We discuss implementation issues of the Singular Vector Kalman Filter (SVKF) and the Local Floquet vector Kalman Filter (LFKF) in a chaotic geophysical system. In these filters, the State Transition Matrix (STM) is projected on subspaces that have smaller dimensions than the state. In the SVKF the projection subspace is spanned by the leading singular vectors of the STM, computed by iterations of forward tangent linear and adjoint models of the system. In LFKF the projection subspace is spanned by the leading Floquet (Schur) vectors of the STM, computed by iterations of nonlinear model of the system. The previous experiments with the SVKF and the LFKF on a chaotic Lorenz 95 system have been promising for their applications in chaotic geophysical systems.

The model is a double-gyre reduced-gravity quasi-geostrophic ocean circulation model, defined on a square domain with no-slip boundary conditions. The system is idealized and forced by known wind stress. The dynamics become chaotic when the viscosity is small enough. Previous work on this model suggests that the ensemble Kalman filter with localization of the ensemble covariance can improve the analysis error to a level that is still much higher than the level of observation noise. Our results suggest that the SVKF and the LFKF can produce better estimates with analysis errors of the level of the observation noise. This is in agreement with our previous results from experiments with the chaotic Lorenz 95 system.

Experiments with various observation networks show that the analysis errors are sensitive to the choice of observation network. When the observation network is large enough and has good spatial distribution over the more turbulent areas, both the SVKF and the LFKF perform well. In contrary, choosing a poor observation network results in poor performance of the filters which otherwise would produce excellent estimates.

To further investigate the sensitivity of the results to the choice of the observation network, an observability metric is proposed. This metric quantifies the projection of retained singular vectors or Floquet vectors on the observations. Therefore, it measures how well the observations can constrain the growing singular vectors or locally unstable Floquet vectors. Our findings with various observation networks are consistent with this metric, suggesting that it can be helpful in designing an adaptive observation network.