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 other is the backward-filter estimate that is derived by reversing the

filter and using the data collected after time t to project backward to

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estimate at time t. For a system to be smoothable, it must also be fully

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observable; a condition that was shown not to be met for the discharge-

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computation problem. However, if xl(t) is ignored temporarily, the remainder

of the system XZ(t) meets the observability and smoothability requirements.

For a reversible process the equations of the forward and backward

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filters are identical; only the direction of time is reversed. Under

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such circumstances the variances of the forward and backward estimates

are symmetrical between equally spaced measurements. Figure 6 illustrates

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the symmetrical nature of the variance of the errors of estimates of xZ(t).

According to Gelb (1974, p. 156-157) the errors of the forward and

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backward estimates at any time tare uncorrelated; that is, the covariance

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of the errors is zero. However, it can be shown that the covariance between

the estimation errors is non-zero for a first-order Markovian process such as

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that used to model xZ(t). A paper giving the deviation of this covariance

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is in the review process now. For uniformly spaced measurements, the

covariance is

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TIt = (1 + 6Z e-ZSA _ 6(e-ZST + e-ZS(A-T)))q/2S

(22)

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where T is t - 1jJi' 1jJi' is the time of the last measurement before ;:;, and
~ - '
F\ r - 1':-ZS2] -1
6 = pi + ~ (23)
PA + l'
where PA is the maximum of P22(t), which occurs just before a discharge
.

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measurement. Optimally combining the forward and backward estimates of

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27

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