|We first illustrate a simple example using constant parameters (constant bi values in equation (1)). These parameters were identified using an instrumental variable method that squeezes the maximum correlation out of temperature or temperatures and past discharge (Ljung, 1995, 1987). As expected, the simulated discharge values are too high in late winter/early spring and too low later (Fig. 3). This is partly because the parameters represent an average over conditions where the days are becoming longer and the nights warmer as spring progresses (the snowpack is also accumulating thermal energy). Therefore the response is first over-and then underestimated.
The next example, a variable parameter Kalman filter analysis, may seem like cheating because we are curve fitting with time-varying parameters. The parameter averages follow different cycles depending on wetness and dryness and lead-lag relations (not shown). The parameters provide predictive power only after a series of air temperature and discharge fluctuations have already occurred (after observed discharge is compared to air temperature). About all that is known in advance is if the previous winter was wet or dry. At least this objectively identifies when the filter gain (coefficient sum) begins to decrease due to a limited snowpack, such as June 1 for the average of the 10 wettest years 1932-1993 (excluding the 2 wettest years 1983 and 1969) and May 16 for the driest (Fig. 4).
Figure 3. Observed and simulated discharge Merced River at Happy Isles 1993 using a constant parameter model.
Figure 4. Upper panel, mean daily discharge 1932-1993 during and following the median day of snowmelt "pulse" (April 19) up to the onset of the "summer transition". Lower panel, the average day of the sum of daily response coefficients, bi, before their continuous decline (the summer transition) in a wet and dry 10-year composite.
Our last example is an initial attempt to use these methods in a prediction mode. At some point only predicted rather than observed air temperatures will be included in the modeling scheme. To keep this simple we are assuming the observed air temperature values are predicted values. Therefore, the results are better than can be expected using true predictions of air temperature. This sumption is of minor significance here because assessment of prediction error in air temperature (which is very small) is a different issue; here we are attempting to predict discharge solely on the basis of air temperature and past estimates of discharge to advance the predictions one day at a time.
When the time series are extended to an 8-day forecast, these initial results appear to be reasonable (Fig. 5). Beyond day 3 the parameters (bi) continue to change but are based solely on predicted values of temperature (assumed) and discharge.
If the parameter changes were small over the 8-day window of forecasting, a constant parameter model may be an adequate method of approximation
(i.e. Fig. 3). But we would only know that after the fact (i.e., hindcasting).
Figure 5. Forecast of Merced River discharge at Happy Isles with a 1-day forecast day 225; 2-day on 226, 3-day on 227, etc. Note divergence around the 5-day forecast between observed and forecast.