Sai Ravela Massachusetts Institute of Technology

1. Sai Ravela Massachusetts Institute of Technology J. Marshall, A. Wong, S. Stransky, C. Hill Collaborators: B. Kuszmaul and C. Leiserson “Planet in a bottle”A Realtime Observatory for Laboratory Simulation of Planetary Circulation

2. Geophysical Fluids in the Laboratory Inference from models and data is fundamental to the earth sciences Laboratory analogs systems can be extremely useful

3. Ravela, Marshall , Wong, Stransky , 07 Planet-in-a-bottle OBS Z DA MODEL

4. Velocity Observations • Velocity measurements using correlation-based optic-flow • 1sec per 1Kx1K image using two processors. • Resolution, sampling and noise cause measurement uncertainty • Climalotological temperature BC in the numerical model

5. Marshall et al., 1997 MIT-GCM (mitgcm.org): incompressible boussinesq fluid in non-hydrostatic mode with a vector-invariant formulation Numerical Simulation • Thermally-driven System (via EOS) • Hydrostatic mode Arakawa C-Grid • Momentum Equations: Adams-Bashforth-2 • Traceer Equations: Upwind-biased DST with Sweby Flux limiter • Elliptic Equaiton: Conjugate Gradients • Vertical Transport implicit.

6. x 23 x 15 (z) {45-8 }x 15cm Domain Cylindrical coordinates. Nonuniformdiscretization of the vertical Random temperature IC Static temperature BC Noslip boundaries Heat-flux controlled with anisotropic thermal diffusivity

7. Estimation from model and data Estimate what? • State Estimation: • NWP type applications, but also reanalysis • Filtering & Smoothing • Parameter Estimation: • Forecasting & Climate • State and Parameter Estimation • The real problem. General Approach: Ensemble-based, multiscale methods.

8. Schedule

9. Producing state estimates Key questions • Where does the ensemble come from? • How many ensemble members are necessary? • What about the computational cost of ensemble propagation? • Does the forecast uncertainty contain truth in it? • What happens when it is not? • What about spurious longrange correlations in reduced rank representations? Ensemble-methods • Reduced-rank Uncertainty • Statistical sampling • Tolerance to nonlinearity • Model is fully nonlinear • Dimensionality • Square-root representation via the ensemble • Variety of approximte filters and smoothers Ravela, Marshall, Hill, Wong and Stransky, 07

11. Approach Snapshots capture flow-dependent uncertainty (Sirovich) BC+IC P(X0|T): IC Perturbation 1 P(T ): Thermal BC Perturbations 4 Deterministic update: 5 – 2D updates 5 – (Elliptic) temperature Nx * Ny – 1D problems P(Xt|Xt-1): Snapshots in time 10 P(Yt|Xt) P(Xt|Xt-1): Deterministic update E>e0? P(Yt|Xt) P(Xt|Xt-1): Ensemble update Ravela, Marshall, Hill, Wong and Stransky, 07

12. EnKF revisited The analysis ensemble is a (weakly) nonlinear combinationof the forecast ensemble. This form greatly facilitates interpretation of smoothing Evensen 03, 04

13. Ravela and McLaughlin, 2007

14. Ravela and McLaughlin, 2007

15. Next Steps • Lagrangian Surface Observations : Multi-Particle Tracking • Volumetric temperature measurements. • Simultaneous state and parameter estimation. • Targeting using FTLE & Effective diffusivity measures. • Semi-lagrangian schemes for increased model timesteps. • MicroRobotic Dye-release platforms.

16. Ravela et al. 2003, 2004, 2005, 2006, 2007 The Amplitude-Position Formulation of Data Assimilation With thanks to K. Emanuel, D. McLaughlin and W. T. Freeman

17. Solitons Many reasons for position error There are many sources of position error: Flow and timing errors, Boundary and Initial Conditions, Parameterizations of physics, sub-grid processes, Numerical integration…Correcting them is very difficult. Hurricanes Thunderstorms

18. 3DVAR Amplitude assimilation of position errors is nonsense!

19. EnKF Distorted analyses are optimal, by definition. They are also inappropriate, leading to poor estimates at best, and blowing the model up, at worst.

20. Key Observations • Why do position errors occur? • Flow & timing errors, discretization and numerical schemes, initial & boundary conditions…most prominently seen in meso-scale problems: storms, fronts, etc. • What is the effect of position errors? • Forecast error covariance is weaker, the estimator is both biased, and will not achieve the cramer-rao bound. • When are they important? • They are important when observations are uncertain and sparse

21. Joint Position Amplitude Formulation Question the standard Assumption; Forecasts are unbiased

22. Bend, then blend

23. Improved control of solution

25. Bend, then Blend Flexible Application • Data Assimilation • Hurricanes , Fronts & Storms • In Geosciences • Reservoir Modeling • Alignment a better metric for structures • Super-resolution simulations • texture (lithology) synthesis • Flow &Velocimetry • Robust winds from GOES • Fluid Tracking • Under failure of brightness constancy • Cambridge 1-step (Bend and Blend) • Variational solution to jointly solves for diplas and amplitudes • Expensive • Cambridge 2-step(Bend, then blend) • Approximate solution • Preprocessor to 3DVAR or EnKF • Inexpensive Students Ryan Abernathy: Scott Stransky Classroom

26. Key Observations • Why is “morphing” a bad idea • Kills amplitude spread. • Why is two-step a good idea • Approximate solution to the joint inference problem. • Efficient O(nlog n), or O(n) with FMM • What resources are available? • Papers, code, consulting, joint prototyping etc.

27. Adaptation to multivariate fields

28. Ravela & Chatdarong, 06 Velocimetry, for Rainfall Modeling Aligned time sequences of cloud fields are used to produce velocity fields for advecting model storms. Velocimetry derived this way is more robust than existing GOES-based wind products.

29. Other applications Magnetometry Alignment (Shell)

30. Super-resolution Example-based Super-resolved Fluids Ravela and Freeman 06

31. Next Steps • Fluid Velocimetry: GOES & Laboratory, release product. • Incorporate Field Alignment in Bottle project DA. • Learning the amplitude-position partition function. • The joint amplitude-position Kalman filter. • Large-scale experiments.