Dynamical Climate Reconstruction Greg Hakim University of Washington Sebastien Dirren, Helga...
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Transcript of Dynamical Climate Reconstruction Greg Hakim University of Washington Sebastien Dirren, Helga...
![Page 1: Dynamical Climate Reconstruction Greg Hakim University of Washington Sebastien Dirren, Helga Huntley, Angie Pendergrass David Battisti, Gerard Roe.](https://reader035.fdocuments.in/reader035/viewer/2022062714/56649d4d5503460f94a2ccc4/html5/thumbnails/1.jpg)
Dynamical Climate Reconstruction
Greg HakimUniversity of Washington
Sebastien Dirren, Helga Huntley, Angie Pendergrass David Battisti, Gerard Roe
![Page 2: Dynamical Climate Reconstruction Greg Hakim University of Washington Sebastien Dirren, Helga Huntley, Angie Pendergrass David Battisti, Gerard Roe.](https://reader035.fdocuments.in/reader035/viewer/2022062714/56649d4d5503460f94a2ccc4/html5/thumbnails/2.jpg)
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
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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
![Page 4: Dynamical Climate Reconstruction Greg Hakim University of Washington Sebastien Dirren, Helga Huntley, Angie Pendergrass David Battisti, Gerard Roe.](https://reader035.fdocuments.in/reader035/viewer/2022062714/56649d4d5503460f94a2ccc4/html5/thumbnails/4.jpg)
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?
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Medieval warm period temperature anomalies
IPCC Chapter 6
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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
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Statistical reconstructions
• “Multivariate statistical calibration of multiproxy network” (Mann et al. 1998)
• Requires stationary spatial patterns of variability
Mann et al. 1998
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Paleoclimate modeling
IPCC Chapter 6
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An attempt at fusion
Data Assimilation through Upscaling and Nudging (DATUN)Jones and Widmann 2003
Multivariate regression
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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
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State Estimation Primer
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Gaussian Update
analysis = background + weighted observations
new obs information
Kalman gain matrix
analysis error covariance ‘<’ background
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Gaussian PDFs
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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.
![Page 15: Dynamical Climate Reconstruction Greg Hakim University of Washington Sebastien Dirren, Helga Huntley, Angie Pendergrass David Battisti, Gerard Roe.](https://reader035.fdocuments.in/reader035/viewer/2022062714/56649d4d5503460f94a2ccc4/html5/thumbnails/15.jpg)
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.
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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?
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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.
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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
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• 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
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RMS instantaneous
Instantaneous states have large errors (comparable to climatology)
Due to lack of observational constraint
(dashed : clim)
![Page 26: Dynamical Climate Reconstruction Greg Hakim University of Washington Sebastien Dirren, Helga Huntley, Angie Pendergrass David Battisti, Gerard Roe.](https://reader035.fdocuments.in/reader035/viewer/2022062714/56649d4d5503460f94a2ccc4/html5/thumbnails/26.jpg)
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
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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
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Snapshot
![Page 29: Dynamical Climate Reconstruction Greg Hakim University of Washington Sebastien Dirren, Helga Huntley, Angie Pendergrass David Battisti, Gerard Roe.](https://reader035.fdocuments.in/reader035/viewer/2022062714/56649d4d5503460f94a2ccc4/html5/thumbnails/29.jpg)
Correlation Patterns as a Function of averaging time (tau)
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Observation Locations
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Average Spatial RMS Error
![Page 32: Dynamical Climate Reconstruction Greg Hakim University of Washington Sebastien Dirren, Helga Huntley, Angie Pendergrass David Battisti, Gerard Roe.](https://reader035.fdocuments.in/reader035/viewer/2022062714/56649d4d5503460f94a2ccc4/html5/thumbnails/32.jpg)
Average Spatial RMS Error
Ensemble used for control
![Page 33: Dynamical Climate Reconstruction Greg Hakim University of Washington Sebastien Dirren, Helga Huntley, Angie Pendergrass David Battisti, Gerard Roe.](https://reader035.fdocuments.in/reader035/viewer/2022062714/56649d4d5503460f94a2ccc4/html5/thumbnails/33.jpg)
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)!
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Observation error Experiments
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Changing o (Observation Error)
• Previously:– o = 0.27 for all τ.
• Now: o ≈ c/3 – (a third of control error).
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Observing Network Design Helga Huntley (U. Delaware)
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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.
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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
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Sensitivity + State Estimation
• Estimate variance change for the i’th observation
• Kalman filter theory gives Ai:
where
• Given at each point, find largest value.
![Page 40: Dynamical Climate Reconstruction Greg Hakim University of Washington Sebastien Dirren, Helga Huntley, Angie Pendergrass David Battisti, Gerard Roe.](https://reader035.fdocuments.in/reader035/viewer/2022062714/56649d4d5503460f94a2ccc4/html5/thumbnails/40.jpg)
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.
![Page 41: Dynamical Climate Reconstruction Greg Hakim University of Washington Sebastien Dirren, Helga Huntley, Angie Pendergrass David Battisti, Gerard Roe.](https://reader035.fdocuments.in/reader035/viewer/2022062714/56649d4d5503460f94a2ccc4/html5/thumbnails/41.jpg)
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.
![Page 42: Dynamical Climate Reconstruction Greg Hakim University of Washington Sebastien Dirren, Helga Huntley, Angie Pendergrass David Battisti, Gerard Roe.](https://reader035.fdocuments.in/reader035/viewer/2022062714/56649d4d5503460f94a2ccc4/html5/thumbnails/42.jpg)
Results for tau = 20
First EOF
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Results for tau = 20
• The ten most sensitive locations (without accounting for prior assimilations)
o = 0.10
![Page 44: Dynamical Climate Reconstruction Greg Hakim University of Washington Sebastien Dirren, Helga Huntley, Angie Pendergrass David Battisti, Gerard Roe.](https://reader035.fdocuments.in/reader035/viewer/2022062714/56649d4d5503460f94a2ccc4/html5/thumbnails/44.jpg)
Results for tau = 20
• The four most sensitive locations, accounting for previously found pts.
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Results for tau = 20; o = 0.10
Note the decreasing effect on the variance.
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Control Case: No Assimilation
Avg error = 5.4484
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100 Random Observation Locations
Avg Error - Anal = 1.0427 - Fcst = 3.6403
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4 Random Observation Locations
Avg Error - Anal = 5.5644 - Fcst = 5.6279
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4 Optimal Observation Locations
Avg Error - Anal = 2.0545 - Fcst = 4.8808
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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
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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.
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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
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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
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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
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Current & Future PlansAngie Pendergrass (UW)
• modeling on the sphere: SPEEDY– simplified physics– slab ocean
• simulated precipitation observations
• ice-core assimilation– annual accumulation– oxygen isotopes
![Page 56: Dynamical Climate Reconstruction Greg Hakim University of Washington Sebastien Dirren, Helga Huntley, Angie Pendergrass David Battisti, Gerard Roe.](https://reader035.fdocuments.in/reader035/viewer/2022062714/56649d4d5503460f94a2ccc4/html5/thumbnails/56.jpg)
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.
![Page 57: Dynamical Climate Reconstruction Greg Hakim University of Washington Sebastien Dirren, Helga Huntley, Angie Pendergrass David Battisti, Gerard Roe.](https://reader035.fdocuments.in/reader035/viewer/2022062714/56649d4d5503460f94a2ccc4/html5/thumbnails/57.jpg)