Analysis-Error Statistics of a Quasigeostrophic Model Using Three-Dimensional Variational Assimilation

Thomas M. Hamill NOAA–CIRES Climate Diagnostics Center, Boulder, Colorado

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Chris Snyder National Center for Atmospheric Research, Boulder, Colorado

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Rebecca E. Morss National Center for Atmospheric Research, Boulder, Colorado

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Abstract

A perfect model Monte Carlo experiment was conducted to explore the characteristics of analysis error in a quasigeostrophic model. An ensemble of cycled analyses was created, with each member of the ensemble receiving different observations and starting from different forecast states. Observations were created by adding random error (consistent with observational error statistics) to vertical profiles extracted from truth run data. Assimilation of new observations was performed every 12 h using a three-dimensional variational analysis scheme. Three observation densities were examined, a low-density network (one observation ∼ every 202 grid points), a moderate-density network (one observation ∼ every 102 grid points), and a high-density network (∼ every 52 grid points). Error characteristics were diagnosed primarily from a subset of 16 analysis times taken every 10 days from a long time series, with the first sample taken after a 50-day spinup. The goal of this paper is to understand the spatial, temporal, and some dynamical characteristics of analysis errors.

Results suggest a nonlinear relationship between observational data density and analysis error; there was a much greater reduction in error from the low- to moderate-density networks than from moderate to high density. Errors in the analysis reflected both structured errors created by the chaotic dynamics as well as random observational errors. The correction of the background toward the observations reduced the error but also randomized the prior dynamical structure of the errors, though there was a dependence of error structure on observational data density. Generally, the more observations, the more homogeneous the errors were in time and space and the less the analysis errors projected onto the leading backward Lyapunov vectors. Analyses provided more information at higher wavenumbers as data density increased. Errors were largest in the upper troposphere and smallest in the mid- to lower troposphere. Relatively small ensembles were effective in capturing a large percentage of the analysis-error variance, though more members were needed to capture a specified fraction of the variance as observation density increased.

Corresponding author address: Dr. Thomas M. Hamill, NOAA–CIRES Climate Diagnostics Center, R/CDC 1, 325 Broadway, Boulder, CO 80305-3328. Email: hamill@cdc.noaa.gov

Abstract

A perfect model Monte Carlo experiment was conducted to explore the characteristics of analysis error in a quasigeostrophic model. An ensemble of cycled analyses was created, with each member of the ensemble receiving different observations and starting from different forecast states. Observations were created by adding random error (consistent with observational error statistics) to vertical profiles extracted from truth run data. Assimilation of new observations was performed every 12 h using a three-dimensional variational analysis scheme. Three observation densities were examined, a low-density network (one observation ∼ every 202 grid points), a moderate-density network (one observation ∼ every 102 grid points), and a high-density network (∼ every 52 grid points). Error characteristics were diagnosed primarily from a subset of 16 analysis times taken every 10 days from a long time series, with the first sample taken after a 50-day spinup. The goal of this paper is to understand the spatial, temporal, and some dynamical characteristics of analysis errors.

Results suggest a nonlinear relationship between observational data density and analysis error; there was a much greater reduction in error from the low- to moderate-density networks than from moderate to high density. Errors in the analysis reflected both structured errors created by the chaotic dynamics as well as random observational errors. The correction of the background toward the observations reduced the error but also randomized the prior dynamical structure of the errors, though there was a dependence of error structure on observational data density. Generally, the more observations, the more homogeneous the errors were in time and space and the less the analysis errors projected onto the leading backward Lyapunov vectors. Analyses provided more information at higher wavenumbers as data density increased. Errors were largest in the upper troposphere and smallest in the mid- to lower troposphere. Relatively small ensembles were effective in capturing a large percentage of the analysis-error variance, though more members were needed to capture a specified fraction of the variance as observation density increased.

Corresponding author address: Dr. Thomas M. Hamill, NOAA–CIRES Climate Diagnostics Center, R/CDC 1, 325 Broadway, Boulder, CO 80305-3328. Email: hamill@cdc.noaa.gov

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