(11)

A first order approximation of the state transition matrix is given by

Φ[1](*k *- 1) ≈

(12)

The estimate of the transition matrix used in this analysis is

*x*3-) (*k *- 1) 1 - *x*3-) (*k *- 1) *x*1-) (*k *- 1) - *x*2-) (*k *- 1) *u*(*k *- 1) - *x*5-) (*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*) *Z*α = 0.1 P1(,-) (*k*) > *z*(*k*) ~ 0.9

^

(14)

1

1.64.

Observational updates were computed for days of direct flow measurement. First, the Kalman

gain matrix was computed as

-1

(15)

Then the covariance matrix was updated as

[

]

(16)

The state vector update was computed as

[

]

^

(17)

And finally the observational update was computed as

^

(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.

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

9