2 Copyright (C) 2008-2015 EDF R&D
4 This file is part of SALOME ADAO module.
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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: ExtendedBlue
25 .. _section_ref_algorithm_ExtendedBlue:
27 Calculation algorithm "*ExtendedBlue*"
28 --------------------------------------
33 This algorithm realizes an extended BLUE (Best Linear Unbiased Estimator) type
34 estimation of the state of a system.
36 This algorithm is a partially non-linear generalization of the
37 :ref:`section_ref_algorithm_Blue`. It is equivalent for a linear observation
38 operator. One can verify the linearity of the observation operator with the help
39 of the :ref:`section_ref_algorithm_LinearityTest`.
41 In case of non-linearity, it is close to the :ref:`section_ref_algorithm_3DVAR`,
42 without being entirely equivalent.
44 Optional and required commands
45 ++++++++++++++++++++++++++++++
47 .. index:: single: AlgorithmParameters
48 .. index:: single: Background
49 .. index:: single: BackgroundError
50 .. index:: single: Observation
51 .. index:: single: ObservationError
52 .. index:: single: ObservationOperator
53 .. index:: single: StoreSupplementaryCalculations
54 .. index:: single: Quantiles
55 .. index:: single: SetSeed
56 .. index:: single: NumberOfSamplesForQuantiles
57 .. index:: single: SimulationForQuantiles
59 The general required commands, available in the editing user interface, are the
63 *Required command*. This indicates the background or initial vector used,
64 previously noted as :math:`\mathbf{x}^b`. Its value is defined as a
65 "*Vector*" or a *VectorSerie*" type object.
68 *Required command*. This indicates the background error covariance matrix,
69 previously noted as :math:`\mathbf{B}`. Its value is defined as a "*Matrix*"
70 type object, a "*ScalarSparseMatrix*" type object, or a
71 "*DiagonalSparseMatrix*" type object.
74 *Required command*. This indicates the observation vector used for data
75 assimilation or optimization, previously noted as :math:`\mathbf{y}^o`. It
76 is defined as a "*Vector*" or a *VectorSerie* type object.
79 *Required command*. This indicates the observation error covariance matrix,
80 previously noted as :math:`\mathbf{R}`. It is defined as a "*Matrix*" type
81 object, a "*ScalarSparseMatrix*" type object, or a "*DiagonalSparseMatrix*"
85 *Required command*. This indicates the observation operator, previously
86 noted :math:`H`, which transforms the input parameters :math:`\mathbf{x}` to
87 results :math:`\mathbf{y}` to be compared to observations
88 :math:`\mathbf{y}^o`. Its value is defined as a "*Function*" type object or
89 a "*Matrix*" type one. In the case of "*Function*" type, different
90 functional forms can be used, as described in the section
91 :ref:`section_ref_operator_requirements`. If there is some control :math:`U`
92 included in the observation, the operator has to be applied to a pair
95 The general optional commands, available in the editing user interface, are
96 indicated in :ref:`section_ref_assimilation_keywords`. Moreover, the parameters
97 of the command "*AlgorithmParameters*" allows to choose the specific options,
98 described hereafter, of the algorithm. See
99 :ref:`section_ref_options_Algorithm_Parameters` for the good use of this
102 The options of the algorithm are the following:
104 StoreSupplementaryCalculations
105 This list indicates the names of the supplementary variables that can be
106 available at the end of the algorithm. It involves potentially costly
107 calculations or memory consumptions. The default is a void list, none of
108 these variables being calculated and stored by default. The possible names
109 are in the following list: ["APosterioriCorrelations",
110 "APosterioriCovariance", "APosterioriStandardDeviations",
111 "APosterioriVariances", "BMA", "CostFunctionJ", "OMA", "OMB", "Innovation",
112 "SigmaBck2", "SigmaObs2", "MahalanobisConsistency",
113 "SimulatedObservationAtBackground", "SimulatedObservationAtOptimum",
114 "SimulationQuantiles"].
116 Example : ``{"StoreSupplementaryCalculations":["BMA","Innovation"]}``
119 This list indicates the values of quantile, between 0 and 1, to be estimated
120 by simulation around the optimal state. The sampling uses a multivariate
121 gaussian random sampling, directed by the *a posteriori* covariance matrix.
122 This option is useful only if the supplementary calculation
123 "SimulationQuantiles" has been chosen. The default is a void list.
125 Example : ``{"Quantiles":[0.1,0.9]}``
128 This key allow to give an integer in order to fix the seed of the random
129 generator used to generate the ensemble. A convenient value is for example
130 1000. By default, the seed is left uninitialized, and so use the default
131 initialization from the computer.
133 Example : ``{"SetSeed":1000}``
135 NumberOfSamplesForQuantiles
136 This key indicates the number of simulation to be done in order to estimate
137 the quantiles. This option is useful only if the supplementary calculation
138 "SimulationQuantiles" has been chosen. The default is 100, which is often
139 sufficient for correct estimation of common quantiles at 5%, 10%, 90% or
142 Example : ``{"NumberOfSamplesForQuantiles":100}``
144 SimulationForQuantiles
145 This key indicates the type of simulation, linear (with the tangent
146 observation operator applied to perturbation increments around the optimal
147 state) or non-linear (with standard observation operator applied to
148 perturbated states), one want to do for each perturbation. It changes mainly
149 the time of each elementary calculation, usually longer in non-linear than
150 in linear. This option is useful only if the supplementary calculation
151 "SimulationQuantiles" has been chosen. The default value is "Linear", and
152 the possible choices are "Linear" and "NonLinear".
154 Example : ``{"SimulationForQuantiles":"Linear"}``
156 Information and variables available at the end of the algorithm
157 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
159 At the output, after executing the algorithm, there are variables and
160 information originating from the calculation. The description of
161 :ref:`section_ref_output_variables` show the way to obtain them by the method
162 named ``get`` of the variable "*ADD*" of the post-processing. The input
163 variables, available to the user at the output in order to facilitate the
164 writing of post-processing procedures, are described in the
165 :ref:`subsection_r_o_v_Inventaire`.
167 The unconditional outputs of the algorithm are the following:
170 *List of vectors*. Each element is an optimal state :math:`\mathbf{x}*` in
171 optimization or an analysis :math:`\mathbf{x}^a` in data assimilation.
173 Example : ``Xa = ADD.get("Analysis")[-1]``
175 The conditional outputs of the algorithm are the following:
177 APosterioriCorrelations
178 *List of matrices*. Each element is an *a posteriori* error correlation
179 matrix of the optimal state.
181 Example : ``C = ADD.get("APosterioriCorrelations")[-1]``
183 APosterioriCovariance
184 *List of matrices*. Each element is an *a posteriori* error covariance
185 matrix :math:`\mathbf{A}*` of the optimal state.
187 Example : ``A = ADD.get("APosterioriCovariance")[-1]``
189 APosterioriStandardDeviations
190 *List of matrices*. Each element is an *a posteriori* error standard
191 deviation matrix of the optimal state.
193 Example : ``E = ADD.get("APosterioriStandardDeviations")[-1]``
196 *List of matrices*. Each element is an *a posteriori* error variance matrix
197 of the optimal state.
199 Example : ``V = ADD.get("APosterioriVariances")[-1]``
202 *List of vectors*. Each element is a vector of difference between the
203 background and the optimal state.
205 Example : ``bma = ADD.get("BMA")[-1]``
208 *List of values*. Each element is a value of the error function :math:`J`.
210 Example : ``J = ADD.get("CostFunctionJ")[:]``
213 *List of values*. Each element is a value of the error function :math:`J^b`,
214 that is of the background difference part.
216 Example : ``Jb = ADD.get("CostFunctionJb")[:]``
219 *List of values*. Each element is a value of the error function :math:`J^o`,
220 that is of the observation difference part.
222 Example : ``Jo = ADD.get("CostFunctionJo")[:]``
225 *List of vectors*. Each element is an innovation vector, which is in static
226 the difference between the optimal and the background, and in dynamic the
229 Example : ``d = ADD.get("Innovation")[-1]``
231 MahalanobisConsistency
232 *List of values*. Each element is a value of the Mahalanobis quality
235 Example : ``m = ADD.get("MahalanobisConsistency")[-1]``
238 *List of vectors*. Each element is a vector of difference between the
239 observation and the optimal state in the observation space.
241 Example : ``oma = ADD.get("OMA")[-1]``
244 *List of vectors*. Each element is a vector of difference between the
245 observation and the background state in the observation space.
247 Example : ``omb = ADD.get("OMB")[-1]``
250 *List of values*. Each element is a value of the quality indicator
251 :math:`(\sigma^b)^2` of the background part.
253 Example : ``sb2 = ADD.get("SigmaBck")[-1]``
256 *List of values*. Each element is a value of the quality indicator
257 :math:`(\sigma^o)^2` of the observation part.
259 Example : ``so2 = ADD.get("SigmaObs")[-1]``
261 SimulatedObservationAtBackground
262 *List of vectors*. Each element is a vector of observation simulated from
263 the background :math:`\mathbf{x}^b`.
265 Example : ``hxb = ADD.get("SimulatedObservationAtBackground")[-1]``
267 SimulatedObservationAtOptimum
268 *List of vectors*. Each element is a vector of observation simulated from
269 the analysis or optimal state :math:`\mathbf{x}^a`.
271 Example : ``hxa = ADD.get("SimulatedObservationAtOptimum")[-1]``
274 *List of vectors*. Each element is a vector corresponding to the observed
275 state which realize the required quantile, in the same order than the
276 quantiles required by the user.
278 Example : ``sQuantiles = ADD.get("SimulationQuantiles")[:]``
283 References to other sections:
284 - :ref:`section_ref_algorithm_Blue`
285 - :ref:`section_ref_algorithm_3DVAR`
286 - :ref:`section_ref_algorithm_LinearityTest`