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
7 modify it under the terms of the GNU Lesser General Public
8 License as published by the Free Software Foundation; either
9 version 2.1 of the License, or (at your option) any later version.
11 This library is distributed in the hope that it will be useful,
12 but WITHOUT ANY WARRANTY; without even the implied warranty of
13 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
14 Lesser General Public License for more details.
16 You should have received a copy of the GNU Lesser General Public
17 License along with this library; if not, write to the Free Software
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", "OMA", "OMB", "CurrentState",
114 "CostFunctionJ", "Innovation", "SigmaBck2", "SigmaObs2",
115 "MahalanobisConsistency", "SimulationQuantiles",
116 "SimulatedObservationAtBackground", "SimulatedObservationAtCurrentState",
117 "SimulatedObservationAtOptimum"].
119 Example : ``{"StoreSupplementaryCalculations":["BMA", "Innovation"]}``
122 This list indicates the values of quantile, between 0 and 1, to be estimated
123 by simulation around the optimal state. The sampling uses a multivariate
124 gaussian random sampling, directed by the *a posteriori* covariance matrix.
125 This option is useful only if the supplementary calculation
126 "SimulationQuantiles" has been chosen. The default is a void list.
128 Example : ``{"Quantiles":[0.1,0.9]}``
131 This key allow to give an integer in order to fix the seed of the random
132 generator used to generate the ensemble. A convenient value is for example
133 1000. By default, the seed is left uninitialized, and so use the default
134 initialization from the computer.
136 Example : ``{"SetSeed":1000}``
138 NumberOfSamplesForQuantiles
139 This key indicates the number of simulation to be done in order to estimate
140 the quantiles. This option is useful only if the supplementary calculation
141 "SimulationQuantiles" has been chosen. The default is 100, which is often
142 sufficient for correct estimation of common quantiles at 5%, 10%, 90% or
145 Example : ``{"NumberOfSamplesForQuantiles":100}``
147 SimulationForQuantiles
148 This key indicates the type of simulation, linear (with the tangent
149 observation operator applied to perturbation increments around the optimal
150 state) or non-linear (with standard observation operator applied to
151 perturbated states), one want to do for each perturbation. It changes mainly
152 the time of each elementary calculation, usually longer in non-linear than
153 in linear. This option is useful only if the supplementary calculation
154 "SimulationQuantiles" has been chosen. The default value is "Linear", and
155 the possible choices are "Linear" and "NonLinear".
157 Example : ``{"SimulationForQuantiles":"Linear"}``
159 Information and variables available at the end of the algorithm
160 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
162 At the output, after executing the algorithm, there are variables and
163 information originating from the calculation. The description of
164 :ref:`section_ref_output_variables` show the way to obtain them by the method
165 named ``get`` of the variable "*ADD*" of the post-processing. The input
166 variables, available to the user at the output in order to facilitate the
167 writing of post-processing procedures, are described in the
168 :ref:`subsection_r_o_v_Inventaire`.
170 The unconditional outputs of the algorithm are the following:
173 *List of vectors*. Each element is an optimal state :math:`\mathbf{x}*` in
174 optimization or an analysis :math:`\mathbf{x}^a` in data assimilation.
176 Example : ``Xa = ADD.get("Analysis")[-1]``
178 The conditional outputs of the algorithm are the following:
180 APosterioriCorrelations
181 *List of matrices*. Each element is an *a posteriori* error correlation
182 matrix of the optimal state.
184 Example : ``C = ADD.get("APosterioriCorrelations")[-1]``
186 APosterioriCovariance
187 *List of matrices*. Each element is an *a posteriori* error covariance
188 matrix :math:`\mathbf{A}*` of the optimal state.
190 Example : ``A = ADD.get("APosterioriCovariance")[-1]``
192 APosterioriStandardDeviations
193 *List of matrices*. Each element is an *a posteriori* error standard
194 deviation matrix of the optimal state.
196 Example : ``E = ADD.get("APosterioriStandardDeviations")[-1]``
199 *List of matrices*. Each element is an *a posteriori* error variance matrix
200 of the optimal state.
202 Example : ``V = ADD.get("APosterioriVariances")[-1]``
205 *List of vectors*. Each element is a vector of difference between the
206 background and the optimal state.
208 Example : ``bma = ADD.get("BMA")[-1]``
211 *List of values*. Each element is a value of the error function :math:`J`.
213 Example : ``J = ADD.get("CostFunctionJ")[:]``
216 *List of values*. Each element is a value of the error function :math:`J^b`,
217 that is of the background difference part.
219 Example : ``Jb = ADD.get("CostFunctionJb")[:]``
222 *List of values*. Each element is a value of the error function :math:`J^o`,
223 that is of the observation difference part.
225 Example : ``Jo = ADD.get("CostFunctionJo")[:]``
228 *List of vectors*. Each element is an innovation vector, which is in static
229 the difference between the optimal and the background, and in dynamic the
232 Example : ``d = ADD.get("Innovation")[-1]``
234 MahalanobisConsistency
235 *List of values*. Each element is a value of the Mahalanobis quality
238 Example : ``m = ADD.get("MahalanobisConsistency")[-1]``
241 *List of vectors*. Each element is a vector of difference between the
242 observation and the optimal state in the observation space.
244 Example : ``oma = ADD.get("OMA")[-1]``
247 *List of vectors*. Each element is a vector of difference between the
248 observation and the background state in the observation space.
250 Example : ``omb = ADD.get("OMB")[-1]``
253 *List of values*. Each element is a value of the quality indicator
254 :math:`(\sigma^b)^2` of the background part.
256 Example : ``sb2 = ADD.get("SigmaBck")[-1]``
259 *List of values*. Each element is a value of the quality indicator
260 :math:`(\sigma^o)^2` of the observation part.
262 Example : ``so2 = ADD.get("SigmaObs")[-1]``
264 SimulatedObservationAtBackground
265 *List of vectors*. Each element is a vector of observation simulated from
266 the background :math:`\mathbf{x}^b`.
268 Example : ``hxb = ADD.get("SimulatedObservationAtBackground")[-1]``
270 SimulatedObservationAtOptimum
271 *List of vectors*. Each element is a vector of observation simulated from
272 the analysis or optimal state :math:`\mathbf{x}^a`.
274 Example : ``hxa = ADD.get("SimulatedObservationAtOptimum")[-1]``
277 *List of vectors*. Each element is a vector corresponding to the observed
278 state which realize the required quantile, in the same order than the
279 quantiles required by the user.
281 Example : ``sQuantiles = ADD.get("SimulationQuantiles")[:]``
286 References to other sections:
287 - :ref:`section_ref_algorithm_ExtendedBlue`
288 - :ref:`section_ref_algorithm_3DVAR`
289 - :ref:`section_ref_algorithm_LinearityTest`
291 Bibliographical references: