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zW{g9!X^!iQs@6mlviw!(3MIaRaXYAe^-OC;#F;f`v9%Yv3mm=ohC{{EUNvePTgnmSsdRzT0~GeufBt+cFr!E_QKF2|^AWxspD418S%# z2}UB6zkYb=IHV+Z+P@5~_|>Wuez8L3wvAJS_mO54(yX$UucLP*_OTCY(L0byLwdXE z)f@xXG&Y@xh)L?%_}9ws7x)?2Y?VFJCVJw*$ntwoYit-Zk43RaPdbJ)P@&U;zTRax zSjRt^E}bLn=N5EcRErsb9#VdmX72!n8m8F8YC5+fXY!HG(SFDz)M$<%QO7~xP#0y1 gzaNwSdRbm!V9{>+)mL@Ehn0h#vOigK{M_~b3r-Zm4FCWD literal 0 HcmV?d00001 diff --git a/doc/en/ref_algorithm_3DVAR.rst b/doc/en/ref_algorithm_3DVAR.rst index bd905fe..9169cd2 100644 --- a/doc/en/ref_algorithm_3DVAR.rst +++ b/doc/en/ref_algorithm_3DVAR.rst @@ -47,6 +47,8 @@ robust formulations are proposed here: pair: Variant ; 3DVAR-VAN pair: Variant ; 3DVAR-Incr pair: Variant ; 3DVAR-PSAS + pair: Variant ; OI + pair: Variant ; Optimal Interpolation - "3DVAR" (3D Variational analysis, see [Lorenc86]_, [LeDimet86]_, [Talagrand97]_), original classical algorithm, extremely robust, which operates in the model space, - "3DVAR-VAN" (3D Variational Analysis with No inversion of B, see [Lorenc88]_), similar algorithm, which operates in the model space, avoiding inversion of the covariance matrix B (except in the case where there are bounds.), @@ -74,6 +76,13 @@ allows to improve the estimation of *a posteriori* error covariances. This extension is obtained by using the "E3DVAR" variant of the filtering algorithm :ref:`section_ref_algorithm_EnsembleKalmanFilter`. +Note that observation and evolution error statistics are assumed to be +Gaussian. So, in the particular case where the observation operator :math:`H` +is linear, this algorithm is strictly equivalent to optimal interpolation (OI). +What's more, it performs both minimum variance estimation (MV or "*Minimum +Variance* estimator") and maximum a posteriori estimation (MAP or "*Maximum A +Posteriori* estimator"), which coincide in this particular case. + .. ------------------------------------ .. .. include:: snippets/Header2Algo12.rst diff --git a/doc/en/theory.rst b/doc/en/theory.rst index 34dffa6..028fee8 100644 --- a/doc/en/theory.rst +++ b/doc/en/theory.rst @@ -810,6 +810,18 @@ follows, particularly appropriate for iterative Kalman filtering algorithms: with **P** the state error covariance and *t* the discrete iterative time. In this scheme, the analysis **(x,P)** is obtained by means of the "*correction*" -by observing the "*prediction*" of the previous state. The concepts described -in this diagram can be directly and simply used in ADAO to elaborate study -cases, and are included in the description and the examples of some algorithms. +by observing the "*prediction*" of the previous state. An another way to +understand data assimilation in dynamics, in the space of measured states and +measured observations, leads to the following representation of a sequential +form of data assimilation: + + .. _schema_d_AD_sequentiel: + .. figure:: images/schema_temporel_sequentiel.png + :align: center + :width: 100% + + **Sequential scheme of states and measures for data assimilation in dynamics** + +The concepts described in this diagram can be directly and simply used in ADAO +to understand and elaborate study cases, and are included in the description +and the examples of some algorithms. diff --git a/doc/fr/images/schema_temporel_sequentiel.png b/doc/fr/images/schema_temporel_sequentiel.png new file mode 100644 index 0000000000000000000000000000000000000000..1d402de5a97b996777f7f3dab51b823e154666f2 GIT binary patch literal 93191 zcmc$_3pmq#_&7XNbQnq^$Ig0ERB{fL;wd3HOfe%x%$YgcbaHr{o+k-8g@nzFoX1!Z z6jyT6~$=RSY#&wbxwubCNxx5;b+ 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extrêmement robuste, opérant dans l'espace du modèle, - "3DVAR-VAN" (3D Variational Analysis with No inversion of B, voir [Lorenc88]_), algorithme similaire, opérant dans l'espace du modèle, permettant d'éviter l'inversion de la matrice de covariance B (sauf dans le cas où il y des bornes), @@ -80,6 +82,15 @@ ces covariances d'erreurs *a posteriori*. On atteint cette extension en utilisant le variant "E3DVAR" de l'algorithme de filtrage :ref:`section_ref_algorithm_EnsembleKalmanFilter`. +On remarque que les statistiques d'erreurs d'observation et d'évolution sont +par hypothèse représentées ici sous forme gaussienne. Alors, dans le cas +particulier où l'opérateur d'observation :math:`H` est linéaire, cet algorithme +est strictement équivalent à l'interpolation optimale (OI ou "*Optimal +Interpolation*"). De plus, il réalise à la fois l'estimation de variance +minimale (MV ou "*Minimum Variance* estimator") et l'estimation de maximum a +posteriori (MAP ou "*Maximum A Posteriori* estimator"), qui coïncident dans ce +cas précis. + .. ------------------------------------ .. .. include:: snippets/Header2Algo12.rst diff --git a/doc/fr/theory.rst b/doc/fr/theory.rst index 40bd280..8dc1bec 100644 --- a/doc/fr/theory.rst +++ b/doc/fr/theory.rst @@ -861,7 +861,18 @@ itératifs de filtrage de type Kalman : avec **P** la covariance d'erreur d'état et *t* le temps itératif discret. Dans ce schéma, l'analyse **(x,P)** est obtenue à travers la "*correction*" par -l'observation de la "*prévision*" de l'état précédent. Les concepts décrits -dans ce schéma peuvent directement et simplement être utilisés dans ADAO pour -construire des cas d'études, et sont repris dans la description et les exemples -de certains algorithmes. +l'observation de la "*prévision*" de l'état précédent. Une autre manière de +comprendre l'assimilation de données dynamique, dans l'espace des états mesurés +et observations mesurées, consiste à représenter la démarche séquentielle +d'assimilation sous la forme suivante : + + .. _schema_d_AD_sequentiel: + .. figure:: images/schema_temporel_sequentiel.png + :align: center + :width: 100% + + **Schéma séquentiel de l'état et des mesures pour l'assimilation de données en dynamique** + +Les concepts décrits dans ce schéma peuvent directement et simplement être +utilisés dans ADAO pour comprendre et construire des cas d'études, et sont +repris dans la description et les exemples de certains algorithmes. -- 2.39.2