SPI and SPEI: Complete Drought Index Guide - Derivation, Formulas, Calculation & Interpretation

The Standardized Precipitation Index (SPI) is a drought index that quantifies precipitation anomalies relative to the long-term statistical distribution at any chosen time scale. A value of 0 means precipitation is exactly at the historical median for that location and season; negative values indicate below-normal precipitation (drought); positive values indicate above-normal conditions (wet). SPI was developed by Thomas B. McKee, Nolan J. Doesken, and John Kleist at Colorado State University and first presented at the 8th Conference on Applied Climatology in 1993 (McKee et al. 1993). The World Meteorological Organization formally recommended SPI as the global standard meteorological drought index in 2012 (WMO-No. 1090). Its primary advantages are: it requires only a precipitation record; it is dimensionless; it is comparable across any climate and season; and it can be computed at multiple time scales simultaneously.

The Standardized Precipitation Evapotranspiration Index (SPEI) was introduced by Vicente-Serrano, Begueria, and Lopez-Moreno (2010) from CSIC Spain. SPEI extends SPI by replacing precipitation alone with the climatic water balance D = P - PET (precipitation minus potential evapotranspiration). This makes SPEI sensitive to temperature-driven increases in evaporative demand, not just precipitation deficits. The key practical difference is that SPEI can identify drought conditions even when precipitation is normal, if temperatures are high enough to create a water balance deficit. Under global warming, SPEI shows increasing drought in many regions even with unchanged precipitation, while SPI shows no trend. SPEI uses a three-parameter log-logistic distribution (because D can be negative), while SPI uses the two-parameter gamma distribution. Both use the same drought classification thresholds (moderate dry: -1.0 to -1.5; severely dry: -1.5 to -2.0; extremely dry: below -2.0) because both are standardised to the same normal distribution.

The complete SPI formula is SPI = Phi_inverse[F(x)], where F(x) = q + (1 - q) x G(x; alpha, beta). Here: q = probability of zero precipitation (proportion of months with P = 0 in the record); G(x; alpha, beta) = cumulative distribution function (CDF) of the two-parameter gamma distribution fitted to the non-zero precipitation accumulations; Phi_inverse = inverse standard normal CDF (probit function). The gamma parameters alpha (shape) and beta (scale) are estimated by the Thom (1958) maximum likelihood approximation: A = ln(x-bar) - mean(ln x); alpha-hat = (1 + sqrt(1 + 4A/3)) / (4A); beta-hat = x-bar / alpha-hat. For regions with no zero-precipitation months (q = 0), the formula simplifies to SPI = Phi_inverse[G(x; alpha, beta)]. Parameters are fitted separately for each calendar month to remove seasonality.

The SPEI formula is SPEI = Phi_inverse[F(D)], where F(D) = [1 + (alpha / (D - gamma))^beta]^(-1) is the CDF of the three-parameter log-logistic distribution, and D = P - PET is the climatic water balance accumulated over the chosen time scale k months. The three log-logistic parameters (alpha = scale, beta = shape, gamma = location) are estimated using L-moments (probability-weighted moments) on the sorted historical D values. L-moment estimation is preferred over maximum likelihood for the log-logistic because it is more robust to outliers and gives better tail behaviour with finite samples. As with SPI, parameters are fitted separately for each calendar month.

The most complete Python library for SPI is the climate-indices package (pip install climate-indices), which implements the full WMO-compliant SPI algorithm. For custom implementations, use scipy.stats: fit the gamma distribution with scipy.stats.gamma.fit(data, floc=0) to get alpha and scale=beta; compute the CDF with gamma.cdf(x, alpha, scale=beta); transform to standard normal with scipy.stats.norm.ppf(cdf_value). Key steps: (1) compute rolling k-month precipitation sums; (2) separate data by calendar month (12 independent fits); (3) compute zero-month probability q; (4) fit gamma to non-zero values using Thom (1958) MLE or scipy MLE; (5) evaluate mixed CDF: F = q + (1-q) x gamma.cdf(x, alpha, scale=beta); (6) apply norm.ppf(F) to get SPI. Clip F to (1e-6, 1-1e-6) before applying norm.ppf to avoid infinite values.

The standard R package is SPEI by Begueria and Vicente-Serrano (install.packages("SPEI")). For SPI: spi(precipitation_ts, scale=3) where precipitation_ts is a ts object with monthly data. For SPEI: first compute PET with thornthwaite(temp_ts, lat=latitude) or hargreaves(Tmin, Tmax, Ra) or penman(...), then compute water balance wb = precipitation_ts - pet, then spei(wb, scale=3). The package returns an object with $fitted (the SPI/SPEI values) and $coefficients (the fitted distribution parameters). Important: the input must be a ts object with frequency=12 (monthly). The package automatically separates by calendar month for parameter fitting. For multiple time scales, call spi() or spei() once per desired time scale; combine results into a data frame for analysis.

Excel SPI calculation requires: (1) computing rolling k-month precipitation sums in a column using SUM over a sliding window; (2) separating values by calendar month using IF or FILTER functions; (3) computing gamma parameters using formulas: mean = AVERAGE, log-mean = AVERAGE(LN(range)), A = LN(mean) - log-mean, alpha = (1+SQRT(1+4*A/3))/(4*A), beta = mean/alpha; (4) computing the gamma CDF for each observation using GAMMA.DIST(x, alpha, beta, TRUE) where TRUE gives the cumulative form; (5) applying the zero-precipitation correction if needed: F = q + (1-q) * GAMMA.DIST(...); (6) converting to SPI using NORM.S.INV(F). The XLSTAT add-in automates the gamma fitting and is recommended for routine use. Note: fit parameters separately for each calendar month, and exclude zeros from the gamma fit while counting them for q.

The choice of time scale depends on the water system component of interest. SPI/SPEI-1 (1-month): surface soil moisture, flash drought detection, very short-term agricultural monitoring. SPI/SPEI-3 (3-month): seasonal growing season drought, rainfed crop stress, pasture conditions, short-term streamflow. SPI/SPEI-6 (6-month): multi-season drought, medium reservoir inflows, deeper soil moisture, irrigation planning. SPI/SPEI-12 (12-month): annual water balance, large reservoir storage, shallow groundwater recharge, hydrological drought declarations. SPI/SPEI-24 (24-month): multi-year drought, deeper groundwater, large lake levels, long-term water supply planning. SPI/SPEI-48 (48-month): decadal variability, deep aquifers. In practice, monitoring agencies compute and report multiple time scales simultaneously, because different water sectors respond to different accumulation periods. WMO recommends computing at a minimum: 1, 3, 6, 12, and 24 months.

WMO (2012) specifies a minimum of 30 years of monthly data to compute reliable SPI values. This is because the gamma distribution parameters (alpha and beta) are estimated from the historical record, and with fewer than 30 observations per calendar month, the parameter estimates are too uncertain to correctly classify extreme events. With 30 years of data, each calendar month provides 30 observations for parameter fitting. With 50 years, parameter uncertainty decreases by approximately 30 percent in the extreme tails (SPI below -2.0 or above +2.0), which is important for drought declaration and insurance applications. The SPEI computation uses the same minimum, but because the water balance D = P - PET is more variable than P alone, somewhat longer records (40 to 50 years) are desirable for the log-logistic three-parameter fitting. Always use a fixed reference period (e.g. 1981 to 2010, the standard WMO climate normal period) for the parameter fitting, and apply those parameters to compute SPI/SPEI for any month within or outside the reference period.

Monthly precipitation totals are non-negative (bounded at zero), right-skewed (occasional heavy rainfall events create a long right tail), and cannot be fitted directly to the normal distribution. The two-parameter gamma distribution captures all of these characteristics: it is defined on [0, infinity), its shape parameter alpha controls skewness (small alpha = highly skewed; large alpha = approximately normal), and its scale parameter beta scales the distribution to match the observed mean. The gamma distribution is also computationally tractable, with well-established MLE parameter estimation formulas (Thom 1958) and a CDF that can be evaluated with standard incomplete gamma function routines available in all scientific computing environments. At long accumulation periods (12+ months), accumulated precipitation becomes approximately normal by the Central Limit Theorem, making the gamma fitting less critical but still applied for methodological consistency across all time scales.

SPEI operates on the water balance D = P - PET, which can take both positive and negative values (PET often exceeds P in arid and semi-arid climates, and during hot summer months in temperate climates). The gamma distribution cannot handle negative values, so an alternative distribution is needed. Vicente-Serrano et al. (2010) evaluated several three-parameter distributions (log-logistic, Pearson Type III, GEV) and found the three-parameter log-logistic provided the best overall fit to water balance data across diverse climate types. The log-logistic also has a closed-form CDF, making computation fast and numerically stable. The three parameters (alpha = scale, beta = shape, gamma = location/shift) are estimated by L-moments (probability-weighted moments), which are more robust than MLE for fitting distributions to climate extremes with limited data.

SPI and SPEI values are dimensionless standard normal deviates. A value of -1.0 means conditions are approximately 1 standard deviation below the historical mean for that location, season, and time scale. In frequency terms: -1.0 is exceeded (i.e. conditions are this dry or drier) approximately 16% of the time under the historical climate. Specific classifications: -0.99 to 0 = near normal (slightly below median); -1.0 to -1.49 = moderately dry (approximately 9.2% probability); -1.5 to -1.99 = severely dry (4.4%); -2.0 and below = extremely dry (2.3%). For practical interpretation: a SPI-3 of -1.5 means the 3-month precipitation total is in the lowest 4.4% of the historical record for that season at that location. SPEI values have the same statistical interpretation but include the additional effect of evaporative demand, so the same value of -1.5 in SPEI represents a water balance deficit (not just a precipitation deficit) that falls in the lowest 4.4% of the historical distribution.

SPI and PDSI are both widely used drought indices but differ fundamentally in approach. SPI is purely statistical: it standardises a physical variable (precipitation) to its historical distribution independently at each location, making it directly comparable across climates. PDSI is physically based: it uses a simple water balance model with soil moisture storage to compute a departure from a calibrated "normal" condition, incorporating temperature effects via PE (Thornthwaite PET) and soil moisture. Key differences: SPI is multi-scale (you choose the time scale); PDSI is approximately equivalent to a 9 to 12-month SPI due to its built-in memory from the soil moisture model. SPI requires only precipitation; PDSI also requires temperature and soil available water capacity. SPI is standardised to the same scale everywhere; PDSI is calibrated locally and values are not directly comparable across regions. WMO recommends SPI over PDSI for global comparisons because of SPI consistency across climates. SPEI combines the best of both: the multi-scale flexibility and statistical standardisation of SPI with the temperature and evapotranspiration sensitivity of PDSI.

Yes. SPI and SPEI are operationally used by many national and regional drought monitoring systems. The US Drought Monitor uses SPI-1, SPI-3, SPI-6, and SPI-12 as primary indicators alongside PDSI and soil moisture data. SADC (Southern African Development Community) Climate Services Centre uses SPI as the primary meteorological drought indicator. India National Disaster Management Authority uses SPI for official drought classification under the Drought Management Manual. FEWS NET (Famine Early Warning Systems Network) uses SPEI for agricultural drought monitoring across Africa and South Asia. For operational use, SPI/SPEI should be computed monthly, reported at multiple time scales, and combined with other indicators (streamflow, groundwater levels, soil moisture, crop conditions) for a comprehensive drought assessment. The onset of a drought is typically defined as the first month where SPI/SPEI falls below -1.0; drought termination is defined as the first month where SPI/SPEI returns to 0 or above.

SPEI (computed with physically sound PET methods like Penman-Monteith) shows a systematic drying trend in many regions under global warming even when precipitation does not change, because rising temperatures increase atmospheric evaporative demand and thus PET. For example, analysis of CMIP6 model projections shows that under SSP5-8.5, SPEI-12 declines by 0.5 to 1.5 standard deviations by 2100 in the Mediterranean, southern Africa, the Sahel, South and Southeast Asia, and western North America. SPI for the same scenarios may show little trend in many of these regions because it does not capture the temperature-driven component of water deficit. This divergence makes SPEI the more appropriate index for climate change impact studies. However, the magnitude of future SPEI drying is strongly sensitive to the PET method: Thornthwaite PET overestimates warming-driven SPEI drying compared to Penman-Monteith. For climate change research, always use FAO-56 Penman-Monteith or Hargreaves-Samani PET.

Ready-computed global SPI and SPEI data are available from several sources. For SPEI: the SPEI Global Drought Monitor (spei.csic.es) provides monthly SPEI at time scales 1 to 48 months on a 0.5-degree global grid using CRU TS precipitation and Hargreaves PET, freely downloadable as NetCDF. For SPI: NOAA/NCEI provides global SPI at 2.5-degree resolution; PRISM provides high-resolution (4 km) SPI and SPEI for the USA. For South Asia and Nepal: the APHRODITE dataset provides 0.25-degree daily precipitation from 1951 to 2015 that can be used as SPI/SPEI input; ERA5 reanalysis provides all required inputs (precipitation, temperature, radiation, humidity, wind) at 0.25-degree global coverage from 1940 to present for SPEI computation. All of these datasets are freely accessible after registration. For station-based SPI, collect 30+ years of monthly station data from national meteorological services and apply the R SPEI package or Python climate-indices.