! Mc Kee Relationship of Drought Frequency

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Eighth Conference on Applied Climatology,17-22 January 1993, Anaheim, California

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THE RELATIONSHIP OF DROUGHT FREQUENCYAND DURATION TO TIME SCALES

Thomas B. McKee, Nolan J. Doesken and John KleistDepartment of Atmospheric Science

Colorado State UniversityFort Collins, CO 80523

1.0 INTRODUCTION Five practical issues become important in anyanalysis of drought. These include: 1) time scale, 2)probability, 3) precipitation deficit, 4) application of thedefinition to precipitation and to the five water supplyvariables, and 5) the relationship of the definition to theimpacts of drought. Frequency, duration and intensityof drought all become functions that depend on theimplicitly or explicitly established time scales. Ourexperience in providing drought information to acollection of decision makers in Colorado is that theyhave a need for current conditions expressed in termsof probability, water deficit, and water supply as apercent of average using recent climatic history (thelast 30 to 100 years) as the basis for comparison. Nosingle drought definition or analysis method hasemerged that addresses all these issues well. Of thevariety of definitions and drought monitoring methodsused in the past, by far the most widely used in theUnited States is the Palmer Drought Index (Palmer,1965), but its weaknesses (Alley, 1984) frequently limitits wise application. For example, time scale is notdefined for the Palmer Index but does inherently exist.

The definition of drought has continually beena stumbling block for drought monitoring and analysis.Wilhite and Glantz (1985) completed a thorough reviewof dozens of drought definitions and identified sixoverall categories: meteorological, climatological,atmospheric, agricultural, hydrologic and watermanagement. Dracup et al. (1980) also revieweddefinitions. All points of view seem to agree thatdrought is a condition of insufficient moisture causedby a deficit in precipitation over some time period.Difficulties are primarily related to the time period overwhich deficits accumulate and to the connection of thedeficit in precipitation to deficits in usable watersources and the impacts that ensue.

Usable water sources include soil moisture,ground water, snowpack, streamflow and reservoirstorage. Any impacts of drought associated with waterdemand exceeding water supply originate with one ormore of these five usable supplies. The time periodfrom the arrival of precipitation until water is availablein each useable form differs greatly. Water uses alsohave characteristic time scales. Consequently, theimpacts of a water deficit are a complex function ofwater source and water use. The time scale over whichprecipitation deficits accumulate becomes extremelyimportant and functionally separates different types ofdrought. Agricultural (soil moisture) droughts, forexample, typically have a much shorter time scale thanhydrologic (groundwater, streamflow and reservoir)droughts.

The purpose of the following discussion is topropose an indicator and definition of drought whichcould serve as a versatile tool in drought monitoringand analysis. This indicator requires only one inputvariable, could be applied in a similar way toprecipitation, snowpack, streamflow, reservoir storage,soil moisture, and ground water, recognizes a varietyof time scales, and provides information onprecipitation deficit, percent of average and probability.

A FUNCTIONAL DEFINITION OF DROUGHT.0any examples are available where


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this simple method is that precipitation is typically notnormally distributed for accumulation periods of 12months or less, but this can be overcome by applyinga transformation to the distribution. The resultingcomputation of standardized precipitation is linearlyproportional to precipitation deficit and allowsspecification of probability, percent of average, andaccumulated precipitation deficit. The basic approachis to use standardized precipitation for a set of timescales which together represent water sources ofseveral types. Soule (1990) used standardizedprecipitation in a study of spatial patterns. Bhalme andMooley (1980) also used the standardized precipitationas a starting point for a drought definition.

An example of the SPI for Fort Collins, CO, is shown inFigure 1 for i = 3, 6, 12, 24, and 48 months for theperiod 1889-1991. When the time periods are small (3or 6 months), the SPI moves frequently above andbelow zero. As the time period is lengthened to 12,24,and 48 months, the SPI responds more slowly tochanges in precipitation. Periods with the SPI negativeand positive become fewer in number but longer induration.

Using the SPI as the indicator, a functionaland quantitative definition of drought can beestablished for each time scale. A drought event fortime scale i is defined here as a period in which theSPI is continuously negative and the SPI reaches avalue of -1.0 or less. The drought begins when the SPIfirst falls below zero and ends with the positive value ofSPI following a value of -1.0 or less. Drought intensityis arbitrarily defined for values of the SPI with thefollowing categories:

The Standardized Precipitation Index (SPI) iscalculated in the following sequence. A monthlyprecipitation data set is prepared for a period of mmonths, ideally a continuous period of at least 30years. A set of averaging periods are selected todetermine a set of time scales of period j monthswhere j is 3, 6, 12, 24, or 48 months. These representarbitrary but typical time scales for precipitation deficitsto affect the five types of usable water sources. Thedata set is moving in the sense that each month a newvalue is determined from the previous i months. Eachof the data sets are fitted to the Gamma function todefine the relationship of probability to precipitation.Once the relationship of probability to precipitation isestablished from the historic records, the probability ofany observed precipitation data point is calculated andused along with an estimate of the inverse normal tocalculate the precipitation deviation for a normallydistributed probability density with a mean of zero andstandard deviation of unity. This value is the SPI forthe particular precipitation data point.

SPI Values Drought Category0 to -0.99

-1.00 to -1.491.50 to -1.99

:s -2.00

mild droughtmoderate drought

severe droughtextreme drought

Time inCategory

-24%9.2%4.4%2.3%-40%

In a long-term climate record each category :s -1.00will be represented a known amount of time which isindicated above.

The definition of drought thus far has includeda beginning date, ending date, and a current droughtintensity. Duration of drought can be either a currentduration since the beginning or the duration of ahistoric drought event from beginning to ending. Peakintensity can easily be determined from the SPI. Ameasure of the accumulated magnitude of the droughtcan be included. Drought M,agnitude (DM) is definedas:

The SPI calculated in this way has thefollowing desirable traits:

The SPI is uniquely related to probability.The precipitation used in SPI can be used tocalculate the precipitation deficit for thecurrent period.

xL SPill1-1The precipitation used in SPI can be used tocalculate the current percent of averageprecipitation for time period of i months.

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DROUGHT CHARACTERISTICS the solid line represents a relationship of droughtduration (D) and time scale i~

The specification of a time scale in thedefinition of drought leads to several basiccharacteristics of drought. Two of the most importantcharacteristics are frequency and duration. Figure 2illustrates series of SPls for each time scale based on120 years of random monthly precipitation based onthe annual cycle and expected variability ofprecipitation at Fort Collins, CO, which has a sprirlgmaximum in precipitation. The most obviouscharacteristics of the drought events in Figure 2 is thatdrought changes as the time scale changes. At longertime scales drought becomes less frequent and oflonger duration very similar to what was noted inFigure 1. A summary of these traits is given inFigure 3 for twenty separate runs of random monthlyprecipitation and for actual observed data for FortCollins, CO. The dashed line is added to illustrate thesimple relationship between the number of droughtsper one hundred years (N) and the time scale i of

(3)=/The general behavior of the number of droughts andthe duration for a random climate is quite similar tothese simple curves. For small time scales of 3months, each new month has a large impact on theperiod sum of precipitation, so it is relatively easy tohave the SPI respond quicklo/ and move from dry towet values. As i becomes larger each new month hasless impact on the total (almc)s1linear with time scale)and the index responds more slowly which leads tofewer droughts of longer duration. The total durationof drought for all time scales is approximately 40% ofthe time. This number is included in the table above.

Examples of questions often asked aboutdrought include the following. How many droughtshave occurred in the past 100 years? How long didthey last? Which was the worst? How do dryconditions now compare with the past? The answersnecessarily depend on definition. Drought has nosimple or unique functional definition. However, adefinition based on the SPI is IDbviously useful becauseis can provide answers to these questions and manyothers. One interesting test is the comparison of thepresent definition to the Palmer Drought Index (PDI).A comparison for Fort Collins, CO, is given in Figure 4.The correlation between the two has a maximum neara time scale of 12 months with a correlation coefficientof 0.85 suggesting that the PDI does indeed have aninherent time scale even though it is not explicitlydefined.

N=.E. I (2)

where c is a constant which in this example is aproduct of the time scale 3 and 96 or 288. Similarly,

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DROUGHT ASSESSMENT ANDMONITORING4.0 they indicate that drought is not likely to end soon andthe impacts must be minimized.Any specific evaluation of drought and itsimpacts requires a specification of time scale since

drought initiation, intensity, duration, magnitude aridending are all dependent on time scale. A time scaleof 3 months has more than 70 droughts per centuryusing the thresholds suggested here. A time scale of24 months is needed to get the total number ofdroughts down to the range of 10-15 per century. Anillustration of the use of the present definition is givenin Figure 5 again using Fort Collins data. Time scalesof 12 and 24 months are included along with theprecipitation deficit accumulated during each drought.The ordinate of the SPI graph has solid lines drawn toSPI = -2, -1, 0, 1, 2. Each of these values of SPI hasa unique value of the probability that the SPI will beequal to or less than the stated value. These values of0.02, 0.16, 0.50, 0.84, and 0.92 are included. Inaddition, each value of SPI has a constant value ofprecipitation as a percent of average which will varywith time scale, geographic location, and period ofrecord of the observations. The index value,probability, and percent of average are all useflJImeasures of dry or wet conditions for each time scale.The accumulated precipitation deficit for each droughtevent is the sum of the monthly departures ofprecipitation from the mean during the drought event.Under each of the larger droughts the OM value for theentire drought is included in parentheses.

5.0 SUMMARYA new definition of drought has been

proposed which explicitly specifies time scales andutilizes a standardized precipitation index. Droughtfrequency decreases inversely and duration increaseslinearly with time scale. Frequency and duration ofrandom climate and actual climate are very similar.The new definition allows a consistent set ofinformation to be calculated including droughtbeginning, ending, intensity, and magnitude. It alsoproduces monitoring information of index values,probability, percent of average, and precipitation deficitduring drought.

Beginning in the fall of 1992, the SPI will beused as an experimental drought monitoring tool inColorado. Operational evaluations will then beperformed as experience is gained relating SPI valuesto actual impacts.

ACKNOWLEDGMENTSThis research was supported by the Colorado

Water Resources Research Institute and the ColoradoAgricultural Experiment Station.

The series for 12 months shows the worstsingle drought to be in the 1950s with other majordroughts in the 1890s, 1960s and 1970s. It also showsthat the 1930s had 4 smaller droughts which, ifcombined, would be more serious than anyone of theothers. The 24-month time scale illustrates that oftenmore than one smaller drought is combined to form asingle large drought as in the 1930s and 1960s. Evenat 24 months the 1930s are not caught as a singledrought event. The depiction of water deficits alsoreveals how in the 1930s and 1960s a deficit rapidlyaccumulated followed by a lull and then a seconddrought phase before any real recovery was achieved.Accumulated water deficit is another candidate to usein defining the magnitude of drought if the initiation isidentified. The choice here to use OM as a measure isbased on the logic that the longer the drought persistswithout a water recharge the worse is the magnitude as

REFERENCESAlley, W.M., 1984: The Palmer drought severity index:Umitation and Assumptions. J. Climate Appl.Meteor.,23,1100-1109.Bhalme, H.N., and D.A. Mooley, 1980: Large-scaledrought/floods and monsoon circulation.Mon. Wea. Rev" 108, 1197.Dracup, J.A., KS. Lee, and E.G. Paulson, Jr., 1980:On the definition of droughts. Water Res.Res., 16, 297.

u.s.almer, W.C., 1965: Meteorological drought.Weather Bureau Res. Paper No. 45, 58.

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! Mc Kee Relationship of Drought Frequency

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