Jérôme I. Mars

Singular value decomposition of quaternion matrices: a new tool for vector-sensor signal processing

We present a new approach for vector-sensor signal modelling and processing, based on the use of quaternion algebra. We introduce the concept of quaternionic signal and give some primary tools to characterize it. We then study the problem of vector-sensor array signals, and introduce a subspace method for wave separation on such arrays. For this purpose, we expose the extension of the Singular Value Decomposition (SVD) algorithm to the field of quaternions. We discuss in more details Singular Value Decomposition for matrices of Quaternions (SVDQ) and linear algebra over quaternions field. The SVDQ allows to calculate the best rank-α approximation of a quaternion matrix and can be used in subspace method for wave separation over vector-sensor array. As the SVDQ takes into account the relationship between components, we show that SVDQ is more efficient than classical SVD in polarized wave separation problem.

Quaternion-MUSIC for vector-sensor array processing

This paper considers the problem of direction of arrival (DOA) and polarization parameters estimation in the case of multiple polarized sources impinging on a vector-sensor array. The quaternion model is used, and a data covariance model is proposed using quaternion formalism. A comparison between long vector orthogonality and quaternion vector orthogonality is also performed, and its implications for signal subspace estimation are discussed. Consequently, a MUSIC-like algorithm is presented, allowing estimation of waves DOAs and polarization parameters. The algorithm is tested in numerical simulations, and performance analysis is conducted. When compared with other MUSIC-like algorithms for vector-sensor array, the newly proposed algorithm results in a reduction by half of memory requirements for representation of data covariance model and reduces the computational effort, for equivalent performance. This paper also illustrates a compact and elegant way of dealing with multicomponent complex-valued data.

MUSIC Algorithm for Vector-Sensors Array Using Biquaternions

In this paper, we use a biquaternion formalism to model vector-sensor signals carrying polarization information. This allows a concise and elegant way of handling signals with eight-dimensional (8-D) vector-valued samples. Using this model, we derive a biquaternionic version of the well-known array processing MUSIC algorithm, and we show its superiority to classically used long-vector approach. New results on biquaternion valued matrix spectral analysis are presented. Of particular interest for the biquaternion MUSIC (BQ-MUSIC) algorithm is the decomposition of the spectral matrix of the data into orthogonal subspaces. We propose an effective algorithm to compute such an orthogonal decomposition of the observation space via the eigenvalue decomposition (EVD) of a Hermitian biquaternionic matrix by means of a newly defined quantity, the quaternion adjoint matrix. The BQ-MUSIC estimator is derived and simulation results illustrate its performances compared with two other approaches in polarized antenna processing (LV-MUSIC and PSA-MUSIC). The proposed algorithm is shown to be superior in several aspects to the existing approaches. Compared with LV-MUSIC, the BQ-MUSIC algorithm is more robust to modelization errors and coherent noise while it can detect less sources. In comparaison with PSA-MUSIC, our approach exhibits more accurate estimation of direction of arrival (DOA) for a small number of sources, while keeping the polarization information accessible.

Modified singular value decomposition by means of independent component analysis

In multisensor signal processing (underwater acoustics, geophysics, etc.), the initial dataset is usually separated into complementary subspaces called signal and noise subspaces in order to enhance the signal-to-noise ratio. The singular value decomposition (SVD) ia a useful tool to achieve this separation. It provides two orthogonal matrices that convey information on normalized wavelets and propagation vectors. As signal and noise subspaces are on the whole well evaluated, usually the SVD procedure cannot correctly extract only the source waves with a high degree of sensor to sensor correlation. This is due to the constraint given by the orthogonality of the propagation vectors, To relax this condition, exploiting the concept of independent component analysis (ICA), we propose another orthogonal matrix made up of statistically independent normalized wavelets. By using this combined SVD-ICA procedure, we obtain a better separation of these source waves in the signal subspace. Efficiency of this new separation procedure is shown on synthetic and real datasets.

Vector-sensor MUSIC for polarized seismic sources localization

This paper addresses the problem of high-resolution polarized source detection and introduces a new eigenstructure-based algorithm that yields direction of arrival (DOA) and polarization estimates using a vector-sensor (or multicomponent-sensor) array. This method is based on separation of the observation space into signal and noise subspaces using fourth-order tensor decomposition. In geophysics, in particular for reservoir acquisition and monitoring, a set of-multicomponent sensors is laid on the ground with constant distance between them. Such a data acquisition scheme has intrinsically three modes: time, distance, and components. The proposed method needs multilinear algebra in order to preserve data structure and avoid reorganization. The data is thus stored in tridimensional arrays rather than matrices. Higher-order eigenvalue decomposition (HOEVD) for fourth-order tensors is considered to achieve subspaces estimation and to compute the eigenelements. We propose a tensorial version of the MUSIC algorithm for a vector-sensor array allowing a joint estimation of DOA and signal polarization estimation. Performances of the proposed algorithm are evaluated.

Estimation of modal group velocities with a single receiver for geoacoustic inversion in shallow water

Due to the expense associated with at-sea sensor deployments, a challenge in underwater acoustics has been to develop methods requiring a minimal number of sensors. This paper introduces an adaptive time-frequency signal processing method designed for application to a single source-receiver sensor pair. The method involves the application of conjugate time-frequency warping transforms to improve the SNR and resolution of the time-frequency distribution (TFD) of the measured field. Such refined knowledge of the TFD facilitates efforts to extract tomographic information about the propagation medium. Here the method is applied to the case of modal propagation in a shallow ocean range independent environment to extract a refined TFD. Given knowledge of the source-receiver separation, the refined TFD is used to extract the frequency dependent group velocities of the individual modal components. The extracted group velocities are then incorporated into a computationally light tomographic inversion method. Simulated and experimental results are discussed.

Applications of autoregressive models and time–frequency analysis to the study of volcanic tremor and long-period events

Volcanic tremor and long-period (LP) events are characterized by sharp spectral peaks that generally result from resonance effects at the source and which concentrate most of the radiated energy. The understanding of these seismovolcanic phenomena requires good descriptions of the distribution in time and frequency of the different spectral components included in the signals, as well as a separation of the resonance effects from less energetic effects such as excitation and propagation. We address the issue of extracting from individual records information as detailed as possible on the physical processes involved at the source. We introduce and compare several time-frequency analysis methods, and we describe the application of autoregressive modeling and deconvolution methods to the characterization and separation of the main spectral components. We propose a signal analysis approach based on the joint use of a set of complementary methods, and we present applications to several examples of volcanic tremor and LP events. The time-frequency analysis of some of the LP events taken as examples reveals short-duration components at the seismogram onsets with energy concentrated at frequencies either higher or lower than the main resonance frequencies. These seismic phases are probably related to the excitation processes of the volcanic resonators. In several cases, the arrival of the main spectral peak has a delay of a few tenths of a second with respect to the first arrival. The residual signals obtained by deconvolving and eliminating the main spectral components contain information about the excitation, such as duration, delay, or frequency band. The residual signals are short for LP events, and continuous for volcanic tremor. The autoregressive modeling of the sample records gives precise estimations of the frequency and quality factor of the main spectral peaks. The measured parameters cover a wide range of values, which is consistent with the great variety of fluids filling resonating cavities in volcanoes.

Matched Representations and Filters for Guided Waves

We propose time-frequency methods to filter dispersive guided wave signals. Guided waves occur in acoustical propagation (oceanic waveguides), geophysics (layered medium), or optics (dielectric optical waveguides). In waveguides, signals can be decomposed into normal modes which contain information on environmental parameters and source localization. As modes present nonlinear time-frequency evolution, modal filtering is not possible with conventional tools. To overcome the difficulty presented by these nonlinearties, we have developed matched tools: matched frequency and time-frequency representations and the modal filterings associated with these representations. The tools developed are based on unitary equivalence principle. Performance and robustness of different proposed modal filters are evaluated and compared. All of these tools can be used for both source localization and environmental inversion.

A Source Separation Technique for Processing of Thermometric Data From Fiber-Optic DTS Measurements for Water Leakage Identification in Dikes

Distributed temperature sensors (DTSs) show real advantages over conventional temperature sensing technology such as low cost for long-range measurement, durability, stability, insensitivity to external perturbations, etc. They are particularly interesting for long-term health assessment of civil engineering structures such as dikes. In this paper, we address the problem of identification of leakage in dikes based on real thermometric data recorded by DTS. Formulating this task as a source separation problem, we propose a methodology based on Principal Component Analysis (PCA) and Independent Component Analysis (ICA). As the first PCA estimated source extracts an energetic subspace, other PCA sources allow to access the leakages. The energy of a leakage being very low compared to the entire data, a temporal windowing approach guarantees the presence of the leakages on these other PCA sources. However, on these sources, the leakages are not well separated from other factors like drains. An ICA processing, providing independent sources, is thus proposed to achieve better identification of the leakages. The study of different preprocessing steps such as normalization, spatial gradient, and transposition allows to propose a final scheme that represents a first step towards the automation of the leakage identification problem.

Single-receiver geoacoustic inversion using modal reversal

This paper introduces a single-receiver geoacoustic-inversion method based on dispersion analysis and adapted to low-frequency impulsive sources in shallow-water environments. In this context, most existing methods take advantage of the modal dispersion curves in the time-frequency domain. Inversion is usually performed by matching estimated dispersion curves with simulated replicas. The method proposed here is different. It considers the received modes in the frequency domain. The modes are transformed using an operator called modal reversal, which is parameterized using environmental parameters. When modal reversal is applied using parameters that match the real environment, dispersion is compensated for in all of the modes. In this case, the reversed modes are in phase and add up constructively, which is not the case when modal reversal is ill-parameterized. To use this phenomenon, a criterion that adds up the reversed modes has been defined. The geoacoustic inversion is finally performed by maximizing this criterion. The proposed method is benchmarked against simulated data, and it is applied to experimental data recorded during the Shallow Water 2006 experiment.