dynamical climate reconstruction greg hakim university of washington sebastien dirren, helga...
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Dynamical Climate Reconstruction
Greg HakimUniversity of Washington
Sebastien Dirren, Helga Huntley, Angie Pendergrass David Battisti, Gerard Roe
Plan
• Motivation: fusing observations & models
• State estimation theory
• Results for a simple model
• Results for a less simple model
• Optimal networks
• Plans for the future
Motivation
• Range of approaches to climate reconstruction.• Observations:
– time-series analysis; multivariate regression– no link to dynamics
• Models– spatial and temporal consistency– no link to observations
• State estimation (this talk)– few attempts thus far– stationary statistics
Goals
• Test new method• Reconstruct last 1-2K years
– Unique dataset for climate variability?– E.g. hurricane variability.– E.g. rational regional downscaling (hydro).
• Test network design ideas– Where to take highest impact new obs?
Medieval warm period temperature anomalies
IPCC Chapter 6
Climate variability: a qualitative approach
North
GRIP δ18O (temperature)
GISP2 K+ (Siberian High)
Swedish tree line limit shift
Sea surface temperature from planktonic foraminiferals
hematite-stained grains in sediment cores (ice rafting)
Varve thickness (westerlies)
Cave speleotherm isotopes (precipitation)
foraminifera
Mayewski et al., 2004
Statistical reconstructions
• “Multivariate statistical calibration of multiproxy network” (Mann et al. 1998)
• Requires stationary spatial patterns of variability
Mann et al. 1998
Paleoclimate modeling
IPCC Chapter 6
An attempt at fusion
Data Assimilation through Upscaling and Nudging (DATUN)Jones and Widmann 2003
Multivariate regression
Fusion Hierarchy
• Nudging: no error estimates• Statistical interpolation• 3DVAR• 4DVAR• Kalman filters• Kalman smoothers
fixed stats}}Today’s talk
} operNWP
The curse of dimensionality looms large in geoscience
State Estimation Primer
Gaussian Update
analysis = background + weighted observations
new obs information
Kalman gain matrix
analysis error covariance ‘<’ background
Gaussian PDFs
Ensemble Kalman Filter
1 2( , ,..., )e
b b b bNX x x x= ' '1
1
Tb b b
e
P X XN
≈−
Crux: use an ensemble of fully non-linear forecasts to model the statistics of the background (expected value and covariance matrix).
Advantages
• No à priori assumption about covariance; state-dependent corrections.
• Ensemble forecasts proceed immediately without perturbations.
Summary of Ensemble Kalman Filter (EnKF) Algorithm
(1) Ensemble forecast provides background estimate & statistics (B) for new analyses.
(2) Ensemble analysis with new observations.
(3) Ensemble forecast to arbitrary future time.
Paleo-assimilation dynamical climate reconstruction
• Observations often time-averaged.– e.g. gauge precip; wind; ice cores.
• Sparse networks.
• Issue:– How to combine averaged observations with
instantaneous model states?
Issue with Traditional Approach
Problem:
Conventional Kalman filtering requires covariance relationships between time-averaged observations and instantaneous states.
High-frequency noise in the instantaneous states contaminates the update.
Solution:
Only update the time-averaged state.
Algorithm
1. Time-averaged of background
2. Compute model-estimate of time-av obs
3. Perturbation from time mean
4. Update time-mean with existing EnKF
5. Add updated mean and unmodified perturbations
6. Propagate model states
7. Recycle with the new background states
• Model (adapted from Lorenz & Emanuel (1998)): Linear combination of fast & slow processes
Illustrative ExampleDirren & Hakim (2005)
“low-freq.”“high-freq.” 450 600lfτ ≈ −
3 4hfτ ≈ −
- LE ~ a scalar discretized around a latitude circle.
- LE has elements of atmos. dynamics:
chaotic behavior, linear waves, damping, forcing
RMS instantaneous
Instantaneous states have large errors (comparable to climatology)
Due to lack of observational constraint
(dashed : clim)
RMS all means
οτ
τ
Obs uncertainty Climatology uncertainty
Improvement Percentage of RMS errors
Total state variable
Constrains signal at higher freq.than the obs themselves!
Ave
ragi
ng
tim
e of
sta
te v
aria
ble
A less simple modelHelga Huntley (U. Delaware)
• QG “climate model”– Radiative relaxation to assumed temperature field– Mountain in center of domain
• Truth simulation– Rigorous error calculations– 100 observations (50 surface & 50 tropopause)– Gaussian errors– Range of time averages
Snapshot
Correlation Patterns as a Function of averaging time (tau)
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Observation Locations
Average Spatial RMS Error
Average Spatial RMS Error
Ensemble used for control
Implications
• State is well constrained by few, noisy, obs.• Forecast error saturates at climatology for tau ~ 30.• For longer averaging times, the model adds little.
Equally good results can be obtained by assimilating the observations with an ensemble drawn from climatology (no model runs required)!
Observation error Experiments
Changing o (Observation Error)
• Previously:– o = 0.27 for all τ.
• Now: o ≈ c/3 – (a third of control error).
Observing Network Design Helga Huntley (U. Delaware)
Optimal Observation Locations
• Rather than use random networks, can we devise a strategy to optimally site new observations? – Yes: choose locations with the largest impact on a
metric of interest.– New theory based on ensemble sensitivity (Hakim &
Torn 2005; Ancell & Hakim 2007; Torn and Hakim 2007) – Here, metric = projection coefficient for first EOF.
Ensemble Sensitivity• Given metric J, find the observation that reduces
uncertainy most (ensemble variance).
• Find a second observation conditional on first.
• Sketch of theory (let x denote the state).– Analysis covariance
– Changes in metric given changes in state
+ O(x2)
– Metric variance
Sensitivity + State Estimation
• Estimate variance change for the i’th observation
• Kalman filter theory gives Ai:
where
• Given at each point, find largest value.
Ensemble Sensitivity (cont’d)
• If H chooses a specific location xi, this all simplifies very nicely:– For the first observation:
– For the second observation, given assimilation of the first observation:
– Etc.
Ensemble Sensitivity (cont’d)
• In fact, with some more calculations, one can find a nice recursive formula, which requires the evaluation of just k+3 lines (1 covariance vector + (k+6) entry-wise mults/divs/adds/subs) for the k’th point.
Results for tau = 20
First EOF
Results for tau = 20
• The ten most sensitive locations (without accounting for prior assimilations)
o = 0.10
Results for tau = 20
• The four most sensitive locations, accounting for previously found pts.
Results for tau = 20; o = 0.10
Note the decreasing effect on the variance.
Control Case: No Assimilation
Avg error = 5.4484
100 Random Observation Locations
Avg Error - Anal = 1.0427 - Fcst = 3.6403
4 Random Observation Locations
Avg Error - Anal = 5.5644 - Fcst = 5.6279
4 Optimal Observation Locations
Avg Error - Anal = 2.0545 - Fcst = 4.8808
SummaryAvg
ErrorFcst Anal
Control 5.4484
100 obs 3.6403 1.0427
4 chosen 4.8808 2.0545
4 random 5.6279 5.5644
4 random 5.5410 5.5091
4 random 5.2942 5.2953Assimilating just the 4 chosen locations yields a significant portion of the gain in error reduction in J achieved with 100 obs.
Percent of ctr error
100
66.8
89.6
103.3
101.7
97.2
19.1
37.7
102.1
101.1
97.2
0 20 40 60 80 100
Control
100 obs
4 chosen
4 random
4 random
4 random
FcstAnal
Experiment: 15 Chosen Observations
• For this experiment, take– 4 best obs to reduce variability in 1st EOF
– 4 best obs to reduce variability in 2nd EOF
– 2 best obs to reduce variability in 3rd EOF
– 2best obs to reduce variability in 4th EOF
– 3 best obs to reduce variability in 5th EOF
• The cut-off for each EOF was chosen as the observation with .
• All obs conditioal on assimilation of previous obs.
15 Obs: Error in 1st EOF Coeff100
66.8
100.7
89.6
87.3
71.7
64.5
19.1
100.1
37.7
34.9
33.1
34.5
0 20 40 60 80 100
Control
100 rand
4 rand (avg)
4 for 1st
8 for 1st
4 + 4
15 total
FcstAnal
EOF1 Control 100R 4R 4O 8O 15 total
Fcst 5.4484 3.6403 5.4877 4.8808 4.7586 3.5138Anal 1.0427 5.4563 2.0545 1.9020 1.8819
15 Obs: Error in 2nd EOF Coeff100
61.4
98.7
80.7
73.8
73.6
66
15.7
94.6
79.1
64.4
38.2
26.9
0 20 40 60 80 100
Control
100 rand
4 rand (avg)
4 for 1st
8 for 1st
4 + 4
15 total
FcstAnal
EOF2 Control 100R 4R 4O 8O 15 total
Fcst 5.4114 3.3212 5.3394 4.3677 3.9937 3.5727
Anal 0.8478 5.1207 4.2796 3.4832 1.4563
15 Observations: RMS Error
100
89.2
102
97
95.6
92.4
89.6
48.4
100.4
87.6
83.7
78.5
69.8
0 20 40 60 80 100
Control
100 rand
4 rand (avg)
4 for 1st
8 for 1st
4 + 4
15 total
FcstAnal
RMS Control 100 R 4 R 4 O 8 O 15 O
Fcst 0.2899 0.2586 0.2957 0.2810 0.2770 0.2596
Anal 0.1402 0.2912 0.2539 0.2425 0.2024
Current & Future PlansAngie Pendergrass (UW)
• modeling on the sphere: SPEEDY– simplified physics– slab ocean
• simulated precipitation observations
• ice-core assimilation– annual accumulation– oxygen isotopes
Summary
• State estimation = fusion of obs & models– Lower errors than obs and model.– Provides best estimate & error estimate
• Idealized experiments: proof of concept– Averaged obs constrain state estimates.– Optimal observing networks
• Moving toward real observations– Opportunity for regional downscaling.