Palaeoenvironmental Transfer Functions in a Bayesian Framework

Transcription

1 Palaeoenvironmental Transfer Functions in a Bayesian Framework. with Application to Holocene Climate Variability in the Near East 1,2, Andreas Hense 1, Frank Neumann 2, Thomas Litt 2 1) Workgroup on Climate Dynamics Meteorological Institute, University of Bonn 2) Workgroup on Palaeobotany Institute of Palaeontology, University of Bonn

2 Overview Aims of this work 1 Development of new statistical methods for palaeoclimate reconstructions, especially climatological-botanical transfer functions 2 Holocene climate reconstructions for the Near East area based on fossil pollen spectra

3 Motivation State of the art Approaching the problem Outline 1 Introduction Motivation State of the art Approaching the problem 2 3 (Ein Gedi) 4 (Birkat Ram) 5

4 Motivation State of the art Approaching the problem Motivation I Relevance of palaeoclimate research Third Assessment Report of the IPCC: Knowledge of the climate history to Forecast natural climate variations Validate climate models Analyse anthropogenic influences Challenges in palaeoclimatology Location of appropriate climate archives Analysis of proxy data The statistical relationship between proxy data and climate parameters: palaeoenvironmental transfer functions

5 Motivation State of the art Approaching the problem Motivation I Relevance of palaeoclimate research Third Assessment Report of the IPCC: Knowledge of the climate history to Forecast natural climate variations Validate climate models Analyse anthropogenic influences Challenges in palaeoclimatology Location of appropriate climate archives Analysis of proxy data The statistical relationship between proxy data and climate parameters: palaeoenvironmental transfer functions

6 Motivation State of the art Approaching the problem E.g. Northern Hemispheric temperature reconstructions by Mann et al. (1998) 1.0 Instrumental data (AD 1902 to 1999) Reconstruction (AD 1000 to 1980) Reconstruction (40 year smoothed) 1998 instrumental value Northern Hemisphere anomaly ( C) relative to 1961 to Year

7 Motivation State of the art Approaching the problem Motivation II Reconstructions by Mann et al. (1998) Von Storch et al. (2004): Test of the used method on climate model simulations Underestimation of past temperature changes by a factor of up to 2.0 Northern Hemisphere anomaly ( C) relative to 1961 to Instrumental data (AD 1902 to 1999) Reconstruction (AD 1000 to 1980) Reconstruction (40 year smoothed) 1998 instrumental value Year Warning (Robertson et al., 1999) At a time when knowledge of past climatic variability is of increasing political and economic importance in world affairs, it is crucial that the statistical techniques employed by climatologists do not give misleading impressions of past climatic change, [... ]

8 Motivation State of the art Approaching the problem Motivation II Reconstructions by Mann et al. (1998) Von Storch et al. (2004): Test of the used method on climate model simulations Underestimation of past temperature changes by a factor of up to 2.0 Northern Hemisphere anomaly ( C) relative to 1961 to Instrumental data (AD 1902 to 1999) Reconstruction (AD 1000 to 1980) Reconstruction (40 year smoothed) 1998 instrumental value Year Warning (Robertson et al., 1999) At a time when knowledge of past climatic variability is of increasing political and economic importance in world affairs, it is crucial that the statistical techniques employed by climatologists do not give misleading impressions of past climatic change, [... ]

9 Motivation State of the art Approaching the problem Palaeoenvironmental transfer functions: state of the art Classical approaches Historically focused on the palaeo archives Various tools for the development of transfer functions: MAT, MCR, WA, WA-PLS, PCR, PFT, PDF, BUM, BUMMER, BIOME,... Capabilities Knowledge of the bio-geochemical processes forming the archives Further improvements / trends Statistical formulation Probabilistic point of view Reconstruction of uncertainties

10 Motivation State of the art Approaching the problem Palaeoenvironmental transfer functions: state of the art Classical approaches Historically focused on the palaeo archives Various tools for the development of transfer functions: MAT, MCR, WA, WA-PLS, PCR, PFT, PDF, BUM, BUMMER, BIOME,... Capabilities Knowledge of the bio-geochemical processes forming the archives Further improvements / trends Statistical formulation Probabilistic point of view Reconstruction of uncertainties

11 Motivation State of the art Approaching the problem Palaeoenvironmental transfer functions: state of the art Classical approaches Historically focused on the palaeo archives Various tools for the development of transfer functions: MAT, MCR, WA, WA-PLS, PCR, PFT, PDF, BUM, BUMMER, BIOME,... Capabilities Knowledge of the bio-geochemical processes forming the archives Further improvements / trends Statistical formulation Probabilistic point of view Reconstruction of uncertainties

12 Motivation State of the art Approaching the problem Approaching the problem: general statistical formulation Basic definitions Climate state vector ( X) Proxy variables ( Y ) Palaeo climate state ( X 0 ) Palaeo proxy variables ( Y 0 ) Recent climate variables, e.g. (T DJF, T JJA, P ANN) Typically bio-geochemical quantities from the biosphere, lithosphere or cryosphere Historical state of X Historical values of Y taken from the palaeo archives Climate system is stochastic X, Y, X 0, Y 0 are random variables and described in terms of probabilities or probability density functions, e.g. X = x, f X ( x) x R n,...

13 Motivation State of the art Approaching the problem Approaching the problem: general statistical formulation Basic definitions Climate state vector ( X) Proxy variables ( Y ) Palaeo climate state ( X 0 ) Palaeo proxy variables ( Y 0 ) Recent climate variables, e.g. (T DJF, T JJA, P ANN) Typically bio-geochemical quantities from the biosphere, lithosphere or cryosphere Historical state of X Historical values of Y taken from the palaeo archives Climate system is stochastic X, Y, X 0, Y 0 are random variables and described in terms of probabilities or probability density functions, e.g. X = x, f X ( x) x R n,...

14 Motivation State of the art Approaching the problem Approaching the problem: general statistical formulation Basic definitions Climate state vector ( X) Proxy variables ( Y ) Palaeo climate state ( X 0 ) Palaeo proxy variables ( Y 0 ) Recent climate variables, e.g. (T DJF, T JJA, P ANN) Typically bio-geochemical quantities from the biosphere, lithosphere or cryosphere Historical state of X Historical values of Y taken from the palaeo archives Climate system is stochastic X, Y, X 0, Y 0 are random variables and described in terms of probabilities or probability density functions, e.g. X = x, f X ( x) x R n,...

15 Motivation State of the art Approaching the problem Approaching the problem: general definition of transfer functions The given data Palaeo proxy data y 0 and recent climate and proxy data X,Y for comparison Desired reconstruction Conditional probability density for the palaeo climate state X 0 f X0 Y 0, X, Y ( x 0 y 0, X, Y) = Z f X0 Y0, θ ( x 0 y 0, ϑ) π θ V θ {z } X, Y ( ϑ X, Y) dϑ {z } Calibration Regression with additional parameters (RV s) θ with values ϑ V θ Solution Approaches to palaeoenvironmental transfer functions reconstructions can be categorised into three successively simplifying classes

16 Motivation State of the art Approaching the problem Approaching the problem: general definition of transfer functions The given data Palaeo proxy data y 0 and recent climate and proxy data X,Y for comparison Desired reconstruction Conditional probability density for the palaeo climate state X 0 f X0 Y 0, X, Y ( x 0 y 0, X, Y) = Z f X0 Y0, θ ( x 0 y 0, ϑ) π θ V θ {z } X, Y ( ϑ X, Y) dϑ {z } Calibration Regression with additional parameters (RV s) θ with values ϑ V θ Solution Approaches to palaeoenvironmental transfer functions reconstructions can be categorised into three successively simplifying classes

17 Motivation State of the art Approaching the problem Approaching the problem: Classification of transfer functions 1 Bayesian approach (HBN) using MCMC Integration f X0 Y 0, X, Y ( x 0 y 0, X, Y) Z f Y, Y0 X, X0, θ (Y, y 0 X, x 0, ϑ) V θ {z } Response/Likelihood 2 Bayesian approach using explicit parameter estimation f X0 Y 0, X, Y ( x 0 y 0, X, Y) f Y X, θ ( y 0 x 0, ϑ opt) π X0 ( x 0 ) {z } {z } Response/Likelihood Prior π X0 ( x 0 ) π θ ( ϑ) {z dϑ } Prior 3 Classical regression techniques Z x 0 = E( X y 0, ϑ opt, f X Y0, θ ) = R n x f X Y0, θ ( x y 0, ϑ opt) d x

18 Motivation State of the art Approaching the problem Approaching the problem: Classification of transfer functions 1 Bayesian approach (HBN) using MCMC Integration f X0 Y 0, X, Y ( x 0 y 0, X, Y) Z f Y, Y0 X, X0, θ (Y, y 0 X, x 0, ϑ) V θ {z } Response/Likelihood 2 Bayesian approach using explicit parameter estimation f X0 Y 0, X, Y ( x 0 y 0, X, Y) f Y X, θ ( y 0 x 0, ϑ opt) π X0 ( x 0 ) {z } {z } Response/Likelihood Prior π X0 ( x 0 ) π θ ( ϑ) {z dϑ } Prior 3 Classical regression techniques Z x 0 = E( X y 0, ϑ opt, f X Y0, θ ) = R n x f X Y0, θ ( x y 0, ϑ opt) d x

19 Motivation State of the art Approaching the problem Approaching the problem: Classification of transfer functions 1 Bayesian approach (HBN) using MCMC Integration f X0 Y 0, X, Y ( x 0 y 0, X, Y) Z f Y, Y0 X, X0, θ (Y, y 0 X, x 0, ϑ) V θ {z } Response/Likelihood 2 Bayesian approach using explicit parameter estimation f X0 Y 0, X, Y ( x 0 y 0, X, Y) f Y X, θ ( y 0 x 0, ϑ opt) π X0 ( x 0 ) {z } {z } Response/Likelihood Prior π X0 ( x 0 ) π θ ( ϑ) {z dϑ } Prior 3 Classical regression techniques Z x 0 = E( X y 0, ϑ opt, f X Y0, θ ) = R n x f X Y0, θ ( x y 0, ϑ opt) d x

20 Motivation State of the art Approaching the problem Approaching the problem: Consequences Guideline for the following two approaches 1 Show how the existing approaches for transfer functions have to be understood in the aforementioned statistical framework 2 Incorporate the valuable ecological knowledge and specify the implicit assumptions within this new context 3 Address the consequential modifications for new approaches

21 Ein Gedi sediment core Birkat Ram sediment core Outline 1 Introduction 2 Ein Gedi sediment core Birkat Ram sediment core 3 (Ein Gedi) 4 (Birkat Ram) 5

22 Ein Gedi sediment core Birkat Ram sediment core Palaeo archives Fossil pollen spectra of lake sediments 1 Western shoreline of the Dead Sea close to Ein Gedi 2 Crater lake Birkat Ram in the Northern Golan Foto: MODIS / Topografie: GLOBE

23 Ein Gedi sediment core Birkat Ram sediment core Ein Gedi sediment core Core material GFZ Potsdam (1997), laminated sediments (21 m, 10,000 years BP) Age-depth model from laminae counting and 14 C dating on plant macro remains (Migowski, 2001) Proxy data (Pollen) Palynological analysis (T. Litt, F. Neumann) Pollen spectra: mixture of four distinct vegetation zones

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25 Ein Gedi sediment core Birkat Ram sediment core Birkat Ram sediment core Core material GFZ Potsdam (1999), composite profile (6.21m, 6,500 years BP) Age-depth model after Schwab et al. (2004), reservoir effect up to 700a Proxy data (Pollen) Palynological analysis (T. Litt, F. Neumann) Always under Mediterranean influence

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27 Climate reconstruction based on biomisation Bayesian approach using mixture models Results and Consequences Outline 1 Introduction 2 3 (Ein Gedi) Climate reconstruction based on biomisation Bayesian approach using mixture models Results and Consequences 4 (Birkat Ram) 5

28 Climate reconstruction based on biomisation Bayesian approach using mixture models Results and Consequences General Concept of biomisation Vegetation zones (biomes*) Special situation of the Dead Sea region Mediterranean territory Irano-Turanian territory Saharo-Arabian territory Sudanian penetration territory (excluded) Holocene climate changes Vegetation: Zohary (1966) / Topografie: GLOBE 1 Relocation of the territories 2 Varying influence on the fossil pollen spectra

29 Climate reconstruction based on biomisation Bayesian approach using mixture models Results and Consequences General Concept of biomisation Vegetation zones (biomes*) Special situation of the Dead Sea region Mediterranean territory Irano-Turanian territory Saharo-Arabian territory Sudanian penetration territory (excluded) Holocene climate changes Vegetation: Zohary (1966) / Topografie: GLOBE 1 Relocation of the territories 2 Varying influence on the fossil pollen spectra

30 Climate reconstruction based on biomisation Bayesian approach using mixture models Results and Consequences Standard approaches to biomisation Standard approaches Maybe a short excursion about definitions (e.g. Prentice et al., 1992) of Biomes Plant functional types Affinity scores

31 Climate reconstruction based on biomisation Bayesian approach using mixture models Results and Consequences Bayesian approach using mixture models Biomisation = Classification problem Probabilities for the presence of each biome P Bk (1) est = b k with biome ratios b k k = 1, 2, 3 taken from the fossil pollen spectra Natural approach Posterior probability density is given by a mixture model f X0,mix ( x 0 b 1, b 2, b 3 ) = 3X k=1 P Bk (1) fb k X (1 x 0) π X0 ( x 0 ) m Bk (1) f Bk X (1 x 0) π X0 ( x 0 ) m Bk (1) likelihood function or biome transfer function prior distribution for the climate state vector marginal probability for the existence of biome B k

32 Climate reconstruction based on biomisation Bayesian approach using mixture models Results and Consequences Bayesian approach using mixture models Biomisation = Classification problem Probabilities for the presence of each biome P Bk (1) est = b k with biome ratios b k k = 1, 2, 3 taken from the fossil pollen spectra Natural approach Posterior probability density is given by a mixture model f X0,mix ( x 0 b 1, b 2, b 3 ) = 3X k=1 P Bk (1) fb k X (1 x 0) π X0 ( x 0 ) m Bk (1) f Bk X (1 x 0) π X0 ( x 0 ) m Bk (1) likelihood function or biome transfer function prior distribution for the climate state vector marginal probability for the existence of biome B k

33 Introduction Climate reconstruction based on biomisation Bayesian approach using mixture models Results and Consequences for estimating the biome transfer functions 30 O E 35 O E 40 O E 45 O E 50 O E 30 O E 35 O E 40 O E 45 O E 50 O E O N 25 O N 30 O N 35 O N O E 40 O E 45 O E O N 30 O N 25 O N 20 O N 20 O N 25 O N 30 O N 35 O N O E 40 O E 45 O E O N 30 O N 25 O N 20 O N Vegetation: Meusel et al. (1965) Climatology: CRU TS 1.1 ( )

34 Climate reconstruction based on biomisation Bayesian approach using mixture models Results and Consequences Estimating the biome transfer functions f Bk X Biome transfer functions Generalised linear model with Logistic regression link function covariate 1 covariate 2 Biome distribution by chorological atlases Climate variables (T DJF, T JJA, P ANN) from CRU TS 1.1 Second order monomials

35 Climate reconstruction based on biomisation Bayesian approach using mixture models Results and Consequences Application to the biome ratios of the Ein Gedi core data depth / time [yrs cal. B.P.] Mediterranean territory Irano-Turanian territory Saharo-Arabian territory

36 Climate reconstruction based on biomisation Bayesian approach using mixture models Results and Consequences Local climate reconstruction (Birkat Ram) posterior pdf P ANN [mm] Time [yrs cal. B.P.] Critical issues The mixture model naturally leads to comparably wide posterior pdf s Biomes span a wider climatic range than single taxa/species Problem of no-modern-analogue situations (not shown)

37 Climate reconstruction using indicator taxa Bayesian approach Results and Consequences Outline 1 Introduction 2 3 (Ein Gedi) 4 (Birkat Ram) Climate reconstruction using indicator taxa Bayesian approach Results and Consequences 5

38 Climate reconstruction using indicator taxa Bayesian approach Results and Consequences Indicator taxa or mutual climatic range method historically Mean July temperature [ C] Picea Alnus Ulmus -5 0 Myrica Define mutual climatic ranges from station data 2 Overlap all taxa found in the palaeo archive Mean January temperature [ C] Figure after Grichuk (1969): Mutual Cliatic Range method

39 Climate reconstruction using indicator taxa Bayesian approach Results and Consequences Characteristics of the indicator taxa approach The clou of this approach Presence / absence instead of abundances indeed an advantage Works simultaneously on pollen and macro fossils Robust against no-modern-analogue situations Mean July temperature [ C] Picea Alnus Ulmus -5 0 Myrica 5 10 Problems due to the graphical solution Implicit assumption of uniform distributions for the MCR s Not very close to plant physiology Boundaries are the decisive factor Graphical definition of MCR s causes overfitting Mean January temperature [ C]

40 Climate reconstruction using indicator taxa Bayesian approach Results and Consequences Characteristics of the indicator taxa approach The clou of this approach Presence / absence instead of abundances indeed an advantage Works simultaneously on pollen and macro fossils Robust against no-modern-analogue situations Mean July temperature [ C] Picea Alnus Ulmus -5 0 Myrica 5 10 Problems due to the graphical solution Implicit assumption of uniform distributions for the MCR s Not very close to plant physiology Boundaries are the decisive factor Graphical definition of MCR s causes overfitting Mean January temperature [ C]

41 Climate reconstruction using indicator taxa Bayesian approach Results and Consequences Reformulation of the indicator taxa approach I Definitions We are looking for the posterior probability density function of X 0 = t (T DJF, T JJA, P ANN ) with values x 0 R 3 Given the existence of a certain subgroup of all n k taxa Y 0 = t (Y 1,..., Y nk ) with values y 0 {0, 1} n k Asumptions The information of presence/absence for each taxon is conditionally independent given the climate state (Caution) We are only using the information of presence

42 Climate reconstruction using indicator taxa Bayesian approach Results and Consequences Reformulation of the indicator taxa approach II Derivation of the posterior pdf We are speaking of existence, especially of coexistence of taxa, which leads to the poster distribution of the climate state vector (details on demand) f X Yi(1),...,Y i(n k ) ( x 0 1,..., 1) π X0 ( x 0 ) n ky k=1 f X Yi( k) ( x 0 1) m X ( x) for all taxa i( k) {1,..., n k } : y i( k) = 1 found in the fossil pollen spectrum f X Yk ( x 0 1) π X0 ( x 0 ) m X ( x) Taxon specific likelihood function Prior distribution of the climate state vector Marginal probability density for the climate state vector

43 Climate reconstruction using indicator taxa Bayesian approach Results and Consequences Estimating the taxon specific likelihood functions f X Yk Taxon specific likelihood functions Recent spatial distribution and the CRU TS 1.1 climate data set pdf gamma component normal component Problem of multivariate non-normal distribution functions Inverse cdf to transform to a three-dimensional standard normal distribution

44 Climate reconstruction using indicator taxa Bayesian approach Results and Consequences Application to the fossil pollen spectra the Birkat Ram core Artemisia Platanus or. Atriplex hal. Ziziphus sp. Plantago lanc. Pistacia lent. Ephedra aph. Ceratonia sil. Sarcopot. sp. Phillyrea la. Olea europ. Quercus ith. Quercus call Depth [cm] Pollen counts above a certain threshold

45 Climate reconstruction using indicator taxa Bayesian approach Results and Consequences Local climate reconstruction (Ein Gedi) posterior pdf T DJF [ C] P ANN [mm] Depth [cm]

46 Summary Outlook Outline 1 Introduction 2 3 (Ein Gedi) 4 (Birkat Ram) 5 Summary Outlook

47 Summary Outlook Summary with respect to the aims of this work Transfer functions Classical concepts for pollen based reconstructions of climate can be transformed to statistical methods, providing probabilistic reconstructions The The Important aspect: Only feasible by interdisciplinary work Climate reconstructions The Dead Sea area (10,000 years BP) The Golan Heights (6,500 years BP) Quantitative climate reconstructions for the Holocene Near East are rare, especially with estimates of the uncertainties in reconstruction.

48 Summary Outlook Summary with respect to the aims of this work Transfer functions Classical concepts for pollen based reconstructions of climate can be transformed to statistical methods, providing probabilistic reconstructions The The Important aspect: Only feasible by interdisciplinary work Climate reconstructions The Dead Sea area (10,000 years BP) The Golan Heights (6,500 years BP) Quantitative climate reconstructions for the Holocene Near East are rare, especially with estimates of the uncertainties in reconstruction.

49 Summary Outlook Outlook Various technical refinements Bayesian Biome Model: Estimation of probability of occurrence of each biome Bayesian Indicator Taxa model: Geostatistical methods for the estimation of likelihoods from geobotanical data Bayesian Indicator Taxa model: Underestimation of extreme climate states Application to less special data DEKLIM project: European climate reconstructions (in progress)

50 Summary Outlook Outlook Various technical refinements Bayesian Biome Model: Estimation of probability of occurrence of each biome Bayesian Indicator Taxa model: Geostatistical methods for the estimation of likelihoods from geobotanical data Bayesian Indicator Taxa model: Underestimation of extreme climate states Application to less special data DEKLIM project: European climate reconstructions (in progress)