Studying Uncertainty in Palaeoclimate Reconstruction

SUPRaNet SUPRModels

SUPR softwareBrian 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 oC

Mean Temp of Coldest Month

Climate over 100,000 yearsGreenland Ice Core

10,000 year intervals

Oxygen isotope – proxy for Greenland tempMedian smooth.

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Age

Past 23000 years

The long summer

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Past 23000 years

Climate over 100,000 yearsGreenland Ice Core

10,000 year intervalsThe long summer

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”

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Climate over 15,000 yearsGreenland 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 tree GISP ice Sediment PollenThanks to Vincent Garreta

coresamples

mult. counts by taxa

Pollen

Data

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

Model Issues

• Climate - Sedimentation - Veg responselatent processes

– Climate smooth (almost everywhere)– Sedimentation non decreasing– Veg response smooth

• Data generating process– Pollen – superimposed pres/abs & abundance– 14C - 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

ModelledUncertaintyDoes 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 14C 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)

• 14C Calibration curve latent process– ‘Smooth’ – in sense of Gaussian Process (Bcal)

• 14C 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?

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Bivariate Gamma Renewal Process

Comp Poisson Gamma wrt x; x incs exponentialComp Poisson Gamma wrt y; y incs exponential

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Compound Poisson Gamma Process

We take y = 1 for access to CPGand x > 2 for continuity wrt x

Slope = Exp / Gamma= Exp x InvGamma infinite var if x > 2

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Modelling with Bivariate Gamma Renewal Process

Data assumed to be subset of renewal pointsImplicitly not smallMarginalised wrt renewal ptsIndep increments processStochastic interpolation by simulation

new y

unknown x

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Stochastic Interpolation

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Monotone piece-wise linear CPG Process

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Stochastic Interpolation

Density

Known Depths

Known age

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

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c(t)y(t)

Modern (c, y ) pairsIn space

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c(t)y(t)

Eg dendroTwo time seriesMuch c data missing

Eg pollenOne time seriesAll 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 presentTaxa not species

Eg yi=0 prob q(c)yi~Poisson (λ(c)) prob 1-q(c)

Thus obs yi=0, yi=1 very diff implications

One-slice-at-a time

• Slice j has count vector yj, depth dj

• Whence – separately - π(cj| yj) and π(tj| dj)

Response Chronmodule module

Uncertainty one-layer-at-a-time

Pollen => Uncertain ClimateDepth => Uncertain depth

But monotonicity

Here showing 10 of 150 layers

Uncertainty one-layer-at-a-time

Uncertainty jointly

Many potential climate histories areConsistent with ‘one-at-a-timeJointly inconsistent with Climate TheoryRefine/subsample

Coherent Histories

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

Coherent Histories

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

Coherent Histories

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

Coherent Histories

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

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

One joint (coherent) history

One joint (coherent) history

One joint (coherent) history

MTCO Reconstruction

One layer at a time, showing temporal uncertainty

Jointly, century resolution, allowing for temporal uncertainty

Marginaltime-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 oC

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