Changes
in
the
Frequency
of
Observed
Temperature
Extremes
Largely
Driven
by
a
Distribution
Shift
Ronak
N.
Patel
1
,
David
B.
Bonan
1
,
and
Tapio
Schneider
1
1
Environmental
Science
and
Engineering,
California
Institute
of
Technology,
Pasadena,
CA,
USA
Abstract
Extreme
heat
poses
significant
threats
to
human
life
and
ecosystems.
Quantifying
the
effects
of
anthropogenic
climate
change
on
extreme
heat
has
remained
challenging,
in
part
due
to
the
short
observational
record.
Here,
we
isolate
the
most
slowly
varying
component
of
the
frequency
at
which
the
historical
90th
and
99th
percentiles
were
exceeded
in
observational
records
from
1955
to
2021
by
using
a
statistical
method
called
low‐frequency
component
analysis.
The
emerging
spatiotemporal
signal
in
the
changing
frequency
of
temperature
extremes
can
be
attributed
to
a
shift
of
the
temperature
distribution
by
local
warming
of
the
annual‐
mean
daily
maximum
temperature.
The
shift
explains
over
80%
of
the
interannual
variability
in
the
frequency
at
which
the
historical
90th
percentile
is
exceeded
in
the
tropics
and
up
to
50%
in
higher
latitudes.
This
work
connects
variability
in
the
frequency
of
extreme
surface
temperatures
to
variability
in
mean
local
warming.
Plain
Language
Summary
Over
the
past
few
decades,
regions
across
the
globe
have
experienced
substantial
increases
in
surface
temperature
extremes,
posing
significant
threats
to
human
life,
as
well
as
critical
agriculture
and
energy
sectors.
Due
to
the
relatively
short
observational
record,
it
has
been
difficult
to
disentangle
the
relative
roles
of
natural
variability
and
anthropogenic
forcing
in
driving
changes
to
temperature
extremes.
Here,
we
introduce
a
simple
framework
for
understanding
the
increasing
frequency
of
surface
temperature
extremes
by
employing
a
statistical
method
to
isolate
the
most
slowly
changing,
and
hence
most
likely
anthropogenic,
component
of
surface
temperature
extremes.
We
find
that
the
emerging
signal
in
the
changing
frequency
of
temperature
extremes
is
largely
driven
by
a
shift
in
the
temperature
distribution
by
mean
local
warming.
The
shift
explains
over
80%
of
the
observed
variability
in
exceeding
the
90th
percentile
in
the
tropics
and
up
to
50%
in
higher
latitudes.
It
also
explains
why
changes
in
the
frequency
of
extremes
appear
to
be
more
rapid
than
changes
closer
to
the
center
of
the
temperature
distribution.
This
work
offers
guidance
for
climate
risk
assessment
and
adaptation
strategies
by
connecting
variability
in
the
frequency
of
extreme
temperatures
to
variability
in
mean
warming
at
a
given
location.
1.
Introduction
Increasingly
frequent
extreme
heat
events
pose
significant
threats
to
human
health,
agriculture,
and
energy
systems
worldwide
(Intergovernmental
Panel
on
Climate
Change
(IPCC),
2023
).
Distinguishing
between
the
effects
of
anthropogenic
climate
change
and
natural
(or
internal)
variability
on
extreme
heat
is
crucial
for
effective
adaptation
strategies
(Diffenbaugh
&
Field,
2013
).
Such
differentiation
has
been
challenging
due
to
the
relatively
short
observational
record
and
regional
climate
variability,
which
exerts
a
strong
control
on
local
changes
in
extremes
(Fischer
&
Knutti,
2014
).
Improving
our
understanding
of
the
mechanisms
driving
changes
in
tem-
perature
extremes
is
essential
for
advancing
climate
resilience
and
informing
decisions
aimed
at
adapting
to
the
impacts
of
extreme
heat
events
in
a
warming
climate.
Over
the
past
decade,
a
series
of
studies
have
shown
that
changes
in
extreme
surface
temperatures
over
land
can
be
related
to
a
shift
in
the
temperature
distribution
due
to
mean
warming
(e.g.,
Donat
&
Alexander,
2012
;
Lau
&
Nath,
2012
;
Loikith
&
Neelin,
2015
;
McKinnon
et
al.,
2016
;
Rhines
&
Huybers,
2013
;
Simolo
&
Corti,
2022
;
Simolo
et
al.,
2011
;
Weaver
et
al.,
2014
).
For
example,
Donat
and
Alexander
(
2012
)
used
global
observations
to
demonstrate
that
a
shift
in
the
temperature
distribution
explains
the
rise
in
surface
temperature
extremes
between
two
historical
30‐year
time
periods.
Similarly,
Rhines
and
Huybers
(
2013
)
illustrated
that
the
increased
frequency
of
summer
temperature
extremes
primarily
reflects
changes
in
the
mean,
rather
than
the
variance,
of
the
tem-
perature
distribution.
Loikith
and
Neelin
(
2015
)
also
show
that
shifting
left‐skewed
(short
right‐tailed)
temper-
ature
distributions
to
the
right
results
in
an
increase
in
temperature
extremes.
Collectively,
these
studies
show
that
mean
local
warming
can
explain
the
observed
increase
in
temperature
extremes.
However,
most
of
these
studies
RESEARCH
LETTER
10.1029/2024GL110707
Key
Points:
•
Shifting
the
surface
temperature
distribution
by
mean
local
warming
explains
much
of
the
frequency
increase
in
observed
temperature
extremes
•
Mean
local
warming
explains
80%
of
the
observed
variability
in
90th
percentile
exceedance
in
the
tropics
and
up
to
50%
in
higher
latitudes
•
Narrower
temperature
distributions
in
the
tropics
are
associated
with
a
larger
increase
in
extreme
heat
frequency
compared
to
midlatitudes
Supporting
Information:
Supporting
Information
may
be
found
in
the
online
version
of
this
article.
Correspondence
to:
R.
N.
Patel,
ronak@caltech.edu
Citation:
Patel,
R.
N.,
Bonan,
D.
B.,
&
Schneider,
T.
(2024).
Changes
in
the
frequency
of
observed
temperature
extremes
largely
driven
by
a
distribution
shift.
Geophysical
Research
Letters
,
51
,
e2024GL110707.
https://doi.org/10.1029/2024GL110707
Received
11
JUN
2024
Accepted
27
NOV
2024
©
2024.
The
Author(s).
This
is
an
open
access
article
under
the
terms
of
the
Creative
Commons
Attribution
License,
which
permits
use,
distribution
and
reproduction
in
any
medium,
provided
the
original
work
is
properly
cited.
PATEL
ET
AL.
1
of
10
examine
specific
historical
time
periods
(e.g.,
Donat
&
Alexander,
2012
)
or
specific
regional
domains
such
as
North
America
(e.g.,
Loikith
&
Neelin,
2015
)
or
Europe
(e.g.,
Simolo
et
al.,
2011
).
It
is
unclear
to
what
extent
a
shift
of
the
temperature
distribution
in
observations
can
explain
surface
temperature
extremes
globally
and
to
what
extent
it
can
explain
the
increasing
frequency
of
surface
temperature
extremes.
Additionally,
the
short
observational
record
and
large
degree
of
interannual‐to‐decadal
variability
have
made
it
challenging
to
quantify
the
anthropogenic
influence
on
observed
temperature
extremes.
In
this
paper,
we
introduce
a
framework
for
quantifying
the
spatiotemporal
increase
in
the
frequency
of
extreme
temperatures
from
1955
to
2021.
We
demonstrate
that
a
shift
of
the
temperature
distribution
due
to
mean
local
warming
largely
explains
the
observed
increase
in
days
surpassing
historical
90th
and
99th
percentiles
of
daily
maximum
temperatures.
This
is
achieved
by
first
isolating
the
most
slowly
changing,
and
hence
most
likely
forced,
component
of
temperature
extremes
using
a
statistical
method
called
low‐frequency
component
(LFC)
analysis
(LFCA,
Schneider
&
Held,
2001
;
Wills
et
al.,
2018
).
By
isolating
the
most
slowly
varying
component
in
the
proportion
of
days
exceeding
historical
temperature
thresholds,
we
observe
a
distinct
difference
between
the
increase
in
days
above
the
historical
90th
and
99th
percentile
thresholds.
A
slow
shift
in
the
temperature
dis-
tribution
by
mean
local
warming
is
consistent
with
the
more
rapid
acceleration
in
the
frequency
of
high
percentile
extremes
(99th
percentile)
compared
to
lower
percentile
thresholds
(90th
percentile)
identified
by
LFCA.
We
then
show
that
interannual
variability
in
mean
local
warming
can
also
explain
interannual
variability
in
the
frequency
of
temperature
extremes
through
a
shift
of
the
temperature
distribution.
While
previous
studies
have
attributed
the
observed
increase
in
extreme
temperatures
between
two
historical
time
periods
to
a
shift
in
the
temperature
distribution,
this
study
comprehensively
illustrates
the
spatiotemporal
structure
of
these
changes
and
connects
variability
in
surface
temperature
extremes
to
variability
in
mean
local
warming.
2.
Data
and
Methods
2.1.
Temperature
Processing
Temperature
data
was
obtained
from
the
Berkeley
Earth
Surface
Temperatures
data
set
(Rohde
&
Haus-
father,
2020
)
through
the
gridded
Experimental
Global
Daily
Land
Average
High
Temperature
(TMAX)
data
set
on
a
1°
×
1°
grid.
Further
details
on
the
data
homogenization
and
gridding
process
used
in
the
Berkeley
Earth
Surface
Temperatures
data
set
can
be
found
in
Rohde
et
al.
(
2013
).
Even
though
the
daily
Berkeley
Earth
data
is
subject
to
gap‐filling
in
the
temperature
reconstruction,
the
coarser
HadGHCND
observational
data
set
(Caesar
et
al.,
2006
)
is
found
to
agree
with
Berkeley
Earth
in
terms
of
decadal
heatwave
trends
in
regions
with
sufficient
data
(Perkins‐Kirkpatrick
&
Lewis,
2020
).
This
gives
us
confidence
in
the
long‐term
trends
presented
by
the
Berkeley
Earth
data
set.
From
the
provided
climatology
and
anomaly
data
in
the
Berkeley
Earth
data
file,
we
first
reconstruct
the
observed
temperature
time
series
at
each
grid
point
from
1880
to
2021.
Specifically,
T
=
̄
T
T
′
where
̄
T
is
the
provided
1951–1980
climatology
and
T
′
is
the
surface
temperature
anomaly
relative
to
the
1951–1980
baseline.
Next,
we
mask
out
regions
of
the
world
with
insufficient
data,
which
are
defined
as
having
more
than
5
days
of
missing
data
in
any
given
year.
We
then
subjectively
determine
that
1955–2021
is
the
period
where
we
can
maximize
the
portion
of
the
world
with
continuous
data
and
have
the
longest
record
possible.
Then,
at
each
location,
we
calculate
the
90th
and
99th
percentiles
for
the
1961–1990
baseline,
which
we
call
the
“historical
period.”
This
is
the
first
three‐decade
long
period
in
the
shortened
time
series.
Finally,
at
each
location,
we
compute
the
percent
of
days
where
the
observed
temperature
is
above
the
90th
and
99th
percentile
temperature
thresholds
in
the
historical
period.
This
produces
a
spatiotemporal
time
series
of
the
percent
(or
number)
of
days
in
a
given
year
above
the
historical
90th
and
99th
percentile,
which
is
later
used
for
LFCA.
To
compute
spatial
averages
of
this
time
series,
we
first
mask
out
locations
with
insufficient
data
for
the
period
1955–2021.
From
that
mask,
we
calculate
an
average
over
each
of
the
land
regions
in
IPCC
AR6
(Iturbide
et
al.,
2020
).
Global
and
regional
averages
mentioned
in
the
text
use
an
area‐weighted
mean.
2.2.
Low‐Frequency
Component
Analysis
To
estimate
the
spatiotemporal
structure
of
the
slow
response
in
the
frequency
of
extreme
temperatures
to
climate
change,
we
use
a
statistical
method
called
LFCA
(Schneider
&
Held,
2001
;
Wills
et
al.,
2018
).
LFCA
calculates
linear
combinations
of
a
set
of
empirical
orthogonal
functions
(EOFs)
that
maximize
the
ratio
of
low‐frequency
to
Geophysical
Research
Letters
10.1029/2024GL110707
PATEL
ET
AL.
2
of
10
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total
variance
in
their
corresponding
time
series.
Those
time
series
are
called
LFCs,
and
the
combination
of
EOFs
produces
low‐frequency
patterns
(LFPs).
We
define
low‐frequency
variance
as
variance
which
makes
it
through
a
10‐year
low‐pass
filter.
Therefore,
the
first
LFP
is
the
linear
combination
of
the
first
15
EOFs
with
the
highest
possible
ratio
of
interdecadal‐to‐intradecadal
variance;
that
is,
LFCA
maximizes
the
ratio
of
multidecadal
vari-
ance
to
higher‐frequency
variance
to
effectively
isolate
modes
of
variability
with
longer
timescales.
LFPs
and
LFCs
are
normalized
such
that
the
LFP
shows
the
anomaly
pattern
corresponding
to
a
one‐standard‐deviation
anomaly
in
the
corresponding
LFC.
The
full
methodology
of
LFCA
is
given
in
Section
5.1
of
Wills
et
al.
(
2018
).
The
major
difference
between
this
methodology
and
that
presented
in
Wills
et
al.
(
2018
)
is
that
we
normalize
the
time
series
to
the
mean
of
the
historical
period
(1961–1990),
whereas
Wills
et
al.
(
2018
)
normalizes
to
the
mean
of
the
entire
time
series.
This
change
allows
us
to
more
easily
calculate
anomalies
in
the
number
of
days
above
the
historical
thresholds.
2.3.
Shifting
the
Temperature
Distribution
To
shift
the
temperature
distribution
at
a
given
location
by
the
“mean
local
warming,”
we
first
calculate
the
mean
daily
maximum
temperature
over
the
historical
period
(1961–1990).
Then
the
annual
mean
daily
maximum
temperature
for
each
year
is
calculated.
From
this,
we
obtain
a
mean
daily
maximum
temperature
anomaly
for
each
year.
This
annual‐mean
anomaly
provides
the
mean
warming
relative
to
the
historical
period
for
any
specific
year
at
any
specific
location
and
is
referred
to
as
“mean
local
warming.”
Then,
we
add
the
annual‐mean
warming
to
the
empirical
cumulative
density
function
(CDF)
for
the
historical
period
at
each
location.
Finally,
we
calculate
the
percent
of
this
new
CDF
that
is
above
the
90th
and
99th
percentile
thresholds
in
the
historical
period.
This
gives
us
the
percent
of
days
in
any
year
above
the
specified
historical
thresholds
obtained
by
shifting
the
historical
temperature
distribution
by
the
amount
of
warming
in
the
mean
experienced
at
that
location.
The
percent
of
days
in
each
year
above
the
historical
90th
and
99th
percentiles
is
then
compared
to
the
observed
incidence
of
those
temperature
extremes.
To
quantitatively
compare
the
skill
of
the
shift
mechanism,
we
calculate
the
percent
of
variance
in
observed
temperature
extremes
that
is
explained
by
a
shift
in
the
historical
temperature
distribution
R
2
)
for
both
the
90th
and
99th
percentiles.
3.
Results
3.1.
More
Extremely
Warm
Days
Over
Time
We
begin
by
using
the
Berkeley
Earth
Surface
Temperatures
data
set
to
analyze
the
time
series
of
the
percent
of
days
on
which
the
daily
maximum
temperature
is
above
the
historical
90th
and
99th
percentiles
in
each
of
the
IPCC
AR6
land
regions
(Figure
1
).
Movies
S1
and
S2
show
higher‐resolution,
gridded
spatiotemporal
changes
in
the
percent
of
days
above
the
historical
thresholds
from
1955
to
2021.
While
the
number
of
extremely
warm
days
has
been
increasing,
we
see
large
spatiotemporal
variability
in
the
trends
of
extreme
temperatures.
The
relative
increase
in
the
number
of
days
above
the
historical
99th
percentile
is
greater
than
the
increase
in
the
number
of
days
above
the
historical
90th
percentile
(Figure
1b
).
The
globally
averaged
fractional
increase
from
the
historical
period
to
the
present
is
4.5x
and
2.1x
for
the
99th
and
90th
percentile
thresholds,
respectively.
While
there
is
substantial
interannual
and
decadal
variability
in
each
time
series,
the
tropical
regions
(Regions
9–12,
21–24,
and
38),
as
well
as
North
Africa
(Region
20)
and
the
Arabian
peninsula
(Region
36),
tend
to
exhibit
larger
increases
in
the
frequency
of
temperature
extremes
when
compared
to
midlatitude
regions,
like
North
America
(Regions
1–5),
which
exhibit
relatively
little
change
(Figure
1a
).
There
also
exists
substantial
interannual
variability
in
the
frequency
of
extremely
warm
days
across
all
regions
(Figure
1b
).
Next,
we
isolate
the
temporal
signal
present
in
these
noisy
observations
across
the
world.
3.2.
The
Slowly
Varying
Change
in
Extremes
To
gain
insight
into
the
spatiotemporal
variability
of
changes
in
extreme
temperatures,
we
use
LFCA
to
extract
the
most
slowly
varying
component
in
the
percentage
of
days
exceeding
historical
temperature
thresholds
by
maximizing
the
ratio
of
low
frequency
to
higher‐frequency
variance.
The
resulting
LFCs
and
LFPs
are
interpreted
in
a
similar
way
as
principal
components:
the
LFP
indicates
the
local
change
associated
with
σ
=
1
change
in
the
LFC.
The
results
in
Figure
2
are
consistent
with
previous
studies
(e.g.,
Hawkins
et
al.,
2020
),
revealing
a
pro-
nounced
climate
change
signal
in
temperature
extremes
from
the
noisy
underlying
data
set,
particularly
in
tropical
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Figure
1.
Observed
increase
in
the
frequency
of
extreme
warm
days
from
1955
to
2021.
(a)
The
linear
trend
in
the
percent
of
days
in
a
year
above
the
historical
90th
percentile
for
each
IPCC
AR6
land
region.
(b)
The
percent
of
days
each
year
from
1955
to
2021
above
the
historical
90th
(orange)
and
99th
(red)
percentiles
in
each
of
the
IPCC
AR6
land
regions
from
panel
(a).
The
historical
period
is
defined
as
1961–1990.
The
vertical
gray
lines
in
each
subpanel
mark
every
20
years
starting
in
1960.
Horizontal
gray
lines
correspond
to
20%
and
40%
of
days
above
the
historical
90th
percentile.
Because
of
the
dual
y
axis,
horizontal
gray
lines
in
each
subpanel
also
correspond
to
2%
and
4%
of
days
above
the
historical
99th
percentile.
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regions
where
the
LFP
loadings
are
over
twice
as
large
as
in
the
extratropics
(Figures
2c
and
2d
).
Using
LFCA,
we
are
able
to
effectively
identify
regions
with
the
highest
signal‐to‐noise
ratio
in
the
change
in
frequency
of
warm
and
extremely
warm
days.
Increases
in
the
frequency
of
temperature
extremes
from
the
historical
period
to
2021
across
the
Arabian
Peninsula
and
North
Africa
are
particularly
prominent.
There
is
a
greater
than
1.5x
and
6x
increase
in
the
frequency
with
which
the
historical
90th
and
99th
percentile
respectively
are
exceeded
(Figures
2c
and
2d
).
In
contrast,
we
observe
relatively
little
change
in
days
above
the
90th
and
99th
percentiles
across
Central
North
America
and
Western
Russia
(Figures
2c
and
2d
).
The
most
slowly
varying
component
of
the
temperature
time
series
closely
follows
the
observed
data
from
Figure
1
for
many
of
the
IPCC
AR6
land
regions
(Figure
S1
in
Supporting
Information
S1
).
The
spatial
patterns
of
the
change
in
frequency
of
temperature
extremes
are
similar
between
the
90th
and
99th
percentiles
(Figures
2c
and
2d
),
which
suggests
a
similar
physical
mechanism.
However,
the
fractional
increase
in
days
above
the
historical
99th
percentile
is
much
larger
than
the
90th
percentile.
The
temporal
structure
of
each
LFC
also
has
a
different
shape.
Focusing
on
the
change
in
daytime
temperature
exceeding
the
historical
90th
percentile
(Figure
2a
),
the
climate
change
signal
already
emerged
from
background
variability
around
the
year
1980,
similar
to
the
emergence
in
the
mean
(Hawkins
et
al.,
2020
).
Subsequently,
the
percent
of
days
exceeding
the
historical
90th
percentile
threshold
increased
approximately
linearly.
In
contrast,
the
climate
change
signal
in
daytime
temperature
exceeding
the
historical
99th
percentile
emerged
in
the
1990s
(Figure
2b
),
followed
by
a
much
more
rapid
increase
in
days
exceeding
the
once‐rare
threshold.
Many
locations
in
the
tropics
and
subtropics
now
experience
a
more
than
factor
3
increase
in
the
frequency
of
this
heat
hazard.
3.3.
Shift
in
the
Temperature
Distribution
To
understand
the
underlying
mechanism
behind
the
spatiotemporal
pattern
of
changes
in
temperature
extremes
isolated
by
LFCA,
we
build
off
previous
work
(e.g.,
Donat
&
Alexander,
2012
;
Loikith
&
Neelin,
2015
;
Rhines
&
Huybers,
2013
)
and
relate
the
observed
changes
in
extremes
to
characteristics
of
the
temperature
distribution.
Tropical
regions
have
narrower
historical
temperature
distributions
(i.e.,
lower
standard
deviation)
compared
to
the
midlatitudes
(Figure
S2
in
Supporting
Information
S1
).
In
areas
where
temperature
distributions
are
narrower,
warming
causes
a
greater
portion
of
the
shifted
temperature
distribution
to
fall
outside
of
the
historical
range.
This
results
in
more
days
in
the
present
climate
to
be
above
a
given
historical
extreme
temperature
threshold,
which
is
why
the
LFP
values
in
Figures
2c
and
2d
are
higher
in
the
tropics.
While
locations
with
a
more
negatively
skewed
Figure
2.
The
slowest
low‐frequency
components
(LFCs)
and
corresponding
low‐frequency
patterns
(LFPs)
in
surface
temperature
extremes.
The
first
(a)
LFC
and
(c)
corresponding
LFP
of
the
percent
of
days
in
a
year
where
the
daytime
maximum
temperature
exceeds
the
historical
(1961–1990)
90th
percentile
threshold.
(b–d)
Same
as
(a–c)
but
for
the
99th
percentile.
Note
different
color
bar
limits.
Locations
with
more
than
5
days
of
missing
data
in
any
year
from
1955
to
2021
are
masked
in
gray.
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temperature
distribution
would
also
theoretically
experience
a
larger
fraction
of
days
in
a
warmer
climate
falling
outside
of
the
historical
range,
it
is
much
harder
to
pick
out
locations
where
this
is
the
case
(Figure
S2
in
Sup-
porting
Information
S1
).
More
negatively
skewed
temperature
distributions
along
the
Gulf
Coast
of
the
United
States,
mainland
Southeast
Asia,
and
near
Paraguay
are
some
of
the
few
locations
where
this
is
notable
(Figure
S2
in
Supporting
Information
S1
and
Figures
2c
and
2d
).
If
we
approximate
the
annual
temperature
distribution
at
a
given
location
as
a
Gaussian,
the
area
to
the
right
of
the
historical
90th
and
99th
percentiles
will
increase
as
we
shift
the
distribution
to
the
right
(Figures
3a
and
3b
).
The
increase
in
the
area
to
the
right
of
the
historical
90th
and
99th
percentile
corresponds
to
a
higher
percentage
of
days
in
a
year
above
those
thresholds.
The
theoretical
increase
in
the
area
to
the
right
of
the
historical
90th
and
99th
percentiles
qualitatively
matches
the
temporal
pattern
of
the
increase
we
see
in
the
LFCs
in
Figures
2a
and
2b
.
Specifically,
we
would
expect
a
more
rapid
increase
in
days
exceeding
the
historical
99th
percentile
compared
to
the
90th,
as
observed.
For
example,
in
the
observed
temperature
distribution
for
Kigali,
Rwanda,
which
is
approximately
Gaussian,
a
mean
shift
effectively
captures
the
changes
in
the
90th
percentile
(
R
2
=
94%)
and
qualitatively
captures
the
changes
in
the
99th
percentile
(
R
2
=
50%)
for
any
given
year
(Figures
3c
and
3d
).
The
temperature
distribution
in
many
locations
is
approximately
Gaussian,
which
explains
why
the
temporal
patterns
of
LFCs
in
Figures
2a
and
2b
match
so
well
to
what
is
expected
from
shifting
a
Gaussian.
More
generally,
we
would
see
such
an
accelerating
increase
in
the
frequency
of
exceeding
the
historical
99th
percentile
for
any
temperature
distribution
f
(
T
)
where
d
f
/
d
T
<
0
in
the
neighborhood
of
the
historical
99th
percentile
tempera-
ture
T
=
T
99
.
Figure
3.
Shifting
surface
temperature
distributions
and
the
effect
on
changing
extremes.
(a)
Shifting
a
Gaussian
probability
density
function
(PDF)
with
mean
μ
and
standard
deviation
σ
,
where
the
solid
PDF
is
the
“historical”
distribution,
and
the
dashed
PDF
is
the
“shifted”
one.
Shifting
a
PDF
causes
the
red
hatched
(days
above
the
99th
percentile)
and
orange
(days
above
the
90th
percentile)
areas
to
get
larger
at
different
rates.
(b)
The
growing
red
and
orange
areas
as
a
function
of
the
shift
(warming)
relative
to
the
standard
deviation
(width)
of
the
distribution.
(c)
Shifting
the
historical
observed
temperature
distribution
in
a
tropical
location
(Kigali,
Rwanda)
by
the
mean
temperature
anomaly
in
2015.
(d)
Same
as
(b)
but
solid
lines
indicate
the
theoretical
increase
in
orange
and
red
hatched
area
based
on
the
distribution
in
panel
(c).
Points
indicate
the
actual
change
in
percent
of
days
above
historical
temperature
thresholds
corresponding
to
the
temperature
anomaly
for
each
year.
The
year
2015,
corresponding
to
the
shift
in
panel
(c),
is
indicated
by
a
star.
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