WSP11570

• File Number: 8111.807
• Description: Arkansas River Compact Administration - Stream Gage Evaluation
• Basin: Arkansas
• Date: 1/1/1980
• Author: USGS
• Title: Cost-Effective Stream Gaging Strategies for the Lower Colorado River
• Water Supply Pro - Doc Type: Publication

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The general equation for error propagation of a continuous ~ystem

between measurements is

l' l'
Prt} = FP(t} + NtJ, + CQc-

(18)

where P(tJ is an n x n matrL~ of error covariances, and Q is a vector of

spectral densities of the random forcing functions, ~(tJ. The discharge-

computation problem has a single forcing function; therefore Q is a scalar,

denoted q. The elements of the matrLx, P(tJ, are as follows: Pll(tJ is

the error variance of xl(tJ, PZZ(tJ is the error variance of xZ(tJ, and

PlZ(t) = P21(tJ is the covariance between errors in xl(tJ and x2(tJ.

As discussed earlier, discharge measurements are considered

to be instantaneous and are obtained at discrete times only. Therefore,

the amount of information about xl(tJ and'x2(tJ changes abruptly upon

completion of a discharge measurement. This abrupt change is measured by a

change in the covariance matrix, P(t}. Denote the value of P(t} just ,prior

to a measurement at time t by PCt-) and the value of P(t) just after the
+ +
measurement as P(t). Gelb (1974, p. 109-110) expresses pet ) as

P(t+) = (I - X(t)H] 2(t-)

(19)

where I is an identity matrix and K(t) is known as the Kalman gain matrix and

is defined

_ _ 1'. _ l' -1
K(t) = dt )lr (HP(t )Ir + iI]

(ZO)

where iI is the covariance matrix of measurement errors, the superscript T

indicates a matrL~ transpose, and the superscript -1 indicates a matru{

inversion. In the discharge computation problem, the measurements pertain

only to x2(t), not xl (t), and R is thus a scalar, denoted!', that specifies

the variance of the measurement error. The combination of equations 15, 19,

and ZO results in

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