EFFECT OF DROUGHT ON URBAN WATER

SUPPLIES. I : DROUGHT ANALYSIS

By David M . Frick,1 Dennis Bode,2 and Jose D. Salas,3

Members , ASCE

ABSTRACT: This paper summarizes studies conducted for the City of Fort Collins, Colorado, to determine the effects of prolonged droughts on the city's water sup­plies. As a result of the analysis, various water-supply strategies were developed to be used by the city for handling the problems associated with extreme droughts. This study utilized stochastic modeling coupled with water-resource-system mod­eling to evaluate drought effects on the water supplies of the city. A stochastic model was developed for the five river basins that account for the city's water supplies. The stochastic model was verified with 400 years of tree-ring data ob­tained for the area. Periods of records were then generated to determine potential droughts that could occur. Representative l-in-20-, l-in-50-, l-in-100-, and 1-in-500-year droughts were selected for the analysis.

INTRODUCTION

The potential for drought and actual drought present all communities, whether located in humid or arid areas, with difficult policy and management prob­lems. For communities in many arid areas, where total annual surface-water withdrawals by urban and agricultural water users approximate (and in some years exceed) average overall water availability, policy, and management problems are more difficult and urgent. The water board and city council of Fort Collins, Colorado, have for some time been interested in analyzing the effects of an extreme and prolonged drought on the city's present and future ability to meet city water demands.

Drought can be viewed simply in terms of water supply and demand. While fluctuating greatly over short periods of time* the water supply, over a long period of time, tends to be relatively constant. Water demand in the city of Fort Collins, however, is related to domestic use, lawn-irrigation require­ments, and industrial demands. Increased population and industrial devel­opment inherently mean a greater demand for water. This, in turn, implies an increasing vulnerability of present water-resource systems to the occur­rence of drought. Increased demand also suggests broader, more severe im­pact of drought when it does occur.

Continued pressures on limited water supplies will make drought problems still more serious in the future as city demand increases. Understanding the drought phenomenon, and particularly determining its frequency and severity as it affects various water users, constitutes the basic information necessary for sound planning and management of control measures to mitigate the im-

'Vice Pres. for Engrg., Resour. Consultants, Inc., P.O. Box Q, Fort Collins, CO 80522.

2Water Resour. Engr., City of Fort Collins, P.O. Box 580, Fort Collins, CO 80522. 3Prof. of Civ. Engrg., Hydro, and Water Resour. Program, Colorado State Univ.,

Fort Collins, CO 80523. Note. Discussion open until November 1, 1990. Separate discussions should be

submitted for the individual papers in this symposium. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on No­vember 24, 1987. This paper is part of the Journal of Hydraulic Engineering, Vol. 116, No. 6, June, 1990. ©ASCE, ISSN 0733-9429/90/0006-0733/$1.00 + $.15 per page. Paper No. 24725.

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pact of droughts for the city of Fort Collins. The water board and the city council decided in the spring of 1984 that

the city should know the degree to which it is drought-proofed as a basis for consideration of future water policies and management practices. Then, in May 1984, the city of Fort Collins contracted for a study to answer these questions. The drought study for the city of Fort Collins involved three major tasks:

1. Determination of prolonged drought characteristics for the Cache la Poudre River (the major water-supply source for Fort Collins).

2. Evaluation of drought effects on the city's water supply for both present and future conditions.

3. Identification and analysis of various water-supply strategies that could be adopted by the city for dealing with extreme droughts.

Following is a brief background of studies and literature related to drought-identification characteristics. The paper then continues summarizing the pro­cedures used, and results for the first task are presented. The results of the remaining two tasks are presented in the companion paper—II. Water-Sup­ply Analysis (Frick et al. 1990). More detailed information can be obtained from the study reports ("Droughts" 1985).

BRIEF REVIEW OF PAST STUDIES

The definition of drought varies with time and place and means different things to different people. It has been difficult in the past to develop a uni­versal and accepted definition. Several writers on droughts have tried to de­fine drought for various conditions, using such terms as the water-supply drought, agricultural drought, climatologic drought, and hydrologic drought (Sabrahmanyam 1967). In very general terms, "drought is a condition of moisture deficit sufficient to have an adverse effect on vegetation, animals, and man over a sizable area" (Warrick 1975). However, prolonged lack of precipitation is often closely associated with drought (Russell et al. 1970).

Every water user has its own concept of drought. In agriculture drought means a shortage of moisture in the root zone of crops. To a hydrologist, it may mean below-average water levels in streams, reservoirs, ground-water aquifers, lakes, or soil. In an economic sense, drought means a water short­age that affects or disturbs the established production and water uses. Al­though these concepts are based on different viewpoints, they depend upon the effects of prolonged weather conditions with deficient moisture. Even for the same water user, the concept of drought may change with time and place. For example, the agricultural concept is closely tied to water needs of various crops. A drought for sugar beet growers may not be a drought for winter wheat producers. For each, this concept can change during the growing season, mainly by climatic variations but also according to soil con­ditions, growth state, and the ways in which the crop is cultivated. Similarly, the existence of a variety of climates over the earth's surface implies that droughts are related to the basic climate (Thornthwaite 1948).

Past studies on droughts around the world have led to several definitions. A drought in Great Britain was considered a period of 15 consecutive days each with less than 0.01 in. of rain (Heathcote 1973). In the United States,

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many definitions of drought relevant to specific water uses or to specific disciplines have been advanced. A commonly used definition was "a period of 21 days when rainfall is less than 30 percent of the normal for the place and time" (Tannehill 1947). Other definitions include a period of strong wind, low precipitation, high temperature, and unusually low relative hu­midity (Condra 1944); a day on which the available soil moisture was de­pleted to some small percentage of available capacity (Van Bavel and Ver-linden 1956); and monthly or annual precipitation less than some specified fraction of normal (McGuire and Palmer 1957). Other similar definitions have been proposed and used (Saarinen 1966).

A commonly used drought definition was suggested by Palmer (1965). It takes into account precipitation, potential evapotranspiration (Thornthwaite method), antecedent soil-moisture conditions, and an estimate of the avail­able soil moisture and runoff. Although Palmer defined his drought index as characterizing a meteorological drought, it is very much related to the agricultural drought. In fact, Warrick (1975) states that the use of this index "may give some indication of the overall impact of drought on society." Warrick cites Booth and Voeller (1967) as having "found that even in more developed societies which have been successful in reducing the effects of the natural environment, the more severe a drought becomes, as measured by the Palmer index, the greater the proportion of the society that will feel its impact." On the other hand, Russell et al. (1970) states that "purely meteorological drought, as defined by the Palmer index, may or may not have an impact on a particular system, depending on the relative inadequacy of that system." This difference in opinion evidences the fact that a unique definition and description of drought properties may not be appropriate for all water users and drought conditions.

The classical approach to drought problems was to find the probability of the instantaneous smallest value by the theory of extremes. This approach does not tell anything about the duration and areal coverage of droughts. Unlike flood problems, the duration and areal coverage are very important in drought problems. Palmer (1964) used his drought index to study the areal distribution of the frequency of occurrence of droughts of various severities in the Great Plains. It showed that most areas in the Great Plains endured severe or extreme drought from 15-25% of the time in the period 1931—60, but most of the area to the east had serious drought less than 10% of the time. From Palmer's results, it may be inferred that the eastern part of the plains of Colorado may have had extreme drought conditions about 15% of the time. However, these results are very rough estimates and may not be useful in practical situations. Recently, Doesken et al. (1983) extended Pal­mer-index studies to the state of Colorado. Likewise, Dickerson and Dethier (1970) applied Palmer's index for determining the drought frequencies in the northeastern United States and Lawson et al. (1971) studied the spatial and temporal characteristics of droughts in the state of Nebraska by using the Palmer indices for the eight climatic divisions of the state.

Very few results of investigations on areal drought coverage are available. Even a descriptive method of areal characteristics of drought has not been well developed. Little has been done on applying the quantitative statistical methods on areal coverage. Pinkayan (1966) studied the probability of oc­currence of wet and dry years over a large area. Gibbs and Maher (1967) analyzed the areal extent of past droughts in Australia by classifying the

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annual precipitation by using the decile range. Investigators such as Spar (1968) used a special precipitation or runoff property to discuss the drought phenomenon, without analyzing it quantitatively.

In the past two decades, drought research has expanded and more powerful statistical tools have been brought into play. In this context, it is worth men­tioning the studies by Yevjevich (1967), Saldarriaga and Yevjevich (1970), Millan and Yevjevich (1971), Guen-ero and Yevjevich (1975), Gupta and Duckstein (1975), Sen (1976, 1980), Santos (1981), Chander et al. (1981), Santos et al. (1983), and Correia et al. (1985). In Colorado, where climatic characteristics are different and streamflow and reservoir storage add a large percentage to the water supply, the problem becomes considerably complex. For this study, a drought is defined as a sustained period of low precipitation (rainfall and snow) such that the water available from the Poudre River and imported waters will not meet the needs of water users in the basin. A major portion of this study is to define the magnitude and duration of this drought.

STOCHASTIC GENERATION OF WATER SUPPLIES AND DROUGHT ANALYSIS

General Description of Approach Since one of the objectives of the study was to determine droughts in the

Cache la Poudre River corresponding to 20, 50, 100, and 500 years of return period, it was necessary to generate long-term synthetic records of virgin flows in the Poudre as well as water imports from which such droughts can be derived. Furthermore, since another aspect of the study was to analyze the impact of such droughts on the city of Fort Collins water-supply system, it was necessary to further generate corresponding virgin flows and water imports on a monthly basis so that they can be routed through the system's simulation model to determine the impact.

To this end, historical streamflow data in the Poudre basin as well as in the Colorado River, North Fork Michigan River, Laramie River, and the Big Thompson River were obtained at the gaging stations shown in Fig. 1. These historical records were adjusted for upstream diversions, imports and changes in reservoir storage. The resulting adjusted records are commonly referred to as virgin flows and are representative of streamflows that would have occurred without any regulation by man-made facilities. As is usually the case, the original data had unequal length of records with samples vary­ing from 32 years (for North Fork Michigan River) to 100 years (for Cache la Poudre River). Thus, prior to further analysis, all records were extended to cover the period 1911-83, although in the case of the Poudre River rec­ords for 1911—83 and 1884-1983 were used in the analysis. Likewise, sta­tistical analysis was carried out in order to detect and adjust for nonhomo-geneities or inconsistencies.

Data of the five sites were statistically analyzed to determine the basic characteristics such as the mean, variances, skewnesses, auto-covariances, as well as cross covariances. Furthermore, historical drought statistics were derived for data of the Poudre River. Then, a multivariate stochastic model was fitted to the annual flows and used to generate a number of synthetic flow traces of various lengths. These synthetic traces were analyzed to en­sure that the historical statistics were reproduced including drought statistics at the Poudre River.' Streamflow of the Poudre River originates not only in

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4*» COLORADO

/

•A

A I.

HOT SULPHUR SPRINGS

lu/

#

NORTH FORK MICHIGAN RIVER

X t LARAMIE

3 2

O NEAR GOULD O

- i DIVERSION

GRAND RIVER J

COLORADO IMPORTS (C)

I DIVERSION

|GRAND iRIVER DITCH IMPORT

MICHIGAN DITCH IMPORT

NEAR GLENDEVEY

- - i DIVERSION

LARAMIE RIVER IMPORT

-_>

%

BIG THOMPSON RIVER (BT)

MOUTH OF CANYON NEAR DRAKE

C-BT IMPORTS

K CACHE LA POUDRE RIVER

0 MOUTH OF CANYON

4?

FIG. 1. Schematic Representation of Cache la Poudre River System and Imports from Other Basins in Rocky Mountains

its own watershed but supplemental water is imported from the Western Slope of the Rocky Mountains. Thus, additional models were used to derive such water imports on an annual basis from the Laramie River, North Fork Mich­igan River, and Colorado River. Subsequently, such annual water imports as well as the generated annual flows of the Big Thompson and Poudre rivers were disaggregated to obtain the corresponding monthly flows. Fig. 2 sum­marizes the approach followed in this part of the study.

STATISTICAL CHARACTERISTICS OF HISTORICAL ANNUAL FLOWS

Basic Statistics The annual historical statistics such as the mean, standard deviation, skew-

ness coefficient, and lag-1 serial correlation coefficient for the five key rivers (Poudre, Laramie, North Fork Michigan, Colorado, and Big Thompson) are shown in Table 1. The coefficients of variation for all sites vary about 0.30, which is normally found in the region. The skewness coefficients are all positive but small or statistically insignificant except for the Poudre River (1.05), which is indicative of a nonnormal distribution. In fact, the plot of the frequency distribution of annual flows of the Poudre on a log-probability paper fits quite well a straight line, suggesting that the log-normal model

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H i s t o r i c a l Data

<< S t a t i s t i c a l C h a r a c t e r i s t i c s of H i s t o r i c a l Dat

' -H i s t o r i c a l Drough t s

a

Multivariate Modeling of Annual Flows

Modeling and Generation of Annual Water Imports

Generation of Synthetic Annual Virgin Flows

_1_ Analysis of Drought Characteristics

Modeling and Generation of Monthly Flows and Imports

FIG. 2. Summariied Approach for Stochastic Modeling and Generation for Drought Analysis of the Cache la Poudre River System

TABLE 1. Comparison of Historical and Generated Basic Statistics of Virgin An­nual Flows

River

(D Poudre Laramie NF Michigan Colorado Big Thompson

Mean"

Historical (2)

287.9 76.0 12.8

480.8 128.1

Generated (3)

288.6 76.1 12.8

480.6 128.2

Standard Deviation8

Historical (4)

104.9 18.5 3.7

126.3 40.2

Generated (S)

106.1 18.6 3.7

126.6 40.3

Skewness Coefficient

Historical (6)

1.046 0.121 0.210 0.167 0.340

Generated (7)

0.976 0.002 0.001

-0.005 0.409

Lag-1 Correlation Coefficient

Historical (8)

0.012b

0.025 0.141

-0.024 -0.113

Generated

(9)

0.100 -0.009

0.123 -0.011 -0.015

Expressed in 1,000 acre-ft. bThis statistic was 0.137 based on period 1884-1983.

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TABLE 2. Comparison of Historical and Generated Lag-Zero Cross Correlations between Annual Virgin Flows of Key Rivers in Cache La Poudre System

River

(1)

Poudre Historical Generated

Laramie Historical Generated

NF Michigan Historical Generated

Colorado Historical Generated

Big Thompson Historical Generated

Poudre (2)

1.000 1.000

— —

— —

— —

— —

Laramie (3)

0.797 0.820

1.000 1.000

— —

— —

— —

NF Michigan (4)

0.691 0.751

0.846 0.851

1.000 1.000

— —

— —

Colorado (5)

0.719 0.793

0.877 0.877

0.898 0.892

1.000 1.000

— —

Big Thompson (6)

0.925 0.921

0.777 0.787

0.684 0.701

0.801 0.815

1.000 1.000

may be a useful distribution for that river. The lag-1 serial correlation coefficients shown in column 8 of Table 1

indicate that in all cases the correlations are not significantly different from zero. However, one must be careful about taking this conclusion purely on statistical grounds, since it is well known that the lag-1 correlation (p^ of annual flows is usually small but positive. Thus, being p! small, because of sampling variability, it is always possible that for a given sample the esti­mates of pi may become zero or even negative as shown in the case of the Colorado and Big Thompson rivers. Another point to note here is the fact that if the 100-year record (1884-1983) of the Poudre is considered, the estimate of p, becomes 0.137. Likewise, the estimate of p! for the North Fork Michigan River is 0.141. These small but positive correlations were considered in the final selection of the parameters of the model for gener­ating annual flows in the system. In addition to serial correlations, lag-zero cross correlations among the stations were determined as shown in the matrix of Table 2. In all cases these correlations are statistically significant at the 95% level.

Drought Statistics A decision made early in the study was to use, to the extent possible,

drought terminology as found in published literature on the subject. A drought is usually defined as a year or series of years with below-normal runoff. The drought must be consecutive, starting in the first year with below-average runoff and continuing through a series of years until the drought is broken by a year with above-average streamflow. An extended dry period that hap­pens to include within it a single year of above-average runoff is not clas­sified as one continuous drought. Instead, it is considered to be two separate droughts, at least according to the commonly accepted definition.

It is also standard to define droughts according to the following three prop­erties: duration, cumulative deficit, and average annual shortage. Duration

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Syr.

-Run Length (Duration)

Syr. 6yr.

• Run Sum (De f i c i t )

O O

o

5 O 200-

<

Long Term Avg.

Annual Flow

— Run Intensity ( Avg. Annual

Shortage, Typical)

YEARS

FIG. 3. Drought Terminology: (a) Drought with Worst Average Annual Shortage; (b) Drought with Longest Duration; (c) Drought with Greatest Total Deficit

is the total number of years spanned by a drought and is also referred to as run length (RL). Cumulative deficit is the total shortage, compared to the average, experienced over the course of the drought; it is also referred to as severity or run sum (RS). The average annual shortage is equal to the total deficit divided by the drought duration; it is also referred to as magnitude or run intensity (RI = RS/RL). For more detailed description of these def­initions, see for instance Yevjevich (1967) and Dracup et al. (1980a, b). For a given sample, the foregoing drought properties can be defined on an av­erage basis or as the maximum or largest value within the sample. Although in the study both average and maximum values were considered, only those referred to maximum values are reported in the present paper.

The drought terminology is shown graphically in Fig. 3. As can be seen from the figure, it is possible in any given period for a drought with the greatest total deficit to be different from a drought with the longest duration and different from a drought with the worst average annual shortage. It is somewhat common, however, for the drought with the worst total deficit to be the same as the drought with the longest duration. The drought with the worst average annual shortage is usually much shorter in length than the other two, often spanning a period of only 1 or 2 years, The drought study focused primarily on droughts with the greatest total deficit (the third type of drought in Fig. 3) as being the most stressful condition for water supplies

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on the Poudre River. However, the other two types of droughts were also considered in the overall drought study.

The time series of the Poudre River annual virgin flows is shown in Fig. 4. It reveals that the three types of drought indicated routinely occur on the Poudre. For instance, it is common for the years with the lowest annual runoff to be grouped together with a number of other below-average years, with many of the years only moderately below average. The average short­age over the course of the drought is large but not as extreme as the average shortage in the worst-magnitude (run intensity) drought, which is normally only 1 or 2 years in length. Historically, the worst-magnitude drought was that of 1919; however, the drought that contained the worst single year was the drought of 1953-1956 (containing the extremely low runoff year of 1954). The latter type of drought puts stress on a water supply that relies only on direct-flow decrees of more than the type experienced in 1919, primarily because such a water supply must provide sufficient water during the worst single year, a more severe case than the worst average series of years.

The other type of drought that routinely occurs on the Poudre is a long-term drought that occasionally contains a year or years with above-average runoff. Such a drought occurred historically from 1930 to 1941 (also shown in Fig. 4). The drought was 12 years long and contained a single above-average year in 1938. This would be considered two separate droughts using the commonly accepted definition of drought; however, the drought could be interpreted as one consecutive dry period by using running averages in­stead of individual years. Such an extended drought puts stress on a water supply that relies heavily on long-term carryover storage. A single above-average year in the middle of the drought may not adequately refill the res­ervoirs to bring an end to the period of stress. It was decided, somewhat arbitrarily, that a drought of this type would end whenever the 4-year run­ning average was above normal. In other words, the average of the runoff in the first wet year and in the following 3 years must be above average in order to insure that the reservoirs would refill and that the long-term drought would be broken. Using the 4-year running average seemed to provide ac­ceptable criteria to make this determination.

Table 3 gives the 20-, 50-, and 100-year historical drought characteristics for the 1884-1983 annual virgin flows of the Poudre River. It includes the maximum duration or run length (RL), the maximum severity or run sum (RS) and the maximum magnitude or intensity (RT) for truncation levels (de­mand levels) corresponding to 100%, 75%, and 50% of the mean annual flows. The historical sample was split into five subsamples of 20 years, two of 50 years, and one of 100 years. Then, from each subsample the drought statistics were obtained and averaged over the number of subsamples. The derived statistics were generally as one might expect, with their numerical values increasing as the size of the subsample increases. However, one must note that the historical 50-year-drought statistics were underestimated be­cause when the 100-year record was split into two 50-year subsamples (1884-1933 and 1934-83) the 8-year drought of 1930-37 was also split into two 4-year droughts (see for instance Table 3 for TL = 287,931 acre-ft), resulting in a significant reduction in the average drought duration and drought se­verity. Of course the effect of this underestimation is less noticeable as the truncation level becomes smaller, as seen in Table 3.

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5 o _1

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0-,

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19

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1941

19

61

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TABLE 3. 20-, 50-, and 100-Year Historical Drought Statistics Based on 1884-1983 Record of Annual Virgin Flows of Poudre River

Truncation level (1)

T = 20 Years

RL (vr) (2)

RS (acre-ft)

(3)

RI (acre-ft/yr)

(4)

T = 50 Years

RL

(yr) (5)

RS (acre-ft)

(6)

RI (acre-ft/yr)

(7)

T = 100 Years

RL

(yr) (S)

RS (acre-ft)

(9)

RI (acre-ft/yr)

(10)

(a) 100% of Mean Annual Flow

287.9* 287.9b

4.2 4.0

303,300 300,600

109,100 4.0 7.6

333,700 515,300

126,200 8.0 8.0

535,400 535,400

137,900

(b) 75% of Mean Annual Flow

215.9* 215.9b

2.0 1.9

84,500 85,900

59,100 2.0 2.7

111,400 127,500

77,400 3.0 3.0

156,800 156,800

88,900

(c) 50% of Mean Annual Flow

144.0* 144.0b

1.0 1.0

18,000 18,800

— 1.0 1.0

22,000 20,000

— 1.0 1.0

22,000 22,000

—

aMean values based on nonoverlapping samples. bMean values based on overlapping samples.

Modeling and Generation of Virgin Annual Flows A multivariate contemporaneous autoregressive (CAR) model as described

by Salas et al. (1980) was used in this study to model the virgin annual flows of the five key stations (shown in Fig. 1) in the Cache la Poudre River system. This model allows for representing the temporal- and spatial-depen­dence structure in a sequential manner in that a univariate or single-site au­toregressive model is first fitted to each series to reproduce the temporal-dependence structure, then the residuals from each series are derived, which are assumed to be uncorrelated in time but correlated in space. Subsequently, a simple (lag-zero) multivariate model is fitted to reproduce the spatial de­pendence of such residuals. The advantages and limitations of this approach have been widely discussed in the literature. See, for instance, Salas et al. (1980), Hipel (1984), Stedinger et al. (1985), Salas et al. (1985), and Bar-tolini and Salas (1985).

First of all, some appropriate transformations were used to transform the original annual flow series xf, i = 1, . . . , 5 into normal. Thus, for the Poudre River the logarithmic transformation was used while the square-root transformation was found to be more suitable for the Big Thompson River. On the other hand, no transformations were necessary for the other three rivers. Let us denote by yf, i = 1 5 the series of transformed or original flows, as the case may be. These transformed series were standardized, i.e., series with mean zero and standard deviation one, and then fitted by a lag-1 autoregressive model as

yf = 4>(;) yf, + zf (1)

where <|>(/) = autoregressive coefficient and zf has mean zero and standard deviation o-0) and is assumed to be a normal series uncorrelated in time but correlated in space (among the different sites). In addition, zf was further standardized by dividing it by CT<0. This new standardized series, denoted by €,<0 has the same correlation properties as the series zf. The parameters of the model were estimated by the method of moments based on the 1884-

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1983 (100 years) record of the Poudre River and the 1911-83 (73 years) record of the other four rivers. Then, the residual series z,w were determined by inverting Eq. 1. To account for the cross correlation among the es the lag-zero multivariate model (Salas et al. 1980)

e, = B& (2)

was used in which B was assumed to be a lower triangular parameter matrix and £, = a standardized normal variable (vector) uncorrelated in time and in space.

The foregoing models were checked by a number of statistical tests, in­cluding the Porte-Manteau lack-of-fit test to check that the es of Eq. 1 are uncorrelated in time, the skewness test of normality to check that the es of Eq. 1 and the es of Eq. 2 are normal, the heteroscedascity test to check that the noises have constant variance, a test to check that the residuals £, are uncorrelated in space, and an overfitting check based on the Akaike Infor­mation Criteria (AIC) to check that the model is parsimonious in the number of parameters.

It is evident that the foregoing model does not incorporate any specific features of long-term persistence that often are important in generating flows. Long-term persistence, in general, becomes evident through correlation func­tions and storage and drought-related properties. However, when one uses the correlation function for fitting a model, often the storage properties (for instance, range of cumulative sums) and drought properties (for instance, drought lengths) derived from such models may not be comparable (usually they are underestimated) to corresponding properties derived from the his­torical sample. In our study, the data of the Poudre River system did not show any evidence for long-term persistence. The historical correlograms show very small correlations and the droughts generated by the aforemen­tioned model compared quite well with those observed in the historical rec­ord. Therefore, the model was considered to be satisfactory for the system of our studies. Further discussion on this topic follows.

Comparison of Historical and Generated Statistics Further verification of the model was made by analyzing the statistical

characteristics of generated flow data. A total of 50,000 years of flow data at the five sites were generated based on the foregoing models, which were arranged into 684 samples of 73 years long (resembling sizes as the period 1911-83) at each site. Then, the basic statistics were computed from each sample and the average of each determined based on the 684 samples. Table 1 shows a comparison of the historical and generated mean, standard de­viation, skewness coefficient, and lag-1 serial-correlation coefficient, and Table 2 shows a comparison of the historical and generated lag-zero cross correlations among the sites. One can observe that the generated statistics resemble very closely those derived from the historical sample. In addition to comparing historical and generated basic statistics for the five rivers of the Poudre system, drought statistics of the Poudre River derived from the historical sample, from the 50,000 years of generated data (see Table 3), and from data generated based on tree-ring data obtained from the Labora­tory of Tree-Ring Research (University of Arizona, Tucson, Ariz.) were compared.

The 50,000 years of generated data were used to determine the 20-, 50-,

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TABLE 4. Drought Properties for Poudre River Determined from 50,000-Year Generated Sampie

Return period (yr)

d) 20 Mean 20 Max 20 Min 50 Mean 50 Max 50 Min

100 Mean 100 Max 100 Min 500 Mean 500 Max 500 Min

Maximum duration (RL) (yr)

(2)

4.7 18

1 6.5

20 3 7.8

20 4

10.8 20

6

Maximum" total deficit (RS)

(acre-ft) (3)

375,000 1,626,000

39,000 516,000

1,927,000 185,000 628,000

1,927,000 243,000 890,000

1,927,000 559,000

Maximum" average annual shortage (RI)

(acre-ft/yr) (4)

112,000 210,000

39,000 130,000 210,000

71,000 142,000 210,000

94,000 166,000 210,000 127,000

"Also referred to as worst-severity drought. bAlso referred to as worst-magnitude drought.

100-, and 500-year worst droughts that could be expected to occur on the Poudre River. In the case of the 500-year drought, the process consisted of subdividing the data record into 100 traces of 500 years each. The worst duration, severity, and magnitude droughts that occurred in each 500-year trace were then determined. The resulting droughts (a total of 100 for each drought property) were averaged to arrive at the worst average drought that could be expected to occur once every 500 years. The process was repeated for the 100-, 50-, and 20-year droughts. The results, summarized in Table 4, are based upon the standard drought terminology and definitions noted at the beginning of this section. An interpretation of the table for just the 500-year drought is as follows: (I) The 500-year longest drought would be 10.8 years, although worst-duration droughts as short as 6 years and as long as 20 years were observed in the 100 traces of 500 years each; (2) the 500-year-worst severe drought (worst severity) gives a total cumulative deficit of 890,000 acre-ft over the duration of the drought. Worst-severity droughts with deficits as little as 560,000 acre-ft to as much as 1,900,000 acre-ft were observed in the 100 traces of 500 years each; and (3) the average 500-year worst-magnitude drought (worst intensity) gives a shortage of 166,000 acre-ft per year for the duration of the drought, which often would only be 1 or 2 years in length. Note that drought magnitudes as small as 127,000 acre-ft and as big as 210,000 acre-ft occurred in the 100 generated samples.

The drought characteristics for a truncation level equal to the mean (288,000 acre-ft) determined from the 50,000-year record are similar to those observed in the historical record and in the 400 years of record generated using tree rings. Table 5 is a comparison of the drought characteristics resulting from the three sources of information. Similar comparisons resulted for drought statistics based on other truncation levels. It should be noted that for the 1-in-50 drought the historical 100-year record was split into two 50-year in-

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TABLE 5. Comparisons of Generated Data with Historical Record for Annual Poudre River Virgin Flows

(1)

Long-term mean Standard deviation about

mean l-in-20 Droughts

Average RL,„ (yr) Average RSm (acre-ft) Average RI,„ (acre-ft/yr)

l-in-50 droughts Average RLm (yr) Average RSm (acre-ft) Average RIm (acre-ft/yr)

l-in-100 Droughts Average RLm (yr) Average RS„, (acre-ft) Average RI„, (acre-ft/yr)

l-in-500 Droughts Average RLm (yr) Average RSm (acre-ft) Average RIm (acre-ft/yr)

50,000 years of stochastically

generated data (2)

289,000

106,000

4.7 375,000 112,000

6.5 516,000 130,000

7.8 628,000 142,000

10.8 890,000 166,000

393 years of data generated from tree rings

(1572-1964) (3)

288,000

98,000

4.2 349,000 114,000

6.1 491,000 141,000

7.3 573,000 158,000

9.0C

777,000 218,000

100 years of historical

data (1884-1983)

(4)

288,000"

105,000

4.2 303,000 109,000

4.0 334,000 126,000

8.0b

535,000 138,000

d

"Historical mean and standard deviation based on 73-year record, 1911-1983, as op­posed to 100-year record.

bHistorical record contains only 100-year drought. cData set generated from tree rings contains only one 500-year drought. dHistorical record is too short for estimating even one 500-year drought.

TABLE 6. Drought Properties, Worst Single Year

Frequency

(1)

1 in 20 years 1 in 100 years 1 in 500 years

Annual virgin flow in worst year (acre-ft/yr)

(2)

140,000 113,000 96,000

Average number of consecutive below average years

(3)

3 4 5

TABLE 7. Drought Properties, 4-Year-Running-Average Drought with Worst Long-Term Deficit

Frequency (D

1 in 20 years 1 in 100 years 1 in 500 years

Average cumulative deficit (acre-ft)

(2)

460,000 840,000

1,300,000

Average number of years in drought

(3)

6 13 19

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-J

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Typ

es

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TABLE 8. Droughts Selected for Simulation of Poudre River Water Supplies

1 in 20a

Year0

(1)

430 431 432

Poudre River virgin flow (acre-ft/yr)

(2)

151,000 234,000 124,000

1 in 50"

Year6

(3)

19 20 21 22 23 24

Poudre River virgin flow (acre-ft/yr)

(4)

208,000 209,000 138,000 257,000 197,000 167,000

1 in 100c

Year0

(5)

87 88 89 90 91 92 93 94

Poudre River virgin flow (acre-ft/yr)

(6)

272,000 180,000 162,000 206,000 255,000 203,000 140,000 201,000

1 in 500b

Year8

(7)

329 330 331 332 333 334 335

Poudre River virgin flow (acre-ft/yr)

(8)

156,000 239,000 180,000 178,000 108,000 115,000 210,000

a3-year duration; total deficit = 360,000 acre-ft. b6-year duration; total deficit = 550,000 acre-ft. c8-year duration; total deficit = 690,000 acre-ft. d7-year duration; total deficit = 830,000 acre-ft. "Years correspond to record in data set 2.

crements covering the years 1884-1933 and 1934-83. This caused the 8-year drought of 1930-37 to be considered two 4-year droughts, resulting in a significant reduction in the average run length and run sum for the his­torical l-in-50 droughts. Hence, the discrepancy between the average run length and run sum for the historical l-in-50 data and the other two data sets is not significant. In addition, the stochastically generated traces were used to determine the drought characteristics based on the drought containing the worst single-year and the 4-year-running-average drought with the worst long-term deficit. The results are summarized in Tables 6 and 7. It may be observed that the 4-year running average drought criteria gives generally more severe droughts than those obtained from the standard drought defi­nitions given earlier in the text.

Selection of Typical 20-, 50-, 100-, and 500-Year Droughts It is cumbersome to select typical 20-, 50-, 100-, and 500-year droughts

from the long-term synthetic record. That is because every drought is dif­ferent in its own way and because a drought that may be extreme in one respect, such as severity (or total deficit) may not be that bad in another, such as magnitude (or low average annual flows). It was decided, from a water-supply standpoint, that three types of drought should be considered: (1) A drought containing the worst single year; (2) a drought with the worst severity (or total deficit over consecutive dry years); and (3) a 4-year-run­ning-average drought with the worst long-term deficit (where an occasional above-average year does not necessarily end the drought). The three types of drought were considered because it was believed that the first would stress direct-flow water rights, the second would stress short-term reservoir stor­age, and the third would stress long-term reservoir storage.

The general characteristics of the three drought types are shown graphi­cally in Fig. 5. The figure was derived from information presented in Tables

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200-

o o o

100-

• IN-20 DROUGHT Deficit' 360,000 A.F. Duration ! 3 Years Data Set Yrs. • 430-432

< a 40o- i UJ

< 300 -UJ z UJ 2 0 0 -o

100-

L

15

50 DROUGHT Deficit' 550,000 A.F. Duration' 6 Yrs. Data Set Yrs.' 19-24

I-IN-100 DROUGHT Deficit' 690,000 A.F. Duration' 8 Yrs. Data Set Yrs. ' 8 7 - 9 4

I - IN-500 DROUGHT Deficit' 830,000 A.F. Duration' 7 Yrs. Data Set Yrs.' 329-335

TIME (Years)

FIG. 6. Representative Droughts Selected from the Stochastically Generated Record

4, 6, and 7. As can be seen, the three drought types are quite different for a l-in-500 return frequency; however, they appear to converge as the drought frequency becomes less and for a frequency of 1 in 20 they are somewhat similar. After discussions with city personnel, it was decided that the worst severity droughts (the middle set of droughts in Fig. 5) should be used in the model simulation of the Poudre River to estimate the yields of the city's water rights. Preliminary model results indicated that these drought types

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stressed the city's present water supply to the greatest extent, especially when relatively extreme years occurred toward the end of a drought.

The stochastically generated virgin flows were reviewed in order to select droughts with the same general appearance as those in the middle portion of Fig. 5 and with the following specific criteria: (1) 20-year drought: total deficit of approximately 380,000 acre-ft; (2) 50-year drought: total deficit of approximately 520,000 acre-ft; (3) 100-year drought: total deficit of ap­proximately 630,000 acre-ft; and (4) 500-year drought: total deficit of ap­proximately 890,000 acre-ft. The droughts selected correspond to the data set 2 (or sample 2 from the 100 samples generated), are listed in Table 8, and are also shown graphically in Fig. 6. It should be stressed that slightly different droughts could have been selected from the long-term synthetic record. In fact, initially a drought of 11 years duration was picked for the 500-year drought as opposed to the 7-year drought noted in Table 8 and shown in Fig. 6. The shorter-term drought was found to be more stressful on the city's water supply and was used in place of the longer-term drought. Despite difficulties with the selection process, it is believed that fairly rep­resentative droughts were chosen for the 20-to-500-year conditions,

It is interesting to note that two droughts, which occurred during the his­toric 100-year record, are generally representative of typical extreme con­ditions. The consecutive drought of the 1930s, which extended from 1930 to 1937, was 8 years long and had a total deficit of 540,000 acre-ft. The deficit is representative of a worst-severity drought with a return frequency of around 1 in 50. The duration of the drought is within the range to be expected for a return frequency of 1 in 100. The drought of the 1950s, which extended from 1953 to 1956, was 4 years long and had a total deficit of 420,000 acre-ft. These criteria correspond approximately to a typical drought with a return frequency of around 1 in 25. If the criterion of a 4-year running average is used, then the drought of the 1930s extends from 1930 to 1941, and the drought of the 1950s extends from 1950 to 1956. The corresponding durations and cumulative deficits would be 12 years/760,000 acre-ft and 7 years/430,000 acre-ft, respectively. These droughts are representative of typical 100-year- and 20-year-running-average droughts as shown in the right-hand portion of Fig. 5. A 4-year-running-average drought with a return period of slightly less than 500 years could be created by combining, back-to-back, the drought of 1930-41 with the drought of 1950-56. Such a drought would have a total long-term deficit of 1,200,000 acre-ft and a duration of 19 years, conditions typical of approximately a 500-year-running-average drought, as shown in the lower-right-hand portion of Fig. 5.

CONCLUSIONS

The results of part I of the study provided "design droughts" for use in water-supply analysis. The procedures presented here can be applied in cal­culating drought characteristics of water-supply sources for other commu­nities. Once "design droughts" have been established, the effects of the droughts on water supplies for the community can be simulated utilizing water-resource-system models.

This analysis showed that historically experienced droughts such as those in the 1930s and 1950s do not represent as severe a drought as previously thought. In fact, tree-ring data indicate that more severe droughts have oc-

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curred. The. use of the, stochastic model seems to adequately produce "design droughts" for the five-basin system that matches historical drought charac­teristics and characteristics of long-term tree-ring data. Also, it appears for this water-supply system that annual droughts stress it more than running-average droughts.

The use of this generated drought data is demonstrated in the companion paper, part II (Frick et al. 1990). The companion paper shows the disag­gregation of annual flows to monthly flows and a simulation of actual water supplies.

ACKNOWLEDGMENTS

The writers wish to thank the Water Board of the City of Fort Collins for funding this study. In addition, acknowledgment is due to the Colorado Ag­ricultural Experiment Station, Projects N. 357 and N. 645, for supporting related precipitation- and drought-prediction studies in the region of the study. Likewise, we wish to acknowledge the contributions of Dr. G. Tabios, Mr. Sam Bryson, and Mr. Andy Pineda for their contributions to various parts of the study.

APPENDIX i. REFERENCES

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Booth, A. W., and Voeller, D. (1967). Meteorological drought and its social impact in Illinois. Dept. of Geography, Water Resour. Ctr., Univ. of Illinois, Urbana, 111.

Chander, S., et al. (1981). "Analysis of surplus and deficit using runs." J. Hydrol., 49(1/2), 193-208.

Condra, G. E. (1944). "Drought: Its effect and measures of control in Nebraska." Nebraska Conservation Bulletin No. 25, Univ. of Nebraska, Conservation and Sur­vey Div., Lincoln, Neb.

Correia, F. N., Santos, M. A., and Rodriguez, R. R. (1985). "Engineering risk in regional drought studies." Paper presented at Advanced Study Inst, on Engrg. Reliability and Risk in Water Resour., Univ. of Arizona, Tucson, Ariz.

Dickerson, W. H., and Dethier, B. E. (1970). "Drought frequency in the north­eastern United States." Bulletin 595, Agricultural Experiment Station, West Vir­ginia Univ., Morgantown, W. Va.

Doesken, N., Kleist, J., and McKee, T. (1983). "Use of the Palmer index and other water supply indexes for drought monitoring in Colorado." Climatology ReportN 83-3, Dept. of Atmospheric Sci., Colorado State Univ., Fort Collins, Colo.

Dracup, J. A., Lee, K. S., and Paulson, E. G., Jr. (1980a). "Statistical character­istics of drought events." Water Resour. Res., 16(2), 289-296.

Dracup, J. A., Lee, K. S., and Paulson, E. G., Jr. (1980b). "On the definition of droughts." Water Resour. Res., 16(2), 297-302.

"Droughts and their effect on the water supplies for the city of Fort Collins, Colo­rado." (1985). Resource Consultants, Inc., Fort Collins, Colo.

Frick, D. M., Bode, D., and Salas, J. D. (1990). "Effect of drought on urban water supplies. II: Water-supply analysis." / . Hydr. Engrg., ASCE, 116(6), 754-764.

Gibbs, W. J., and Maher, J. V. (1967). "Rainfall deciles as drought indicators. Bulletin No. 48, Bureau of Meteorology, Melbourne, Australia.

Guerrero, P., and Yevjevich, V. (1975). "Analysis of drought characteristics by the theory of runs." Hydrol. Paper No. 80, Colorado State Univ., Fort Collins, Colo.

Gupta, V. K., and Duckstein, L. (1975). "A stochastic analysis of extreme droughts." Water Resour. Res., 11(2), 221-118.

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Heathcote, R. L. (1973). "Drought perception." Environmental, economic, and so­cial significance of drought, J. V. Lovett, ed., Angus and Robertson, Cremorne Junction, Australia.

Hipel, K. W. (1985). "Stochastic research in multivariate analysis, multivariate anal­ysis of hydrologic processes," Proc, 4th Int. Hydrologic Symp., Colorado State Univ., Fort Collins, Colo.

Lawson, M. P., et al. (1971). "Nebraska droughts: a study of their past chronological and spatial extent with implications for the future." Occasional Papers No. 1, Dept. of Geography, Univ. of Nebraska, Lincoln, Neb.

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APPENDIX II. NOTATION

The following symbols are used in this paper:

run length; run sum; magnitude or run intensity; annual flow time series for time t and site i; normalized annual flows for time t and site i\ residual series for time t and site i; standard normal variable uncorrelated in time for site ;'; lag-one serial-correlation coefficient; standard deviation of zf standard normal variable (vector) uncorrelated in time and space; and autoregressive model parameter for site ;'.

RL = RS = RI = x? = yf = z? = ei° = Pi =

CT«-> =

^ =

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