Studying Uncertainty in Palaeoclimate Reconstruction SUPRaNet SUPRModels SUPR software Brian Huntley, Andrew Parnell Caitlin Buck, James Sweeney and many others Science Foundation Ireland Leverhulm Trust

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

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

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

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

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)

Proxy Data Collection Oak treeGISP iceSedimentPollen Thanks to Vincent Garreta

core samples mult. counts by taxa Pollen

Data

Data Issues Pollen 150 slices – 28 taxa (not species); many counts zero – Calibrated with modern data 8000 locations 14 C5 dates – worst uncertainties ± 2000 years Climate `smoothness’ – GISP data 100,000 years, as published

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 – 14 C - Bcal Priors - Algorithms …….

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

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

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

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

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

Chronology example

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) 14 C Calibration curve latent process – ‘Smooth’ – in sense of Gaussian Process (Bcal) 14 C Lab data generation process – Gaussian errors

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 Unit Square Monotone piece-wise linear CPG Process

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 Density Known Depths Known age Calendar age

Data

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

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

Calibration Model Simple model of Pollen Data Generating Process ‘Response’ y depends smoothly on clim c Two aspects Presence/Absence Rel abundance if present Taxa not species Eg y i =0 probq(c) y i ~Poisson (λ(c)) prob1-q(c) Thus obsy i =0, y i =1very diff implications

One-slice-at-a time Slice j has count vector y j, depth d j Whence – separately - π(c j | y j ) and π(t j | d j ) ResponseChronmodule

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

Uncertainty one-layer-at-a-time

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

Coherent Histories One-slice-at-a-time samples => {c(t 1 ), c(t 2 ),……c(t n )}

Coherent Histories One-slice-at-a-time samples => {c(t 1 ), c(t 2 ),……c(t n )}

Coherent Histories One-slice-at-a-time samples => {c(t 1 ), c(t 2 ),……c(t n )}

Coherent Histories One-slice-at-a-time samples => {c(t 1 ), c(t 2 ),……c(t n )}

GISP series (20 years)

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

One joint (coherent) history

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

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

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

Rapid Change in GDD5 Study uncertainty in non linear functionals of past climate 1000 histories Identify 100 yr period with greatest change

Result: one pollen core in Ireland 95% of plausible scenarios have at least one 100 year +ve change > 5 o C Mean Temp of Coldest Month

Communication Scientist to scientist Exeter Workshop – Data Sets – With Uncertainty Associated with what precise support?

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 Modular Algorithms – Sample paths, ensembles – Monte Carlo – Marginalisation to well defined random vars and events