Waves, Information and Local Predictability IPAM Workshop Presentation By Joseph Tribbia NCAR

Waves, information and local predictability: Outline History Motivation Goals of targeted observing (Un)certainty prediction and flow Analysis of simple basic flows Conclusions and ramifications Some general problems for the future

Brief history of data assimilation NWP requires initial conditions Interpolation of observations (Panofsky,Cressman, Doos) Statistical interpolation (Gandin, Rutherford, Schlatter) Four-dimensional assimilation (Thompson, Charney, Peterson, Ghil, Talagrand)

4D method of assimilation

Recently: variant of Kalman filter

Motivation Lorenz and Emanuel (1998): invented the field of adaptive observing Suppose one wants to improve Thursday’s forecast in LA, where should one observe the atmosphere today?

Goals of Targeted Observing ‘Better’ forecast in a local domain-difficult to achieve because of random errors Reduced forecast uncertainty in domain-achievable Need a metric for increased reliability-relative entropy (G,S,M,K,DS,N,L)

Baumhefner experiments:

The wave perspective: models 1D Barotropic 1D Baroclinic 2D Spherical

Uncertainty propagation Compare two initial covariances One with uniform uncertainty, the other with locally smaller variance

How does relative certainty propagate? Simplest example: 1D Rossby wave context compare pulse (mean) propagation (group velocity) with (co)variance propagation pulse t=0 var t=o

Evolution after 10 days pulse at t=10d variance t-=10d

Unstable 1D Linear 2-level QG Pulse at t=10d Variance at t=10d

Add downstream U variation to 2-level model x variation of U Pulse at t=3d Variance at t=3d

Add downstream U variation to 2-level model Pulse at t=10d Variance at t=10d

Relative uncertainty: x-varying U pulse t=3d relative variance t=3d pulse t=10d relative variance t=10d

Barotropic vorticity equation with solid body rotation Relative variance at t=4d streamfunction Relative variance at t=20d streamfunction

Conclusions and ramifications Pulse perturbations and error variance differences propagate similarly if weighted properly Aspects of variance propagation ascribed to nonlinearity may be ‘weighted ‘ wave dispersion Group velocity gives a wave dynamic perspective to adaptive observing strategies

Future: nonlinear problem (Bayes)

Parameter estimation