A lag‐1 smoother approach to system‐error estimation: sequential method

Starting from sequential data assimilation arguments, the present work shows how to use residual statistics from filtering and lag‐1 (6 h) smoothing to infer components of the system (model) error covariance matrix that project on to a dense observing network. The residuals relationships involving the system‐error covariance matrix are similar to those available to derive background, observation and analysis‐error covariance information from filter residual statistics. An illustration of the approach is given for two low‐dimensional dynamical systems: a linear damped harmonic oscillator and the nonlinear Lorenz system. The application examples consider the important case of evaluating the ability to estimate the model‐error covariance from residual time series obtained from suboptimal filters and smoothers that assume the model to be perfect. The examples show the residuals to contain the necessary information to allow for such estimation. The examples also illustrate the consequences of estimating covariances through time series of residuals (available in practice) instead of multiple realizations from Monte Carlo sampling. A recast of the sequential approach into variational approach language will appear in a companion article.