Localized ensemble-based tangent linear models and their use in propagating hybrid error covariance models

Hybrid error covariance models that blend climatological estimates of forecast error covariances with ensemble-based, flow-dependent forecast error covariances have led to significant reductions in forecast error when employed in 4DVAR data assimilation schemes. Tangent linear models (TLMs) designed to predict the differences between perturbed and unperturbed simulations of the weather forecast are a key component of such 4DVAR schemes. However, many forecasting centers have found that TLMs and their adjoints do not scale well computationally and are difficult to create and maintain-particularly for coupled ocean-wave-ice-atmosphere models. In this paper, the authors create ensemble-based TLMs (ETLMs) and test their ability to propagate both climatological and flow-dependent parts of hybrid error covariance models. These tests demonstrate that rank deficiency limits the utility of unlocalized ETLMs. High-rank, time-evolving, flow-adaptive localization functions are constructed and tested using recursive application of short-duration ETLMs, each of which is localized using a static localization. Since TLM operators do not need to be semipositive definite, the authors experiment with a variety of localization approaches including step function localization. The step function localization leads to a local formulation that was found to be highly effective. In tests using simple one-dimensional models with both dispersive and nondispersive dynamics, it is shown that practical ETLM configurations were effective at propagating covariances as far as four error correlation scales.