1. Dynamical Climate Reconstruction Greg Hakim University of Washington Sebastien Dirren, Helga Huntley, Angie Pendergrass David Battisti, Gerard Roe

2. 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

3. 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

4. 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?

5. Medieval warm period temperature anomalies IPCC Chapter 6

6. Climate variability: a qualitative approach North GRIP δ 18 O (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

7. Statistical reconstructions “Multivariate statistical calibration of multiproxy network” (Mann et al. 1998) Requires stationary spatial patterns of variability Mann et al. 1998

8. Paleoclimate modeling IPCC Chapter 6

9. An attempt at fusion Data Assimilation through Upscaling and Nudging (DATUN) Jones and Widmann 2003 Multivariate regression

10. Fusion Hierarchy Nudging: no error estimates Statistical interpolation 3DVAR 4DVAR Kalman filters Kalman smoothers fixed stats } } Today’s talk } oper NWP The curse of dimensionality looms large in geoscience

11. State Estimation Primer

12. Gaussian Update analysis = background + weighted observations new obs information Kalman gain matrix analysis error covariance ‘<’ background

13. Gaussian PDFs

14. Ensemble Kalman Filter 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.

15. 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.

16. 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?

17. 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.

18. 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

24. Model (adapted from Lorenz & Emanuel (1998)): Linear combination of fast & slow processes Illustrative Example Dirren & Hakim (2005) “low-freq.” “high-freq.” - LE ~ a scalar discretized around a latitude circle. - LE has elements of atmos. dynamics: chaotic behavior, linear waves, damping, forcing

25. RMS instantaneous Instantaneous states have large errors (comparable to climatology) Due to lack of observational constraint (dashed : clim)

26. RMS all means Obs uncertaintyClimatology uncertainty Improvement Percentage of RMS errors Total state variable Constrains signal at higher freq.than the obs themselves! Averaging time of state variable

27. Potential Problems 1.Possible covariance contamination 2.Extra observation noise

28. A less simple model Helga 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

29. Snapshot

30. Correlation Patterns as a Function of averaging time (tau)

31. Observation Locations

32. Average Spatial RMS Error

33. Ensemble used for control

34. 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)!

35. Observation error Experiments

36. Changing  o (Observation Error) Previously: –  o = 0.27 for all . Now: –  o ≈  c /3 –(a third of control error).

37. Observing Network Design Helga Huntley (U. Delaware)

38. 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.

39. 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(  x 2 ) –Metric variance

40. Sensitivity + State Estimation Estimate variance change for the i’th observation Kalman filter theory gives A i : where Given  at each point, find largest value.

41. Ensemble Sensitivity (cont’d) If H chooses a specific location x i, this all simplifies very nicely: –For the first observation: –For the second observation, given assimilation of the first observation: –Etc.

42. 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.

43. Results for tau = 20 First EOF

44. Results for tau = 20 The ten most sensitive locations (without accounting for prior assimilations)  o = 0.10

45. Results for tau = 20 The four most sensitive locations, accounting for previously found pts.

46. Results for tau = 20;  o = 0.10 Note the decreasing effect on the variance.

47. Control Case: No Assimilation Avg error = 5.4484

48. 100 Random Observation Locations Avg Error - Anal = 1.0427 - Fcst = 3.6403

49. 4 Random Observation Locations Avg Error - Anal = 5.5644 - Fcst = 5.6279

50. 4 Optimal Observation Locations Avg Error - Anal = 2.0545 - Fcst = 4.8808

51. Summary Avg Error FcstAnal Control5.4484 100 obs3.64031.0427 4 chosen4.88082.0545 4 random5.62795.5644 4 random5.54105.5091 4 random5.29425.2953 Assimilating 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

52. 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.

53. 15 Obs: Error in 1st EOF Coeff EOF1Control100R4R4O8O15 total Fcst5.4484 3.64035.48774.88084.75863.5138 Anal 1.04275.45632.05451.90201.8819

54. 15 Obs: Error in 2nd EOF Coeff EOF2Control100R4R4O8O15 total Fcst5.41143.32125.33944.36773.99373.5727 Anal0.84785.12074.27963.48321.4563

55. 15 Observations: RMS Error RMSControl100 R4 R4 O8 O15 O Fcst0.28990.25860.29570.28100.27700.2596 Anal0.14020.29120.25390.24250.2024

56. Current & Future Plans Angie Pendergrass (UW) modeling on the sphere: SPEEDY – simplified physics – slab ocean simulated precipitation observations ice-core assimilation – annual accumulation – oxygen isotopes

57. 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.