MONTHLY WEATHER REVIEW Editor, ALFRED J. HENRY
VOL. 66 No. 12
W. B. Ao. 974 DECEMBER, 1928 CLOSED FEBRUARY 2, 1929 ISSUED MARCH 2, 1929
A CRITIQUE ON THE CONSTRUCTION AND USE OF MINIMUM-TEMPERATURE
FORMULAS
By ECKLEY S. ELLISON
[Weather Burean. Pomona, Calif, October 21, 19281
INTRODUCTION
Considerable attentibn during the last decade has been directed toward the development of methods for forecast-
ing the minimum temperature using empirical mathe- matical formulas. Many formulas have been suggested,
tempts to consider the subject as an entity to ascertain
the relative merits of the different systems proposed, If we denote the minimum temperature by y, the and to delineate the limitations that necessarily must general equation for its evaluation under ideal frost
exist in the construction and application of minimum temperature formulas in actual forecasting practice.
(1)
5. Topographical situation of station.-Points on the valley floor, in general, are colder than points on the
surrounding hillsides due to the effects of air drainage.
(1) The minimum temperature a t any point, then, would
depend upon the degree of temperature inversion pre- vailing and the relative location of this oint in the
as a function of the temperature inversion, f (I ).
conditions becomes:
some of which are inherently faulty. This paper at- stratified lower atmosphere. Symbolical P y expressed
=f(n +m +f(A> +m) +f(4 GENERAL CONSIDERATIONS
It is reasonable to assume that the factor (t) can be
seasons, since the average length of the night would remain practically constant over the period of extreme
frost danger in the short spring and autumn frost periods, while the longer winter frost period is centered over the
winter solstice when the nights become progressively longer to the solstice and then become progressively .).
shorter at the same rate.
Frost usually occurs under conditions of high baro- neglected provided the data are develope f by frost
metric pressure or when the weather is passing to anti-
cyclonic control following the progression of a cyclone.
Atmospheric stability is pronounced. Weather changes are due chiefly to insolation and rising temperature in the
’ daytime coupled with free radiation and falling tem era-
in temperature under ideal frost conditions, i. e., clear
skies, no wind to disturb the stratification of the lower
under the influence of free radiation to a point below freezing a t sunrise, it is but a step to the enumeration Humphreys states (2): “* * * the temperature of
and mathematical expression of the various factors the surface layer of the atmosphere is chiefly controlled.
that operate collectively to produce these chpiges. by the temperature of the greedily absorbing and freely Under ideal frost conditions the more important factors radiating surface of the earth.” If a thermometer is
that influence the minimum temperature a t any given exposed 4% feet above the ground in a fruit region (4) or
point can be enumerated : other standard thermometer shelter, it has been found
1. Temperature of the radiating surface.-The rate of by experiment that even where the surface texture of radiation varies as the 4th power of the absolute tempera- the ground is made to vary between the extremes of
ture of the radiating surface. Let this factor be es- bare black soil and luxuriant vegetation, there is very
pressed symbolically as T. littler difference in temperature that can be attributed to
2. Length of night, or interval during which free radiation the condition of the ground or radiating surface. (5) (6)
takes place.-Symbolically, t. (7). For the purpose of this paper it can be inferred that 3. Absolute humidity.-The atmosphere would be the free air temperature at a point 4% feet above the practically diathermanous were i t not for the presence ground is a direct function of the temperature of the
in it of water vapor, which possesses the property of freely absorbing and radiating ground surface, and as absorbing to some extent the outgoing waves of heat such can be used in lieu of the temperature of the radi-
radiation. The rate of radiation is inversely propor- ating surface. tional to the absolute humidity. Symbolically expressed Now when it is considered that the rate of radiation
as A. at any given instant during the night chiefly depends
4. Liberation of latent heat Tom condensation and fusion upon the pair of values to be associated with the temper-
and that for every pair of values so associated there is a bolically expessed as L.
ture a t night. From the regularity of the nocturna 7 .fall
layers of the atmosphere, and the temperature falling INDEX TO TEMPERATURE OF RADIATING SURFACE
of atmospheric moisture un 2 er specid conditions.-Sym- ature of the radiating surface and the absolute humidity,
33761-20-1 485
486 MONTHLY WEATHER REVIEW DECEMBER, 1928
corresponding air of values for the temperature of the dew point an$ the relative humidity, it is possible to
express equation 1, empirically, as
(2)
where d and h are the temperature of the dew point and the relative humidity, respectively.
The factor f (L ) is considered to be taken care of by
the new factor f (h ), since the higher the numerical value of the relative humidity a t sunset, the greater is the
probability of the dew point temperature being reached,
and conversely. Some error necessarily is introduced here since the factor f(L) is only indirectly represented
by the factor f (h ), but when it is considered that the
air temperature often remains quite stationary for
several hours after the temperature of the dew point has been reached, it is seen that this error usually is of
small order. This function, now discarded, will be
considered again later. Let us now leave equation 2 in its present form and
proceed to the development of the argument which
ultimately will lead to the evaluat'ion of the factorf(1).
Y =m +m) +f (0
INFLUENCE OF EXPOSURE
Temperature inversion is a phenomenon met with wherever frost occurs. It is due to the fact that the
thermal conductivity of air is so poor that superstrata
of different temperature may exist. At night, under ideal frost conditions, the ground surface is cooled by
the free radiation of its heat below the temperature of the air immediately adjacent. This surface layer of
air becomes chilled through the conduction of its heat to the colder ground surface, while its increased density
holds it to the ground; in fact, even is responsible for
its motion or drift to the surface point of lowest elevation.
Other conditions being the same, the amount of tem- perature inversion will be greater following the day
with the higher maximum temperature. Also will the
.Y degree of temperature inversion be greater on those nights when the values of dew point and relative hu-
midity are relatively low, for it is on such nights that the rate of effective radiation approaches its maximum and
minimum temperatures are lowest. Recent studies in temperature inversion have brought
out the fact that very large differences in temperature
often exist between hill and valley stations (8) (9) (10)
(11)) or a t different elevations on towers erected in cold
orchards (12). Young (9) has found a site in southern
California where the minimum temperature a t a hillside station less than one-half mile distant and 225 feet
higher than a base station was on the average nearly
17O F. higher than a t the base station, with estreine
differences as great as 28' F. and as small as 8' F. Let us assume that temperature formulas were to be constructed for each of these stations, using equation 2.
It is evident that the absolute humidity a t both stations
would be subject onl to minute local variations and that
in the values assigned to the factor j(1). The adiabatic
temperature difference is here neglected owing to its
doubtful existence and its small theoretical value. For
the base station, f(I)=O, and would vanish. For the hill station, f (I) = + 17O, with extreme errors of applica-
tion ranging from - 9' to f 11') or more than SI&-
cient to nullify any accuracy to be obtained by compu-
tation with the other factors.
the only practical d ifference in the formulas would be
In every air drainage system there is some one point
where the slowly movin air f i s t gathers to form the
lowlands. Such a point is called a key point and the temperature station a t the key point is called a key
station. It is obvious that the temperature a t the key
point is unaffected by the degree of prevailing tempera- ture inversion, since it is the point from which the tem-
perature inversions are measured and the one point where
the degree of temperature inversion is always zero. Thus the value of the factor j (I ) can be made to vary more or less a t will merely by moving from one point to another, and can be made to vanish a t the key point.
The general equation for minimum temperature under
ideal frost condit,ions a t t,he key station can now be writ ten :
nucleus of the pool of col % air that later covers the valley
Y =.m +j(h). (3 1
FACTORS DIFFICULT OF MATHEMATICAL EXPRESSION
The ideal frost conditions occur, but rarely in nature.
Other factors are required if Equation 3 is to be made
apphable to all nights.
I . Ejects &e to changing weather conditions.--If warmer or colder air were brought in from surrounding
regions by the general atmospheric circulation, or if the absolute humidity were changed by this wind move-
ment, it is obvious that the minimum temperature would be affected.
2. Ejects due to mechanical action of night winds.-
Such winds are of frequent occurrence. They disturb t,he air drainage, even to the estent of shifting the key
point; raise the surface teniperature by m i n g the warmer air aloft with the colder air near the ground (13);
and sometimes completely prevent the inversion of
temperature. 3. Eflecects due to cloudiness.-Clouds of any type com-
posed of water droplets interfere with free radiation, but
dense lower clouds have greater effect than any other kind (14) (15). High thin clouds, composed of ice
crystals, have little or no effect on the rate of cooling
t,hrough rsdiation. Any one of the factors just named could have suf-
ficient influence by itself to prevent, to a lar e extent,
the usual nocturnal fall in temperature, by %srupting the normal pr0cesse.s under whkh the surface tempera-
ture is made to fall. Many nights which would prove damagingly cold under ideal frost conditions are char-
actjerked by temperatures well above the danger point,
due to the action of winds. Clouds overspreading the sky in the latter portion of the night have been respon-
sible for an abatement of threatened frost on many
occasions, and fog oft'en prevents it. Changes in abso-
lute humidity during the night sometimes alter the rate of free radiat,ion and cause minimum temperatures other
t,han would have been experienced without such change.
These factors are difficult of mathematical ex res-
sion. Their eflects can be approsimated from a stu of current weather maps, and esoterically interprete x8 in
terms of the number of degrees by which the free radia- tion minimum temperature formula estimate must be
modified. Minimum temperature formulas, no matter how accurate they may be on nights when the ideal frost condition prevails, can not be used indiscriminately; nor
has it been found possible to construct minimum tempera-
ture formulas of any practical value for all nights. The best solution to the problem of minimum temperature forecasting is to express mathematically the factors that
DECEMBER, 1928 MONTHLY WEATHER REVIEW 487
can be so expressed and approximate the others. The minimum temperature formula is thus a means to an end
and not an end in itself.
SELECTION OF DATA
Theoretically only the temperature data obtained on
radiation nights should be used in the construction of the minimum temperature formula. In actual practice, how-
ever, it is found that there are very few nights during the
frost season when ideal radiation conditions obtain. On many ni hts when frost gives real concern-there is mofe or
But the fact remains that on most nights when frost does occur, ideal .radiation conditions are present during. the
greater portion of the night, and under these conditlons the minimum temperature closely approximates that
which would have been experienced under ideal frost
less win c f or cloudiness or both over a portion of the night.
conditions.
It would appear that when the normal f a l l in tempera-
ture is interruDted bv adverse radiation conditions of a local nature, such a; intermittent breezes 01: occasional
cloudiness, with a return to the ideal conditions after a
short interval, the fall in temperature is accelerated until tge effects of the temporary temperature rise are obliter- ated and the normal fall is again resumed. There is no
ready way of explaining this phenomenon except by coii- sidering that the nonradiation conditions are intensely
local during any given interval and that the great system of air drainage is not permanently affected to any material
extent. Data from which minimum temperature formulas are
to be derived should include all nights when the minima at the key station were 32’ F. or lower, with the occa-
sional rejection of data obtained on cold nights when
radiation conditions did not preponderate, and some additional data taken under ideal frost conditions even
though the key station minima were above freezing.
POINTS OF SIMILARITS I N FORMULAS
It is a point in common with all minimum temperature
formulas that after the factors to be correlated have been
selected, the actual construction of the formula is pat- terned after the same general method. A “dot chart” is prepared by plotting one factor against another. The
roblem is then concerned with a determination of the
E ne of “best fit” (17) to the data on the dot chart. The
finished formula itself is simply the mathematical expres- sion of this line. The construction of the formula from data accumu-
lated over a number of years presupposes an average condition in the moisture content of the surface soil.
That the extreme condition in soil moisture I? an. impor-
tant factor in minimum temperatufe forecastmg 1s qdite
evident from the practical apphcation of the hygrometric formula during periods of extreme drought or immediately after periods of heavy rainfall. When the surface soil is
abnormally dry, minimum temperatures lower by from
lo F. to 3 O F. than the hygrometric formula estimate usually are experienced. When the surface soil is thor-
oughly soaked With rain, the minimum temperature es- perienced usually is from 1’ E”. to 3 O F. hi her than the
cious use of tbs pnnciple it often is possible to improve upon the accuracy of the formula estimate.
indicated minimum temperature by formu K a. By judi-
TEMPERATURE FORMULAS SUGGESTED
Many investi ators have suggested from time to time
temperature from factors which can be assigned definite
values in the late afternoon or early evening. These formulas can be placed, roughly, into three mathematical
groups :
empirical formu 5 as designed to evaluate the minimum
Group 1. y=f (2‘). Group 2. y=f (d ). Group 3. y=f (d )+f (h).
It is now proposed to deal with each group by itself,
first listing by numbered paragraphs the various formulas as they have been proposed by their originators, and later
critically discussing each formula in a paragraph num-
bered to correspond. The following mathemahcal con- ventions will be used throughout the remainder of the paper:
y IS the minimum temperature,
d is the temperature of the dew point a t an afternoon observation,
h is the relative humidity coincident with d, n is a number deduced from study of data,
V,, is a variable depending on d , and,
Vh is a variable depending on h.
FORMULAS IN GROUP 1
1. “Median-hour ” relationship suggested by Beds
(18). The average time of occurrence of the temperature halfway between the maximum and minimum tempera-
tures is found. The temperature reading taken a t the time of the median is then subtracted from the maximum temperature and the remainder is used to indicate the
approximate fall that will occur between the median and
minimum temperatures.
3. “Post-median-hour ” relationship suggested by Thomas (19). The number of degrees in temperature between the maximum and the temperature registered at
10 p. m. is considered to be two-thirds of the fall between maximum and minimum.
3. “Pre-median-hour)) method originated by Alter (21).
The trend of temperature fall in the early evening is used
by the forecaster to predict the median-hour temperature
by estrapolation, and thus to arrive a t an earlier approxi-
mation of minimum temperature by the true median- hour method.
4. “ Maximum-minimum ’) relationship proposed by Nichols (22) (23). The minimum temperature is con- sidered to be a direct function of the preceding maximum
temperature, so that when the maximum temperature is
known, the minimum temperature can be determined.
5. “Daily temperature range” method formulated by
Smith (26). The average, reatest, and least daily
and the values used in minimum temperature forecasting
after the maximurn temperature is known.
temperature range is compute 5 by semimonthly periods,
DISCUSSION OF FORMULAS IN GROUP 1
1. The median-hour formula is based upon the general
principle of assumed harmonic relationship between time and tem erature under ideal frost conditions. Unless a
special &velopment of the principle is made, however,
there is one type of ni ht that occurs rather frequently under ideal frost con8tions wherein the median-hour
488 MONTHLY WEATHER REVIEW DECEMBER, 1928
formula would not apply. This type of night occurs
when the dew-point temperature is reac.hed, or dosely approached, near the hour of the median. How long thc
air temperature will then rema.in nearly stationary, or
continue to fall a t a constantly decreasing ratme, or
whether the air tempe.rature will fall lower than the
dew point a t all, are matters not related to the. rat,e of effective radiation pwceding the median hour. Since
there is no evidence of any such special development
having been made, i t is probable that some of the in-
accuracies in the application of the median-hour formula are due to this necessity for segre.gation of data.
Many investigators who have tried t,o utilize this type
of formula have reported unsatisfactory results. It would appear that the median-hour formula is open to
some serious practical objections.
To give satisfactory results any type of ininimum- temperature formula must have some application on
nights when ide.al frost conditions do not obtain a t all times between sunset and sunrise. Many cold nights
are preceded by cloudy afte.rnoons, as frost is obse.rved
frequently following the passage of a c,yclone, and the
time of occurrence of the maximum temperature is affected to such an extent as to impair the applicnt,ion of
the median-hour method. A rapid drop in air te.mpe.rature in the early part of
the night is quite often followed, in many clist.rict’s, by local winds which cwse fluctuating temper a t, ures over
short intervals. In fact, in some places, the t,opography
is such that the fall in temperature must be conside,red
as a causative agent in the production of loc.al winds, as,
for example, the well-known phenomenon of mountain and valley winds. The median-hour method stakes all
on the air temperature a t a certain instant, yet the
tem erature a t this instant is often affecked by local
conitions.
The time of occurrence of the median hour in nmny
sections of the country is so late that it is impracticabl? to use the formula in the preparation of forecasts.
Changes in absolute humidity near the time of the median will seriously impair the application of this
method by changing the rate of effective radiation afte,r the median temperature occurs. Such absolute hunlidity
changes are frequenFly observed in mountainous country
due to reversal of mnds aloft under special condibions.
2. The post-median-hour relationship is open to the
same objections as those just listed for the median-hour
formula, except that the lateness of the zero hour detracks
even more from its practical use.
3. The pre-median-hour method also is subject to the.
same criticisms. The method was originally derisecl to
overcome one of the principal objections to the median-
hour formula, namely, the delay in the preparation of the minimum temperature forecast while wa.iting for the
median hour to be reached. It is evident that the chances
for error in the approximation of t,he median-hour t,ein-
perature would make the forecasts by this method subject
to more risk than by the median-hour method itse,lf,
without any hope of attaining greater accuracy in the predictions.
4. The maximum-minimum temperature formula sim- ply.states that with a given maximum a certain range will ensue and a gven minimum be reached. When it is con-
sidered that with any given maximum temperature a
variety of values for absolute humidity are observed in
practice, it is evident that differences in the rates of free radiation of the earth’s heat will occur during different
nights and consequently a variety of minimum tempern-
tures are experienced. These formulas, then, appear to be faulty a t the.ir source. Nichols, who is the chief proponent of the maximum-
minimum type of formula, in the endeavor to improve
the usefulness of the basic relationship, has extended the application by a complex system of type classification
wherein five classes of weather conditions are recognized
(24). He concludes, however, that the relationship ie inferior to the hygrometric correlation and that
“* * * greater inaccuracy is likely to result from incorrect classification than from inaccuracies in the for-
mulas when correctly applied” (25).
5. The daily temperature range method is a variation
of the maximum-minimum method previously discussed and is open to the same objections.
From a purely abstract consideration of values inherent
in temperature formulas of the form y =f (T ) it would seem that in so far as accuracy is concerned they would
range in the following order:
1. Postmedian-hour formulas.
2. Median-hour formulas.
3. Premedian-hour formulas.
4. Maximum-minimum formulas. The postmedian-hour formulas would be the most accu-
ra.te and the maximum-minimum formulas the le@ accurate, owing to the variation in the time to elapse
be.fore the occurrence of the minimum temperature
after the determination of T. A forecast prepared one or two hours in advanc.e of an event certainly should be expected to ha,re greater accuracy than a forecast pre-
pared 12 to 14 hours in advanc.e.
In actual pract>ice, however, the minimum tempera- ture forecast must be made and disseminated long before
the time of occurrence of the median hour in most parts
of the country. This fact alone excludes from serious consideration all the formulas of this group except the maximum-nlinimum and certain of the premedian-hour
formulas. Thus, the only formulas of the form y=f(T) available for practical use are the least accurate of the
group.
FORMULAS IN GROUP 2
I . “Evening dew-point ” relationshp proposed by
Humphreys (3): The temperature is assumed not to fall below the value of the coincident dew-point tem er-
ature, which, for forecasting purposes, is considerea to be closely the same as the evening dew point. In other words, the minimum temperature is determined by the value of the evening dew point.
2. “ Wet-bulb” method originated by Angstrom (27):
A constant is subtra.cted from the wet-bulb temperature a t sunset and the remainder indicates the ensuing mini-
mum temperature.
3. “ Wet-bulb-minimum temperature ” method pro-
posed by Heyser (28): The average difference between the u-et-bulb temperature a t 5: 00 p. m., and the ensuing
minimum temperature is found. The 5 : 00 p. in. wet-bulb temperature, when decreased by the amount of average difference, is the estimated minimum temperature.
4. “Depression of the evening dew-point ” method
tmried by Smith (29): The difference obtained by sub- tracting the evening dew-point temperature from the coincident air temperature a t an evening observation
is used in correlation with the difference obtained by
subtracting the evening dew-point temperature from
the ensuing minimum temperature. The relationship thus determined enables the minimum temperature to
be computed from evening observational data.
DECEMBER, 1928 MONTHLY WEATHER REVIEW 489
5 . “Depression of the dew oint below maximum
difference obtained by subtracting the temperature of
the evening dew point from the preceding maximum temperature is argued against the ensuing range in
temperature between maximum and minimum. The
relationship determined enables the minimum temper-
ature to be computed from the evening observational
data.
temperature,” method originate B by Nichols (31) : The
DISCUSSION OF FORMULAS IN GROUP 2
1. Soon after its initial formal statement, Smith (33) provided a refutation based on observational fact of the
principle inferred in the evening dew-point formula,
namely that the minimum temperature would not be lower than the temperature of the evening dew point.
Other investigators (11) and the meteorological records
taken in connection with fruit-frost work establish the soundness of Smith’s argument. Generally speaking,
the relationship can not be consistently demonstrated except for elevated stations, well placed in the inversion
layer. At key stations the minimum temperature often is lower than the evening dew point by more than So to
loo F., with occasional extremes of more than 20° F. The formula, therefore, is inherently faulty.
2. The wet-bulb temperature is determined when the coincident dr -bulb temperature and the dew point arc
when the relative humidity is 100 per cent,, for it is then
that the wet and dry bulb temperatures coincide, and its value becomes progressively less with decreasing
relative humidity. I n other words, wet-bulb tempera-
tures and relative humidity change in direct ratio with unchanging dew point.
The wet-
bulb temperature can now be made to vary by causing
that of the dry bulb to change. If the difference between the minimum temperature and the wet-bulb readings
a t an evening observation is always to be a constant,
then we are justified in assuming that the difference between the minimum and the dew point, which remains
fixed, will likewise be a constant for all values qf rt77ati2.e
humidity. But when the dew point is fixed, Equation 3 asu cures us
that the variation of the nlinimum temperature from the
evening dew point depends upon the relative humidity,
or,
and investigators are universally agreed that the right-
hand member of this equation is not a constant. The
hygrometric dot charts that comprise Supplement 16,
Monthly Weather Review, offer ample proof of this
known. Un d er any given condition its value is greatest
Assume the dew point to remain constant.
,?/ - d =J(h)
~~
c,onten.tion. During the winter of 1922-23, Dague experimented
with &uzstrorn’s formula in a district in southern Cali- fornia a t d found in this season that, the variation of the
minimum temperature from the evening wet-bulb tem-
perature was not a constant, but had valws ranging
from +12 P. to +23’f’. (35). We conclude that ngstrom’s wet-bulbi formula is
fundamentally in error and must be rejected:%
3. This formula is of the same order as the one pre-
viously discussed.>% Keyser,-the proposer- himself, admits its inferiority (28). I *$ ,i’:f
4. Some rather 4‘unsatisfactory attempts have been
made to correlate1 the depression* of the eveningxdew point with the variation of the minimum -temperature
from the evening dew point (29). The difficulty with the relationship here expressed lies in the fact that when
the depression of the evening dew point is computed a pure number is obtained which may be the same for
widely differing values of absolute humiditfly or air tem- perature. Thus, for example, the number 21, which
here is taken to represent the difference between the air
temperature and dew-point temperature at an evening observation, mn-y occur under any condition of absolute
humidity, as the air temperature may vary in such manner that a differential of 21 always is maintained; or the same number 21 may be obtained with any value of
air temperature, depending on the absolute humidity.
The appearance of the differential 21, therefore, is no
indication whatever of bhe amount of moisture in the air. Many widely diflering rates of free radiation, con-
sequently with widely differing minimum temperatures,
can occur with any designated differential.
5 . The depression of the evening dew-point temper-
ature below the maxinium temperature formula simply states that a certain range in temperature will ensue
following the occurrence of a certain differential between
masimum temperature and evening dew point. But the teniperature difference between afternoon maximum and
evening dew point is not a ineasure of t,he absolute humidity, for these latter are independent factors and almost any difference can obtain between them when
one or the other is re arded as being fixed. With any designated differentiay between afternoon maximum
temperature and evening dewpoint, therefore, a variety
of values for absolute humidity may exist, and with them, a variety of nocturnal temperature ranges instead of but one.
FORMULAS IN GROUP 3
Strictly speaking, there has been but one fundamental
relationship of the form y=f(d) +f(h) set forth by inves- tigators, although a t first, glance several different for-
mulas seem to appear. The difference lies not in the basic relationship itself but in the methods used to
express this relationship mathematically. The differ- ence in formulas, therefore, is one of form rather than of
concept.
1. ‘(Hygrometric” method. The fundamental con-
cept underlying all hygrometric formulas in this group is that the ensuing minimum temperature will be greater
or less than the evening dew-point temperature by an
amount depending on the relative humidity. In this method the difference obtained by subtracting the eve-
ning clew-poin t temperature from the minimum tem- perature is argued against the evening relative humidity.
The relationship is used to predict the minimum tem-‘
perature from evening observational data.
DISCUSSION OF FORMULAS IN GROUP 3
1. The hygrometric relationship is the only one so far
proposed that conforms strictly to equation 3, evolved
for predicting the minimum temperature at the key sta- tionAunder ideal frost conditions. It is to be expected
that the hygrometriciformulas should be the most satis- factory. The greater part of theipublished literature on
miiiimurn temperature forniulas has to,,deal with those based on;this relationship, and t,he consensus of investi-
gators is,ealniost unaminous in awarding to them the wreath ;of superiority. These formulas, then, deserve
more than a passing glance. The hygrometric relationship was f i s t proposed by
Donne1 in 1910, while working on Boise, Idaho, frost
490 MONTHLY WEATHER REVIEW DECEMBER, 1928
records (36) (20) (37). Donnel developed a hygrometric
formula for Boise based on psychrometric observations taken a t 6 :O O p. m. through the assumption that the line of best fit to the hygrometric dot chart was a straight line
whose equation is of the form
So far as known, no extended practical use in forecast- ing was ever made of this equation and it was regarded as unsatisfactory except for a cert,ain narrow range of rela-
tive humidity and dew point, where fairly consistent
results were obtained.
In August, 1917, Smith published in the Monthly
Weather Review the results of an investigation of the hygrometric relationship based on fruit-frost work in Ohlo since 1915 (34). He followed a different line of
attack than Donnel and determined from the correlation
was cloudy a t observation, but indicated values too high when the weather a t observation was part’ly cloudy or
clear. This defect was remedied by writing separate equations for these two classes of nights. In effect, this
amounted to subtracting 2%’ F. and 5’ F., respective1 ,
cloudy and clear nights. In using these equations it was found that when the
5:OO p. in. dew point was below 30’ F. or above 40’ F., the minimum temperature indicated by the formula
consistently varied from the actual minimum temperature by an amount that was nearly constant with t’he same dew
point. Also, when the relative humidity a t 5:OO p. m. was above 67 per cent the formula needed revision
upward by an amount depending on the relat,ive humidity. Thus was developed the device called the “method of arbitrary corrections,” by means of which the basic
strai ht-line formula was made to take on an irregular curvi 7 inear form. Later, the results of t’his invesihgation
from the original formula to make it applicable on par d y
0
FIGURE I.-Hygrometric dot chert for key station et Medford. Oreg., based on spring frost records from 1917 to 1024. State of weather at ohwrstion is shown by symbob. The parabolic curve of formula 1 and the straight lines of formula 3 hare heen ulut ted on the chart
coefficient the probable existence of a linear relationship,
later calculating the straight-line formula by the method of least squares (38) and expressing t,he hygrometric
were published in Supplement 16 (16).
his formula after this form:
h-n
4
Young expressed
formula in the following convent,ion: y =d - -+ 176+ 17h.
Y=a-bR .where
Y is the difference between the minimum tempera-
ture and the evening dew point, or Y = y-d;
R the evening relative humidity, or R = K ; and a and b are two numbers deduced from study of data,
a = nl/% and b = l/nz.
Marvin has already demonstrated the mathematical identity of these original Smith and Donnel equations
(36). In this case both investigators arrived at exactly the same place by following different routes; one of which was considerably longer and more difficult t’han the other.
In the spring of 1917, Young, while engaged on fruit-
frost work a t Medford, Oreg., investigated the hygro- metric formula proposed by Donnel for Boise (39). He
used data taken a t 5 :O O p. m. instead of 6 :O O p. m., and found that the original Donnel equation gave fairly
accura,be results a t Medford on cold nights when the sky
.
or
He, therefore, was the b s t to use in actual minimum temperature forecasting a hydrometric formula which expressed a curved line of best fit to the data from which
i t was constructed.
In July, 1019, Smith published in Supplement 16 a
met,hod for fitting parabolic curves to hygrometric dot charts after the Marvin “star-point” system (17) and the
mathematical expression of hygrometric formulas in the form of parabolic equations :
y - d = n1 + n,h + n3h2.
Nichols found that a rectilinear hyperbola would, in some cases, give better correlation than the parabolic
type curve (32). Nichols (25) and Keyser (40) brought forward t,he idea
that dot-chart data in all cases might not best be repre- senhed by any of the simpler mathematical curves and
suggest,ed t,he extension of t,he “star-point” system to the
DECEMBER, 1928 MONTHLY WEATHER REVIEW 491
Partly cloudy or
Clear _______________
cloudy.
production of a curve drawn by eye alone. This free- hand curve could than be used directly from the dot chart without recourse to further mathematical methods.
Since these various hygrometric formulas and fore- casting methods are all based on the fundamental rela-
tionshp expressed in Equation 3, it is inipossible to make selection of the best one of the group without some
investigation into the relative merits of nll. Perhaps the best method of making this comparison would be to
develop all the formulas from the same data, and then to determine the relative wort4 by applying the fromulas
back on the data from which they were derived. If one formula were superior to another this method surely
would bring it to attention.
An intensive study of hygrometric data taken a t the
Medford, Oreg., key station during the spring frost
seasons 1917 to 1928, inclusive, is now proposed. I n this study only the finished dot charts, formulas, and
frequency diagrams of the investigation will be presented.
Reference is made to the hygrometric, dot chart shown in Figure 1 from which a number of formulas are now to
be derived. On this chart are plotted data secured on
106 nights a t the Medford, Oreg., key station during
the spring seasons over a 12-year interval. The data
have been selected to include all nights when the mini-
Any value------
45% or more ____
i}y=d-s -
and its position shown on the dot chart. (See figure 1.)
When this formula is applied back on the data from
which it was derived the results shown in the frequency diagram are obtained.
If, now, t'he method of arbitrary corrections be applied
to the Smith formula as a base, the new formiila becomes
(See figure 3.)
y-d=34-1.6h+.015h2+ Va+ Vh. Formula 2.
with values for the variables shown in the table below:
d vd I h v h
I -__
186% to
When Formula 2 is applied back
the results shown in the frequency
are obtained.
FIGWEE 2.-Hygrometric dot chart for key station at Medford. Oreg., hased on spring frost records from 191: to 1928. State of weather at observation is shown by sym- bols. The free-hand curve of formula 5 and the h.:sic straight lines of formula 7 have been plotted on the chart
mum temperature fell to 34O F. or lower, provided the
sky was clear over the greater portion of the night. Symbols on the chart show the atrate of the sky a t the
time of observation. Naturally, a large portion of the data were taken on nights not ideal for free radiation and
the wide scattering of the dots may be attributed to this
reason. The rejection of data on nights when the sky did not clear until after midnight would result in a much
more compact arrangement of the dots. This was not
done because it was desired to deal with the situation under conditions as they occur in actual practice.
SMITH FORMULA
The f i s t formula to be considered is the parabolic type patterned after the Smith met,hod. Three "Star
points" have been selected as indicated on Figure 1 ;
one point for each 35 dots. The hygrometric formula is expressed :
y - d = 34 - 1.6h + .015ha. ' Formula 1.
It is obvious from the frequency diagrams that Formula
2 gives much better application to the data than Formula
1. Not only is the amplitude of variation greatly re-
duced, but also the accuracy is materially increased.
DONNEL FORMULA
Examination of the dot chart in Figure 1 shows a t once
that it would not be possible to select a single straight line to represent the data faithfully. Accordingly a sub-
division of the data has been made in order to apply the Donne1 method.
Foqula 3 State of weather at Relative humidity at observation 1 observation I 1 &-30 Clear _______________ 44% or less _____ y=d-- 3
492 MONTHLY WEATHER REVIEW DECEMBER, 1928
The position of these lines is shown in Figure 1, and the
application of the formula is pmsented in the frequency diagram in Figure 5.
Applying the method of arbitrary corrections to For- mula 3 changes it to this form:
State observation of weather at Relative observation humidity at I 1 Formula 4
where the variables take on
d vd
he following values :
The application of Formula 4 is shown in Figure 6.
Again has the aniplitude of variation been reduced and
the accuracy and dependability of the formula greatly
increased.
DECEMBER, 1928 MONTNLY WEATHER REVIEW 493
above 52 per cent. It is in reality a combination of the
better parts of both Formula 1 and Formula 3 and the frequency diagram showing its application assures us
that it is better than either the Smith of the Donnel formulas taken separately. See Figure 7.
The addition of the method of arbit,raiy correctmioils to
Formula 5 changes it to this form :
y=cz+ v,+ V h . Formula 6 .
where the variables take on values a,s follows:
h v h
+IS%
+ 15%
+ 13% +11:.4
+ 10
Z Y + 5%
+ 4%
+3% +4 +3
+2
+ 1%
? -1
- 1% - 1>$ -2 -2
- 2% -3 -3
- 3:a -4
-4
-4% -5 -5
- 5%
Vh
-6 -6 -6 - 654 -7
-- 7 5
- ih -734 -7
C '
l r d
The application of Formula 6 is shown in Figure 8. As
, before, an improvement of the orignal formula has been effected by these simple means.
1-OUNG FORMULA
Young uses a base equation of the Donnel ttype for
each of three classes of nights, depending on the state of weather a t observation. The method of arbitrary cor- rections is applied to the formula as a whole. The base
equations in this case are plotted in Figure 2 and the
hygrometric formula expressed :
State of weather a t ohserration Formula i
i k -
The application of this formula is shown in Figure 9.
It is a fact worth noting that the amplitude of variation in Figure 9, roughly from 5' too high to 5 O too low, is
nearly the same as shown in figures 4, 6, and 8. The types of frequency curves in these figures, to, are not, entirely dissimilar. If the mass of data were greater the similarity undoubtedly would be more pronounced.
Assuming that the data on the dot chart have but
one correct interpretation, only one line of best fit can be drawn. The evidence in this case points strongly
to the idea that the line of best fit is irregular, since the
addition of, the arbitrary corrections to any base formula
results in an irregular line unless the base formula itself
is the line of best fit, in which case the corrections vanish.
In each case he.re considered the irregular line formula
produced by the corrections gives better application to
the data than 6he regular base formula by itself. And in each case the irregular line formula when applied
to the original data produces the same general type of
frequency curve, regardless of the type of base formula
used in the derivation. We c.onc.lude, therefore, that the addition of the method of arbitrary corrections to
any base line or curve will produce the same irregular
line of best fit to any hygrometric dot chart'.
We are now in position to select the best type of hygro-
metric formula. Since the formulas are iclentical, we may select any in which the method of arbitrary corrections has been considered, and since the base formula can be any
line or curve, we are justified in choosing the simplest, which is, of course, the straight line. We make selection, then, of the formula which uses a straight line as a base
but changes to an irregular curve, if necessary, by the
addition of the arbitrary corrections. These spec,ifications fit the formula constructed by Young's method.
I n seeking out the hygrometric formula to apply to different key stations, however, the investigator must
always be on the alert to recognize any special conditions
with which he may be confronted. While a formula developed after the methods just outlined will give good
application a t any well chosen key station, a different construction placed on the data ma result in an even
in such form as to give greater weight to local conditions of cloudiness.
better formula. For instance, Formu i a 7 can be rewritten
State of weather at observation 1 Formula 8
d v dl
d Yd2
h v h l
h V bE
494 MONTHLY WEATHER REVIEW DECEMBER, 1928
The frequency diagram in Figure 10 the applica-
tion of this formula shows that the amp tude of variation
has been reduced to 3' for 103 of the 106 cases of frost used to construct the formula.
METHOD OF CONSTRUCTING YOUNG FORMULAS
Since it is a conclusion of this study that the Young type of formula is superior to the others, it may be well to state briefly, step b step, an easy method of develop-
1. Construct a dot chart after the methods outlined by
Smith (30). This is the initial step in the mnstruction of any ninimum temperature formula. Either make a
separate chart for clear, partly cloudy, and cloudy nights,
or better, place all data on one chart, denoting segrega- tion by use of inks of different color. A small indes
number keyed to the original data sheet is placed near
each dot for identification purposes.
2. Draw a straight line on the chart to represent, as nearly as a straight line may, the data. To simplify
the formula, care should be taken to select a line capable
of whole number expression, even though the position of the true line is shifted sli htly. Draw a line for each
ing this type of formu T a.
class of data if one line wil B not represent the whole.
3. Express this line in the form:
where n, is the number measured on the horizontal relative humidity scale where the, line intersects the
X-axis, or zero line of departure, and, n2 is the number of
units measured by projecting the line on the horizontal relative humidity scale, that the line changes in order to
pass through one unit of variation by projection on the
vertical scale of departures.
4. To facilitate the calculation of the arbitrary cor-
rections, the data for each day should be written on a
card. The example shown below indicates the data to be considered and the method of preparing the cards. The
headings, of course, can be omitted.
Evening Evening Minimum I 1 umidity I ture 1
No. dew point hrelative tempera- State of weather Index I
116 1 30.5' I 76% 1 28.4
~
Cloudy. Cleared9:30p. m.
5. Using the straight line formula just calculated, go through the cards and calculate the formula minimum
temperature. Find the departure between this and the
actual minimum temperature.
6. Now arrange the cards in the order of increasing
relative humidity. At more or less regular intervals-say,
about 10 cards-calculate the average departure of the
formula minimum temperature estimate from the actual
minimum temperature. This is the arbitrary correction to be applied to the formula over the range of relative
humidity indicated. Apply this correction to the original formula estimate and calculate the new departure.
7. After the relative humidity corrections have been
determined and applied, arrange the cards in the order of increasing dew point. Repeat the process just out-
lined, thus determining the arbitrary correction to be
applied to the formula over the range of dew point indicated.
- ~-
CONCLUSIONS
Under ideal frost conditions the air temperature at the
surface falls during the night,, due to t'he influence of
severa.1 factors operatsing simultaneously. The cumula-
tive effects of these several factors can be given mathe- matical expression, and the minimum temperature closely
calculated from hygrometric data taken near sunset on
the preceding day, provided care is taken to locate the point for which forecasts are to be prepared with due
regard for local topography. The forecast or key station should be placed as nearly
as possible to the point where the cold air that drains
from the surrounding slopes first gathers to form the
nuceleus of the pool of cold air that covers the lowlands in the morning. By so locating the key station it is possible
to avoid entirely the effects due to temperature inversion. Minimum temperature forecasts by formula are not
satisfactory for elevated stations. The errors introduced
by the factor of temperature inversion are usually so
great as to void any accuracy 60 be obtained by compu-
tation from the other factors. As a general rule it may be stated that the greater the influence of inversion on
t,he minimum temperature a t any statmion, the poorer will be the application of the minimum temperature formula.
The formula must have a t least as great avariation in appli- cation as the difference between the average and extremes
of temperature departure due to inversion; Formula con-
struction for an elevated city station is a hopeless task. As the means of accomplishment are always less im-
portant than the ends to be attained, so is the hy m- metric formula minimum temperature indications of K ess importance than the final forecast of minimum tempera-
ture, which properly is the result of processes both mathe-
matical and empirical. But the fact remains that the hygrometric formula is an indispensible tool in the hands
of the forecaster. Under any of the conditions where frost
is formed a skillfully designed hygrometric formula based
on key station records over at least a 10-year period offers a purely mathematical method of placing the ininimum
temperature estimate within 3' F., of the actual to be ex-
pe.rienc.ed fully 90 per c,ent of the time, and in most c.asee
within 6' F., in the remaining 10 pe,r cent. The skill of the forecaster is then direcked toward an improvement of the
original mathematical estimate within this narrow range.
general, i t is easy to recognize the cases where the depar-
true w-ill be in ewess of 3' F. This occurs during senous freeze types of weather where the maximum temperature and dew point are unusually low, with an 'accompaniment
of winds of decided force; or else when local winds and cloudiness are to be experienced during a portion of the
night. The experience of the previous morning often is
a t hand. The amount of moisture in the surface soil
sometimes offers a dependable means of correction. For the rest, the personal esperience of the forecaster in hie own particular district, and his personal ability to deter-
mine accurately the immediate weather from the current
weather chart and translate these indications into the
terms of degrees in temperature) is put to test. The forecaster also must, be able to differentiate between
the condit,ions where frost will or will not oc.cur. It is a
fac.t' that the majority of nights during the period when frost ditnger is most ac.ute would be frosty if the sky re-
mained clear and there was no surfac.e wind. The hygro- metric formula, being constructed from data taken on
cold nights only, invariably indicates this tendency. But all nights during the frost season are not actually frosty.
Some are cloudy; some are windy; and some are rainy. I t is the forec.aster's own job t,o segregate the nights when
the formula mill have application. Properly used, the formula will indicate about how low the t,empera,ture at
the keg station will fall under ideal frost condit,ions, and
gives t'lie forecaster a base upon which to build, but it has
The forecaster has many things to guide him. In .
DECEMBER, 1928 MONTHLY WEATHER REVIEW 495
nothing but theoretical application on nights when frost is not experienced. The hygrometric formula is not c.on-
structed for use on nights like these Of the many temperature formulas that have been
suggested, * some are inherently faulty and others have but linlited use. The fainily of temperature formulas
based upon the hygrometric relationship has been shown in this paper,to be out'standing!y superior. It so happens,
however, that within this family of hygrometric formulas
there is one that maintains a flexibility in construction and accuracy in practical application in a degree not attained by the others. This is the Young formula wherein the
basic relationship is linear and close construction given to the data by a series of arbitrary corrections.
Much in useful accuracy is lost if the variables V , and
Vh are not computed. It is not enough to carry the formula developnient only to the point where the base
formulas are determined, for although in such c.ases t,here. is likely to l e escellent, correlation between the factors,
values usually can be assigned the variables which will result in even c.loser correlation. The variables provide :
1. For the influence of local topography, and,
2. ,4n enipiric evaluation of j (Lj. If only a short record is available for a key station where it is desired to develop a formula, it will be found
that not enough data are a t hand from which to construcl a satisfactory formula of the Young type. I n this case
the Smith or Nichols type of formula is recommended, as
these formulas lend theinselves readily to int,erpolat,ion.
Later, a formula after Young's method can be constructed. Any of t,he other formulas in the hygrometric family, when given the same treatment of arbitrary correction as
used in Young's formula, have the same flexibility and
accuracy. The smooth inathematical line of be.st fit is
altered by the corrections to approach the posit,ion of an
irregular line of best fit. There is no evidence to support
the idea that the data on t,he dot chart must conform t'o some standard mathematical line. or c.urvc?. The evi-
dence, rather, points to the contrary vkw t,ha.t, in most ca.ses, the line of best fit is an irregular curve. By t,he
applica.tion of the method of arbitratly c.orrec.t>ions
all the forniulas in the hygrometric group are reduced 6o the same general irregular form regardless of the form of
the base formula, whether parabola, hyperbola, fre.e-hand curve, or .straight line. In fact, all hygroinetric. formulas
wherein the arbitrary corrections have been considered can be shown to be mat~heinatically identical.
Selection of the Young straight-line formula as the
superior formula of the hygrometric group is made for the
reason that, since all the formulas when given similar treatment by arbitrary correc,tions are mathemrtt,ically
identical, it is proper to select the simplest inea,ns of
accomplishing a desire.d end.
LITERATURE CITED
(1) Air Drainage Explained. Charles F. Marvin. Monthly Weather Review. October, 1914. 42: 553-585.
(2) Frost Protection. W. J. Huniphreys. Monthly Weather Review. October, 1914. 42: 562-569. Page 562. (3) Same citation. Page 565. (4) Influence of Exposure on Temperature Observations. Floyd D. Toung. hlonthly Weather Review. December,
(5) Influence of Cover Crops on Orchard Temperatures. Floyd D. Young. Monthly Weather Review. October, 1923.
(6) An Experiment tp Determiiie the Effects of Cover Crops on the Frost Hazard. Ecliley S. Ellison. Abstracted in Monthly Weather Review. October, 1922. 50: 526.
(7) Further Study of the Relation between Cover Crops and
Orchard Temperatures. Floyd D. Young. Monthly Weather Review. September, 1925. 53: 387-391.
1920. 48: 709-711.
50: 521-526.
(8) Effect of Topography on Temperature Distribution in Southern California. Floyd D. Young. Monthly Weather Re- view. August, 1920. 48: 462.
(9) Nocturnal Temperature Inversions in Oregon and California. Floyd D. Young. Monthly Weather Review. March,
Floyd D. Young. Monthlv Weather Review. November, 1923.
1921. 49: 138-148. Notes on the 1922 Freeze in Southern California.
61: 581-585. "
Temperature Survey of the Salt River Valley, Ariz. James H. Gordon. Monthly Weather Review. May, 1921.
Rate of Increase in Temperature with Altitude During Frosty Nights in Orange Groves in Southern California. Floyd D. Young. California Citrograph. March, 1920. Pa e 136. Frost trotection by Artificial Mixing of Air. Floyd D. Young. California Citrograph. February, 1924. Page 125. Smoke Cover and Direct Radiation in Frost Protection Floyd D. Young. California Citrograph. November, 1920. Pa e 6. Forecasting hfinimum Temperatures in Oregon and Cali- fornia. Floyd D. Young. Monthly Weather Review, Su plement 16. Pages 53-60. Page 57. Same &tation. Pages 53-60. Introduction. Charles F. Marvin. Monthly Weather Re- view, Supplement 16. Page 5. Forecasting Frost in the North Pacific States. Edward A. Beals. Bulletin No. 41-W. B. No. 173. February 27, 1912. Page 39. Same Citation. Page 36. Same Citation. Page 17. Forecasting Minimum Temperatures in Utah. J. Cecil Alter. Monthly Weather Review, Supplement 16. Page 49. Predicting Minimum Temperatures in Grand Valley, Colo. Esek S. Nichols, Monthly Weather Review. May,
Notes on Damage to Fruit by Low Temperatures; Prediction of Minimum Tern eratures. Esek S. Nichols. Monthly Weather Review, gupplement 16. Page 41. Same citation. Page 39. Same citation. Page 42. Frost Warnings and Orchard Heating in Ohio.
Studies of the Frost Problem, I. Anders Angstrom.
49: 271-274.
1918. 46: 213-228.
J. Warren Smith. Monthly Weather Review. October, 1914. 42:579. Geo- grafiska Annaler. January, 1920, pa es 2&32. Ab- stracted in Monthly Weather Review, aovember, 1920. 48: 640-641 by J. Warren Smith. Damaging Temperatures and Their Calculation in Advance By Simple Arithmetic. Elgie M. Keyser. Proceedings of the Washington State Hort.icultura1 Association for 1922. Olympia, Wash. Pages 97-103. Predicting Minimum Temperatures from Hygrometric Data. J. Warren Smith. Monthly Weather Review, Supple- ment 16. Page 18. Same citation. Pages G l 9 . Notes on Formulas for Use in Forecasting Minimum Temper- atures. E. s. Nichols. Monthly Weather Review. December, 1926. 54:49%501. Page 500. Same citation. Page 499. Predicting Minimum Temperatures. J. Warren Smith. Monthly Weather Review. August. 1917. 45: 402-407. -I
Page 402. Same citation. Pages 402-407. Report of Temperature Survey and Fruit-Frost Investiga- $ions, San Gabriel Valley District, Southern California. Charles I. Dague. Special report on file in Division of Agricultural Meteorology, Washington. Page 15. Historical Note. Charles F. Marvin. Monthly Weather Review, August, 1917. 46:407. Report on Fruit-Frost Work in Southern Idaho. Edward L. Wells. Special report on file in Division of Agricultural Meteorology, Washington. June, 1910. Element.ai-y Notes on Least Squares, The Theory of Statistics and Correlations, for Meteorology and Agriculture. Charles F. Marvin. Monthly Weather Review, October,
Report on the Frost Warning Service in the Rogue River Valley, Oreg., for the Spring of 1917. Floyd D. Young. Special report on file in Division of Agricultural Meteor- ology, Washington, D. C. Pages 14-34.
Calculating Temperature Extremes in Spokane County, Wash. E. M. Keyser. Monthly Weather Review, October, 1922. 50: 526-528. Page 527.
1916. 44: 551-569.