|Air temperature data are a composite of mean daily values from Sacramento, Tahoe, Nevada City and Hetch Hetchy (Fig. 1), 1932-1993 (Riddle, unpublished). This index is used to represent air temperature variations in the upper Merced Basin. Daily averaged discharge is from the USGS gaging station at Happy Isles.
On average 95% of the discharge in Merced River above Happy Isles is snowmelt driven (Clow and others, 1996; Cobb and Biesecker, 1971) as illustrated in Fig. 2 by the wide separation between peaks in precipitation (winter) and discharge (spring). See Cayan and others, (1993); Cayan (1996); and Dettinger and Cayan (1995), for details on the temporal- spatial relations between snowmelt driven (high elevation) and rainfall driven (low elevation) discharge; an overview of snowmelt is provided by Morris (1985). Water year days 165, March 13 through 285, July 12 were selected for study; this period generally encompasses the rise and decline in snowmelt-driven discharge.
Figure 1. Study area, meteorological and river gage sites.
Figure 2. Low-pass filtered air temperature, precipitation and discharge (double-pass boxcar filter to preserve phase; 15 days for air temperature, 25 days for precipitation and 9 days for discharge). Precipitation and air temperature from a mean-daily index: Sacramento, Tahoe, Nevada City, and Hetch Hetchy. Discharge from Merced River, Happy Isles, Yosemite National Park, California.
Our focus is on the air temperature-discharge linkage and mostly for the high snowpack -high discharge years. Given the system complexity, it may seem unexpected that linear methods capture most of the discharge variance using only air temperature as input. One statistical approach is the use of a difference equation to estimate output (discharge) from filtered input (temperature) and, possibly, past output:
Where [n] is the time index, ai are the past discharge (Q) coefficients and bi the present and past temperature (T) coefficients. In the simplest case discharge is estimated by multiplying the corresponding coefficients with air temperature and summing. Each day's discharge is based on the temperature for the present and past 2 or 3 days (ultimately the past temperature signal fades into model noise).
In a more realistic model, the coefficients, bi, vary with time. A Kalman filter (Ljung, 1995,1987; Brown and Hwang, 1977), is one approach to estimate the time-varying parameters. Kalman filter methods are a well-defined way to pick optimum coefficients; for details, see the above references. The filter recursively estimates how past values of air temperature and discharge should be weighted to produce an optimal estimate of discharge given errors in both the observations and the filter (the model). The Kalman filter method used here is a 1-day forecast (Ljung, 1995 and 1987).
In attempting to predict discharge beyond 1 day, the new estimates of the coefficients must be based on simulated (by equation (1)) rather than observed discharge. The Kalman filter gives daily estimates of discharge and time-varying parameters. The first step in prediction is to estimate the coefficients used to calculate discharge. This starting point was arbitrarily selected here as day 225. The coefficients for day 225 are used in the difference equation (1) with day 226 temperature to estimate day 226 discharge. This estimated discharge is then fed back into the Kalman filter to estimate a new set of coefficients, (and discharge) for day 226 which are again fed into equation (1). This procedure is repeated, increasing the forecast time in this example to 8 days.