Implicit particle methods and their connection to weak and strong 4D-Var

The implicit particle filter is a Monte Carlo method for data assimilation. The idea is to guide the particles to remain in the high-probability regions of the posterior pdf. This is done in two steps. First, the high-probability regions are identified by a numerical minimization; second, samples within the high-probability regions are obtained by solving algebraic equations with a random right-hand-side. Specifically, the implicit particle filter finds the mode of the posterior pdf and then solves algebraic equations to obtain samples in the neighborhood of this mode. In variational data assimilation, one finds the mode of the posterior pdf. There is thus a connection between the implicit particle filter and variational data assimilation. In particular, one can turn variational codes into implicit particle filters by adding a sampling step (i.e. solving simple algebraic equations). The benefit can be that the implicit filter improves the state estimate by approximating the conditional mean, which is the minimum mean square error estimate. We present an example in detail to explain the implicit particle filter in its variational implementation.