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14 Probable Effects of the Proposed Sulphur Gulch Reservoir on Colorado River Quantity and Quality near Grand Junction, Colorado Nonlinear Regression and Residual Analysis In this study, a nonlinear least-squares approach (Cooley and Naff, l 982) is used to estimate best-fit parameters to predic- tive equations that compute reservoir surface area as a function of reservoir volume (fig. 7), probability of exceedance as a function of streamflow at Palisade (fig. 8), reservoir evapora- tion as function of time (fig. 9), streamflow at the Plateau Creek Qage near Cameo as a function of streamflow at the Colorado River gage near Cameo (fig. 10), salinity as a function of Colorado River streamflow at Cameo and Colorado River near Palisade (figs. I 1 and 13), dissolved solids (salinity) as a func- tion of runoff salinity at Sulphur Gulch (fig. 14), and salinity (dissolved solids) as a function of streamflow at Plateau Creek near Cameo (fig. 12). The equations and fitted-parameters for these functions are summarized in table 5. Because salinity as a function of streamflow and evaporation as a function of time are stochastic, residual analysis must be performed and the results incorporated into the regression equation. By virtue of its formulation, regression renders an other- wise stochastic process deterministic through the estimation of a single set of best-fit parameters. As described, use of these best-fit parameters in the predictive equation results in a deter- ministic outcome; that is, a given input always produces the same output. Whereas deterministic equations are appropriate for describing nonrandom variables, such as exceedance proba- bilities and reservoir surface area, these equations are inappro- priate for predicting the range of behavior attributed to random variables such as streamtow, evaporation, and salinity. To con- vert from deterministic to stochastic equations, the process vari- ability and (or) uncertainty are reintroduced. This variability is reintroduced into the mixing model by adding residuals to the deterministic equation following random sampling of probabil- ity distribution functions describing the set of differences between the measured and predicted values (residuals). In this study, the various residuals are fit to Logistic and Weibull prob- abilitydistributions, as summarized in table 6. The Monte Carlo method used to select random residuals from these residual probability distribution functions is discussed in the next section. Q 300 W 0: Q 250 W Q ~ 200 W W 0= ~ ~ Q 150 ~ Z 100 W 50 W ~ 0 Morrte Carlo Method The Monte Carlo method is an efficient technique that overcomes analytical challenges associated with devising and implementing stochastic equations through the use of a random number generator. In general, the Monte Carlo method builds up successive model scenarios (realizations) using input values that are randomly selected to reduce the likelihood for bias from probability distributions already defined. In this study, the Monte Carlo method (Sargent and Wainwright, 1996) is used to draw random values from probability distributions for each model input variable used in the calculation. For example, by incorporating residual probability distributions into the predic- tive equations derived through regression, repeated sampling and calculation of the associated dependent variable results in alternate realizations (equally likely simulations known as sto- chastic modeling). Examples of stochastic modeling for evapo- ration, streamflow, and salinity (dissolved solids) are shown in figures 9-14. Two realizations are shown for many model parameters to illustrate the random nature introduced through residual analysis and Monte Carlo method. In general, the stochastic modeling reasonably replicated the random character for streamflow, evaporation, and salinity (dissolved solids) variables throughout the year. In some cases, the realizations did not appear to replicate certain extreme events. Examples of extreme events include the measured value of evaporation on day 143 that was 104x 10-~ in. (1.04 in.) (fig. 9), salinity (dissolved solids) of 340 mg/L at 13,100 ft~/s (fig. 1 l ), and 140 mg/L at l 3,200 ft3/s (fig. l l ). The reason for not replicating the full range of events is attributed to limited random sampling. Statistical evaluation of random variables computed for 1,500 Monte Carlo trials better matched the range associated with extreme events than for a fewer number of trials. Whereas stochastic simulation ofdissolved-solids concen- trations in runoff at Sulphur Gulch replicated the variability associated with the Dry Fork at Upper Station measurements, it is interesting and important to note that the actual dissolved- solids and streamflow measurements at Sulphur Gulch in 2002 tend to support the hydrograph separation approach used herein 4,000 8,000 12,000 16,000 RESERVOIR VOLUME, IN ACRE-FEET Figure 7. Comparison of reservoir volume and reservoir surface area.