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

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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

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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

Snapshot

Correlation Patterns as a Function of averaging time (tau)

QuickTime™ and aTIFF (LZW) decompressor

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QuickTime™ and aTIFF (LZW) decompressor

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QuickTime™ and aTIFF (LZW) decompressor

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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

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QuickTime™ and aTIFF (LZW) decompressor

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QuickTime™ and aTIFF (LZW) decompressor

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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.