with the formula of Stirling (^{Davis, 1972}):

Calculated the three fit parameters (γ, α, u) of each duration k analyzed, equation 5 is applied with x = Dtk to estimate the non-exceedance probabilities F(x) corresponding to each difference. Then, the rational numerical approximation is used, developed by ^{Zelen and Severo (1972)}, and exposed by ^{Campos-Aranda (2015)}, to convert F(x) into the standardized normal variable Z of zero mean and unit variance, which corresponds to SPEI. Light, moderate, severe and extreme droughts are defined when the SPEI ranges from zero to -1.00, from -1.00 to -1.50, from -1.50 to -2.00 and when it is less than -2.00, respectively.

Monthly ETP calculation

^{Mavromatis (2007)} found that the use of simple or complex methods for estimation of ETP _j,i , leads to similar results when applying drought indices such as that of Palmer. Based on this result, ^{Vicente-Serrano et al. (2010)} adopt a simple approach to estimate the ETP _{j, i} , through the Thornthwaite formula, whose expression is:

in which, Fc is a corrective factor function of the latitude of place (LAT) and of number of days in the month (ndm), its formula is:

where, N is the maximum sunshine or maximum number of hours with average monthly sun. For its estimation in the Mexican Republic, ^{Campos-Aranda (2005)} developed the following empirical expression:

where nm is the number of month, with 1 for January and 12 for December; A and B are constants function of LAT in degrees, with the following expressions:

Tt _j,i is the average monthly temperature in °C and IC _i is an annual heat index, equal to the sum of the 12 monthly indices, which are:

Finally, the exponent m is function of IC _i with the following empirical equation:

values of Tt _j,i higher than 26.5 °C there is no influence of IC _i , so ETP _j,i is only function of Tt _j,i and is tabulated in ^{Campos-Aranda (2005)}.

Hypothetical scenarios of climate change

^{Vicente-Serrano et al. (2010)} and ^{Ma et al. (2014)} have suggested that as a consequence of climate change processes, at least two scenarios must be studied, the first is a reduction in precipitation and the second, an increase in average temperature. Based on quantitative analyses done by these authors and the climatic projections for Mexico by ^{Montero, Martínez, Castillo and Espinoza (2010)}, three scenarios were established for analysis: (1) a progressive and linear reduction of 20% in annual precipitation of the historical record; (2) a progressive and linear increase of 4 °C in the average annual temperature record and (3) the superimposition of both changes in the historical record. The changes mentioned have a direct impact in the severity and duration of droughts (^{Fuchs et al., 2014}). The correction of the monthly precipitation record (PM _j,i ) is made based on the following equation:

in which, Δ _P is the slope of reduction and therefore equals to the quotient of 0.20 between the number of years NA and i is the year counter, ranging from 1 to NA. The average monthly temperature correction (TM _ij ) is carried out with equation:

now, Δ_T is the slope of the increase and therefore equal to the quotient of 4 °C between NA.

Historical records available

The climatological station Zacatecas is located in the city of the same name, which is the capital of the state of Zacatecas, Mexico, which according to historical information provided by the Local Office of the National Water Commission (CONAGUA), has operated continuously and has not undergone changes of location, so the records can be considered reliable. Its geographical coordinates are as follows: latitude 22° 45' N, longitude 102° 34' W.G. and altitude 2485 m.a.s.l. Its available monthly precipitation (mm) records and average temperature (°C) in the Excel files of CONAGUA of Zacatecas, started in January 1953 and are available until December 2015, with missing data in April 1986 and several months of the years 2010 to 2013.

The first missing data were adopted equal to the monthly averages and the rest was considered equal to the values recorded in the same months in the climatological station Guadalupe, which is approximately 6 km in a straight line and is located inside of the same geographical sub-region. This was considered acceptable due to the similarity that both records show at annual level, both in precipitation and in average temperature. On the other hand, in the Climatological Bulletin No. 3 (^{SARH, 1980}) of the Hydrological Region No. 37 (El Salado), the available records of the Zacatecas station of the monthly precipitation and average temperature begin in January of 1930 and go up to December of 1978. Then, the period from January 1930 to December 1952, without missing data, was taken from such Bulletin and with it a joint record of NA = 86 years was integrated. The twelve monthly average values of the integrated precipitation record are: 16.2, 9.6, 5.7, 7.5, 17.7, 80.0, 102.1, 97.9, 85.4, 35.4, 12.8 and 12.0, whose sum is 482.2 mm, magnitude corresponding to the average annual precipitation. Values of the average monthly temperature record are: 11.7, 12.7, 14.8, 17.1, 19.0, 19.0, 17.5, 17.6, 17.0, 16.1, 14.1 and 12.3, with an annual average value of 15.7 °C.

Homogeneity tests

Starting from the monthly historical records, the annual values of precipitation and average temperature were integrated, to which three basic tests (Helmert, Sequences and Von Neumann) and six specific ones were applied: two of persistence (Anderson and Sneyers), two of trend (Kendall and Spearman), one of them of change in the variability (Bartlett) and another of change in the mean, that of Cramer. These tests can be consulted in WMO (¹⁹⁷¹); ^{Campos-Aranda (2005)}, and ^{Machiwal and Jha (2012)}.

The annual precipitation record is perfectly homogeneous, since no general or specific test detected any deterministic components. On the other hand, the mean annual temperature record was not homogeneous because, according to the basic and specific tests, shows persistence and change in mean. This evidence of lack of homogeneity justifies the approach of scenarios related to climate change.

SPEI values with the historical records

For this study it was decided to analyze the following nine durations k of droughts: 3, 6, 9, 12, 18, 24, 30, 36 and 48 months. In Table 1 the results obtained with available historical records were concentrated for each duration according to four concepts: (1) those related to the statistical properties of the sequences formed, (2) those corresponding to the fitting of the Log-Logistics distribution, (3) those associated with the statistical indicators of SPEI and (4) those belonging to the defined drought types.

The first group of results gives an idea of the variability and bias of the sequences formed with the criterion of moving sums; in such a way that the second group, which are the fit parameters of the distribution, are related to such behavior. The third group of results is the most important, because it defines the quality of the fit achieved and therefore the reliability of the estimates made with the SPEI, since its mean, variance and percentage of droughts that it defines should be zero, the unit and 50%. As such indicators approximate the values quoted, the estimate will be better or more reliable.

As Table 1 shows, the most accurate estimates were those of durations of 12, 24 and 36 months, defining approximate percentages of 32.2, 11.5, 5.0 and 1.3 of light, moderate, severe and extreme droughts, respectively.

Figure 1a shows the historical evolution of SPEI with duration of 12 months, observing that in general there is alternation of wet and dry periods. The longest duration drought began towards the sequence 153 (September 1941) and ended in the 416 (August 1963). The date corresponding to a sequence is obtained by clearing from Equation 1 the value of NA in years, which is added to the initial year of the record. In this period the value of the most extreme SPEI occurs, with -2.742 in sequence 321 (October 1954).

Figure 1 (a) SPEI evolution of 12 months duration in the climatological station Zacatecas, of the state of Zacatecas, Mexico; (b) SPEI evolution (k = 12 months) with reduction of 20% in the annual precipitation and increase of 4 °C in the average annual temperature and (c) Differences between the SPEI shown (b - c). 

SPEI values with altered climate records

Based on equations 17 and 18, the historical records were modified to incorporate a progressive and linear decrease of 20% in annual precipitation and a progressive and linear increase of 4 °C in the average annual temperature. Table 2 shows various relevant values of the historical records and those of the altered or modified records according to possible climate change, as well as the annual values of the ETP estimated with the Thornthwaite method.

Similar to Table 1, in Table 3, the SPEI results for the altered record according to the third scenario are presented, i.e., considering the combined effect of both modifications. Due to space limitation the results of the SPEI with reduction of precipitation, and with increase in average temperature are not shown, but they are available with the author. Based on all the SPEI results, the duration of 24 months was defined as the ideal for the contrast between the values obtained with the historical record and the altered ones. This contrast is shown in Table 4.

The results shown in Table 4 indicate that, in general, light and extreme droughts will increase as a result of probable climate change; in contrast, moderate and severe droughts will decrease. In these conclusions there is an anomaly in the severe droughts of the record with increase of temperature, which also increase. Regarding the extreme values of the SPEI, they are considered in agreement with the modifications imposed to the historical records. For the third scenario, the most critical, it is defined in Table 3 as more accurate results those of the 18-month duration, with the following percentages for the four types of droughts sought: 33.1, 9.9, 5.0 y 2.1.

Figure 1b shows the evolution of the SPEI of 12- month duration in the record that includes both effects of climate change; it is clear how droughts will increase in duration and severity towards the end of the record, significantly reducing the wet periods. Finally, Figure 1c shows the SPEI differences of Figure 1b minus 1a.