Discretization error and signal/error correlation in atmospheric data assimilation : (I). All scales resolved

Abstract

Both the numerical models used in atmospheric data assimilation and the forward interpolation from the analysis mesh to the observations are subject to discretization errors. To examine the effect of these errors, a generalized Kalman filter, in which both model and observation errors are functions of the signal, is formulated. Far from the red model error spectrum assumed in many studies, the formulation yields a model error spectrum which increases with wavenumber to reach a maximum at the truncation limit. The resulting second-moment equations are studied in the context of the one-dimensional linear advection equation and, even for this simple equation, found to be quite complex. For example, it is found that signal/error correlations, not normally considered in standard Kalman filter theory, can play an important rale. Two types of (semi-Lagrangian) model discretization and two types of forward interpolation are examined in this paper. The first type uses Fourier interpolation (and has no error), while the second type uses cubic spline interpolation (and has amplitude and phase errors). To facilitate understanding of the general case, various simpler cases are considered first, e.g., the case of a uniform observation network with the same number of observations as analysis meshpoints reveals important analogies between the forward interpolation error and the model discretization error. It is found that the perfect-model assumption can result in degeneracy for any observation network when the signal/error correlation is properly accounted for. The case of a single observation (coinciding with an analysis gridpoint) strikingly illustrates the importance of the signal/error correlations and suggests that simple model error parametrization based purely on the model discretization error, and neglecting these correlations, would seriously underestimate the forecast and analysis errors. In this paper, it is assumed that the analysis mesh can resolve all scales in the signal. The effect of unresolved scales is considered in a companion paper.