STATISTICAL INVESTIGATION OF ANOMALIES
IN THE TEMPERATURE RECORD OF BOSTON, MASSACHUSETTS
by
ELLIOT NEWMAN
B.S., The Pennsylvania State University
(1963)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF
SCIENCE
at the
MASSACHUSETTS INSTITUTE OF
4
TECHNOLOGY
February, 1965
Signature of Author ............
Department of Meteorology, November 15, 1964
Certified by.................
Thesis Supervisor
Accepted by
.......
Chairma
............
...
Depa~tmental Committee on Graduate Students
STATISTICAL INVESTIGATION OF ANOMALIES
IN THE TEMPERATURE RECORD OF BOSTON, MASSACHUSETTS
by
Elliot Newman
Submitted to the Department of Meteorology on November 15, 1964
in partial fulfillment of the requirements
for the degree of Master of Science
ABSTRACT
Normal maximum and minimum temperatures are computed for each date
of the winter season from the 92-year Boston temperature record. Trend
is removed from the normal and individual yearly values and standard
deviations and errors are computed. Statistical significance levels are
calculated and a count of significant values is made. The probability
of obtaining these by chance is computed and found to be quite high.
Average maximum and minimum values are computed for each date for each
phase of the double sunspot cycle in addition to a major-half phase and
minor-half phase. A procedure similar to that mentioned earlier is
followed and four phases out of 20 are found to have 1% significance
counts with a high degree of confidence. The standard deviations of
both maximum and minimum temperatures for the eight phases exhibit cycles
that are parallel to the double sunspot cycle. The result of applying
a statistical "F" test indicates that the maximum and minimum cycles
are real. An apparent division of the winter season is examined in which
anomalous values tend to occur in the early part of the winter during
the minor-half phases and in the latter part of the winter during the
major-half phases.
Thesis Supervisor: Hurd C. Willet
Title:
Professor of Meteorology
ACKNOWLEDGMENT
I would like to thank Professor Hurd C. Willett for his scholarly
aid and direction of this study.
Without his enthusiastic support,
this study would not have progressed so satisfactorily.
I would also
like to thank Dr. Edward M. Brooks and Mr. John Prohaska for their
critical comments.
The aid of M.I.T.'s Weather Bureau Research Project's
staff is greatly appreciated.
The assistance of the Climatological
Office of the U. S. Weather Bureau, Boston, in obtaining the data was
invaluable.
Mrs. Laura B. Shafran skillfully drafted all figures and
Mrs. Theresa B. Wajda most adeptly typed this text. All computational
work was done at the M.I.T. Computation Center.
Finally, I would like
to express gratitude to my wife, Nancy, for her help in data handling,
typing the drafts of this text, and for her wonderful composure during
the entire period of this study.
iii
TABLE OF CONTENTS
Section
Title
INTRODUCTION
A.
B.
Historical Background
Object
INVESTIGATION OF NORMAL DATA
III
INVESTIGATION OF RELATIONSHIP TO
DOUBLE SUNSPOT CYCLE
CONCLUSIONS
REFERENCES
Page
LIST OF TABLES
Page
Title
Table
Values of
A , o,
and
A
14
TN
II
Values of "t" and corresponding levels of
significance.
17
III
Significance counts.
19
IV
Values of "T" and significance levels for
various counts.
19
V
Significance levels (%) of selected dates
for normal maximum and minimum temperatures.
21
VI
Classification of years 1872-1964 by phases
of Double Sunspot Cycle.
27
VII
Standard deviations and standard errors of
maximum and minimum temperatures for phases
of Double Sunspot Cycle.
48
VIII
Significance counts.
51
IX
Values of "T" and significance levels for
various counts.
52
X
Significance levels (%) of selected dates
for phases of Double Sunspot Cycle.
53
XI
Average absolute value of significance levels
(%) of selected dates for phases of Double
Sunspot Cycle.
56
XII
Average reciprocal of significance levels (%)
of selected dates for phases of Double Sunspot Cycle.
58
LIST OF FIGURES
Figure
Caption
Page
1
Example of a temperature record.
9
2
Example of a temperature record and associated trend.
9
3
Example of trend-free temperature record.
9
4
Average normal minimum temperature values
with trend, 5%, and 1% significance levels.
11
5
Average normal maximum temperature values
with trend, 5%, and 1% significance levels.
12
6
Schematic representation of idealized 20-24
year double sunspot cycle.
25
7
Average minimum temperature values for
major-half phase with trend, 5%, and 1%
significance levels.
28
8
Average minimum temperature values for
minor-half phase with trend, 5%, and 1%
significance levels.
29
9
Average minimum temperature values for N-MM
phase with trend, 5%, and 1% significance
levels.
30
10
Average minimum temperature values for MM
phase with trend, 5%, and 1% significance
levels.
31
11
Average minimum temperature values for
NM-NN phase with trend, 5%, and 1% significance levels.
32
12
Average minimum temperature values for NN
phase with trend, 5%, and 1% significance
levels.
33
13
Average minimum temperature values for NN-M
phase with trend, 5%, and 1% significance
levels.
34
LIST OF FIGURES (Continued)
Figure
Caption
Page
14
Average minimum temperature values for M
phase with trend, 5%, and 1% significance
levels.
35
15
Average minimum temperature values for M-N
phase with trend, 5%, and 1% significance
levels.
36
16
Average minimum temperature values for N
phase with trend, 5%, and 1% significance
levels.
37
17
Average maximum temperature values for
major-half phase with trend, 5%, and 1%
significance levels.
38
18
Average maximum temperature values for
minor-half phase with trend, 5%, and 1%
significance levels.
39
19
Average maximum temperature values for N-MM
phase with trend, 5%, and 1% significance
levels.
40
20
Average maximum temperature values for MM
phase with trend, 5%, and 1% significance
levels.
41
21
Average maximum temperature values for
MM-NN phase with trend, 5%, and 1% significance levels.
42
22
Average maximum temperature values for NN
phase with trend, 5%, and 1% significance
levels.
43
23
Average maximum temperature values for NN-M
phase with trend, 5%, and 1% significance
levels.
44
24
Average maximum temperature values for M
phase with trend, 5%, and 1% significance
levels.
45
vii
LIST OF FIGURES (Continued)
Figure
Caption
Page
25
Average maximum temperature values for M-N
phase with trend, 5%, and 1% significance
levels,
46
26
Average maximum temperature values for N
phase with trend, 5%, and 1% significance
levels.
47
viii
SECTION I
INTRODUCTION
A.
HISTORICAL BACKGROUND
An investigation of the folklore of the world will reveal many
references to the annual recurrence of weather phenomena characterized
by periods of unseasonable warmth
Thaw",
"January Frost",
or cold.
Ice Saints",
Names such as -
"Indian Summer",
"January
"Old Wives'
Summer" and "Christmas Thaw", to mention only a few, are common terms
throughout the world, although some may apply only to certain geographic
areas.
Most amateur forecasters and meteorologists will stand fast and
unambiguously maintain that these "singularities" are real and will
undoubtedly recur in the future; while among professional meteorologists,
the subject has always and is presently quite controversial.
At this point, it is necessary to define the terms, "1singularity"
and"anomaly", and to adopt these meanings throughout this text, although
it
may be necessary to supersede the definition when a direct quote is
used.
The term "anomaly" will be used to denote a value which appears
to be a peak or trough in the parameter's time profile, while the term
'singularity" will be used to denote an "anomaly" which can be shown to
be statistically significant.
A review of the literature shows that the concept of temperature
singularities dates as far back as 1860 when Forbes (1860) discussed
"Periodic Anomalies" in the 40 year temperature record of Edinburgh.
Talman (1919) published a comprehensive bibliography of work done on
this subject up to that date.
In the past half century, the majority
of contributions have come from the European scene (Bayer, 1956; Blagoev
and Lalovski, 1958; Brooks and Mirrlees, 1930; Kramer, Post, and
Scharringer, 1952; Lamb, 1950; McIntosh, 1953; Peczely, 1958; Reynolds,
1955; and Shinya, 1958).
In general, results have been quite contro-
versial with some agreement among authors and considerable disagreement.
The work in America has been chiefly confined to investigations of
the "January Thaw".
Since the origin of this "Thaw" is folklore, there
is no precise, universal definition of this phenomenon.
In New England,
the "January Thaw" is thought to be a warm spell occurring approximately
on 21-22 January.
There is another warm period during the first week
of January that has also been given this name in other parts of the
country.
In 1919, Marvin, who was then chief of the United States Weather
Bureau, made an examination of the annual temperature record of several
North American cities.
He concluded that:
"Each striking feature on
a long record is therefore no evidence of the persistent recurrence of
peculiar irregularities, but is simply a residual scar or imprint of
some unusual event or a few which have fortuitously combined at about
the time in question.
Time will inevitably efface these,...."
There is no doubt that these stern and quite conclusive results
seemed to close the case of "singularities" and hindered any further
2
undertakings on this subject.
Although research was hindered, it was
not entirely stopped by Marvin's conclusions.
In contrast, Slocum (1941) investigated the temperature records
in Washington, D. C.,and a few other selected stations, and concluded
that there was a definite "warm anomaly" during the period 20-23
January followed by a colder than normal period about 5 February.
In Marvin's (1919) paper, he stipulates that for a singularity
to exist, it must be supported by physical reasoning,
Further research
along these lines was performed by Wahl (1952) who studied the temperature record of Boston, Massachusetts and other selected stations in
addition to records of other parameters such as gradient wind direction,
pressure gradient, thundershower occurrence, and snow depth.
He
attempted to relate the January Thaw to a general circulation indexcycle pattern.
When he compared the normal sea-level pressure chart for 20
January with that of 27 January, he found a striking contrast.
On 20
January, the Bermuda High is stronger and further North than normal.
In addition, the normal trough along the east coast of the United States
is displaced far out to sea and another trough is located in the Midwest
creating a strong southwesterly flow in the eastern United States.
the 27th, the pattern has completely changed.
By
The trough that was in
the Midwest is now off the East Coast and a large anticyclone dominates
most of the United States.
Thus Wahl concludes that a low zonal index-
cycle pattern favors the anomalous warmth in January.
Another investigation seeking a link between a January Thaw and
the general circulation was performed by Duquet (1963).
Using 700 mb
and mean weekly surface temperature data of many stations, he concludes:
"The course of the winter ...
United States.
involves two stages in the northeastern
The stages are separated by a 'warm spell' which occurs
in early January.
The geographical extent of the warm spell indicates
that Gulf Coast cyclones tracking along the Appalachians are the vehicle
by which warm air is transported into the region.
the amplitude of the warm spell ...
Secular changes in
are related to simultaneous changes
in the planetary circulation and are not independent fluctuations."
Duquet also notes that at the time of the thaw the potential energy in
the atmosphere over North America is at a maximum due to the fact that
over the northwestern part of North America, the temperature is at a
minimum.
This would lead one to expect to find the thaw at stations in
the West at a different time than its occurrence in the East.
This is
confirmed by Wahl, who found a rise in temperature at Columbia, Missouri,
between 17 and 21 January.
Kangresses (1957) also found a possible
singularity in the minimum temperature at Phoenix, Arizona.
Rebman
(1954) and Schick (1944) found similar results in other western stations.
In each of these two cases, either normal or below normal temperatures
occurred during 21-22 January.
In another investigation, Lautzenheiser (1957) tabulated dates of
thaw periods during January in Boston in order to find a preferential
period of occurrence.
He concludes that:
"This failed to show any
particular date as being much favored over another,
fact that the period January 21-23 ...
any other three day period."
...
it is a curious
actually had fewer thaws than
He attributes the fact that more thaws
occur during the earlier part of January than the latter to the annual
march of temperature.
le further concludes that:
"one cannot pick out
a preferred date for a 'January Thaw' other than it should occur in
January.
They do not occur every year ...
(but) on the other hand,
over 50% of the January's had two or more thaws."
As can be seen by the brief discussions above of previous work on
this subject that conclusions are inconsistent or contradictory.
In
addition, due to the lack of universal definition of the January Thaw,
different time periods have been used for the occurrence of the thaw in
many of the investigations.
B.
OBJECT
The primary object of this study is to examine the winter-season
temperature records of Boston, Massachusetts, and to investigate the
existence of the January Thaw.
By comparison with other singularities
in the temperature record, the actual significance will be determined.
It is necessary to emphasize the use of the term significance rather
than reality.
It is almost impossible, when dealing with data such as
this, to prove that a singularity is real and exists, but it is possible
to show that the probability of the chance occurrence of these anomalous
points in the temperature record is very small.
It is not intended that a physical cause-and-effect relationship
should be formed between the "Thaw" and some other parameter, but
rather to determine when and how significant are any singularities
during winter.
It is also intended that a comparison between the
"January Thaw" with other anomalies be made and a determination of its
relationship to other time periods.
The possible relationship between
the eight phases of the double sunspot cycle and occurrence of anomalies
will also be investigated.
SECTION II
INVESTIGATION OF NORMAL DATA
The official Weather Bureau temperature record for Boston,
Massachusetts dates back to 1872.
Although this consists of data
taken at different locations, the errors introduced by different
locations may be considered negligible because in this study only
deviationsfrom the normals are used.
Although most previous studies
examined either the mean temperature record of a station or the maximum
temperature, it was decided to investigate both the maximum and minimum
temperature records and treat each as an independent entity.
The data
consist of the 92-yearly values for each date of the winter season,
Since 29 February occurs only
defined as 1 December to 28 February.
approximately every four years and would therefore comprise a small
sample, it was omitted from this study.
A 92-year arithmetic mean was computed for both maximum and
minimum temperatures for each date.
Let us call this the daily "normal",
and denote it by X.,
92
X.
=
z
n= 1
-
n
92
(i =
1,
2,
3,
...
90)
Since maximum and minimum temperatures are treated separately, the
notion may be applied to either.
(1)
Since the data contain a trend due to the annual march of
temperature, it was decided that all computations would be done on
data that are trend-free, i.e.,
actual data values minus the smoothed
daily normal which represents the trend.
If the trend were not removed
and deviations, A., were computed from the grand normal, X,
-
90
2
(2)
- X
(3)
X =
i=1
A.
1
= X.
1
unrealistic results would be obtained and would undoubtedly lead to
incorrect conclusions concerning singularities and their significance.
If a trend curve were used instead of the grand normal, the seasonal
influence would be removed and all deviations would be relative to a
straight-line mean.
Figures 1 through 3 illustrate this point.
As can be seen in
Figure 1, all deviations computed from point A to point C will have
positive values while those computed from C to D will be negative.
Figure 2 illustrates a hypothetical "normal" value curve and an associated trend curve.
Figure 3 shows the result of taking deviations with
respect to the trend.
The mean X
-
X in this figure equals zero.
is quite evident, there are positive and negative values present
throughout the entire time period.
The actual method used to determine the seasonal trend was
arbitrary.
Some authors (Duquet, 1963; McIntosh, 1953; and others)
As
LU
0
LJ
X
aLU
T IM E
Figure 1.
A
Example of a temperature record.
B
TREND
TIME
-j
Figure 2.
Example of a temperature record and associated trend.
PLUS
0-
X;-X
MINUS
TIME
-
Figure 3.
Example of trend-free temperature record.
9
used the first two harmonics to determine the periodic fluctuations
of their data.
But, in contrast to their data which covered 12-month
periods, these data consist of a record of only three months.
Therefore,
since harmonic analysis can only be used on data which contains at least
one cycle, another method would have to be used to determine an analytic
Panofsky (1958) suggests that when the trend
expression for the trend.
shows definite curvature, a parabola may be fitted to the data by the
method of least squares.
Figures 4 and 5 illustrate the 92-year normal curves and the
In the
second degree fitted polynomial, denoted in the term'trend".
mathematical process of curve fitting, the convention of numbering the
dates consecutively from 1 to 90 was adopted.
The equations of the
polynomials are as follows:
for maximum temperatures,
0.270D + 0.0025D 2
(4)
X = 29.6 - 0.318D + 0.0027D 2
(5)
X = 43.3 -
for minimum temperatures,
where
X =
Temperature
D =
Date
(1 December = 1, 28 February = 90).
The temperature value on the trend curve (Xti, i = 1, 2, 3,
...
subtracted from each of 92 data points for each date (xi..,
i = 1, 2,
3,
...
90,
j
= 1, 2,
...
92) and a standard deviation of these
90) was
3-~- N
__TA-+,L4
I
IN, -
i
W- X
A, '' k;, i IN
-ee
I --
311f
i-T
N
j
i-44a-4+++A-44±
XI-~
'don I I I I I I I 1 1 1
I
TTTT
_64
I
III
I
.
IIn!-
L_
-H44
+-1-i-+
A *iU~4
LUI I i I I
4111
,.-
.1-...,
M104
I-
a
IF
11
- 1-1
~1i1~4~
PPT
4HI1
L__444-XIf
-1-44-
,-i-~
i &E
'4I~
44 -
-44±~+HIA
ft
-
2~i
~
Ihtl4#tl1
2Z2
H+++t
++
-
-t
w--+
1.1Lj
F-I
-
5W
11~
i i i Iii i
-
44±jW1
~111
+-I
4
I
J iI iii i i i
+ -1144Ii i i i L
T
±#
-4+
Eh~CE~B~
t
_L
i4
,
- Iiij
Figure 4.
AKill
fillII!it
II
I
.
.............
Average normal minimum temperature values with trend, 5%, and 1% significance levels.
...
KEUrFEL & LSSLR CO.
III'
+r
221
u''n
Ii
ijf'
---
i'
-L1
714
~I
B.
itn,~InIt
* Ii
+/
;
rf1 ti
-TREiNI
7 r-
4-
t4 t I
-00-
41
%JT ,1
K
. V
I
-1
ii
t 1 119I
n
I
T4I
T
T
Ii
DEICEMBER-
4K;
25
~~~1~~~
Figure 5.
-
I-
-7
-FE.B UA-R~
T
I
- T'-
I
t1
-
-- T
T+
~3
Average normal maximum temperature values with trend, 5%, and 1% significance levels.
departures was computed according to the following:
90
92
2
-2'I
8280
(A)
(6)
i=l j=l
where
A. . = x.
(i = 1,2,3,...90; j
.- X.
=
1,2,3.. .92) (7)
tl
1,]
1,]
In addition, a standard deviation of the departures of the 92-year normal
Let us call this af:
values from the trend was computed.
2
90
i -2
-2f)
(8)
i= 1
where
(i = 1,2,3,...90)
f. = X. - X .
(9)
When working with a distribution of mean values of a sample,
statistical theory states that the standard deviation of these means, or
the standard error, as it is called, may be defined as:
A
IN
C
A
(10)
where N equals the number of A's in each A.
Thus the effect of averag-
ing random data is to reduce the standard deviation of the raw data by
a factor of
In terms of the terminology used in
.
-
this paper,
1N
crA
OA
(TA
A
O
Table I lists values of
N
*A' c0f, and the standard error,
from the above calculations.
(11)
a-_ , obtained
A
the values
that
note
to
is
interesting
It
13
Table I
Values of OA , cr
N
and
,N
r (deg
2)
0f(deg )
N
Maximum Temperatures
10.22
1.11
1.07
Minimum Temperatures
10.33
1.21
1.08
N = 92
(deg)
of a-f were slightly greater than the standard error.
This can be
interpreted to mean that the grouping of the data into date means is not
random, but there is an indication of a more than random tendency towards
formation of singularities.
The question now arises as to how significant are the ridges and
valleys in the normal temperature curves presented in Figures 4 and 5.
In order to answer this question, statistical significance tests must
be employed.
In general, these tests can only be applied to data which
consist of independent samples.
This infers that there should be a
low autocorrelation between points.
It is a well known fact that a
daily temperature record does not constitute an independent-data set.
The interdependence of daily temperature has been called "coherence"
by some authors (Walker, 1946).
similar problem.
McIntosh (1953) was faced with a
He made a comparison of the frequency distributions
of a non-coherent random series (very small autocorrelation), a coherent
random series and the theoretical normal Gaussian distribution.
The
number of points falling in intervals > 2 t, where:
t =
(12)
x = point value of data
a
= standard deviation of data
were quite similar in all instances.
McIntosh concludes that a coherent
series of random temperature departures can legitimately be assumed to
approximate the normal distribution.
A significance level represents the probability of a random number
taking on a certain value different than the mean.
Generally signifi-
cance levels are computed from the relation:
t
x
a
=
(13)
=
point value of data
=
mean of data
=
standard deviation of data
and the normal distribution function.
Table II presents various levels
of significance and their respective values of "t".
To illustrate
this point, take for example, the case of a point falling outside the
1% level.
This means that the ratio of its deviation from the mean
value to the standard deviation is greater than the value of "t" for
one percent.
It can be interpreted as saying that there is less than
one chance in one hundred that this value is really not significantly
different from the mean value and is only a random occurrence.
It is quite evident that the larger the value of "t", the greater
is the significance of any values found at that distance from the mean,
and the probability that the value has been obtained by a random
fluctuation is smaller.
In the case of the data being investigated,
the significance bands representing the various levels are parabolic
curves that are parallel to the trend curve.
These are presented in
Figures 4 and 5.
A significance count was made using the criterion that if two
dates which are significant on any level less than 5% are not three
16
Table II
Values of "t" and corresponding levels of significance.
Level of
Significance
(M)
t
1.0
2.576
2.5
2.041
5.0
1.960
10.0
1.650
20.0
1.282
50.0
0.674
days apart, only one would be included in the "count".
This is necessary
since the normal distribution assumes that the data are independent
and as was discussed before, temperature data are not.
Table III
presents the "counts" for the normal maximum and minimum curves.
The significance of the counts obtained may also be found in order
to determine if "k" values obtained at a particular level are different
from what one could expect by random chance.
by (14),
In this case, "T" as defined
corresponds to a significance level similar to "t" used above.
T
k - E
0(15)
E = Np
(16)
-dNpq
where
k = number of values obtained at a particular level
N = total number of points
p = probability of an event occurring
q = probability of an event not occurring = (1-p)
E = probable number of values obtained at a particular level.
Table IV presents values of "T" for the values obtained in Table
III and their corresponding levels of significance.
As one can notice immediately, the probability of obtaining these
counts from a random sample by chance is quite large.
Therefore,
nothing conclusive can be said about the occurrence of singularities in
the 92-year normal maximum and minimum temperature record.
Table III
SIGNIFICANCE COUNTS
2 %
5%
1%
Maximum Temperatures
5
3
1
Minimum Temperatures
5
3
2
Table IV
Values of "T" and significance levels for various counts.
Significance
Level of Data
(M)
Significance
Count
N
T
Significance
Level of Counts
5
5
90
0.242
81
22
3
90
0.500
62
1
1
90
0.105
92
1
2
90
1.160
25
The eleven maximum and eight minimum values which exhibited the
largest significance are presented in Table V.
Upon investigating
the maximum curve values, one finds a sharp contrast in the levels of
significance that appear during the early half of the winter (1 December21 January) as compared to the latter half.
From 7 December to 20
January, the most significant level present is the 4% level, while in
the period from 21 January to 15 February all values are equal to or
more significant than the 4% value.
occurs on 21-22 January.
The most outstanding singularity
The percentages for the two days cannot be
thought of as separate entities due to their proximity in time.
The minimum curve values are quite different, with all but one
below the 10% level.
The 2-3 February period appears as the most
significant and, as a matter of fact, even more significant than the
21-22 January period in the maximum values.
To conclude, it is quite evident from this type of analysis that
the reality of the January Thaw during 21-22 January cannot be accepted
with confidence since the chance probability of obtaining only one
value at the 1% level with a total of 90 data points is rather high.
Secondly, it appears that there is at least one other singularity
that is equally as or more significant than the January Thaw.
Evidently
this type of averaging does not clearly bring out the singularities.
It is interesting to note that Wahl (1952) presents a similar conclusion
and suggests that since averaging over allyears shows only the residual
effects, some other method of averaging must be devised.
Table V
Significance levels (%) of selected dates for normal maximum and minimum
temperatures.
DATE
MAXIMUM
TEMPERATURES
Level of
Significance
D7
D16
+9
-4
Jl
+13
J2
+11
J7
J21
J22
+9
+0.9
+0.9
J29
F2
F15
F20
-3
-2
+4
+9
(M)
DATE
MINIMUM
TEMPERATURES
Level of
Significance
(%)
J3
J7
J21
J22
+8
+1
+3
+1
J29
-16
F2
-0.2
D = December
+ above normal
J = January
- below normal
F = February
F3
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F14
+9
It was with this thought in mind that it was decided to extend
this analysis in order to determine another way of classifying the data.
SECTION III
INVESTIGATION OF RELATIONSHIP
TO DOUBLE SUNSPOT CYCLE
Since another averaging scheme was needed, and changes in temperature can be related to variationsin the general circulation, it was
decided to classify the data into groups which reflected these variations.
Willett (1949) has related the 20-24 year double sunspot cycle to changes
in the general circulation.
Willett has suspected that there might be
a predominance of singularities during certain phases of this cycle.
The classification of the data according to this cycle appeared to be
a convenient and logical manner to further investigate this problem.
The 20-24 year double sunspot cycle is based on the fact that
alternate (major and minor) sunspot maxima have significant physical
differences.
The double sunspot cycle consists of the following eight
three-year phases:
1)
The three years of most rapid increase in average
N-MM -
value of the Zurich relative sunspot number (RSS) following a minor
maximum.
2)
MM (Major Maximum) -
The three years of maximum average value
of RSS.
3)
MM-NN -
The three years of most rapid decline in average
value of RSS following the major maximum.
4)
NN -
The three years of minimum average value of RSS following
the major maximum.
NN-M -
5)
The three years of most rapid increase in average
value of RSS following a major maximum.
M (Minor Maximum) -
6)
The three years of maximum average RSS
at the alternate maximum, usually but not always lower in number than
the major maximum.
7)
M-N -
The three years of most rapid decline in average value
of RSS following a minor maximum.
8)
N -
The three years of minimum average value of RSS following
a minor maximum.
In addition to a division into these eight phases, a division
into two other phases were considered:
half phase.
a major-half phase and a minor-
The major-half phase was an average of phases 1-4 and the
minor half phase was an average of phases 5-8.
Figure 6 illustrates graphically the above definition of the eight
phases.
time.
It must be noted that this is an idealized curve of RSS vs
Differences from this and in general between the two halves of
the cycle are discussed by Willett (1964): "Although the basic cycle
of sunspot numbers is the so-called eleven-year cycle, which has
actually varied in length from seven to seventeen years between successive sunspot maxima, and from nine to fourteen years between minima,
the so-called Hale or double sunspot cycle is much more clearly reflected
in solar-climatic relationships, at least outside of the tropics.
That
there is a physical reality in the double sunspot cycle on the sun is
indicated by a tendency for the sunspot number to be alternately lower
MM
I
0
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3
4
5
6
7
8
9
10
11
I
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13
TIME (RELATIVE
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Schematic representation of idealized 20-24 year
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19
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21
22
L
23
24
and higher with successive maxima, for the polarity of the magnetic
fields associated with sunspot groups on the sun's surface to reverse
from one maximum to the next, and for the corpuscular (charged particle)
radiations reaching the earth from the sun to be quite differently
related to alternate maxima."
The shortening or lengthening of the cycle may occasionally cause
one year's overlap or an extra year between the three-year phases of the
cycle.
Table VI contains a summary of the classification of the years
1872-1964 according to the phases of the double sunspot cycle.
The data were classified according to Table VI and averages,
standard deviations and standard errors were computed in a manner
similar to that described in Section II.
It is necessary to emphasize
that the above calculations were made on trend-free data.
Figures 7 through 26 show the average maximum and minimum temperature values for each phase of the double sunspot cycle and also the
trend curve.
The 5% and 1% significance bands are also shown and will
be discussed below.
Table VII lists the computed values of the standard deviations of
each phase (computed from all data points in phase, not from average
phase values) and their standard errors.
The cycle that is present
in the standard deviations appears very interesting and rather significant.
If one considers a large group of random data and divides them
into eight equal subgroups, one would expect to find equal standard
Table VI
Classification of years 1872-1964 by phases of Double Sunspot
Cycle.
N-MM
MM
MM-NN
NN
NN-M
M
M-N
N
1890
1872
1872
1877
1879
1882
1885
1888
1891
1892
1873
1878
1880
1883
1886
1889
1892
1893
1874
1879
1881
1884
1887
1890
1915
1894
1895
1900
1903
1905
1908
1911
1916
1917
1896
1901
1904
1906
1909
1912
1917
1918
1897
1902
1905
1907
1910
1913
1935
1919
1919
1922
1924
1927
1930
1932
1936
1937
1920
1923
1925
1928
1931
1933
1937
1938
1921
1924
1926
1929
1932
1934
1955
1939
1940
1942
1945
1947
1950
1952
1956
1957
1941
1943
1946
1948
1951
1953
1957
1958
1942
1944
1947
1949
1952
1954
1959
1960
1963
1961
1964
1962
30
28
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4
Figure 26.
i-j
!
t
I I
I
I
I
I
'tj I
Average maximum temnerature values for N Dbase with trend, 5%, and 1% si,-mificance levels.
-T
-1
Table VII
Standard deviations and standard errors of maximum and
minimum temperatures for phases of Double Sunspot Cycle.
STANDARD 2ERROR
STANDARD DEVIATIONS
(deg2
(deg2)
Maximum
Temperatures
Minimum
Temperatures
Phase
Maximum
Temperatures
Minimum
Temperatures
10.09
10.28
N-MM
2.91
2.97
9.90
10.15
MM
2.86
2.93
9.51
9.93
MM-NN
2.75
2.87
9.43
9.80
NN
2.72
2.83
9.73
10.05
Major
1.35
1.39
9.97
9.98
NN-M
2.88
2.88
10.28
10.24
M
2.97
2.96
10.62
10.53
M-N
3.07
3.04
10.97
10.58
N
3.17
3.05
10.51
10.42
Minor
1.52
1.50
deviations for each group or values which were not greatly different.
If this were not the case, the question of the validity of the
assumption that the data is random can be raised.
But if the data are
classified in such a way that the variability of each group was not
random, then one could expect widely differing values of the standard
deviations.
To determine the chance that this cycle is a random occurrence
or significant result, a statistical "F" Test (see Panofsky, 1958) was
applied to the data.
An "F" value is a ratio of the variance between
the groups being tested and the variance within the groupsbeing tested.
If the variation within the groups is larger, then the variation
between groups may be just a random fluctuation.
There are tables
available which give different significance levels and corresponding
values of "F".
If the value of "F" exceeds this tabulated value then
the group-mean values are significantly different at that level.
For the maximum curve values, an "F" value was obtained which
states that this cycle was significant on the 0.0000015% level while
the cycle of minimum curve standard deviations was significant on the
0.06% level.
Thus the variation of the variability of temperature
between the eight phases of the double sunspot cycle
is much more
significant than the variation within each phase and thus the cycle is
significant and, in view of the level of significance, undoubtedly
real.
The significance bands placed on the individual phase curves are
shown in Figures 7 through 26.
Upon inspection, it will be found that
anomalous values occur on certain dates during some phases, but not
Even the preferred date of the January Thaw on 21-22
during others.
January appears insignificantly different from the mean value during
a few phases.
Table VIII contains a significance count for each of the
ten phases for both maximum and minimum values.
Table IX shows the
level of significance of the counts as was shown in Section II.
These
results, similar to those in Section II, are not too impressive.
Looking at the 5% and 2 % category, one cannot find a phase which
has a count that would not be expected just by chance.
The 1% category
has three maximum and one minimum phases that can be considered significant.
If the categories are summed individually for maximum and
minimum values, only the 1% maximum category shows any semblance of
significance.
Table Xa andXb contain
a tabulation of the significance
certain dat'es during each of the phases.
levels of
The dates were chosen from
the 92-year normal curves as representing the most probable dates of
Most dates are significant only during a particular
singularities.
phase or two at most.
opposing values.
The January Thaw period has interesting and
The 21 January date appears to be significant during
the minor-half phases while 22 January is significant during the majorhalf phases.
As a matter of fact, there appears to be a tendency for
anomalous dates from 1 December to 21 January to show more significance
Table VIII
Significance counts.
MAXIMUM TEMPERATURES
Phase
<5%
< 2%
MINIMUM TEMPERATURES
< 1%
< 5%
<2 %
< 1%
N-MM
4
4
2
5
2
1
MM
7
1
0
3
1
1
MM-NN
4
3
0
1
0
0
NN
3
1
1
1
1
1
Major
6
4
3
4
3
3
NN-M
1
0
0
3
0
0
M
5
4
3
5
4
2
M-N
7
3
2
4
1
0
N
5
3
1
7
1
1
Minor
7
5
3
6
2
1
Normal
5
3
1
5
3
2
54
31
16
44
18
12
Sum
Table IX
Values of "T" and significance levels for various
counts.
Significance Level
Being Counted
Count
N
T
Significance Level
of Count
(%)
(M)
5
7
90
1.210
22.6
5
54
990
0.685
49.0
5
44
990
-0.840
40.0
2
4
90
1.170
24.0
21
31
990
1.270
20.0
2
18
990
-1.380
16.8
1
3
90
2.220
2.6
1
16
990
1.930
5.4
1
12
990
0.665
51.0
Table Xa
Significance levels (%) of selected dates for phases of Double Sunspot
Cycle.
MAXIMUM TEMPERATURES
D7
D16
Ji
+23
-55
-86
+30
MM
+65
-63
+73
MM-NN
+69
-25
NN
-21
Major
NN-M
J21
J22
J29
+23
+39
+ 3
-58
+15
+48
+78
+ 3
+61
+16
-88
+83
-11
-44
-69
-14
+66
-5
-94
+ 6
+27
-48
+94
M
+ 9
-92
M-N
+24
N
Minor
DATE
J2
J7
F2
F15
F20
-17
+19
-83
+73
-13
+6
+42
+22
- 1
- 1
+47
-56
+28
+11
-38
-97
-42
+56
+83
+21
+0.01
- 8
-0.5
+10
+76
-42
+42
+60
-98
-38
- 6
-92
-92
+20
-81
+21
+33
+50
-31
-73
+12
+44
- 4
+ 3
+13
+10
+ 2
+20
+47
+82
+52
+52
+11
+ 6
+ 3
+0.2
+22
+11
-59
-41
+68
+19
+ 8
+0.6
-10
+0.4
+ 6
+ 1
+0.7
+51
-32
-47
+ 9
+11
PHASE
N-MM
D = December
+ Above Normal
J = January
-
F = February
Below Normal
Table Xb
Significance levels (%) of selected dates for phases of Double Sunspot
Cycle.
MINIMUM TEMPERATURES
DATE
J21
J22
J29
F14
PHASE
N-NM
+11
+ 7
+11
+ 2
-94
- 2
-0.3
+54
MM
+61
+34
+56
+ 3
+44
-31
-79
+32
MM-NN
+ 4
-78
+67
+ 6
-
- 9
-31
+36
NN
-13
-48
+26
+57
-36
+86
-48
-19
Major
+20
+35
+ 7
+.08
-42
-2
-1
+42
NN-M
-27
+47
-62
-47
-28
-6
-8
+67
M
-73
+14
+52
+87
-79
-76
-16
+ 1
M-N
+11
+26
+12
+26
+47
-65
-32
+43
+0.2
+ 3
+24
+98
- 7
+98
+41
-58
+ 7
+0.3
+11
+72
-20
-17
- 8
+10
N
Minor
15
J = January
+ Above Norma 1
F = February
-
Below
Normal
during the minor half with the exception of the 14 February minimum and
20 February maximum.
For a closer look at this, average values of significance were
computed for each phase by using the absolute value of the individual
levels.
In addition, an average was computed for each of the two
apparent halves of the winter season.
The results of these calculations
are shown in Table XIa and XTh. They illustrate the apparent differences
in the winter season noted above, although not unambiguously.
The
significance levels involved are all rather large and it is very difficult to show distinctly by this method the time preference for the
major- and minor-half phases.
Another method to illustrate this discrepancy in the winter season
involves the use of a weighted mean.
If the reciprocals of the values
in Table Xa and Xb are computed and averages of these values are computed in the same manner as was used for Table XIa and XIb, the occurrence of a value at the 1% level is weighted 10 times more than an
occurrence at the 10% level.
Although this might not be the best
weighting scheme, it will suffice to illustrate this point.
Table XIIa and XIIb present
the results of applying this scheme.
Large values indicate high significance.
It is possible to conclude from these results that there is a
tendency for anomalies to occur in the first half of the winter during
Table XIa
Average absolute value of significance levels (%)
of selected dates for phases of Double Sunspot
Cycle.
MAXIMUM TEMPERATURES
DATE
A4l Davs
J22-F15
D7-J21
F20
PHASE
N-MM
39.6
24.2
48.4
MM
43.5
23.8
54.9
MM-NN
42.6
17.8
56.9
NN
39.2
47.0
34.7
Major
33.6
4.6
50.1
NN-M
58.1
58.5
57.9
M
42.4
41.5
42.9
M-N
28.1
50.2
15.4
N
22.6
46.8
8.7
Minor
15.3
34.8
4.2
Normal
6.0
2.5
8.0
D = December
J
January
F = February
Table XIb
Average absolute value of significance levels (%)
of selected dates for phases of Double Sunspot
Cycle.
MINIMUM TEMPERATURES
DATE
All Days
J22-F3
PHASE
D15-J21
F14
N-MM
23.6
24.6
22.8
MM
48.7
39.2
56.2
MM-NN
32.6
15.2
46.4
NN
39.1
56.8
25.0
Major
23.7
11.3
33.6
NN-M
35.8
22.2
46.6
M
53.6
64.5
44.8
M-N
39.3
42.5
36.8
N
46.8
61.0
35.4
Minor
21.9
29.2
16.1
Normal
7.3
4.4
D = December
J = January
F = February
9.6
Table XIIa
Average reciprocal of significance levels (%) of
selected dates for phases of Double Sunspot Cycle.
MAXIMUM TEMPERATURES
DATE
All Days
J22-F15
D7F21
PHAS
N-MM
0.06
0.12
0.03
MM
0.07
0.15
0.02
MM-NN
0.20
0.52
0.02
NN
0.04
0.04
0.04
Major
9.34
25.6
0.07
NN-M
0.03
0.05
0.02
M
0.04
0.04
0.04
M-N
0.13
0.03
0.19
N
0.54
0.03
0.84
Minor
0.65
0.05
0.99
Normal
0.12
0.55
0.27
D = December
J,= January
F = February
Table XIIb
Average reciprocal of significance levels (%) of
selected dates for phases of Double Sunspot Cycle.
MINIMUM TEMPERATURES
DATE
All Days
J22-F3
D15-J21
F14
PHASE
N-MM
0.52
1.09
0.08
MM
0.06
0.10
0.02
MM-NN
0.08
0.09
0.07
NN
0.04
0.02
0.05
Major
1.59
3.51
0.52
NN-M
0.05
0.09
0.02
M
0.14
0,02
0.22
M-N
0.04
0.03
0.05
N
0.62
0.05
1.08
Minor
0.44
0.06
0.74
Normal
1.41
2.77
0.32
D = December
J = January
F = February
the minor half of the double sunspot cycle and to occur in the second
half during the major half.
SECTION IV
CONCLUSIONS
To summarize, there are four conclusions that may be drawn from
this investigation:
1)
That it is impossible on the basis of Boston temperature data
to establish the reality of the"January Thaw"or any other singularities
using normal curves due to the small difference between the actual
count of significant anomalies and the number to be expected by chance.
2)
When the data are classified according to the 20-24 year
double sunspot cycle it is once again not possible to prove any presence
of singularities.
3)
The variation in the variability of temperature is quite cyclic
This
in character and parallels the 20-24 year double sunspot cycle.
is the one clearly significant relationship between the temperature
regime at Boston and the double sunspot cycle that emerges from this
study.
4)
There appears to be a division in the winter season.
Higher
anomalous values appear during the minor half of the double sunspot
cycle for the period from 1 December to 21 January while for the
remainder of the winter they occur during the major half of the cycle.
This is consistent with the fact that major blocks in the general
circulation tend to occur most strongly and regularly during the
latter half of the winter and during the major half of the double
sunspot cycle.
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pp. 8-15.
Meteorologicke Zpravy, Prague,
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