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<br /> <br />dependent variable for model calibration; therefore, the estimated concentra- <br />tions correspond to flow-weighted values. Monthly dissolved-solids discharge <br />was computed as the product of the estimated concentration and streamflow. <br /> <br />Weighted Regression <br /> <br />N <br />o <br />F'... <br />'-I' <br /> <br />Extension of dissolved-solids records commonly is made by regression of <br />dissolved solids on streamflow. Lane (1975) reviewed dissolved-solids- <br />streamflow relations that had been proposed by many researchers and <br />recommended using a power equation: <br /> <br />b <br />C = aQ , <br /> <br />(1) <br /> <br />where C = dissolved-solids concentration; <br />Q = streamflow; and <br />a and b = empirical parameters. <br /> <br />This equation subsequently was used by Steele (1976) and DeLong (1977) in <br />analyzing Colorado River basin data. <br /> <br />Equation 1 can be transformed into a linear model so the empirical <br />constants can be evaluated by least-squares regression: <br /> <br />In(C) = a* + b In(Q) + e, <br /> <br />(2) <br /> <br />where a* = In(a); and <br />e = random error, which is assumed to be normally distributed <br />with a mean value of zero and a constant variance. <br /> <br />If the parameters in equation 2 are estimated using the method of least <br />squares, the calibrated model is: <br /> <br />In(C) = a* + 0 In(Q), <br /> <br />(3) <br /> <br />where In(C) = the estimated value of In(C), and <br />a* and 0 = regression coefficients. <br /> <br />For any particular observation, the residual is the difference between the <br />observed value of In(C) and the corresponding estimate from the calibrated <br />model (eq. 3). If the calibrated model is correct, the residuals should tend <br />to exhibit the properties assumed for random errors (Draper and Smith, 1981). <br /> <br />Lane (1975) and DeLong (1977) suggested that the coefficients a* and 0 be <br />evaluated as harmonic functions of time, with a period of one year, to account <br />for seasonal hysteresis in the relation. Mueller and Osen (1987) reported <br />that allowing both coefficients to vary with time introduced significant col- <br />linearity into the model. This collinearity, which can cause poor estimation <br />of regression coefficients, can be eliminated by evaluating only the intercept <br /> <br />8 <br />