The diffusion kernel filter applied to Lagrangian data assimilation

The diffusion kernel filter is a sequential particle-method approach to data assimilation of time series data and evolutionary models. The method is applicable to nonlinear/non-Gaussian problems. Within branches of prediction it parameterizes small fluctuations of Brownian-driven paths about deterministic paths. Its implementation is relatively straightforward, provided a tangent linear model is available. A by-product of the parameterization is a bound on the infinity norm of the covariance matrix of such fluctuations (divided by the grid model dimension). As such it can be used to define a notion of "prediction" itself. It can also be used to assess the short time sensitivity of the deterministic history to Brownian noise or Gaussian initial perturbations. In pure oceanic Lagrangian data assimilation, the dynamics and the statistics are nonlinear and non-Gaussian, respectively. Both of these characteristics challenge conventional methods, such as the extended Kalman filter and the popular ensemble Kalman filter. The diffusion kernel filter is proposed as an alternative and is evaluated here on a problem that is often used as a test bed for Lagrangian data assimilation: it consists of tracking point vortices and passive drifters, using a dynamical model and data, both of which have known error statistics. It is found that the diffusion kernel filter captures the first few moments of the random dynamics, with a computational cost that is competitive with a particle filter estimation strategy. The authors also introduce a clustered version of the diffusion kernel filter (cDKF), which is shown to be significantly more efficient with regard to computational cost, at the expense of a slight degradation in the description of the statistics of the dynamical history. Upon parallelizing branches of prediction, cDKF can be computationally competitive with EKF.