Multi-Temporal Interferometric Synthetic Aperture Radar (MT-InSAR) exploits Synthetic Aperture Radar images time series (SAR-TS) for surface deformation monitoring via phase difference (with respect to a reference image) estimation. Most of the actual state-of-the-art MT-InSAR rely on temporal covariance matrix of the SAR-TS, assuming Gaussian distribution. However, these approaches become computationally expensive when the time series lengthens and new images are added to the data vector. This paper proposes a novel approach to sequentially integrate each newly acquired image using Phase Linking (PL) and Maximum Likelihood Estimation (MLE). The methodology divides the data into blocks, using previous images and estimations as a prior to sequentially estimate the phase of the new image. Actually, this framework allows to consider non Gaussian distributions, such as a mixture of scaled Gaussian distribution, which is particularly important to consider when dealing with urban areas.

Traditional Phase-Linking (PL) algorithms are known for their high cost, especially with the huge volume of Synthetic Aperture Radar (SAR) images generated by Sentinel-1 SAR missions. Recently, a COvariance Fitting Interferometric Phase Linking (COFI-PL) approach has been proposed, which can be seen as a generic framework for existing PL methods. Although this method is less computationally expensive than traditional PL approaches, COFI-PL exploits the entire covariance matrix, which poses a challenge with the increasing time series of SAR images. However, COFI-PL, like traditional PL approaches, cannot accommodate the efficient inclusion of newly acquired SAR images. This paper overcomes this drawback by introducing a sequential integration of a block of newly acquired SAR images. Specifically, we propose a method for effectively addressing optimization problems associated with phase-only complex vectors on the torus based on the Majorization-Minimization framework. The proposed approach demonstrates comparable performance to the offline COFI-PL method while achieving a reduction in computation time , of approximately 15% in both simulations and real data experiments. Moreover, it outperforms state-of-the-art sequential approaches in terms of both accuracy and speed.

Interferometric phase linking (IPL) has become a prominent technique for processing images of areas containing distributed scatterers in SAR interferometry. Traditionally, IPL consists in estimating consistent phase differences between all pairs of SAR images in a time series from the sample covariance matrix (SCM) of pixel patches on a sliding window. This article reformulates this task as a covariance fitting problem; IPL appears then as a form of projection of an input covariance matrix so that it satisfies the phase closure property. This approach yields a systematic methodology to frame IPL as an optimization problem on the torus of phase-only complex vectors. On the modeling side, the formulation is modular and allows for a flexible choice of covariance matrix estimates, regularization options, and matrix distances. In particular, we demonstrate that most existing IPL algorithms appear as special instances of this framework. In addition, we propose some new options, which were not covered by the state of the art, whose merits are illustrated through simulations and a real-world case study. On the computational side, another contribution of this article is the derivation of generic and computationally efficient algorithms for IPL using majorization-minimization (MM) and Riemannian optimization.

This paper introduces a novel approach for sequential estimation of the interferometric phase in the context of long Synthetic Aperture Radar (SAR) image time series. When newly acquired data arrive, the data set expands and can be partitioned into two distinct blocks. One represents the previous SAR images and the other represents the newly acquired data. The proposed approach (S-MLE-PL) exploits sequential maximum likelihood estimation of the covariance matrix of the whole data set, taking the existing data set as prior information. This approach facilitates the continuous interferometric phase estimation by incorporating the new data into the previous context. In addition, it presents the advantage of reduced computation time compared to the traditional approaches, making it a more efficient solution for operational displacement estimation.

Mountain glaciers play a critical role for mountain ecosystems and society with major concerns related to their future evolution and related water resources. Modeling glacier future evolution allows anticipating climate change impacts and informing policy decisions. It relies on accurate ice thickness estimation at regional scales. This paper proposes a deep learning based approach in a supervised learning framework for ice thickness estimation at a regional scale from surface ice velocity measurements and a digital elevation model. A neural network model built upon a ResNet architecture is proposed based on the trade-off between the model complexity and the prediction efficiency. Promising results are obtained from data including 1400 glaciers in the Swiss Alps, highlighting the potential of deep learning based approach for large scale ice thickness estimation. The incorporation of expert’s knowledge into the neural network model further helps refine the model prediction and improve the model relevance. The ice volume difference between the reference issued from GPR measurements and the predictions by the proposed neural network model varies between 0.5% and 16% of the reference volume. Larger ice volume difference is mainly related to over-deepening of the bedrock resulting from past larger extent of the glacier, which information is not included in the data.

Machine learning has demonstrated potentiality for challenging physical tasks, such as inverting complex mechanisms with important data limitations. It is now competing with traditional methods that involve statistical and physical modeling. These methods face significant challenges, including long computation time, extensive prior knowledge requirements, and sensitivity to scarce and noisy data which limit their ability to generalize. Regarding these difficulties, this article aims to explore the potential deployment of a deep learning-based method to solve an inverse problem in volcanology, that is, to estimate the volume change and depth of a Mogi-type source model from surface displacement measurements. Simulated displacement samples are used to get rid of insufficient amounts of real data and a lack of ground truth. Particular efforts are devoted to proper data preparation, including proposing a semi-automatic technique for training, validation, and testing data sampling and investigating the impact of data distribution, data diversity, and noise. Real data over the Suswa volcano are also used to further assess the performance of the proposed deep learning method. Results with both synthetic and real data provide evidence to consider deep learning-based methods for geophysical inverse problems.

Glaciers play a critical role in the Earth’s climate system, and accurate estimates of their behaviours are essential for understanding the impacts of climate change and informing policy decisions. One of the most important parameters for such a task is ice distribution, which is difficult to measure and predict using traditional physics-based models. In this study, we propose a deep learning approach to predict glacier thickness by learning directly from ice velocity and topography. Our approach overcomes the limitations of traditional physics-based models, such as computational cost and the need for expert knowledge to calibrate the models. In addition, deep learning models are flexible enough to explore the relevance of multimodality and multitasking to address the physical problem. Our results demonstrate the feasibility of quickly training a neural network model with sufficient training data and producing stable, high-quality ice thickness estimates. We highlight the importance of some specific input features suggested by geophysicists that have a positive impact on model stability.

In this paper, two SAR image time series acquired by PAZ and Sentinel-1 satellites in 2020 (29 and 60 images respectively) are used to investigate surface changes of different ice/snow-covered areas in the Mont-Blanc massif. The evolution of the backscatter coefficient and several statistical parameters in both X and C band SAR images is analyzed on ice aprons, on valley glacier accumulation and ablation areas, and on ice-free areas. Dry and wet snow changes are observed and correlated with meteorological data (temperature at 4 different elevations and snow height) acquired by a weather station.

This paper proposes an algorithm for phase differences estimation in multi-temporal InSAR. The proposed approach is based on covariance fitting estimation and the majorization-minimization algorithm. Experiments with Sentinel-1 images of Mexico City demonstrate that the proposed approach compares favorably to the state-of-the-art phase linking (i.e., maximum likelihood-based approaches) when the sample support is low (i.e., when the number of pixels in the multi-look window cannot scale with the number of SAR images). Hence, the proposed approach can improve the spatial resolution of phase difference estimation in case of large SAR image time series.

Phase linking is a prominent methodology to estimate coherence and phase difference in interferometric synthetic-aperture radar. This method is driven by a maximum likelihood estimation approach, which allows to fully exploit all the possible interferograms from a time series. Its performance is, however, known to be affected by the accuracy of the covariance matrix estimation step, which usually requires to introduce additional prior information on its structure when there is a small sample support (spatial window). Moreover, most phase linking algorithms are built upon the sample covariance matrix, due to the assumption of an underlying Gaussian distribution. In a scenario where SAR data is high resolution, or when the study area is spatially heterogeneous (e.g., urban area), this assumption can also limit the accuracy of the covariance matrix estimation step. Considering the two aforementioned issues, we introduce alternative statistical models, whose maximum likelihood estimators then yield new phase linking algorithms. In order to be robust to non-Gaussian data, we consider the use of a more general model of scaled mixture of Gaussian. To address small sample support issues, we also generalize this approach to a possibly low-rank structured covariance matrix. A unified algorithm to perform phase linking given these models is then derived and validated by simulations and a real data case (Sentinel-1 data).

This paper focuses on understanding the temporal behaviour of Ice Aprons (IAs) in the Mont-Blanc massif using high resolution SAR coherence matrix. InSAR coherence is estimated between all possible pairs of TerraSAR-X images acquired in 2009 and 2011 as well as between PAZ images acquired in 2020, both at 11-day interval. The results show that coherence values in summer are higher in 2020 than in 2009 and 2011. Coherence matrices are also computed for different regions of a glacial system. The results are compared with those of IAs to understand the differences in their temporal and physical behaviours. In summer, all IAs show an increase in coherence values, while other glacier regions show very low or no coherence. This information could be useful for automatic classification methods, where IAs could be classified separately as a different class from the other types of glaciers.

This paper presents a new algorithm for improving the estimation of interferometric SAR (InSAR) phases in the context of time series and phase linking approach. Based on maximum likelihood estimator of a multivariate Gaussian model, the estimation of the InSAR phases is solved using a Block Coordinate Descent algorithm. Compared to the state-of-the-art approaches, the main improvement lies on the joint estimation of the covariance matrix and the InSAR phases instead of using a plug-in coherence estimate obtained from the sample covariance of the data or the modeling of the temporal decorrelation of the target under observation. Results of synthetic simulations confirm the improvement brought by the proposed estimator.

Numerous glacier velocity observations, derived from spaceborne imagery, are available online, but it remains difficult to analyze them because they are measured with different temporal baselines, by various sensors. In this study, we propose a novel formulation of the temporal closure to fuse multi-temporal multi-sensor velocity observations without prior information on the displacement behavior and the data uncertainty. We establish a system of linear equations between combinations of displacement observations and fractions of estimated displacements. The proposed approach provides a velocity time-series with a regular and optimal temporal sampling, the latter representing a compromise between the temporal resolution and the signal-to-noise ratio. The proposed approach is first evaluated on synthetic datasets and second on Sentinel-2 and Venμs velocity observations over the Fox glacier in New Zealand. The results show the intra-annual variability of Fox glacier surface velocity with a reduced uncertainty and complete temporal coverage.

Today, more and more velocity observations are available online or on-demand. However, this amount of data is complex to analyze since velocity observations span different temporal baselines. Velocities obtained from a small temporal baseline are close to the derivative of the displacement but are more likely to be contaminated by noise. Velocities obtained from a long temporal baseline approximate the mean velocity between two dates but can be affected by temporal decorrelation. Having short and long temporal baselines provides a data redundancy that needs to be properly considered. In this article, we propose a method that aims to extract short-term velocity time series with regular temporal sampling from all available displacement observations. The proposed method relies on a temporal inversion based on an improved temporal closure of the displacement observation network. Two criteria are proposed to determine the optimal temporal sampling to study short-term variations. To take the unequal data uncertainty into account, the temporal inversion is done by an iterative reweighted least square using a well-established weighting function, without preprocessing. The proposed method results in velocity time series with an optimal temporal sampling, improved temporal coverage, reduced uncertainty, and no redundancy. The studied area is the Kyagar glacier, in the North of the Karakoram range that is characterized by strong velocity variations originated from a glacier surge and additional seasonal variability.

In the above article[1],(2)was incorrectly provided. The correct equation should be:

This article proposes a method based on the temporal closure of the displacement measurement's network. The aim is to extract short-term glacier velocities and to use data redundancy to reject outliers and reduce uncertainty. By using all the available displacement measurements, we retrieve a displacement time series between consecutive observation dates by means of an inversion. The proposed inversion method is an Iterative Weighted Least Square (IWLS) with a regularization on the discrete derivative of displacements. We apply our method to a glaciers velocity data-set covering Fox Glacier in the Southern Alps of New Zealand.

This paper focuses on the visibility of glacier surfaces in the Mont Blanc Massif (Western European Alps) on TerraSAR-X/PAZ images and the identification of geometric distortions (GDs) based on the SAR acquisition geometry. Small glaciers which exist in complex topographies like steep slopes are most prone to GDs. We built a visibility map for both ascending/descending orbits by utilizing previously documented algorithms like the R-Index (RI) and the Layover Shadow (LS) simulations, combined with an analysis of the angle between the steepest slope direction (SSD) and line of sight (LOS) vectors. The visibility map allows us to identify glaciers on steeper slopes which should be considered for further analysis using TerraSAR-X/PAZ images.

Missing data is a critical pitfall in the investigation of remotely sensed displacement measurement because it prevents from a full understanding of the physical phenomenon under observation. In the sight of reconstructing incomplete displacement data, this letter presents a data-driven spatiotemporal gap-filling method, which is an extension of the expectation–maximization-empirical orthogonal function (EM-EOF) method. The presented method decomposes an augmented spatiotemporal covariance of a displacement time series into EOF modes and then selects the optimal set of EOF modes to reconstruct the time series. This selection is based on the cross-validation root-mean-square error and a confidence index associated with each eigenvalue. The estimated missing values are then iteratively updated until convergence. Results on displacement time series derived from cross correlation of Sentinel-2 optical images over Fox Glacier in New-Zealand’s Alps show that the reconstruction accuracy is improved compared with the EM-EOF method. The proposed extension can tackle challenging cases, i.e., short time series with heterogeneous displacement behaviors corrupted by a large amount of missing data and noise.

An iterative method, namely expectation maximization-empirical orthogonal functions (EM-EOF) is proposed for the first time to retrieve missing values in synthetic aperture radar (SAR) displacement time series. This method decomposes the temporal covariance of a displacement measurement time series into different EOF modes by solving the eigenvalue problem, and then selects the optimal number of EOF modes to reconstruct the time series. After an appropriate initialization of missing values, the EM-EOF method performs: 1) a cross-validation root-mean-square error (cross-RMSE) minimization to find an estimate of the optimal number of EOF modes used in the reconstruction and 2) an iterative update of missing values which gives the best estimate of missing data points according to the cross-RMSE. Synthetic simulations have been first performed to highlight the efficiency of EM-EOF in the case of various displacement signal complexities and different types of noise and gaps, and a thorough error analysis has been conducted to determine the sensitivity of the method to signal-to-noise ratio (SNR), quantity of gaps, and type of noise and gaps. Then, EM-EOF is applied to three displacement measurement time series computed from Sentinel-1 A/B SAR images: two interferograms time series over Gorner and Miage glaciers, and one offset time series over the Argentière Glacier covering a period extending from September 2016 to December 2017. Both synthetic simulations and real data applications demonstrate the ability of EM-EOF to retrieve missing values, even in the cases of frequent data gaps, limited size of the time series, and spatio-temporally correlated noise and gaps.

An iterative method, namely extended EM-EOF (Expectation Maximization - Empirical Orthogonal Functions) is proposed to retrieve missing values in satellite derived displacement time series. The method constructs the spatio-temporal covariance of a displacement time series, decomposes it into different EOF modes by solving the eigenvalue problem and then selects an optimal number of EOF modes to reconstruct the time series based on cross validation errors. The latter are also used as convergence criterion in an EM-like algorithm. A confidence index associated with each eigenvalue, estimated from the eigenvalue uncertainty, is introduced as a metric for refining the estimated optimal number of EOF modes. Results on simulated displacement time series perturbed by spatially correlated noise demonstrate the potential of extended EM-EOF to impute spatio-temporal missing values even in case of a reduced size of time series.

An iterative method, namely EM-EOF (Expectation Maximization-Empirical Orthogonal Functions) is proposed for the first time to retrieve missing values in InSAR displacement time series. The method decomposes the temporal covariance into different EOF modes by solving the eigenvalue problem, and then selects an optimal number of EOF modes to reconstruct the time series. After an appropriate initialization of missing values, the proposed method performs (i) a cross-validation error minimization to find an estimate of the optimal number of EOF modes used in the reconstruction and (ii) an iterative update of missing values which gives the best estimate of missing data points according to the cross-validation error. Results on a time series of Sentinel-1 A/B unwrapped interferograms over the Gorner glacier from November 2016 to March 2017 demonstrate the high efficiency of the proposed method at retrieving missing values and denoising the time series, even in case of time series with limited size and spatio-temporally correlated gaps.

In this paper, a data-adaptive method, namely, principal modes (PM) method, based on the spatially averaged temporal covariance of a time series of InSAR displacement measurement obtained from consecutive SAR acquisitions is proposed to retrieve the displacement signal for decorrelating targets. On wrapped interferogram time series, the PM method can highlight and restore coherent fringe patterns where they are more or less significantly hindered by decorrelation noise, whereas on unwrapped interferogram time series, the PM method provides a satisfactory separation of the displacement signal from the spatially correlated perturbations. A two-stage application of the PM method to both wrapped and unwrapped interferogram time series can significantly improve the retrieval of the displacement signal. Synthetic simulations are first performed to investigate the impact of the choice of the appropriate number of modes to retain in the empirical orthogonal function decomposition and of the time series size on the performance of the PM method, as well as to highlight the efficiency of the PM method. Then, the PM method is applied to time series of wrapped and unwrapped Sentinel 1 A/B interferograms over the Gorner glacier between October 2016 and April 2017. The main characteristics of the PM method, such as realistic assumptions, ease of implementation, and high efficiency, are highlighted.

In this paper, we present new results of the use of Sentinel-1 data to monitor Alpine glacier displacement by SAR differential interferometry (D-InSAR) in Chamonix-Mont-Blanc Valley. Two time series of Sentinel-1 A/B images acquired from October 2016 to early April 2017 (including 31 ascending and 25 descending acquisitions) are used to form 6-day interferograms and to evaluate their potential for displacement measurements over small fast moving Alpine glaciers. Results show that, even at low latitudes as in the French Alps, fringe patterns can be observed over the glaciers during the cold season with favorable anti-cyclonic meteorological conditions. Different processing steps to derive final displacement fields are presented and discussed and the results are compared with ERS-Tandem results obtained on the same glaciers in winter 1996.

In this paper, a data-adaptive method, namely Principal Modes (PM) method, based on the spatially averaged temporal covariance of a time series is proposed to extract the displacement signal from a time series of Sentinel 1 A/B inter-ferograms over the Gorner glacier during the period between October 2016 and April 2017. On unwrapped interferogram time series, the PM method provides a satisfactory separation of the displacement signal from the spatially correlated perturbations, while on wrapped interferogram time series, the PM method can highlight fringe patterns where they are more or less significantly hindered by the decorrelation noise.

At the end of the 20th century, the development of spatial geodetic techniques (optical & SAR imagery, GPS) has allowed for drastic improvement of the spatial coverage and the resolution of the displacement measurements. The arrival of these techniques has caused an effective revolution by significantly improving our ability to measure the ground movement, as well as their temporal evolutions with great precision over large areas. Spectacular results have been obtained in numerous applications with displacement of various characteristics (in terms of magnitude, duration, spatial distribution): the study of subsidence in urban areas, of the co-seismic, inter-seismic and post-seismic motions, of glacier flows, of volcanic deformation, etc. Nowadays, the displacement maps obtained by remote sensing techniques cover almost the whole world, with a precision within millimetres per year. Therefore, they are considered as the predominant sources for studies of the terrestrial deformation, from which geophysical models of the deformation have been retrieved to further understand the deformation source in depth. To this end, good knowledge of the reliability of the remote sensing data, as well as of the physical models accordingly obtained is crucial for all the researches and applications that use these sources of information. A perspective of significant improvement in the accuracy of the displacement measurement appears with the growing availability of remote sensing data. Methodological development in fusion of displacement measurements and of the integration of a physical model based on the supercomputer facilities seems necessary to reduce the uncertainty and to improve the accuracy of the displacement measurement. In this context, this paper addresses the current status, challenges and perspectives of the remotely sensed displacement measurement fusion.

In this paper, an attempt of Alpine glacier flow modeling is performed based on a series of high resolution TerraSAR-X SAR images and a Digital Elevation Model. First, a glacier flow model is established according to the fluid mechanics theory in a simplified framework. Second, the displacement field over the glacier obtained from the sub-pixel image correlation of a series of TerraSAR-X SAR images is used to refine the model obtained previously. The comparison between the data observation and the model prediction allows for the validation of the established model. According to the obtained results, despite the simplifications made in the modeling, the established glacier flow model can provide general satisfactory results. Further investigation and improvement of this glacier flow model seem promising.

In this paper, a dedicated phase unwrapping approach, taking a priori information into account and combining multi-scales local frequencies of the interferometric phase, is developed in order to get around of the discontinuity and aliasing problems. In this approach, the interferogram is characterized by local frequency of the phase. The multi-scales local frequencies of the interferometric phase are estimated and fused to the local frequency at the optimal scale. This optimal scale, the lowest resolution scale allowing phase unwrapping without aliasing problem, is determined from the a priori displacement information. The application is performed on the displacement measurement of the 2005 Kashmir earthquake. The a priori displacement information is issued from a deformation model. The advantages of this approach are highlighted by the obtained results.

In this paper, 2 data fusion strategies from SAR images are investigated through application to measurement of displacement field due to the Kashmir earthquake (Mw=7.6, 2005). Firstly, the 3D displacement field at the Earth's surface is retrieved by a linear inversion, using the measurements from sub-pixel image correlation and differential interferometry. In addition to the generalized least square method, a fuzzy approach is applied to represent the measurement uncertainty. Secondly, the geometry of the fault is optimized by a non linear inversion, using the same measurements. The inter-comparisons between strategies and approaches are performed in order to highlight the advantages and disadvantages of each strategy and approach.

International audience