Assimilation of Stratospheric Chemical Tracer Observations Using a Kalman Filter. Part I: Formulation

The first part of this two-part article describes the formulation of a Kalman filter system for assimilating limb-sounding observations of stratospheric chemical constituents into a tracer transport model. The system is based on a two-dimensional isentropic approximation, permitting a full Kalman filter implementation and a thorough study of its behavior in a real-data environment. Datasets from two instruments on the Upper Atmosphere Research Satellite with very different viewing geometries are used in the assimilation experiments. A robust chi-squared diagnostic, which compares statistics of the observed-minus-forecast residuals with those calculated by the filter algorithm, is used to help formulate the statistical inputs to the filter, as well as to tune covariance parameters and to validate the assimilation results. Two significant departures from the standard (discrete) Kalman filter formulation were found to be important in this study. First, it was discovered that the standard Kalman filter covariance propagation is highly inaccurate for this problem. Spurious and rapid loss of variance and increase of correlation length scales occur as a result of diffusion of the small-scale structures inherent in tracer error covariance fields. A new formulation based on well-understood properties of the continuum error covariance propagation was therefore introduced. Second, validation diagnostics suggested that the initial error, model error, and representativeness error are all more appropriately assumed to be relative than absolute in this problem. A filter formulation for relative errors was therefore devised. With these two modifications, this Kalman filter assimilation system has only three tunable variance parameters and one tunable correlation length-scale parameter. 1