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Proxy-Climate Transfer Methods
Proxy WorkshopJuly 4th 2011, Cologne, Germany
Christian Ohlwein1 and Andreas Hense1
in collaboration with: Thomas Litt2 Norbert Kuhl2, and Eugene Wahl3, . . .
1) Meteorological Institute at the University of Bonn, Germany2) Steinmann-Institute of Geology, Mineralogy and Paleontology, Bonn, Germany3) National Oceanic and Atmospheric Administration (NOAA), Boulder, CO, USA
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Requirements
Requirements
Find proxy-climate relationships
Combine multiple proxy variables
Account for spatio-temporal processes
Estimate uncertainties
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Objectives
Objectives
1 Explain proxy-climate transfer methods in a probabilistic framework
2 Show examples of probabilistic pollen-climate transfer methods
3 Discuss the aspect of uncertainty
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Outline
1 Proxy-climate transfer methodsClassical transfer functionsProbabilistic (Bayesian) frameworkDevelopment of new methods
2 Example I: mutual climatic ranges
3 Example II: modern analogues
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Classical transfer functions1. Proxy-climate transfer methods
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Classical transfer functions1. Proxy-climate transfer methods
Deterministic point of view?
Stochastic climate system
Limited knowledge of the processes
Spatio-temporal representation
Measurement errors
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Probabilistic (Bayesian) framework1. Proxy-climate transfer methods
Random variables
Palaeo climate state ~X0
Parameters ~θ
Palaeo proxy variables ~Y0
Climate state ~XProxy variables ~Y
ReconstructionConditional probability density
[~X0|~Y0, ~X , ~Y ] or [~X0, ~θ|~Y0, ~X , ~Y ]
(c.f. classical regression: ~µ~X0= E(~X0| . . . ))
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Development of new methods1. Proxy-climate transfer methods
Classical concepts
Historically focused on the palaeo archives
Knowledge of the bio-geochemical processes
Various “tools” for transfer functions
New approaches
1 Rewrite in a probabilistic framework
2 Specify the implicit assumptions
3 Address the consequential modifications
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Outline
1 Proxy-climate transfer methods
2 Example I: mutual climatic rangesPollen proxiesMutual climatic range approachProbabilistic indicator taxa model
3 Example II: modern analogues
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Pollen proxies2. Example I: mutual climatic ranges
Holocene pollen spectrum from lake Holzmaar
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Pollen proxies2. Example I: mutual climatic ranges
Relation between pollen counts and climate complex
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Mutual climatic range approach2. Example I: mutual climatic ranges
Mutual climatic range (MCR)
Overlap MCR of all taxain the fossil spectrum
Characteristics
Only presence
Pollen and macro fossils
Robust againstno modern analogues
Probabilistic point of view
Implicit assumption of uniform distributions
Graphical definition of MCR causes overfitting
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Probabilistic indicator taxa model2. Example I: mutual climatic ranges
Assumptions
Only use “taxon present”
Conditional independence
Probabilistic indicator taxa model
For all taxa i(k) ∈ {1, . . . , nk} : yi(k) = 1in the fossil pollen spectrum (.. appendix):
f~X |Yi(1),...,Yi(nk )(~x0 |1, . . . , 1) ∝
nk∏k=1
f~X |Yi(k)(~x0|1)
m~X (X)· π~X0
(~x0)
f~X |Yk(~x0|1) Taxon-specific likelihood functionπ~X0
(~x0) Prior distribution of the climate state vectorm~X (X) Marginal distribution of the climate state vector
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Probabilistic indicator taxa model2. Example I: mutual climatic ranges
Climate reconstructionBivariate probability density functions of winter temperature and annualprecipitation for three sample layers
Temporal variation of the expectated value
Temporal variation of the (co)variance (note: Var(~X0), not Var(~µ~X0)!)
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Outline
1 Proxy-climate transfer methods
2 Example I: mutual climatic ranges
3 Example II: modern analoguesModern analogue techniquePollen-ratio model
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Modern analogue technique3. Example II: modern analogues
Modern analogue technique (MAT)
Assignment: pollen spectrum metric←→ modern analogue direct←→ climate
Problems: no modern analogue situations and uncertainty estimates?
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Pollen-ratio model3. Example II: modern analogues
Probabilistic point of view
MAT relates to draws from a multinomial distribution
Pollen-ratio model
Reduced complexity MAT 2 taxa
Temperature as covariate
Logit link with binomial error (GLM)
Probabilistic reconstruction
Sample GLM parameters (MCMC)
Sample from pollen counts
Reconstruction as sample viainverse model
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Pollen-ratio model3. Example II: modern analogues
Single-site reconstruction for one of three nearby lakes in Wisconsin, USA
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Pollen-ratio model3. Example II: modern analogues
Ensemble reconstruction for three nearby lakes in Wisconsin, USA
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Outline
1 Proxy-climate transfer methods
2 Example I: mutual climatic ranges
3 Example II: modern analogues
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Summary
1. Explain proxy-climate transfer methods in a probabilistic framework
No deterministic relationship
Conditional probability densities
2. Show examples of probabilistic pollen-climate transfer method
Derived from classical methods (MCR/MAT/. . . )
Account for as many random effects as possible
3. Discuss the aspect of uncertainty
Added value of uncertainty reconstructions
Uncertainties most likely to be often underestimated
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Further reading
Piecing Together the Past: Statistical Insights into PaleoclimaticReconstructions(Tingley et al., 2010)
A Bayesian Algorithm for Reconstructing Climate Anomalies inSpace and Times(Tingley and Huybers, 2010)
The value of multi-proxy reconstruction of past climate(Li et al., 2010)
Reconstruction of Quaternary temperature fields by dynamicallyconsistent smoothing(Gebhardt et al., 2007)
Review of probabilistic pollen-climate transfer methods(Ohlwein and Wahl, 2011)
Proxy-Climate Transfer Methods
Proxy WorkshopJuly 4th 2011, Cologne, Germany
Christian Ohlwein1 and Andreas Hense1
in collaboration with: Thomas Litt2 Norbert Kuhl2, and Eugene Wahl3, . . .
1) Meteorological Institute at the University of Bonn, Germany2) Steinmann-Institute of Geology, Mineralogy and Paleontology, Bonn, Germany3) National Oceanic and Atmospheric Administration (NOAA), Boulder, CO, USA
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Outline
1 Proxy-climate transfer methodsClassical transfer functionsProbabilistic (Bayesian) frameworkDevelopment of new methods
2 Example I: mutual climatic rangesPollen proxiesMutual climatic range approachProbabilistic indicator taxa model
3 Example II: modern analoguesModern analogue techniquePollen-ratio model
II Appendix
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Appendix
4 General statistical formulationClassification of transfer functions“Regression” and “Calibration”Probabilistic indicator taxa model
5 Dead Sea (Biomisation) and lake Birkat Ram (Indicator)Probabilistic indicator taxa model for lake Birkat RamBiome model for the Dead Sea area
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Classification of transfer functions5. General statistical formulation
1 Classical regression techniques
~µ~X0= E(~X0| . . . )
2 Bayesian approach with plug-in estimator
[~X0|~Y0, ~X , ~Y ] ∝ [~Y0|~X0, ~X , ~Y ] · [~X0]
f~X0|~Y0,~X ,~Y (~x0|~y0,X,Y)︸ ︷︷ ︸
posterior
∝ f~Y |~X (~y0|~x0; ~ϑ)︸ ︷︷ ︸response / likelihood
π~X0(~x0)︸ ︷︷ ︸
prior
3 Bayesian approach (BHM) using MCMC integration
[~X0, ~θ|~Y0, ~X , ~Y ] ∝ [~Y , ~Y0|~X , ~X0, ~θ] · [~X , ~X0|~θ] · [~θ]
f~X0|~Y0,~X ,~Y (~x0|~y0,X,Y)︸ ︷︷ ︸
posterior
∝∫Vθ
f~Y ,~Y0|~X ,~X0,~θ(Y, ~y0|X, ~x0, ~ϑ)︸ ︷︷ ︸
data stage
π~X ,~X0|~θ(~x , ~x0|~ϑ)︸ ︷︷ ︸
process stage
π~θ(~ϑ)︸ ︷︷ ︸
prior
d~ϑ
.. “regression” & “calibration”
Christian Ohlwein Meteorological Institute ▪ University of Bonn
“Regression” and “Calibration”5. General statistical formulation
ReconstructionConditional probability density for the palaeo climate state ~X0
f~X0|~Y0,~X ,~Y (~x0|~y0,X,Y) =
∫Vθ
f~X0|~Y0,~θ(~x0|~y0, ~ϑ)︸ ︷︷ ︸
Calibration
π~θ|~X ,~Y (~ϑ|X,Y)︸ ︷︷ ︸
Regression
d~ϑ
with parameter space ~ϑ ∈ Vθ
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Probabilistic indicator taxa model5. General statistical formulation
fY1,...,Ynk |~X (y1, . . . , ynk |~x) = fY1|~X
(y1|~x) ·nk∏
k=2
fYk |Y1,...,Yk−1,~X (yk |yk , . . . , yk−1, ~x)
=
nk∏k=1
fYk |~X(yk |~x)
=
nk∏k=1
f~X |Yk(~x |yk ) · πYk (yk )
m~X (~x)
f~X |~Y (~x |y1, . . . , ynk ) =f~Y |~X (y1, . . . , ynk |~x) · π~X (~x)
m~Y (y1, . . . , ynk )
=π~X (~x)
m~Y (y1, . . . , ynk )·
nk∏k=1
f~X |Yk(~x |yk ) · πYk (yk )
m~X (~x)
f~X |Yi(1),...,Yi(nk )(~x |1, . . . , 1,C) ∝ π~X (~x) ·
nk∏k=1
f~X |Yi(k)(~x |1,C)
m~X (~x)
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Probabilistic indicator taxa model5. General statistical formulation
Taxon specific likelihood functions f~X |Yk
Precipitation component not Gaussian
Problem of multivariate non-normallydistributed populations
Solved by a copula approach
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Probabilistic indicator taxa model5. General statistical formulation
f−1N (0,Σ) c~X f~X
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Probabilistic indicator taxa model for lake Birkat Ram6. Dead Sea (Biomisation) and lake Birkat Ram (Indicator)
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Probabilistic indicator taxa model for lake Birkat Ram6. Dead Sea (Biomisation) and lake Birkat Ram (Indicator)
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Biome model for the Dead Sea area6. Dead Sea (Biomisation) and lake Birkat Ram (Indicator)
Vegetation: Zohary (1966) / Topografie: GLOBE
Vegetation zones (biomes*)Special situation of the Dead Searegion
Mediterranean territory
Irano-Turanian territory
Saharo-Arabian territory
Sudanian penetration territory(excluded)
Holocene climate changes
1 Relocation of the territories
2 Varying influence on the fossilpollen spectra
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Biome model for the Dead Sea area6. Dead Sea (Biomisation) and lake Birkat Ram (Indicator)
Christian Ohlwein Meteorological Institute ▪ University of Bonn
Outline
1 Proxy-climate transfer methodsClassical transfer functionsProbabilistic (Bayesian) frameworkDevelopment of new methods
2 Example I: mutual climatic rangesPollen proxiesMutual climatic range approachProbabilistic indicator taxa model
3 Example II: modern analoguesModern analogue techniquePollen-ratio model
II Appendix