Dynamical Properties of Error Statistics in a Shallow-Water Model

Abstract

Numerical ocean diagnoses and predictions rely on two types of information:model information and data information. Sequential estimation theoryshows that the most probable state is a linear combination of thetwo, weighted according to their error statistics. A Kalman filtertechnique is applied to a one-layer reduced-gravity linear oceanmodel in a rectangular midlatitude basin. The model reproduces themain features of the subtropical wind-driven gyre; the filter isused to study the dynamical behavior of the error statistics. Ona midlatitude f plane, the error-correlation patterns among the statevariables revealed by the Kalman filter are isotropic and homogeneousand satisfy a geostrophic relation. Introducing the beta effect breaksthe isotropy and homogeneity of the correlations, inducing behaviorthat is in agreement with two observational facts: 1) the latitudinaldependence of horizontal correlations and 2) the elliptic correlationshape of the mass field, elongated along the southwest-northeastorientation in the Northern Hemisphere. When a meridional line ofobservations is assimilated intermittently, the correlation patternsare dynamically adjusted to be wider to the east of the observingline than to the west. This is due to the westward propagation oferrors by the model's Rossby wave dynamics. The influence functionof observations, based on the gain matrix of the Kalman filter, issubjected to polar decomposition into an amplitude part and a vectornormalized by the amplitude-that is, a solid angle. The amplitudepart contains the current observational information and determinesthe absolute weight given to an observation. The angular part isrelated to the previous observations only and reflects the structureof relative weights, whose behavior is similar to that of error correlations.A criterion measuring the relative importance of different typesof observations is defined, using Kalman filter techniques and geostrophic-errorassumptions. The results from numerical experiments to examine thecorrectness of this criterion resolve apparent contradictions amongthe recent results of R. Daley, M. Ghil, and N. A. Phillips.