Based on the close determination of po, it is useful to show the effect of B on calculated water level. The water level zi+l in year i+1 is calculated from the water level zi and precipitation pi in year i from the relation:
where precipitation for water year 1962 (October 1, 1961 to September 30, 1962) is used to predict the water level on October 1, 1962. Figure 6 shows calculated water-level histories for values of B of 1.2 and 1.5 spanning the likely range from the results in Table 2. Although one curve agrees with measured water levels better in some years, the other curve agrees better in other years. Both model calculations have substantial mismatches in the periods 1972-77 and 1984-85. The available range of B does not seem to be able to adequately model the actual variation in lake level. The basic determinant of the match is the average precipitation po rather than the slope B.
In order to improve the prediction of lake level it is necessary to develop a more representative measure of precipitation. Fortunately, yearly precipitation measurements made at the North Rim make that possible. It has long been known that there is a substantial gradient of precipitation across Crater Lake, and adding a second data set makes it possible to more adequately sample this variation. It is also possible that some storms are not regionally extensive and will preferentially effect one site over the other. If there is experimental variability in measurements from one precipitation gage, adding a second gage provides the opportunity to better sample the true value in a given year. Figure 7 shows the North Rim measurement plotted against the Park Headquarters value for the period 1964-88. The two measures are closely correlated, and the slope relating the two is 0.718. The average precipitation for Park Headquarters during this period is 168.6 cm, and the predicted precipitation for Park Headquarters is 168.5 cm using the best slope and the average for the North Rim of 121.0 cm. Thus it seems reasonable to use the precipitation from the North Rim gage times 1.392 = 1/0.718 as another measure of precipitation on the lake. The scatter of the two data sets in Figure 7 indicates that a single measure of precipitation represents precipitation over the entire lake with an uncertainty of about± 10 cm.
Figure 8 shows both data sets and Table 3 gives the correlation parameters. The North Rim gage was not installed until the 1964 water year, and two points on Figure 8 have only Headquarters data. Because there are two measures of precipitation, change in water level is used as the independent variable. The standard errors are somewhat smaller with the combined data set, but the results are not significantly different from using Headquarters data alone (Table 3). However, when the average of the two measures is used to model lake level, the improved correlation is striking (compare Figures 6 and 9).The average absolute deviation of the modeled water level from the measured water level is reduced from 25 cm for Headquarters data alone to 13 cm for the combined data set. Although the correlation of precipitation with change in water level is not notably improved, the addition of a second measure of precipitation has dramatically improved the modeled water-level variation.
