JULY 1955 MONTHLY WEATHER REVIEW 143
PERSISTENCE OF WARM WEATHER AT DENVER, COLORADO
R. K. LEATHERWOOD
Weather Bureau Airport Station, Denver, Colo.
[Manuscript received April 2, 1954; revised May 18. 19551
ABSTRACT
A 40-year record of maximum temperatures at Denver is examined to identify sequences of days which show
a maximum temperature 10' F. or more above the normal. The actual persistence of a warm-weather regime is
compared with that on chance, on an annual basis, and also during particular seasons in which comparable magnitudes
of persistence might be expected. The skill scores of persistace forecasts for "tomorrow" and "tomorrow and the
day after tomorrow" continuing a arm spell are computed for November-March and September-October regimes.
1. INTRODUCTION broken sequence of occurrences. In table 1
The location of Denver in the rain shadow of the Con-
tinental Divide and in an area favorable for chinooks [l],
while at the same time high enough to escape many of the
polar outbreaks whichsweep southward over the Great
Plains, is reflect.ed in t'he length of periods of warm
weather. Warm weather spells are closely tied to t,he
persistence of the Great Basin High which prevents in-
fluxes of Pacific air from reaching Denver with major
force, and they counteract outbreaks of polar air over the
Plains [2]. The purpose of this study is to determine the
relative importance of the factor of persistence in the
length of warm spells at Denver.
2. PROCEDURE
For the purpose of this study, a "warm" day was arbi-
trarily defined as one on which the maximum temperature
reached a value 10' F. or more above normal for that day
of the year. A 40-year record (1911-1950) of maximum
temperatures was examined. Daily normal maxima were
derived from smoothed curves of monthly normal maxima
and each day of record was compared with the normal.
Table 1 shows the totals of runs of consecutive warm
days for the period of record. Each run was assigned to
the month in which the greatest number of warm days in
the run fell; if the run was equally divided between two
months it wasassigned to the month in which the run
started. The total number of days in the study was
14,610, of which 2,880 were warm days, as defined above.
Brooks and Carruthers [3] give the following formula
for computing the number of runs of different lengths
expected in a series in which there is no persistence:
f(N, p , p, T ) -Nqp', where N is the total number of days
considered, p is the probability of occurrence of a type
under consideration in a unit period, p is the probability
of non-occurrence (1 -p ), and T is the length of the un-
p=2880/14610=.197, q=.803, hT=14610,
and T takes on the successive values 1, 2, 3, . . . 12.
I t was recognized that the variation of the weather
with the seasons wouldgivediffering persistencies, so a
division was made into comparable weather regimes.
During the late spring and summer, when temperatures
are relatively high and thunderstorm activity is a potent
factor in keeping maximum temperatures down, the
incidence of warm days (as defined in this study) may
be expected to be small, and this is borne out by the
data in table 1. This also corresponds to the persistence
variation noted by Blair [4] for Lincoln, Nebr. In late
fall, winter, and early spring there are low average tem-
peratures and a rapid rate of movement of pressure
TABLE 1.-Number of warm-weather sequences of various lengths by
months, totals of those sequences both separately and cumulatively from longest to shortest, and number of those runs expected on chance (Brooks and Carruthers [SI). Data for the years 1911-1960.
I FULL YEAR-LEWGTE OF WARM PEBIOD IN DAYS
""""""
January ______._..____
--.--- .--_ _- ______ ______ ____ 1 1 5 13 26 43 57 April. _________._.___.
____-_ ______ ._____ ______ ___. 2 4 8 10 25 31 64 March. ______._...._._
-__--_ ______ ______ 1 1 1 3 4 12 18 30 59 February __.__ ~
_..___.
1 1 ______ ______ 1 2 4 8 4 14 44 64
June ..___._._.__._.... 66 18 7 3 _-._-_ ______ ______ ______ ____ .___ ____ 3 5 14 32 62 May ..._________...__.
--__-_ ______ ._____ ______ ._._ ____ 1 3 July _____._.__._
~ _.___ 26 8 6 2 1 .___
______ _____. 1 ___. .___ 1 ____ 5 16 22 54 September ...________.
--_-__ ______ ._____ ______ ____ ____ ____ ____ 1 4 11 27 August __._._.___.__..
___--- ______ ______ ______ .___ _.__
------
November _.__._______ 55 37 25 13 6 4 ___. ____ ------ 1 .___.. ______
October .._.._________ 68 43 10 10 4 3 4 1 1 ______
1 3 2 5 5 13 26 49 87 175 349 667 Total .______________
------ ______ 2 2 2 3 5 7 9 10 30 65 December ____._______
--_-__ 1
""""""
""_""" -
Chance frequency ex-
Cumulative total (re-
""""
pectation __._.......
1 4 6 11 16 29 55 104 191 366 715 1,382 verse) _____________.
O.Ol<O.Ol<O.Ol<O.O1 3.50.680.140.03 18 Qo 455 2,311
Total days in the 40-year period: 14,610. Total days of maximum temperature loo F. or more above normal: 2.880. Probability of random occurrence of such a day: 0.197.
144 MONTHLY WEATHER REVIEW JULY 1965
TABLE 2.-Totals, both separately and cumulatively from longest to shortest, of sequences of various lengths for warm-weather days for the period NOVEMBER-MARCH, and number of those runs
expected on chance (Brooks and Carruthers [SI). Data for the years 1911-1960.
NOVEMREB-DECEMREBJANUAR~-~ERRUARY-UA~E-LENQTH
OP WARY PERIOD IN DAY8
1 1 2 l 3 1 4 1 5 1 6 1 7 1 8 1 9 I 1 0 I l l I 1 2
-
Chance frequency ex- Total _______________ 1 2 2 3 4 8 20 33 48 92 172 307
pectation ___________ 1,1134 303
1 3 5 8 12 20 40 73 121 213 386 692 verse) ________________
Cumulative total (re- <o.o1 <o.ol <O.OI ~0.01 1.40.360.01 6.3 20 79
I I ,I I I I I ! I I I
Total day8 In the CO-year period: 6 OM. Total da s of maximum temperat& 10” F. or more above normal: 1,673. Probabilfty of random occurrence of such a day: 0.280.
TABLE 3.-Totals, both separately and cumulatively from longestto sequences of various lengths for warm-weather days for SEPTEMBER-OCTOBERl and number ofthose runs
chance (Brooks and Carruthers [SI). Data for the years
191 1-1 960.
EEPTEMRER-OCTORER-LENGTH 01 WARM PERIOD IN DAYS
6 7 8 11 10 9
“-__“-
Total ______________ 122 66 28 chance freqne7lcy
Cumulative total
4
1 1 3 1 2 8 4 (reverse) __________ 244122 67 31
<0.01 <O.Ol <0.01 0.01 0.03 0.14
1
”___”
2 1 4
Total days in the 40-year period: 2,440. Total da of maximum temperature IOo F. or more above normal: 499. Pmbab8y of random omurrence of such a day: 0.204 or approximately 1/5.
systems, while in the early fall the weather is generally
fair with occasional storms. Tables 2 and 3 give data
derived by separating the two weather regimes just
mentioned.
Figures 1 and 2 indicate the percentage of occurrence
of various lengths of sequences of warm days following
warm spells of different lengths, for the November-March
and September-October seasons respectively. The per-
centages a t the tops of the figures indicate those expected
on mere chance. The percentages in the bodies of the
figures are derived from the “cumulative totals” in
tables 2 and 3. For example, taking data from table 2,
of the 121 warm spells which lasted for at least 4 days 20,
or 17 percent, went on to at least the seventh day, or
3 more days. It will be noted that in figure 1 there is a
maximum percentage of persistence expectance of a t
least one more day of warm weather after a preceding
spell of 8 warm days, with a second maximum after 4
days, and there is a bias of maximum expectancies down-
ward and to the right, emphasizing a tendency toward
warm periods of about 5 to 6 or of 9 to 10 days duration.
In figure 2 it is seen that the maximum persistence per-
centage for one more warm day is after the fifth or the
eighth day, and there is again a bias downward and to
the right indicating a tendency toward warm periods
26Ye 7% 2% 41% (Chance frequency expectation) 3 2 I I 1 I I I I I I I I 1
e
0 2 -
21- ,, ,, 1
5 0 I 1 I I 1 I I I I I I
0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2
Minimum length of continuation of warm spell (days)
FIGURE 1.-Percent of time a given sequence of warm days waa
followed by at least 1, at least 2, etc., successive warm days.
Isolines are drawn a t 10 percent intervals. Data for period
November-March, 1911-1950.
In
: I I
ZI 2 0 % 4 % I% <tv0 (Chance frequency expectation)
& IO 100 SEPTEMBER-OCTOBER
I I I I I I I I I I
E
IT
2
I
I
I
<I
C
f o d ; h ; 1 ;:;;: I ::,
Minimum length of continuatlon of warm Spell (days)
FIGURE 2.-Percent of time a given sequence of warm days waa
followed by at least 1, at least 2, etc., successive warm days.
Isolines are drawn a t 10 percent intervals. Data for period
September-October, 1911-1950.
of 6 to 7 or of about 9 days duration. Whiie there were
only 4 cases of 8-day spells of “warm” weather during
September and October for the 40-year period, 3 out of
the 4 did extend for a t least 1 more day.
Figures 3 and 4 indicate the percent of the time that
“tomorrow” should be warm following sequences ofwarm
periods of increasing lengths, on a persistence basis alone,
that is, the percent due to pure chance has been eliminated.
For example, we note in figure 1 that while an initial day
of warm weather was followed 55 percent of the time by
a t least 1 more warm day, the random chance expectancy
of such an occurrence (as given at the top of the table)
is 26 percent, the remaining 29 percent being the percent
JULY 1955 MONTHLY 145
55 I I 1 I I I I I I I
50 -
45 -
a 40 -
0
c
2 35 - -
L a
X 30
- 25
-
.c
0
c
c
a
a a
20 -
I 5 -
IO -
5 -
NOVEMBER-DECEMBER-JANUARY
FEBRUARY-MARCH
\
\
\
\
I
I
1
40
.70
35
.IO
.40
1
E 0
NOVEMBER-DECEMBER .IO _”
JANUARY-FEBRUARY-MARCH v) Y
I I I I I I I I n
I 2 3 4 5 6 7 a
Length of preceding warm period (days)
9”
v - J O 0 1 2 3 4 5 6 7 8 9 l O 1 1 FIGURE 5.-Percent of time that “tomorrow and day after to-
Length of preceding warm period (days) morrow” will be warm following a warm period of given length, according to persistence factor alone, and the skill score t o be
FIGURE 3.-Percent of time that “tomorrow’s’’ weather will be expected on a persistence forecast. on November-March
warm following a warm period of given length, according to per- data, 1911-1950.
sistence factor alone, and the skill score to be expected on a per-
sistence forecast. Beeed on November-March data, 1911-1950.
Trend indicated by dashed line is considered to be based on too
few cmes to be of meteorological significance.
L 20
a f
IO
t SEPTEMBER
- OCT OBER
.70
45
.60
40
.IO
5
c O L ’ I I I I I I I I I 10 0 1 2 3 4 5 6 7 8 9 l O 1 1
Length of preceding warm period (days)
FIGURE 4.-Percent of time that “tomorrow’s” weather will be
warm following a warm period of given length, according to per-
sistence factor alone, and the skill score to be expected on a per-
sistence forecast. Rased on September-October data, 1911-1950.
Trend indicated by dashed line is considered to be based on too
few cases to be of meteorological significance.
SEPTEMBER-OCTOBER
- .40
c
0
E 0 .c
W
- .30 E c
In
In
w
.-
L
n
- .20
0 0
u)
- - .-
- .IO 0-l
x
0 0 1 2 3 4 5 6 7 6 9
Length of preceding warm period (days)
FIGURE 6.-Percentof time that “tomorrow and day after to-
morrow” will be warm following a warm period of a given length,
according to persistence factor alone, and the skill score t o be
expected on a persistence forecast. Based on September-October
data, 1911-1950.
to be ascribed solely to the persistence effect in figure 3. In September and October the maximum expectancy of
It is seen that the persistence in the November-March persistence is found after the fifth and eighth days, dis-
season is highest after a warm spell of 8 days with another regarding the one case of a 10-day spell of warm weather
peak for a warm day following a sequence of 4 warm days. which went on to 11 before breaking.
146 MONTHLY JULY 1955
Figures 5 and 6 indicate the percent of time that a
warm spell should continue through “day after tomorrow”
following sequences of warm periods of increasing lengths,
on a persistence basis alone. It is seen that the persistence
in the November-March season is highest after a warm
spell of 8 days with another peak for 2 warm days follow-
ing a sequence of 3 warm days. In September and October
the maximum expectancy of persistence for a t least 2
more days is found after the fifth warm day.
Jorgensen [5] has shown in his study of rain and no-rain
periods during the winter a t San Francisco that the skill
score of a persistency forecast may be obtained by multi-
plying the percentage of persistenc.y by a common factor.
The skill score is given by S,=- C-E, I where Cis the num- T- E,
ber of correct forecasts, T the t,otal number of forecasts,
and E, the number which would be expected correct on a
chance basis alone. If persistence is expressed as a per-
centage P which may be expected to be correct solely due
to persistence, we may say PT=C-E,. From figures
1 and 2 we note that E, takes on the values .26T and
.20T respectively for the case of at least one more day of
warm weather, hence the values of T-Ec become .74T
and .80T, respectively. Substituting in the equation
for S,, for the November-March data we get 3,-
and for bhe September-October data S --- Similar
reasoning may be applied to the skill score of a persist-
ency forecast for “tomorrow and day after,” and the skill
scores may be referred to the same graph (with a suitable
change in the vertical scale) as the persistence in figures
3,4, 5, and 6.
P
-3’
P
““.80
3. CONCLUSIONS
1. There is a tendency, especially during the colder
portions of the year, for the weather a t Denver to remain
in a warm regime longer than mere chance would indicate.
This tendency to persist seems to favor especially certain
lengths of periods of warm weather.
2. Persistence may be assumed to be a meteorological
factor inasmuch as it may be a combination of many
weighted factors, some of them so obscure as to escape
the forecaster’s notice.
3. The skill scores for forecasts based upon persistence
of warm weather indicate that the forecaster must have a
high confidence factor if he is to differ with and verify
in excess of a forecast based upon such persistence.
REFERENCES
1. A. W.Cook and A. G. Topil, “Chinooks East of the
Mountains in Colorado”, U. S. Weather Bureau,
Denver, Colo., 1947, (Unpublished)
2. A. W. Cook and Denver Forecasting Staff, “Forecast-
ing Rules from Denver”, U. s. Weather Bureau,
Denver, Colo., 1944, (Unpublished)
3. C. E. P. Brooks and N. Carruthers, Handbook of Stu-
tiscal Methods in Meteorology, Her Majesty’s Sta-
tionery Office, London, 1953, pp. 310-311.
4. T. A. Blair, ‘(The Coefficient of Persistence”, Monthly
Weather Review, vol. 52, No. 7, July 1924; p. 350.
5. Donald L. Jorgensen, (‘Persistency of Rain and No-
Rain Periods During the Wint,er a t San Francisco”,
Monthly Weather Review, vol. 77, No. 11, November
1949, pp. 303-307.