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<br /> <br />~ <br />~ <br />~~ <br />~ <br /> <br />mass-balance estimates, the recalibrated model results for natural dissolved- <br />solids discharge were expected to be sensitive to the mass-balance estimates. <br />A sensitivity analysis was performed to test the effect of variations in the <br />mass-balance estimate on the results of the selected model. Mass-balance <br />estimates of natural dissolved-solids discharge at the test sites were varied <br />within a range of iSO percent of the estimates reported in table 6. Prede- <br />velopment data computed from these mass-balance estimates then were included <br />in the original data set for that site and the model was recalibrated, The <br />! resulting model estimates of natural dissolved-solids discharge were changed <br />by almost the same percentage as the change in the mass-balance estimate, <br />This confirmed the importance of accurate mass-balance estimates. <br /> <br /> <br />Revisions to the Selected Model <br /> <br />After its selection as the best model form, the exponential model was <br />tested further to ensure that it satisfied the necessary conditions for <br />adequate regression results, Also, because the actual regression analysis was <br />based on logarithms of , dissolved-solids discharge and streamflow, the re- <br />sultant equation had to be detransformed to estimate natural dissolved-solids <br />discharge. <br /> <br />Equalization of Residual Variance <br /> <br />One necessary condition for obtaining accurate results from least-squares <br />regression is that the random errors (e in eq. 13) have a constant variance <br />throughout the range of the dependent variable. Violations of the constant- <br />variance assumption can be detected by plotting the predicted values of In(DH) <br />against the residuals, <br /> <br />Initial residual plots for the three test sites are shown in figure 2. <br />Three different patterns can be observed, For site 1, Colorado River near <br />Glenwood Springs, Colo" the variance of the residuals seems to be reasonably <br />constant throughout the entire range of prediction. However, for site 3, <br />Gunnison River near Grand Junction, Colo" the variance of the residuals is <br />small at small predicted values of dissolved-solids discharge. Most of these <br />small discharge values are associated with the predevelopment estimates. For <br />site 9, Yampa River near Maybell, Colo., the variance of the residuals seems <br />to increase slightly with increasing discharge. <br /> <br />The problem of unequal variance can be decreased by differential weight- <br />ing of the residuals in the least-squares solution. Residuals that tend to be <br />large are given a small weight, and residuals that tend to be small are given <br />a large weight, Weighting factors commonly are computed proportional to the <br />inverse of the residual variance for discrete values of the independent <br />variable, For a continuous variable, such as streamflow, measurements can be <br />divided into classes having a discrete range of values, and the weighting <br />factor can be computed proportional to the inverse of the residual variance <br />within the class (Mongtomery and Peck, 1982, p. 101), In the Colorado River <br />basin, where dissolved-solids discharge has a distinct seasonal pattern, <br />measurements can be classified by month. The maximum discharge occurs during <br />snowmelt runoff, April through July, and the minimum discharge occurs during <br /> <br />24 <br />