Filter degeneracy is the main obstacle for the implementation of particle filters in nonlinear high-dimensional models. A new scheme, the implicit equal-weights particle filter (IEWPF), is introduced, in which samples are drawn implicitly from proposal densities with a different covariance for each particle, such that all particle weights are equal by construction.We test and explore the properties of the new scheme using a 1000 dimensional simple linear model and the 1000 dimensional nonlinear Lorenz96 model and compare the performance of the scheme with that of a local ensemble transformed Kalman filter (LETKF). The new scheme is never degenerate and shows good and consistent performance in all experiments. The LETKF has lower root-mean-square errors at observed grid points, but its ensemble spread is too low at unobserved grid points, where the IEWPF performs better. Furthermore, the IEWPF has a consistent spread in all experiments.This new filter opens up a new class of particle filters that, by construction, do not suffer from the curse of dimensionality.

Two recent works have adapted the Kalman-Bucy filter into an ensemble setting. In the first formulation, 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 Kalman filter. In the second formulation, 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 analyse the behaviour of the ODEs involved in these formulations. We demonstrate that they stiffen for large magnitudes of the ratio of background error to observational error variance, and that using the integration scheme proposed in both formulations can lead to failure. A numerical 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 known as SPEEDY. The results from all three models are encouraging and warrant further exploration of these assimilation techniques.

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