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<br />I <br />I, <br />1 <br />1 <br />1 <br />I <br /> <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />I' <br />1 <br />1 <br />I <br /> <br />y ~ (Qc - QR) - (QT - QR) (7) <br />For notational convenience, the following identities will be used for the <br />remainder of this paper: <br /> <br /> t <br />'-::1(1';) ~ ! (qT(1';J - qR(t))dt <br /> 0 <br />and t <br />^ <br />'-::1(1';) ~ ! (qC(t) - qR(t))dt <br /> 0 <br /> <br />(8) <br /> <br />(9) <br /> <br />The error in annual discharge can be redefined <br /> <br />^ <br />y ~ '-::1(1) - xl(l) (10) <br />which is assumed to be a function of the frequency of discharge measurement. <br /> <br />The variable '-::1(1) cannot be observed; thus equation 10 cannot be directly <br /> <br /> <br />evaluated. However the techniques of Kalman filtering (Gelb, 1974) yield an <br /> <br />^ <br /> <br />estimate of the variance of y. If '-::1(1) meets the assumption of being an <br /> <br /> <br />unbiased estimator of =1(1), minimization of the variance of y is equivalent <br /> <br /> <br />to the stated objective of minimizing its root mean square. <br /> <br />The general form of the state equation for a system that can be <br /> <br />described by a linear differential equation with a random forCing function <br /> <br />is <br /> <br />~(t) = F ~(t) + e ~(t) (11) <br /> <br /> <br />where ~(t) is a vector of length n of the first derivatives with respect <br /> <br /> <br />to time of the state variables, F and e are matrices of coefficients, <br /> <br /> <br />~(t) is a vector of the state variables, and ~(t) is a vector of the <br /> <br /> <br />random noise or forcing variables. The governing differential equation <br /> <br /> <br />for the discharge-computation problem is assumed to be of the form <br /> <br />d2.-::1(t) aldrl(i:) <br />+ <br />,,2 at <br />a-z; <br /> <br />+ aO'-::l (1';) <br />21 <br /> <br />= <J(t) <br /> <br />(12) <br />