## Abstract

An offline approach is proposed for the estimation of model and data error covariance matrices whereby covariance matrices of model data residuals are “matched” to their theoretical expectations using familiar least-squares methods. This covariance matching approach is both a powerful diagnostic tool for addressing theoretical questions and an efficient estimator for real data assimilation studies.

Provided that model and data errors are independent, that error propagation is approximately linear, and that an observability condition is met, it is in theory possible to fully resolve covariance matrices for both model and data errors. In practice, however, due to large uncertainties in sample estimates of covariance matrices, the number of statistically significant parameters that can be estimated is two to three orders of magnitude smaller than the total number of independent observations.

The covariance matching approach is applied in the North Pacific (5°–60°N, 132°–252°E) to TOPEX/Poseidon sea level anomaly data, acoustic tomography data from the Acoustic Thermometry of Ocean Climate Project, and a GCM. A reduced state linear model that describes large-scale internal (baroclinic) error dynamics is constructed. Twin experiments suggest that altimetric data are ill suited to estimating the statistics of the vertical GCM error structure, but that such estimates can in theory be obtained using acoustic data.

The particular GCM integration exhibits a warming trend relative to TOPEX/Poseidon data of order 1 cm yr^{−1} corresponding to a peak warming of up to 0.2°C yr^{−1} in the acoustic data at depths ranging from 50 to 200 m. At the annual cycle, GCM and TOPEX/Poseidon sea level anomaly are in phase, but GCM amplitude is 2 cm smaller, with the error confined above 200-m depth. After removal of trends and annual cycles, the low-frequency/wavenumber (periods >2 months, wavelengths >16°) TOPEX/Poseidon sea level anomaly is order 6 cm^{2}. The GCM explains about 40% of that variance. By covariance matching, it is estimated that 60% of the GCM–TOPEX/Poseidon residual variance is consistent with the reduced state linear model.

* Current affiliation: Jet Propulsion Laboratory, Pasadena, California.

*Corresponding author address:* Dimitris Menemenlis, Jet Propulsion Laboratory, Mail Stop 300/323, 4800 Oak Dr., Pasadena, CA 91109.

Email: dimitri@pacific.jpl.nasa.gov