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,J~ ~ ." ,:.t;J .. '~i ~::.~ '. 1180 ,..~:.....,...l;o;l:...... . JOURNAL OF APPLIED METEOROLOGY VOLUME 19 y I t . e 2.0 2.0 La~ 5 I Lare. half 1.5 All 103 CC'll ~. 7 1.0 An 103 CC'. I .a 1.& SMail half Sill. II half ~ o ,', o 10 100 1000 AREA PER GAGE (km9.) FIG. 3. Means j, standard deviations s( 10" m3). and coefficients of variation S"ly2. for rain swath accumulation versus lraingage spacing, which represent among-storm.s plus sampling vari- ances, also decreased for the same reason. The de- crease of y at large gage spacings is due to increas- ingly more CC's being missed entirely. From about 100-800 km2 per gage, y increases for the larger CC's (also reflected in combined 103 samples) be- cause the rainfall per gage changes little while the area per gage increases. The variances generally show an increase for wide gage spacings. The coef- ficient of variation, S2/y2, which is an index of re- quired sample size, shows a sharp increase for gage spacings greater than about 300 km2. For the smaller half of the CC' s this occurs before the 100 km2 point. 4. CC's required to detect a treatment effect The Wilcoxon (Mann-Whitney) rank-sum test (Noether, 1967) was applied to test for significance of treatment effect. Sample CC's were generated by randomly choosing a storm and then randomly seed- ing or not seeding it (1: I ratio). Each CC was then "returned" to the data base so it could be chosen, . at random, again. The rank-sum test was first applied after 15 samples were generated in this fashion, and then in increments of five samples until the required one-tailed a-level was reached. If more than 350 samples were needed, the process was halted to con- I serve computer time. The convergence of four simulated experiments . to an a-level of 0.05 (= 1.645 standard-normal deviates of Wilcoxon rank-sum test statistic) is ; . 2.5 2.5 o 0/0 100 1000 " shown in Fig. 4. Rainfall accumulations summed from all range bins were used in this figure. An as- sumed treatment effect of i5 = 25% was added to the total rainfall volume of each randomly "seeded" CC. This figure exemplifies the hazards of detecting a seeding treatment with a large noise level. In one case only 25 storms were required, whereas one reGJ..1ired over 350. If two of these experiments were terminated early, the conclusion would be a nega- tive treatment effect! This process of achieving a specified a-level was repeated' 100 times for each assigned value of 0 to give a frequency distribution of the number of CC' s required. This was used to estima.te {3( = 1 - power). The number of storms required to reach a {3-level was the value separating I OO( 1- {3)% of the totals from the remainder of the 100 n~plications. Figs. 5 and 6 iIIustrate the number of CC's re- quired to satisfy various a, {3 and I) constraints. In several examples, more than the maximum allow- able 350 storms were needed. Fig. 5 presents exam- ples for all 103 cases for a = 0.05 and 0 = 100%. It can be seen that increasing th(~ area per gage also !increases the number of CC's required over most of the range <:onsidered. However, the increase in required CC::'s is rather gradual until about 300 km2 per gage is exceeded. For Fig. 6 the 103 CC's were, stratified into two halves according to total rainfall volume. As might iQe anticipated, gage spacing is more important with lthe smaller storms, becoming critical for values q'r 7 0.5 ,', 010 , tOO .,..-- 11000 "'. '''- _ .".'1 ~. r-~"~~''':;;,~ ,,:~?:~~~~-~'~~-~.~~~~~~~'~-'" -~~J~::~0~'~.'..J~:~?;.~"?:::'~"~'~ !