**Climate Divisions** - To simplify the national picture somewhat, the
United States has been divided into 344 climate divisions, with no more
than 10 per state. The divisional precipitation and temperature data are
averages of typically 10-50 individual stations. Monthly divisional climate
information for the 48 contiguous states is available from 1895 onward.
Data from the most recent 1-3 months is provisional and based on a smaller
number of stations.

**Period used** - The WRCC SPI products all use the entire divisional
period of record from January 1, 1895 through the latest month available.

**Time scale** - The number of months extending through the end of
the current month. Some processes are rapidly affected by atmospheric behavior,
such as dryland agriculture, and the rate at which grasses and brush dry
out, and the relevant time scale is a month or two. Other processes have
longer time scales, typically several months, such as the rate at which
shallow wells, small ponds, and smaller rivers become drier or wetter.
Some processes have much longer time scales, such as the rate at which
major reservoirs, or aquifers, or large natural bodies of water rise and
fall, and the time scale of these variations is on the order of several
years.

Five quantities are computed as part of the Standardized Precipitation Index procedure. They follow as a consequence of an observation made by Dr. Tom McKee, State Climatologist for Colorado, that users are interested in one or several among the following types of information:

- What is the absolute amount of precipitation?
- What is the absolute departure from climatology?
- What is the relative departure from climatology?
- How frequent are such departures from climatology?
- How can this be expressed as a single number comparable across climates?

**1. Accumulated Precipitation** - The total precipitation that has
fallen during the indicated number of months, through the end of the month
displayed.

**2. Accumulated Precipitation Departure** - The amount by which
the indicated accumulated precipitation is above or below the long term
average for exactly the same set of months. The local seasonal cycle of
long-term average precipitation is automatically accounted for. A departure
of 0 indicates totals are exactly equal to climatological values.

**3. Accumulated Precipitation Percent of Average** - The observed
accumulated precipitation, over the time scale of interest and extending
through the end of the last month indicated, divided by the long-term average
precipitation which would be expected to accumulate over the same set of
months, and then multiplied by 100. A value of 0 indicates no precipitation
at all, and a value of 100 percent indicates that the amount is equal to
the climatological average.

**4. Percentile, or "Probability of Non-Exceedance"** - This quantity
indicates how often a value of the magnitude observed is seen, its degree
of "unusualness". A value of 0 means that zero percent of the other values
in the record do not exceed that value, or in other words, that all other
values exceed that value, so that the value in question is so low that
it seldom if ever occurs. A value of 50 indicates that half of the historical
values are higher and 50 percent are lower. A value of 75 indicates that
75 percent of the values are as low as this value, or conversely, that
only 25 percent of the values are higher than the given value. A value
of 99 means that 99 percent of the observed values are lower, and that
this value is in the top 1 percent of all values. Values near 50 are not
unusual; values near 0 or 100 are very unusual.

**5. Standardized Precipitation Index** - The SPI was formulated
by Tom Mckee, Nolan Doesken and John Kleist of the Colorado Climate Center
in 1993. The purpose is to assign a single numeric value to the precipitation
which can be compared across regions with markedly different climates.
Technically, the SPI is the number of standard deviations that the observed
value would deviate from the long-term mean, for a normally distributed
random variable. Since precipitation is not normally distributed, a transformation
is first applied so that the transformed precipitation values follow a
normal distribution.

The Standardized Precipitation Index was designed to explicitly express the fact that it is possible to simultaneously experience wet conditions on one or more time scales, and dry conditions at other time scales, often a difficult concept to convey in simple terms to decision-makers. Consequently, a separate SPI value is calculated for a selection of time scales, covering the last 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 18, 24, 30, 36, 48, 60, and 72 months, and ending on the last day of the latest month.

**Methdology** - First, a time series of the precipitation
value of interest is generated. Then, a frequency distribution is
selected and a statistical fit to the data is determined. The cumulative
distribution is formed from the fitted frequency distribution. The
percentile for the particular time series element of interest, usually
the latest one, is selected from the cumulative distribution. For
"ties" (multiple instances of the same value), the upper value is used
(probability of non-exceedance). For any other theoretical probability
distribution, the analogous point on its associated cumulative frequency
distribution can be determined. Here, the normal distribution is
used, with mean zero and standard deviation of one, and value in standardized
units of a given percentile is found can be readily determined. For
the normal distribution, these are exactly the same as units of standard
deviations. The Standardized Precipitation Index can be thought of
as the number of standard deviations that the precipitation value of interest
would be away from the mean, for an equivalent normal distribution and
adequate choice of fitted theoretical distribution for the actual data.
In effect, the method consists of a transformation of one frequency distribution
to another frequency distribution, in this case the widely used normal,
or Gaussian, distribution.

In this case, following McKee et al. (1993, 1995), we have chosen to use the incomplete beta distribution (see, for example, Wilks, 1995, p 95-97). This distribution is very robust and can deal with the wide range of extreme climates found in the western United States, especially those where monthly and seasonal precipitation of zero is common and expected. Guttman (1998, 1999) has examined the properties of the SPI in great detail, and has determined that the Pearson III distribution is likely to give essentially equivalent results, and in some instances slightly better. We have not yet modified the code (as of May 2001) to use the Pearson III, but the incomplete beta distribution in use is quite close.

- Guttman, N.B., 1998. Comparing the Palmer Drought Index and the Standardized Precipitation Index. Journal of the American Water Resources Association, 34(1), 113-121.
- Guttman, N.B., 1999. Accepting the Standardized Precipitation Index: A calculation algorithm. Journal of the American Water Resources Association, 35(2), 311-322.
- McKee, T.B., N.J. Doesken, and J. Kleist, 1993. The relationship of drought frequency and duration ot time scales. Eighth Conference on Applied Climatology, American Meteorological Society, Jan 17-23, 1993, Anaheim CA, pp. 179-186.
- McKee, T.B., N.J. Doesken, and J. Kleist, 1995. Drought monitoring with multiple time scales. Ninth Conference on Applied Climatology, American Meteorological Society, Jan 15-20, 1995, Dallas TX, pp. 233-236.
- Wilkes, D.S., 1995. Statistical methods in the atmospheric sciences: An introduction. Academic Press, 467 pp.

Western Regional Climate Center, wrcc@dri.edu