Sai Ravela Massachusetts Institute of Technology J. Marshall, A. Wong, S. Stransky, C. Hill Collaborators: B. Kuszmaul and C. Leiserson

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

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

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

Numerical Simulation MIT-GCM (mitgcm.org): incompressible boussinesq fluid in non-hydrostatic mode with a vector-invariant formulation 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. Marshall et al., 1997

Domain 120x 23 x 15 (z) {45-8 }x 15cm 1.Cylindrical coordinates. 2.Nonuniform discretization of the vertical 3.Random temperature IC 4.Static temperature BC 5.Noslip boundaries 6.Heat-flux controlled with anisotropic thermal diffusivity

Estimate what? Estimation from model and data 1.State Estimation: 1.NWP type applications, but also reanalysis 2.Filtering & Smoothing 2.Parameter Estimation: 1.Forecasting & Climate 3.State and Parameter Estimation 1.The real problem. General Approach: Ensemble-based, multiscale methods.

Schedule

Producing state estimates Ravela, Marshall, Hill, Wong and Stransky, 07 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 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?

Approach Ravela, Marshall, Hill, Wong and Stransky, 07 P(T ): Thermal BC Perturbations 4 P(X0|T): IC Perturbation1 P(Xt|Xt-1): Snapshots in time 10 E>e 0 ? P(Yt|Xt) P(Xt|Xt-1): Ensemble update P(Yt|Xt) P(Xt|Xt-1): Deterministic update BC+IC Deterministic update: 5 – 2D updates 5 – (Elliptic) temperature Nx * Ny – 1D problems Snapshots capture flow-dependent uncertainty (Sirovich)

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

Ravela and McLaughlin, 2007

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.

Ravela et al. 2003, 2004, 2005, 2006, 2007 With thanks to K. Emanuel, D. McLaughlin and W. T. Freeman

Thunderstorms Hurricanes 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.

Amplitude assimilation of position errors is nonsense! 3DVAR

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

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

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

Bend, then blend

Improved control of solution

Flexible Application Students Ryan Abernathy: Scott Stransky Classroom  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 Bend, then Blend

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.

Adaptation to multivariate fields

Velocimetry, for Rainfall Modeling Ravela & Chatdarong, 06 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.

Other applications Magnetometry Alignment (Shell)

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

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.