Climate change detection and
attribution methods
Exploratory Workshop DADA, Buenos Aires, 15-18 Oct 2012
Francis Zwiers, Pacific Climate Impacts Consortium, University of Victoria, Canada
Photo: F. Zwiers
Introduction
• Two types of
approaches currently in
use - non-optimal and
“optimal”
• Both rely heavily on
climate models
• The objective is always
to assess the evidence
contained in the
observations
• Methods are simple, yet
complex
Photo: F. Zwiers
Optimal approach
• Originally developed in a couple of different ways
– Optimal filtering (North and colleagues, early 1980’s)
– Optimal fingerprinting (Hasselmann,1979; Hegerl et al, 1996; 1997
• They are equivalent (Hegerl and North, 1997) and amount
to generalized linear regression
• Subsequently have OLS and EIV variants
– OLS; Allan and Tett (1999)
– TLS; Allan and Stott (2003)
– EIV; Huntingford et al (2006)
Photo: F. Zwiers
• Recently development concerns regularization of the
regression problem
– Ribes et al., 2009, 2012a, 2012b
Outline
•Optimal filtering
•Optimal
fingerprinting
•Recent
developments
Photo: F. Zwiers
An early detection study - Bell (1982)
• “Signal in noise” problem
T FN
Observed field T(x,t)
at locations x and times t
Climate’s deterministic
response to an
“external” forcing
(such as G, S, Sol, Vol)
Natural internal variability
(I.e., the “noise”)
Bell (1982)
• simple linear space-time “filtering” to remove the noise
l
m
At   w( x, )T ( x, t    1)
 1 x 1
w( x, )  1


x
• notation
T
At  w T
t
T
t
- extended temperature field
w
- weights
F
t
N
- signal field
t
- noise field
Optimal detection statistic ...
• Maximizes the signal to noise ratio
2
T
2
E( At )
(w Ft )
R 
 T
V( At ) w Σ NN w
2
subject to the constraint that the weights sum to one
w e  1 where e  (1, 1, , 1)
T
T

1
NN
w  cΣ Ft
T
• Constant c is unimportant so can set c=1
wΣ
1
NN
5 4 2
 R  31.25
Ft  
 5 
Tt ~ N (Ft , ΣNN )
Tt ~ N (0, ΣNN )
After Hasselmann (1979)
 5
Ft   
 5
 4 0

Σ NN  
0 1
A simple detection test
Assume that At is Gaussian

can test
H 0 : E( At )  0
with
1
2
NN t
1
NN t
( At )
(F Σ F )
Zt 

V( At )
F Σ F
2
2
T
t
T
t
Z ~  when H0 is true
2
t
2
1

reject when Z2>4
Bell’s application
• Estimate S/N ratio for NH seasonal mean
temperature circa 1972
• divided NH into 3 latitude zones
– equator to 30N, 30-60N, 60-90N
• assumed covariance between zones is zero
 ΣNN  diagV75 V45 V15 
• Got signal from a 4xCO2 equilibrium run
– DCO2 = 1200 ppm (Manabe and Stouffer, 1980)
– estimated warming for 1972 (10% increase in CO2)
DT1972
ln(
1
.
1
)



DT4 CO 2

ln( 4) 

GHG
Signal
Estimate of expected warming2
due to 10% increase in CO2
DJF
JJA
After Bell (1982)
Si
Optimal Average
Area Weighted Average
2
1
Winter
Estimated
S/N ratio
• ~25% gain
• S/N ratio is
large, but
signal not
detected in
1972 - why?
• poor estimate
of variance
• ocean delay
• other signals
Spring
Summer
Autumn
Optimal
Area weighted
average
R  (F Σ
T
t
After Bell (1982)
1
NN
1/ 2
Ft )
Outline
•Optimal filtering
•Optimal
fingerprinting
•Recent
developments
Photo: F. Zwiers
Observations
Model
1946-56
1986-96
Filtering
and projection
onto reduced
dimension space
Y
Evaluate
amplitude
estimates
X
Y  X  
ˆ
Total least squares regression
in reduced dimension space
ˆ
Evaluate
goodness of
fit
Weaver and Zwiers, 2000
The regression model
• Evolution of methods since the IPCC SAR (“the balance
of evidence suggests…”)
• Most studies now use an errors in variables approach
Y  (X  ξ)β  ε
Observations
Signals
(estimated Signal
errors
from
Scaling
climate
factors
models)
Errors
Y  (X  ξ)β  ε
• Observations represented in a
dimension-reduced space
– Typically
• Filtered spatially (to retain
large scales)
• Filtered temporally (to retain
decadal variability - 5-11
decades)
• Projected onto low-order
space-time EOFs
• Signals estimated from
– Multi-model ensembles of
historical simulations
• With different combinations of
external forcings
– Anthropogenic (GHG,
aerosols, etc)
– Natural (Volcanic, solar)
IPCC WG1 AR4 Fig. TS-23
Examples
of signals
20th
century
response
to forcing
simulated
by PCM
IPCC WG1 AR4 Fig. 9.1
Solar
Volcanic
GHGs
Ozone
Direct SO4 aerosol
All
Y  (X  ξ)β  ε
• Signal error term represents effects of
– Internal variability (ensemble sizes are finite)
– Structural error
• Know that multi-model mean often a better presentation of current climate
• Do not know how model space has been sampled
• Scaling factor
– Alters amplitude of simulated response pattern
• Error term
– Sampling error in observations (hopefully small)
– Internal variability (substantial, particular at smaller scales)
– Misfit between model-simulated signal and real signal (hopefully small …
a scaling factor near unity would support this)
• Ultimate small sample inference problem:
Observations provide very little information
about the error variance-covariance structure
Typical D&A problem setup
• Typical approach in a global
analysis of surface temperature
– Often start with HadCRU data
(5°x 5°), monthly mean anomalies
– Calculate annual or decadal mean
anomalies
– Filter to retain only large scales
• Spectrally transform (T4  25
spectral coefficients), or
• Average into large grid boxes
(e.g., 30°x40°  up to 6x9=54
boxes)
– For a 110-yr global analysis
performed with T4 spectral
filtering and decadal mean
anomalies dim(Y) = 25x11 = 275
Photo: F. Zwiers
• The OLS form of the estimator of the scaling factors β is
ˆβ  (XT Σ
ˆ 1X)1 XT Σ
ˆ 1Y
ˆ is the estimated variance-covariance matrix of
where Σ
the observations Y
ˆ would be 275x275
• Even with T4 filtering, Σ
 Need further dimension reduction
• Constraints on dimensionality
ˆ
– Need to be able to invert covariance matrix Σ
– Covariance needs to be well estimated on retained space-time
scales
– Should only keep scales on which climate model represents
internal variability reasonably well
– Should be able to represent signal vector reasonably well
• Further constraint
– To avoid bias, optimization and uncertainty analysis should be
performed separately
 Require two independent estimates of internal variability
ˆ
Σ
– An estimate
for the optimization step and to
1
estimate scaling factors β
ˆ
Σ
– An estimate2 to make estimate uncertainties and
make inferences
εˆ  Y  Xβˆ
• Residuals from the regression model
are used to assess misfit and model based estimates of
internal variability
Basic procedure
1. Determine space-time scale of interest (e.g., global, T4
smoothing, decadal time scale, past 50-years)
2. Gather all data
•
•
Observations
Ensembles of historical climate runs
•
•
Might use runs with ALL and ANT forcing to separate effects of
ANT and NAT forcing in observations
Control runs (no forcing, needed to estimate internal variability)
3. Process all data
•
Observations
•
•
homogenize, center, grid, identify where missing
Historical climate runs
•
•
•
“mask” to duplicate missingness of observations,
process each run as the observations (no need to homogenize)
ensemble average to estimate signals
Basic procedure ….
Observations
1946-56
Model
1986-96
Y
X
3. Process all data - continued
•
Control run(s), within ensemble variability for individual models
•
•
•
Divide into two parts
Organize each part into “chunks” covering the same period as the
observations – typically allow chunks to overlap
• 2000 yr run  2x1000 yr pieces  2x94x60 yr chunks
Process each chunk as the observations
Basic procedure …
4. Filtering step
•
Apply space and time filtering to all processed data sets
•
suppose doing a 1950-2010 analysis using observations, ALL and
ANT ensembles of size 5 from one model, 2000 yr control
 1 obs + 2x5 forced + 2x94 control = 200 datasets to process
5. Optimization step
•
•
•
ˆ
Use 1st sample of control run chunks to estimate Σ
1
Select an EOF truncation
ˆ 1
Calculate Moore-Penrose inverse Σ
1
6. Fit the regression model in the reduced space
•
ˆ  (XT Σ1X)1 XT Σ1Y
OLS scaling factor estimates are β
1
1
Basic procedure …
7. Rudimentary residual diagnostics on the fit
•
Is residual variance consistent with model estimated internal
variability?
•
Allen and Tett (1999)
ˆ 1εˆ  (Y  Xβˆ )T Σ
ˆ 1 (Y  Xβˆ ) ~ (k  m) F
εˆ T Σ
2
2
k  m,v
•
Ignores sampling variability in the optimization (Allen and Stott,
2003).
•
Ribes et al (2012a) therefore show that
 ( k  m)
1
ˆ
εˆ Σ 2 εˆ ~
Fk  m,v  k 1
v  k 1
T
would be more appropriate
Basic procedure ….
8. Repeat 6-7 for a range of EOF truncations k=1,2,….
Residual consistency test as a function of EOF truncation
Space-time analysis of transformed extreme precipitation
Obs are 5-year means for 1950-1999 averaged over Northern mid-lat and tropic bands
Dashed  estimate of internal
variance doubled
Min et al, 2011, Fig S8b (right)
Basic procedure ….
9. Make inferences about scaling factors
•
OLS expression that ignores uncertainty in the basis looks like…
ˆ 1 (βˆ  β) ~ m F
(βˆ  β)T Σ
β
m ,v
ˆ  FT Σ
ˆ 1F and F  ( XT Σ
ˆ 1 X) 1 XT Σ
ˆ 1
w hereΣ
β
1
2 1
1
1
1
A “typical” detection result
Scaling factor estimates as a function of EOF truncation
Space-time analysis of transformed annual extreme precipitation
Obs are 5-year means for 1950-1999 averaged over Northern mid-lat and tropic bands
*
Residual consistency test fails
O Residual consistency test fails with doubled internal variance
Min et al, 2011, Fig S8a (right)
Outline
•Optimal filtering
•Optimal
fingerprinting
•Recent
developments
Photo: F. Zwiers
How should we regularize the problem?
• Approach to date has been adhoc
– Filtering + sample covariance matrix (may not be well
conditioned) + EOF truncation (Moore-Penrose inverse)
– Neither EOF nor eigenvalues well estimated
– Truncation criteria not clear
 Results can be ambiguous in some cases
 Filtering occurs both external to the analysis, and within
the analysis
How should we regularize the problem?
• Ribes (2009, 2012a, 2012b) has suggested using
the well-conditioned regularized estimator of Ledoit
and Wolf (2004)
ˆ  ˆI
ˆ  ˆC

• Weighted average of the sample covariance matrix
and a structured covariance matrix, which in this
case is the identify matrix
• This estimate is always well conditioned, is
consistent, and has better accuracy than the
sample estimator
• Separates the filtering problem from the D&A
analysis.
How should we regularize the problem?
• Ledoit and Wolf (2004) point out that the weighted
average
ˆ  ˆI
ˆ  ˆC

has a Bayesian interpretation (with I
corresponding to the prior, and ˆ a posterior
estimate)
• Perhaps convergence could be improved by using
? a more physically appropriate structured estimator
in place of I? Perhaps the other DA can help?
What about other distributional settings?
Y | X ~ GEV( X , , )
T
Y
X



Space-time vector of annual extremes
Space-time signal matrix (one column per signal)
Vector of scaling factors
Vector of scale parameters
Vector of shape parameters
Note that these
are vectors
Conclusions
• The method continues to evolve
• Thinking hard about regularization is a good
development (but perhaps not most critical)
• Some key questions
– How do we make objective prefiltering choices?
– How should we construct the “monte-carlo” sample of
realizations that is used to estimate internal variability?
– Similar question for signal estimates
– How should we proceed as we push answer questions
about extremes?
Photo: F. Zwiers
Thank you