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<br />1476
<br />
<br />JOURNiL OF APPLIED METEOROLOGY
<br />
<br />VOLUME 20
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<br />TABLE 3, Sample size, N,v, due to sampling variance alone re-
<br />quired to detect a 25% change in mean precipitation amount due
<br />to seeding.
<br />
<br /> GPR
<br />G (A,o) 100 50 10 5 2 0,5
<br />2,25 (10) 18 70 389 1,270 3,368
<br />1.40 (15) 10 39 226 769 2,172
<br />0,90 (20) 7 28 160 550 1,617
<br />0,60 (25) 6 23 126 438 1,327
<br />
<br />Z is the normal standard deviate, a is the probability
<br />of a type I error, f3 is the probability of a type II
<br />error, s is the standard deviation (of the log-trans-
<br />formed variable for a log-normal distribution) of the
<br />nonseeded sample, x is the mean of the nonseeded.
<br />sample, and d is the fractional difference in means
<br />that it is desired to detect. We first calculate the
<br />sample size requirements due to network sampling
<br />variance alone, Nsv, assuming that all the storms are
<br />identical but pass over the gage networks in a n;mdom
<br />manner. Each combination of G and GPR that is
<br />selected yields a value of s/x from Eq. (2) which is
<br />then inserted into Eq. (3) to yield the required Nsv.
<br />A one-sided test with a = 0.05 (or two-sided test
<br />with a = 0.10) and f3 = 0.10 were assumed. Table
<br />3 presents the sample size, Nsv, as a function of GPR
<br />and G, required to detect a 25% change in precipi-
<br />tation means.
<br />It can be seen from Table 3 that the sample size
<br />increases as the gage density (GPR) decreases and
<br />the precipitation gradient increases (increasing G or
<br />decreasing Aso). For reasonable gage densities and
<br />nonuniform precipitation gradients, the sample size
<br />due to network sampling variance alone is apprecia-
<br />ble. Convective rain showers of short duration like
<br />those reported for Florida, Arizona, and Montana
<br />(see Table 3) appear to be most problematical be-
<br />cause they are relatively small in area, implying a
<br />high gage density for a given value of GPR, and they
<br />are characterized by high spatial precipitation gra-
<br />dients. The convective rains in the Midwest appear
<br />to be associated with larger scale systems that have
<br />more uniform precipitation gradients. This type of
<br />
<br />weather system poses less of a problem due to sam-
<br />pling variance.
<br />We shall now place the sample sizes due to net-
<br />work s~mpling variance in perspective by comparing
<br />them to the sample sizes required as a result of nat-
<br />ural storm variability. To do this we consider the
<br />statistical characteristics of total precipitation from
<br />112 convective storms that were observed during the
<br />1976 and 1977 Montana HIPLEX program by dig-
<br />itized 5.4 cm radar (Schroeder and K1azura, 1978),
<br />The rainfall for each convective storm was estimated
<br />from the radar reflectivity factor measured at the 10
<br />elevation scan every 5 min and accumulated in each
<br />0.5 km by 10 radar bin over the lifetime of the storm
<br />to obtain the total precipitation footprint. The Mar-
<br />shall-Palmer (1948) Z-R relationship, Z = 200 R1.6,
<br />was used to determine the radar-derived rainfall pat-
<br />terns with a 25 dBZ threshold being applied to ac-
<br />count for evaporation in the dry, subcloud layer of
<br />the Montana environment (Hildebrand et ai" 1979).
<br />These isohyetal maps were then analyzed to deter-
<br />mine storm sizes, rain volumes, durations, raincell
<br />composition, and the sizes and rain volumes of the
<br />raincells. The statistical characteristics of these
<br />storm precipitation parameters, which were ex-
<br />tremely well~fitted to a log-normal distribution, are
<br />shown in Table 4. The sample sizes Nnv, required to
<br />detect a 25% change in mean storm precipitation due
<br />to seeding when only natural variability is considered
<br />was calculated using the log-normal version of Eq.
<br />(3). For a = 0.05 and f3, = 0.10, it was found that
<br />Nnv = 2,237, Table 5 gives the percentage of the total
<br />sample size (resulting from network sampling vari-
<br />ance plus natural variability) that is due to the sam-
<br />pling variance contribution by various gage densities,
<br />that is 100 Nsv/(Nsv + Nnv)' The median area of all
<br />raincells of the Montana storms is 90 km2. However,
<br />the median area of the primary (largest contributor
<br />to the total rain volume) rain cells is 160 km2. The
<br />primary raincells contribute an average of 88% of
<br />the storm's total rainfall.
<br />Viewed in this manner it is clear that the natural
<br />variability of convective precipitation is mainly re-
<br />sponsible for the large sample size requirements in
<br />evaluating the hypothetical precipitation augmen-
<br />
<br />TABLE 4. Statistical characteristics of Montana convective storm precipitation,
<br />
<br /> Log-normal
<br /> Standard Log-normal standard
<br />Precipitation parameter Mean deviation Median mean deviation
<br />Storm volume (m3) 4,18 X 10' 10,32 X 10' 0,86 X 10' 11.40 1.80
<br />Storm area (km2) 444.2 738.1 167.0 5,18 1.40
<br />Storm duration (min) 117,3 66.4 100,0 4,62 0,54
<br />Number of raincells per storm 2.03 1.98 1.0
<br />Raincell volume (m3) 2.11 X 10' 5.24 X 10' 0,397 X 10' 10.75 1.73
<br />Raineell area (km2) 223,3 350,7 90,0 4.57 1.23
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