long-range weather forecasting

SMITHSONIAN MISCELLANEOUS COLLECTIONS VOLUME

138,

NUMBER

³

EoetJling J'unb

LONG-RANGE WEATHER FORECASTING

By

C. G.

ABBOT

Research Associate of the Smithsonian Institution

(Publication 4352)

CITY OF WASHINGTON

PUBLISHED BY THE SMITHSONIAN INSTITUTION

FEBRUARY

16, 1959

THE LORD BALTIMORE PRESS, INC.

BALTIWIORE, WID., U. S. A.

LONG-RANGE WEATHER FORECASTING

By

C. G.

abbot

Research Associate, Smithsonian Institution

Weather

comprises departures

from

climate,

which

represents the average

march

ofweatherconditions over long intervals.

For

purposes of

my

long-range forecasting,weather fallsintothree classes

:

A. Weather

of this half-century separated

from

that of the next preceding.

B.

Weather

attending high sunspot activity separated

from

that attending low,

C.

Weather

influenced

by

different seasons of the year, separately tabulated.

Subject to these separations, precipitation

and

temperature at all stations,

and

in all times within the past century,

and

probably for

many

years to come, are determined in their principal trends by one family of^periods,

which

are exact submultiples of ^/j months.

About ₇₀

harmonics belonging to this family are

known

to be oc- curring in one or another of several kinds of

phenomena. For

pur- poses of long-range weather forecasting 27 of ^these periods are sufifi- cient.

By

using

them

correctly, forecasts

and

backcasts for as

much

as

60

years have resulted successfully.

No

data other than records of

monthly mean

weather observations

and Wolf

sunspot

numbers

for the past centuryare required for these

forecasts or backcasts.

Table i gives the 27 ^periods employed.

They

are tabulated as fractions of 273

months and

as periods in

months and

fractions thereof.

One

thousand

months

or

more

of consecutive records are to be employed.

The

following groups are

made

for forecasting

:

As

regards seasons:

January

to April;

May

to

August

^;

September

to

December.

As

regards sunspotactivity:

Wolf numbers

lessthan 20;

Wolf num-

bers above 20.

SMITHSONIAN MISCELLANEOUS COLLECTIONS, VOL. 138, NO. 3

2

SMITHSONIAN MISCELLANEOUS COLLECTIONS

VOL. 138

As

regards secular time^:

Years

of the first half of the record_; years of the second half.

With

thisgrouping, approximately i,ooo

months

of records will be divided into

220

groups.

Many

groups willtherefore contain too

few columns

in tabulation to give trustworthy means.

To remedy

this difficulty, I

combine

sixgroups into one,

by

shiftingphases to agree-

ment and

taking a general

mean.

I

make

such combinations for all periods

up

to 15^ months, throwing records with sunspots less than

20 Wolf numbers

into one combination,

and

those above 20 into another.

For

periods above 15^

months

I forego division for the three periods of the year.

When

using the

combined forms

in fore- castingthey

must

beusedinthe originalphasesofconstituents.

These

arrangements for the shorter periods are illustrated

by

figure i.

By

the process just described

and

illustrated inthefigure,onegreatly reducesbelow

220

the actual

number

of

mean forms and

amplitudesto be

combined

in

making

aforecast

from

₂₇periods.

But when

usingthe

combined mean forms

described, one

must

readjust theirphasestothe phases their constituent parts

had

originally. Besides ^this fruitful source of mistakes, one

must

be careful to apply correctly at the proper intervals allowances to take care of the fractional parts of

months found

in

24

^{of the}27^periodsof table i,

and must

also regard plus

and minus

signs.

Details of

my method

of long-range weather forecasting are ex- tensively published in the Journal of Solar

Energy and

Engineering, vol. 2,

No.

^I,

January

1958. I shall confine the remainder of this paper to evidences that support the validity of the

method and

the soundness of its results.

I.

The

exact master period or

fundamental

is 2yj months.

This period (within existing uncertainty) is double the

well-known

"sunspot period" of iif months. ^{It is also associated} with Hale's magnetic period in sunspots.

Having

discovered that approximately

10% months

is a strong period in

Washington

precipitation, I deter-

mined

its amplitude for several periods differing slightly

from 10%

months.

For

this purpose I used about

790 monthly mean

values of

Washington

precipitation, all observed

when Wolf

sunspot

numbers

exceeded 20.

The

values

were smoothed by 3-month

consecutive means,

which

of course reduces the ranges of percentage departures

from normal

to about two-thirds of their actual

monthly

values.

Table 2

and

figure 2

show

theresults.

NO. 3

LONG-RANGE WEATHER FORECASTING — ^ABBOT

Phase ApJustment.^The.6^^mo.Period.

CATtcoRY2Assemble. CatesorvIAssembl-^.

b,0« C,t3 flA"" t;OK CjOK McmJ a. a.Tj lli'>l C'

MO

LO:r

IJi C,<:

120 gi' 100 lOS S6

108 fe<3 (00 51 54

100 (02 104 103

no 86

92 97

a^Of.

3 + S 6 I 2 3 4

Fig. I.

—

^Sixfold combinations of weather periods.

4 SMITHSONIAN MISCELLANEOUS COLLECTIONS

^{VOL. 138}

The

irregularities of run are naturally caused by irregularities in the rainfall

from month

to month.

The

phases of the

4

^periods

differ, of course, because the periods are unequal.

Figure 2 clearly

shows

that a value of the master period between 273

and

275

months

is definitely indicated.

I have preferred 273

months

rather than 275

months

because it is

anintegral multiple of the strong periods 7, 13, 39,

and

91 months. It

cannot be

much more

than one-third percent

from

the true master period.

Table i.

—

Harmonicsof275months Fractionsof 273.

.

Lengthinmonths

.

Fractions of273^.

.

Lengthinmonths

.

Fractions of273^.

.

Lengthinmonths

.

1/3 1/4 1/5 1/6 1/7

91 68-1/4 54-3/5 45-1/2 39

1/12 1/14 1/15 1/18 1/20

22-3/4 9-1/2 18-1/5 15-1/6 13-13/20

1/27 1/28 1/30 1/33 1/36

10-1/9 9-3/4 9-1/10 8-3/11 7-7/^2

1/8 1/9 i/io i/ii 34-1/8 30-1/3 27-3/10 24-9/1

I

1/21 1/22 1/24 1/26

13 12-9/22 11-3/8 10-1/2 1/39 1/45 1/54 1/63 7 6-1/15 5-1/18 4-1/3

Period Months 271.2

27 2730

27 275-0

27 277.0

27

Table ^2.

—

Percentage amplitudes of proposed periods Percentageruns

105.7 103.4 102.5 100.7 100.9 96.3 97.3 97.9 98.0 97.7

95-7 95-8 93.4 96,1 99.3 102.0 103.7 108.0 104.8 loi.i

109.8 102.4 103.3 99-3 95-4 92.9 96.2 97.6 98.8 104.5

94.6 104.4 106.2 101.3 105.8 105.5 194-6 97-5 96.9 93-3

Ranges Percent 9.4

14.6

16.9

12.9

2. All discovered periods in weather are integrally related to ^/j months.

As

toperiods greater than 273

months

^:

About

1936 ^I pointed out thatthe fluctuations of level of the Great

Lakes

indicated periods of severe drought in the

Northwest

following 1837, 1885,

and

1929.

Lesser droughts occurred about halfway betweenthese dates.

Know-

ing the solar period of about 22| years, I predicted drought in the decades 1950-60, 1975- 1985,2000-2010,

and

2020-2030.

The

drought following 1952 has ^{already occurred.}

Though more

severe than I expected, it ended in 1957, lasting only half as long as the great de- pressions of the

Lakes

following 1837, 1885,

and

1929.

NO. 3

LONG-RANGE WEATHER FORECASTING ABBOT

Exact

evidence

on

the lengths of the periods less than 273

months

is at hand.

When

one tabulates

monthly

precipitation values to dis- coverthe

forms and

amplitudesof periods ranginginlength

from

_15^

to91 months,thedirecttabulationsoften

show no

convincing evidence

16

14-

Fig. 3.—The 6

-month period in St. Louis precipitation cleared of shorter integrally related periods.

NO. 3

LONG-RANGE WEATHER FORECASTING ABBOT

₇

ure3, Iquote

from my

^paper (Journ. Solar

Energy and

Eng.,^vol. 2,

No.

I, p. 32), already ^cited.

As

a graphicalillustration, I reproduce hereas figure_3, figure

9

^of

my

paper "Sixty-year

Weather

Forecasts." ^{^} It represents the clear- anceof the

68^-month

period in St.Louis precipitation.

A and B

are

mean

curves,

A

of

8

repetitions prior to 1900,

and B

^of 7 repetitions subsequentto 1900.

The

phaseof

A

^appearstobe ₃

months

laterthan thephase of B.

Moving A

back3

months and

taking the

mean

ofthe two, curve

C

results. This

shows

plainlythe presence of aperiod of

5

of

68^

^{months. Obtainingits average}

form and

amplitude

and

sub- tracting

from

curveC,curve

D

isobtained.

Curve D

presents 7

humps

of about equal length.

Taking

their

mean form and

amplitude

and

subtracting,curve

E

remains. In curve

E

halvesare clearly indicated,

though

not exclusively. Evaluatingthe average half

and

subtracting, curve

F

appears.

And now

obviously there remains a regular period of ^ of

68^

months. Evaluating

and

subtracting,

we

find curve G,

which

well represents the

68^-month

period sought. Its range is 13 percent of St. Louis

normal

precipitation. This single

diagram

pre- sents five

members

of the family related integrally to 273 months.

These

are_{^, ^,}M2,V20,

and Hs

of273 months.

Concerning

figure _4, I continue quoting

from

the paper just cited, pages 32

and

_33.

Infigure

4

^{are given theresults}

and

^detailsof theremoval of over- riding periods

from

the curve of the period of

45^ months

^{in the} precipitationat Natural Bridge, Arizona, for

SS

^{^}

<

^{20, between the}

years 1890

and

1920. In curve

Ai

^{are the} averagevalues of the pre- cipitation for allrepetitions of the

45^-month

period

when SS <

20.

Curve

Ai remains after the removal of the average ordinates of the period (452)

/4

^months.

Curve

Ag is

what

remains after

removing from

curve Ai

(45i)/6

^months.

Curve

A3 remains after

removing

(45^) /7 months.

Curve

A4 remains after

removing (45i)/5

^months.

Curve

Ag remains after

removing (452)/3

^months,

and

it

shows

ap- proximately the true

form

of the

45^-month

period.

The

amplitude of the

smooth

curve through curve A5 is 21 percent of

normal

^pre-

cipitation at Natural Bridge.

The

reader should understand that in threeother cases the

45^-month

period

was

similarly cleared of over- riders.

These

cases are for

SS <

20, in theyears 1921 to 1950; ^for

SS >

^20, between 1890

and

1920;

and

for

SS > 20 from

1921 to 1950.

So

this

one

period,

45^

months, involved

much

thought

and

1Smithsonian Misc. Coll., vol. 128, No. 3, Publ. 4211, 1955.

2

SS

^is

my

abbreviationfor Sunspot number.

SMITHSONIAN MISCELLANEOUS COLLECTIONS

VOL. 138

Fig. 4.

—

^The 4Si-month period in Natural Bridge precipitation, between 1890 and 1920, for intervals ofWolf numbers below 20, cleared of shorter integrally related periods.

NO. 3

LONG-RANGE WEATHER FORECASTING ABBOT 9

work, requiring three additional treatments similar to figure 4. All the periods above 22

months

in length required

them

also.

Merely

referring to the

two

examples given above, of the clearing of overriders of lesserlengthintegrally relatedtotheoriginalperiodic curves, the readerhasproof that notlessthan 11 integral submultiples of

273 months

^exist as regular periods in precipitation.

Those

dis- covered in figures 3

and

4 ^are ^,^, ^,M2,Vis, %o, %4, %8,Vzo, Vse,

and

^/42 of 273 months. Their ranges ⁱⁿ nocase are lessthan 10percent of

normal

precipitation.

Yet

^all of

them

are unrecognized by meteor- ologists.

Besides these evidences I have in

my

records

hundreds

of similar examples

which

prove unquestionably that all of the 27 periods in weather

named

ⁱⁿ table i are exact integral submultiples of 273 months.

Not

only so, but I have

many

similar examples

from

other fields of research

which

prove that notonly these 27 weather periods but

many

others are exact submultiples of 273 months.

I suggest to students of

hydrodynamics

that this family

may

be a fruitful field for mathematical research in its relation to solar varia- tions

and

weather.

3.

Approximately

sine

forms

of the longer periodsin precipitation.

Not

only

do

the curves of the sun's variation representing the family integrally related to 273

months

approximate to sine curves, notwithstandingtheir large accidental errors

compared

to their small percentage range,but also the longer weatherperiods ⁱⁿ precipitation

and

temperature,

when

cleared of integrally related shorter periods, are closely like sine forms.

Figure ₅ is taken

from my

paper "Periodic Solar Variation."^{^}

Though none

of thecurves ofvariation inthe solar-constantmeasures

shown

in that figure exceeds 0.25 percent of the solar constant in amplitude, 20 of the

26

curves are wellrepresented

by smooth

curves of nearly simple sine form.

In figure

6

I

show some

of the longer periods in precipitation for several weather stations, first as given by direct tabulation

from

the weather records, second after the removal of overriding shorter pe- riods integrally related,

and

third

by smooth

curves following the

forms

of the cleared originals.

The

approximate sine

forms

are apparent in the

end

results,

though

quite impossible to descry in the original tabulations.

3Smithsonian Misc. Coll., vol. 18, No. 4, Publ.4213, fig. 3, 1955.

lO

SMITHSONIAN MISCELLANEOUS COLLECTIONS

^{VOL. I38}

Fig.6a.

—

Periods inprecipitationascomputed andas cleared ofsuperposed integrallyrelated shorter periods.

12

SMITHSONIAN MISCELLANEOUS COLLECTIONS

VOL. 138

4. Inequality of ^periodic amplittides.

A

curious observation has presented itself

which may

have

some

bearing

on any

theoretical treatment of the

problem

of identity of the families of periods relatedto273

months found

inthe solarvaria-

FiG. 6b.

—

^Periodsin precipitationascomputed andas cleared ofsuperposed integrally related shorter periods.

tion, the weather,

and

in other fields.

The

observation I refer to is this:

A

^{great majority} of the scores of periods I have determined for various cities, at times of sunspot

numbers

less than

20 Wolf num-

bers, exceed considerably in amplitude the corresponding periods in the weatherof thesecitiesas investigated for times of

Wolf numbers

exceeding 20. Table _{3 gives a} sample of such excesses.

NO. 3

LONG-RANGE WEATHER FORECASTING ABBOT

1

3

5.

Weather

trends

may

be well forecasted or hackcasted for

many

years

forward

or backward.

I have given above

some

hints about forecasting

from

the average

background found

in a thousand months, if used together with the

knowledge

of a family of periods integrally related to 273 months.

Details of the

method

are given in the paper cited above. Since that paper

was

prepared,

much

additional experience has led to slight

Table 3.

—

Percentage amplitudes of totalnormal precipitation of periods com- pared,forfirsthalfandsecondhalf ofyearsof records,andfor

Wolf sunspot numbers greater and less than 20 A.

Mean

precipitation, St. Louis data, many cases

en

^ ^{Period,months: 4-1/3 5-i/iS 6-1/ 15 7 8-3/11 g-i/io 34-1/8}

•9 o ^{First half 6 15 17 20 22 20} 17

g ^,- Second half 12 12 22 19 27 34 23

-M

Mean

9 14 19 19 29 27 20

p<

S First half 8 ₇ 16 15 20 'i 20

P P

/JO

" ^ Secondhalf 6 16 11 15 17 20 15

'3'^

Mean

7 11 13 15 18 25 17

Ratio means 1.3 1.3 1.5 1.3 1.6 i.i 1.2

B.

Mean

precipitation, Natural Bridge, Ariz., data, many cases

" _{Period,months}

: 4-1/3 5-i/i8 6-1/15 ₇ 8-3/11 9-1/10 34-1/8

First half 19 18 35 27 43 53

3

w

^{Second half} 29 19 40 42 43 59

- Second half 29 19 40 42 43 59 ^{^}

Mean

24 19 27 34 43 56

o

g ^{Firsthalf 15 16 23 23} 26 42

S ^ _{Secondhalf 13 18 26 32 26 33} ''

^

^{/^}

^Mean

^{14 17 25 27 26 27}

'^ Ratio means 1.7 ^i.i 1.5 1.2 1.7 1.4 1.7

modifications

and improvements

in the procedure,

which

I will not here relate, but which, as will appear, have given still better results in forecasting for

my

^latest work.

Lest readers

may

suppose that nothing is a true veritable forecast or backcast

which

deals with time included ⁱⁿ the 1,000-month back- ground, Ineedonly

remind them

thatevery

month

forecastedorhack- casteddepends

on

^the

combined

influence of 1,000

months

of records.

In

any

one year there are but ¹² months.

Hence

the 12

monthly

records of that year can have little

more

than i percent influence in dictating the

form and

amplitudeof the predictedcurve for that year.

14

SMITHSONIAN MISCELLANEOUS COLLECTIONS

VOL. 138

To

Speak plainly, every year's curve that I

am

about to

show

is a real prediction,

whether

within orbeforeor after theintervalof about 1,000

months

included in its basis.

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^-3i'.z?4jo a.i^;»^'ar//y Fig. 7.

—

^Facsimile of computation of St. Louis precipitation, 1875-1879, compared with the observed precipitation as percentage of normal. Data from monthly mean

precipitationsmoothed by S-monthrunning means. Dottedcurvefrom summationof22 regular periodicities, determined as averages over the 86-year epoch, 1854- 1939. Full curve, the event.

First of all I

show

figures _7, 8,

and 9

^taken

from

Smithsonian Publ.

No.

421_1,prepared in 1954^totestthe

method

for theprecipita- tion in St. Louis

and

Peoria

and

temperature in St. Louis.

A

con- siderable part of these curves give correlation coefficients between

3 I 3 ^o S

-^o.Sa -a c v su u

§i

'a"I .STO

o) E

.22 S

J^ §2

cJ2<

8

C/2

IS

-ivwuaiMJ09&

i6

NO. 3

LONG-RANGE WEATHER FORECASTING ABBOT

17

prediction

and

event ranging above

80

percent.

The

results

shown

in figure

9

^{for 6-year intervals}

40

years

from

their

mean

^basis,

show

correlation coefficients

from

50to70percent. I repeat thelittle figure representing St. Louis temperature

more

distinctly in figure 10.

Although

I have stated coefficients of correlation, the

method

of correlations is not a fair test of the usefulness of the forecast.

For when

a curve is delineated

from

the average conditions of 1,000

months

ata distance of

40

years

from

its average source,

one

cannot fairly expect that the phase of the observed event will never be affected by unusual or accidental displacements due to

temporary

/934 35 35 37 38 55 40

Fig. 10.

—

Enlargement of temperature forecast for St. Louis shown ⁱⁿfigure 9.

causes.

Moreover

3- or

5-month

smoothing

may

displace

maxima by

one or

two

months.

When

such displacementsof even only one

month

occur, the curves of prediction

and

event

may

often separate widely in ordinates,

though

to the eye it is apparent that they ^relate to the

same

features. In such cases the correlation coefficient is unjustly greatly diminished.

I

now show

in figure 11 the prediction

and

event for

Washington,

D.

C, and Spokane

precipitation

made

ⁱⁿ 1958

from

records centering in 1898

and

1915 respectively,

and compared

with the observations

from

₁₉₅₀to 1956available to

me

atthe time. Certain

improvements

introduced after the

Washington work was done

helped to

make

the

fit withobservation better for

Spokane

than for

any

city I have thus

(X4

i8

NO. 3

LONG-RANGE WEATHER FORECASTING — ABBOT

IQ far

worked

upon. I call attention in figure ii to very

abnormal

pre- cipitation in

January and March

1950 ^at Spokane.

6.

The

_{future of} themethod.

In 1955 ^the Association for Applied Solar

Energy

requested

me

to test the

method on

precipitation atthe station Natural Bridge near Flagstaff, Ariz. I did so,

and

the result appears as figure ₅ of thepaperintheJournalof Solar

Energy and

Engineeringabovecited.

It

seemed

to the Association of such value that an

arrangement was made

with Prof. Charles

Wexell

of

Arizona

State College to

have

electronic computations

made

of precipitation at 30 stations well dis- tributed over the United States. Prof. Wexell's son, Johnathan, cleverly

programmed

the electronic computer,

and

has prepared the

220

tables required foreach of the 30^stations.

Already

I have

made

forecasts for several of the

30

^cities with the aid of

my

secretary, Mrs.

Windom, and

of

Mrs. James

Hill, a rapid

worker

assigned to

me

^by Secretary Carmichael of the Smithsonian Institution.

We hope

to complete this large

program

in the

coming

winter. I propose to publish three

maps

of the United States for each year, 1959-1967, about April 1959, ^to delineate the regions of equal departure

from normal

precipitation in the

United

States, so far as they can be indi- catedby these 30 forecasts.

A

thorough study

would

requireperhaps

two

or three times as

many

stations. If thispreliminaryessayappears useful as time passes, doubtless such a thorough expansion of the

work

will come.

In the Bulletin of the

American

Meteorological Society, vol. 39,

No.

6,

June

1958,p.296,

we

read^:

"The

petroleumindustry estimates thata

modest improvement

inlong-range temperature forecasts

would

be

worth

one

hundred

milliondollarsa yearthrough

more

economical operations."

I wrote to Dr.

Waterman,

the head of the National Research Foundation, that I

am

^{sure this} can be done,

and

that with suitable financialaid forelectroniccomputations

and

otherassistance,I believe I could undertake to supply it in 1959, as soon as

my

present taskis

over. I

would

undertake lo-year temperature forecasts for

any

se- lected 10cities inthe United States orabroad

where

about 1,000con- secutive

monthly

records of

mean

temperature are available.

Dr. Carmichael, Secretary of the Smithsonian Institution, has al- lotted funds for a lo-year temperature forecast for 10 cities,

which

I hopetocomplete during 1959^as a Smithsonian project.