because the state vector was formulated to include both a signal element (the streamflow ratio) and
unknown parameters. In addition to the unknown parameters in the state vector, five threshold
parameters were estimated externally to the extended Kalman filter.
A discrete-time formulation was used for consistency with the availability of hydrological and
climatological data and for filter simplicity. In this report, the length of the discrete time step was
1 day, a length that facilitated analysis of extensive historical periods of record.
A discrete-time extended Kalman filter was developed to account for the effects of ice on stream-
flow. The filter consists of two models, a nonlinear process model and a linear measurement model.
The general form of the nonlinear process model is
x(k) = f [x(k - 1), k - 1] + w(k - 1)
(1)
where
x(k) = state vector. In this report, the state vector is partitioned into two components. The
first is the streamflow ratio and is called the state signal element. The second is the
state parameter vector. The total number of elements in the state vector is the dimen-
sion of the state space.
f[x(k1), k1] = nonlinear function of the state at the previous time step plus other information on
auxiliary variables available at time k1.
w(k1) = value from a random sequence representing process noise* at time k1. The
sequence w is assumed to be independent and normally distributed, with a mean of
zero and a diagonal covariance structure Q(k1) commonly written w~N[0, Q(k1)].
In this application, only the variance of x1(k) was assumed be nonzero; no process
noise was associated with the state parameters.
The time-varying linear measurement model is of the form
z(k) = H(k) x(k) + v(k)
(2)
where z(k) = streamflow at time k.
H(k) = time-varying measurement sensitivity matrix that is represented by the vector
[h(k) 0 0], where h(k) is the apparent streamflow.
v(k) = value from a random sequence representing measurement noise at time k. The
sequence v is assumed to be independent and normally distributed, with a mean of
zero and variance of R(k). The subset of days indexed by k on which direct measure-
ments of streamflow were obtained is denoted as k′. For developing a projection, the
variance R(k′) was assumed to be proportional to the published streamflow on days
of direct measurement. In open-water conditions, the standard error was assumed to
be 2.5% of the published values; during ice-affected conditions, the standard error
was assumed to be 8.0% of the published streamflow. On days without direct meas-
urement, published flow data were not used to update the estimate of the
streamflow ratio.
Formulation
Based upon analysis of historical streamflow characterizations developed by traditional meth-
ods for estimating ice-affected flow, the dynamics of ice effects were classified into three modes.
Mode 1 dynamics are generated by the sudden formationablation of ice, as indicated by abrupt
changes in apparent streamflow. Mode 2 dynamics are caused by stable ice conditions and are
approximated by a first-order difference equation relating the streamflow ratio to air temperature
* In the analysis of dynamic systems, noise is random inputs that cannot be directly measured or controlled.
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