MARCH 1959 MONTHLY WEATHER REVIEW 107
A NOMOGRAM TO DETERMINE
MONTHLY POTENTIAL EVAPOTRANSPIRATION
T. E. A. VAN HYLCKAMA
Laboratory of Climatology, Drexel Institute of Technology, Centerton, N.J.’
[Manuscrip! received November 20, 1958; revised January 16, 19591
ABSTRACT
A nomogram f o r the solution of Thornthwaite’s empi rival relationships among mean nlorlthly temperature, lati-
tude, and potential evapotranspiration is constructed. Zn the form of alignment charts, the nomogram is simple to
use and has the added advantages of the temperature scale being labeled in both degrees Celsius and degrees Fahren-
heit ant1 of the potential evapotranspiration scale being labeled in both centimeters and inches.
1. INTRODUCTION
Potential evapotranspiration is defined a.s the anmunt
of water that would be evaporated from the ground or be
transpired by the vegetation if there were no shortage of
water at any time during the period of observation. The
concept has proven to be :L valuable tool in the, classifica-
tion of climates as well as in the computation of water
balances [ 11, [ 21, [ 31.
Thornthwaite [4] found an empirical relationship
among the mean monthly tenlperatures, the latitude of
the location, and the potential evapotrallspirat,ion. The
determination of the latter involves the use of two tables
and a nomogram and is rather time consunling, especially
if series of years for different locations hare to be
computed.
The nomogram presente(1 here IIXS tllr result of tile
desire to have a fast, accurate, and convenirllt metllotl to
determine the potential eval’Otl.:tnspil.ation according to
Thornthwaite’s method. T l ~e 11l:muscript \vas ready f o ~
publicatiol~ whell the autllor’s attention M’RS drawn to a
recent article by Palmer :~nd Havens [5], dealing with
the same subject. The alignnlent charts described in the
present paper are easier t,o use and either alo~le or in corn-
bination with the graphiwl tec.hnique of 1’;tlmer and
Havens will expedite tlle conlpntntions of potential
evs~potraasy)iratio~~.
2. THE THORNTHWAITE FORMULA
in which
0,=0.000000675 J“.- 0.0000771 J2+ 0.01792 Jf0.49239.
(2)
Third, E must be multiplied by a factor for daylight (f ) ,
adjusting P to the mean possible duration of sunlight in
each mohth of the ye’ar. This factor depends on the lati-
tude of the location.
The heat index (J ) can be computed from Thorn-
thwaite’s [4] table IV, but it is simpler to construct a
conversion scale where both O F. and O C. are inserted.
Such a scale is presented a t t h e t o p of figure 1. By
adding the i’s for each of the twelve months of the year
J is determined. This complet,es the first step.
The second step involves the use of equation (1) which,
if we take logarithms and put a =F (J ), becomes:
log E-log l.A=F(J) log lot-F(J) log J. (3)
Tllis equat,ion can be simplified t,o:
logE=P(J) log t’-P’(J) (4)
1)s ignoring the constant log 1.6, putting 10t=t’ and P (J )
log J= F’ (J ) . Equation (4) in determinant form is :
-1ogP 1 0
log t‘ 0 1 =o. (5)
I F ’ (J ) 1 P (J )
Finally the potentia1 evapotranspiration (PE) is cal-
Ef= (PE) (6)
culated according to :
The computation of potential evapotr:lllspiratio11.at,iolL :tc-
cording to Thornthmde‘s [4] formula consists of three
steps. First, the heat index (J ) has to be found for t,he
year and consists of the summation of 12 monthly indices
(i), where i= ( t/5)1.514 and t= the mean temperature of wit11 k, 77% and 12 constants.
the month in “C. Second, the unadjusted potent,ial evapo-
transpipation (E ) must. be computet1 according to the
formula : For the purpose of constructing the nomogram the de-
which can be writt,en :
-?)& k7n log E 1
91 lcn log f 1 =o (7) 0 kmn log (PE) m f n
3. CONSTRUCTING THE NOMOGRAM
li’= l .A ( l O f /J ) ll (1) ternlinaut (5) must be multiplied by the non-zero de-
*Present Rfeliation: U.S. Geological Survey, Washington 25, D.C. terminant (see [SI)
108 M O N T H L Y W E A T MARCH 1959
pl q1
p3 q 3 r3
(8)
which gives :
-p1 log E+pz "q1 log E+ pz
p1 log t' + p3 p1 log t' + p3
PlF'(J) +p z +p 3 P (J ) q1F'(J) +q z +p 3 F (J )
-rl log E+r2
rlF' (J) + r2 + r,F (J )
rl log t'+r3 =o. (9)
Inspection shows that it is permissible to put p2=pJ=p1
=q3=r1=0, but p ,, p2, rz, and r3 must be non-zero.
Therefore :
-p,log E 4 2 rz
p,logt' 0 r3
p l P ' (J ) qz r2+r3P(J)
=o (10)
in which r, may be equal to r3. By multiplying each row
by the reciprocal of its element in the third column, ignor-
ing subscripts, rearranging, and inserting the constant, log
1.6,we find the parametric equations:
The nomogram can now be constructed and consists of
two parallel logarithmic scales of equal modulus ruming
in opposite direction and a third line not parallel to these.
The slanted line is not straight b'ecause F (J ) is a poly-
nomial. Ho'wever, within the range of J's for which
Thornthwaite's formula has proven to give adequate re-
sults, the line is straight, except for slight curve's below
J= 10 and above J= 160. Below J= 5 the line aplm~aclles
zero asymptotically toward o,= (1/1.49239) (q /r ) since
by equation (2), P (J ) =0.49239 when J= 0 ; and above
J=160 the line curves similarly toward xl=O. Between
J= 10 and J= 160 the slope of the J line is determined by
t=26.5" C. on the t scale and E=13.5 cm. on the E' scale.
At this temperature E is 13.5cm. regardless of index,
For E= 1.6 cm., equation (3) becomes :
P (J ) log 1 0 t =P (J ) l o g J (12)
which, for P (J ) # 0, gives log J=log lot. The t scale
can therefore be used to find the J's on the sloping line by
connecting 1.6 cm. on the E scale with the different values
(in cm.) on the t scale.
For temperatures above 26.5" C. (79.7" F.) values of
E are given by Thornthwaite [4]. These are added to the
E scale above 13.5 as indioated in figure 1.
A convenient size of nomogram is obtained by making
the distanqe between two vertical scales q/r=17 cm. and
the length of one logarithmic cycle on the vertical lines
p/r= 12.5 cm.
I n order to find the actual pot.entia1 evapotranspiration
a second nomogram is added to the first. Determinant
(7 ) becomes with moduli 12.5 for t,he h' line and 10 for the
1'E line :
-2.512.5 log E 1
0 1 0 l o g PE 1
10 50 log f 1 1 =O (13)
so the f line has a 50-cm. cycle, while the distance between
t,lLe K and PE lines is 2.5 cm. and between the PE and f
lines 10 cm. Since t,he B line now functions only as a
pivot line it does not need to be marked. Markings for
"F. and inches can be atldetl :IS desired to t,he t and PE
1 illes.
4. ACCURACY AND PRACTICAL USE OF THE .
NOMOGRAM
The nomogram gives results at least as accurate as those
obtained by the old method and when used in routine pro-
cedures has proven to be much faster, especially if one
metan d value is established for any one station land a series
of years of record is being processed. The deviations oc-
curring in consequellee of the use of a mean J value are
negligible.
Speed and accuracy both are increased even more if the
nonlogram is drawn on "hard board?' or similar material.
It can then be equipped with a system of pins and slots
(for an illustration, see [7] ). The J and the pivot line
shoultl be grooved. A pin is fastened at the appropriate
place on the J line. A slotted transparent arm is made to
slide over this pin and is connected via a sliding pin on
the pivot line with another transparent arm. When the
left arm is placed on the appropriate place on the t line,
the pin on the pivot line slides automatically into the right
position and the right arm can be placed immediately on
the proper place along tjhe f line to obtain the PE.
The f values depend on the month of the year and the
latitude of the location. They must be obtained from
tables as published by Thornthwaite ([4], p. 93) or
Thornthwaite and Mather ( [ 11, p. 98). Although it
would be possible to arrange these tables in the form of a
graph, it would make a nomogram unnecessarily cluttered
and awkward to handle.
This nomogram, as any nomogram, is of principal ad-
vantage if series of computations have to be made, such
as series of years of monthly PE's at one station or mean
monthly PE's at digerent stations. (In the latter case the
stations should. be grouped by latitude.)
It is, therefore, convenient to mark the f values belong-
ing to the latitude of the station, or stations, before taking
a series of readings. As an example the mean temperature
for May in Bridgeton, X..J., is 17.5" C.; J=58.3; f , taken
from Thornthwaite's table, is 1.23 at latitude 39" N.; ant1
the PE is read to be 9.2 cm., as illustrated in figure 1.
APPENDIX
Table 1, as taken from Thornthwaite [4] presents tlie f
values for I:~t,it,~ldes 50" N. to 50" 8. for the 12 months of
MARCH 1959
t
32.2 .
OF
32.5 -
33 -
34 -
35 -
36
38 -
-
-
40 -
45-
50-
60-
76:
80-
1
109
-.5
-1
-
-2
-3
3
"5
-
-
-
-
- IO
./ "15 ./
-
- 30
E
m r r )
If OC > 26.5
/ ' ---
/'
/
I
I
- IO (9.2)
4
3
2 ' I : - .5
L.9
Feb/Nov
k.7
FIGUKE 1.-A l~o~r~ogram to deterluine the lmtential evapotranspiration according to Thornthwaite's formula.
110 MONTHLY WEATHER REVIEM7 MARCH. 1959
TARLE 1.-Mean possible duration of sunlight i n the Northern and the year. For latitudes f:lrther north or south the f’s of Southern Hemispheres expressed in units of SO days of 16 hours each.
(After Thornthwaite [Q].) the 50th parallel should be used, presumably because the
ON.Lat 0
10
5
15
20 26
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40 41
42
43
44
I
45
48
47
49
50
S. Lat. 5
10
16
25
30
35
40 42
44
48
46
60
20
-
- -
ran.
-
I. 04
1.02
1. 00
.95
.97
.93 .92
.92
.91
.91
.w .w .89
.88 .I
.87
.87
.86 .85
.85
.84
.83 .82
.81
.81
.79 .80
.77
.76
.75
.74
!. 06 I. 08
I. 12
I. 14
I. 17
1. 23
I. 27
I. 28
1. 30 1. 32
I. 34
I. 37
I. 20
- -
Feb.
-
.94 .93 .91
.91
.90 .89
.88 .88 .88 .87
.87
.86
.87
.86 .85
.85
; 85
.84 .84 .84 .83 .83 .83 .82
.82
.81
.81
.80
.80 .79
.78
.95
.97
.98
1.00
1.01
1.03
1.04
1. 06
1.07
1.08
1. 10
1. 11
1. 12
-
- -
Mar.
-
1.04
1.03
1.03
1.03
1.03
1.03
1.03
1.03
1.03
1.03
1.03
1.03
1.03
1.03
1. 03
1.03
1.03
1.03
1.03
1.03
1. 03
1.03
1. 03
1.02
1.02
1.02
1.02
1. 02
1. 02
1.02
1. 02
1.04
1.05
1.05
1.05 1.05
1.06
1.06
1.07
1.07
1.07 1.07
1.08
1.08
-
- -
kpr.
-
1. 01
1. 02
I. 03
I. 04
1. 05
I. 06
I. 06
I. 07
I. 07
1. 07
1. 08
1. 08
1. 08
1.09
1. 09
I. 09
1. 10
1. 10
1. 10
I. 11
1. 11
I. 11 1. 12
I. 12
I. 13
1. 13
1. 13
I. 14
I. 14
I. 14
I. 15
I. 00
.99
.98
.97
.96 .95
.94 .93 .92
.92 .91
.!M .89
- -
-
1.04
1.06
1.08
1. 11
1. 15
1. 13
1. 15
1. 16
1. 16
1. 17
1. 18
1. 18
1. 19
1. 19
1. 20 1. 21
1. 21
1. 22
1.23
1.23
1.24
1.25
1.26
1.26
1. 27
1. 28
1. 29
1.30
1. 31
1.32
1.33
1.02
1.01
.98
.96 .94 .92
.89
.86 .8S .83 .82
.80 .77
- -
‘me
-
I. 01
I. 03
I. 06 I. 08
I. 11
1. 14
1. 15
1. 15
1. 16
1. 16
1. 17
1. 18
1. 19
I. 20 1. 20 I. 21
1. 22
1. 23
1. 24
I. 24
I. 2s
1. 26 1. 27
1. 28
I. 29 I. a 1. 31
1. 32
I. 33
1. 34
I. 36
.99 .96 .94 .91
.E% .85
.82
.78
.74
.76
.72
.70 .67
- -
July
-
1.04
1.06
1.08
1. 12
1. 14
1. 17
1. 17
1. 18
1. 18
1. 19
1.20
1.20
1. 21
1.22
1.22
1. 23
1. 24
1. 25
1. 25
1. 26
1. 27
1.27
1. 28
1.29
1. 30
1. 31
1.32
1. 33
1. 34
1.35
1.37
1.02
1. 00 .97
.95
.93
.90
.87
.84
.82
.79
.8 1
. 76
.74
- -
hlg.
-
1.04
1. 05
1. 07
1. 08
I. 11
1. 12
1. 12
1. 13
1. 13
1.13
1. 14
1. 14
1. 15
I. 15
I. 16
I. 16
1. 16
1. 17
1. 17
I. 18
1. 18
I. 19
I. 19
1. u)
I. 20
I. 21
I. 22
I. 22
I. 23
1. 24
I. 2s
I. 03
I. 01
.99
.98
.96
.94
.92
.92
.91
.90 .89
.88
I. on
-
- -
lept
__
1.01
1.01
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.03
1.03
1. 03
1.03
1.03
1. 03
1. 03
1.03
1.03
1.04
1.04
1. 04
1. 04
1. 04
1.04
1. 04
1.04
1.04
1. 04
1. os
1.06
I. os
1.00
1.00
1.00
1.00 1.00
1.00
1.00
1.00
1.00
.99
.99
.99
.99
- __
3ct.
”
1. 04
1. 03
I. 02
I. 01
I. 00
.99 .99
.99
.98
.98
.98 .98
.98
.97
.97
. 97
.97
.97
.96 .96
.!xi .%
.9s
.95
.9s
.94 .94 .93
.93 .93
.92
1. 05
1. oc, 1. 07
1. 08
1. 10
1. 12
I. 13
I. 15
I. 16
I. 17
1. 17
I. 18
1. 19
- -
VOV
-
1.01
.99 .98
.95
.93
.91
.91
.90 .90
.90
.89
.89
.88
.88
.87
.86 .86
.84
.85
.84
.83
,232
.82
.80
.8 1
.79
.79
. 78
.77
.76
.76
1. 03
1. 05
1. 07
1. n9
1.11
1. 14
1. 17
1.20
1. 22
1. 23
1. 27
1. 2s
1. 29
- -
lee.
-
I. 02
1. 04
.99
.97
.94
.91
.91
. 90
.89
.88
.88
.8 i
.%
.84 .85
.R3
.8 4
. 83 . 82
.81
. 80
. i 9
.7 i
. 76
.74 . 7s
.73
.72
.il
. i o
. 90
1. 06
1. 12
1. 10
1.16
1. 18
1. 21
1. 2s
1. 29
1.31
1.33
1. 3.5
1.37
1. 41
effect of longer daylight is offset by the diminishing effec-
tiveness of radi a t’ ,Ion.
REFERENCES
1. C. W. Thornthwaite and .1. R. Jlather, “The \\’ater Balance,”
I)rc.rcl Iwstitlrtc of Tecllnologjl Publications in Climato7ogg,
rol. VIII, No. 1. Laboratory of Climatology, Centerton, N.J.,
1934, pp. 1-104.
2. T. E. A. ran Hylvkanla, “The Water Balance of the Earth,”
Drercl Ilrxfitute of Technologg Y?tblicatiown in Clin~atologl/,
rol. VIII, No. 1, Laboratory of Climatology, Centerton, N.J.,
1154, pi). 105-203.
2. L). B. Carter, “‘l’he Water Balance of the Mediterranean and
Black Seas,” Drexel Institute of Tec?i.nology Publications in
Clir~ato7oqy, rol. VIII, No. 1, Laboratory of Climatology, Cen-
terton, N.J., 1954, pp. 20G227.
4. C. W. Thornthwaite, “An Approach Toward a Ieational Classi-
fication of Climate,” Geopvphical Reaiew, vol. XXXVIII,
194S, pp. 55-94.
3. W. C. Palmer and -4. Vaughn Havens, ‘‘A Graphical Technique
for Determining Evapotranspiration by the Thornthwaite
Method,” it1011thly Weatlter RCC~CLL;, vol. 56, No. 4, April 1958,-
6. &I. d‘Ocagne, TlraitC de Nonzographie, Gauthier-Villars et Cie.,
Paris, 1921, Ch. IV.
7. Irwin Remson and T. E. A . ran Hylckama, “Nonlograms for the
Rapid Analysis of Aquifer Tests,” Journal of the Americcts
Water Works Association. vol. 48. 1936. DD. 511-516.
pp. 125-128.