Statistical Methods for Meteorology and Climate Change

The value of multi-proxy reconstruction ofpast climate

Bo Li

Department of Statistics, Purdue University

Based on joint work with:

Doug Nychka and Caspar Ammann

National Center for Atmospheric Research (NCAR)

Montreal, Canada Jan 14, 2011

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http://news.bbc.co.uk/2/hi/science/nature/8618024.stm

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Outline

• Introduction-Why care about past climate?

• Various data sources

• BHM to integrate data from different sources

• Numerical analysis from climate model output

• Summary and discussion

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Why care about the PAST temperature?

• Long time series of climate variables including tem-perature are required to understand the dynamics ofclimate change

• Direct observations of surface temperature is onlyavailable from 1850

• Validate climate models - Atmosphere/Ocean Gen-eral Circulation Model (AOGCM)

How to get past temperatures?Reconstruct the past temperature from indirect obser-vations (proxies) such as Tree Ring, Pollen and Boreholeand Radiative Forcings.

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Data - Tree Ring

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Data - Pollen

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Data - Borehole

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Forcings

1000 1200 1400 1600 1800 2000

a

b

c

a: Volcanism (contains substantial noise)

b: solar irradiance

c: green house gases

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Formulate the problem

Skill of each proxy and forcings

• Tree ring (Dendrochronology): annual to decadal

• Pollen: bi-decadal to semi-centennial

• Borehole: centennial and onward

• Forcings: external drivers

Goal: Reconstruct the 850-1849 temperature by allproxies, forcings and the 1850-1999 temperature

Bayesian Hierarchical Model (BHM) to thread allproxies, forcings and temperatures

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Bayesian Hierarchical Model (BHM)

A distribution rule:[P, T, θ] = [P |T, θ][T |θ][θ]

Three hierarchies:

• Data Stage: [Proxies|Temperature, Parameters]

Likelihood of Proxies given temperatures

• Process Stage: [Temperature|Parameters]

Physical model of temperature process

• Parameter Stage: [Parameters]

Specify the prior of parameters

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BHM

• Data Stage: [Data|Geophysical Process, Parameters]

Di|(T′1,T′2)′ = µiD1+ βiDMD(T′1,T′2)′+ εiD, εiD ∼ AR(2)

Pj|(T′1,T′2)′ = µjP1+ βjPMP(T′1,T′2)′+ εjP , εjP ∼ AR(2)

Bk|(T′1,T′2)′ = MB{µkB1+ βkB(T′1,T′2)′+ εkB}, εkB

iid∼N(0, σ2B)

V|V0 = (1+ εV )V0, εViid∼ N(0,1/64)

– Di,Pj,Bk: tree-ring (Dendrochronology),

Pollen, Borehole [different length]

– V0, V: Volcanism, Volcanic series with error

– MD, MP , MB: transformation matrices to link tem-perature series to tree-ring, pollen and borehole

– T1: unknown temperatures; T2: observed temper-atures (1850-present)

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BHM

• Process Stage: [Geophysical Process|Parameters]

(T′1,T′2)|(S,V0,C) = β01+β1S+β2V0+β3C+εT , εT ∼ AR(2)

– S, V0, C: Solar irradiance, Volcanism,

greenhouse gases (CO2)

• Parameter Stage: [Parameters]

Specify the prior of parameters

Target: estimate T1 given T2 (1850-present),

proxies and forcings

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Apply BHM to Climate from Models

National Center for Atmospheric Research (NCAR) Com-munity Climate System Model (CCSM) Version 1.4

• Provide test beds to evaluate our reconstructionmethod

• Climate model is highly nonlinear and substantialcomplexity

Resolution:

– 3.750 × 3.750/400km×400km for atmosphere and land

• Generate synthetic proxies (15 Tree-Ring, 10 Pollenand 5 Borehole) from model climate based on theirown characteristics

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Climate from Models

Pollen: 7.50 × 7.50; Borehole: 200 × 200

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Synthetic Proxies Generation

Generate synthetic proxies:

• 15 tree rings: subtract the 10-year smoothing aver-age from each of the local temperature series

• 10 pollen: Sample a 10-year smoothing average tem-perature series at 30 year intervals

1000 1200 1400 1600 1800 2000

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tempe

rature

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aly

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Tree-rings: black line; Pollen: red dots

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• 5 borehole:

– POM-SAT Forward Model

– describe the diffusion process ofsurface temperature

year

tempe

rature

0.00.20.40.60.81.0

1000 1200 1400 1600 1800 2000

pulse at year 8500.00.20.40.60.81.0

pulse at year 6500.00.20.40.60.81.0

pulse at year 4500.00.20.40.60.81.0

pulse at year 2500.00.20.40.60.81.0

pulse at year 50

0.00 0.04 0.08 0.12

depth temperature

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pth

(m

)

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00

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Interesting Questions

• What is the optimal skill of our BHM model?

• What is the skill of each proxy?

• What is the role of forcings?

• What if we only model T1 in the process stage?

• How does the noise in proxies affect the reconstruc-tion?

– Perfect proxies: proxies directly generated fromthe local/regional temperatures

– Contaminated proxies: The signal to noise ratio ischosen 1:4 in terms of their variance

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Numerical Study

Five models with different subsets of proxy orrelated data:

• local/regional Temperature series (T)

Tl|(T′1,T′2)′ = µl1+ βl(T′1,T

′2)′+ εl, εl ∼ AR(2)(σ2, φ1, φ2)

• Tree ring (Dendrochronology) only (D)

• Tree ring + Pollen (DP)

• Tree ring + Borehole (DB)

• Tree ring + Borehole + Pollen (DBP)

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Numerical Study

A 23 factorial design for each of the five models (40reconstructions):

• with/without forcing covariates

(T′1,T′2)′ = β01+ εT , εT ∼ AR(2)(σ2

T , φ1T , φ2T).

• with/without proxy noise

• modeling T1/T in the process model

T1|(S,V0,C) = β01+ β1S+ β2V0 + β3C+ εT

εT ∼ AR(2)(σ2T , φ1T , φ2T).

orT1 = β01+ εT , εT ∼ AR(2)(σ2

T , φ1T , φ2T)

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Bias, Variance and Rmse

• Bias = E(T1 − T1)

• Variance = var(T1 − T1)

• Rmse =√E(T1 − T1)2 = sqrt(Bias2+Variance).

model

bias

−0.2

0.0

0.2

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T D DP DB DBP

C

C

C

C

CF F F F F

TNoNoise

T D DP DB DBP

C C C

C

CFF

FF

F

T1NoNoise

T D DP DB DBP

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CC

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F F FF

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TNoise

T D DP DB DBP

C CC

C

CF

F

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T1Noise

“C” and “F” are the reconstructions without forcings (with con-

stant mean function) and with forcings incorporated, respectively.

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model

varia

nce

0.0200.0250.0300.0350.0400.045

T D DP DB DBP

C

C

C

C

C

F

FF

FF

TNoNoise

T D DP DB DBP

C

C

C

C

C

F

FF

FF

T1NoNoise

T D DP DB DBP

C

C

C

C

C

F

F F

F

F

TNoise

T D DP DB DBP

C

CC

CC

F

F

F

F

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T1Noise

model

rmse

0.2

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T D DP DB DBP

C

C

C

C

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FF F F F

TNoNoise

T D DP DB DBP

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C C CC

FF F F F

T1NoNoise

T D DP DB DBP

C

C

C

C

C

FF

FF F

TNoise

T D DP DB DBP

C

C

C CC

F

F

FF

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T1Noise

“C” and “F” are the reconstructions without forcings (with con-

stant mean function) and with forcings incorporated, respectively.

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A Formal Way for Reconstruction Comparison

Posterior predictive loss criterion (Gelfand and Ghosh,1998)

Dk(m) = P (m) +k

k+1G(m)

• Dk(m): loss function for each reconstruction m,

m = 1,2, . . . ,40

• k ≥ 0: a weighting parameter

• P (m): sum of predictive variances that imposes penaltyon the complexity of models

• G(m): sum of squared errors that measures the

goodness-of-fit

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0.00

50.

050

0.50

0

period (year)

spec

trum

1000 400 200 100 50 30 20 10 5 3

T

DBP

DP

DDB

Using smoothed spectrum of reconstruction residuals from the five

data models to illustrate the frequency band at which proxies cap-

ture the variation of the temperature process.

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The Value of Forcings and Proxies

• Forcing covariates dramatically reduce the bias, thevariance and thus rmse

– if the included proxy data do not well represent thelow frequency variability

• Not surprisingly

– tree-rings retain the high frequency variability

– pollen captures the variability at about a 30 yearperiod and onward

• Kind of surprising: Borehole helps to reduce the biasonly a bit

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Other Inferences

• Cause of Bias

The reconstruction with T = (T′1,T′2)′ in the process

stage carries systematic positive bias

– in particular when forcing is not included, i.e.,

(T′1,T′2)′ = β01+ εT

– Reason: T1 and T2 do not have the same mean

– Solution: Incorporate forcings

• Sensitivity to Noise in Proxies

The reconstruction deteriorates somewhat but notterribly

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1000 1200 1400 1600 1800 2000

tem

pera

ture

targetreconstruction

targetreconstruction

targetreconstruction

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60.

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6−

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60.

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c

b

a

The reconstructions using tree-rings and pollen together with forc-

ings in three scenarios. a: modeling T and without noise; b: mod-

eling T1 and without noise; c: modeling T and with noise.

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Summary and Future Work

• Systematically investigate the role of proxies and forc-ings to provide a guide for temperature reconstruc-tion

• Suggest multi-proxy reconstructions with tree-ringsand pollen assemblages and also including externalforcings

• Bayesian analysis provides a rigorous and easy methodto quantify the uncertainty of the reconstruction

• Explicitly modeled measurement errors

(Ammann, Genton and Li, 2010)

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Summary and Future Work

• More formally consider dating errors in pollen data(Haslett and Parnell, 2008)

• Extend the results to real world conditions

• Spatio-temporal reconstruction of multiple climatevariables (Tingley and Huybers, 2010)

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Thanks!

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