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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