Treatment of observation error due to unresolved scales in atmospheric data assimilation

Observations of the atmospheric state include scales of motion that are not resolved by numerical models into which the observed data are assimilated. The resulting observation error due to unresolved scales, part of the "representativeness error," is state dependent and correlated in time. A mathematical formalism and algorithmic approach has been developed for treating this error in the data assimilation process, under an assumption that there is no model error. The approach is based on approximating the continuum Kalman filter in such a way as to maintain terms that account for the observation error due to unresolved scales. The two resulting approximate filters resemble the Schmidt-Kalman filter and the traditional discrete Kalman filter. The approach is tested for the model problem of a passive tracer undergoing advection in a shear flow on the sphere. The state contains infinitely many spherical harmonics, with a nonstationary spectrum, and the problem is to estimate the projection of this state onto a finite spherical harmonic expansion, using observations of the full state. Numerical experiments demonstrate that approximate filters work well for the model problem provided that the exact covariance function of the unresolved scales is known. The traditional filter is more convenient in practice since it requires only the covariance matrix obtained by evaluating this covariance function at the observation points. A method for modeling this covariance matrix in the traditional filter is successful for the model problem.