Transcript ppt
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) • Recently development concerns regularization of the regression problem – Ribes et al., 2009, 2012a, 2012b
Photo: F. Zwiers
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 • notation
A t
m l
1
x
1
w
(
x
, )
T
(
x
,
t
1 )
x w
(
x
,
)
1
A t
w
T
T
t
T
t
w F
t
N
t - extended temperature field - weights - signal field - noise field
Optimal detection statistic ...
• Maximizes the signal to noise ratio
R
2
E(
A t
)
2
V(
A t
)
(
w
T
F
t
)
2
w
T
Σ NN w
subject to the constraint that the weights sum to one
w
T
e
1 where
e
( 1 , 1 ,
, 1 )
T
w
T
c
Σ
1
NN F
t
• Constant
c
is unimportant so can set
c=1
w
Σ
1
NN F
t
5 5 4
R
2 31.25
T
t
~
N
(
0
,
Σ NN
)
T
t
~
N
(
F
t
,
Σ NN
)
F
t
5 5
Σ NN
4 0 0 1 After Hasselmann (1979)
A simple detection test
is Gaussian Assume that A t
can test
H
0
: E(
A t
)
0
with
Z t
2
(
A t
)
2
V(
A t
)
(
F
t T
F
t T
Σ
1
Σ NN
1
NN F
t
F
t
)
2
Z t
2 ~ 1 2 when
H
0 is true
reject when Z 2 >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
Σ NN
diag
V
75
V
45
V
15 • Got signal from a 4xCO 2 equilibrium run – D CO 2 = 1200 ppm (Manabe and Stouffer, 1980) – estimated warming for 1972 (10% increase in CO 2 ) D
T
1972 ln( 1 .
1 ) ln( 4 ) D
T
4 CO 2
GHG Signal
1.0
oC 0.75
Warming due to 10% increase in CO 2 Winter (DJF) Summer (JJA) 0.50
0.25
0.0
0 30 Latitude 60 90 due to 10% increase in CO 2 DJF JJA After Bell (1982)
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 6 5 Hemispheric Seasonal Averages 4 3 2 1 Winter Optimal Average Area Weighted Average Spring Summer Autumn
R
Optimal After Bell (1982) Area weighted average (
F
t T
Σ
1
NN F
t
) 1 / 2
Outline
•Optimal filtering •Optimal fingerprinting •Recent developments
Photo: F. Zwiers
Observations Model 1946-56 1986-96
Filtering and projection
Y
onto reduced dimension space
Evaluate amplitude estimates X Y
X
Total least squares regression in reduced dimension space ˆ 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 from climate models) Signal errors Scaling factors Errors
Y
(X
ξ)β
ε
•
Observations
dimension-reduced space – Typically • Filtered spatially (to retain large scales) • Filtered temporally (to retain decadal variability - 5-11 decades) represented in a • 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 Solar
GHGs
Direct SO 4 aerosol Volcanic Ozone
All
IPCC WG1 AR4 Fig. 9.1
• • •
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 spectral coefficients), or 25 • 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 (
X
T
1
X
) 1
X
T
1
Y
where is the estimated variance-covariance matrix of the observations
Y
• Even with T4 filtering, would be 275x275 Need further dimension reduction • Constraints on dimensionality – Need to be able to invert covariance matrix – 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 – An estimate for the optimization step and to estimate scaling factors β make inferences
ε
ˆ
Y
X
ˆ • 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
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
X
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 • • • Use 1 st sample of control run chunks to estimate Select an EOF truncation Calculate Moore-Penrose inverse ˆ 1 1 1 6. Fit the regression model in the reduced space • OLS scaling factor estimates are (
X
T
Σ 1
1 X
) 1
X
T
Σ 1
1 Y
Basic procedure …
7. Rudimentary residual diagnostics on the fit • Is residual variance consistent with model estimated internal variability?
• Allen and Tett (1999)
T
1 2
(
Y
X
ˆ )
T
2
1
(
Y
X β
) ~ (
k
m
)
F k
m
,
v
• Ignores sampling variability in the optimization (Allen and Stott, 2003).
• Ribes et al (2012a) therefore show that
ε
ˆ
T
2
1 ε
ˆ ~
v
(
k
k
m
1 )
F k
m
,
v
k
1 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… where
β
(
β F
1
T
β
)
T
2
1 F
1
1 β
(
β
β
) ~
mF m
,
v
and
F
1 (
X
T
ˆ
1
1 X
)
1 X
T
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
C
ˆ
I
• 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
C
ˆ
I
has a Bayesian interpretation (with
I
ˆ 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