Reconstructing carbon dioxide for the last 2000 years: a ... · How to trade a paper clip for a...

Reconstructing carbon dioxidefor the last 2000 years:a hierarchical success story.

Douglas NychkaNational Center for Atmospheric Research

National Science Foundation UBC October 2014

Outline

D. Nychka Tour of an inverse problem 2

• A 19th Century Inverse Problem• CO2 reconstruction problem• Hierarchical models (Data, process, priors)• Variational problems• back to ice cores

Main ideas:Solve an ”inverse problem” by

• separating what is observed from what is to be estimated.• using Bayesian statistics as a strategy for inference.

222b Baker Street


From “The Adventure of the Speckled Band”

”Good-morning, madam,” said Holmes cheerily. ”My nameis Sherlock Holmes.

222b Baker Street


Her features and figure were those of a woman of thirty, buther hair was shot with premature grey, and her expressionwas weary and haggard.

222b Baker Street


”We shall soon set matters right, I have no doubt. You havecome in by train this morning, I see. ...and yet you had a good drive in a dog-cart, along heavyroads, before you reached the station.... ”


”The left arm of your jacket is spattered with mud in noless than seven places. The marks are perfectly fresh.

There is no vehicle save a dog-cart which throws up mud inthat way, and then only when you sit on the left-hand sideof the driver.”

Holmes’ calculation


Before meeting Ms. Helen Stoner:

• A PRIOR probability of type of vehicle

Knowledge of vehicles effects:• LIKELIHOOD of observation given type of vehicle

Combine prior with observation:• POSTERIOR is a product:Likelihood of mud stains given type of vehicle

× Probability of type of vehicle

Maximize over vehicle


Holmes’ conclusion – the highest probability• vehicle = dog cart

Some differences


Holmes’ genius:Uses observations without assuming much prior information!

Weather and climate applications:

• Our observations are not as decisive – we must rely more on the priorinformation.

• Have to use a computer to do the computation!

NCAR/U Wyoming, Yellowstone Supercomputer, 70,000+ processors.

Convenient notation


X and Y are random variables

• [X] probability density function for X

• [X,Y ] joint probability density function for X and Y

• [Y |X] conditional probability density function for Y given X

Properties• By definition: [X,Y ] = [Y |X][X]

• Iterating: [X,Y, Z] = [Y |X,Z][X|Z][Z]


Bayes Theorem (1764)

[X|Y ] =[Y |X][X]

[Y ]

From “Y given X” to “X given Y” !

Another view of Holmes’ deductionFrom knowledge of type of given the transport → kind of vehicle giventhe observations

Our problemFrom knowledge of how CO2 is trapped in ice given the atmospheric

concentration

→ past CO2 concentrations given ice core observations.

Law Dome, East Antarctica


Ice core drill and glaciologists from the Australian Antarctic Division and

Antarctic Climate and Ecosystems CRC, Law Dome, East Antarctica

What is the past history of CO2 concentrationsin the atmosphere?

Observations


CO2 concentrations as a function depth.

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●●

280 300 320 340

ppm CO2

dept

h (m

)

800

600

400

200 Conceptually: Depth is related to the

time air was trapped in the core.

y(depth) = F(c(time))

F developed by Trudinger et al. (2013)

Inverse problem: Invert CO2 concentra-tions by depth to concentrations by time.

c(time) = F−1(y(depth)

Firn Ice Inverse problem harder for upper ice layers that have notcompletely consolidated.

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Back to Hierarchical Models


Y - data, c = c1, c2, ..., cn process at closely spaced times(yearly, θ parameters

Data level: [Y |c, θ] What we observe if we knew the true process.

Process level: [c|θ] The dynamics of the process (regardless of whatwe measure)

Priors: [θ] Uncertainty in the statistical parameters used to describethe data and process levels.

Joint distribution:

[Y |c, θ][c|θ][θ]

Where we are going: Bayes theorem


Inversion (also called posterior)

[c, θ|Y ] proportional to [Y |c, θ][c|θ][θ]

take logs

log([c, θ|Y ]) = log([Y |c, θ]) + log([c|θ]) + log([θ]) + constants

fit to data weak constraint on c parameters

• This is a complete probability density that quantifies un-

certainty in result.

• Large values (high probability) indicate good estimates of

the parameters and process.

The first part: Data model


c: a vector of the n concentrations at an annual time scale for last2000+ years

y = y1, y2, ..., ym: a vector of the concentrations at an annual timescale for last 2000 years

W : a m × n matrix where each row maps the concentrations to themeasurements at the depths.

Wc: Expected concentrations at observed depths– the forward model F


The measurements:

y = Wc+ e

e = e1, e2, ..., em : measurement errors – assumed to be N(0, σ2)

What does W look like?


W = 101× 2026

Rows are weights applied to concentrations over time.

Note: W is not invertible!

Effect of indirectly observing CO2.


Rows of W for most recent ice layers.

Second Part: Process level


Annual concentrations based on a linear trend, human emissions covari-

ate, and autoregressive “noise”.

ct = α1 + α2t+ α3Emissions(t) + ut

ut = βut−1 + vt

vt are uncorrelated normals and the variance of ut is ρ

This can be written as c is N(Xα, ρΣ(β))

Recent carbon emissions: Emissions(t)

1700 1750 1800 1850 1900 1950 2000

020

0050

00

Year CE

Car

bon

emis

sion

s

more on the autoregressive noise


AR 1 process:ut = βut−1 + white noise – same as corr(ut, us) = e−|t−s|/δ

β = .8

0 200 400 600 800 1000

−3

−1

01

23

proc

ess

β = .99

0 200 400 600 800 1000

−2

−1

01

2

proc

ess

Third part: Priors – parameters


• σ2 measurement error variance• α1, α2, α3 regression parameters• β autoregressive parameter (correlation)• ρ variance of autoregressive process.• W

Empirical BayesFor this application we will estimate these parameters by maximum

likelihood and just use these values.

Maximum likelihood ≡ uniform ( or flat) prior distribution

Uncertainty in W


Rows of W : mean and 20 member ensemble

1800 1820 1840 1860

0.00

0.04

0.08

0.12

Years CE

Wei

ghts

Contribution to an ice core layer

Turning the crank


Finding the concentrations


Posterior probability density:

[c(.), θ|Y ] ∼ [Y |c(.), θ][c(.)|θ][θ]

log([c(.), θ|Y ]) = log([Y |c(.), θ]) + log([c(.)|θ]) + log([θ]) + constant

Data + Process + Prior

Using data and process level models:

−(1/2)(y −Wc)T (y −Wc)/σ2−(1/2)ρ(c−Xα)TΣ(β)−1(c−Xα)+ stuff

fit to data weak constraint on c parameters


The program:

• Maximize over c and parameters to get “ best ” estimate.

• Sample from the posterior density to get an ensemble.

Example of likelihood surface


Likelihood can be maximized analytically over all parameters except for

correlation and “ noise to signal ” ratio ( variance of measurement error

to variance of concentrations ).

Maximize these last two numerically.

log Likelihood surface

0.2 0.4 0.6 0.8

0.90

0.92

0.94

0.96

0.98

noise/signal variance

Pro

cess

cor

rela

tion

−285

−280

−275

−270

−265

−260

−255


Full reconstruction Just trend and emissions.

Inference for decrease during Little Age


1580 1600 1620 1640 1660

265

270

275

280

285

290

cTime

outM

LE$c

Hat

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1580 1600 1620 1640 1660

265

270

275

280

285

290

cTime

outMLE$cHat

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●

Minima and locations of the posterior reconstructions help to pin down

the timing and size of the decrease during this cooler period of Northern

Hemisphere climate.

Onset of industrialization.


1650 1670 1690 1710 1730 1750 1770 1790 1810 1830 1850 1870 1890 1910 1930 1950 1970 1990

27

02

80

29

03

00

31

03

20

33

03

40

35

03

60

Years CE

CO

2 C

on

ce

ntr

atio

n (

pp

m)

Reconstructed CO2 Concentration: 1650−2000

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Reconstructing carbon dioxide for the last 2000 years: a ... · How to trade a paper clip for a...

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Transcript of Reconstructing carbon dioxide for the last 2000 years: a ... · How to trade a paper clip for a...