Searcharxiv⌕ Search

arXiv subjects

Bertrand Chapron

Publications and source records attributed to Bertrand Chapron.

At least 37 records · Page 2Linked to original sources
arXiv2023

Multimodal 4DVarNets for the reconstruction of sea surface dynamics from SST-SSH synergies

Due to the irregular space-time sampling of sea surface observations, the reconstruction of sea surface dynamics is a challenging inverse problem. While satellite altimetry provides a direct observation of the sea surface height (SSH), which relates to the divergence-free component of sea surface currents, the associated sampling pattern prevents from retrieving fine-scale sea surface dynamics, typically below a 10-day time scale. By contrast, other satellite sensors provide higher-resolution observations of sea surface tracers such as sea surface temperature (SST). Multimodal inversion schemes then arise as an appealing strategy. Though theoretical evidence supports the existence of an explicit relationship between sea surface temperature and sea surface dynamics under specific dynamical regimes, the generalization to the variety of upper ocean dynamical regimes is complex. Here, we investigate this issue from a physics-informed learning perspective. We introduce a trainable multimodal inversion scheme for the reconstruction of sea surface dynamics from multi-source satellite-derived observations. The proposed 4DVarNet schemes combine a variational formulation involving trainable observation and a priori terms with a trainable gradient-based solver. We report an application to the reconstruction of the divergence-free component of sea surface dynamics from satellite-derived SSH and SST data. An observing system simulation experiment for a Gulf Stream region supports the relevance of our approach compared with state-of-the-art schemes. We report relative improvement greater than 50% compared with the operational altimetry product in terms of root mean square error and resolved space-time scales. We discuss further the application and extension of the proposed approach for the reconstruction and forecasting of geophysical dynamics from irregularly-sampled satellite observations.

eess.IV↗
arXiv2022

Guided Unsupervised Learning by Subaperture Decomposition for Ocean SAR Image Retrieval

Spaceborne synthetic aperture radar (SAR) can provide accurate images of the ocean surface roughness day-or-night in nearly all weather conditions, being an unique asset for many geophysical applications. Considering the huge amount of data daily acquired by satellites, automated techniques for physical features extraction are needed. Even if supervised deep learning methods attain state-of-the-art results, they require great amount of labeled data, which are difficult and excessively expensive to acquire for ocean SAR imagery. To this end, we use the subaperture decomposition (SD) algorithm to enhance the unsupervised learning retrieval on the ocean surface, empowering ocean researchers to search into large ocean databases. We empirically prove that SD improve the retrieval precision with over 20% for an unsupervised transformer auto-encoder network. Moreover, we show that SD brings important performance boost when Doppler centroid images are used as input data, leading the way to new unsupervised physics guided retrieval algorithms.

cs.CV↗
arXiv2022

Guided deep learning by subaperture decomposition: ocean patterns from SAR imagery

Spaceborne synthetic aperture radar can provide meters scale images of the ocean surface roughness day or night in nearly all weather conditions. This makes it a unique asset for many geophysical applications. Sentinel 1 SAR wave mode vignettes have made possible to capture many important oceanic and atmospheric phenomena since 2014. However, considering the amount of data provided, expanding applications requires a strategy to automatically process and extract geophysical parameters. In this study, we propose to apply subaperture decomposition as a preprocessing stage for SAR deep learning models. Our data centring approach surpassed the baseline by 0.7, obtaining state of the art on the TenGeoPSARwv data set. In addition, we empirically showed that subaperture decomposition could bring additional information over the original vignette, by rising the number of clusters for an unsupervised segmentation method. Overall, we encourage the development of data centring approaches, showing that, data preprocessing could bring significant performance improvements over existing deep learning models.

cs.CV↗
arXiv2022

Multimodal learning-based inversion models for the space-time reconstruction of satellite-derived geophysical fields

For numerous earth observation applications, one may benefit from various satellite sensors to address the reconstruction of some process or information of interest. A variety of satellite sensors deliver observation data with different sampling patterns due satellite orbits and/or their sensitivity to atmospheric conditions (e.g., clour cover, heavy rains,...). Beyond the ability to account for irregularly-sampled observations, the definition of model-driven inversion methods is often limited to specific case-studies where one can explicitly derive a physical model to relate the different observation sources. Here, we investigate how end-to-end learning schemes provide new means to address multimodal inversion problems. The proposed scheme combines a variational formulation with trainable observation operators, {\em a priori} terms and solvers. Through an application to space oceanography, we show how this scheme can successfully extract relevant information from satellite-derived sea surface temperature images and enhance the reconstruction of sea surface currents issued from satellite altimetry data.

cs.CV↗
arXiv2022

Eigenvalues of Autocovariance Matrix: A Practical Method to Identify the Koopman Eigenfrequencies

To infer eigenvalues of the infinite-dimensional Koopman operator, we study the leading eigenvalues of the autocovariance matrix associated with a given observable of a dynamical system. For any observable $f$ for which all the time-delayed autocovariance exist, we construct a Hilbert space $\mathcal{H}_f$ and a Koopman-like operator $\mathcal{K}$ that acts on $\mathcal{H}_f$. We prove that the leading eigenvalues of the autocovariance matrix has one-to-one correspondence with the energy of $f$ that is represented by the eigenvectors of $\mathcal{K}$. The proof is associated to several representation theorems of isometric operators on a Hilbert space, and the weak-mixing property of the observables represented by the continuous spectrum. We also provide an alternative proof of the weakly mixing property. When $f$ is an observable of an ergodic dynamical system which has a finite invariant measure $μ$, $\mathcal{H}_f$ coincides with closure in $L^2(X,dμ)$ of Krylov subspace generated by $f$, and $\mathcal{K}$ coincides with the classical Koopman operator. The main theorem sheds light to the theoretical foundation of several semi-empirical methods, including singular spectrum analysis (SSA), data-adaptive harmonic analysis (DAHD), Hankel DMD and Hankel alternative view of Koopman analysis (HAVOK). It shows that, when the system is ergodic and has finite invariant measure, the leading temporal empirical orthogonal functions indeed correspond to the Koopman eigenfrequencies. A theorem-based practical methodology is then proposed to identify the eigenfrequencies of $\mathcal{K}$ from a given time series. It builds on the fact that the convergence of the renormalized eigenvalues of the Gram matrix is a necessary and sufficient condition for the existence of $\mathcal{K}-$eigenfrequencies.

math.OC↗
arXiv2022

Bounded nonlinear forecasts of partially observed geophysical systems with physics-constrained deep learning

The complexity of real-world geophysical systems is often compounded by the fact that the observed measurements depend on hidden variables. These latent variables include unresolved small scales and/or rapidly evolving processes, partially observed couplings, or forcings in coupled systems. This is the case in ocean-atmosphere dynamics, for which unknown interior dynamics can affect surface observations. The identification of computationally-relevant representations of such partially-observed and highly nonlinear systems is thus challenging and often limited to short-term forecast applications. Here, we investigate the physics-constrained learning of implicit dynamical embeddings, leveraging neural ordinary differential equation (NODE) representations. A key objective is to constrain their boundedness, which promotes the generalization of the learned dynamics to arbitrary initial condition. The proposed architecture is implemented within a deep learning framework, and its relevance is demonstrated with respect to state-of-the-art schemes for different case-studies representative of geophysical dynamics.

stat.ML↗
arXiv2022

Bridging Koopman Operator and time-series auto-correlation based Hilbert-Schmidt operator

Given a stationary continuous-time process $f(t)$, the Hilbert-Schmidt operator $A_τ$ can be defined for every finite $τ$\cite{Vautard1989SingularSA}. Let $λ_{τ,i}$ be the eigenvalues of $A_τ$ with descending order. In this article, a Hilbert space $\mathcal{H}_f$ and the (time-shift) continuous one-parameter semigroup of isometries $\mathcal{K}^s$ are defined. Let $\{v_i, i\in\mathbb{N}\}$ be the eigenvectors of $\mathcal{K}^s$ for all $s\geq 0$. Let $f = \displaystyle\sum_{i=1}^{\infty}a_iv_i + f^{\perp}$ be the orthogonal decomposition with descending $|a_i|$. We prove that $\displaystyle\lim_{τ\to\infty}λ_{τ,i} = |a_i|^2$. The continuous one-parameter semigroup $\{\mathcal{K}^s: s\geq 0\}$ is equivalent, almost surely, to the classical Koopman one-parameter semigroup defined on $L^2(X,ν)$, if the dynamical system is ergodic and has invariant measure $ν$ on the phase space $X$.

math.OC↗
arXiv2021

Exploitation of error correlation in a large analysis validation: GlobCurrent case study

An assessment of variance in ocean current signal and noise shared by in situ observations (drifters) and a large gridded analysis (GlobCurrent) is sought as a function of day of the year for 1993-2015 and across a broad spectrum of current speed. Regardless of the division of collocations, it is difficult to claim that any synoptic assessment can be based on independent observations. Instead, a measurement model that departs from ordinary linear regression by accommodating error correlation is proposed. The interpretation of independence is explored by applying Fuller's (1987) concept of equation and measurement error to a division of error into shared (correlated) and unshared (uncorrelated) components, respectively. The resulting division of variance in the new model favours noise. Ocean current shared (equation) error is of comparable magnitude to unshared (measurement) error and the latter is, for GlobCurrent and drifters respectively, comparable to ordinary and reverse linear regression. Although signal variance appears to be small, its utility as a measure of agreement between two variates is highlighted. Sparse collocations that sample a dense grid permit a first order autoregressive form of measurement model to be considered, including parameterizations of analysis-in situ error cross-correlation and analysis temporal error autocorrelation. The former (cross-correlation) is an equation error term that accommodates error shared by both GlobCurrent and drifters. The latter (autocorrelation) facilitates an identification and retrieval of all model parameters. Solutions are sought using a prescribed calibration between GlobCurrent and drifters (by variance matching). Because the true current variance of GlobCurrent and drifters is small, signal to noise ratio is near zero at best. This is particularly evident for moderate current speed and meridional current component.

stat.AP↗
arXiv2021

Learning Runge-Kutta Integration Schemes for ODE Simulation and Identification

Deriving analytical solutions of ordinary differential equations is usually restricted to a small subset of problems and numerical techniques are considered. Inevitably, a numerical simulation of a differential equation will then always be distinct from a true analytical solution. An efficient integration scheme shall further not only provide a trajectory throughout a given state, but also be derived to ensure the generated simulation to be close to the analytical one. Consequently, several integration schemes were developed for different classes of differential equations. Unfortunately, when considering the integration of complex non-linear systems, as well as the identification of non-linear equations from data, this choice of the integration scheme is often far from being trivial. In this paper, we propose a novel framework to learn integration schemes that minimize an integration-related cost function. We demonstrate the relevance of the proposed learning-based approach for non-linear equations and include a quantitative analysis w.r.t. classical state-of-the-art integration techniques, especially where the latter may not apply.

math.NA↗
arXiv2021

The speed of breaking waves controls sea surface drag

The coupling between wind-waves and atmospheric surface layer turbulence sets surface drag. This coupling is however usually represented through a roughness length. Originally suggested on purely dimensional grounds, this roughness length does not directly correspond to a measurable physical quantity of the wind-and-wave system. Here, to go beyond this representation, we formalize ideas underlying the Beaufort scale by quantifying the velocity of breaking short waves that are the most coupled to near-surface wind. This velocity increases with wind speed, reflecting the fact that stronger winds can be visually identified by longer (and faster) breakers becoming predominant on the sea surface. A phenomenological turbulence model further shows that this velocity is associated with breaking waves that impede the most the formation of turbulent eddies. Scales of such eddies are then constrained inside a so-called roughness sub-layer. Unlike previous theoretical developments, the proposed breaker velocity is a directly measurable quantity, which could be used to characterize the coupling between wind and waves using remote sensing techniques. This work provides a physical framework for new formulations of air-sea momentum exchange in which the effects of surface currents and slicks on surface drag can also be incorporated. Finally, it provides a long-sought physical explanation for the Beaufort scale: a universal link between wave breaking, wind speed and surface drag.

physics.flu-dyn↗
arXiv2020

Learning Variational Data Assimilation Models and Solvers

This paper addresses variational data assimilation from a learning point of view. Data assimilation aims to reconstruct the time evolution of some state given a series of observations, possibly noisy and irregularly-sampled. Using automatic differentiation tools embedded in deep learning frameworks, we introduce end-to-end neural network architectures for data assimilation. It comprises two key components: a variational model and a gradient-based solver both implemented as neural networks. A key feature of the proposed end-to-end learning architecture is that we may train the NN models using both supervised and unsupervised strategies. Our numerical experiments on Lorenz-63 and Lorenz-96 systems report significant gain w.r.t. a classic gradient-based minimization of the variational cost both in terms of reconstruction performance and optimization complexity. Intriguingly, we also show that the variational models issued from the true Lorenz-63 and Lorenz-96 ODE representations may not lead to the best reconstruction performance. We believe these results may open new research avenues for the specification of assimilation models in geoscience.

physics.comp-ph↗
arXiv2019

Learning Latent Dynamics for Partially-Observed Chaotic Systems

This paper addresses the data-driven identification of latent dynamical representations of partially-observed systems, i.e., dynamical systems for which some components are never observed, with an emphasis on forecasting applications, including long-term asymptotic patterns. Whereas state-of-the-art data-driven approaches rely on delay embeddings and linear decompositions of the underlying operators, we introduce a framework based on the data-driven identification of an augmented state-space model using a neural-network-based representation. For a given training dataset, it amounts to jointly learn an ODE (Ordinary Differential Equation) representation in the latent space and reconstructing latent states. Through numerical experiments, we demonstrate the relevance of the proposed framework w.r.t. state-of-the-art approaches in terms of short-term forecasting performance and long-term behaviour. We further discuss how the proposed framework relates to Koopman operator theory and Takens' embedding theorem.

stat.ML↗
arXiv2019

Wind, wave and current interactions appear key for quantifying cross-shelf transport and carbon export; new knowledge and the potential of SKIM to enable monitoring

The highly heterogeneous and biologically active continental shelf-seas are important components of the oceanic carbon sink. Carbon rich water from shelf-seas is exported at depth to the open ocean, a process known as the continental shelf pump, with open-ocean surface water moving (transported) onto the shelf driving the export at depth. Existing methods to study shelf-wide exchange focus on the wind or geostrophic currents, often ignoring their combined effect, spatial heterogeniety or any other ageostrophic components. Here we investigate the influence that wind, wave and current interactions can have on surface transport and carbon export across continental shelves. Using a 21 year global re-analysis dataset we confirm that geostrophic and wind driven Ekman processes are important for the transport of water onto shelf seas; but the dominance of each is location and season dependent. A global wave model re-analysis shows that one type of ageostrophic flow, Stokes drift due to waves, can also be significant. A regional case study using two submesocale model simulations identifies that up to 100% of the cross-shelf surface flow in European seas can be due to ageostrophic components. Using these results and grouping shelf-seas based on their observed carbon accumulation rates shows that differences in rates are consistent with imbalances between the processes driving atmosphere-ocean exchange at the surface and those driving carbon export at depth. Therefore expected future changes in wind and wave climate support the need to monitor cross-shelf transport and the size of the continental shelf-sea carbon pump. The results presented show that the Sea Surface Kinematics Multiscale monitoring satellite mission (SKIM), will be capable of providing measurements of the total cross-shelf current, which are now needed to enable routine monitoring of the global continental shelf-sea carbon pump.

physics.ao-ph↗
arXiv2017

Stochastic modelling and diffusion modes for proper orthogonal decomposition models and small-scale flow analysis

We present here a new stochastic modelling in the constitution of fluid flow reduced-order models. This framework introduces a spatially inhomogeneous random field to represent the unresolved small-scale velocity component. Such a decomposition of the velocity in terms of a smooth large-scale velocity component and a rough, highly oscillating, component gives rise, without any supplementary assumption, to a large-scale flow dynamics that includes a modified advection term together with an inhomogeneous diffusion term. Both of those terms, related respectively to turbophoresis and mixing effects, depend on the variance of the unre-solved small-scale velocity component. They bring to the reduced system an explicit subgrid term enabling to take into account the action of the truncated modes. Besides, a decomposition of the variance tensor in terms of diffusion modes provides a meaningful statistical representation of the stationary or nonstationary structuration of the small-scale velocity and of its action on the resolved modes. This supplies a useful tool for turbulent fluid flows data analysis. We apply this methodology to circular cylinder wake flow at Reynolds numbers Re = 300 and Re = 3900, respectively. The finite dimensional models of the wake flows reveal the energy and the anisotropy distributions of the small-scale diffusion modes. These distributions identify critical regions where corrective advection effects as well as structured energy dissipation effects take place. In providing rigorously derived subgrid terms, the proposed approach yields accurate and robust temporal reconstruction of the low-dimensional models.

physics.flu-dyn↗
arXiv2017

Geophysical flows under location uncertainty, Part I Random transport and general models

A stochastic flow representation is considered with the Eulerian velocity decomposed between a smooth large scale component and a rough small-scale turbulent component. The latter is specified as a random field uncorrelated in time. Subsequently, the material derivative is modified and leads to a stochastic version of the material derivative to include a drift correction , an inhomogeneous and anisotropic diffusion, and a multiplicative noise. As derived, this stochastic transport exhibits a remarkable energy conservation property for any realizations. As demonstrated, this pivotal operator further provides elegant means to derive stochastic formulations of classical representations of geophysical flow dynamics.

physics.geo-ph↗
arXiv2016

Geophysical flows under location uncertainty, Part II: Quasi-geostrophy and efficient ensemble spreading

Models under location uncertainty are derived assuming that a component of the velocity is uncorrelated in time. The material derivative is accordingly modified to include an advection correction, inhomogeneous and anisotropic diffusion terms and a multiplicative noise contribution. In this paper, simplified geophysical dynamics are derived from a Boussinesq model under location uncertainty. Invoking usual scaling approximations and a moderate influence of the subgrid terms, stochastic formulations are obtained for the stratified Quasi-Geostrophy (QG) and the Surface Quasi-Geostrophy (SQG) models. Based on numerical simulations, benefits of the proposed stochastic formalism are demonstrated. A single realization of models under location uncertainty can restore small-scale structures. An ensemble of realizations further helps to assess model error prediction and outperforms perturbed deterministic models by one order of magnitude. Such a high uncertainty quantification skill is of primary interests for assimilation ensemble methods. MATLAB code examples are available online.

physics.ao-ph↗
arXiv2016

Geophysical flows under location uncertainty, Part III: SQG and frontal dynamics under strong turbulence conditions

Models under location uncertainty are derived assuming that a component of the velocity is uncorrelated in time. The material derivative is accordingly modified to include an advection correction, inhomogeneous and anisotropic diffusion terms and a multiplicative noise contribution. This change can be consitently applied to all fluid dynamics evolution laws. This paper continues to explore benefits of this framework and consequences of specific scaling assumptions. Starting from a Boussinesq model under location uncertainty, a model is developed to describe a mesoscale flow subject to a strong underlying submesoscale activity. As obtained, the geostrophic balance is modified and the Quasi-Geostrophic (QG) assumptions remarkably lead to a zero Potential Vorticity (PV). The ensuing Surface Quasi-Geostrophic (SQG) model provides a simple diagnosis of warm frontolysis and cold frontogenesis.

physics.ao-ph↗