Role of Global Warming on Frequency of Record Temperature ... · IPCC 2001: global warming 1.1℉ over past century 3.8℉ 1.4℉ 0 100 200 300 day of the year!20 0 20 40 60 80 100

collaborator: Mark Petersen (LANL)

Role of Global Warming on Frequency of Record Temperature Events

magnitude of successive records

Facts about urban temperatures

Tutorial: evolution of record temperature eventsmagnitude of successive recordstime between successive records

annual pattern & long-term trends

Role of global warming on recordsSummary & Outlook

daily temperature distribution

role of inter-day temperature correlationsseasonal effectsasymmetry in high & low temperatureschanging variability

Large Deviations Conference, MCTP, June 2007thanks to CHMI & Phil Jones (EAU) for data

Philadelphia Temperatures

record high

record low

av. high

av. low

Apr 22, 1985 90FTemple U. colloquium Aug 7, 1918 106F

Feb 9, 1934 -11F

annual temperature pattern

IPCC 2001: global warming 1.1℉ over past century

3.8℉

1.4℉

0 100 200 300day of the year

−20

0

20

40

60

80

100

daily

tem

pera

ture

annual middle

annual high

annual low

long-term temperature trends

1880 1920 1960 2000year

40

50

60

70

aver

age

tem

pera

ture

!15 !10 !5 0 5 10 15

ΔT

10!3

10!2

10!1

100

P(Δ

T)

Daily Temperature Distribution

Jan. 5

Apr. 5

Oct. 5

July 5

9-day averages

!10 0 101

10

!8 !4 0 4 81

10

100

!10 !5 0 51

10

100Milan1763

Brussels1833

CET1878

Uppsala1722

JulyOctAprJan

!10 !5 0 5 101

10

100

T3

t3

T2

t2

T1

t1

T0

t0

Evolution of temperature records

year

temperature on a fixed day

(Arnold et al., Records, 1998)

Goal: compute tk, pk(t)Tk,Pk(T )

basic assumption: daily temperatures are iid continuous variables

distribution independent!

}

Disclaimers

126-285 years of data---climatologically puny

Antarctic Ice Core Data(Petit et al., Nature,1999)

Disclaimers

record temperatures → most data discarded

data from a limited number of stations (12)

no control for possible urban heat island effect

unknown data quality and accuracy: only daily high & low are reportedreported accuracy of 1℉ (or 1℃ or even 0.1℃)

126-285 years of data---climatologically puny

.....

few records → asymptotic analysis questionable

Evolution of Typical Record Temperature

p(T ) ! daily temperature distribution

T0 =

!!

0

T p(T ) dT

Tk

Tk+1

!!

Tk

T p(T ) dT

!!

Tk

p(T ) dT

Tk+1 =

Record Temperature Distributions

previous record T’ < Tnext n temperatures < T’

last temperature = T

Pk(T ) =

!

" T

0

Pk!1(T")

##

n=0

[ p<(T ") ]n dT "

$

p(T ) ,

=

!

" T

0

Pk!1(T ")

p>(T ")dT "

$

p(T ) .

probability distribution of kth record

p>(T ) !

!!

T

p(T ") dT "

prob. temperature > T prob. temperature < T

p<(T ) = 1 ! p>(T )

Record Time Evolution

= p<(Tk)n!1 p>(Tk)

qn(Tk) ! prob. (k + 1)st record n years after kth at Tk

tk+1 ! tk =!!

n=1

n pn"1< p> =

1

p>(Tk)

expected time between kth and (k + 1)st records at Tk:

n-1 non-records record

infinite waiting time in general

waiting time distribution for kth record:

Qn(k) !

!!

0

Pk(T ) qn(T ) dT

=

!!

0

Pk(T ) p<(T )n"1 p>(T ) dT

averaged over Tk

for given Tkfinite

Record Time Statistics(Glick 1978; Sibani et al 1997, Krug & Jain 05, Majumdar)

!i =

!

1 if record in ith year0 otherwise

define

P (n) =(ln t)n

n!e! ln t

probability that current record is broken in ith year:

!!i!j" = !!i"!!j"

therefore !n(t)" =t!

i=1

!!i" ! ln t

then !!i" =1

i + 1

...

0 1 2 i

largest

...

0 1 2 i

1st

2nd

! time until next record = "=1

i(i + 1)

...

j

newlargest

2nd

Records with Gaussian Temperature Distribution

p(T ) =1

!

2!"2e!T 2/2!2

Tk+1 =

!!

TkT p(T ) dT

!!

Tkp(T ) dT

T0 = 0 T1 =

!

2

!"

!T

!k!

"2

T" Tk !

#2k"2

Tk+1 =

! !

Tk

1"2!"2

T e#T 2/2"2

dT! !

Tk

1"2!"2

e#T 2/2"2 dT

=Tk e#T 2

k/2"2

erfc"

Tk/!

2!2#

" Tk

$

1 +!2

T 2k

%

k # $

Philadelphia Record Temperature Values

0 2 4 6 8 10

k

0

1

2

Tk/σ

!

k

min. temp

max. temp

Records If Global Warming Is Occurring

p>(Tk; tk + j) =

!!

Tk

e"[T"v(tk+j)] dT

= e"(Tk"vtk) ejv! X ejv

exceedance probability

prob of record at year n

qn(Tk) ! p>(Tk) p<(Tk)n!1

" env X

n!1!

j=1

(1 # ejv X)" #$ %

when 0

record must occur

p(T ; t) =

!

e!(T!vt) T > vt

0 T < vt

assume temperature distribution as a soluble example only

(over)estimate of record time

ejvX = 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

100 101 102 103

10−3

10−2

10−1

probability

time

v=0

0.003

v=0.003

0.006

v=0.0060.012

v=0.012

qn(Tk) !

!

(1 " Y )n e!nw Y n < n# ! w!1

(1 " Y )1/w e!nw Y n > n#

qn(Tk) ! p>(Tk) p<(Tk)n!1

" e!nw Y

n!1!

j=1

(1 # e!jw Y )

Global Cooling (low temperature records in warming)

in product

1/w j

1!Y

1

each term

with w = |v| > 0

Y = e!(Tk+wtk)

(eTk for w = 0)! tk+1 " tk # e[Tk+wtk]

!t

!k! e

k+wtcontinuum limit:

no solution for

tk >1

w

(1!e!wt) = w(ek

!1)! wt

.! "# $

<0

Frequency of Record Low Temperatures

100 101 102 103

10−3

10−2

10−1

probability

time

v=0

0.003

v=0.003

0.006

v=0.0060.012

v=0.012100 101 102 103

10−3

10−2

10−1

probability

time

v=0

v=0.003v=0.006v=0.012

1 10 1001

10

100

years after 1893

Potsdam

1 10 1001

10

100

years after 1878

CET

1 10 1001

10

100

years after 1833

Brussels

More Record Frequencies

highs

lows

1 10 1001

10

100

years after 1763

Milan

1 10 1001

10

100

years after 1775

Prague

Summary & Outlook

Current global warming rate seems to significantly affect record temperature statistics

Extreme value considerations give record temperature and time statistics

Inter-day temperature correlations do not appear to affect record temperature statistics

1 10 100

t

10!3

10!2

10!1

100

Cα(t

)

Inter-day Temperature Correlations

low

high

middle

slope -4/3

t

C!(t) =1

365

365!

i=1

!TiTi+t" # !Ti"!Ti+t"

!T 2i" # !Ti"2

! = low, middle, high

(other work: slope ~0.7 e.g., Bunde et al.)

Summary & Outlook



Open issues:


Seasonal effectsmore record highs in spring

low

high

average

Seasonal Variance

0 100 200 300day of the year

4

6

8

10

12da

ily v

aria

nces

(10-

day

aver

ages

)

number of record highs(20-day average)

3

5

7

& Record Numbers

Summary & Outlook


Open issues: Seasonal effects


Day/night asymmetrymore record highs in spring


1880 1920 1960 20000

100

200

# d

ays w

ith

hig

hs c

old

er

tha

n T

70

50

30

& fewer cold days....

more hot days....

1880 1920 1960 20000

50

100

150

# da

ys w

ith h

ighs

exc

eedi

ng T

75

80

85

90

1880 1920 1960 2000year

0

50

100

150

# da

ys w

ith lo

ws c

olde

r tha

n T 35

30

25

20

....and more cold nights

1880 1920 1960 20000

100

200

300

# da

ys w

ith lo

ws e

xcee

ding

T

40

50

60

70

....but fewer hot nights

Summary & Outlook






Changing variability

Annual Variability Slowly Increasing

1880 1920 1960 20008

9

10

11

12

0.15C

0.09C

0 0.2 0.4 0.6 0.8 10

20

40

60

80

0.5!1!

1.5!

2!

0 0.2 0.4 0.6 0.8 10

20

40

602!

1.5!

1!

0.5!

Decreasing Time Between Threshold Events

days

temporal decile temporal decile

low excursionshigh excursions

Summary & Outlook





Comprehensive data & theory validation


Changing variability

Documents

Role of Global Warming on Frequency of Record Temperature ... · IPCC 2001: global warming 1.1℉ over past century 3.8℉ 1.4℉ 0 100 200 300 day of the year!20 0 20 40 60 80 100