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DESCRIPTION OF THE MODEL 9
<br />1.00
<br />H
<br />J_
<br />C0 0.75
<br />a
<br />m
<br />0
<br />a
<br />LLI 0.50
<br />U
<br />Z
<br />Q
<br />O
<br />W 0.25
<br />U
<br />x
<br />w
<br />Dry
<br />Average
<br />Wet
<br />1,000,000 2,000,000 3,000,000
<br />STREAMFLOW, IN
<br />CUBIC FEET PER SECOND PER YEAR
<br />Figure 3. Annual flow exceedance at Colorado River near Cameo
<br />(09095500-for1974-2001.
<br />Random Variables
<br />Most real-world problems involve elements of variability
<br />or uncertainty called random variables (Kalos and Whitlock,
<br />1986). Random variables are those variables that do not have a
<br />fixed value at a particular point in space and time but instead are
<br />described by probability distributions that account for a range of
<br />possible values. There are two types of random variables: dis-
<br />crete and continuous. A discrete random variable may take on
<br />only a countable number of distinct values, such as the number
<br />of reservoir discharge days. A continuous random variable
<br />takes an infinite number of possible values. Examples of contin-
<br />uous random variables include measurements such as stream-
<br />flow, evaporation, diversions, and salinity. Whereas a random
<br />variable has either an associated probability distribution (dis-
<br />crete random variable) or probability distributions (continuous
<br />random variable), all random variables (discrete and continu-
<br />ous) have a cumulative distribution function. The cumulative
<br />distribution function for a discrete random variable is deter-
<br />mined by summing the probabilities, whereas the cumulative
<br />distribution function for a continuous random variable is deter-
<br />mined by integrating the probability density function. The
<br />cumulative distribution function represents the probability that
<br />a variable will occur at or below a given value, whereas a
<br />reverse cumulative distribution function represents the proba-
<br />bility that a variable will occur at or above a given value
<br />(exceedance probability). The cumulative and reverse cumula-
<br />tive distribution functions both are presented in this report by
<br />using tables of percentile so that the likelihood of selected fore-
<br />cast variables can be evaluated. A percentile is a number
<br />between 0 and 100 that indicates the percentage of a probability
<br />distribution that is equal to or below a value (cumulative distri-
<br />bution function) or equal to or above a value (reverse cumula-
<br />tive distribution function).
<br />By using historical streamflow measurements, daily
<br />streamflow values were aggregated by year for the entire period
<br />of record, calendar years 1974 to 2001. This data aggregation
<br />resulted in 365 random variables, each with about 27 samples
<br />(2001 was a partial year). These daily values then were fit to one
<br />of 14 continuous probability distribution functions: Beta, Bino-
<br />mial, Exponential, Extreme value, Geometric, Hypergeometric,
<br />Logistic, Lognormal, Normal, Pareto, Poisson, Triangular, Uni-
<br />form, and Weibull (Werckman and others, 2001). The quality of
<br />distributional fit was judged by using one of several goodness-
<br />of-fit criteria that included Chi-square, Kolmogorov-Smirnov,
<br />and Anderson-Darling (Werckman and others, 2001). Multiple
<br />goodness-of-fit criteria were used because one (or more) of
<br />these criteria yielded a best model that was judged infeasible. In
<br />general, daily streamflow measurements were best character-
<br />ized using Extreme value, Logistic, Lognormal, Normal,
<br />Pareto, Triangular, Uniform, or Weibull probability distribu-
<br />tions. The actual distribution selected to represent a daily
<br />streamflow random variable was based on the distributional fit
<br />with the highest rank.
<br />To investigate the operational dependency between diver-
<br />sions and hydrologic conditions, correlation coefficients were
<br />computed between daily streamflow for the period of record
<br />(calendar year 1986 to 2001) for Colorado River near Cameo
<br />(09095500), Plateau Creek near Cameo (09105000), Colorado
<br />River near Palisade (0916000), and diversions at the Grand Val-
<br />ley Irrigation Canal (GVIC) and Government Highline Canal
<br />(GHC). This 15-year period of record includes three wet, nine
<br />typical, and three dry hydrologic periods. A summary of corre-
<br />lation coefficients between these hydrologic entities is pre-
<br />sented in table 3.
<br />Whereas the dependency appears high between individual
<br />streamflow-gaging stations (Cameo, Palisade, and Plateau) and
<br />between individual streamflow diversions (GVIC and GHC),
<br />correlation between streamflow and diversion records is rela-
<br />tively weak. The relatively weak correlation between stream-
<br />flow and diversion records underscores the relative indepen-
<br />dence between diversion operations and hydrologic conditions.
<br />This high degree of independence between streamflow diver-
<br />sion and hydrologic condition is further illustrated by observing
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