Neural Ordinary Differential Equations (NODEs) serve as continuous-time analogs of residual networks. They provide a system-theoretic perspective on neural network architecture design and offer natural solutions for time series modeling, forecasting, and applications where invertible neural networks are essential. However, these models suffer from slow performance due to heavy numerical solver overhead. For instance, a popular solution for training and inference of NODEs consists in using adaptive step size solvers such as the popular Dormand-Prince 5(4) (DOPRI). These solvers dynamically adjust the Number of Function Evaluations (NFE) as the equation fits the training data and becomes more complex. However, this comes at the cost of an increased number of function evaluations, which reduces computational efficiency. In this work, we propose a novel approach: making the parameters of the numerical integration scheme trainable. By doing so, the numerical scheme dynamically adapts to the dynamics of the NODE, resulting in a model that operates with a fixed NFE. We compare the proposed trainable solvers with state-of-the-art approaches, including DOPRI, for different benchmarks, including classification, density estimation, and dynamical system modeling. Overall, we report a state-of-the-art performance for all benchmarks in terms of accuracy metrics, while enhancing the computational efficiency through trainable fixed-step-size solvers. This work opens up new possibilities for practical and efficient modeling applications with NODEs.

When considering the data-driven identification of non-linear differential equations, the choice of the integration scheme to use is far from being trivial and may dramatically impact the identification problem. In this work, we discuss this aspect and propose a novel architecture that jointly learns Neural Ordinary Differential Equations (NODEs) as well as the corresponding integration schemes that would minimize the forecast of a given sequence of observations. We demonstrate its relevance with numerical experiments on non-linear dynamics, including chaotic systems.

The monitoring of waves at the ocean surface is critical for both operational needs (e.g., maritime traffic) and scientific studies (e.g., air-sea interactions). Synthetic aperture radar (SAR) Satellites provide one of the only remote sensing observations to retrieve ocean wave information on a global scale. However state-of-the-art SAR processing schemes often lead to poor inversion performance due to overly-simplistic assumptions. Here we leverage deep learning schemes to address these shortcomings. We state the targeted measurement of the ocean wave spectrum at sea surface as a neural mapping from SAR satellite observations. We exploit supervised deep learning schemes trained from a large-scale collocation dataset between real SAR observations and Wavewatch III model data. Our results emphasize for the first time how deep learning schemes can outperform the state-of-the-art analytical SAR-based inversion with an improvement in terms of mean square error greater than 65%. We analyse and discuss further the key features of the trained neural processing.

The reconstruction of sea surface geophysical variables typically relies either on Optimal Interpolation (OI) or on model-based approaches which explicitly exploit a dynamical model. While the optimal interpolation suffers from smoothing issues making it unreliable in retrieving fine scale variability, the selection and parametrization of a dynamical model, when considering model based data assimilation strategies, remains a complex issue since several trade-offs between the models complexity and its applicability in sea surface data assimilation need to be carefully addressed. In this work, and motivated by the success of artificial intelligence algorithms in various signal processing fields as well as the increasing amounts of observations and simulation datasets, we explore a data driven Koopman model and formulate the reconstruction problem as solution of the classical Kalman filter in a space of observables. The proposed architecture although linear, is shown to outperform state-of-the-art non linear data-driven filtering schemes such as the Analog Data Assimilation (AnDA).

The forecasting and assimilation of sea surface dynamics from satellite-derived data is a challenging issue. Data-driven approaches have arisen as promising schemes to fully exploit satellite observations. A key feature of sea surface dynamics is that they relate to partially-observed dynamics. Here, guided by physical and mathematical considerations of the underlying dynamics of a given system, we propose a novel neural networks architecture for the identification of ordinary differential equations (ODE) of partially observed systems. Numerical experiments for toy models and a sea level anomaly dynamics illustrate the relevance of the proposed scheme for forecasting and assimilation issues w.r.t. other state-of-the-art data-driven models.

The Chinese-French oceanography satellite, CFOSAT, was launched on October 2018. Two Ku-band scatterometers are on-board: SCAT for the wind observation and SWIM for the wave observation. This paper presents the most recent results on the SWIM data quality analysis a few months after the end of the CAL/VAL phase.

In the last years, Neural Networks have enriched the state-of-the-art in probabilistic modeling. This is principally due to the advances in deep learning which allow a better understanding of complex systems. However, the stochastic representation of spatio-temporal fields is still an open challenge that may benefit from the recent advances in probabilistic modelization. In this work, we explore neural network to derive a stochastic representation of spatio-temporal dynamical systems based on ensemble forecasting. Trough the implementation of our stochastic model in a classical Kalman filtering scheme, we demonstrate the relevance of the proposed architecture in the reconstruction of geophysical fields with respect to the state-of-the-art approaches.

The "Sea surface KInematics Multiscale monitoring" (SKIM) concept is a satellite mission designed to measure the world ocean surface currents, ice drift and ocean waves at spatial scales of 40 km [1] . It has been recently selected by the European Space Agency (ESA) as a finalist of the Earth Explorer 9 (EE9) call of proposal alongside with the FORUM concept. The onboard sensor will be a coherent rotating Ka-band radar providing Doppler measurements at 6 and 12 degrees incidence. This configuration allows for measurement of the directional wave spectrum, which yields accurate corrections of the wave-induced bias in the current measurements. Near nadir Doppler Ka-band measurements over the ocean have shown sensibilities to sea surface kinematics [2] , [3] and the surfacic motions due to waves and current are the main geophysical contributors to the measured microwave Doppler shift. However, these measured velocities have complex relations to oceanic motion and theoretical as well as numerical inspections are necessary to fully understand the undelying physical interactions. The understanding of this relation is at the heart of the SKIM mission measurement concept whose main objective is to provide the best estimate of the underlying sea surface current. The purpose of this work is to show how the strongest contribution of the geophysical Doppler shift is due to waves motion (the so called wave bais) and how it can be estimated to finally extract the current contribution. A theroretical approach is presented and compared to experimental results obtained from airborne campain (not presented in this abstract). To complete this two approaches, results of complex numerical simulations, processed on the High Performance Computer of Ifremer, involving a maximum of physical phenomena (satellite, sensor and surface kinematics, complex ocean electromagnetic scattering, ...) as well as onboard processings (range migration, pulse-pair algorithm, ...) are presented to corroborate theoretical predictions and experimental results.

In this work, we propose an alternative to the Ensemble Kalman filter through the implementation of a neural networks filtering scheme based on a parametric stochastic model. From our numerical experiment, we prove the relevance of the proposed architecture in the reconstruction of geophysical fields with respect to the state-of-the-art schemes.

Recent studies have investigated the identification of governing equations of geophysical systems from data. Here, we investigate such identification issues for ocean surface dy-namcis from ocean remote sensing data. From a methodological point of view, we address the learning of data-driven dynamical models when only provided with a noisy training dataset. We propose a novel architecture that relies on data assimilation schemes to learn the underlying dynamical model through the minimization of a reconstruction cost. We demonstrate the relevance of the proposed architecture with respect to the state-of-the-art approaches in the identification and forecasting of synthetic and real case-studies.

This paper presents the goal, organization and first results of the calibration and validation activities devoted to the Ku-band wave scatterometer SWIM in operation since the end of October 2018 after the launch of the China-France Oceanography SATellite CFOSAT.

Sentinel-1 ‘s SARs operate since April 2014 (S1-A) and April 2016 (S1-B) and routinely acquire images over coastal and open waters. Compared to its predecessor ENVISAT/ASAR, Sentinel-1 SAR offers several improvements such as a better Wave Mode imagette coverage, more systematic dual-polarizations, a new TOPSAR acquisition mode over coastal regions and improved Doppler estimator allowing higher resolution Doppler grid. Based on these additional capabilities, many algorithmic and use case scenario improvements have been tested and validated to provide a more complete ocean state view and first direct assessment of extreme events. Instrument level issues like the accuracy of the satellite restituted attitude could also be highlighted.

The GlobCurrent project (http://www.globcurrent.org) aimed to: (i) advance the quantitative estimation of ocean surface currents from satellite sensor synergy; and (ii) demonstrate impact in user-led scientific, operational and commercial applications that, in turn, would improve and strengthen the uptake of satellite measurements. It is often demonstrated that sharp gradients in the sea surface temperature (SST) and current fields and the ocean surface chlorophyll-a distribution are spatially correlated with the sea surface roughness anomaly fields at small spatial scales, in the sub-mesocale (1-10 km) to the mesoscale (30-80 km). The 2-dimensional structures manifested in the satellite observations represent evidence of the upper ocean dynamics. Whereas the quasi geostrophic assumption is valid for the upper ocean dynamics at the larger scale (> 100 km), possible triggering mechanisms for the expressions at the mesoscale-to-submesoscale may include spiraling tracers of inertial motion and the interaction of the wind-driven Ekman layer with the quasi-geostrophic current field. This latter, in turn, produces bands of downwelling (convergence) and upwelling (divergence) near fronts. A regular utilization of the sensor synergy approach with the combination of Sentinel-3, Sentinel-2 and Sentinel-l together with other satellite missions will provide a highly valuable data set for further research and development to better relate the 2-dimensional surface expressions and the upper ocean dynamics.

The reconstruction of ocean surface wave characteristics, i.e. directional energy distribution and dispersion, from ocean sunglitter images is discussed. Wave spectral analysis is performed, through the definition of linear transfer functions, to relate image brightness variations and sea surface slopes. Surface-current Doppler shift modifies the wave dispersion relation and can be obtained from cross-correlating two images and tracking glint changes over a known time. The technique is tested on photographs, taken from an aircraft in the Gulf of Mexico, and from a drone in the Black sea. The method is further extended to satellite sunglitter imagery, taking advantage of the technical peculiarities of Sentinel- 2 sensors helping to infer both the brightness gradient in the along-track direction. Wave dispersion information through the inter-channel time delay allows to reconstruct the current vector. The results demonstrate encouraging prospects of using the sunglitter imagery for the sea state monitoring, including formation of abnormally high waves and the study of wave-current interactions at different scales.

Doppler radars at all incidence angles measure mean velocities and spreads that have complex relations to oceanic motions, with opportunities to measure winds, waves, and currents. Here, we extend previous theoretical models of backscatter and Doppler using a Kirchhoff approximation and the physical optics model. We show that in Ka-band, around 12° incidence, range-resolved measurements of Doppler and backscatter provide unambiguous estimations of the wave spectrum and surface current. This property is illustrated with numerical examples and airborne data from the Air Surface Water Ocean Topography instrument. The same measurement conditions can be exploited for global ocean mapping from the low Earth orbit sensor satellite configuration.

Sentinel-1's SAR instrument offers a number of improvements with respect to its predecessor ENVISAT/ASAR such as a much better Wave Mode imagette coverage, improved Doppler estimator allowing higher resolution Doppler grid, more systematic dual-polarizations and a new TOPSAR acquisition mode. In the context of SEOM program, the Sentinel-1 ocean study offers to take advantage of these new capabilities to improve the retrieval of ocean sea state parameters: surface wind fields, directional wave spectrum, total significant wave height and surface currents. The study also tackles the ability to conduct a synergetic retrieval scheme in which the mutual effects of sea state components are taken into account.

Processing and analysis of Doppler information from Sentinel 1A Interferometric Wide (IW) and Extra Wide (EW) modes are performed for assessing the capabilities of mapping ocean surface current field. Data from Agulhas (South-Africa) and Norwegian Coast are used in combination with numerical models, higher-order satellite products, and Lagrangian drifters. Results show strong Doppler signal and dynamics from coastal areas caused by a mixture of surface current and wind/wave induced drifts at a spatial resolution of around 2 km2in IW mode and 4km2in EW mode. Doppler values of up to 70 Hz are observed, corresponding to a surface drift velocity of 3.5 m/s. The Sentinel 1 retrieved surface current component is in reasonable agreement with the circulation models and drifter measurements. Surface current values up to 1.5 m/s are observed in the central Agulhas current, with a standard deviation of around 0.39 m/s with respect to Lagrangian drifters.

At the horizon 2020, 2-D ocean wave measurements from 4 different satellites will be available. It will be a world first. Anticipating the large number of available data and the specificity of each sensor, we will have to define new methodologies to combine all different observations. This will offer new opportunities to describe ocean waves at global scale with improved time-space resolutions and will open new perspectives to study ocean waves. This paper deals with the perspectives for combining and exploiting ocean wave spectra measured from different space missions such as Sentinel-1, CFOSAT and HY-3. An example of application is shown on waves generated by hurricane Kilo.

An effective methodology using satellite high-resolution polarized information to interpret and quantitatively assess various surface ocean phenomena is suggested. Using a sample RADARSAT-2 quad-polarization ocean synthetic aperture radar (SAR) scene, the dual co-polarization (VV and HH) radar data are combined into polarization difference, polarization ratio, and nonpolarized components. As demonstrated, these field quantities provide means to distinguish Bragg scattering mechanism and radar returns from breaking waves. As shown, quantitative characteristics of the surface manifestation of ocean currents, slicks, and wind field features in these dual co-polarization properties are very different and may be effectively used in the development of new SAR detection and discrimination algorithms.

The synthetic aperture radar (SAR) Doppler centroid has been used to estimate the scatter line-of-sight radar velocity. In weak to moderate ocean surface current environment, the SAR Doppler centroid is dominated by the directionality and strength of wave-induced ocean surface displacements. In this paper, we show how this sea state signature can be used to improve surface wind retrieval from SAR. Doppler shifts of C-band radar return signals from the ocean are thoroughly investigated by colocating wind measurements from the ASCAT scatterometer with Doppler centroid anomalies retrieved from Envisat ASAR. An empirical geophysical model function (CDOP) is derived, predicting Doppler shifts at both VV and HH polarization as function of wind speed, radar incidence angle, and wind direction with respect to radar look direction. This function is used into a Bayesian inversion scheme in combination with wind from a priori forecast model and the normalized radar cross section (NRCS). The benefit of Doppler for SAR wind retrieval is shown in complex meteorological situations such as atmospheric fronts or low pressure systems. Using in situ information, validation reveals that this method helps to improve the wind direction retrieval. Uncertainty of the calibration of Doppler shift from Envisat ASAR hampers the inversion scheme in cases where NRCS and model wind are accurate and in close agreement. The method is however very promising with respect of future SAR missions, in particular Sentinel-1, where the Doppler centroid anomaly will be more robustly retrieved.

In this paper a method for calculating the antenna contribution to the Doppler Centroid is presented. The Antenna Doppler Contribution (ADC) is originated by the non idealities affecting the TR modules of the active antenna arrays. It is an undesired effect for SAR applications requiring stringent performances on the Doppler Centroid control capability. The first part of this paper introduces the problem and shows how it is possible to predict the ADC by exploiting a suitable Antenna Model. The second part of the paper presents a comparison between the ADC calculated with an Antenna Model and the ADC retrieved from real ASAR data.

The processing steps and error corrections needed to retrieve estimates of sea surface range Doppler velocities from Envisat Advanced Synthetic Aperture Radar Wide Swath Medium resolution image products are presented. Retrieval accuracies based on examination of the corrected Doppler shift measurements are assessed. The root-mean-square errors of the Doppler shift after bias corrections are found to be 4.7 and 3.9 Hz in VV and HH polarizations, respectively. At 35° incidence angle, this corresponds to horizontal Doppler velocities of 23 and 19 cm/s.

C-band SAR is well adapted to detect ocean pollution because backscatter is reduced by oil slicks. They appear as dark patches on the image as the increase of viscosity due to the presence of oil damps gravity-capillarity waves. In order to detect these dark patches, we use algorithms based on filters, gradients, and morphological mathematics, and a new approach based on ocean surface characterization. We have tested these methods on ERS and ENVISAT images acquired during Prestige tanker wreckage and the results are compared with aircraft surveys. We conclude that slicks with high contrast and simple shape are easily detected using basic algorithms, but most of the time, other methods are needed. The ocean characterization method is a way to follow for improving oil slick detection and providing decision aids for classification step