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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computation problem, measurements are made only at specific times (not

continuously), and the measurements result only in estmates of ~i(tJ

(not of xl(t)). Thus ,the specific measurement equation is
z(t) = [0 1] [~l(t)] + v(t)
x2(tJ

(17)

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where z(t) is the measured discharge, ~V(t), minus the rated discharge,

qR(t) , at time t. Equation 17 is applicable only at the specific times

of the discharge measurements. Between measurements v(t) can be assumed

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to have infinite variance, which is the equivalent of no new information

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being collected.
The proper use of Kalman-filtering techniques requires both w(t) and v(t)

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be independent, Gaussian random variables. In the case of measurement error,

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v(t), this requirement probably is not violated grossly. On the other hand,

w(t) can be influenced by the choice of the parameters, aD and aI' that are
used to describe the time series model of x2(t). These two parameters must

be evaluated on their own merits at each site at which discharge accuracy

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is to be modeled by a Kalman filter.

As stated earlier, the primary interest of this study is the definition

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of the variance of the error of estimate of Xl(l) as a function of the

frequency of discharge measurement. On the surface this would seem to

entail a classical implementation of optimal fixed-point smoothing as

described by Gelb (1974, p. 157). However, to perform optimal smoothing,

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the system must be fully observable. The discharge-computation problem

does not meet this criterion because xl(t) is never measured directly. Its

error can be estimated only from that of =2(t).

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