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How do millennial proxy reconstructions methods stack up?
Terry Lee1, Francis Zwiers2, Min Tsao1
1Dept. of Math and Stats, University of Victoria, Victoria, BC2Climate Research Division, Environment Canada, Toronto, Ontario
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σ= 10, r=28, b=8/3
Allen and Stott, 2003, Climate Dynamics
Section 4
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Distribution of1-day averages Distribution of500-day averages
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Outline• Background
• Reconstruction calibration techniques
• Their evaluation
• Conclusions
• Two extensions
• Comments??
Photo: F. Zwiers
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The “hockey stick” (TAR SPM)
• “It is likely that the 1990s have been the warmest decade and 1998 the warmest year of the millennium”
IPCC TAR Fig. 1b
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The “hockey-stick” debate …
• Iconic figure and bold statement drew lots of interest from the sceptics
• Some Canadians particularly vocal
• Criticism focused on
– Statistical method (influence of centering on EOFs)
– Choice of proxies (use of Bristlecone Pines)
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There was another figure in the TAR …
IPCC TAR Fig. 2.21
• Indicating that the general form was robust,
• and that some reconstructions were more variable than others
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Reconstructions at time of AR4
After IPCC WG1 AR4, Fig. 6.10
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Interpreting the reconstructions with models
• There now exist several long climate simulations with reconstructed volcanic, solar and GHG forcing
– With AOGCMs (ECHO-G, CSM 1.4, HadCM3 …)
– EMICS
– EBMS
– (Note – now even more as a consequence of
CMIP5)
• These seem able to reproduce some of the features
seen in the reconstructions…
• Detection studies confirm the presence of natural and
anthropogenic signals…
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Temperature of last millenium
All
Nat
After IPCC WG1 AR4, Fig. 6.14
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IPCC AR4 Assessment
• Average Northern Hemisphere temperatures during the second half of the 20th century were very likelyhigher than during any other 50-year period in the last 500 years and likely the highest in at least the past 1300 years.
• Assessment refers to longer time scales than in the TAR
• Many more reconstructions available
• Some detection results indicating external influence, detectable in all reconstructions that were examined
• Some inter-comparison of reconstruction techniques
• We have now undertaken a more complete assessment of available techniques (although still far from exhaustive)
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The reconstruction techniques
Photo: F. Zwiers
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The Reconstruction Problem
NH mean
temperature
Proxy series
Need to reconstruct Known
1860 20001000
Known 20001000 1860Known
Calibration period
Year
Training period
Identify a statistical relationship between a collection proxies and NH temperature
Apply that relationship to reconstruct past NH temperature
Reconstruction period
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Two types of reconstruction techniques
• CPS – composite plus scale
– Average (or composite) proxies into some index
(e.g., just average, and make dimensionless)
– Calibrate the composite to hemispheric mean
temperature from instrumental data
• CFR – climate field reconstruction
– EOF regression, or other technique, to reconstruct hemispheric temperature field
• Used, for example, to reconstruct SSTs back
into 1800’s using sparse instrumental data
– Spatially average the reconstructed field to
estimate hemispheric mean temperature
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Reconstruction techniques
ttt PT εβ +=• Ordinary Least Squares
tttt PT εηβ +−= )(• Total Least Squares
2/1)](/)([ˆ PVTV
PT ttt
=
+=
β
εβ• Variance Matching
t
NH k
kktkt dAvuT ελ += ∫ ∑ ,
• MBH (1998)
ttt TP εβ +=• Inverse Regression
ttt-t
ttt
Fφ TT
TP
ωδ
εβ
++=
+=
1
• Kalman Filter/Smoother
• RegEM
CPS
CFR
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Proxy data (Pt)e.g., thickness of tree ring
State-space model
known known
Year
unknown knownNH temperature (Tt)
known known
1850 20071000
Estimated influence of external forcings (Ft)
Estimated by Kalman filter (Kalman 1960)
Observation equation:
State equation:
UNKNOWN !
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Standard parameter estimation approach (STD)
• Maximum likelihood of state process given proxies and estimated responses to forcing.
known knownProxy data (Pt)
unknown knownState process:NH temperature (Tt)
known known
1850 20071000
Estimated influence of external forcings (Ft)
Kalman filter estimates:
non-linear function of parameters
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Improved parameter estimation technique (ALL)
Kalman filter estimates
known knownProxy data (Pt)
unknown knownState process:NH temperature (Tt)
known known
1850 20071000
Estimated influence of external forcings (Ft)
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known knownProxy data (Pt)
knownState process:NH temperature (Tt)
known known
1850 20071000
Estimated influence of external forcings (Ft)
Competing parameter estimation technique (CAL)
• Uses only calibration period data.
• MLEs can be solved explicitly, for example,
,
simple
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• Likelihood too complicated to derive asymptotic
properties in all cases � used Monte-Carlo
• Generate data under assumed model:
– Specified reasonable values for A, φ, δ, R and Q.
– Used Ft from a simple EBM of the climate.
– Generated noise terms et and wt from normal.
),0(~ , T T
),0(~ ,T P
1QNwwF
RNeeA
tttt-t
tttt
++=
+=
δφ
Comparing estimator properties
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phi
N=200 N=500 N=998
4.88
1.68
0.72
2.1
1.31
0.67
2.412.07
2.6
phi
N=200 N=500 N=998
4.12
2.81
1.93
1 1 11.29
1.88
3.38
Estimated efficiency for φ (relative to ALL)
---- STD (n=N)---- ALL (n=N)---- CAL (n=100)
Smaller value in efficiency= smaller estimator variance
= better estimator
n = sample size
Relative efficiency of persistence parameter
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Known in simulation/ Estimated by Kalman filter
known known
known
known known
1850 20071000
N=200 N=500 N=998
1.11
1.041.02
1 1 10.99
1 1.01
Reconstruction error (relative to ALL)
---- STD (n=N)---- ALL (n=N)---- CAL (n=100)
Smaller error = better approach
Reconstruction error
Proxy data
NH temperature
Estimated influence of external forcings
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),0(~ , T T
),0(~ ,T P
1QNwwF
RNeeA
tttt-t
tttt
++=
+=
δφ
N=200 N=500 N=998
2.52.79 2.86
1.14 1.24 1.331.11 1.12 1.11
e t =α et-1+κ t , κ t ~N(0, R)
Change in reconstruction error
(relative to iid case)
---- STD (n=N)---- ALL (n=N)---- CAL (n=100)
Smaller change = more robust
Estimators from CAL are asymptotically unbiased!
Robustness
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• STD approach should be avoided.
– Larger bias.
– Lower efficiency.
– Less robust.
• CAL approach is more robust
– Reconstruction errors similar to ALL approach
– Resistant to misspecification of state equation
– MLEs easily obtained
• ALL approach if desire is to estimate state equation
parameters
Summary
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Evaluating the reconstruction techniques
Photo: F. Zwiers
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Evaluating Reconstruction techniques
• von Storch et al (2004) proposed using long climate simulations
• Idea was to
– Sample model output at locations coinciding with proxies
– Degrade by adding white noise
– Apply reconstruction method
– Compare with the known model simulated hemispheric mean temperature
• There are nuances to consider such as,
– how much noise should be added?
– what colour of noise should be added?
– should one detrend prior to diagnosing the obs/proxy relationship?
– what part of the spectrum should be used?
• The latter three questions are all broadly equivalent
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Proposed evaluation technique
• Proposed by von Storch et al. (2004)
Simulated NH mean temperature time series
100 pseudo-proxy series
2) Apply reconstruction method
Reconstructed NH mean temperature time series
1) Add noise to 100 grid-box temperature series 3) Compare
Climate model output
(~2300 grid boxes)
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Some details
• We did this
– Two different climate models
– Two signal to noise ratios
– White and red noise
– Different sizes of proxy networks
– Different lengths of calibration period
– Effects of detrending prior to calibration
• We did NOT condition on a fixed proxy network
– Repeated reconstruction 1000 times
• A different proxy network at each time
• A different realization of added noise each time
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100 pseudo proxies – 1860-1970 calibration – mean abs deviation
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ECHO-G
15 pseudo proxies – 1860-1970 calibration – mean abs deviation
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ECHO-G
15 pseudo proxies – 1880-1960 calibration – mean abs deviation
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15 point network – 11 year moving average – CSM - SNR = 0.5
One particular reconstruction from the sample of 100
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Conclusions
• Some reconstruction methods to should be avoided
• Several other methods compare well, particularly at low frequency
• Assessment is not sensitive to which “reality” one compares against
• CFR methods do not have a particular predilection to producing hockey stick like reconstructions
• The new Mann et al CFR technique works well, but CFR does not appear to be consistently better that CPS
• Kalman filter approach appears to produce consistently good results across time scales
• There is probably greater sensitivity to the choice of proxy than to the choice of reconstruction method (see also Juckes et al, 2007, Climate of the Past Discussions, vol 2)
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Two Extensions using Kalman smoother …
Photo: F. Zwiers
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ttt-t
ttt
Fφ TT
TP
ωδ
εβ
++=
+=
1
State space model
)()()()( tStGtVoltSolt SulrGHGrVolrSolrF αααα +++=
where
• Data assimilation (of sorts) using a climate model to
assimilate proxy information and observed hemispheric
temperatures
• Fitting the state space model provides estimates of
coefficients and their uncertainty � allows “detection”
• Integrate climate model forward in time � allows
“prediction”
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Climate change detection:
• Using real-world proxy data (1510-1960):
– Effect of greenhouse gas, aerosol and volcanic forcing has significantly affected our past climate.
– Consistent with other detection studies.
Temperature prediction:
• Result with real-world proxy data (1510-1960):
Year
Te
mp
era
ture
an
om
alie
s (
De
gre
ss K
)
1920 1940 1960 1980
0.0
0.2
0.4
0.6
ObservedState-space model forecast
Observation
Temperature forecast for 1961-1990
Results …
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The End
Photo: F.Zwiers
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References• Lee, T.C.K., F.W. Zwiers and M. Tsao, 2008: Evaluation of
millennial proxy reconstruction methods. Climate Dynamics, 31, 263-281, doi 10.1007/s00382-007-0351-9.
• Lee, T.C.K., M. Tsao, F.W. Zwiers, 2010: State-space model for proxy-based millennial reconstruction. Canadian Journal of Statistics, 38, 488-505.