n. w. watkins with d. a. stainforth , s. c. chapman · 3 papers •chapman, stainforth, watkins,...

Quantifying the time dynamics of the full local

distribution of daily (temperature &) precipitation

and its uncertainties

N. W. Watkins1,3,5 with

D. A. Stainforth3,4,1, S. C. Chapman1,2

1CFSA, Physics, Univ. of Warwick; 2Mathematics and Statistics, UIT; 3CATS, LSE, 4Grantham Institute LSE;

5Open University, UK

We acknowledge the E-OBS dataset from the EU-FP6 project ENSEMBLES (http://ensembles-eu.metoffice.com) and the data providers in the ECA&D project (http://eca.knmi.nl)

http://eca.knmi.nl/

3 Papers• Chapman, Stainforth, Watkins, Phil. Trans. R.

Soc. A. 2013

• Stainforth, Chapman, Watkins, Environ. Res. Lett., 2013

• Chapman, Stainforth, Watkins, Environ. Res. Lett., 2015

• All at Sandra’s webpage, my Researchgate etc

Temperature

Pap

Temperature

P Precip

Temperature

TEMPERATURE: Is it just hot today, or is it climate change?- why this particular question?

Perception of past climate change as experienced locally

A local phenomenon- varies with geographical location, quantile/variable (less cold days or hotter warm days?)

Even locally, trend in the extremes may not follow trend in the mean (average days are no warmer, but the hottest days are much hotter?)

Is local trend very different to global mean temperature trend?

[Later on, more heavy tailed climate variables (precipitation)]

Planning thresholds- crops, ecosystems, building overheating…

Fundamental statistical constraints- to what extent can we answer this question using only the data?

Is it just hot today, or is it climate change?

Quantify likelihood (of daily temperature) and how it is changingcumulative density function (cdf) C(T)

Measure a daily temperature T

Fraction of days observed to be colder than T

1

0T

Compare two different time periods t1 and t2

C(T,t1) C(T,t2)

Fundamental constraints on the number of observations

Quantify likelihood (of daily temperature) and how it is changing

Fraction of days observed to be colder than T

Only 92 days per summer season

multistation data only from 1950

C(T,t1), C(T,t2) have uncertainties

1

0

T

C(T,t1)C(T,t2) Can we measure how

fast C is changing?

-uncertainties depend on unknown functional form of C!

warming

Constraint on number of observations → different samples of C will fluctuate in the absence of time dependence‘random’ error in C

The daily NAO index is constructed by projecting the daily

(00Z) 500mb height anomalies over the Northern

Hemisphere onto the loading pattern of the NAO (shown)

defined as the first leading mode of Rotated Empirical

Orthogonal Function (REOF) analysis of monthly mean500mb height during 1950-2000 period.

NAO-Hurrell et al Science 1995, plots courtesy

UCAR, NOAA

NAO

AO

AMO

Natural oscillations (which may also be changing with the climate): North Atlantic Oscillation (NAO), Arctic Oscillation (AO), Atlantic MultidecadalOscillation (AMO), El Nino… index is deviation from mean pattern

Plotted ~1860-2009

precip

temp

Probability of daily temperature, precip. and how it’s changing-depends on where you live

Distribution of stations in Eobsgridded set

Data 1950-now

Coverage not uniform. Not always the same stations. Data not continuous.

Stations are moved, or city ‘heat islands’ move near them.

Not always the same instrumentation.

Observation constraints→ ‘random’ and systematic errors

pdf P(T,t):Likelihood of T observed in the range T-T+dT

cdf C(T,t):Fraction of days observed to be colder than T

4 locations in E-OBSis C changing with time?at what temp T is it changing?

T

T

0

1

0

<1

<T>

0.5

0.75

0.25

each cdf, pdf is made using 9 successive summers9x92 days in each 1950- now, so 63 years…

Remember this plot you will see it again!

MethodParameterize how different quantiles are changing with time (forcing)

Observe for example daily temperature Tfrom a given location over τ seasons (years) centred on an epoch t (year).

Form cdf C for different epochs giving C(t,T) for each t

Now, cdf C can change with t on timescales that are much longer than τ*separation of timescales*This climate state has some unknown forcing parameter g(t). Then:

C(t,T)≡C(g(t),T)

C(g,T)

T

C(g(t1))C(g(t2))

warming

At t2 we observe a temperature T*> Tq

Does this observation arise from forcing, i.e. the change in C with g(t)?

q

Tq

( , )

set 0

q

q

q

q xg

q C x g

C Cdq dx dg

x g

dq

since the pdf is the differential of the cdf w.r.t.

x this is:

1 (

( ))

)

(q

q

q x

Cdx dg

x

x

PP

C

xg

Provided C can be treated as a smooth, differentiable

function we can write, with

dxq=x*-xq

The change dxq=x*-xq due solely to

forcing is obtained by setting dq=0.

Estimating the change at a quantile (temps)

Now to estimate ∆x and its uncertainty…

N yearly seasons dataper cdf

( )T TQ Q P

Estimate ∆Q=Q(t2)-Q(t1)

Each pair of cdfsgives one estimate

We can repeat this procedure for different t1,t2 to get several estimates of ∆(Q=any quantity that depends on the distribution).

Estimate changes and uncertainties solely from dataConsider some Q(x) that captures a property of the distribution C(x)…

pdf<threshold

1

0 x

C(x,t1)C(x,t2)

time

Use all 10 estimates to quantify uncertainty

TQ

x

Show you “light tailed” example-temperaturedaily mean, max, min

-show you how uncertainties can be due to restrictions on the observations-look at some data(temperatures: E-OBS gridded dataset since 1950, Europe)

STEP

1Aggregate

observations over several consecutive

years. STEP

2

Construct the empirical CDF and PDF for each of the time slices. ST

EP 3

Estimate the change either in a specific threshold or

specific quantile.

STEP

4

The significance of

the trend is estimated

based on the statistical

uncertainty in the input data.

2

0 0

For each 'year' , that is, each 100 values

specify shift, scale parameters ( ), ( ).

Mean , varian

no time

ce

v

,skewness 2 / ,excess Kurtosis 6 / .

(i) : ariation

mean changes:

3; 5

(ii) as (

t x

a t b t

ab ab b b

a a b b

0

0

i) with shift 5 /100

(iii) ( ) [1 1 / 2( /100)]; consmean, variance change:

mean, variance, skew, kurtosis change

tant

(iv) ( ) [1 1 / 2( /100)]; constant:

x x t

a t a t b

b t b t a

Test how well this works:Illustrativemodel dataSame size sample as actual data‘fair dice’

pdf

cdf

time

( ) 1 ( )

( )

1( , )

( ) ( ( ))

x

a t b t

a tP x t x e

b t a t

Gamma distributed iid

Technical details:

Change dTq=T*-Tq

Shown:∆C=C(t=80)-C(t=20)

P all ‘years’, estimate:

pdf

cdf

( )

(

0

)

)

0

(P T

T

C

C

T

constant varies , varies , , , variesS k Model data3 ‘years’ data(3 consecutive ‘seasons’)per pdf, cdf

( )

( )q

C

P TT

T

pdf

cdf

( )

(

0

)

)

0

(P T

T

C

C

T

Change dTq=T*-Tq

( )

( )q

C

P TT

T

constant varies , varies , , , variesS k

So maybe ok to use 9 ‘years’ data (9 consecutive ‘seasons’) per cdf…

Model data9 ‘years’ data(9 consecutive ‘seasons’)per pdf, cdf

constant varies , varies , , , variesS k

P>0.005

( )

( )q

C

P TT

T

3 years data per cdf

9 years..

Model data10 estimates of ∆TIllustrates likely ‘spread’ in results just due to sample size and rate of convergence

∆T

∆T

Data: E-OBS max daily temperatures at 4 locations

look at four locations at longitude and latitude(i)[4.75 52.25] Leiden Holland (ii) [-4.75 51.75] west Wales (iii) [-4.75 42.75] Leon, north Spain(iv) [11.25 43.75] Florence, Italy.summer aggregate daily temperatures over 3 month intervals within each year, June, July and August, 92 samples per year.

pdf

cdf

E-OBS dataset (http://eca.knmi.nl)

http://eca.knmi.nl/

pdf

cdf

( )

(

0

)

)

0

(P T

T

C

C

T

Change dTq=T*-Tq

Shown:∆C=C(t=2005)-C(t=1955)


( )

( )q

C

P TT

T

E-OBS data3 ‘years’ data(3 consecutive ‘seasons’)per pdf, cdf

Leiden, NL W-Wales N-Spain Florence, IT


pdf

cdf

( )

(

0

)

)

0

(P T

T

C

C

T

( )

( )q

C

P TT

T

Change dTQ=T*-TQ

Shown:∆C=C(t=2005)-C(t=1955)


E-OBS data9 ‘years’ data(9 consecutive ‘seasons’)per pdf, cdf

Combine the results and uncertainties on one map…2 kinds of uncertainty (i) ‘random’ fluctuations (ii) systematic trends, cycles

big trend, small spread

small trend, small spread

big trend, big spreadsmall trend, big spread

plot the minimum ∆T from the 10 estimates for each T and location

∆C 1955-1995, 1956-1996 …


P>0.005

( )

( )q

C

P TT

T

9 years..


E-OBS data10 estimates of ∆TIllustrates likely ‘spread’ in results due to all data issues

Indication of where results are robust and where not!

∆C 1955-1995, 1956-1996 …


P>0.005

( )

( )q

C

P TT

T

9 years..


E-OBS data10 estimates of ∆TIllustrates likely ‘spread’ in results due to all data issues

Indication of where results are robust and where not!

Use minimum of all 10 samples

Summer daily max temp minimum change at quantile

Summer daily maximum temperature signed smallest abs(∆T) of 10 estimates

• Tendency for upper quantilesto show largest ∆T

• Highest ∆T across a band from northern France to Denmark at the highest quantiles.

• At mid-to-high quantiles, largest ∆T in a band slightly further south.

• Scandinavia- little robust signature.

Winter daily min temp minimum change at quantile

Winter daily minimum temperaturesigned smallest abs(∆T) in 10 estimates

Winter night time temperatures- smaller changes in ∆T EXCEPT:• At the very lowest

quantiles in central western Europe, and

• At low to mid-quantiles in Scandinavia

NOW PRECIP: Estimate change in the extremes-‘heavy tailed’ pdfs, daily precipitation

-look at uncertainties just due to restrictions on the observations-look at some data(temperatures-Eobs gridded dataset since 1950, Europe)

II: Capturing the behaviour of the extremes (precip)mean total season precip above a threshold (on all days wetter than) x=PT-how wet are the downpour days

Change in mean total season precip on all days wetter than PT-how much more rain now falls on downpour days

% change in total … on all days wetter than PT-is that a big change?

Change in fraction of total season precip that occurs on days wetter than PT-all days are wetter? or same total season precip, now in downpours?

PT

( )T TP x x P

Generalizes indices at fixed threshold eg:Alexander, L. V. et al. Global observed changes in daily climate extremes of temperature and precipitation. JGR-Atmos. 111, doi:D05109 10.1029/2005jd006290 (2006).Climdex. Datasets for Indices of Climate Extremes, http://www.climdex.org/indices.html (2015).

TP

% %TT

T

PP

P

( 0)

TT

T T

PF

P P

TP

x

http://www.climdex.org/indices.html

Shifting mean

Shifting mean and changing

shapeChanging

shape

Explore these scenarios:

Can these changes be distinguished in the data?(effect of fundamental constraints/uncertainties)

(a) Change in precip at a quantile

(b) Change in mean total season precip on all days wetter than PT

(c) % change in total season precipon all days wetter than PT

(d) Change in fraction of total season precip that occurs on days wetter than PT

Modelling uncertainties: Gamma distributed iid RV

Use the min of 10 samples..Throw away if not all of one sign(zero spanning values)









E-OBS daily precipitation4 locations

(i) SW Scotland(ii) Central Scotland(iii) Dordogne(iv) SW Wales

Precipitation: consider wet days only x>1mm





E-OBS daily winter precipitation4 locations


more total precip

and shift to more intense days

and shift to more intense days







Not much more total precip

but shift to more intense days







Clearly identifiable drying







No clear single trend



E-OBS daily winter precipitationMaps of min of all 10 estimates, zero spanning values not plotted




+2mm on a q=0.95 day

-2mm on a q=0.95 day

+40% on all days >15mm

(a) % change in total season precip on all days wetter than PT

(b) Change in fraction of total season precip that occurs on days wetter than PT


1

In time observe events which in ascending size are

.. ..

in there are 1 events of si

,

'Return Time' for event i

ze

( )( 1) ( )

Now ( ) /

so (1 )

or in term

s

k N

k

k

k

t N

x x x

t N k x x

x x

t tR x

N k N k

C x x k N

RR C

C

s of the change at quantile

( 1)( 0)

C C Rdx dq

P P R

Finally, this can all be recast in terms of conventional ‘Return Times’…

t

Summary and Next Steps

Our method is entirely data-based: not about prediction or attribution, but rather characterising recent rate of change at particular quantiles.

Focus at particular quantiles in particular locations.

With temperature looked a d(temperature) at given temp. Can generalise idea as we did in precip to derived quantities which are application specific e.g. all days which exceed a threshold for flood management.

Talking to end users will enable us to find these quantities.

Summary and Next Steps

Decisions are usually threshold-specific. Planning will be sensitive to how likely such a threshold crossing is in future e.g. potential collaboration with Zaid Chalabi of Public Health England.

Uncertainties are subtle and depend on where you are geographically and in the distribution.

Need for modelling that goes beyond iid into dependence (and even long range dependence), and into more realistic, data-determined distributions-not just Gaussian and gamma.

n. w. watkins with d. a. stainforth , s. c. chapman · 3 papers •chapman, stainforth, watkins,...

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