2 Copyright (C) 2008-2015 EDF R&D
4 This file is part of SALOME ADAO module.
6 This library is free software; you can redistribute it and/or
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14 Lesser General Public License for more details.
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18 Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
20 See http://www.salome-platform.org/ or email : webmaster.salome@opencascade.com
22 Author: Jean-Philippe Argaud, jean-philippe.argaud@edf.fr, EDF R&D
24 .. index:: single: Blue
25 .. _section_ref_algorithm_Blue:
27 Calculation algorithm "*Blue*"
28 ------------------------------
33 This algorithm realizes a BLUE (Best Linear Unbiased Estimator) type estimation
34 of the state of a system. More precisely, it is an Aitken estimator.
36 This algorithm is always the fastest of all the assimilation algorithms of ADAO.
37 It is theoretically reserved for observation operator cases which are linear,
38 even if it sometimes works in "slightly" non-linear cases. One can verify the
39 linearity of the observation operator with the help of the
40 :ref:`section_ref_algorithm_LinearityTest`.
42 In case of non-linearity, even slightly marked, it will be easily prefered the
43 :ref:`section_ref_algorithm_ExtendedBlue` or the
44 :ref:`section_ref_algorithm_3DVAR`.
46 Optional and required commands
47 ++++++++++++++++++++++++++++++
49 .. index:: single: AlgorithmParameters
50 .. index:: single: Background
51 .. index:: single: BackgroundError
52 .. index:: single: Observation
53 .. index:: single: ObservationError
54 .. index:: single: ObservationOperator
55 .. index:: single: StoreSupplementaryCalculations
56 .. index:: single: Quantiles
57 .. index:: single: SetSeed
58 .. index:: single: NumberOfSamplesForQuantiles
59 .. index:: single: SimulationForQuantiles
61 The general required commands, available in the editing user interface, are the
65 *Required command*. This indicates the background or initial vector used,
66 previously noted as :math:`\mathbf{x}^b`. Its value is defined as a
67 "*Vector*" or a *VectorSerie*" type object.
70 *Required command*. This indicates the background error covariance matrix,
71 previously noted as :math:`\mathbf{B}`. Its value is defined as a "*Matrix*"
72 type object, a "*ScalarSparseMatrix*" type object, or a
73 "*DiagonalSparseMatrix*" type object.
76 *Required command*. This indicates the observation vector used for data
77 assimilation or optimization, previously noted as :math:`\mathbf{y}^o`. It
78 is defined as a "*Vector*" or a *VectorSerie* type object.
81 *Required command*. This indicates the observation error covariance matrix,
82 previously noted as :math:`\mathbf{R}`. It is defined as a "*Matrix*" type
83 object, a "*ScalarSparseMatrix*" type object, or a "*DiagonalSparseMatrix*"
87 *Required command*. This indicates the observation operator, previously
88 noted :math:`H`, which transforms the input parameters :math:`\mathbf{x}` to
89 results :math:`\mathbf{y}` to be compared to observations
90 :math:`\mathbf{y}^o`. Its value is defined as a "*Function*" type object or
91 a "*Matrix*" type one. In the case of "*Function*" type, different
92 functional forms can be used, as described in the section
93 :ref:`section_ref_operator_requirements`. If there is some control :math:`U`
94 included in the observation, the operator has to be applied to a pair
97 The general optional commands, available in the editing user interface, are
98 indicated in :ref:`section_ref_assimilation_keywords`. Moreover, the parameters
99 of the command "*AlgorithmParameters*" allows to choose the specific options,
100 described hereafter, of the algorithm. See
101 :ref:`section_ref_options_Algorithm_Parameters` for the good use of this
104 The options of the algorithm are the following:
106 StoreSupplementaryCalculations
107 This list indicates the names of the supplementary variables that can be
108 available at the end of the algorithm. It involves potentially costly
109 calculations or memory consumptions. The default is a void list, none of
110 these variables being calculated and stored by default. The possible names
111 are in the following list: ["APosterioriCorrelations",
112 "APosterioriCovariance", "APosterioriStandardDeviations",
113 "APosterioriVariances", "BMA", "CostFunctionJ", "OMA", "OMB", "Innovation",
114 "SigmaBck2", "SigmaObs2", "MahalanobisConsistency",
115 "SimulatedObservationAtBackground", "SimulatedObservationAtOptimum",
116 "SimulationQuantiles"].
118 Example : ``{"StoreSupplementaryCalculations":["BMA","Innovation"]}``
121 This list indicates the values of quantile, between 0 and 1, to be estimated
122 by simulation around the optimal state. The sampling uses a multivariate
123 gaussian random sampling, directed by the *a posteriori* covariance matrix.
124 This option is useful only if the supplementary calculation
125 "SimulationQuantiles" has been chosen. The default is a void list.
127 Example : ``{"Quantiles":[0.1,0.9]}``
130 This key allow to give an integer in order to fix the seed of the random
131 generator used to generate the ensemble. A convenient value is for example
132 1000. By default, the seed is left uninitialized, and so use the default
133 initialization from the computer.
135 Example : ``{"SetSeed":1000}``
137 NumberOfSamplesForQuantiles
138 This key indicates the number of simulation to be done in order to estimate
139 the quantiles. This option is useful only if the supplementary calculation
140 "SimulationQuantiles" has been chosen. The default is 100, which is often
141 sufficient for correct estimation of common quantiles at 5%, 10%, 90% or
144 Example : ``{"NumberOfSamplesForQuantiles":100}``
146 SimulationForQuantiles
147 This key indicates the type of simulation, linear (with the tangent
148 observation operator applied to perturbation increments around the optimal
149 state) or non-linear (with standard observation operator applied to
150 perturbated states), one want to do for each perturbation. It changes mainly
151 the time of each elementary calculation, usually longer in non-linear than
152 in linear. This option is useful only if the supplementary calculation
153 "SimulationQuantiles" has been chosen. The default value is "Linear", and
154 the possible choices are "Linear" and "NonLinear".
156 Example : ``{"SimulationForQuantiles":"Linear"}``
158 Information and variables available at the end of the algorithm
159 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
161 At the output, after executing the algorithm, there are variables and
162 information originating from the calculation. The description of
163 :ref:`section_ref_output_variables` show the way to obtain them by the method
164 named ``get`` of the variable "*ADD*" of the post-processing. The input
165 variables, available to the user at the output in order to facilitate the
166 writing of post-processing procedures, are described in the
167 :ref:`subsection_r_o_v_Inventaire`.
169 The unconditional outputs of the algorithm are the following:
172 *List of vectors*. Each element is an optimal state :math:`\mathbf{x}*` in
173 optimization or an analysis :math:`\mathbf{x}^a` in data assimilation.
175 Example : ``Xa = ADD.get("Analysis")[-1]``
177 The conditional outputs of the algorithm are the following:
179 APosterioriCorrelations
180 *List of matrices*. Each element is an *a posteriori* error correlation
181 matrix of the optimal state.
183 Example : ``C = ADD.get("APosterioriCorrelations")[-1]``
185 APosterioriCovariance
186 *List of matrices*. Each element is an *a posteriori* error covariance
187 matrix :math:`\mathbf{A}*` of the optimal state.
189 Example : ``A = ADD.get("APosterioriCovariance")[-1]``
191 APosterioriStandardDeviations
192 *List of matrices*. Each element is an *a posteriori* error standard
193 deviation matrix of the optimal state.
195 Example : ``E = ADD.get("APosterioriStandardDeviations")[-1]``
198 *List of matrices*. Each element is an *a posteriori* error variance matrix
199 of the optimal state.
201 Example : ``V = ADD.get("APosterioriVariances")[-1]``
204 *List of vectors*. Each element is a vector of difference between the
205 background and the optimal state.
207 Example : ``bma = ADD.get("BMA")[-1]``
210 *List of values*. Each element is a value of the error function :math:`J`.
212 Example : ``J = ADD.get("CostFunctionJ")[:]``
215 *List of values*. Each element is a value of the error function :math:`J^b`,
216 that is of the background difference part.
218 Example : ``Jb = ADD.get("CostFunctionJb")[:]``
221 *List of values*. Each element is a value of the error function :math:`J^o`,
222 that is of the observation difference part.
224 Example : ``Jo = ADD.get("CostFunctionJo")[:]``
227 *List of vectors*. Each element is an innovation vector, which is in static
228 the difference between the optimal and the background, and in dynamic the
231 Example : ``d = ADD.get("Innovation")[-1]``
233 MahalanobisConsistency
234 *List of values*. Each element is a value of the Mahalanobis quality
237 Example : ``m = ADD.get("MahalanobisConsistency")[-1]``
240 *List of vectors*. Each element is a vector of difference between the
241 observation and the optimal state in the observation space.
243 Example : ``oma = ADD.get("OMA")[-1]``
246 *List of vectors*. Each element is a vector of difference between the
247 observation and the background state in the observation space.
249 Example : ``omb = ADD.get("OMB")[-1]``
252 *List of values*. Each element is a value of the quality indicator
253 :math:`(\sigma^b)^2` of the background part.
255 Example : ``sb2 = ADD.get("SigmaBck")[-1]``
258 *List of values*. Each element is a value of the quality indicator
259 :math:`(\sigma^o)^2` of the observation part.
261 Example : ``so2 = ADD.get("SigmaObs")[-1]``
263 SimulatedObservationAtBackground
264 *List of vectors*. Each element is a vector of observation simulated from
265 the background :math:`\mathbf{x}^b`.
267 Example : ``hxb = ADD.get("SimulatedObservationAtBackground")[-1]``
269 SimulatedObservationAtOptimum
270 *List of vectors*. Each element is a vector of observation simulated from
271 the analysis or optimal state :math:`\mathbf{x}^a`.
273 Example : ``hxa = ADD.get("SimulatedObservationAtOptimum")[-1]``
276 *List of vectors*. Each element is a vector corresponding to the observed
277 state which realize the required quantile, in the same order than the
278 quantiles required by the user.
280 Example : ``sQuantiles = ADD.get("SimulationQuantiles")[:]``
285 References to other sections:
286 - :ref:`section_ref_algorithm_ExtendedBlue`
287 - :ref:`section_ref_algorithm_3DVAR`
288 - :ref:`section_ref_algorithm_LinearityTest`
290 Bibliographical references: