  Two recent works have adapted the Kalman-Bucy filter into an ensemble
setting. In the first formulation, BR10, the full ensemble is updated in the
analysis step as the solution of single set of ODEs in pseudo-BGR09, the
ensemble of perturbations is updated by the solution of an ordinary
differential equation (ODE) in pseudo-time, while the mean is updated as in the
standard KF. In the second formulation, BR10, the full ensemble is updated in
the analysis step as the solution of single set of ODEs in pseudo-time. Neither
requires matrix inversions except for the frequently diagonal observation error
covariance.
  We analyze the behavior of the ODEs involved in these formulations. We
demonstrate that they stiffen for large magnitudes of the ratio of background
to observational error covariance, and that using the integration scheme
proposed in both BGR09 and BR10 can lead to failure. An integration scheme that
is both stable and is not computationally expensive is proposed. We develop
transform-based alternatives for these Bucy-type approaches so that the
integrations are computed in ensemble space where the variables are weights (of
dimension equal to the ensemble size) rather than model variables.
  Finally, the performance of our ensemble transform Kalman-Bucy
implementations is evaluated using three models: the 3-variable Lorenz 1963
model, the 40-variable Lorenz 1996 model, and a medium complexity atmospheric
general circulation model (AGCM) known as SPEEDY. The results from all three
models are encouraging and warrant further exploration of these assimilation
techniques.
