Studying Uncertainty in Palaeoclimate Reconstruction SUPRaNet SUPRModels SUPR software

1. Studying Uncertainty in Palaeoclimate ReconstructionSUPRaNet SUPRModelsSUPR software Brian Huntley, Andrew Parnell Caitlin Buck, James Sweeney and many others Science Foundation Ireland Leverhulm Trust

2. Result: one pollen core in Ireland Mean Temp of Coldest Month 95% of plausible scenarios have at least one “100 year +ve change” > 5 oC

3. Climate over 100,000 yearsGreenland Ice Core 10,000 year intervals The long summer Past 23000 years Oxygen isotope – proxy for Greenland temp Median smooth.

4. Climate over 100,000 yearsGreenland Ice Core 10,000 year intervals The long summer Past 23000 years Int Panel on Climate Change WG1 2007 “During the last glacial period, abrupt regional warmings (probably up to 16◦C within decades over Greenland) occurred repeatedly over the North Atlantic region”

5. Climate over 15,000 yearsGreenland Ice Core Younger Dryas Holocene What’s the probability of abrupt climate change? Ice dynamics? Ocean dynamics? Transition

6. Modelling Philosophy Climate is – • Latent space-time stoch process C(s,t) • All measurements are • Indirect, incomplete, with error • ‘Regionalised’ relative to some ‘support’ • Uncertainty • Prob (Event) • Event needs explicitly defined function of C(s,t)

7. Proxy Data Collection Oak tree GISP ice Sediment Pollen Thanks to Vincent Garreta

8. samples mult. counts by taxa Pollen core

9. Data

10. Data Issues • Pollen 150 slices • 28 taxa (not species); many counts zero • Calibrated with modern data 8000 locations • 14C 5 dates • worst uncertainties ± 2000 years • Climate `smoothness’ • GISP data 100,000 years, as published

11. Model Issues • Climate - Sedimentation - Veg response latent processes • Climate smooth (almost everywhere) • Sedimentation non decreasing • Veg response smooth • Data generating process • Pollen – superimposed pres/abs & abundance • 14C - Bcal • Priors - Algorithms …….

12. SUPR-ambitions • Principles • All sources of uncertainty • Models and modules • Communication • Scientist to scientist • to others • Software Bclim • Future • SUPR tech stuff • non-linear • non-Gaussian • multi-proxy • space-time • incl rapid change • dating uncertainty • mechanistic system models • fully Bayesian • fast software

13. Modelling Approach • Latent processes • With defined stochastic properties • Involving explicit priors • Conditional on ‘values’ of process(es) • Explicit stochastic models of • Forward Data Generating Processes • Combined via conditional independence • System Model

14. Modelling Approach • Modular Algorithms • Sample paths, ensembles • Monte Carlo • Marginalisation to well defined random vars and events

15. Progress in Modelling Uncertainty Modelled Uncertainty Does it change? In time? In space? • Statistical models • Partially observed space-time stochastic processes • Bayesian inference • Monte Carlo methods • Sample paths • Thinning , integrating • Communication • Supplementary materials

16. SUPR Info • Proxy data: typically cores • Multiple proxies, cores; multivariate counts • Known location(s) in (2D) space • Known depths – unknown dates, some 14C data • Calibration data – modern, imperfect • System theory • Uniformitarian Hyp • Climate ‘smoothness’; Sedimention Rates ≥ 0 • Proxy Data Generating Processes

17. Chronology example

18. Bchron Models • Sedimentation a latent process • Rates ≥ 0, piecewise const • Depth vs Time - piece-wise linear • Random change points (Poisson Process) • Random variation in rates (based on Gamma dist) • 14C Calibration curve latent process • ‘Smooth’ – in sense of Gaussian Process (Bcal) • 14C Lab data generation process • Gaussian errors

19. Bchron Algorithm Posterior – via Monte Carlo Samples • Entire depth/time histories, jointly • Generate random piece-wise linear ‘curves’ • Retain only those that are ‘consistent’ with model of data generating system • Inference • Key Parameter; shape par in Gamma dist • How much COULD rates vary?

20. Bivariate Gamma Renewal Process Comp Poisson Gamma wrt x; x incs exponential Comp Poisson Gamma wrt y; y incs exponential

21. Compound Poisson Gamma Process We take y= 1 for access to CPG and x> 2 for continuity wrt x Slope = Exp / Gamma = Exp x InvGamma infinite var if x> 2

22. Modelling with Bivariate Gamma Renewal Process Data assumed to be subset of renewal points Implicitly  not small Marginalised wrt renewal pts Indep increments process Stochastic interpolation by simulation new y unknown x

23. Stochastic Interpolation Monotone piece-wise linear CPG Process Unit Square

24. Stochastic Interpolation Monotone piece-wise linear CPG Process

25. Stochastic Interpolation Monotone piece-wise linear CPG Process

26. Stochastic Interpolation Monotone piece-wise linear CPG Process

27. Stochastic Interpolation Monotone piece-wise linear CPG Process

28. Stochastic Interpolation Monotone piece-wise linear CPG Process

29. Stochastic Interpolation Known age Known Depths Calendar age Known age Density

30. Data

31. Time-Slice “Transfer-Function”via Modern Training Data Glendalough Hypothesis Modern analogue Climate at Glendalough 8,000 yearsBP “like” Somewhere right now The present is a model for the past

32. Calibration Modern (c, y ) pairs In space Eg dendro Two time series Much c data missing Eg pollen One time series All c data missing c(t) y(t) c(t) y(t) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Over-lapping time series Space for time substitution

33. Calibration Model Simple model of Pollen Data Generating Process • ‘Response’ y depends smoothly on climc • Two aspects Presence/Absence Rel abundance if present Taxa not species Egyi=0 probq(c) yi~Poisson (λ(c)) prob 1-q(c) Thus obsyi=0, yi=1 very diff implications

34. One-slice-at-a time • Slice j has count vector yj, depth dj • Whence – separately - π(cj| yj) and π(tj| dj) Response Chron module module

35. Uncertainty one-layer-at-a-time Here showing 10 of 150 layers Pollen => Uncertain Climate Depth => Uncertain depth But monotonicity

36. Uncertainty one-layer-at-a-time

37. Uncertainty jointly Many potential climate histories are Consistent with ‘one-at-a-time Jointly inconsistent with Climate Theory Refine/subsample

38. Coherent Histories One-slice-at-a-time samples => {c(t1), c(t2),……c(tn)}

39. Coherent Histories One-slice-at-a-time samples => {c(t1), c(t2),……c(tn)}

40. Coherent Histories One-slice-at-a-time samples => {c(t1), c(t2),……c(tn)}

41. Coherent Histories One-slice-at-a-time samples => {c(t1), c(t2),……c(tn)}

42. GISP series (20 years)

43. Climate property? Non-overlapping (20 year?) averages are such that first differences are: • adequately modelled as independent • inadequately modelled by Normal dist • adequately modelled by Normal Inv Gaussian • Closed form pdf • Infinitely divisible • Easily simulated, scale mixture of Gaussian dist

44. One joint (coherent) history

45. One joint (coherent) history

46. One joint (coherent) history

47. One joint (coherent) history

48. MTCO Reconstruction Marginal time-slice: may not be unimodal Jointly, century resolution, allowing for temporal uncertainty One layer at a time, showing temporal uncertainty

49. Rapid Change in GDD5 Identify 100 yr period with greatest change One history

50. Rapid Change in GDD5 Identify 100 yr period with greatest change One history