SPI drought index and SPEI explained

A district reports 380 mm of rain for the season, and someone asks whether that is a drought. The answer depends on where you are: 380 mm is a flood-free dry year in Kutch and a disaster in a rainforest valley. What is needed is a number that says how unusual the rain was for that place and that time of year. The SPI drought index does exactly this, and the SPEI extends it by adding temperature. This article explains how both are calculated, how to read the SPI drought classification, and where each index misleads.

What is the SPI drought index?

The Standardized Precipitation Index (SPI) expresses accumulated rainfall as a number of standard deviations from the long-term normal for the same place and the same months. A value of 0 means typical rainfall, negative values mean drier than usual, and positive values mean wetter. Because the scale is the same everywhere, an SPI of −1.5 in Jaisalmer and −1.5 in Cherrapunji describe equally rare dryness, even though the rainfall totals differ many times over [1][2].

Three American climatologists, T. B. McKee, N. J. Doesken and J. Kleist, proposed the index in 1993. Their starting point was that drought is a deficit that builds up over a period, and that the period matters: soil moisture responds to rain within weeks, while streamflow, groundwater and reservoirs respond over many months [1][2]. The index is therefore always quoted with a time scale, such as the 3-month SPI (the rain of the last three months, compared with the same three months in all earlier years).

In 2009 the Lincoln Declaration on Drought Indices recommended the SPI to all National Meteorological and Hydrological Services for characterising meteorological drought, and in 2011 the World Meteorological Congress endorsed it. The World Meteorological Organization (WMO) then published the Standardized Precipitation Index User Guide (WMO-No. 1090) in 2012 [2]. In India, the Manual for Drought Management (2016) of the Ministry of Agriculture and Farmers Welfare lists rainfall deviation, SPI or dry spell as the mandatory rainfall indicators for declaring drought, and points States to IMD for SPI data [3].

What is the SPI drought index formula?

The SPI drought index formula has no single closed form. It is a three-step procedure: accumulate, fit, transform [1][4].

1. Accumulate. Sum the rainfall over the chosen window, for example July–September, for every year on record.
2. Fit. Fit a probability distribution to those totals. McKee and colleagues used the gamma distribution, which suits rainfall because it is skewed and cannot go below zero [1][4]. The shape and scale parameters are estimated for each time scale and each calendar month separately.
3. Transform. Convert the probability of the observed total into the value of a standard normal variable with the same probability. That value is the SPI.

x   = rainfall total for the window (mm)
G(x) = gamma cumulative probability, fitted to the long record
q   = fraction of years with zero rainfall in the window
H(x) = q + (1 − q) × G(x)          (probability of x or less)
SPI  = Φ⁻¹( H(x) )                 (Φ⁻¹ = inverse standard normal)

Gamma parameters (Thom's maximum-likelihood estimate):
A = ln(mean of x) − mean of ln(x)
α = (1 / 4A) × ( 1 + √(1 + 4A/3) ),   β = mean of x / α

The zero-rainfall correction q matters in dry climates and dry seasons, where some totals are exactly zero and the gamma function is undefined [4]. The estimates of α, β and the zero fraction come from the record, so the record must be long enough: the WMO guide says ideally at least 20–30 years of monthly values, with 50–60 years or more being preferable, and the Manual for Drought Management asks for at least 30 years [2][3].

A worked example (illustrative)

Suppose the gamma fit for July–September rainfall at a station gives a probability of 0.070 for this year's total, meaning only 7 % of years in the record were as dry or drier. The SPI is the point on the standard normal curve with 7 % of the area below it:

Φ(−1.48) = 0.069,  Φ(−1.47) = 0.071   →   SPI ≈ −1.48

That falls in the "moderately dry" class, close to the "severely dry" boundary at −1.5. Roughly one year in 14 is this dry (1 / 0.07), for that station and that season. The same procedure applied to a wet year with H(x) = 0.93 gives +1.48.

What is the SPI drought classification?

The SPI drought classification used by WMO follows McKee and colleagues [2]:

The WMO guide quotes the dry-side shares (9.2, 4.4 and 2.3 %) and notes that, because the SPI is standardised, these percentages follow from the normal distribution at any place. That is why an "extreme" SPI drought is, by design, rare everywhere [2].

McKee and colleagues also defined a drought event: a spell in which the SPI stays negative and reaches −1.0 or lower. The event begins when the SPI first falls below zero and ends when it turns positive [1][2]. The India Manual for Drought Management uses SPI below −1.0 as one of its first-stage triggers, alongside rainfall deviation and dry spells, and then checks impact indicators (crops, vegetation, soil moisture, hydrology) before a drought is declared [3]. For rainfall alone, IMD's own categories for departure from normal are different: −20 to −59 % is "deficient" and −60 to −99 % "large deficient" [3]. See normal, excess, deficient: how IMD classifies monsoon rainfall for how those percentages work.

Try it yourself

Which time scale should you use?

The WMO guide gives a rule of thumb: a 1- or 2-month SPI for meteorological drought, 1 to 6 months for agricultural drought, and 6 to 24 months or more for hydrological drought, because streamflow, reservoirs and groundwater integrate rainfall over longer periods [2]. In the Indian monsoon setting, a 3-month SPI ending in September compares June–August or July–September rain with the same window in earlier years, and answers "was this a poor monsoon season?". A 12-month SPI ending in September says whether the hydrological year was short of water, which is closer to what a reservoir or aquifer feels. The WMO guide notes that longer SPIs "gravitate toward zero unless a distinctive wet or dry trend is taking place" because they average out shorter swings [2].

Three cautions apply to every time scale:

• Short windows can mislead in dry seasons. In a month that is normally nearly dry, a trivial departure in millimetres can produce a large SPI [2].
• Windows shorter than a month are not recommended. Days with no rain make the index behave erratically [2].
• Windows beyond 24 months need very long records. WMO notes the tails become unreliable unless 80–100 years of data are available [2].

For the soil-side story that follows rainfall deficits, see soil moisture and runoff.

What is SPEI and how is it different from SPI?

SPEI stands for the Standardized Precipitation Evapotranspiration Index. It keeps the SPI's idea (accumulate, fit a distribution, standardise) but applies it to a climatic water balance, rainfall minus potential evapotranspiration (PET), instead of rainfall alone. Vicente-Serrano, Beguería and López-Moreno introduced it in 2010 [5]. PET is the water the atmosphere could evaporate from a well-watered surface, and it rises with temperature.

D  = P − PET            (monthly water balance, mm; negative = deficit)
Accumulate D over the chosen time scale, fit a log-logistic
distribution, and convert to a standard normal value → SPEI

The reason for the change is warming. The SPI cannot respond to temperature, and the WMO guide says plainly that it "is not suitable for climate change analysis because temperature is not an input parameter" [2]. In the 2010 paper, tested at observatories in different climates, only the SPEI and the self-calibrated Palmer index detected increasing drought severity under warming, driven by higher evaporative demand [5]. The SPEI keeps the SPI's multi-scale design, which the authors argue is its advantage over the Palmer index [5].

A worked example (illustrative)

Take three pre-monsoon months, April to June, at a hill station. All numbers are illustrative.

Month   P (mm)   PET (mm)   D = P − PET (mm)
Apr       40        110          −70
May       55        140          −85
Jun       80        150          −70
Total    175        400         −225

Same rainfall, PET 10 % higher in a warmer year:
PET total = 440 mm     →   D = 175 − 440 = −265 mm

The SPI sees 175 mm in both years and gives the same value. The SPEI sees a deficit that is 40 mm larger in the warm year and reports a drier condition. This is the mechanism by which warmer years can be drier on paper without any change in rain.

How is PET calculated for SPEI?

The original SPEI used a simple temperature-based method that needs only monthly mean temperature and latitude, as in the thornthwaite function of the open-source SPEI package; the same package also offers Hargreaves and Penman options [6]. Later work by the same group examined how fitting methods and different evapotranspiration models affect the index [7]. The choice of PET method matters when stations lack humidity, wind and radiation records. Two analysts who use different PET methods will get different SPEI values from identical data, so the method must always be stated with the result. For the physics of evaporation itself, see evaporation and the reservoir water balance.

SPI or SPEI: which one should you use?

Our own work uses both indices for India. In a study of drought hazard, vulnerability and risk in the Krishna River basin, we compared SPI, SPEI and two other indices over 1901–2002 and found that one of them, the Reconnaissance Drought Index, differed markedly from the rest [8]. In projections for India under climate change, we found evapotranspiration likely to dominate future drought dynamics even where rainfall increases, with SPEI indicating the highest risk [9].

Limits to keep in mind

• Both indices measure the weather, not the impact. A meteorological drought becomes an agricultural or hydrological drought only through soils, crops, irrigation and reservoirs, which is why India's manual adds impact indicators [3].
• The reference period matters. Values change as the record grows, and values based on preliminary data may change [2].
• Missing data and short records weaken the tails. The WMO guide accepts some missing data but warns about the loss of confidence in extremes [2].
• Gauge quality sets the ceiling. Any rainfall-based index inherits the errors and gaps of the gauges behind it, which matters most in sparsely gauged Himalayan catchments. See tipping-bucket rain gauge errors, siting and calibration.

Key points

• SPI converts accumulated rainfall into standard deviations from normal, using a fitted gamma distribution and a transformation to the standard normal. An SPI of −1.0 or below is moderately dry or worse; −2.0 or below is extreme.
• Always quote the time scale: 1–6 months for soil and crops, 6–24 months for streamflow, reservoirs and groundwater.
• SPEI replaces rainfall with rainfall minus potential evapotranspiration and uses a log-logistic fit, so it responds to warming where SPI cannot.
• A record of 30 years is a minimum; 50–60 years is better, and the SPI is not recommended beyond 24 months without very long data.
• State the index, time scale, reference period and, for SPEI, the PET method whenever a value is reported.

Where ClimMaTech fits

Both indices are only as good as the rainfall and temperature records beneath them. At ClimMaTech we work on the field-station side of hydromet data, with attention to siting, quality control and continuous records. VARSHAQ, our weather station, is in development and is intended to add rainfall and other weather variables at the site where a local decision is taken, so that a local record can sit beside IMD's district-scale figures.

References

1. T. B. McKee, N. J. Doesken, J. Kleist, The relationship of drought frequency and duration to time scales, Eighth Conference on Applied Climatology, American Meteorological Society, Anaheim, California, 17–22 January 1993. climate.colostate.edu
2. World Meteorological Organization, Standardized Precipitation Index User Guide (M. Svoboda, M. Hayes, D. Wood), WMO-No. 1090, Geneva, 2012. library.wmo.int
3. Department of Agriculture, Cooperation & Farmers Welfare, Ministry of Agriculture & Farmers Welfare, Government of India, Manual for Drought Management, December 2016. ksndmc.org (copy)
4. D. C. Edwards, T. B. McKee, Characteristics of 20th century drought in the United States at multiple time scales, Climatology Report No. 97-2, Department of Atmospheric Science, Colorado State University, 1997. mountainscholar.org
5. S. M. Vicente-Serrano, S. Beguería, J. I. López-Moreno, A multiscalar drought index sensitive to global warming: the Standardized Precipitation Evapotranspiration Index, Journal of Climate 23(7), 1696–1718, 2010. doi:10.1175/2009JCLI2909.1
6. S. Beguería, S. M. Vicente-Serrano, SPEI: calculation of the Standardised Precipitation-Evapotranspiration Index, R package documentation, "Drought indices". search.r-project.org
7. S. Beguería, S. M. Vicente-Serrano, F. Reig, B. Latorre, Standardized precipitation evapotranspiration index (SPEI) revisited: parameter fitting, evapotranspiration models, tools, datasets and drought monitoring, International Journal of Climatology 34(10), 3001–3023, 2014. doi:10.1002/joc.3887
8. G. R. Singh, M. K. Jain, V. Gupta, Spatiotemporal assessment of drought hazard, vulnerability and risk in the Krishna River basin, India, Natural Hazards 99(2), 611–635, 2019. doi:10.1007/s11069-019-03762-6
9. V. Gupta, M. K. Jain, Investigation of multi-model spatiotemporal mesoscale drought projections over India under climate change scenario, Journal of Hydrology 567, 489–509, 2018. doi:10.1016/j.jhydrol.2018.10.012