values. Mode 3 dynamics, finally, come from the eventual elimination of ice effects at higher air
temperatures. Each mode corresponds to a unique process model; however, the measurement
model was consistent for each mode.
The process model for mode 1 dynamics is an algebraic equation of the form
h(k - 1)
x1(k) =
x1(k - 1).
(3)
h(k)
Mode 1 dynamics were applied for either of two conditions:
If the 1 day change in apparent streamflow increased by more than q_dl percent and the average
air temperature was less then t_lo.
If the one day change in apparent streamflow decreased by more than q_dl percent when the
streamflow ratio was less than 1.
Because of the lack of derivative information in eq 3, q_dl and t_lo were included among five thresh-
old parameters estimated outside the extended Kalman filter. Thus, for mode 1 dynamics, the state
vector only included the streamflow ratio and the corresponding dimension of the state space was 1.
The process model for mode 2 dynamics was a first-order difference equation, driven by air
temperatures u, that was similar to one developed and implemented in a previous investigation
(Holtschlag 1996). The process model for mode 2 dynamics is of the form
x1(k) = x2 (k - 1) + x3[x1(k - 1) - x2 (k - 1)]+ x4 (k - 1)[u(k - 1) - x5 (k - 1)]
~
(4)
which says that at times of ice effects and constant air temperature x5, the streamflow ratio (x1) is in
ference equation that includes an autoregressive component with parameter x3 and a forcing func-
tion term driven by daily air temperature values. Air temperatures that vary from a nominal value
of x5 change the streamflow ratio from its nominal value of x2 at a rate of x4. This form of a difference
equation is nonlinear in parameters because both rate parameters (x3 and x4) and offset parameters
(x2 and x5) are estimated simultaneously.
Prior information on the distribution of streamflow ratios during periods of ice effects was used
in hope of facilitating the solution of this inherently difficult estimation problem. Air temperature
values u indicate an exponentially weighted average of temperatures from the 3 previous days. An
~
exponential weighting factor t_wt, used in computing u , was included among threshold para-
~
meters, because it did not occur explicitly in eq 4. The dimension of the state space for mode 2
dynamics is 5.
The process model for mode 3 dynamics is an algebraic equation of the form
u(k - 1) - t _ hi
~
x1(k) = x1(k - 1) +
[1 - x1(k - 1)].
(5)
t _ ou - t _ hi
Mode 3 dynamics are applied when the exponentially weighted air temperature value exceeds t_hi.
Then, the streamflow ratio increases from its value at time k 1 to a value of 1 when the exponen-
tially weighted air temperatures equal t_ou. Because t_hi and t_ou occur explicitly in the process
model, they could be included as parameters in the state vector. However, for simplicity in this
analysis, they were included among threshold parameters estimated outside of the extended Kal-
man filter. As in the case of mode 1 dynamics, the dimension of the state space for mode 3 dynamics
was treated as 1. Estimates of streamflow ratio for all dynamic modes were constrained to the inter-
val between the minimum ratio of published to apparent streamflow determined from historical
record and 1.
The process models for the three dynamic modes were developed so that the first (only) element
in the state vector was the signal element (the estimated streamflow ratio). For this application,
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