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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P(t+) =

P1l(t-) -

2 ,-
P12 (t )
P22(t ) + r

P12(t-) - P12(t-)P22(t-)
P22(t ) + l'

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(21)

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P12(t-) -

P12(t-)P22(t-)
P22(t ) + l'

PZ2(t-) -

Z -
P22 (t )
P22(t ) + l'

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An example of the time trace of Pll(t) and PZZ(t) for the case of

six equally spaced discharge measurements during a water year is shown

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in figure 5. The ith measurement during the water year is made at time

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~i' The variance, PZ2(t), of the error of estimate of discharge rate
can be seen to rise to a peak just before a discharge measurement is made

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at which time uncertainty is a maximum. Immediately after completion of

a discharge measurement, uncertainty is a minimum but begins its increase

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that ends only when the next measurement is made. If measurements are

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equally spaced in time, P22(t) is a periodic function.
The variance of total discharge since the beginning of the water year,

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Pll(t), can be seen to increase from a value of zero at the beginning of

the year to a maximum at the end of the year.

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Figure 5 and equations 18 and Zl pertain to the real-time computation

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of discharge; that is, the computation of discharge at time t is performed

with only discharge measurements and correlative data up to time t. In most

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cases real-time discharge esimates are not the only requirement at a streamflow

station; data from before, during, and after the water year of interest can

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be used to obtain better estimates than those that may be computed in real

time. The process of including all pertinent information in the estimation

procedure is known as smoothing in filter-theory parlance. Smoothing is the

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equivalent of optimally combining two estimates of the unknown states at

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time t. One estimate is the real-time or forward-filter described above,
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