Drought Bulletin

The SPI (McKee et al., 1993) is a statistical index based on the comparison between the precipitation total in a given area of interest for a chosen time interval \(t\) (where \(t\) = 3, 6, 12 e 24 months in the present bulletin) and the cumulative probability distribution for the precipitation data of that area for the identical interval \(t\). In other words, the SPI statistical interpretation of the one-month precipitation total registered in June in a particular location and time of the year is judged against June data from other years, in the same location; whereas the 6-month SPI in June is calculated considering the precipitation accumulated from January to June against the time series of the same 6-month accumulated precipitation from previous years, and so on and so forth.

Very long time series are necessary for the SPI calculation. WMO (2012) suggested to consider at least 30 years of continuous monthly precipitation data. Daily NCEP/NCAR precipitation rate reanalysis (kg m^-2 s^-1), which is available online from 1948 up to now over the entire terrestrial globe, is suitable to provide adequate time series. Reanalysis data are archived at ISPRA, and updated monthly with newly-available reanalyses. In order to calculate the SPI at different timescales, time series of 3-monthly, 6-monthly, 12-monthly and 24-monthly averaged precipitation are built for any gridpoint in the considered domains.

For each grid point, the long-term time series is fitted to a probability distribution. Thom (1966) found that the gamma distribution fits well this climatological precipitation time series.

Given \(X\) the long-term time series of precipitation accumulations over the desired timescale \(t\) (= 3, 6, 12 or 24 months), for each \(x \gt 0\) the gamma distribution \(g(x)\) is defined as:\[g(x)=\frac{1}{\beta^{\alpha}\,\Gamma(\alpha)}\,x^{\alpha-1}\,e^{-x/\beta}\]where \(\alpha\) \((\gt 0)\) is a shape parameter, \(\beta\) \((\gt 0)\) is a scale parameter and \(\Gamma(\alpha)\) is the gamma function. The fitting is performed by optimally estimating the alpha and beta parameters (indicated with \(\hat{.}\)) by means of the maximum likelihood method:\[\hat{\alpha} = \frac{1}{4A}\left(1+\sqrt{1+\frac{4A}{3}}\right)\] \[\hat{\beta} = \frac{\bar{x}}{\hat{\alpha}} \]where \(A=\ln\left(\bar{x}\right)-\frac{1}{n}\Sigma_{n}\ln(x)\) and \(\bar{x}\) is the average of the \(n\) precipitation data. 

Thus, the longer the period used to calculate the distribution parameters, the more robust the estimation of the \(g(x)\) parameters is. For this reason, unless to have long-term series homogeneously distributed over the area of interest, the NCEP/NCAR precipitation reanalysis data, which is available since 1948 (more than 70 years), seem to be an optimal choice to perform the drought monitoring at European and national scale. The cumulative probability is then given by:\[G(x)=\int_0^x g(x)dx=\frac{1}{\hat{\beta}^{\hat{\alpha}}\,\Gamma(\hat{\alpha})}\int_0^x x^{\hat{\alpha}-1}\,e^{-x/\hat{\beta}}dx,\] which can be easily estimated using the numerical approximations provided in literature (see, e.g, Abramowitz and Stegun, 1965, Press et al., 2007). However, since the gamma distribution is not defined for \(x\) equal to zero and the precipitation time series may contain zeros, the cumulative distribution is redefined as follows:\[H(x)=q+(1-q)G(X)\] where \(q\) is the probability of a zero precipitation that can be estimated as the ratio between the number of zeros in the precipitation time series (\(m\)) and the total number of precipitation observations, i.e.: \(q=m/n\).

The cumulative distribution \(H(x)\) is then transformed into a normal distribution (see Panofsky and Brier, 1958) so that the mean SPI for the location and desired period is zero (Edwards and McKee, 1997). The transformation allows maintaining the probability of being less than a given value of the variate from the gamma distribution the same of the probability of being less than the corresponding value of the transformed normally distributed variate.
Computationally, the SPI value can be obtained by using the approximation proposed by Abramowitz and Stegun (1965) that converts cumulative probability to the standadrd normal random variable Z:\[
Z=SPI=\begin{cases}-\left(h-\frac{c_0+c_1h+c_2h^2}{1+d_1h+d_2h^2+d_3h^3}\right)&\mbox{for } 0 \lt H(x) \le 0.5\\+\left(h-\frac{c_0+c_1h+c_2h^2}{1+d_1h+d_2h^2+d_3h^3}\right) &\mbox{for } 0.5 \lt H(x) \le 1 \end{cases}\]where:\[h=\begin{cases}\sqrt{ln\left(\frac{1}{(H(x))^2}\right)}&\mbox{for } 0 \lt H(x) \le 0.5\\\sqrt{ln\left(\frac{1}{(1-H(x))^2}\right)}&\mbox{for } 0.5 \lt H(x) \le 1 \end{cases}\]and:\[\begin{matrix}c_0 = 2.515517 & c_1 = 0.802853 & c_2 = 0.010328 \\ d_1 = 1.432788 & d_2 = 0.189269 & d_3 = 0.001308. \end{matrix}\]The SPI quantifies the relationship between the precipitation occured in a given timescale and the corresponding climatological norm, addressing the intensity of drought (precipitation deficit) or abnormal wetness of the area investigated. Since the SPI is normally distributed, both dry and wet periods can be monitored. Negative values indicate less than median precipitation (drier periods), positive SPI values indicate greater than median precipitation (wetter periods). The standardized departure from the climatological norm for the location and season considered quantifies the magnitude (severity) of the dry or wet event. Moreover, it allows for comparisons between different locations in different climates. 

Further details on the SPI and its computation can be found in the bibliographical references reported below.

 

Bibliography

Abramowitz, M., and I.A. Stegun (eds.), 1965: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, Inc., New York, New York, 1046 pp.

Edwards, D.C., and T.B. McKee, 1997: Characteristics of 20th century drought in the United States at multiple time scales. Climatology Rep. 97–2, Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado, 155 pp.

McKee, T.B., N.J. Doesken, and J. Kleist, 1993: The relationship of drought frequency and duration of time scales. In Proc. of Eighth Conference on Applied Climatology, American Meteorological Society, January 17–23, 1993, Anaheim CA.

Panofsky, H. A., and G.W. Brier, 1958: Some applications of statistics to meteorology. Pennsylvania State University, University Park, 224 pp.

Press, W.H., S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, 2007: Numerical Recipes: The Art of Scientific Computing, Third Edition. Cambridge University Press, 1256 pp.

Thom, H.C. S., 1966: Some methods of climatological analysis. WMO N. 199. Technical Note N. 81., Ginevra, 53 pp.

WMO–World Meteorological Organization, 2012: Standardized Precipitation Index User Guide (M. Svoboda, M., Hayes, M., Wood, D.). WMO-No. 1090, Geneva, 24pp.