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, xi 2 Illustration des deux étapes du filtrage bayésien: mutation et sélection, p.3
,
,
, xvii 7 Organisation des quatre étapes d'une itération de la méthode de reconstruction avec pour chaque étape les paramètres qui l'influence, Évolution du vecteur d'état des particules lors d'une itération
, Forme du majorant de l'espérance de N G0 suivant ? add et ? obs
, xxii 14 Indices de Sobol d'ordre 1 pour le nombre de potentiels nuls
, Graphe d'interaction de la sortie globale
, xxviii 21 Exemple de rééchantillonage par inversion de la fonction de répartition
,
, 24 Evolution of the world passenger air traffic from 1950 to, 2016.
, Illustration of horizontal wind shear on a runway
, MODIS image of the Canaries and Madeira islands creating a turbulent flow downstream. Characteristic length and speed involved in the Reynolds number estimation are highlighted (original image from NASA, public domain), p.14
, Kolmogorov spectrum with -5/3 slope
, Illustration of Lagragian model use for downscaling, p.24
,
, , p.32
, Particles at the same vertical at t and trajectory of one of them during the 50 next time steps
, can be seen as the projection of X t onto a plan accessible to measure. 37 2.2 The stochastic kernel K associates each point x in the departure space to a probability in the arrival space
, Mutation and selection steps in a Bayesian filter, p.41
, Illustration of the genetic selection algorithm
, , p.60
, Backscatter of light by layers of atmosphere, p.61
, Basic overview of major differences in lidars
, Example of chunked signal coming back from one pulse, p.64
, Space-time ambiguity in the lidar measurement, p.65
, Principle of heterodyne detection
, Example of spectrum of the output signal for one pulse, p.67
, Average of 1000 spectra, corresponding to 1000 pulses, p.67
,
, , p.71
, Lidar measurements of IOP8 and part used
,
, Illustration of irregular measurement issues (gap filling), p.78
, Illustration of missing data processing
, Variographic swarm and empirical variogram for the output b, p.101
, , p.101
, Empirical variogram and Gaussian variogram fitting for the output r V, p.102
, Empirical variogram and Gaussian variogram fitting for the output N G0, p.102
, Empirical variogram and Gaussian variogram fitting for the output r k, p.102
, Empirical variogram and Gaussian variogram fitting for the output T exe, p.102
, Example of 1-dimensional kriging of the function x ? x sin(x) with a Matèrn 5/2 variogram. Blue dots are the observations, dotted black line is the target function, red solid line is the kriging estimation
, 110 5.2 Geometry of the 1D turbulence reconstruction problem and vocabulary, p.111
, Evolution of particle state vector within a time step
, , p.115
, Diagram of input/outputs and parameters for the mutation step, p.119
, Illustration of the conditioning step on 1D examples, p.119
, 123 5.10 Diagram of input/outputs and parameters for the conditioning step, p.123
, Illustration of the selection step
, Diagram of input/outputs and parameters for the selection step, p.126
, Diagram of input/outputs and parameters for the estimation step, p.128
,
,
,
,
,
, 134 5.21 Example of histogram of maximum potential. This histogram has been shown to have a shape which can help to diagnose degeneracy of the filter, Amount of particles out, redistributed and rejected, p.135
, Pie chart of time spent at each step
, Typical shape of a wind time series
, Shape of maximum weight histogram
, Typical shape of a TKE time series (vertical component, p.138, 2015.
, Typical shape of a EDR time series (vertical component, p.138, 2015.
, Inputs and outputs of turbulence reconstruction
, Illustration of RMSE of the wind r V
, Illustration of RMSE of the TKE r k
, Illustration of the wind spectrum slope b
, Theoretical bound for the average of N G0 against ? add and ? obs, p.149
, 152 6.2 Situation of the input parameters in the program. The dotted outer circle stand for the time loop in the reconstruction algorithm, Diagram of the system on which is done the sensitivity analysis, p.153
, , p.161
,
,
, 166 6.10 Proportion of 2 nd and 1 st order for b
,
,
,
,
, First order Sobol indices for r k and uncertainty, p.171
, , p.171
,
,
,
,
, First order Sobol indices for r V and uncertainty, p.176
, Proportion of 2 nd and 1 st order influence for r, p.176
,
,
,
,
, , p.180
, , p.181
,
,
,
,
, Proportion of variance in the vector output
, Proportion of first and second order for all outputs, p.188
, Tile of 2 nd order for all outputs
, Graph of interaction for all outputs
, Sobol indices of ? add and uncertainty
, , p.194
,
, Sobol indices of C 0 and uncertainty
, Sobol indices of C 1 and uncertainty, p.195
,
,
,
,
, 203 7.2 Evolution of b when only ? add and ? obs vary. The sampling grid has 20 values of ? obs and 20 values of ? obs (400 points in total). The red plan is at the level b = ?5/3 (theoretical expected value)
, Evolution of b when ? add vary, for different values of ? obs . Horizontal dotted line is b = ?5/3. Vertical dashed lines signalize when ? add = ? obs for each value of ? obs
, Evolution of b when ? obs vary, for different values of ? add . Horizontal dotted line is b = ?5/3. Vertical dashed lines signalize when ? add = ? obs for each value of ? add
, Evolution of r V when only N and ? add vary
, Evolution of r V when only N and ? obs vary
, Evolution of r V when only ? add and ? obs vary
, Well set case: evolution of r V when only N and ? obs vary with ? obs = ? add, p.210
, Evolution of r V with N when ? obs = ? add . Regressions (dashed lines) show the observed decrease is close the square root, as predicted by the theory, p.211
, Evolution of r k when only and ? add vary
, Evolution of r k when only ? add and ? obs vary
, Evolution of N G0 when only C 0 and C 1 vary
, Evolution of r V when only C 0 and C 1 vary, p.219
, Evolution of r k when only C 0 and C 1 vary
, Evolution of N G0 when only N and ? add vary
, Evolution of N G0 with N . Regressions show an exponential decrease, p.222
, Evolution of N G0 when only ? add and ? obs vary
, 222 8.1 Illustration of admissible area with L 1 and L 2 norms. The extreme points are located on an axis for the L 1 norm, thus one of the coefficient is null. Credit: Par LaBaguette -Travail personnel
, Soft threshold: link between the Lasso estimator and the least square estimator.233 8.3 Hard threshold: link between the best subset estimator and the least square estimator
, MSE against the penalty in Lasso (soft threshold), p.238
, MSE against the penalty in Lasso (from minimization), p.238
, Value of the coefficients estimated by Lasso regression against the penalty (in log-scale). The coefficients are all at 0 for large penalty and raise in order of importance up to their value as obtained with ordinary least square, p.240
, Comparison of Monte Carlo, least squares, Lasso and best subset estimators, p.241
, 245 8.10 First order Sobol indices with Matérn 3/2 variogram, First order Sobol indices with Gaussian variogram, p.245
, , p.246
, Indices de Sobol d'ordre 1 moyens des cinq sorties
, Reynolds axioms for the average operator
,
, Basic characteristics of the Doppler lidar that provided the data, p.73
, Recap of Sobol index estimates calculated in this work, p.96
, Results of the evaluation of the meta-model by K-fold cross-validation (K = 5), p.105
, Nomenclature of discrete stochastic Lagrangian model, vol.118
, Summary of input parameters for the sensitivity analysis, p.154
, Summary of output parameters for the sensitivity analysis, p.155
, Information displayed on each figure type
, Range of variations of the outputs (from original response surface), p.187
, Average first order Sobol indices for each input
, Couples of inputs experimented
, Range of variation and nominal value for each input, p.203
,
, Definition (?-algebra), p.251
, Definition (Measure)
, Definition (Probability)
, Definition (Random variable)
, Definition (Law of a random variable), A.5
, Definition (Probability density function)
, Definition (L p space)
, Definition (L p space)
, Expected value)
, Definition (Momenta)
, A.11 Definition (Conditional probability)
, Stochastic process)
, Definition (Brownian motion), A.13
, Gaussian process)
, A.15 Definition (Stationarity)
, A.16 Definition (Stationarity at order p)
, 17 Definition (Intrinsic process)
, A.18 Definition (Ergodicity)
, Definition (Fourier transform)
, Definition (Power spectral density)
, xix 4.1 Theorem (ANOVA decomposition)
, Theorem (Hoeffding decomposition)
, Theorem (Stone decomposition)
, Theorem (Bayes)
, Theorem (Law of total probability)
, Theorem (ANOVA decomposition)
, Theorem (Hoeffding decomposition)