.. .. Satellite, 111 6.2.2 Surface Water Ocean Topography altimetry mission, p.115

. .. Geophysical-field-interpolation, 118 6.3.1 Exemplar-based methods

.. .. Conclusion, 128 7 Chapter Locally-adapted convolutional models for the reconstruction of SLA fields from satellite along-track data "Think global, act local

, 2.4 Locally-adapted dictionary-based convolutional models

.. .. Conclusion,

.. .. Case,

.. .. Method,

, Nadir along-track vs. SWOT data

.. .. Additional,

.. .. Conclusion,

. .. , Normalized mean squared estimation error (nMSE) final median value (at convergence) for linear modes ? k as a function of the number of classes K considered. Results presented for the ALS algorithm using a gradient descent approach (ALSgradient), the ALS algorithm using a direct minimization (ALS-direct) and the dictionary-based decomposition of local linear operators (LLOD), p.71

, Normalized mean squared y n reconstruction error (nMSE) final median value (at convergence) as a function of the number of classes K considered. Results presented for the ALS algorithm using a gradient descent approach (ALS-gradient), the ALS algorithm using a direct minimization (ALS-direct) and the dictionarybased decomposition of local linear operators (LLOD)

. .. , Normalized mean squared estimation error (nMSE) final median value (at convergence) for mixing coefficients ? nk as a function of mixing coefficient signal-to-noise ratio (SNR) when Gaussian noise is added to cluster-specific mixing coefficients. Results presented for the ALS algorithm using a gradient descent approach (ALSgradient), the ALS algorithm using a direct minimization (ALS-direct) and the dictionary-based decomposition of local linear operators (LLOD), p.73

. .. , Normalized mean squared estimation error (nMSE) final median value (at convergence) for linear modes ? k as a function of mixing coefficient signal-to-noise ratio (SNR) when Gaussian noise is added to cluster-specific mixing coefficients. Results presented for the ALS algorithm using a gradient descent approach (ALSgradient), the ALS algorithm using a direct minimization (ALS-direct) and the dictionary-based decomposition of local linear operators (LLOD)

, Results presented for the ALS algorithm using a gradient descent approach (ALS-gradient), the ALS algorithm using a direct minimization (ALS-direct) and the dictionary-based decomposition of local linear operators (LLOD)

, Normalized mean squared estimation error (nMSE) final median value (at convergence) for mixing coefficients ? nk as a function of the ratio between cluster standard deviation ? x and minimal distance d min (parameter ? = 6 ?x d min ), for initially non-overlapping clusters. Results presented for the ALS algorithm using a gradient descent approach (ALS-gradient), the ALS algorithm using a direct minimization (ALS-direct) and the dictionary-based decomposition of local linear operators (LLOD)

. Ground-truth, S. Sst, and . Fields-on, SST-derived predictions of the SSS fields for each mode of the considered dictionary-based local linear operator decomposition considering K = 2 modes, 2011.

, Distribution of SST-SSS correlation when mode 1 (resp. mode 2) of the two-class dictionary-based local linear operator decomposition dominates, i.e. ? n1 > ? n2 (resp. ? n2 > ? n1 )

, Reported results correspond to both considered models, namely our proposed non-negative decomposition model and a latent class regression model, p.95, 2004.

, denoted by x, and the SSH field (in meters represented by contour lines), denoted by y, at locations s i and s j and time t i, 8 Sketch of the considered patch-based representation of the SST field (in degrees represented in false colors), 2014.

, Reported results correspond to our proposed non-negative decomposition model considering K = 2 classes, 2004.

, The Jet colormap was explicitly chosen to enhance contrast between positive and negative reconstruction error values

, Time series of the daily FVE (Fraction of Variance Explained) for model (3.3): top, SSH prediction for the first mode (SQG-like mode); middle, global SSH prediction; bottom, global SSH gradient prediction. FVE is computed as 1 ? F V U , with F V U being the fraction of variance unexplained, i.e., the ratio between the reconstruction error variance and the ground-truth field variance, p.101

, Mean annual SSH field prediction for each dynamical mode of model (3.3): top, first mode (SQG-like mode); bottom, second mode

, Mean annual SSH prediction error standard deviation for the first mode (SQG-like mode) of model (3.3)

, Mean annual mapping of mixing coefficients ? for each dynamical mode of model (3.3): top, mode 1 (SQG-like mode); bottom, mode 2

, Time series of the daily mean of mixing coefficients ? of model (3.3): top, mode 1 (SQG-like mode); bottom, mode 2

, Spatio-temporal variability of mixing coefficients ? of model (3.3) with respect to four contrasted zones

, Comparison of conventional pulse-limited radar altimetry (left) and synthetic aperture radar (SAR) altimetry (right)

, Credits: CLS. Source: duacs. cls. fr 2 . Used with permission, Different quantities derived from altimetric measurements with reference to the Earth's Geoid and the Ellipsoid

. .. Ieee, Comparison of classic nadir along-track observations (6.3a) and off-nadir wideswath SWOT observations (6.3b), p.115, 2019.

, Each antenna receives the backscatter from both swaths, which are illuminated by a single antenna with two different polarizations (to have two separate simultaneous measurements). Interferometry is used to estimate the OST for both swaths from the received backscatter on both antennas, SWOT satellite scheme. A 10 m boom separates two SAR antennas

, Comparison of along-track nadir-looking altimeters (left) and off-nadir wide-swath interferometry altimeters (right). The off-nadir track allows for a wider footprint of the instrument, while a higher resolution can be achieved by exploiting interferometry between two off-nadir radiometers. Credits: AVISO 3 . Used with permission, p.117

, Illustration of the irregular sampling of high-resolution observations associated with ocean remote sensing data: sea level anomaly image with the sampled alongtrack positions by satellite altimeters (cyan squares) in a ±10-day time window around, 2012.

, } t , for a global convolutional model and for locally-adapted decompositions of a global convolutional model using principal component analysis (PCA) [206], KSVD [3] and non-negative decomposition (NN) and considering K = 10 classes. The probability distribution of the nRMSE for daily low-resolution SLA images {Y LR (t)} t is given as reference (noted as SLA LR ), Probability distribution for the normalized root mean square reconstruction error (nRMSE) for daily high-resolution SLA images {Y (t)

, first row, from left to right, real high-resolution SSH image Y , low-resolution SSH image Y LR (noted as SSH LR ), reconstruction of high-resolution SSH image Y using global convolutional model (7.1); second row, reconstruction of high-resolution SSH image Y using a 10-class locally-adapted decomposition (7.4) of global convolutional model (7.1) using, from left to right, High-resolution SSH image Y reconstruction, 2012.

. .. Ieee, 136 8.1 Comparison of nadir along-track observations (8.1b) and SWOT observations (8.1c) generated from ground-truth high-resolution SLA fields (8.1a) using real satellite tracks spatio-temporal locations and the SWOT simulator, respectively, 2019.

, Synthesis of constrained blind source separation algorithms. Comparison criteria include enforced constraints, convergence properties, flexibility and code availability, p.48

, Computational cost of the different steps of the proposed algorithms, number of operations

, Error statistics (after convergence) for the different algorithms considered, computed over 100 runs of the algorithms with randomly generated data, p.68

, Reported results include mixing coefficients ? nk normalized mean squared estimation error (e ? nk ), modal matrices ? k normalized mean squared estimation error (e ? k ), final log-likelihood (L (? nk , ? k )) for the latent class initialization using model (3.4), and final relative mean squared error (rMSE, presented as a percentage) for the reconstruction of variables {y n }. Results presented for both the EM and pseudo-EM initialization variants of the considered algorithm, Estimation features for the single observation version of the ALS-direct algorithm (Algorithm 1) applied to a single-cluster ground-truth synthetic dataset without parameter sharing between observations, 2016.

, Relative mean square error (rMSE, given as a percentage) for the different models considered

, the two-class latent regression model introduced in [244] and our proposed nonnegative decomposition model considering K = 1 and K = 2 classes, 2016.

, Normalized root mean squared error (nRMSE) values for global SSH and SSH gradient predictions for a two-mode model with and without external SQG forcing, and for a single-mode model without SQG forcing.nRMSE values where calculated by normalizing the spatio-temporal global prediction RMSE for SSH and SSH gradient fields by the spatio-temporal standard deviation of the real SSH and SSH gradient fields

, } t , for a global convolutional model and for locally-adapted decompositions of a global convolutional model using principal component analysis (PCA) [206], KSVD [3] and non-negative decomposition (NN), considering K = 2, K = 5 and K = 10 classes. The nRMSE value for daily low-resolution SLA images {Y LR (t)} t is given as reference (noted as SLA LR ). Best results for each number of classes K considered are presented in bold. Results that outperform a global convolutional model are underlined, Normalized root mean square reconstruction error (nRMSE) for daily high-resolution SLA images {Y (t)

, ?SLA) reconstruction from nadir along-track observations for the different algorithms considered, namely OI, vol.57

, Root mean squared error (Correlation) for AnDA SLA and SLA gradient (?SLA) reconstruction from nadir along-track observations accumulated on a window t 0 ± List of Tables 8.7 Root mean squared error (Correlation) for AnDA SLA and SLA gradient (?SLA) reconstruction when assimilating daily SWOT observations and/or their corresponding numerically-resolved SWOT observation gradients, p.172

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