272 
TaBLeE 1—Persistence factor f for various amounts 
of precipitation, Hamburg area, 
Germany, winter 
Limits Dry |Traces| 0.01 vats Lene i ze 
inch 
Dry 1.46 0.89) 0.79) 0.68) 0.59) 0.44) 0.33 
Traces to | 0.71} 2.09| 1.77} 1.31] 1.21} 0.94] 0.73 
0.01 
0.02-0.03 | 0.72] 1.17] 1.27] 1.28] 1.21] 1.32] 0.58 
0.04-0.09 | 0.58] 0.81) 1.24) 1.33) 1.40) 1.53) 1.48 
0.10-0.40 | 0.55] 0.72) 0.91] 1.06] 1.45) 1.72) 2.19 
>0.40 | 0.45] 0.68} 0.26) 0.71] 1.36) 2.16] 3.81 
daily precipitations. Table 1 shows a persistence 
factor f, which is the observed frequency of oc- 
currence divided by the expected value without 
persistence (discussed in detail by the author 
[Essenwanger, 1956a]) . 
This factor is listed for several classes of daily 
amounts and it is easy to see that this is not a 
constant for the different rainfall groups. Thus, 
departures between theory and observation may 
be expected from this result, even when it would 
be possible to establish an adequate discrete 
scale. The parameter of persistence d, which can 
be derived from this negative binomial theory 
may perhaps lead to some climatological rela- 
tions, although in general, the result is influenced 
by the discrepancy mentioned above. 
This leaves the incomplete T function, which 
has been successfully applied by Barger and 
Thom [1949; Thom 1958, 1957ab]. This fune- 
tion renders good fits for amounts of approxi- 
mately five days and more [Barger, 1957] de- 
pending on the proportion of near zero rainfall 
days, while for shorter periods the application 
is restricted to special collectives like the storm 
series [Thom, 1957ab]. Therefore, the question 
is still open: How should we treat precipitation 
data for periods shorter than five days? Before 
giving some answers to this problem, a few facts 
about transformation of scales for precipitation 
data should be discussed. 
Transformation of scales—The author dis- 
tinguishes between two sorts of transformation. 
In the first case one applies a pure formal math- 
ematical function, developed to reduce the data 
into the desired form, such as a gaussian dis- 
tribution. In the second case we derive the trans- 
formation function from the physical back- 
ground. Either may be justified for particular 
purposes; the second one should be considered 
OSKAR ESSENWANGER 
at least, as not every physical law renders a lin- 
ear relationship. 
By formal transformation any curve can be 
reduced into a normal distribution. For example, 
when the data follow an incomplete gamma 
function, one may transform them as is illus- 
trated by the lower curve of Figure 1. The ordi- 
nate is the probability scale of a normal dis- 
tribution, and the accumulated frequency has 
to be a straight line when it follows the gaussian 
law. The lower curve in Figure 1 shows the 
storms for Santa Barbara in February 1931-1950, 
[Thom, 1957a] transformed from an incomplete 
gamma function. The data follow a fairly straight 
line when we neglect about five and ten per cent 
at the upper and lower ends, respectively. This 
seems permissible. 
The upper curve is constructed for observed 
daily precipitation sums at Asheville in Septem- 
ber (1907-1956) in a linear scale. Suppose we 
are interested in computing the transformation 
function which reduces the curved line to a 
straight line. Taking the smallest possible class 
interval (0.01 inch) the first interval includes 
12% of the total observations and for the range 
from 0.01 inch to 3.20 inches we would have to 
take more than 300 class intervals into account. 
By this method we would have to assume that 
12% follow the same law which we derive for 
the other S8% of the observations (0.01 inch is 
the least measurable amount recorded). If we 
would take 0.1 inch as the first class, in order to 
save the laborious work and reduce the values 
to 30 class intervals, then we do not consider 
more than 40%, which is remarkably high. In 
practice it has been found that those small 
amounts do not fit well in the theoretical curve 
derived for the larger amount. Further discus- 
sion follows in connection with Figure 2. 
As the frequency in those lower classes in- 
creases, the shorter the time period of measure- 
ment becomes, such as hourly totals, it is natu- 
ral that the formal transformation is not strictly 
valid for the whole curve. This may be demon- 
strated by Figure 2. It represents the frequency 
distribution of daily precipitation December- 
February at Munich for a cubic scale. The theo- 
retical normal distribution and the observed 
values agree fairly for sums =>1.0 mm (= 0.04 
inch). However, the part less than 1.0 mm is 
drastically different. This is the part in the 
small precipitation amounts. In the accumulated 
frequency the part in the hatched area is com- 
pensated at the 40th percentile. 
