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## Methods for orthogonality formulations

Principal Component Analysis To provide some historical context and background, orthogonal decomposition techniques have their origin in social sciences, going back to the works of Pearson [206]. The seminal work by Hotelling [104–106] formally introduced Principal Component Analysis (PCA), probably the most widespread orthogonal decomposition technique. In the geosciences domain, EOFs where introduced in the 1940s [77,140,174,197,198]. PCA/EOFs has also been used in several other domains. To cite a few, it is known as the Hotelling transform [104,104,106] in multivariate quality control and also as proper orthogonal decomposition (POD) [178, 179] in mechanical engineering. Moreover, it also relates closely to techniques such as the discrete Karhunen–Loève transform (KLT) [164] in signal processing, and factor analysis [95]. Even though PCA is mainly an exploratory analysis technique, it can be used for both dimensionality reduction [92] and filtering [197] in a straightforward manner. Generally speaking, PCA aims at decomposing a set of observations of possibly correlated variables as a linear combination of an ensemble of uncorrelated variables (or principal components).

To achieve this, principal components are chosen to ensure that each one of them accounts for the largest possible variability while being orthogonal to each other. In this respect, PCA closely relates to ICA, and on may think of PCA as particular implementation of ICA involving an independence criterion that enforces independence up to the second-order moments, thus involving the maximization of the variance explained by each component under the assumption that components are orthogonal [39]. Conversely, ICA may be viewed as an extension of PCA, with the particularity that PCA imposes independence up to the second-order moments only (given the maximization of variance of orthogonal components), whereas ICA may impose independence up to higher orders, depending on the considered criterion [39].

Formally, given a dataset of observations yn ∈ RI , n = 1, . . . ,N (which we can assume to be centered without loss of generality), PCA will resort to a decomposition of the covariance structure underlying the data in order to discover the directions of maximal variance. It is important to note that, for geophysical applications, observations yn may contain values of a geophysical field at I distinct locations si at time tn, in which case the covariance matrix elements [C]ij contains the covariances between full time series at location pairs (si, sj ). Alternatively, observations yn may reflect a geophysical field value at I distinct times ti at a fixed location sn; covariance matrix elements [C]ij will then be the covariances between full spatial maps of y at different times (ti, tj ).

### Methods for Tikhonov regularization

The greatest advantage of Tikhonov regularization is that it allows for the direct derivation of an analytic expression for parameter estimators, given that the ℓ2-norm exploited is a continuous and derivable function for all real valued vectors α ∈ RK. Importantly, this means that no indirect methods need to be developed to compute the Tikhonov-regularized solution for the general blind source separation problem (2.6). Indeed, in the case of blind source separation, one may reformulate Equation (2.6) as follows: Y = SA +W, subject to: ||A||2 F < ǫ.

#### Methods for non-negative formulations

Active-set algorithms Active sets algorithms are based on the assumption that only some of the n inequality constraints involved in minimization (2.10) are active (i.e. the associated coefficient αk is zero). The main idea is that constrained minimization problem (2.10) boils down to an unconstrained minimization problem on the passive set only (i.e. on coefficients αk which are not null), assuming that the active set is known. Active set coefficients αk are then simply set to zero to obtain the full solution of the constrained problem.

The simplest version of an active set method will then exploit an alternating least squares scheme to approximate the solution. In this respect, an initial feasible solution is found and all n constraints are considered to be in the passive set. At each iteration, the unconstrained minimization problem on the passive set is solved, variables in the active set are identified and removed from the passive set, and the process is repeated. The algorithm stop is given by aconvergence condition, usually on some measure of the optimality of the solution.

Methods for non-negative matrix factorization Numerically, algorithms for non-negative matrix factorization rely mostly on alternating least squares and multiplicative update rules (introduced in [155]). Given that minimization problem (2.12) involves a matrix product, it is non-convex if a joint minimization over both A and S is attempted. However, the sub-problem of optimizing A (resp. S) while considering S (resp. A) as fixed is indeed convex. This strongly motivates the application of ALS-type algorithms. At each iteration, one may then exploit gradient-splitting algorithms using proximal operators or similar optimization techniques. Alternatively, Lee and Seung introduced in [155] a set of multiplicative update rules that ensure that the cost function introduced in constrained minimization problem (2.12) decreases at each iteration (even thought such update rules provide no assurance of convergence towards a global minimum).

**Methods for sparse formulations**

KSVD KSVD [3] is a sparse dictionary learning algorithm designed to iteratively solve problem (2.15) by alternating optimizations over sparse mixing coefficients αk and signals sk. In this respect, KSVD closely relates to alternating least squares approaches. Specifically, the optimization of reference signals sk is performed by rewriting problem (2.15) in matricial form: Y = SA +W, subject to: ||[A]:n||0 ≤ M, ∀n ∈ J1,NK.

**Table of contents :**

Acknowledgments

Résumé long en français

Abstract

Acronyms

Notation

**1 Introduction and thesis summary **

1.1 Context

1.2 Geophysical motivations for data-driven approaches

1.3 Contributions

1.4 Dissertation outline

1.5 Publications and Valorization

1.5.1 International Conference papers

1.5.2 National Conference papers

1.5.3 Journal papers

1.5.4 Poster presentations

1.5.5 Invited talks

1.5.6 Other activities

**I Non-negative decomposition of local operators **

**2 State of the art and related work **

2.1 Introduction

2.2 Blind source separation

2.2.1 Proposed formulations

2.2.2 Associated numerical methods

2.3 Operator decomposition

2.3.1 Orthogonality-based decompositions

2.3.2 Dynamic mode decomposition

2.3.3 Tikhonov-regularized decompositions

2.4 Conclusion

**3 Proposed models and algorithms **

3.1 Introduction

3.2 General model

3.3 Non-negative linear superposition of linear modes

3.3.1 Latent class model

3.3.2 Parameter estimation

3.3.3 Alternating least squares algorithm

3.4 Reformulation based on local linear operators

3.4.1 Parameter Estimation

3.4.2 Dictionary-based decomposition of local linear operators

3.4.3 Model training and application

3.4.4 Computational complexity analysis

3.5 Conclusion

**4 Algorithm evaluation and benchmarking **

4.1 Introduction

4.2 Synthetic dataset generation

4.3 Estimation performance under ideal settings

4.4 Estimation performance with noisy mixing coefficients

4.5 Latent class initialization evaluation

4.6 Conclusion

**5 Applications **

5.1 Introduction

5.2 Data-driven forecasting of dynamical systems

5.3 Multi-tracer synergies for the characterization of upper ocean dynamics

5.3.1 Segmentation of upper ocean dynamics from SST/SSS relationships in the Alboran Sea

5.3.2 Segmentation of upper ocean dynamics from SST/SSH relationships in the Agulhas region

5.4 Conclusion

**II Data-driven interpolation of sea level anomaly fields from satellitederived ocean remote sensing data **

**6 Context and related work **

6.1 Introduction

6.2 Satellite altimetry

6.2.1 Along-track nadir altimetry

6.2.2 Surface Water Ocean Topography altimetry mission

6.3 Geophysical field interpolation

6.3.1 Exemplar-based methods

6.3.2 Projection-based methods

6.3.3 Data assimilation

6.3.4 Analog Forecasting

6.3.5 AnDA: The analog data assimilation framework

6.4 Conclusion

**7 Locally-adapted convolutional models for the reconstruction of SLA fields from satellite along-track data **

7.1 Introduction

7.2 Model formulation

7.2.1 Problem statement

7.2.2 Unconstrained model calibration

7.2.3 Dictionary-based decompositions

7.2.4 Locally-adapted dictionary-based convolutional models

7.2.5 Data and Methodology

7.2.6 Results

7.3 Conclusion

**8 Data-driven fusion of nadir along-track and SWOT data **

8.1 Introduction

8.2 Benchmarked methods

8.3 Case study and data

8.3.1 Data

8.3.2 Observing System Simulation Experiment

8.3.3 Pseudo-observations

8.4 Method comparison and benchmarking

8.5 AnDA Parameter Sensitivity Analysis

8.5.1 Patch size

8.5.2 Number of neighbours

8.5.3 Assimilation lag

8.5.4 Pseudo-observations half-window size

8.6 AnDA Noise sensitivity analysis

8.6.1 Nadir along-track data

8.6.2 SWOT data

8.7 Nadir along-track vs. SWOT data

8.8 Altimetric data fusion

8.9 Additional regressors for analog forecasting

8.10 Assimilation of observation gradients

8.11 Conclusion

**III Conclusion and Perspectives **

**9 Conclusion and Perspectives **

9.1 Conclusions

9.2 Perspectives

9.3 Final thoughts

List of Figures

List of Tables

**Bibliography**