The first matching scheme assumes that the value for B determined from the yearly water balance should hold through the year, and the average absolute deviation is minimized between water supply calculated from equations (9) and (10) in order to determine a. Based on the comparison of Park Headquarters and North Rim precipitation made above, it is clear that there is significant variability in the precipitation measured at Park Headquarters that is unrelated to the water balance. In order to make a consistent calculation for finding a, daily precipitation is adjusted so that the total precipitation for the year is what is calculated from equation (5) with B = 1.325 and po = 169.2 cm/y. This adjustment factor Ba is introduced into equation (10) when calculating the water supply from precipitation data. The maximum adjustment is 5 %. This assumption is relaxed below.
Figure 13 shows the two calculated water supplies for the medium precipitation case for a =1.0, and Figure 14 shows the water supplies for a such that the average absolute deviation of the two calculations is minimized. As one would expect, the calculation for a =1.0 in Figure 13 shows a substantial excess in water supply calculated from precipitation that grows in the winter, becomes approximately constant during the late spring, and decreases to zero during the summer. The results in Figure 14 for a = 0.84 show that the two calculations for water supply match reasonably well throughout the year. The average absolute deviation has been reduced from 7.6 cm to 1.7 cm (Table 5). The simple model of storage and then uniform release explains the first order variation in water supply. Although one would like to improve the model, the use of a constant adjustment factor for precipitation makes it difficult to assess relative contributions of additional details. The next step would be to include nonconstant release of the snow melt and nonconstant evaporation. Since these two effects overlap for a significant part of the year, defining which effect is dominant would require more consistent daily-precipitation data than are available. There is a suggestion in the difference plot in Figure 14 that the rate of snow melt may be higher in the the late spring than the early summer and that the rate of evaporation may be higher in summer (larger B in equation (9)) than winter.
Figures 15, 16, and 17 show the results for low, high, and average precipitation. The average absolute deviations are substantially reduced by the model from values calculated using a =1.0 (Table 5). The values of a obtained range from 0.81 to 0.90. The quantity B a is a measure of the fraction of precipitation falling directly on the lake compared to that measured at Park Headquarters. The lake appears to receive about 10%more precipitation than-that at Park Headquarters, though the range is from 7% to 19% (Table 5). Phillips (1968, p. E15) estimates that precipitation on the lake is 7% greater than at Park Headquarters. The appearance of the difference curves in Figures 15, 16, and 17 tends to confirm the notion that the rate of snow melt may be higher in the the late spring than the early summer and that the of evaporation may be higher in summer than winter, but uncertainties in precipitation amounts make this a weak conclusion. The matches with adjusted precipitation are compatible with the yearly water balance but do not lend any support to those values because they are used in the adjustment process.
Another method for matching the two calculations of water supply is to search parameter space for values of B and a that minimize the average absolute deviation between the curves. Table 6 gives the results of such a search for the four cases. The search was conducted to the number of significant figures given for the two parameters, and no adjustment was made to the precipitation data. Values of a only range from 0.81 to 0.86.Values of B range from 1.2 to 1.5 and span the entire range of reasonable values based on the yearly water supply. Average absolute deviations (Table 6) are similar for each case to the values given in Table 5. Although daily precipitation and water-level data are numerous, uncertainty in the precipitation data limit the ability of the data to determine the best value of B. Visual comparisons provide no obvious rationale for choosing one match versus another. For example, Figure 18 shows the calculated water supplies for unadjusted precipitation for the high-precipitation case. Comparing Figures 16 and 18, neither is visually superior to the other in modeling the calculated water supply. Thus both the yearly and daily water balances are unable to fix B with precipitation data only from Park Headquarters. The values for the average-precipitation case are quite similar to the values for the adjusted precipitation calculation (Tables 5 and 6). This agreement can be interpreted as showing that one needs the larger number of years in the average precipitation case to get a good determination of B and a. Alternatively, one could contend that that the agreement is fortuitous, because the adjustment factor Ba is so close to one.
