On the Role of Global Warming on the
Frequency of Record-Temperature Events
collaborator: Mark Petersen (LANL)
Empirics: Philadelphia temperature (why Philadelphia?)
annual pattern & long-term trends
daily temperature distribution
Tutorial: Evolution of record temperature events
magnitude of successive records
time between successive records
Comparison with Philadelphia data
role (or non-role) of global warming
Summary & Outlook
role of inter-day temperature correlations
seasonal effects
asymmetry between high & low temperatures
Philadelphia Climate
annual temperature pattern
Apr 22, 1985 90F
Temple U. colloquium
long-term temperature trends
Aug 7, 1918 106F
IPCC 2001: global warming 0.6! over past century
70
100
annual high
record high
3.8!
average temperature
daily temperature
80
60
av. high
40
av. low
20
record low
60
annual middle
50
1.4!
0
annual low
Feb 9, 1934 -11F
!20
0
100
200
day of the year
300
40
1880
1920
1960
year
2000
Daily Temperature Distribution
0
10
−1
P(!T)
10
Jan. 5
Apr. 5
−2
10
9-day averages
Oct. 5
July 5
−3
10
−15
−10
−5
0
!T
5
10
15
Tutorial:Evolution of Temperature Records
(Arnold et al., Records, 1998)
temperature
on a fixed day
basic assumption: daily temperatures are iid
continuous variables
T3
T2
T1
T0
t0
t1
t2
Tk , Pk (T ) tk , pk (t)
}
Goal: compute
t3
distribution
independent!
year
Disclaimers
record temperatures " most data discarded
data from a single station only
no control for possible urban heat island effect
only 126 years of data---climatologically puny
unknown data quality and accuracy:
only daily high & low are reported
reported accuracy of 1!
few records " asymptotic analysis questionable
.....
Evolution of Typical Record Temperature
p(T ) ≡ daily temperature distribution
T0 =
!
∞
T p(T ) dT
0
!
∞
T p(T ) dT
Tk+1 = !
Tk
Tk+1
∞
Tk
p(T ) dT
Tk
Record Temperature Distributions
prob. temperature > T
! ∞
p> (T ) ≡
p(T " ) dT "
prob. temperature < T
p< (T ) = 1 − p> (T )
T
probability distribution of k th record
previous record T’ < T
Pk (T )
=
!"
T
"
Pk−1 (T )
0
=
!"
0
next n temperatures < T’
∞
#
"
[ p< (T ) ] dT
n=0
T
n
last temperature = T
Pk−1 (T " )
"
dT
p> (T " )
$
p(T ) .
"
$
p(T ) ,
Record Time Evolution
st
qn (Tk ) ≡ prob. (k + 1)
n−1
= p< (Tk )
n-1 non-records
record n years after k
th
at Tk
p> (Tk )
record
expected time between k th and (k + 1)st records at Tk :
∞
!
1
n−1
n p < p> =
tk+1 − tk =
finite for given Tk
p
(T
)
>
k
n=1
waiting time distribution for k th record: averaged over Tk
! ∞
Qn (k) ≡
Pk (T ) qn (T ) dT
in general
0
infinite
! ∞
n−1
Pk (T ) p< (T )
p> (T ) dT waiting time
=
0
Record Time Statistics
(Glick 1978; Sibani et al 1997, Krug & Jain 05, Majumdar)
define
σi =
!
1 if record in ith year
0 otherwise
1
then !σi " =
i+1
largest
...
...
!σi σj " = !σi "!σj "
i
0 1 2
therefore
!n(t)" =
t
!
i=1
!σi " ∼ ln t
j
(ln t)n − ln t
e
P (n) =
n!
1st
probability that current record is broken in ith year:
2nd
...
1
→ time until next record = ∞
=
i(i + 1)
0 1 2
i
Records with Gaussian Temperature Distribution
p(T ) = √
1
2πσ 2
e
T1 =
T0 = 0
Tk+1
−T 2 /2σ 2
=
!
Tk+1 =
2
σ
π
2
2
1
−T
/2σ
√
T
e
dT
2
Tk
! ∞ 2πσ1
2 /2σ 2
−T
√
e
dT
Tk
2πσ 2
!∞
−Tk2 /2σ 2
T1 e
√
=
"
#
erfc Tk / 2σ 2
%
$
2
σ
∼ Tk 1 + 2
Tk
2
σ
∂T
∼
∂k
T
→
Tk ∼
k→∞
√
2kσ 2
!∞
T p(T ) dT
!Tk∞
Tk
p(T ) dT
Philadelphia Record Temperature Values . . .
min. temp
2
Tk/!
√
k
max. temp
1
0
0
2
4
6
k
8
10
. . . and Record Times
min. temp
2
max. temp
k
0
10
1
k=5
k=4
1/t
!1
0
0
2
4
6
k
probability
Tk/!
√
10
8
10
k=
3
simulation
k=
2
min. temp
!2
10
max. temp
0
10
1
k=
1
2
10
10
time
3
10
Records If Global Warming Is Occurring
assume daily
temperature
distribution
exceedance
probability
p(T ; t) =
!
e−(T −vt)
0
p> (Tk ; tk + j)
=
!
∞
e−[T −v(tk +j)] dT
Tk
−(Tk −vtk ) jv
= e
prob of record
at year n
qn (Tk )
e
≡ X ejv
n−1
p> (Tk ) p< (Tk )
n−1
!
nv
jv
→ e X
(1 − e X)
"
#$
%
≡
j=1
overestimate of
record time
as a soluble
example only
T > vt
T < vt
when 0
record must occur
ejv X = 1 → (tk+1 − tk )v = Tk − vtk
→ tk ∼
k
v
records ultimately
occur at constant rate
(Ballerini & Resnick, 85; Borokov, 99)
Frequency of Record High Temperatures
!1
probability
10
v=0.012
v=0.006
0.012
!2
10
0.006
0.003
v=0.003
v=0
!3
10
0
10
1
2
10
10
time
3
10
Global Cooling (low temperature records in warming)
qn (Tk )
p> (Tk ) p< (Tk )n−1
with w = |v| > 0
n−1
!
−(Tk +wtk )
Y
=
e
−nw
−jw
→ e
Y
(1 − e
Y)
≡
j=1
each term
in product
1
qn (Tk ) ∼
!
(1 − Y )n e−nw Y
(1 − Y )1/w e−nw Y
[Tk +wtk ]
→ tk+1 − tk ∼ e
n < n∗ ∼ w−1
n > n∗
1−Y
(eTk for w = 0)
1/w
∂t
k+wt
continuum limit:
∼e
∂k
−wt
k
)
=
w(e
−1)
(1−e
.! "# $
≈ wt
<0
no solution for
1
tk > w
j
Frequency of Record Low Temperatures
!1
!2
10
v=0
!3
!1
10
v=0.003
10
0
10
1
10
probability
probability
10
v=0.006
v=0.012
!2
10
time
0.012
2
10
0.006
v=0.012
v=0.006
3
10
0.003
!3
10
v=0.003
v=0
Summary & Outlook
Record temperature and time statistics are obtainable
from basic extreme value considerations
Current global warming rate does not yet seem to
significantly affect record temperature statistics
Inter-day temperature correlations also do not
affect record temperature statistics
Inter-day Temperature Correlations
365
!
1
!Ti Ti+t " − !Ti "!Ti+t "
α
=
low,
middle,
high
Cα (t) =
2
365 i=1
!Ti " − !Ti "2
0
10
slope -4/3
(other work: slope 0.7
e.g., Bunde et al.)
−1
10
C!(t)
low
middle
−2
10
high
−3
10
1
10
t
t
100
Summary & Outlook
Record temperature and time statistics are obtainable
from basic extreme value considerations
Current global warming rate does not yet seem to
significantly affect record temperature statistics
Inter-day temperature correlations also do not
affect record temperature statistics
Open issues: Difference between high & low records
1705 record highs; 1346 record lows
Seasonal effects
more record highs in spring
daily variances (10-day averages)
Seasonal Variance & Record Numbers
12
7
high
10
number of record highs
(20-day average)
average
5
8
low
6
3
4
0
100
200
day of the year
300
Summary & Outlook
Record temperature and time statistics are obtainable
from basic extreme value considerations
Current global warming rate does not yet seem to
significantly affect record temperature statistics
Inter-day temperature correlations also do not
affect record temperature statistics
Open issues: Difference between high & low records
1705 record highs; 1346 record lows
Seasonal effects
more record highs in spring
Day/night asymmetry
more hot days....
150
100
80
and fewer cold days....
85
50
90
0
1880
70
1920
1960
200
2000
# days with highs colder than T
# days with highs exceeding T
75
50
100
30
0
1880
1920
1960
2000
....but fewer hot nights
300
200
50
....and more cold nights
60
100
150
70
0
1880
1920
1960
# days with lows colder than T
# days with lows exceeding T
40
35
2000
100
30
50
25
20
0
1880
1920
1960
year
2000
Summary & Outlook
Record temperature and time statistics are obtainable
from basic extreme value considerations
Current global warming rate does not yet seem to
significantly affect record temperature statistics
Inter-day temperature correlations also do not
affect record temperature statistics
Open issues: Difference between high & low records
1705 record highs; 1346 record lows
Seasonal effects
more record highs in spring
Day/night asymmetry
Role of changing variability
Serious data collection & model validation
Thank you Newton Institute!
Thank you for participating!
Have a safe trip home!
I hope to see you again!