There are three questions that are useful in understanding the differences between the three studies: 1) What is the effect of water year on the results? 2) Is precipitation the appropriate independent variable for the least squares correlation? 3) Are the differences between the results statistically significant? Each of these questions can be addressed by analysis of the data set for the 1962-1988 time period. Figure 2 shows the average daily precipitation for all years with complete data The average water level was calculated by setting the water level at zero on September 30 and thus is at an arbitrary datum. Phillips(1968), Simpson (1970) and Redmond (1990) used Oct 1 – Sep 30 as their water year. The storage precipitation gage is measured on July 1 for a water year of July 1 to June 30. Because these data are needed to improve the measure of precipitation, the significance of changing the water year on the water balance must be determined. Only a small amount of precipitation occurs between July 1 and Sep 30 (Figure 2), and there should be little difference in precipitation for water years starting on July 1 or October 1. The water level on July 1 is nearly the highest level for the year while that on Oct 1 is nearly the lowest level for the year (Figure 2). Thus the July- I start date tends to emphasize the level aftermost of the precipitation for the year has taken place, and the October-i start date emphasizes the level after the level has dropped from the summer evaporation period. Most of the snow melt is completed by July 1 according to the data in Figure 4 of Redmond (1990), and the major inflow should be about the same for the two starting dates.
Water balances performed for the two starting dates show that the difference is not statistically significant. Figure 3 shows the yearly precipitation (top) and water-level data (bottom) for the Oct 1 – Sep 30 water year for the period 1961-1988 used by Redmond(1990). Although there are quite large excursions of water level during this period, there is no systematic evolution of water level. Figure 4 reproduces the correlation of Redmond(1990). Years with high precipitation appear to have an especially variable relationship between change in water level and precipitation. Results for the correlation in Figure 4 as well as for the July 1 – June 30 water year are given in Table 2. The slope B that I calculate for the same data set used by Redmond (1990) is slightly different (Tables 1 and 2). The long-term average precipitation that keeps the lake level unchanged is identical for the two starting dates for water years. The differences for the parameter values are not statistically significant (Table 2) although the range is similar to that between Phillips and Redmond’s values (Table 1). Thus it appears that these data without some additional constraint cannot distinguish between these results.
Even though the values for B and B po are different for the two starting dates for water years, the value of p, is identical. This suggests that nearly all of the uncertainty is inB, and one should consider using equation (5) as a basis for correlation. Another reason for using water-level change as the independent variable is that normal least squares minimizes the difference between the y-variable and the least-squares line in order to establish the best slope and intercept. Although precipitation is the source of water-level change, the variable with uncertainty is precipitation not water level. Water levels are
*The convention for water year is that the period Oct 1, 1987 to Sep 30, 1988 is known as the 1988 water year.
known to 0.3 cm which is generally a small fraction of the change in water level. Although precipitation is measured to a similar number of significant figures, the gage at Park Headquarters is being used to represent precipitation over a 68 km2 water shed. This is necessarily a noisy measurement. Table 2 gives the results for redoing the correlations with change in water level as the independent variable, and Figure 5 shows the correlation for the Oct 1 – Sep 30 water year. The standard error for po is 2 cm (Table 2), and this confirms that it is a well-determined quantity. The fractional uncertainties in 8 and 1/B are similar for both choices of independent variable, and there is no improvement in the uncertainty of this quantity by changing the independent variable. The values of B are higher than the previous correlation resulting in increased total water supply estimates. Differences between values for the two starting dates for water years are far from statistical significance.
