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## Global and local analogs

The global analog strategy is the direct application of the introduced analog forecasting strategies to the entire state vector. We also introduce a local analog forecasting operator. For a given state x(t), the analogs ak(xl(t)) in the reference catalog, and their associated successors sk(xl(t)) for each component l of the state x(t) are defined according to a component-wise local neighborhood. The evaluation of the kernel function and the computation of the associated normalized weights ωk (xl(t)) involve only a portion of the state vector x(t) defining some component-wise local neighborhood around the lth component of the state vector (typically {xl−ν(t), . . . , xl(t) . . . , xl+ν(t)} with ν the width of the considered component-wise neighborhood).

The idea of using local analogs is motivated by the fact that points tends to scatter far away from each other in high dimensions, which make the search for skillful analogs nearly impossible. For instance, [151] has shown that finding a relevant analog at synoptic scale over the Northern Hemisphere for atmospheric data would require 1030 years of data to match the observational errors at that time. Conversely, he also hinted that lower degrees of freedom of the states lead to better analog forecasting performance. Following this analysis, the analog forecasting of the global state is split as a series of local and low-dimensional analog forecasting operations. Note that such local analogs also help reducing possibly spurious correlations.

### Analog data assimilation

The analog data assimilation is stated as a sequential and stochastic assimilation scheme, using Monte Carlo methods. It amounts to estimating the so-called filtering and smoothing posterior likelihoods, respectively p(x(t)|y(1), . . . , y(t)) the distribution of the current state knowing past and current observations and p(x(t)|y(1), . . . , y(T)) the distribution of the current state knowing past, current and future observations. We investigate both Ensemble Kalman filter/smoother and particle filter.

#### Analog Ensemble Kalman Filter and Smoother (AnEnKF/AnEnKS)

The AnEnKF and AnEnKS equations are equivalent to those of the EnKF and EnKS described in 1.2.1, except for the update step where we use the analog forecasting operator. Algorithm 5 The Analog Ensemble Kalman Filter algorithm.

1: Input: xb and B parameters of the prior Gaussian distribution.

2: Generate vectors xf i (1) ∀i ∈ {1, …,N} using a multivariate Gaussian random generator with mean vector xb and covariance matrix B. The index i of the state vector corresponds to the ith realization of the Monte Carlo procedure (called member or particle).

3: Set t = 1.

4: Prediction step.

• Apply the analog forecasting A operator to each member of the ensemble following (2.1) to generate xf i (t).

• The forecast state is represented by the sample mean xf (t) and the sample covariance Pf (t).

**Experiments with Lorenz-96 model**

Experiment 1: The first numerical experiment consisted only in the application of analog forecasting (without assimilation) from a catalog. We build a database using Lorenz-96 equations, then we split the samples randomly to 2/3 for training the analog forecasting operators and 1/3 for test. Finally, we compare the RMSE w.r.t ground truth data as a function of Lorenz-96 time. For local analogs, we consider ν = 2 the width of the considered component-wise neighborhood. Figure 2.3 shows the results of this experiment using the three choices for the analog forecasting operator A. The locally-linear approach outperforms the two other approaches confirming that its forecasts are with lower bias compared to the other approaches. However, it also involves more parameters which increases the variance of the forecasts. This bias-variance trade-off supports the greater generalization capabilities of the locally-linear operator, when the dynamics can well be approximated locally by a linear operator.

**Table of contents :**

Acknowledgments

Abstract

Résumé

Acronyms

Thesis summary

Introduction and problem statement

Contributions

Publications

Invited conferences and workshop talks

**I Analog methods for state-space problems: Analog Data Assimilation**

**1 Data Assimilation and Analog methods **

1.1 State-Space models

1.1.1 The Kalman Filter

1.1.2 The Particle Filter

1.1.3 Hidden Markov Models

1.2 Data assimilation in geoscience

1.2.1 Ensemble Kalman filters (EnKF) and smoothers (EnKS) as an example of stochastic data assimilation

1.2.2 Optimal Interpolation

1.3 Analog forecasting

1.4 Discussion and conclusion

**2 The Analog Data Assimilation **

2.1 Data-driven data assimilation

2.2 Analog forecasting strategies

2.2.1 Analog forecasting operator

2.2.2 Global and local analogs

2.3 Analog data assimilation

2.3.1 Analog Ensemble Kalman Filter and Smoother (AnEnKF/AnEnKS)

2.3.2 Analog Particle Filter (AnPF)

2.3.3 Analog Hidden Markov Models (AnHMM)

2.4 Numerical Experiments

2.4.1 Chaotic models

2.4.2 Experimental details

2.4.3 Experiments with Lorenz-96 model

2.4.4 Experiments with Lorenz-63 model

2.5 Conclusions and perspectives

**II Dealing with high-dimensional fields: The Multiscale Analog Data Assimilation **

**3 Interpolation of missing data in Sea Surface Temperature maps **

3.1 The Multiscale Analog Data Assimilation

3.1.1 Motivation

3.1.2 Multi-scale data-driven priors

3.2 Missing data interpolation in Sea Surface Temperature maps

3.3 Problem statement and related work

3.3.1 Model-driven approaches

3.3.2 Data-driven approaches

3.4 Application of the patch-based AnDA

3.5 Results

3.5.1 Experimental setting

3.5.2 Interpolation performance

3.6 Conclusion

**4 Analog Spatio-Temporal Interpolation of Sea Level Anomalies from Altimeterderived Data **

4.1 Motivation

4.2 Introduction

4.3 Data: OFES (OGCM for the Earth Simulator)

4.3.1 Model simulation data

4.3.2 Along track data

4.4 Analog reconstruction for altimeter-derived SLA

4.4.1 Patch-based state-space formulation

4.4.2 Patch-based analog dynamical models

4.4.3 Numerical resolution

4.5 Results

4.5.1 Experimental setting

4.5.2 SLA reconstruction from noise-free along-track data

4.5.3 SLA reconstruction from noisy along-track data

4.5.4 Conditioning by auxiliary variables

4.6 Discussion and conclusion

**III Conclusion **

**5 Conclusions and Perspectives **

5.1 Conclusion

5.2 Perspectives and Future Work

5.2.1 The Analog data assimilation and its applications

5.2.2 Machine Learning for dynamical systems

5.2.3 Deep Learning for detection and classification of eddies from SSH maps .

List of Figures

List of Tables

**Bibliography **