Section: Scientific Foundations
Data Assimilation and Inverse Methods
Despite their permanent improvement, models are always characterized by an imperfect physics and some poorly known parameters (e.g. initial and boundary conditions). This is why it is important to also have observations of natural systems. However, observations provide only a partial (and sometimes very indirect) view of reality, localized in time and space.
Since models and observations taken separately do not allow for a deterministic reconstruction of real geophysical flows, it is necessary to use these heterogeneous but complementary sources of information simultaneously, by using data assimilation methods. These tools for inverse modelling are based on the mathematical theories of optimal control and stochastic filtering. Their aim is to identify system parameters which are poorly known in order to correct, in an optimal manner, the model trajectory, bringing it closer to the available observations.
Variational methods are based on the minimization of a function measuring the discrepancy between a model solution and observations, using optimal control techniques for this purpose. The model inputs are then used as control variables. The Euler Lagrange condition for optimality is satisfied by the solution of the “Optimality System" (OS) that contains the adjoint model obtained by derivation and transposition of the direct model. It is important to point out that this OS contains all the available information: model, data and statistics. The OS can therefore be considered as a generalized model. The adjoint model is a very powerful tool which can also be used for other applications, such as sensitivity studies.
Stochastic filtering is the basic tool in the sequential approach to the problem of data assimilation into numerical models, especially in meteorology and oceanography. The (unknown) initial state of the system can be conveniently modeled by a random vector, and the error of the dynamical model can be taken into account by introducing a random noise term. The goal of filtering is to obtain a good approximation of the conditional expectation of the system state (and of its error covariance matrix) given the observed data. These data appear as the realizations of a random process related to the system state and contaminated by an observation noise.
The development of data assimilation methods in the context of geophysical fluids, however, is difficult for several reasons:

the models are often strongly nonlinear, whereas the theories result in optimal solutions only in the context of linear systems;

the model error statistics are generally poorly known;

the size of the model state variable is often quite large, which requires dealing with huge covariance matrices and working with very large control spaces;

data assimilation methods generally increase the computational costs of the models by one or two orders of magnitude.
Such methods are now used operationally (after 15 years of research) in the main meteorological and oceanographic centers, but tremendous development is still needed to improve the quality of the identification, to reduce their cost, and to make them available for other types of applications.
A challenge of particular interest consists in developing methods for assimilating image data. Indeed, images and sequences of images represent a large amount of data which are currently underused in numerical forecast systems. For example, precursors of extreme meteorological events like thunderstorms, for which an early forecast is required, are visible on satellite images.
However, despite their huge informative potential, images are only used in a qualitative way by forecasters, mainly because of the lack of an appropriate methodological framework. In order to extend data assimilation techniques to image data we need to be able to:

identify and extract from images dynamics the relevant information (for instance structures) about the model state variables evolution;

link images dynamics with the underlying physical evolution processes;

define functional spaces for images which have good topological properties;

build observation operators which permit to map the model state variables space onto the aforementioned image space.
The use of images dynamics in numerical forecast systems is not restricted to meteorological or oceanographic applications: other scientific disciplines like hydrology (spatial observation of the main river bed during a flood), glaciology (radar exploration of polar ices, ice cover), medicine, etc. are interested in the development of such techniques.