DISCRETIZATION ERROR AND SIGNAL ERROR CORRELATION IN ATMOSPHERIC DATAASSIMILATION .1. ALL SCALES RESOLVED/

Both the numerical models used in atmospheric data assimilation and th e forward interpolation from the analysis mesh to the observations are subject to discretization errors. To examine the effect of these erro rs, a generalized Kalman filter, in which both model and observation e rrors are functions of the signal, is formulated. Far from the red mod el error spectrum assumed in many studies, the formulation yields a mo del error spectrum which increases with wavenumber to reach a maximum at the truncation limit. The resulting second-moment equations are stu died in the context of the one-dimensional linear advection equation a nd, even for this simple equation, found to be quite complex. For exam ple, it is found that signal/error correlations, not normally consider ed in standard Kalman filter theory, can play an important role. Two t ypes of (semi-Lagrangian) model discretization and two types of forwar d interpolation are examined in this paper. The first type uses Fourie r interpolation (and has no error), while the second type uses cubic s pline interpolation (and has amplitude and phase errors). To facilitat e understanding of the general case, various simpler cases are conside red first, e.g., the case of a uniform observation network with the sa me number of observations as analysis meshpoints reveals important ana logies between the forward interpolation error and the model discretiz ation error. It is found that the perfect-model assumption can result in degeneracy for any observation network when the signal/error correl ation is properly accounted for. The case of a single observation (coi nciding with an analysis gridpoint) strikingly illustrates the importa nce of the signal/error correlations and suggests that simple model er ror parametrization based purely on the model discretization error, an d neglecting these correlations, would seriously underestimate the for ecast and analysis errors. In this paper, it is assumed that the analy sis mesh can resolve all scales in the signal. The effect of unresolve d scales is considered in a companion paper.