NCAR PROJECTS

1. Forecasting and Data Assimilation. Collaborators: Doug Nychka (GSP) and Chris Snyder (UCAR/MMM).

Several aspects of numerical weather prediction (NWP) make forecasting and data assimilation particularly challenging:  very high-dimensional systems, strongly non-linear (possibly chaotic) dynamics, and real-time requirements for assimilating data and physical models.   In practice one must address multi-modal forecast distributions, specify spatial covariance structures, use severely rank-deficient matrices, devise sampling schemes, and understand the properties of the sample based filtering algorithms.  In this project we use the ensemble Kalman eilter (ensKf) as a tool to track atmospheric states and make numerical weather predictions.  The procedure involves a data assimilation step based on the Kalman filter.
    One important goal of the project is to make statisticians aware of a rich area for new research on spatio-temporal methods that is distinct from non-linear filtering problems addressing simpler, low-dimensional systems.  To view the physics involved in NWP, run the movie of the Lorenz attractor.   Although a theoretical system, the Lorenz (Lorenz, E.  1963: Deterministic non-periodic flow.  J. Atmospheric sciences. 20, 130-141) attractor describes the convective flow of a fluid heated from below in a gravitational field. A more realistic system is given by Lorenz's 40-variable model, designed to mimick atmospheric flow on a latitude circle.  The movie of the 40-variable system shows a 10 member ensemble derived using the ensKf.

Publications:

Proceedings METMA workshop

Toward a nonlinear ensemble filter for high dimensional systems        Journal of Geophysical research, 108(D24), 8775

2.    Estimating Upper Ocean Current Fields. Collaborators: Ralph Milliff (Colorado Research Associates), Peter Niiler, (Scripps Institution of             Oceanography, CA), and  Richard Jones (GSP, University of Colorado-Denver).                                                                 

The dynamics of the upper ocean currents in the Labrador Sea have implications for understanding the massive air-sea exchanges that drive the thermohaline circulation.  To this end we estimate basic spatial and temporal correlation scales for the Labrador Sea based on several months of telemetry data from 21 Minimet drifting buoys.  A state-space approach is used where the observed buoy's position is related to an underlying state equation based on the ocean current and surface wind vector.  Parameter estimates in the state equation are consistent with the damping and the inertial time scales of the ocean and spatial scales of the Labrador Sea basin.  The figures below show buoy tracks over the period October 1996 to January 1997 used to estimate the ocean current dynamics and the coupling to surface winds.  Note the so-called inertial loops in the buoy path trajectories, indicative of a dynamical balance between surface wind forcing, surface friction, and the Coriolis acceleration.

Publications: 

A state-space model for ocean drifter motions dominated by inertial oscillations (submitted)

 

 

 

3. The effects of sampling variability in the Ensemble Kalman filter. Collaborators: Reinhard Furrer (GSP).

Many Monte-Carlo based NWP updating algorithms make use of large scale covariance structures and pose difficult questions in random matrix theory. One particular challenge of such techniques is the fact the available sample size is typically a few orders of magnitude less than the system dimension (e.g., sample size  = O(10^3) << O(10^6) = system dimension), producing highly rank-deficient sample matrices. Moreover, non-linear transforms of such sample matrices are commonly needed. This project aims to understand the effects of employing sample based versions of the Kalman filter in (extremely) high-dimensional systems. Our results provide theoretical justification for many commonly applied ad-hoc NWP-procedures devised to address small sample effects, and further delineate accuracy among competing updating algorithms.  We propose a shrinkage-type estimator for the error covariance based on the implicit spatial structure of the involved physical systems and measurement networks. In particular, they establish analytical results relating the eigenvalue structure of forecast error covariance matrices, employed sample size, and the mean squared error of sample based posterior mean estimates.

Estimation of Prior and Post. Cov. Matrices in Kalman Filter Variants (submitted)