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", "OMA", "OMB", "CurrentState",
112 "CostFunctionJ", "Innovation", "SigmaBck2", "SigmaObs2",
113 "MahalanobisConsistency", "SimulationQuantiles",
114 "SimulatedObservationAtBackground", "SimulatedObservationAtCurrentState",
115 "SimulatedObservationAtOptimum"].
117 Example : ``{"StoreSupplementaryCalculations":["BMA", "Innovation"]}``
120 This list indicates the values of quantile, between 0 and 1, to be estimated
121 by simulation around the optimal state. The sampling uses a multivariate
122 gaussian random sampling, directed by the *a posteriori* covariance matrix.
123 This option is useful only if the supplementary calculation
124 "SimulationQuantiles" has been chosen. The default is a void list.
126 Example : ``{"Quantiles":[0.1,0.9]}``
129 This key allow to give an integer in order to fix the seed of the random
130 generator used to generate the ensemble. A convenient value is for example
131 1000. By default, the seed is left uninitialized, and so use the default
132 initialization from the computer.
134 Example : ``{"SetSeed":1000}``
136 NumberOfSamplesForQuantiles
137 This key indicates the number of simulation to be done in order to estimate
138 the quantiles. This option is useful only if the supplementary calculation
139 "SimulationQuantiles" has been chosen. The default is 100, which is often
140 sufficient for correct estimation of common quantiles at 5%, 10%, 90% or
143 Example : ``{"NumberOfSamplesForQuantiles":100}``
145 SimulationForQuantiles
146 This key indicates the type of simulation, linear (with the tangent
147 observation operator applied to perturbation increments around the optimal
148 state) or non-linear (with standard observation operator applied to
149 perturbated states), one want to do for each perturbation. It changes mainly
150 the time of each elementary calculation, usually longer in non-linear than
151 in linear. This option is useful only if the supplementary calculation
152 "SimulationQuantiles" has been chosen. The default value is "Linear", and
153 the possible choices are "Linear" and "NonLinear".
155 Example : ``{"SimulationForQuantiles":"Linear"}``
157 Information and variables available at the end of the algorithm
158 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
160 At the output, after executing the algorithm, there are variables and
161 information originating from the calculation. The description of
162 :ref:`section_ref_output_variables` show the way to obtain them by the method
163 named ``get`` of the variable "*ADD*" of the post-processing. The input
164 variables, available to the user at the output in order to facilitate the
165 writing of post-processing procedures, are described in the
166 :ref:`subsection_r_o_v_Inventaire`.
168 The unconditional outputs of the algorithm are the following:
171 *List of vectors*. Each element is an optimal state :math:`\mathbf{x}*` in
172 optimization or an analysis :math:`\mathbf{x}^a` in data assimilation.
174 Example : ``Xa = ADD.get("Analysis")[-1]``
176 The conditional outputs of the algorithm are the following:
178 APosterioriCorrelations
179 *List of matrices*. Each element is an *a posteriori* error correlation
180 matrix of the optimal state.
182 Example : ``C = ADD.get("APosterioriCorrelations")[-1]``
184 APosterioriCovariance
185 *List of matrices*. Each element is an *a posteriori* error covariance
186 matrix :math:`\mathbf{A}*` of the optimal state.
188 Example : ``A = ADD.get("APosterioriCovariance")[-1]``
190 APosterioriStandardDeviations
191 *List of matrices*. Each element is an *a posteriori* error standard
192 deviation matrix of the optimal state.
194 Example : ``E = ADD.get("APosterioriStandardDeviations")[-1]``
197 *List of matrices*. Each element is an *a posteriori* error variance matrix
198 of the optimal state.
200 Example : ``V = ADD.get("APosterioriVariances")[-1]``
203 *List of vectors*. Each element is a vector of difference between the
204 background and the optimal state.
206 Example : ``bma = ADD.get("BMA")[-1]``
209 *List of values*. Each element is a value of the error function :math:`J`.
211 Example : ``J = ADD.get("CostFunctionJ")[:]``
214 *List of values*. Each element is a value of the error function :math:`J^b`,
215 that is of the background difference part.
217 Example : ``Jb = ADD.get("CostFunctionJb")[:]``
220 *List of values*. Each element is a value of the error function :math:`J^o`,
221 that is of the observation difference part.
223 Example : ``Jo = ADD.get("CostFunctionJo")[:]``
226 *List of vectors*. Each element is an innovation vector, which is in static
227 the difference between the optimal and the background, and in dynamic the
230 Example : ``d = ADD.get("Innovation")[-1]``
232 MahalanobisConsistency
233 *List of values*. Each element is a value of the Mahalanobis quality
236 Example : ``m = ADD.get("MahalanobisConsistency")[-1]``
239 *List of vectors*. Each element is a vector of difference between the
240 observation and the optimal state in the observation space.
242 Example : ``oma = ADD.get("OMA")[-1]``
245 *List of vectors*. Each element is a vector of difference between the
246 observation and the background state in the observation space.
248 Example : ``omb = ADD.get("OMB")[-1]``
251 *List of values*. Each element is a value of the quality indicator
252 :math:`(\sigma^b)^2` of the background part.
254 Example : ``sb2 = ADD.get("SigmaBck")[-1]``
257 *List of values*. Each element is a value of the quality indicator
258 :math:`(\sigma^o)^2` of the observation part.
260 Example : ``so2 = ADD.get("SigmaObs")[-1]``
262 SimulatedObservationAtBackground
263 *List of vectors*. Each element is a vector of observation simulated from
264 the background :math:`\mathbf{x}^b`.
266 Example : ``hxb = ADD.get("SimulatedObservationAtBackground")[-1]``
268 SimulatedObservationAtOptimum
269 *List of vectors*. Each element is a vector of observation simulated from
270 the analysis or optimal state :math:`\mathbf{x}^a`.
272 Example : ``hxa = ADD.get("SimulatedObservationAtOptimum")[-1]``
275 *List of vectors*. Each element is a vector corresponding to the observed
276 state which realize the required quantile, in the same order than the
277 quantiles required by the user.
279 Example : ``sQuantiles = ADD.get("SimulationQuantiles")[:]``
284 References to other sections:
285 - :ref:`section_ref_algorithm_Blue`
286 - :ref:`section_ref_algorithm_3DVAR`
287 - :ref:`section_ref_algorithm_LinearityTest`