EXTENDED-RANGE WEATHER FORECASTING* 
By FRANZ BAUR 
University of Frankfurt am Main 
SCIENTIFIC BASES OF EXTENDED-RANGE 
WEATHER FORECASTING 
The Basic Problems of Macrometeorology. The First 
Basic Problem. Macrometeorology forms the scientific 
basis for extended-range forecasting. The first basic 
problem of this new branch of meteorology is whether 
or not a real Grosswetter exists at all. In other words, 
can the observed longer periods of persistent cold or 
warm, dry or wet weather be attributed to some major 
variable influences, or are they merely the consequence 
of the so-called ‘persistence tendency” in connection 
with random developments and the annual variation of 
meteorological elements? Among the older school of 
meteorologists, there are, at least in Europe, those who 
are convinced that Grosswetter takes its course accord- 
ing to the principle: “small causes, large effects.” They 
try, for instance, to explain the development of a severe 
winter by the fact that clearing takes place at the be- 
ginning of the winter after the first widespread snow- 
storm, thus causing the temperature to drop consider- 
ably. Hence the cold air is “maintained” so that with 
the next upgliding of warm air, a new snowfall will occur 
and thus the wintry cold will gradually be amplified. 
Their reasoning is as follows: If it had “accidentally” 
been just two degrees warmer on that first day of winter, 
there would have been rain instead of snow. Thus the 
radiative heat loss during the following night would 
not have been so great. Less cold air would have 
formed and, finally, the winter would have been less 
severe. However, the following reasoning can likewise 
be applied to this case: Durmg every winter in the 
temperate zone, there will be one case of clearing 
after a snowstorm. Whether the winter followimg a 
snowfall with subsequent clearing will, in the long run, 
turn out to be severe or mild depends on whether the 
probability of the occurrence of cold days is larger or 
smaller than normal as a consequence of governing 
influences. If there should be a severe cold wave, there 
are nevertheless sufficient possibilities for the cold to 
recede after some time, when there exist no governing 
influences that would cause a tendency toward new 
cold outbreaks and thus a greater probability of cold 
days. 
The winter of 1876-77 in central Europe is an ex- 
cellent example, showing that even after very severe 
winter days mild weather can recur even before the 
end of the winter season. The month of December 1876 
started with especially mild weather; the departure of 
the temperature from a well-established average was 
+4.9C for the first half of December in Berlin. Then 
cooling, snowfall, and clearing occurred, and the daily 
*Translated from the original German. 
mean temperature for December 24 dropped to —15.8C 
(corresponding to a departure from normal of —16C), 
and it remained below —10C until December 27. How- 
ever, the daily mean temperature of December 30 was 
again 6.4C above normal, the month of January was 
3.5C above normal, and the month of February was 
2.2C above normal. 
Such isolated examples, however, are not sufficient 
to enable one to decide whether Grosswetter, in the sense 
mentioned above, exists in reality. Rather, it is neces- 
sary to establish statistically from extensive observa- 
tional data whether or not the fundamental probabil- 
ities of the occurrence of the meteorological phenomena 
that determine weather are (aside from the annual 
trend) constant or subject to variations. 
Lexis’ theorem [36] serves as the method of finding 
fluctuations of the fundamental probability. If the 
probability p of the occurrence of a phenomenon during 
an arbitrary number n of test series (each consisting 
of s independent trials) remains constant, the resulting 
distribution of the frequencies mi, m2,...m, of the 
occurrence of the phenomenon in s trials is a so-called 
“Bernoulli distribution,’ whose standard deviation 1s 
on = Vsp(1 — p). If, however, the probability of the 
occurrence of the phenomenon remains constant from 
test to test within a series, but varies from series to 
series, then the observed standard deviation o;, is larger 
than oz. In this case, the quotient Q: = o;/co2, known 
as Lexis’ ratio, is greater than unity. If, however, 
the probability p varies from test to test, whereas po = 
(p1 + po +...+ p;)/s remains constant from series 
to series, then Q; < 1. Lexis’ ratio, therefore, is a 
criterion for determining whether or not the funda- 
mental probability is subject to variations. 
In its original form, however, Lexis’ theory cannot 
be applied to meteorological observations, since in these 
—owing to their persistence tendency—the condition 
of the, independence of consecutive terms is not ful- 
filled. Baur [12], therefore, introduced a new criterion, 
the “divergence coefficient of the second kind” which 
is defined by the quotient D, = o;/o%, where Markoft’s 
standard deviation cy is used instead of Bernoulli’s. 
By cy we mean the standard deviation of the dis- 
tribution w,(x), where w,(x) designates the probability 
with which the phenomenon having the (constant) fun- 
damental probability p will occur x times among the 
first m observations of a Markoff alternating chain 
(i.e., an alternating series where each term depends 
on the preceding term). If the fundamental probability 
remains unchanged from series to series and from test 
to test, the mathematical expectation of D, again 
equals unity. If, however, the fundamental probability 
varies from series to series, but remains unchanged 
from test to test, then D, > 1. 
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