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

Abstract

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.

Applications of information theory in ensemble data assimilation

“…The values of the parameter α are tuned to the ensemble size to approximately satisfy the expected chi-square innovation statistic, calculated for optimized innovations and normalized by the analysis error (e.g. Dee, 1995;Ménard et al, 2000;Zupanski, 2005). Instrument errors and the values of the parameter α used in data assimilation experiments of this study are listed in Table I.…”

“…The localization was also found beneficial in the full-rank KF filter applications due to spurious loss of variance in the discrete KF covariance evolution equation (e.g. Ménard et al, 2000). Since covariance localizations are typically achieved by employing arbitrary covariance functions (e.g.…”

A long‐term stratospheric ozone data set from assimilation of satellite observations: High‐latitude ozone anomalies

“…For a more detailed mathematical treatment, the reader is referred to the cited references and, e.g., Ménard et al [2000].…”

Data assimilation in atmospheric chemistry models: current status and future prospects for coupled chemistry meteorology models

“…Paradoxically, they have slowly percolated in air quality data assimilation, where they should be crucial given the uncertainty in most forcings or the sparsity of observations for in situ concentration measurements. The error covariance matrices can be parameterized with a restricted set of hyper-parameters, and those hyperparameters can be estimated through maximum-likelihood or L-curve tests (Ménard et al, 2000;Davoine and Bocquet, 2007;Elbern et al, 2007). Alternatively, with sufficient data, the whole structure of the error covariance matrices in the observation space can be diagnosed using consistency matrix identities; see for example Schwinger and Elbern (2010), who applied the approach of Desroziers et al (2005) to a stratospheric chemistry 4D-Var system.…”