T
P(-) (k) = Φ[1](k - 1)P(+) (k - 1) Φ[1] (k - 1) + Q(k - 1).
(11)
A first order approximation of the state transition matrix is given by
f [x, k - 1] x = x(-) (k -1) .
Φ[1](k - 1) ≈
(12)
x
The estimate of the transition matrix used in this analysis is
x3-) (k - 1) 1 - x3-) (k - 1) x1-) (k - 1) - x2-) (k - 1) u(k - 1) - x5-) (k - 1) x(-) (k - 1)
(
(
(
(
(
~
4
0
1
0
0
0
.
Φ[1](k - 1) =
(13)
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
The value of Q was determined such that
[
]
Prob z() (k) Zα = 0.1 P1(,-) (k) > z(k) ~ 0.9
^
(14)
1
1.64.
Observational updates
Observational updates were computed for days of direct flow measurement. First, the Kalman
gain matrix was computed as
K (k ′) = P(-) (k ′) H T (k ′)[H(k ′) P(-) (k ′)H T (k ′) + R(k ′)] .
-1
(15)
Then the covariance matrix was updated as
[
]
P(+) (k ′) = I - K (k ′) H(k ′) P(-) (k ′).
(16)
The state vector update was computed as
[
]
x(+) (k ′) = x(-) (k ′) + K (k ′) z(k ′) - z(-) (k ′) .
^
(17)
And finally the observational update was computed as
z(+) (k) = H(k)x(+) (k) = [h(k) 0 0]x(+) (k).
^
(18)
On days without direct flow measurement, observational updates were set equal to the temporal
updates computed for that time step. Thus, projected values computed by use of the extended Kal-
man filter were equal to the observational updates on days of direct streamflow measurement and
were equal to the temporal updates otherwise. No adjustment was included for uncertainty in the
apparent streamflow values.
Computation
Computational procedures were developed and tested in the MATLAB programming environ-
ment (The MathWorks, Inc., Natick, Massachusetts) on the basis of algorithms developed by Grewal
and Andrews (1993). To provide numerical stability, the temporal update of the error covariance
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