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DESCRIPTION OF THE MODEL 13 Autocorrelation Two random variables are said to be independent if, and only if, the value of one variable has no influence on the other variable. When one random variable has influence on another random variable, the random variables are correlated. Random variables can be correlated temporally and (or) spatially. Exam- ples of temporally correlated time-series variables include daily streamflow, stream diversions, and evaporation. For these ran- dom variables, observations related (correlated) through hydro- logic processes (natural and anthropogenic) may persist over many days. Whereas each daily random variable could be inde- pendently sampled and used in a model calculation, neighbor- inadaily values influence each other. For this reason, model calculations based on independently sampled daily distributions are valid only on an annual basis. To more accurately predict the hydrologic response on a daily basis, some means of identifying and incorporating temporal correlation of random variables is needed. Autocorrelation is a mathematical technique that com- monly is used to reveal the correlation between elements of time-series observations (Werckman and others, 2001). Auto- correlation also can be present when residual error terms from observations of the same variable at different times are corre- lated. One example of an autocorrelated random variable is daily streamflow. In this study, the calculated autocorrelation function, for the complete record of daily values of streamflow, indicates correlated (related) streamflow over a lag period of as much as about 75 days (fig. 6), whereas the autocorrelation function for annual streamflow measurements are correlated over a period of about 57 days. By aggregating daily streamflow values over the period of record (daily record with an annual frequency), the correlation coefficient matrix was computed between daily streamflow val- ues. As expected, streamflow correlation between adjacent days 1.00 was greatest (correlation coefficient of about 0.99) with corre- lation between streamflow observations diminishing for mea- surements separated by increasing number of days (over longer periods of time). Whereas an autocorrelation function provides a characteristic (lumped) value for all time-series values at a given lag, a correlation matrix provides discrete information between individual time-series values. For example, inspection of the correlation matrix between streamflow reveals that successive daily values are variable and highly correlated with correlation coefficients between 0.883 (day 280 to 282) and 0.996 (many daily combinations). The findings based on the autocorrelation function and correlation matrix underscores the fact that daily streamflow should not be modeled as indepen- dentrandom variables, but rather the interdependency between daily random streamflow variables should be incorporated into the stochastic mixing model. When defining correlation coefficients for use in the sto- chastic mixing model, practical limitations relating to the num- ber of correlated variables exist, and for that reason only corre- lation coefficients between streamflow random variables associated with a single daily lag are introduced into the model. Implementation of correlated random variables is handled using the method described by Yastrebov and others (1996). In gen- eral,the Yastrebov method generates a sample from a multivari- ate random distribution using a rotation algorithm that has appropriate correlations, and then these variables are trans- formed so that they have the specified distributions. Other ran- dom variables that exhibited autocorrelation include stream- flow diversions, evaporation, and salinity. Whereas streamflow correlation coefficients are introduced into the stochastic model using correlation coefficients associated with daily lags, sto- chastic nonlinear equations are developed and used to predict evaporation and salinity. streamflow diversions, which are con- sidered random variables, are handled on a seasonal basis (see table 4). W 0.75 J Z O W 0.50 ~_ Z Q 0.25 a J W ~ 0 O U fl0.25 25 50 75 100 LAG, DAY Figure 6. Autocorrelation function for streamflow at Cameo during 1974-2001.