Reconstructing geophysical fields from noisy and partial remote sensing observations is a classical problem well studied in the literature. Data assimilation is one class of popular methods to address this issue, and is done through the use of classical stochastic filtering techniques, such as ensemble Kalman or particle filters and smoothers. They proceed by an online evaluation of the physical modelin order to provide a forecast for the state. Therefore, the performanceof data assimilation heavily relies on the definition of the physical model. In contrast, the amount of observation and simulation data has grown very quickly in the last decades. This thesis focuses on performing data assimilation in a data-driven way and this without having access to explicit model equations. The main contribution of this thesis lies in developing and evaluating the Analog Data Assimilation(AnDA), which combines analog methods (nearest neighbors search) and stochastic filtering methods (Kalman filters, particle filters, Hidden Markov Models). Through applications to both simplified chaotic models and real ocean remote sensing case-studies (sea surface temperature, along-track sea level anomalies), we demonstrate the relevance of AnDA for missing data interpolation of nonlinear and high dimensional dynamical systems from irregularly-sampled and noisy observations. Driven by the rise of machine learning in the recent years, the last part of this thesis is dedicated to the development of deep learning models for the detection and tracking of ocean eddies from multi-source and/or multi-temporal data (e.g., SST-SSH), the general objective being to outperform expert-based approaches.