Article

# Analog Data Assimilation of Along-Track Nadir and Wide-Swath SWOT Altimetry Observations in the Western Mediterranean Sea

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## Abstract

The growing availability of ocean data brought forth by recent advancements in remote sensing, in situ measurements, and numerical models supports the development of data-driven strategies as a powerful, computationally efficient alternative to model-based approaches for the interpolation of high-resolution, gap-free, regularly gridded sea surface geophysical fields from partial satellite-derived observations. In this paper, we investigate such data-driven strategies for the spatio-temporal interpolation of sea level anomaly (SLA) fields in the Western Mediterranean Sea from satellite-derived altimetry data. We introduce and evaluate the analog data assimilation (AnDA) framework, which exploits patch-based analog forecasting operators within a classic Kalman-based data assimilation scheme. With a view toward the upcoming wide-swath surface water and ocean topography (SWOT) mission, two different types of altimetry data are assimilated: along-track nadir data and wide-swath SWOT altimetry data. Using an observing system simulation experiment, we demonstrate the relevance of AnDA as an improved interpolation method, particularly for mesoscale features in the 20- to 100-km horizontal scale range. Results report an SLA reconstruction RMSE (correlation) improvement of 42% (14%) with respect to optimal interpolation, and show a clear gain when the joint assimilation of SWOT and along-track nadir observations are considered.

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... Lguensat et al. (2017), who used machine learning (ML) methods in the Mediterranean, South China sea area (Lguensat et al., 2019b), and the Gulf of Mexico (Zhen et al., 2020) through principal component analysis (PCA), K nearest neighbor (KNN), and k-dimensional tree (KD-Tree) technologies, introduced a data-driven mapping method named AnDA (Lguensat et al., 2017). Furthermore, Lopez-Radcenco et al. (2019) extended the AnDA method to the mapping for multi-satellitecombined observations, SWOT observing system simulation experiments (OSSEs)-simulated data, as well as the combination of observations of nadir altimeter satellites and SWOT, and then obtained higher-accuracy SSH grid data products than the result of the OI method (Lopez-Radcenco et al., 2019). By utilizing deep learning (DL) for the DI theory verification in North Atlantic regions, Lguensat et al. (2019a) proved that the combination of the DI method and DL is feasible for data mapping. ...
... Lguensat et al. (2017), who used machine learning (ML) methods in the Mediterranean, South China sea area (Lguensat et al., 2019b), and the Gulf of Mexico (Zhen et al., 2020) through principal component analysis (PCA), K nearest neighbor (KNN), and k-dimensional tree (KD-Tree) technologies, introduced a data-driven mapping method named AnDA (Lguensat et al., 2017). Furthermore, Lopez-Radcenco et al. (2019) extended the AnDA method to the mapping for multi-satellitecombined observations, SWOT observing system simulation experiments (OSSEs)-simulated data, as well as the combination of observations of nadir altimeter satellites and SWOT, and then obtained higher-accuracy SSH grid data products than the result of the OI method (Lopez-Radcenco et al., 2019). By utilizing deep learning (DL) for the DI theory verification in North Atlantic regions, Lguensat et al. (2019a) proved that the combination of the DI method and DL is feasible for data mapping. ...
... Being different from the classical model-driven method, the data-driven methods rely on the spatial-temporal relationship of the observations so that a data-driven method can capture the ocean phenomena that may not be accounted for in purely numerical models (Lopez-Radcenco et al., 2019). Moreover, the DI method (one of the model-driven methods) is based on the PV conservation theory. ...
Preprint
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Two-dimensional mapping of sea surface height (SSH) for future wide-swath satellite altimetry (WSA) is a challenge at present. So far, considering the utilization of data-driven methods is a new researching direction for SSH mapping. In general, the data-driven mapping methods rely on the spatial-temporal relationship of the observations. These methods require training in large volumes, and the time cost is high, especially for the WSA observations. This paper proposed the prediction neural networks for mapping (Mapping-PNN) method to improve the training efficiency and maintain stable data and mapping capabilities. By 10-year wide-swath satellite along track observing system simulation experiments (OSSEs) on the HYCOM data, the experiment results indicate that the method introduced in this paper can improve the training efficiency and meet the grid mapping expectations. Compared with other methods, the root mean squared error (RMSE) of the mapping-PNN method can be limited within the range of ∼1.8 cm, and the new method can promote the observation of the ocean phenomena scale with < ∼40 km, which reaches state of the art.
... In this context, a very active field of research now consists in taking advantage of the big amount of data and numerical simulations available to overcome these limits of conventional altimetric products, which motivate complementary developments combining high resolution remote sensing and numerical simulations. Over the last years, purely data-driven and artifical intelligence (AI)-based algorithms have just been proposed [1], [2], [3], [4], [5] to deal with problems directly related to data assimilation and operational oceanography. More specifically, promising preliminary results have been seen for the sea surface reconstruction and prediction from partial and noisy satellite observations. ...
... Because the spacetime interpolations will focus on a daily-basis temporal resolution, we also build nadir pseudo-observations with an additional strategy by accumulating observations over a time window t k ± d days centered at time t k in order to increase the daily nadir spatial sampling. As in [4], we investigate the response of the different interpolation techniques when parameter d is either set to 0 or 5, see Figures 2a and 2c. ...
... More precisely, dx f k is sampled from the Gaussian prior dx f k |dx a k−1 ∼ N (µ k , Σ k ), where the mean µ k and the covariance matrix Σ k are estimated using the so-called locally linear model [1], i.e. a weighted linear regression between the K nearest analogs and their successors. As in [4], a patch-based version of AnDA coupled with an EOF-based representation of the individual patches is used. The anomaly field dx is splitted into 169 vectorized patches p(s, t) of sizes 1 • × 1 • , corresponding to 20 pixels × 20 pixels, with overlapping areas of 5 pixels. ...
Conference Paper
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Over the last years, a very active field of research aims at exploring new data-driven and learning-based methodologies to propose computationally efficient strategies able to benefit from the large amount of observational remote sensing and numerical simulations for the reconstruction, interpolation and prediction of high-resolution derived products of geophysical fields. In this paper, we investigate how they might help to solve for the oversmoothing of the state-of-the-art optimal interpolation (OI) techniques in the reconstruction of sea surface height (SSH) spatio-temporal fields. We focus on a small region, part of the GULFSTREAM and mainly driven by energetic mesoscale dynamics. Based on an Observation System Simulation Experiment (OSSE), we will use the NATL60 high resolution deterministic ocean simulation of the North Atlantic to generate two types of pseudo altimetric observational dataset: along-track nadir data for the current capabilities of the observation system and wide-swath SWOT data in the context of the upcoming SWOT mission. We briefly introduce the analog data assimilation (AnDA), an up-to-date version of the DINEOF algorithm, and a new NN-based end-to-end learning framework for the representation of spatio-temporal irregulary-sampled data. We evaluate how some of these methods are a significant improvements, particularly by catching up the small scales ranging up to 30-40km, inaccessible by the conventional methods so far. A clear gain is also demonstrated when assimilating jointly wide-swath SWOT and (agreggated) along-track nadir observations.
... (a) Conventional along-track altimetry (b) Wide-swath SWOT altimetry Figure 6.3 -Comparison of classic nadir along-track observations (6.3a) and off-nadir wide-swath SWOT observations (6.3b). Adapted from [170]. c 2019 IEEE. ...
... For all the reported experiments, performance is measured by means of the root mean squared error (RMSE, in m) (8.1c) generated from ground-truth high-resolution SLA fields (8.1a) using real satellite tracks spatio-temporal locations and the SWOT simulator, respectively. Adapted from [170]. c 2019 IEEE. ...
... c 2019 IEEE. [170]. c 2019 IEEE. ...
Thesis
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In the last few decades, the ever-growing availability of multi-source ocean remote sensing data has been a key factor for improving our understanding of upper ocean dynamics. In this regard, developing efficient approaches to exploit these datasets is of major importance. Particularly, the decomposition of geophysical processes into relevant modes is a key issue for characterization, forecasting and reconstruction problems. Inspired by recent advances in blind source separation, we aim, in the first part of this thesis dissertation, at extending non-negative blind source separation models to the problem of the observation-based characterization and decomposition of linear operators or transfer functions between variables of interest. We develop mathematically sound and computationally efficient schemes. We illustrate the relevance of the proposed decomposition models in different applications involving the analysis and forecasting of geophysical dynamics. Subsequently, given that the ever-increasing availability of multi-source datasets supports the exploration of data-driven alternatives to classical model-driven formulations, we explore recently introduced data-driven models for the interpolation of geophysical fields from irregularly sampled satellite-derived observations. Importantly, with a view towards the future SWOT mission, the first satellite mission to produce complete two-dimensional wide-swath satellite altimetry observations, we focus on assessing the extent to which SWOT data may lead to an improved reconstruction of altimetry fields.
... Indeed, some recent studies, (e.g. Lopez-Radcenco et al., 2019), demonstrated that accumulating wideswath observations over multiple days may lead to higher reconstruction errors and reduced forecasting performance in the Western Mediterranean due to wide-swath observations, which might be capturing moving/changing structures and features multiple times. This may lead to the reconstruction/forecast of spurious structures. ...
... Due to its wide swath coverage, SWOT has a definite advantage in capturing the synoptic oceanic features over the nadir altimeters. This data when assimilated in numerical ocean models should definitely help to improve upon the existing accuracies in ocean state forecast (Lopez-Radcenco et al. 2019). ...
Article
Conventional nadir looking altimeters make along track measurements on a line and mapped sea level anomaly (SLA) information is obtained using a combination of several such altimeters (Jason, SARAL, Cryosat etc.). Mapping techniques, in general, introduce a lot of uncertainties in sea level representation and sub-mesoscale variability. Surface Water and Ocean Topography (SWOT) mission, based on radar interferometry, will measure SLA along wide swath thus providing detailed ocean information. This study aims to evaluate the advantages of SWOT measurements over nadir looking altimeters by making use of SWOT-simulator tool in the Bay of Bengal (BoB) region. Although, BoB is a small basin but interestingly it is full of mesoscale and sub-mesoscale features. The study performs several sensitivity experiments to allow a comparison of gridded SLA product from SWOT with the product from a constellation of nadir altimeters. Space-time scales for mapping the SLA from SWOT were obtained by performing a series of sensitivity experiments involving different spatial resolutions and temporal sampling. Sensitivity to different type of errors on the quality of mapped SLA fields from nadir-altimeters and SWOT is also carried out. In case of SWOT, mapped SLA fields generated using correlated noise were better as compared to the maps that were generated by making an assumption that the noise is uncorrelated. It is found that gridded SLA from SWOT have less error in the eddy dominant (high variability) regions as compared to the mapped SLA field from nadir altimeters, which perform better in the regions of low SLA variability. Apart from this, the position and strength of mesoscale eddies is well resolved by SWOT-mapped SLA fields as compared to nadir-altimeter mapped fields.
... The interpolation and difference computation algorithms should be studied further to improve accuracy of vertical deflection computation. Methods of [41,42] may be helpful on this topic. ...
... According to Figure 5, gravity anomaly errors have a nearly linear relationship with InRA phase errors, which validates Equations (12) and (18). In order to obtain gravity anomalies with high accuracy, wide-swath altimetry systems should have high interferometric phase accuracy, which can be achieved by multi-looking [24,41]. Multi-looking reduces phase error by averaging a squared number of pixels. ...
Article
Full-text available
The traditional altimetry satellite, which is based on pulse-limited radar altimeter, only measures ocean surface heights along tracks; hence, leads to poorer accuracy in the east component of the vertical deflections compared to the north component, which in turn limits the final accuracy of the marine gravity field inversion. Wide-swath altimetry using radar interferometry can measure ocean surface heights in two dimensions and, thus, can be used to compute vertical deflections in an arbitrary direction with the same accuracy. This paper aims to investigate the impact of Interferometric Radar Altimeter (InRA) errors on gravity field inversion. The error propagation between gravity anomalies and InRA measurements is analyzed, and formulas of their relationship are given. By giving a group of possible InRA parameters, numerical simulations are conducted to analyze the accuracy of gravity anomaly inversion. The results show that the accuracy of the gravity anomalies is mainly influenced by the phase errors of InRA; and the errors of gravity anomalies have a linear approximation relationship with the phase errors. The results also show that the east component of the vertical deflections has almost the same accuracy as the north component.
... The space-time sampling of satellite altimeters will 20 however still remain scarce for a long time, which has motivated a recent research literature towards the improvement of the interpolation of satellite-derived SSH fields, see e.g. (Lopez-Radcenco et al., 2019;Lguensat et al., 2017;Beauchamp et al., 2021;Ballarotta et al., 2019). ...
Preprint
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The reconstruction of sea surface currents from satellite altimeter data is a key challenge in spatial oceanography, especially with the upcoming wide-swath SWOT (Surface Ocean and Water Topography) altimeter mission. Operational systems however generally fail to retrieve mesoscale dynamics for horizontal scales below 100 km and time-scale below 10 days. Here, we address this challenge through the 4DVarnet framework, an end-to-end neural scheme backed on a variational data assimilation formulation. We introduce a parametrization of the 4DVarNet scheme dedicated to the space-time interpolation of satellite altimeter data. Within an observing system simulation experiment (NATL60), we demonstrate the relevance of the proposed approach both for nadir and nadir+swot altimeter configurations for two contrasted case-study regions in terms of upper ocean dynamics. We report relative improvement with respect to the operational optimal interpolation between 30 % and 60 % in terms of reconstruction error. Interestingly, for the nadir+swot altimeter configuration, we reach resolved space-time scales below 70 km and 7 days. The code is open-source to enable reproductibility and future collaborative developments. Beyond its applicability to large-scale domains, we also address uncertainty quantification issues and generalization properties of the proposed learning setting. We discuss further future research avenues and extensions to other ocean data assimilation and space oceanography challenges.
... The space-time sampling of satellite altimeters will however still remain scarce for a long time, which has motivated a recent research literature towards the improvement of the interpolation of satellite-derived SSH fields, see e.g. (Lopez-Radcenco et al., 2019;Lguensat et al., 2017;Beauchamp et al., 2021;Ballarotta et al., 2019). ...
Preprint
Full-text available
The reconstruction of sea surface currents from satellite altimeter data is a key challenge in spatial oceanography, especially with the upcoming wide-swath SWOT (Surface Ocean and Water Topography) altimeter mission. Operational systems however generally fail to retrieve mesoscale dynamics for horizontal scales below 100km and time-scale below 10 days. Here, we address this challenge through the 4DVarnet framework, an end-to-end neural scheme backed on a variational data assimilation formulation. We introduce a parametrization of the 4DVarNet scheme dedicated to the space-time interpolation of satellite altimeter data. Within an observing system simulation experiment (NATL60), we demonstrate the relevance of the proposed approach both for nadir and nadir+swot altimeter configurations for two contrasted case-study regions in terms of upper ocean dynamics. We report relative improvement with respect to the operational optimal interpolation between 30% and 60% in terms of reconstruction error. Interestingly, for the nadir+swot altimeter configuration, we reach resolved space-time scales below 70km and 7days. The code is open-source to enable reproductibility and future collaborative developments. Beyond its applicability to large-scale domains, we also address uncertainty quantification issues and generalization properties of the proposed learning setting. We discuss further future research avenues and extensions to other ocean data assimilation and space oceanography challenges.
... Missing data problems are very common in SWoT [31,32]. The main reasons behind data loss are limited resources, noise, collision, and accidental damage to the wireless system. ...
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Sea Surface Height (SSH) observations describing scales in the range 10 - 100 km are crucial to better understand energy transfers across scales in the open ocean and to quantify vertical exchanges of heat and biogeochemical tracers. The Surface Water Ocean Topography (SWOT) mission is a new wide-swath altimetric satellite which is planned to be launched in 2022. SWOT will provide information on SSH at a kilometric resolution, but uncertainties due to various sources of errors will challenge our capacity to extract the physical signal of structures below a few tens of kilometers. Filtering SWOT noise and errors is a key step towards an optimal interpretation of the data.The aim of this study is to explore image de-noising techniques to assess the capabilities of the future SWOT data to resolve the oceanic fine scales. Pseudo-SWOT data are generated with the SWOT simulator for Ocean Science, which uses as input the SSH outputs from high-resolution Ocean General Circulation Models (OGCMs). Several de-noising techniques are tested, to find the one that renders the most accurate SSH and its derivatives fields while preserving the magnitude and shape of the oceanic features present. The techniques are evaluated based on the root mean square error, spectra and other diagnostics.In Chapter 3, the pseudo-SWOT data for the Science phase is analyzed to assess the capabilities of SWOT to resolve the meso- and submesoscale in the western Mediterranean. A Laplacian diffusion de-noising technique is implemented allowing to recover SSH, geostrophic velocity and relative vorticity down to 40 - 60 km. This first step allowed to adequately observe the mesoscale, but space is left for improvement at the submesoscale, specially in better preserving the intensity of the SSH signal.In Chapter 4, another de-noising technique is explored and implemented in the same region for the satellite's fast-sampling phase. This technique is motivated by recent advances in data assimilation techniques to remove spatially correlated errors based on SSH and its derivatives. It aims at retrieving accurate SSH derivatives, by recovering their structure and preserving their magnitude. A variational method is implemented which can penalize the SSH derivatives of first, second, third order or a combination of them. We find that the best parameterization is based on a second order penalization, and find the optimal parameters of this setup. Thanks to this technique the wavelengths resolved by SWOT in this region are reduced by a factor of 2, whilst preserving the magnitude of the SSH fields and its derivatives.In Chapter 5, we investigate the finest spatial scale that SWOT could resolve after de-noising in several regions, seasons and using different OGCMs. Our study focuses on different regions and seasons in order to document the variety of regimes that SWOT will sample. The de-noising algorithm performs well even in the presence of intense unbalanced motions, and it systematically reduces the smallest resolvable wavelength. Advanced de-noising algorithms also allow to reliably reconstruct SSH gradients (related to geostrophic velocities) and second order derivatives (related to geostrophic vorticity). Our results also show that a significant uncertainty remains about SWOT's finest resolved scale in a given region and season because of the large spread in the level of variance predicted among our high-resolution ocean model simulations.The de-noising technique developed, implemented and tested in this doctoral thesis allows to recover, in some cases, SWOT spatial scales as low as 15 km. This method is a very useful contribution to achieving the objectives of the SWOT mission. The results found will help better understand the ocean's dynamics and oceanic features and their role in the climate system.
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Over the last few years, a very active field of research has aimed at exploring new data-driven and learning-based methodologies to propose computationally efficient strategies able to benefit from the large amount of observational remote sensing and numerical simulations for the reconstruction, interpolation and prediction of high-resolution derived products of geophysical fields. In this paper, we investigate how they might help to solve for the oversmoothing of the state-of-the-art optimal interpolation (OI) techniques in the reconstruction of sea surface height (SSH) spatio-temporal fields. We focus on two small 10°×10° GULFSTREAM and 8°×10° OSMOSIS regions, part of the North Atlantic basin: the GULFSTREAM area is mainly driven by energetic mesoscale dynamics, while OSMOSIS is less energetic but with more noticeable small spatial patterns. Based on observation system simulation experiments (OSSE), we used a NATL60 high resolution deterministic ocean simulation of the North Atlantic to generate two types of pseudo-altimetric observational dataset: along-track nadir data for the current capabilities of the observation system and wide-swath SWOT data in the context of the upcoming SWOT (Surface Water Ocean Topography) mission. We briefly introduce the analog data assimilation (AnDA), an up-to-date version of the DINEOF algorithm, and a new neural networks-based end-to-end learning framework for the representation of spatio-temporal irregularly-sampled data. The main objective of this paper consists of providing a thorough intercomparison exercise with appropriate benchmarking metrics to assess whether these approaches help to improve the SSH altimetric interpolation problem and to identify which one performs best in this context. We demonstrate how the newly introduced NN method is a significant improvement with a plug-and-play implementation and its ability to catch up the small scales ranging up to 40 km, inaccessible by the conventional methods so far. A clear gain is also demonstrated when assimilating jointly wide-swath SWOT and (aggregated) along-track nadir observations.
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An Ocean System Simulation Experiment is used to quantify the observing capability of the Surface Water and Ocean Topography (SWOT) mission and its contribution to higher-quality reconstructed sea level anomaly (SLA) fields using optimal interpolation. The paper focuses on the potential of SWOT for mesoscale observation (wavelengths larger than 100 km and time periods larger than 10 days) and its ability to replace or complement altimetry for classical mesoscale applications. For mesoscale variability, the wide swath from SWOTprovides an unprecedented sampling capability. SWOT alone would enable the regional surface signal reconstruction as precisely as a four-altimeter constellation would, in regions where temporal sampling is optimum. For some specifics latitudes, where swath sampling is degraded, SWOTcapabilities are reduced and show performances equivalent to the historical two-altimeter constellation. In this case, merging SWOT with the two-altimeter constellation stabilizes the global sampling and fully compensates the swath time sampling limitations. Benefits of SWOT measurement are more important within the swath. It would allow a precise local reconstruction of mesoscale structures. Errors of surface signal reconstruction within the swath represent less than 1% (SLA) to 5% (geostrophic velocities reconstruction) of the signal variance in a pessimistic roll error reduction. The errors are slightly reduced by merging swath measurements with the conventional nadir measurements.
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Objective analysis of altimetric data (sea level anomaly) usually assumes that measurement errors are well represented by a white noise, though there are long-wavelength errors that are correlated over thousands of kilometers along the satellite tracks. These errors are typically 3 cm rms for TOPEX/Poseidon (T/P), which is not negligible in low-energy regions. Analyzing maps produced by conventional objective analysis thus reveals residual long-wavelength errors in the form of tracks on the maps. These errors induce sea level gradients perpendicular to the track and, therefore, high geostrophic velocities that can obscure ocean features. To overcome this problem, an improved objective analysis method that takes into account along-track correlated errors is developed. A specific data selection is used to allow an efficient correction of long-wavelength errors while estimating the oceanic signal. The influence of data selection is analyzed, and the method is first tested with simulated data. The method is then applied to real T/P and ERS-1 data in the Canary Basin (a region typical of low eddy energy regions), and the results are compared to those of a conventional objective analysis method. The correction for the along-track long-wavelength error has a very significant effect. For T/P and ERS-1 separately, the mapping difference between the two methods is about 2 cm rms (20% of the signal variance). The variance of the difference in zonal and meridional velocities is roughly 30% and 60%, respectively, of the velocity signal variance. The effect is larger when T/P and ERS-1 are combined. Correcting the long-wavelength error also considerably improves the consistency between the T/P and ERS-1 datasets. The variance of the difference (T/P-ERS-1) is reduced by a factor of 1.7 for the sea level, 1.6 for zonal velocities, and 2.3 for meridional velocities. The method is finally applied globally to T/P data. It is shown that it is tractable at the global scale and that it provides an improved mapping.
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This study represents a first attempt to combine new glider technology data with altimetry measurements to understand the upper ocean dynamics and vertical exchanges in areas with intense horizontal density gradients. In July 2008, just two weeks after Jason-2 altimeter was launched, a glider mission took place along a satellite track in the Alboran Sea (Western Mediterranean). The mission was designed to be almost simultaneous with the satellite passage. Dynamic height from glider reveals a sharp gradient (˜15 cm) and corresponds very well with the absolute dynamic topography from Jason-1 & Jason-2 tandem mission (r > 0.97, rms differences < 1.6 cm). We blend both data sets (glider and altimetry) to obtain a consistent and reliable 3D dynamic height field. Using quasi-geostrophic dynamics, we diagnose large-scale vertical motions (˜1 m day-1) which may provide a local mechanism for the subduction of the chlorophyll tongue observed by the glider.
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Missing data in very high spatial resolution (VHR) optical imagery take origin mainly from the acquisition conditions. Their accurate reconstruction represents a great methodological challenge because of the complexity and the ill-posed nature of the problem. In this letter, we present three different solutions, with all based on the inpainting approach, which consists in reconstructing the missing regions in a given image by propagating the spectrogeometrical information retrieved from the remaining parts of the image. They rely on the idea to enrich the patch search process by including local image properties or by isometric transformations or to reformulate it under a multiresolution processing scheme, respectively. Thorough experiments conducted on two different VHR images are reported and discussed.
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High‐resolution (2 km and hourly) observations of surface currents from High‐Frequency Radars are analyzed in terms of sea level anomalies (SLA) and compared with data from two satellite altimeter ground tracks. Purpose is to investigate whether ocean submesoscale processes can be observed with satellite altimetry. Our results highlight two major problems that must be overcome before being able to resolve submesoscale processes with altimetry: (i) signal contamination from high‐frequency motions and in particular from incoherent internal tides (near‐inertial oscillations have no effect on SLA), and (ii) measurement noise which prevents the computation of accurate cross‐track currents on scales $\cal{O}$ (10 km). The latter may be overcome by future satellite altimeter missions, but the former will require taking into account the effect of mesoscale variability on internal tide propagation in regions where internal tides are significant.
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Most of the kinetic energy of ocean circulation is contained in ubiquitous mesoscale eddies. Their prominent signatures in sea surface height have rendered satellite altimetry highly effective in observing global ocean eddies. Our knowledge of ocean eddy dynamics has grown by leaps and bounds since the advent of satellite altimetry in the early 1980s. A satellite’s fast sampling allows a broad view of the global distribution of eddy variability and its spatial structures. Since the early 1990s, the combination of data available from two simultaneous flying altimeters has resulted in a time-series record of global maps of ocean eddies. Despite the moderate resolution, these maps provide an opportunity to study the temporal and spatial variability of the surface signatures of eddies at a level of detail previously unavailable. A global census of eddies has been constructed to assess their population, polarity, intensity, and nonlinearity. The velocity and pattern of eddy propagation, as well as eddy transports of heat and salt, have been mapped globally. For the first time, the cascade of eddy energy through various scales has been computed from observations, providing evidence for the theory of ocean turbulence. Notwithstanding the tremendous progress made using existing observations, their limited resolution has prevented study of variability at wavelengths shorter than 100 km, where important eddy processes take place, ranging from energy dissipation to mixing and transport of water properties that are critical to understanding the ocean’s roles in Earth’s climate. The technology of radar interferometry promises to allow wide-swath measurement of sea surface height at a resolution that will resolve eddy structures down to 10 km. This approach holds the potential to meet the challenge of extending the observations to submesoscales and to set a standard for future altimetric measurement of the ocean.
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We propose a new measure, the method noise, to evaluate and compare the performance of digital image denoising methods. We first compute and analyze this method noise for a wide class of denoising algorithms, namely the local smoothing filters. Second, we propose a new algorithm, the nonlocal means (NL-means), based on a nonlocal averaging of all pixels in the image. Finally, we present some experiments comparing the NL-means algorithm and the local smoothing filters.
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Most data assimilation algorithms require the inverse of the covariance matrix of the observation errors. In practical applications, the cost of computing this inverse matrix with spatially correlated observation errors is prohibitive. Common practices are therefore to subsample or combine the observations so that the errors of the assimilated observations can be considered uncorrelated. As a consequence, a large fraction of the available observational information is not used in practical applications. In this study, a method is developed to account for the correlations of the errors that will be present in the wide-swath sea surface height measurements, for example, the Surface Water and Ocean Topography (SWOT) mission. It basically consists of the transformation of the observation vector so that the inverse of the corresponding covariance matrix can be replaced by a diagonal matrix, thus allowing to genuinely take into account errors that are spatially correlated in physical space. Numerical experiments of ensemble Kalman filter analysis of SWOT-like observations are conducted with three different observation error covariance matrices. Results suggest that the proposed method provides an effective way to account for error correlations in the assimilation of the future SWOT data. The transformation of the observation vector proposed herein yields both a significant reduction of the root-mean-square errors and a good consistency between the filter analysis error statistics and the true error statistics.
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The transition scale Lt from balanced geostrophic motions to unbalanced wave motions, including nearinertial flows, internal tides, and inertia-gravity wave continuum, is explored using the output from a global 1/488 horizontal resolution Massachusetts Institute of Technology general circulation model (MITgcm) simulation. Defined as the wavelength with equal balanced and unbalanced motion kinetic energy (KE) spectral density, Lt is detected to be geographically highly inhomogeneous: it falls below 40 km in the western boundary current and Antarctic Circumpolar Current regions, increases to 40-100 km in the interior subtropical and subpolar gyres, and exceeds, in general, 200 km in the tropical oceans. With the exception of the Pacific and Indian sectors of the Southern Ocean, the seasonal KE fluctuations of the surface balanced and unbalanced motions are out of phase because of the occurrence of mixed layer instability in winter and trapping of unbalanced motion KE in shallow mixed layer in summer. The combined effect of these seasonal changes renders Lt to be 20 km during winter in 80% of the Northern Hemisphere oceans between 258 and 458N and all of the Southern Hemisphere oceans south of 258S. The transition scale's geographical and seasonal changes are highly relevant to the forthcoming Surface Water and Ocean Topography (SWOT) mission. To improve the detection of balanced submesoscale signals from SWOT, especially in the tropical oceans, efforts to remove stationary internal tidal signals are called for.
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Two global ocean models ranging in horizontal resolution from 1/12° to 1/48° are used to study the space- and time-scales of sea surface height (SSH) signals associated with internal gravity waves (IGWs). Frequency-horizontal wavenumber SSH spectral densities are computed over seven regions of the world ocean from three simulations of the HYbrid Coordinate Ocean Model (HYCOM) and two simulations of the Massachusetts Institute of Technology general circulation model (MITgcm). High-wavenumber, high-frequency SSH variance follows the predicted IGW linear dispersion curves. The realism of high-frequency motions (>0.87cpd) in the models is tested through comparison of the frequency spectral density of dynamic height variance computed from the highest resolution runs of each model (1/25° HYCOM and 1/48° MITgcm) with dynamic height variance frequency spectral density computed from 9 in-situ profiling instruments. These high-frequency motions are of particular interest because of their contributions to the small-scale SSH variability that will be observed on a global scale in the upcoming Surface Water and Ocean Topography (SWOT) satellite altimetry mission. The variance at supertidal frequencies can be comparable to the tidal and low-frequency variance for high-wavenumbers (length scales smaller than ∼50km), especially in the higher resolution simulations. In the highest resolution simulations, the high-frequency variance can be greater than the low-frequency variance at these scales.
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NASA's Surface Water and Ocean Topography (SWOT) satellite, scheduled for launch in 2020, will provide observations of sea surface height anomaly (SSHA) at a significantly higher spatial resolution than current satellite altimeters. This new observation type is expected to improve the ocean model mesoscale circulation. The potential improvement that SWOT will provide is investigated in this work by way of twin-data assimilation experiments using the Navy Coastal Ocean Model four-dimensional variational data assimilation (NCOM-4DVAR) system in its weak constraint formulation. Simulated SWOT observations are sampled from an ocean model run (referred to as the "nature" run) using an observation-simulator program provided by the SWOT science team. The SWOT simulator provides realistic spatial coverage, resolution, and noise characteristics based on the expected performance of the actual satellite. Twin-data assimilation experiments are run for a two-month period during which simulated observations are assimilated into a separate model (known as the background model) in a series of 96-h windows. The final condition of each analysis window is used to initialize a new 96-h forecast, and each forecast is compared to the nature run to determine the impact of the assimilated data. It is demonstrated here that the simulated SWOT observations help to constrain the model mesoscale to be more consistent with the nature run than the assimilation of traditional altimeter observations alone. The findings of this study suggest that data from SWOT may have a substantial impact on improving the ocean model forecast of mesoscale features and surface ocean velocity.
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The Met. Office has developed a Variational assimilation for its Unified Model forecast system, which contains a grid-point model that is run operationally in global, mesoscale, and stratospheric configurations. Key characteristics of the design are: a development path from three-dimensional to four-dimensional variational assimilation; global and limited-area configurations; variational analysis of perturbations; and a carefully designed, well conditioned background term. The background term is implemented using a sequence of Variable transforms to independent balanced and unbalanced variables, to vertical modes, and to spectral coefficients. The coefficients used are based on statistics from differences of one- and two-day forecasts valid at the same time. The covariance model represents many of the features seen in the covariances of forecast differences. The three-dimensional Variational data assimilation (3D-Var) system was implemented in the operational global forecast system on 29 March 1999. In parallel trials, the 3D-Var system gave a 2.7% improvement in a composite skill score (verified against observations and weighted according to the importance of each field).
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Many issues may challenge standard interpolation techniques to produce high-resolution gridded maps of sea surface height in the context of future missions like Surface Water and Ocean Topography (SWOT). The present study proposes a new method to address these challenges. Based on the conservation of potential vorticity, the method provides a simple dynamic approach to interpolation through temporal gaps between high spatial resolution observations. For gaps shorter than 20 days, the dynamic interpolation is extremely efficient and allows for the reconstruction of the time evolution of small mesoscale eddies (below 100 km) that would be smoothed out by conventional methods based on optimal mapping. Such a simple approach offers some perspectives for developing high-level products from high-resolution altimetry data in the future.
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The non persistent phase relationship between internal tides and astronomical forcings, also known as incoherence, has been identified as a major question in the context of future wide-swath satellite altimetry. This study addresses this issue using a novel set of numerical experiments where a plane-wave/low-mode internal tide propagates through a turbulent mesoscale eddy field. These experiments demonstrate the emergence of internal tide incoherence as the eddy turbulence is strengthened. In strongly turbulent situations, the internal tide signature on sea level forms complex interference patterns with large amplifications of the initial internal wave. These patterns evolve more rapidly than the signature of the turbulent eddy field on sea level. The implications of such idealized numerical simulations for wide-swath altimetry are discussed.
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This article studies the regularization of inverse problems with a con- vex prior promoting some notion of low-complexity. This low-complexity is obtained by using regularizers that are partly smooth functions. Such functions force the solution of variational problems to live in a low-dimension manifold which is stable under small perturbations of the functional. This property is crucial to make the underlying low-complexity model robust to small noise. We show that a simple criterion implies the stability of the active manifold to small noise perturbations of the observation when the regularization parameter is tuned proportionally to the noise level. This unifies and generalizes several previous works, where this theorem is known to hold for sparse, group sparse, total variation and low-rank regularizations.
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[1] We present an innovative approach to the generation of remotely sensed high-resolution sea surface topography that improves coastal and mesoscale dynamic characterization. This new method is applied for the period 2002–2010 in the northwestern Mediterranean Sea, an area marked by a small Rossby radius. The spectral content of the new mapped data is closer to that of the along-track signal and displays higher levels of energy in the mesoscale bandwidth with the probability distribution of the new velocity fields 30% closer to drifter estimations. The fields yield levels of eddy kinetic energy 25% higher than standard altimetry products, especially over regions regularly impacted by mesoscale instabilities. Moreover, qualitative and quantitative comparisons with drifters, glider, and satellite sea surface temperature observations further confirm that the new altimetry product provides, in many cases, a better representation of mesoscale features (more than 25% improvement in correlation with glider data during an experiment).
Article
Observations made by satellite altimeters since the 1980s have provided progressively improved views of the global ocean mesoscale eddy field, which contains most of the kinetic energy of the ocean circulation. Along with these improved views, ocean models have progressed from coarse-resolution, highly dissipative mesh grids to higher resolutions where mesoscale eddies dominate the model simulations. We are now able to produce simulations of the present state of the ocean that compare increasingly well with observations. However, the skill of these models in making long-range predictions of the ocean is still very limited, because the models lack a physically based representation of the submesoscales, i.e., scales of 1-100 kilometers, that are important for turbulent transport and energy dissipation. Ocean models running at sufficient resolutions to address submesoscale dynamics have just recently begun to emerge [e.g., Capet et al., 2008], but we need global observations at these scales to guide the model development.
A technique for the objective analysis of oceanic data has been developed and used on simulated data. The technique is based on a standard statistical result—the Gauss-Markov Theorem-which gives an expression for the least square error linear estimate of some physical variable (velocity, stream function, temperature, etc.) given measurements at a limited number of data points, the statistics of the field being estimated in the form of space-time spectra, and the measurement errors. An expression for the r.m.s. error expected in this estimate is also derived and illustrated in the form of ‘error maps’.Efficient sampling arrays can be designed through trial-and-error adjustment of array configurations until a suitable balance of mapping coverage and accuracy, as measured by the error maps, is achieved. Examples of the mapping ability of some simple arrays are given.Using statistics inferred from the preliminary Mid Ocean Dynamics Experiments various realizations of likely flow fields were simulated. The 16 element MODE-I array was tested by comparison of the simulated fields and the objective maps based on inferred ‘measurements’ at the array points. The reliability of statistics inferred from observations was estimated by comparing correlations derived from limited observations of the simulated fields with the known statistics. Correlations derived from two realizations differed significantly but most calculations reproduced the known statistics moderately well.An intercomparison of Eulerian measurements (current meters) and Lagrangian measurements (neutrally buoyant drifters) was also carried out using the objective interpolation method.
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Five years of twice-daily height values of 200- , 500- , and 850- mb surfaces at grid of 1003 points over Northern Hemisphere are procured; weighted root-mean square height difference is used as measure of difference between two states, or error; for each pair of states occurring within one month of same time of year, but in different years, error is computed; there are numerous mediocre analogues but no truly good ones; likelihood of encountering any truly good analogues by processing all existing upper-level data appears to be small.
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The classical filtering and prediction problem is re-examined using the Bode-Sliannon representation of random processes and the “state-transition” method of analysis of dynamic systems. New results are: (1) The formulation and methods of solution of the problem apply without modification to stationary and nonstationary statistics and to growing-memory and infinitememory filters. (2) A nonlinear difference (or differential) equation is derived for the covariance matrix of the optimal estimation error. From the solution of this equation the coefficients of the difference (or differential) equation of the optimal linear filter are obtained without further calculations. (3) The filtering problem is shown to be the dual of the noise-free regulator problem. The new method developed here is applied to two well-known problems, confirming and extending earlier results. The discussion is largely self-contained and proceeds from first principles; basic concepts of the theory of random processes are reviewed in the Appendix.
Article
Four altimeter missions [Jason-1, ERS-2, TOPEX/POSEIDON interleaved with Jason-1 (T/P) and Geosat Follow-On (GFO)] are intercalibrated and merged in an objective analysis scheme with the aim of improving the estimation of mesoscale surface ocean circulation in the Mediterranean Sea. A validation with independent altimetric data shows that, with the combination of three altimeters, in regions of large mesoscale variability, the sea level and velocity can be mapped with a relative accuracy of about 6% and 23%, respectively, which is a factor of 2.2 less than the results derived from Jason-1 alone, and a factor of about 1.5 less than the results obtained from Jason-1 + ERS-2. Mean eddy kinetic energy (EKE) is computed from the different altimeter configurations. It shows that the combination of Jason-1 + ERS-2 fails to reproduce some intense signals. On the contrary, when T/P is added, these features are well recovered and the EKE does not show significant discontinuities due to sampling effects. The impact of the fourth mission (GFO) is less critical but it also improves the representation of energetic structures. In average, the merged Jason-1 + ERS-2 + T/P + GFO maps yield EKE levels 15% higher than Jason-1 + ERS-2. Finally, we show that the consistency between altimetry and Sea Surface Temperature, drifting buoys and tide gauges, is significantly improved when four satellites are merged compared to the results derived from the two-satellite configuration. This study demonstrates that, at least three, but preferably four, altimeter missions are needed for monitoring the Mediterranean mesoscale circulation.
Article
This paper presents a detailed diagnostic analysis of hydrographic and current meter data from three, rapidly repeated, fine-scale surveys of the Almeria–Oran front. Instability of the frontal boundary, between surface waters of Atlantic and Mediterranean origin, is shown to provide a mechanism for significant heat transfer from the surface layers to the deep ocean in winter. The data were collected during the second observational phase of the EU funded OMEGA project on RRS Discovery cruise 224 during December 1996. High resolution hydrographic measurements using the towed undulating CTD vehicle, SeaSoar, traced the subduction of Mediterranean Surface Water across the Almeria–Oran front. This subduction is shown to result from a significant baroclinic component to the instability of the frontal jet. The Q-vector formulation of the omega equation is combined with a scale analysis to quantitatively diagnose vertical transport resulting from mesoscale ageostrophic circulation. The analyses are presented and discussed in the presence of satellite and airborne remotely sensed data; which provide the basis for a thorough and novel approach to the determination of observational error.
Article
A technique previously developed for assessing the effects of sampling errors on sea surface height (SSH) fields constructed from satellite altimeter data is extended to include measurement errors, thus providing estimates of the total mean-squared error of the SSH fields. The measurement error contribution becomes an important consideration with the greater sampling density of a coordinated tandem satellite mission. Mean-squared errors are calculated for a variety of tandem altimeter sampling patterns. The resolution capability of each sampling pattern is assessed from a subjectively chosen but consistent set of criteria for the mean value and the spatial and temporal inhomogeneity of the root-mean-squared errors computed over a representative large collection of estimation times and locations. For a mean mapping error threshold tolerance criterion of 25% of the signal standard deviation, the filter cutoff wavelength and period defining the resolution capability of SSH fields constructed from a tandem TOPEX/Poseidon (T/P) and Jason satellite sampling pattern with evenly spaced ground tracks are about 2.2° by 20 days. This can be compared with the resolution capability of about 6° by 20 days that can be obtained from a single altimeter in the T/P orbit. A tandem T/P-Jason mission with 0,75° spacing between simultaneously sampled parallel tracks that has been suggested for estimating geostrophic velocity yields an SSH mapping resolution capability of about 3.7° by 20 days. For the anticipated factor-of-2 larger orbit errors for ENVISAT compared with Jason, the resolution capability of a tandem Jason-ENVISAT scenario is about 3° by 20 days. For mapping the SSH field, the tandem T/P-Jason sampling patterns with evenly spaced, interleaved ground tracks and either a 5-day or a 0-day offset is far better than the other tandem altimeter mission scenarios considered here. For the highest-resolution mapping, the 5-day offset is preferable to the 0-day offset. The scientific benefits of such a tandem mission are discussed in the context of two specific examples: Rossby wave dispersion and investigation of eddy-mean flow interaction.
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Climate and weather constitute a typical example where high dimensional and complex phenomena meet. The atmospheric system is the result of highly complex interactions between many degrees of freedom or modes. In order to gain insight in understanding the dynamical/physical behaviour involved it is useful to attempt to understand their interactions in terms of a much smaller number of prominent modes of variability. This has led to the development by atmospheric researchers of methods that give a space display and a time display of large space-time atmospheric data.
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An algorithm, the bootstrap filter, is proposed for implementing recursive Bayesian filters. The required density of the state vector is represented as a set of random samples, which are updated and propagated by the algorithm. The method is not restricted by assumptions of linearity or Gaussian noise: it may be applied to any state transition or measurement model. A simulation example of the bearings only tracking problem is presented. This simulation includes schemes for improving the efficiency of the basic algorithm. For this example, the performance of the bootstrap filter is greatly superior to the standard extended Kalman filter
SWOT simulator documentation
• L Gaultier
• C Ubelmann
• L.-L Fu
Non-local regularization of inverse problems
• G Peyré
• S Bougleux
• L Cohen
SWOT simulator documentation
• gaultier