Information presented during the CLIM Undergraduate Workshop

1. Undergraduate Students Workshop
Statistical Developments and Challenges in
Past Climate Reconstruction
Bo Li
Department of Statistics
University of Illinois at Urbana-Champaign
October 24, 2017

2. Acknowledgement:
A mixed group of climate scientists and statisticians:
• Caspa Ammann, Luis Barboza, Peter Craigmile, Murali Haran,
Elizabeth Manshardt, Doug Nychka, Bala Rajaratnam,
Jason Smerdon, Martin Tingley, Frederi Viens,
Sooin Yun, Xianyang Zhang

3. Why care about the PAST climate?
• Accurate and precise reconstructions of past climate
help to characterize natural climate variability on longer
time scales.
• Spatially wide-spread instrumental temperature ob-
servations extend back to only about 1850.
• Validate climate models - Atmosphere/Ocean Gen-
eral Circulation Model (AOGCM)

4. How to recover past climate?
• Earth’s climate history written in ice, wood and stone.
• Reconstruct the past temperature from indirect ob-
servations (proxies) such as
– tree-ring width and densities
– Pollen
– Borehole
– Speleothems (cave deposits)
– Coral records, etc.
• Radiative Forcings: Solar, Volcanic eruption and
Greenhouse gases.

5. Tree-rings
This cross-section of a giant sequoia tree shows some of the tree-rings and fire
scars. The numbers indicate the year that a particular ring was laid down by
the tree

6. Tree-rings
Seasonal variations in density of calcium carbonate skeletons produce growth
rings similar to those in trees

7. Data - Pollen

8. Speleothem
As water runs through the ground, it picks up minerals, the most common
of which is calcium carbonate. When the mineral-rich water drips into caves,
it leaves behind solid mineral deposits. The mineral deposits accumulate into
speleothems.

9. Data - Borehole

10. Forcings
1000 1200 1400 1600 1800 2000
a
b
c
a: Volcanism (contains substantial noise)
b: Solar irradiance
c: Green house gases

11. A quiz for you
Start with a simple reconstruction
• Assume we observe Y and a bunch of X’s.
• Now given new X’s, how to estimate Y?

12. A quiz for you
Start with a simple reconstruction
• Assume we observe Y and a bunch of X’s.
• Now given new X’s, how to estimate Y?
Solution: Linear regression
Y = Xβ +
Y = Xβ

13. Long Climate Proxies and NH temperatures
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−404
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−0.60.2
DegreesC
Exploit where there
is overlap in two
sets of data.
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1800 1850 1900 1950 2000
−4−2024
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−0.6−0.20.20.6
DegreesC

14. A stats perspective
The problem:
• Quantify the uncertainty in the temperature esti-
mates from other kinds of observations (i.e. proxies).
• e.g. What is the uncertainty in the maximum decadal
temperature estimates for the last 1000 years?
A statistical solution:
• Find the distribution of the temperatures given the
other observations.
• Represent this distribution by an ensemble of pos-
sible reconstructions all statistically valid.

15. An ensemble of Santa’s and the ensemble mean
Santa
Don’t send your Xmas letter to Santa !
Credit: Cabinet 15

16. Statistical Relationship between
NH temperature and the proxies.
A minimal recipe for generating ensembles:
• A stationary linear relationship between the temper-
ature and proxies
• the conditional distribution of temperature given prox-
ies is normally distributed.
• Prediction errors are correlated in time
• Adjustment made for overfitting during calibration
period using cross-validation
• Also include the uncertainty in the parameters.

17. Linear prediction of the expected temperature
based on proxies.
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−0.6−0.20.20.6
DegreesC
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1800 1850 1900 1950 2000
−0.40.00.40.8
DegreesC

18. Linear prediction of the expected temperature
based on proxies.
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−0.40.00.40.8
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1860 1880 1900 1920 1940 1960 1980
−0.3−0.10.1
Residual
Clearly the errors are correlated.

19. Linear Model of NH temperature on proxies
The statistical model:
Tt = ptβ + et
Tt, pt: the temperature and proxies at time t.
β: a vector of regression coefficients
The vector of error et follows an AR(2) process:
et = φ1et−1 + φ2et−2 + t, t ∼ iid Normal(0, σ2
)
Model fitting procedure: Generalized Least Squares

20. Overfitting and
uncertainty of model parameter estimates
• Symptom of overfitting: The variance of observed
prediction errors is greater than the prediction vari-
ability derived from the model.
Solution: 10-fold cross-validation to estimate the in-
flation adjustment
• Uncertainty of model parameter estimates:
Parametric bootstrap to estimate the sampling dis-
tribution of the parameter estimates.
• An ensemble:
T = Pβ + (e1000, ..., e1849) |(e1850, ..., e1980)

21. Reconstructed temperatures
An ensemble member
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1000 1200 1400 1600 1800 2000
−0.50.00.5
DegreeC
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A draw from the the conditional distribution of
temperature given the values of the proxies.

22. Reconstructed temperatures
A second ensemble member
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1000 1200 1400 1600 1800 2000
−0.50.00.5
DegreeC
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
qq
q
q
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q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
A draw from the the conditional distribution of
temperature given the values of the proxies.

23. Reconstructed temperatures
Still another ensemble member
q
q
q
q
q
q
q
qq
q
q
qq
q
q
q
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1000 1200 1400 1600 1800 2000
−0.50.00.5
DegreeC
q
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qq
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q
q
q
q
q
q
q
q
q
q
q
q
q
A draw from the the conditional distribution of
temperature given the values of the proxies.

24. Reconstructed temperatures
Decadal means – three ensembles
1000 1200 1400 1600 1800 2000
−0.8−0.40.00.4
Year
Temperature(DegreeC)
q qq

25. Reconstructed temperatures
95 % uncertainty bounds decadal means
DegreeC
1000 1150 1300 1450 1600 1750 1900
−0.6−0.30.10.4

26. Reconstructed temperatures
Maxima uncertainty
DegreeC
q
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1000 1150 1300 1450 1600 1750 1900
−0.6−0.30.10.4

27. Reconstructed temperatures
Maxima uncertainty using box plots
DegreeC
q
q
q
q
q
q
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−0.6−0.30.10.4

28. Reconstructed temperatures
Maxima uncertainty 95% upper bound
DegreeC
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1000 1150 1300 1450 1600 1750 1900
−0.6−0.30.10.4

29. A statistical perspective.
The Program:
• Develop a statistical (perhaps complicated) relation-
ship between temperatures and proxies.
• The prediction is the distribution of the temperatures
given the observed values of the proxies.
What can not be addressed:
Errors in the proxies and analysis outside the statistical
model. (We don’t know what we don’t know.)
e.g. Proxies change in their relationship to temperature
over time

30. How to integrate different data sources
Skill of each proxy and forcings
• Tree ring (Dendrochronology): annual to decadal
• Pollen: bi-decadal to semi-centennial
• Borehole: centennial and onward
• Forcings: external drivers
Goal: Reconstruct the 850-1849 temperature by all
proxies, forcings and the 1850-1999 temperature

31. Bayesian Hierarchical Model (BHM)
Bayesian Hierarchical Model (BHM) to thread all
proxies, forcings and temperatures
Baye’s theorem:
P(A|B) =
P(B|A)P(A)
P(B)
Named after Thomas Bayes (1701-1761)

32. Bayesian Hierarchical Model (BHM)
A distribution rule:
[P, T, θ] = [P|T, θ][T|θ][θ]
Three hierarchies:
• Data Stage: [Proxies|Temperature, Parameters]
Likelihood of Proxies given temperatures
• Process Stage: [Temperature|Parameters]
Physical model of temperature process
• Parameter Stage: [Parameters]
Specify the prior of parameters

33. Apply BHM to Climate from Models
National Center for Atmospheric Research (NCAR) Com-
munity Climate System Model (CCSM) Version 1.4
• Provide test beds to evaluate our reconstruction
method
• Climate model is highly nonlinear and substantial
complexity
Resolution:
– 3.750 × 3.750/400km×400km for atmosphere and land
• Generate synthetic proxies (15 Tree-Ring, 10 Pollen
and 5 Borehole) from model climate based on their
own characteristics

34. Climate from Models
Pollen: 7.50
× 7.50
; Borehole: 200
× 200

35. Synthetic Proxies Generation
Generate synthetic proxies:
• 15 tree rings: subtract the 10-year smoothing aver-
age from each of the local temperature series
• 10 pollen: Sample a 10-year smoothing average tem-
perature series at 30 year intervals
1000 1200 1400 1600 1800 2000
−1.5−1.0−0.50.00.5
temperatureanomaly
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Tree-rings: black line; Pollen: red dots

36. • 5 borehole:
– POM-SAT Forward Model
– describe the diffusion process of
surface temperature
year
temperature
0.0
0.2
0.4
0.6
0.8
1.0
1000 1200 1400 1600 1800 2000
pulse at year 850
0.0
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1.0
pulse at year 650
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1.0
pulse at year 450
0.0
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1.0
pulse at year 250
0.0
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0.8
1.0
pulse at year 50
0.00 0.04 0.08 0.12
depth temperature
depth(m)
400350300250200150100500

37. Numerical Study
Five models with different subsets of proxy or
related data:
• local/regional Temperature series (T)
• Tree ring (Dendrochronology) only (D)
• Tree ring + Pollen (DP)
• Tree ring + Borehole (DB)
• Tree ring + Borehole + Pollen (DBP)

38. 1000 1200 1400 1600 1800 2000
temperature
target
reconstruction
target
reconstruction
target
reconstruction
−0.60.20.6−0.60.20.6−0.60.20.6
c
b
a
The reconstructions using tree-rings and pollen together with forc-
ings in three scenarios. a: modeling T and without noise; b: mod-
eling T1 and without noise; c: modeling T and with noise.

39. 0.0050.0500.500
period (year)
spectrum
1000 400 200 100 50 30 20 10 5 3
T
DBP
DP
D
DB
Using smoothed spectrum of reconstruction residuals from the five data models
to illustrate the frequency band at which proxies capture the variation of the
temperature process Li, Nychka and Ammann (2012).

40. Messages from numerical analyses
• BHM enables integrating information from forcings
and different proxies
• Forcing covariates dramatically reduce the bias, the
variance and thus rmse
– if the included proxy data do not well represent the
low frequency variability
• Tree-rings help to retain the high frequency variabil-
ity, while the pollen captures the variability at about
a 30 year period and onward
• With noise the reconstruction deteriorates somewhat
but not terribly

41. Challenges: Space-time reconstruction
– Multivariate linear regression (Smerdon, 2017):
T = BP + ,
T: instrumental climate records; P: proxy values.
• Covariance-based matrix factorizations:
PCA and canonical correlation analysis.
• Penalized regressions: ridge or lasso regression
• Ordinary least square vs total least square
• Hybrid: split the random field at 20 year time period.
– Bayesian reconstruction (Tingley and Huybers, 2010ab)

42. Challenges – Space-time reconstruction
Many issues remained:
• Nonstationary in both mean and dependency
• Modeling teleconnection (Choi, Li, Zhang and Li, 2015)
• Extensive computation for global scale
• How to evaluate the space-time reconstruction
(Li and Smerdon, 2012; Li, Zhang and Smerdon, 2016; Yun, Li,
Smerdon and Zhang, 2017)
• Multivariate spatial reconstruction
– Non-Gaussian of some climate variable
– Identifiability issue, e.g., temperature and precipi-
tation in general not identifiable