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<br /> <br /> <br />In all cases, t was computed such that the period of th~ harmonic function was <br />1 year: <br /> <br />N <br />~,. <br />o <br />-J <br /> <br />where <br /> <br />t = 2n(m) <br />12 ' <br /> <br />m = month of the calendar year. <br /> <br />(12) <br /> <br />In general, when seasonality was included only in the 'coefficient a, all <br />periodic terms were significant at the 9S-percent level. Also, the VIF's for <br />all terms normally were less than 10. This indicated there was no significant <br />multicollinearity in the models (Montgomery and Peck, 1982). However, when <br />both the coefficient a and the exponent b included seasonality, the periodic <br />terms usually were not significant and the VIF's generally were greater than <br />100, which indicated poor estimation of the regression coefficients. <br /> <br />To avoid the problems associated with multicollinearity, seasonal vari- <br />ation was restricted to the coefficient a, and the exponent b was held <br />constant. The effect of this restriction on model fit was negligible, as is <br />indicated by the data in table 5, The standard error and the coefficient of <br />determination (R2) are given for models with periodic exponents and constant <br />exponents. For three test sites, the largest decrease in R2 caused by removal <br />of seasonal variation from the exponent is 0.006, The largest increase in <br />standard error is 400 tons per month (7 percent). <br /> <br />Comparison of Model Estimates to Mass-Balance Estimates <br /> <br />The best model form was selected from among the additive, power, and <br />exponential models by comparing model estimates of natural dissolved-solids <br />discharge to the mass-balance estimates, Values of monthly natural dissolved- <br />solids discharge were computed using the calibrated models with the develop- <br />ment terms set to zero and the natural-streamflow estimates provided by the <br />U.S. Bureau of Reclamation (written commun., 1983). The mean annual natural <br />dissolved-solids discharge estimated by each model for the water years 1914-57 <br />was then compared to the mass-balance estimate for the same period, <br /> <br />The initial model estimates of mean annual natural dissolved-solids <br />discharge did not compare well to the mass-balance estimates for three test <br />sites (table 6), Also, estimates varied considerably among the models. For <br />example, the model estimates for site 3 varied from 200 to 1,218,000 tons per <br />year, and none were close to the mass-balance estimate of 463,000 tons per <br />year. This variation probably occurred because the true values of natural <br />dissolved-solids discharge were outside the range of the historical data used <br />to fit the models. Since water-resources development in the Upper Colorado <br />River Basin began long before collection of water-quality data and development <br />of irrigated areas was similar to current (1986) conditions by the 1920's, the <br />period of record used to fit the models was not representative of conditions <br />prior to development. To estimate natural conditions, extrapolation beyond <br />the range of the historical data WaS necessary; such extrapolation Can produce <br />large errors, <br /> <br />18 <br />