Continuous Nonlinear Data Assimilation

An important goal for data assimilation is a posterior probability density that does not have large jumps at observation times (filters) or at the start of an assimilation window (smoothers). Another goal is to find efficient ways to represent the prior probability density of the state, for non-Gaussian, but also for Gaussian data assimilation (the so-called B matrix). A solution is a smoother with an ever-extending window, such that the prior is far back in the past. In that way the influence of the prior and of possible jumps is greatly reduced. Because observations in the far future will not influence the present state in any significant manner we can use fixed-lag smoothers that update the model trajectory over a finite time window. While every time window will have large overlap with its previous window helping convergence, the problem is still challenging.

By starting the window with the state that is insensitive to the new observation (because of the window length), that prior state does not have to be updated. In a weak-constraint formulation only the likelihood and the model error terms remain, even in the non-Gaussian case. The resulting solution will not have jumps because the prior pdf of the state at the start of the window is not part of the data assimilation problem anymore. This moves the problem to estimating model errors. There are several reasons why that is quite advantageous. First, the community can move away from trying to estimate prior model state covariances (B matrix), freeing up the work force for more interesting parts of the problem. Second, methods to estimate model errors are maturing, and this will be accelerated significantly by the increased work force dedicated to this problem. Third, model error estimation leads directly to model improvement. The model bias attribution problem that has plagued the community for decades disappears. We will discuss the theory and provide results from simplified models using a nonlinear Continuous Particle Flow Smoother, with emphasis on feasibility to real geophysical systems.