## Data asimilation in highly nonlinear systems

### Scientists

- Dr. Ross Bannister (NCEO): State covariances and model error covariances
- Dr. Alison Fowler (NCEO): Observation covariances and representation errors
- Dr.Javier Amezcua (NCEO): Nonlinear data assimilation and itarative ensemble smoothers
- Dr. Gernot Gappert (NCEO): Data assimilation frame work (EMPIRE and PDAF and Pyanda)
- Flavia Rodrigues Pinhero: Synchronisation and nonlinear data assimilation
- David Sursham (NCEO PhD student, co-supervised with Stefano Ciavatta and Torres): Nonlinear data assimilation for marine biogeochemistry models
- Oana Lang (PhD student, co-supervised with Dan Crisan and Roland Potthast): Stochatic nonlinear high-dimensional data assimilation
- Lea Oljaca (PhD student, co-supervised with Jochem Broecker and Tobias Kuna): Evaluating data assimilation algorithms
- Anne Walter (PhD student, co-supervised with Roland Potthast): Nonlinear data-assimilation for operational NWP
- Dr Mineto Satoh (visitor from NEC, Japan): State and parameter estimation in highly nonlinear high-dimensional systems
- Prof Peter Jan van Leeuwen (NCEO): Project coordinator, Nonlinear data assimilation, proposal-density particle filters and optimal transportation

Under construction !!!

The following Powerpoint presentation gives a short introduction to data assimilation: Intro to Data Assimilation

Typically, the standard data-assimilation methods used in the geosciences look for 'best estimates', e.g. the mean or the mode of the posterior probability density. The same tends to be true for so-called inverse problems.

However, present-day problems ask for nonlinear data assimilation in which mean and mode are not enough to describe the posterior probability density satisfactorily. A new paradigm is needed on data asimilation in the geosciences, and that paradigm is there, and already quite old.

At the left-hand side of this page you can find three applications of a new particle filter that does solve the full nonlinear data assimilation problem, while allowing for high-dimensional dynamical systems with large numbers of independent observations, as described in this paper , see also my publication list .

It is based on the following observations:

- Data-assimilation and inverse problems can be
brought back to Bayes theorem (which can be derived from maximum entropy principles).
The general idea is that your knowledge of the system at hand, represented by a probability density function,
is updated by observations of the system. The observations are drawn from another known probability density function.
Bayes theorem tells us that these two probability densities should be multiplied to find the probability density
that describes our updated information.
- SO THE SOLUTION TO THE DATA ASSIMILATION PROBLEM IS THIS POSTERIOR PROBABILITY DENSITY FUNCTION,
AND DATA ASSIMILATION IS A MULTIPLICATION PROBLEM, NOT AN INVERSE PROBLEM.
- Unfortunately, due to the efficiency of inverse methods for linear Gaussian data-assimilation problems
the notion that data assimilation is an inverse problem managed to keep hold of people's minds.
The full nonlinear problem, however, does let us realise that data assimilation is NOT an inverse problem.
- This is true even for parameter estimation. We just have to multiply our prior probability densiity function (pdf)
of the parameters with the pdf of the observations to obtain the updated pdf of the parameters. Really, this is all!!!

In the 'near' future nonlinear data-assimilation methods will be presented here, with emphasis on particle filtering (for some reason). A review on the latter for geoscience applications can be found here.