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<br />l\) <br />....4 <br />o <br />0') <br /> <br />The additive model is appealing because a physical interpretation could <br />be given to each term, If the relation between the dissolved-solids discharge <br />and streamflow under natural conditions is assumed to be a power function, <br />then the additive model indicates that development in the basin increases the <br />dissolved-solids discharge or, where there are exports from the basin, de- <br />creases the discharge. The historical discharge, therefore, is the sum of <br />the natural dissolved-solids discharge from the basin and the discharge due to <br />development within the basin. <br /> <br />,-,'", <br /> <br />In the power and exponential models, the development variables are <br />empirical factors that change the historical relation between dissolved-solids <br />discharge and streamflow as development increases. Historical streamflow was <br />used to calibrate these models. An important attribute of both these models <br />is that they can be rewritten in linear form and the coefficients evaluated <br />using linear regression: <br /> <br /> <br />Power model: In (Da) = In(a) + bln(Qa) + LC, In(X, + 1) and (7) <br /> ~ ~ , <br />Exponential model: In (Da) = In(a) + bln(Qa) + LC ,X. (8) <br /> ~ ~ <br /> Evaluation of Seasonal Variation <br /> <br />To account for seasonal variation in the relation between dissolved- <br />solids and streamflow (eq. 1), Lane (1975) and DeLong (1977) incorporated <br />periodic functions of time into the empirical coefficient (a) and empirical <br />exponent (b). DeLong (1977) used the formulation: <br /> <br />a = exp(ao + ai sin(t) + a2 cos(t)) , and <br /> <br />(9 ) <br />(10) <br /> <br /> <br />b = bo + bi sin(t) + b2 cos(t) , <br /> <br />, <br />'5 '~ <br /> <br />where <br /> <br />t = a function of time, and <br />ao, ai, a2, bo, b2 = empirical parameters. <br /> <br />'.i <br /> <br />a = ao + ai sin(t) + a2 cos(t) <br /> <br />(11) <br /> <br /> <br />Lane (1975) found a strong linear correlation between the periodic values <br />of a and b that were determined using this formulation. Such a correlation <br />can result in poorly estimated regression coefficients and limit the useful- <br />ness of the model for extrapolation (Montgomery and Peck, 1982), <br /> <br />For all the models used in the present study, parameters a and b were <br />evaluated as periodic functions. Then the VIF's of the periodic terms were <br />checked for indications of collinearity, In the linearized power and <br />exponential models (eq. 7 and 8), the parameters a and b were of the same <br />form used by DeLong (1977), given previously in equations 9 and 10. In the <br />additive model (eq. 4), the empirical coefficient (a) was computed as a <br />summation rather than an exponential function: <br /> <br />17 <br />