Palaeoclimate Reconstruction:Modelling Temporal Uncertainty

Many collaborators:Stats: Bhattacharya, Gelfand, Salter-Townshend, Parnell, Whiley, Wilson, others

Botany: Allen, Huntley, Mitchell,

Support: SFI and previously by Enterprise Ireland and PRTLI

Glendalough Co Wicklow

Sunday Times

“Gobsmacking”says Haslett

“Not what I said”says Haslett

Courtesy of Sunday Times graphics dept

Reconstruction of GDD5

coresamples

mult. counts by taxa

Changing pollen composition

10,000 BP

Use Matts pollen diagramObserved pollen proportions vs 14C y BP

Pollen composition changes Climate changesRecent Shallow

Ancient Deep

ScienceChanging pollen composition in carefully

selected sites• Reflects changing vegetation, which• Reflects changing climate

– GDD5 Growing Deg Days > 5o

– MTCO Mean Temp Coldest Monthwhence

• Can reconstruct climate quantitatively• Can reduce uncertainty about past climate

GDD5 eg Avg temp on four successive days

5 8 4 10

Excess over 5 0 3 0 5

Thus GDD5 = 8

Data

• Pollen Data – multivariate counts – 14 distinguishable taxa– 115 samples at Sluggan Moss, Lough Neagh– 115 depths of which 32 radiocarbon dated– Climate unknown (2D GDD5, MTCO)

• Modern data – 7815 modern sites – Counts known – surface pollen (depth = 0)– Climate known

Statistical Tasks

Aspects• Pollen response to climate

– Use modern data– Transfer functions/response surfaces– Climate one sample at a time

• Climate smoothness in time– Climate history – Greenland ice cores– Dating uncertainties

Statistical Tasks

Given • Modern data (pm,cm) (7815 records); and• Fossil data (pf,?) at one depth

seek post dist π(cf | pm,cm ,pf) at that depth

Additionally, given • ‘Climate smoothness’; and• Fossil data (pf,?, d) at 115 depths• Radiocarbon dating info at 32 depths

seek post dist π(cf | pm,cm ,pf) entire climate history

GlendaloughModern Training Data

• Data on modern pollen compositions pm

7815 sites in Eur/ N. America

• Modern climate cm known.

• Hence relationship

π(pm | cm)

• Adopt for fossil data

π(pf | cf)

Sluggan Moss

Physical and (2D) climate spaces

Climate space grid

Glendalough

An extreme climate?

Impossible climates?Unknown climates?

Growing Deg Day > 5oMea

n T

emp

Col

dest

Mon

th

7815 data locations - grey points.Computational grid - black points

7815 data locations

Pollen response to (2D) climate

•π(p | c) pdfp 14 dim comp vectc 2 dim climate (here)

•Latent Gaussian proc•mixture of multinomials•zero inflation•MCMC

MT

CO

GDD5Small change in climate c Small change in vegetation p = p(c) smooth multivariate function

7815 data locations - grey points.Computational grid - black points

Two stage implementation

1 MCMC creates/stores many realisations of p(c)

2(a) Draw one at random

2(b) MCMC Climate recon

2(c) Repeat (a,b)

Reconstruction of GDD5

Here depth to radiocarbon age presumedknown

Later addressdatinguncertainty

Note dates in Radio-carbon YBP

One sample at a time

Post Dist cf given pf

Given vector of counts at given depth, whence pf

Findπ(GDD5f,MTCOf | pf) by MCMC for each depth

Here concentrate on GDD5

eg depth 10 m

Differential Response to 1D Climate

0 500 1000 1500 2000 2500

C being 1D Climate

A

B

Inverse relationship

• Model taxon productivity response to ‘climate’• Multi-modal climate posteriors natural• Toy example; two taxa one climate dimension

Prop of A high

Post prob C given A high

0 500 1000 1500 2000 2500

Prop of A low

0 500 1000 1500 2000 2500

Post prob C given A low

Implied Climate Histories

Pointwiserealisations of climate

Climateapparently

volatile

Implied Climate Histories

Pointwiserealisations of climate

Climateapparently

volatile

Implied Climate Histories

Pointwiserealisations of climate

Climateapparently

volatile

Implied Climate Histories

Pointwiserealisations of climate

Climateapparently

volatile

Implied Climate Histories

Pointwiserealisations of climate

Climateapparently

volatile

Climate Smoothness

• Climate changes δi = c(timei) - c(timei-1)– Mostly small/sometimes large “smooth”– Depends on increments |timei - timei-1|– Reject (most) volatile climates

• Issues– How smooth?

• Greenland ice cores

– Uncertainty in 14C dating?• Random chronology

Temporal uncertainty

• 115 samples at Sluggan MossFor 115 : core depths di

For 32 : reported 14C ages yi ± σi

• Seek θi true calendar age θi all di

“chronology model”– r(θ) 14C calibration curve

• yi ~ N( r(θi), σi2) (outliers, so long tails)

• r(θ)~ N( μ(θ), σ2(θ)) prior

– Piecewise constant sedimentation rate• Gaussian random walk

ChronologyGiven complete knowledge of sedimentation history, age may be determined from depth

Calendar age θ

d = depth of accumulated sediment

ButOnly know 14C age at some depthsSeek realisations of sediment history,conditional on dataPrior: Gaussian random walk with driftconstrained to be monotonePiecewise const iid sedimentation rate

Temporal uncertainty:single dated sample

Lab report3180 ± 30

Implied post dist

Schematic of Bayesian 14C calibration curveBuck

Temporal uncertainty:all dated samples

Prior:Discrete time (20 year intervals)Random Walk with drift (monotone)

• Draw random θi | yi σi each of 32 di

– Order constraint θi > θk if di > dk

• Stoch. interpolation to undated samples– Sample θm (undated)| θi (dated), all depths

Draw set of random dates for 14C dated samples

x

x

x

x

Calendar age θ

Depth d

Realisations of order constrained radio-carbon dates

drift

drift

Complete random chronology

x

x

x

x

Realisations of order constrained stochastic chronology, conditional on radio-carbon derived dates

Monotone random walk with drift

Depth d

Calendar age θ

Given set of depths

Realisation of a set of calendar dates

Climate Smoothness

• Climate changes δi = c(timei) - c(timei-1)

– Mostly small/sometimes large

– Depends on increments |timei - timei-1|

• Prior for smoothness rejection of histories with large |δi |

implicit smoothing / borrowing strength

• Issues– Prior for δi long tail random walk

Climate over 100,000 yearsGreenland Ice Core

Temporal structure for climate (20 yr. resolution)Frequent small changes, occasional large changes

Ice Core data time series Greenland Ice Core Data10,000 year intervals

Irish study periodOxygen isotope – proxy for Greenland temp

Climate over 100,000 yearsGreenland Ice Core

-6

-4

-2

0

2

4

6

-4 -3 -2 -1 0 1 2 3 4 5

Normal prob plot

First diffs

First Differences

-4

-2

0

2

4

0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000

Years BP

Climate Smoothness

Ice Core data time series Greenland Ice Core Data10,000 year intervals

Long-tailed Random Walk Prior

Model δ = c(t) - c(t-20) as iid NIGNormal Inverse Gamma Random Walk

Sampling Climate Histories

Given – Realisation of pollen response surfaces– Sample pollen at each of 115 depths– Realisation of complete chronology

• 115 dates given 14C dates for 32 samples

– Model for climate smoothness

Sample realisations of climate at 115 dates

Sample climate history every 20 years

Modelled Climate Histories

ClimateSmoothmostly

Modelled Climate Histories

ClimateSmoothmostly

Modelled Climate Histories

ClimateSmoothmostly

Modelled Climate Histories

ClimateSmoothmostly

Modelled Climate Histories

ClimateSmoothmostly

Better Reconstruction of GDD5

Reconstruction of GDD5

Note dates in Radio-carbon YBP

Monte Carlo ModulesResp Surface

Randomset of surfaces

Modern dataClimate and pollen

Random set of 115 dates

Depths andradiocarbon

dates

Dating

RandomClimate History

length 115

Temporal Stochastic

Smoothness

Stochastic Interpolation

Random Climate History 12,600y by

20y step

Summaries

Fossil Pollen

Point wise Recon-

struction

Randompoint-wisehistories

Next Stages

• Multiple sites– Joint reconstruction of two sites– Probable synchronicity of climate change

• Borrow more strength– for dates, for climate smoothness

– Joint reconstruction of many sites in space

• More climate dimensions and taxa– Many high dim response surfaces

• Other proxies, covariates• Confront General Circ. Models

Methodological Issues

• MCMC - the way forward?– Speed and convergence– Approximations esp for response surfaces

– Model checking and model choice

• Technical issues– Zero inflation– Fast high-dim non-parametric smoothing– Long tailed space-time prior for climate– Latent (mixtures of) Gaussian processes