Combining Observations with Models: Penalized Likelihood and Related Methods in Numerical Weather Prediction

We will look at variational data assimilation as practiced by atmospheric scientists, with the eyes of a statistician. Recent operational numerical weather prediction models operate on what might be considered a very grand penalized likelihood point of view: A variational problem is set up and solved to obtain the evolving state of the atmosphere, given heterogenous observations in time and space, a numerical model embodying the nonlinear equations of motion of the atmosphere, and various physical constraints and prior physical and historical information. The idea is to obtain a sequence of state vectors which is ‘close’ to the observations, close to a trajectory satisfying the equations of motion, and simultaneously respects the other information available. The state vector may be as big as 107, and the observation vector 105 or 106, leading to some interesting implementation questions. Interesting non-standard statistical issues abound. 1 Outline 1. What is numerical weather prediction? 2. 3-D VAR, (Three Dimensional Variational Analysis) 4-D VAR (Four Dimensional Variational Analysis) 3. Wind, Divergence and Vorticity, Spherical Harmonics. 4. The NCEP (National Centers for Environmental Prediction), and ECMWF (European Center for Medium Range Weather Prediction) Global Scale NWP (Numerical Weather Weather Prediction Models. 5. Model variables, ζ,D, T, Ps, q. Analysis variables. ‘Balance’. 2 6. The analysis state vector. 7. The variational problem to be solved in 3D-VAR models. Weighting, smoothing and tuning parameters. 8. 4D-VAR Models 9. A toy experiment to examine the feasibility of tuning 4D-VAR models via Generalized Cross Validation. Today’s forecast A copy of the surface temperature and pressure forecast plot from a daily newspaper goes here.