Frequency Distributions of Precipitation 
Oskar ESSENWANGER 
National Weather Record Center, Asheville, N. C. 
Abstract—Considerable difficulty is involved in the physical interpretation of the 
frequency distribution of precipitation. Such distributions usually do not follow the 
law of a gaussian distribution in a linear scale. This brings up the question for trans- 
formation to normality. The problem is rendered more difficult as the usually ob- 
served data have to be considered as truncated on the dry side. 
The author suggests the use of a logarithmic scale. A frequency distribution of 
annual precipitation generally consists of one collective, while in monthly values and 
shorter amounts the mixture of the rainfall processes becomes obvious. A sample for 
the frequency distribution of daily amounts at Asheville, N.C., is discussed. The col- 
lective of excessive daily rain in autumn could be explained in connection with hurri- 
canes on the east coast of the United States and the movement of extratropical 
cyclones through North Carolina. A two-dimensional analysis of intensity and dura- 
tion for single rainfalls in Braunlage (Germany) is also discussed. The sample demon- 
strates that it is necessary in order to read more details from the precipitation amount 
either to spht the rainfall data into intensity groups (such as convective and ad- 
vective types) or to keep the observational interval as short as possible. Hourly data 
may be sufficient. 
Introduction—The physical interpretation of 
the observed data is one of the primary topics of 
climatological work. If it were possible by some 
method to determine all physical processes of 
rainfall a priori, and collect the data into sepa- 
rate and distinct groups, then probably most of 
the following discussion would not be necessary. 
However, the basic physical processes leading to 
precipitation have not been completely investi- 
gated as yet and in many cases the set of ob- 
served data available consists of a complex of 
physical events which we call weather or climate. 
This mixture exists also for other meteorologi- 
cal elements, for which interpretation may be 
easy. Considerable difficulty is involved, how- 
ever, with frequency distributions of precipita- 
tion. In a linear seale they usually do not follow 
the law of a gaussian distribution and represent 
a mixture of different processes of rainfall, at 
least in the form of hourly, daily, monthly, ete., 
amounts. 
The kind of distribution we may expect from 
daily and shorter periods of observations is well 
known. It shows a maximum in the class of 
smallest precipitation amount and decreases to- 
ward higher amounts. A typical sample is illus- 
trated by Schneider-Carius [1954], who smoothed 
the frequency distribution in taking four German 
stations (Berlin, Bremen, Karlsruhe, and Mu- 
271 
nich) and all months of the year together (100,- 
000 values). 
This summation merely serves to demonstrate 
the type of frequency distribution or to derive 
an average rainfall probability for an area, 
though its applicability for climatological de- 
tails is very limited. 
Data in linear scale—Using a linear seale for 
frequency distributions of precipitation, several 
statistical approaches may be made. The best 
fit, to date, under the limitations discussed be- 
low, may be obtained by applying an incomplete 
gamma function though the possibility also exists 
to employ a hyperbola, e* function (= also the 
limiting form of incomplete T function) or the 
negative binomial. 
The hyperbola or a form e-* may solve prob- 
lems of rainfall probability, but furnishes practi- 
cally no physical result. Wanner [1939] tried to 
use the negative binomial series for curve fitting 
in recognition of the persistence involved in me- 
teorological elements. However, this applies a 
discrete function to continuous data and the au- 
thor [Hssenwanger, 1956a] has previously dis- 
cussed the difficulties connected in arranging pre- 
cipitation data in discrete form to meet the 
requirement of the negative binomial distribu- 
tion. Besides this, the persistence for rainfall 
data is not a constant. This may be proved for 
