matrix was computed by use of the Thornton UD factorization algorithm (Grewal and Andrews
1993, p. 255) rather than by directly using eq 613. Similarly, the Bierman observational update algo-
rithm (Grewal and Andrews 1993, p. 245) with corrections provided by D. Flynn* was used rather
than directly using eq 1517. The MATLAB code was converted to code in the C++ programming
language (Weiss 1996) for improved portability (a program listing is available from the first author,
DJH).
PROJECTING ICE-AFFECTED STREAMFLOW
Filter estimates generally are at time k, based on measurements up to and including time k. In
contrast, forecast estimates are made at time k, based on data up to, but not including, time k. In this
report, extended Kalman filter estimates of streamflow are projections, forecasts determined on the
basis of the temporal updates. However, on days of direct measurement, more accurate filter esti-
mates are computed by use of observational updates.
St. John River at Dickey, Maine
The extended Kalman filter was initialized to St. John River data by manually adjusting prelimi-
nary estimates of threshold parameter values. This minimized the sum of squared errors in the
extrapolated streamflow ratio x1-) (k ′) - x1(k ′) , where k′ indicates days of direct streamflow measure-
(
ments. Once satisfactory estimates of the threshold parameters were obtained, they were fixed (Table
1). Then the filter was run repetitively, using the state vector and error covariance matrix computed
on the last iteration of the previous run to initialize the subsequent run. The filter was run repeatedly
until elements in the state parameter vector were essentially constant. In this process, initial esti-
mates for the state error covariance matrix converged from an initially specified diagonal matrix
with nonzero components of [0.5 0.5 0.01 10] to the standard errors shown in Table 1.
Final estimates for the state parameters indicate that mode 1 dynamics are highly autoregressive,
as shown by the parameter x3 = 0.981, about a streamflow ratio offset of x2 = 0.544. Streamflow ratios
increase at a rate x4 = 0.000855C1 from the temperature offset x5 = 3.19C. Although the value for x2
Table 1. Threshold and filter parameters for the extended Kalman filter for the St. John
River at Dickey, Maine.
Estimated
Parameter
standard
symbol
Parameter description
Estimate
Sensitivity
error
t_hi
High temperature at which ice breakup begins
(in degrees Celsius).
4.50
0.0824
--
t_lo
Low temperature at which sudden increases in apparent
streamflow indicates ice accumulation (in degrees Celsius).
2.25
0.00868
--
t_ou
High temperature at which ice breakup is complete
(in degrees Celsius).
10.0
0.0427
--
t_wt
Exponential weighting factor for daily temperatures.
0.95
0.0424
--
q_dl
Threshold at which changes in apparent daily streamflows
are considered large.
0.35
16.1
--
x2
Offset streamflow ratio.
0.544
--
0.0937
x3
Autoregressive parameter for streamflow ratio.
0.981
--
0.000153
x4
Parameter relating air temperature to changes in
streamflow ratios (in degrees Celsius1).
0.000855
--
0.00000459
x5
Offset temperature (in degrees Celsius).
3.19
--
2.08
* Personal communication, California State University at Fullerton, 1996.
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