+..
+ Copyright (C) 2008-2015 EDF R&D
+
+ This file is part of SALOME ADAO module.
+
+ This library is free software; you can redistribute it and/or
+ modify it under the terms of the GNU Lesser General Public
+ License as published by the Free Software Foundation; either
+ version 2.1 of the License, or (at your option) any later version.
+
+ This library is distributed in the hope that it will be useful,
+ but WITHOUT ANY WARRANTY; without even the implied warranty of
+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+ Lesser General Public License for more details.
+
+ You should have received a copy of the GNU Lesser General Public
+ License along with this library; if not, write to the Free Software
+ Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+
+ See http://www.salome-platform.org/ or email : webmaster.salome@opencascade.com
+
+ Author: Jean-Philippe Argaud, jean-philippe.argaud@edf.fr, EDF R&D
+
.. _section_theory:
-================================================================================
-A brief introduction to Data Assimilation and Optimization
-================================================================================
+=================================================================================
+**[DocT]** A brief introduction to Data Assimilation and Optimization
+=================================================================================
.. index:: single: Data Assimilation
.. index:: single: true state
.. index:: single: fields reconstruction
.. index:: single: measures interpolation
+.. index:: single: fields interpolation
-Fields reconstruction consists in finding, from a restricted set of real
-measures, the physical field which is the most *consistent* with these measures.
+**Fields reconstruction (or interpolation)** consists in finding, from a
+restricted set of real measures, the physical field which is the most
+*consistent* with these measures.
-This consistency is to understand in terms of interpolation, that is to say that
-the field we want to reconstruct, using data assimilation on measures, has to
-fit at best the measures, while remaining constrained by the overall field
+This *consistency* is to understand in terms of interpolation, that is to say
+that the field we want to reconstruct, using data assimilation on measures, has
+to fit at best the measures, while remaining constrained by the overall field
calculation. The calculation is thus an *a priori* estimation of the field that
we seek to identify.
---------------------------------------------------------
.. index:: single: parameters identification
+.. index:: single: parameters adjustment
.. index:: single: models adjustment
.. index:: single: calibration
.. index:: single: background
.. index:: single: regularization
+.. index:: single: inverse problems
-The identification of parameters by data assimilation is a form of state
-calibration which uses both the physical measurement and an *a priori*
-parameters estimation (called the "*background*") of the state that one seeks to
-identify, as well as a characterization of their errors. From this point of
-view, it uses all available information on the physical system (even if
-assumptions about errors are relatively restrictive) to find the "*optimal
+The **identification (or adjustment) of parameters** by data assimilation is a
+form of state calibration which uses both the physical measurement and an *a
+priori* parameters estimation (called the "*background*") of the state that one
+seeks to identify, as well as a characterization of their errors. From this
+point of view, it uses all available information on the physical system, with
+restrictive yet realistic assumptions about errors, to find the "*optimal
estimation*" from the true state. We note, in terms of optimization, that the
-background realizes a mathematical regularization of the main problem of
-parameters identification.
+background realizes a "*regularization*", in the mathematical meaning of
+Tikhonov [[Tikhonov77]_ [WikipediaTI]_, of the main problem of parameters
+identification. One can also use the term "*inverse problem*" to refer to this
+process.
-In practice, the two observed gaps "*calculation-background*" and
-"*calculation-measures*" are combined to build the calibration correction of
+In practice, the two observed gaps "*calculation-measures*" and
+"*calculation-background*" are combined to build the calibration correction of
parameters or initial conditions. The addition of these two gaps requires a
relative weight, which is chosen to reflect the trust we give to each piece of
information. This confidence is depicted by the covariance of the errors on the
.. index:: single: background error covariances
.. index:: single: observation error covariances
.. index:: single: covariances
+.. index:: single: 3DVAR
+.. index:: single: Blue
We can write these features in a simple manner. By default, all variables are
-vectors, as there are several parameters to readjust.
+vectors, as there are several parameters to readjust, or a discrete field to
+reconstruct.
According to standard notations in data assimilation, we note
:math:`\mathbf{x}^a` the optimal parameters that is to be determined by
In the simplest case, which is static, the steps of simulation and of
observation can be combined into a single observation operator noted :math:`H`
(linear or nonlinear). It transforms the input parameters :math:`\mathbf{x}` to
-results :math:`\mathbf{y}` to be directly compared to observations
-:math:`\mathbf{y}^o`. Moreover, we use the linearized operator
-:math:`\mathbf{H}` to represent the effect of the full operator :math:`H` around
-a linearization point (and we omit thereafter to mention :math:`H` even if it is
-possible to keep it). In reality, we have already indicated that the stochastic
-nature of variables is essential, coming from the fact that model, background
-and observations are all incorrect. We therefore introduce errors of
-observations additively, in the form of a random vector
-:math:`\mathbf{\epsilon}^o` such that:
+results :math:`\mathbf{y}`, to be directly compared to observations
+:math:`\mathbf{y}^o`:
+
+.. math:: \mathbf{y} = H(\mathbf{x})
+
+Moreover, we use the linearized operator :math:`\mathbf{H}` to represent the
+effect of the full operator :math:`H` around a linearization point (and we omit
+thereafter to mention :math:`H` even if it is possible to keep it). In reality,
+we have already indicated that the stochastic nature of variables is essential,
+coming from the fact that model, background and observations are all incorrect.
+We therefore introduce errors of observations additively, in the form of a
+random vector :math:`\mathbf{\epsilon}^o` such that:
.. math:: \mathbf{y}^o = \mathbf{H} \mathbf{x}^t + \mathbf{\epsilon}^o
The errors represented here are not only those from observation, but also from
-the simulation. We can always consider that these errors are of zero mean. We
-can then define a matrix :math:`\mathbf{R}` of the observation error covariances
-by:
+the simulation. We can always consider that these errors are of zero mean.
+Noting :math:`E[.]` the classical mathematical expectation, we can then define a
+matrix :math:`\mathbf{R}` of the observation error covariances by the
+expression:
.. math:: \mathbf{R} = E[\mathbf{\epsilon}^o.{\mathbf{\epsilon}^o}^T]
-The background can also be written as a function of the true value, by
-introducing the error vector :math:`\mathbf{\epsilon}^b` such that:
+The background can also be written formally as a function of the true value, by
+introducing the errors vector :math:`\mathbf{\epsilon}^b` such that:
.. math:: \mathbf{x}^b = \mathbf{x}^t + \mathbf{\epsilon}^b
-where errors are also assumed of zero mean, in the same manner as for
-observations. We define the :math:`\mathbf{B}` matrix of background error
-covariances by:
+The errors :math:`\mathbf{\epsilon}^b` are also assumed of zero mean, in the
+same manner as for observations. We define the :math:`\mathbf{B}` matrix of
+background error covariances by:
.. math:: \mathbf{B} = E[\mathbf{\epsilon}^b.{\mathbf{\epsilon}^b}^T]
The optimal estimation of the true parameters :math:`\mathbf{x}^t`, given the
background :math:`\mathbf{x}^b` and the observations :math:`\mathbf{y}^o`, is
then the "*analysis*" :math:`\mathbf{x}^a` and comes from the minimisation of an
-error function (in variational assimilation) or from the filtering correction (in
-assimilation by filtering).
+error function, explicit in variational assimilation, or from the filtering
+correction in assimilation by filtering.
In **variational assimilation**, in a static case, one classically attempts to
minimize the following function :math:`J`:
-.. math:: J(\mathbf{x})=(\mathbf{x}-\mathbf{x}^b)^T.\mathbf{B}^{-1}.(\mathbf{x}-\mathbf{x}^b)+(\mathbf{y}^o-\mathbf{H}.\mathbf{x})^T.\mathbf{R}^{-1}.(\mathbf{y}^o-\mathbf{H}.\mathbf{x})
-
-which is usually designed as the "*3D-VAR*" function. Since :math:`\mathbf{B}`
-and :math:`\mathbf{R}` covariance matrices are proportional to the variances of
-errors, their presence in both terms of the function :math:`J` can effectively
-weight the differences by confidence in the background or observations errors.
-The parameters vector :math:`\mathbf{x}` realizing the minimum of this function
-therefore constitute the analysis :math:`\mathbf{x}^a`. It is at this level that
-we have to use the full panoply of function minimization methods otherwise known
-in optimization (see also section `Going further in the state estimation by
-optimization methods`_). Depending on the size of the parameters vector
-:math:`\mathbf{x}` to identify, and of the availability of gradient or Hessian
-of :math:`J`, it is appropriate to adapt the chosen optimization method
-(gradient, Newton, quasi-Newton...).
+.. math:: J(\mathbf{x})=\frac{1}{2}(\mathbf{x}-\mathbf{x}^b)^T.\mathbf{B}^{-1}.(\mathbf{x}-\mathbf{x}^b)+\frac{1}{2}(\mathbf{y}^o-\mathbf{H}.\mathbf{x})^T.\mathbf{R}^{-1}.(\mathbf{y}^o-\mathbf{H}.\mathbf{x})
+
+:math:`J` is classically designed as the "*3D-VAR*" functional in data
+assimlation (see for example [Talagrand97]_) or as the generalized Tikhonov
+regularization functional in optimization (see for example [WikipediaTI]_).
+Since :math:`\mathbf{B}` and :math:`\mathbf{R}` covariance matrices are
+proportional to the variances of errors, their presence in both terms of the
+function :math:`J` can effectively weight the gap terms by the confidence in the
+background or observations errors. The parameters vector :math:`\mathbf{x}`
+realizing the minimum of this function therefore constitute the analysis
+:math:`\mathbf{x}^a`. It is at this level that we have to use the full panoply
+of function minimization methods otherwise known in optimization (see also
+section `Going further in the state estimation by optimization methods`_).
+Depending on the size of the parameters vector :math:`\mathbf{x}` to identify,
+and of the availability of gradient or Hessian of :math:`J`, it is appropriate
+to adapt the chosen optimization method (gradient, Newton, quasi-Newton...).
In **assimilation by filtering**, in this simple case usually referred to as
"*BLUE*" (for "*Best Linear Unbiased Estimator*"), the :math:`\mathbf{x}^a`
The advantage of filtering is to explicitly calculate the gain, to produce then
the *a posteriori* covariance analysis matrix.
-In this simple static case, we can show, under the assumption of Gaussian error
-distributions, that the two *variational* and *filtering* approaches are
-equivalent.
+In this simple static case, we can show, under an assumption of Gaussian error
+distributions (very little restrictive in practice), that the two *variational*
+and *filtering* approaches give the same solution.
It is indicated here that these methods of "*3D-VAR*" and "*BLUE*" may be
extended to dynamic problems, called respectively "*4D-VAR*" and "*Kalman
.. index:: single: data smoothing
To get more information about the data assimilation techniques, the reader can
-consult introductory documents like [Argaud09]_, on-line training courses or
-lectures like [Bouttier99]_ and [Bocquet04]_ (along with other materials coming
-from geosciences applications), or general documents like [Talagrand97]_,
-[Tarantola87]_, [Kalnay03]_, [Ide97]_ and [WikipediaDA]_.
+consult introductory documents like [Talagrand97]_ or [Argaud09]_, on-line
+training courses or lectures like [Bouttier99]_ and [Bocquet04]_ (along with
+other materials coming from geosciences applications), or general documents like
+[Talagrand97]_, [Tarantola87]_, [Kalnay03]_, [Ide97]_, [Tikhonov77]_ and
+[WikipediaDA]_.
Note that data assimilation is not restricted to meteorology or geo-sciences,
but is widely used in other scientific domains. There are several fields in
minimization* extented form, obtained by adding a regularizing term using
:math:`\mathbf{x}-\mathbf{x}^b`, and by weighting the differences using
:math:`\mathbf{B}` and :math:`\mathbf{R}` the two covariance matrices. The
-minimization of the :math:`J` function leads to the *best* `\mathbf{x}` state
-estimation.
+minimization of the :math:`J` function leads to the *best* :math:`\mathbf{x}`
+state estimation. To get more information about these notions, one can consult
+reference general documents like [Tarantola87]_.
State estimation possibilities extension, by using more explicitly optimization
methods and their properties, can be imagined in two ways.
error measures are not differentiables, but some optimization methods can deal
with: heuristics and meta-heuristics for real-valued problem, etc. As
previously, the main drawback remain a greater numerical cost to find state
-estimates, and no guarantee of convergence in finite time. Here, we point also
-the following methods as it is available in the ADAO module: *Particle swarm
-optimization* [WikipediaPSO]_.
+estimates, and no guarantee of convergence in finite time. Here again, we only
+point the following methods as it is available in the ADAO module: *Particle
+swarm optimization* [WikipediaPSO]_.
The reader interested in the subject of optimization can look at [WikipediaMO]_
as a general entry point.
