Vector-Sensor Array Processing for Polarization Parameters and DOA Estimation
© C. Paulus and J. I. Mars. 2010
Received: 13 July 2009
Accepted: 2 May 2010
Published: 31 May 2010
This paper presents a method allowing a complete characterization of wave signals received on a vector-sensor array. The proposed technique is based on wavefields separation processing and on estimation of fundamental waves attributes as the state of polarization state and the direction of arrival. Estimation of these attributes is an important step in data processing for a wide range of applications where vector sensor antennas technology is involved such as seismic processing, electromagnetic fields studies, and telecommunications. Compared to the classic techniques, the proposed method is based on computation of multicomponent wideband spectral matrices which enable to take into account all information given by the vector sensor array structures and thus provide a complete characterization of a larger number of sources.
Over the past decade, the use of vector-sensor array (VSA) technology for source localization has significantly increased allowing a better characterization of the recorded phenomena in a wide range of applications (e.g., acoustics, electromagnetism, radar, sonar, geophysics, etc.) [1–5]. For instance in seismic acquisition case, vector sensors are nowadays widely used, allowing a better characterization of the layers thanks to the state of polarization dimension added to detection process. With a vector sensor, we can have access to the particle-displacement vector that describes the particle motion in 3D at a given point in space. As the state of polarization is wavefield dependent, it can be used as an essential attribute to separate waves in addition to their different DOAs. To resume, multicomponent acquisitions provide more detailed information on the recorded wavefield and VSA-recorded signals allow the estimation of the directions of arrival (DOA) and the polarizations of multiple waves (or sources) impinging the array. In the case of elastic and acoustic seismic surveys, the VSA-recorded signals are a mixture of various wave types (body waves, surface waves, converted waves, multiples, noise, etc.). Combined multicomponent acquisition and multicomponent processing and analysis provide better wave characterizations and enhance the imaging resolution of geological features. In order to perform the characterization of each wave, separation of interfering wavefields is a crucial step. In the case of multicomponent sensor arrays, methods of filtering, of source localization, and of polarization estimation have already been developed for acoustics and electromagnetic sources. In the last decade, many array processing techniques for source localization and polarization estimation using vector sensors have been developed, mainly in electromagnetics. Nehorai and Paldi  proposed the Cramer-Rao bound and the vector cross-product DOA. Li and Compton Jr  developed the ESPRIT algorithm for a vector-sensor array. MUSIC-based algorithms were also proposed by Wong and Zoltowski [6–8], who also developed vector-sensors versions of ESPRIT [9–13]. These approaches represent a highly popular subspace-based parameter estimation method and use matrix techniques directly derived from scalar-sensor array processing. Such a method is based on the long-vector approach, consisting in the concatenation of all components of the vector-sensor array in a long vector .
The originality of our method consists in keeping multidimensional structures of data organization for processing. These structures are more adapted to the nature of seismic polarized signals, allowing data organization closer to its multimodal intrinsic structure. This paper presents a novel subspace separation method performing wavefields separation. This method issued from the Multicomponent Wideband Spectral Matrix Filtering (MWSMF) technique [14, 15] is a subspace separation algorithm derived from the classic spectral matrix filtering presented in [16, 17]. After a separation step where each wavefield has been isolated, we propose polarization and DOA estimations for each separated wavefield that takes all frequencies and all components into account. The algorithm treats the various components as a whole rather than individually. In Section 2, we summarize the noise filtering and wavefields separation principles. In Sections 3 and 4, we present the technique using the estimated multicomponent wideband spectral matrices of sources leading to the estimations of the polarization and of the DOA parameters for each wavefield. Finally in Section 5, we present the performances of the algorithm on several simulated 2C-datasets.
2. Noise Filtering and Wavefield Separation
In this section, the proposed subspace separation technique based on Multicomponent Wideband Spectral Matrix Filtering (MWSMF) is briefly explained (for more details, the reader might refer to [14, 15]).
2.1. Model Formulation and Hypothesis
In case of multicomponent acquisition with vector sensor array, seismic data depend on three parameters: time ( samples), distance ( sensors), and number of components ( components). These components, recording signals in three directions as (for in-line axis), (for cross-line axis), and for vertical axis, allow to express the state of polarization of the different wavefields.
, a vector of size which corresponds to the additive noises supposed to be additive, temporally and spatially white, uncorrelated with sources, nonpolarized and with identical power spectral density .
2.2. Estimation of the Multicomponent Wideband Spectral Matrix
To avoid the fact that is noninvertible (not full rank) and to decorrelate sources from noise and from themselves, we perform smoothing operators to estimate the matrix . This step is crucial since the effectiveness of filtering depends on this estimation. In practice, mathematical expectation operator is an averaging operation, like spatial or frequential smoothing or both of them [19–22]. Objective of these averaging operations is to reduce the influence of terms corresponding to the interactions between different sources in order to uncorrelate them, and to uncorrelate sources and noise, making the inversion of the spectral matrix possible. The spatial smoothing could be done by averaging spatial sub-bands. The uniform linear array with sensors is subdivided into overlapping subarrays in order to have several identical arrays, which will be used to estimate spectral matrices in order to build a smoothed matrix. Shan et al.  have proven that if the number of subarrays is greater than or equal to the number of sources Nw, then the spectral matrix of the sources is nonsingular. However, one assumption is that the wave does not vary rapidly over the number of sensors used in the average, in particular, amplitude fluctuations must be smoothed out. To introduce frequency smoothing, two ways can be performed: either by weighting the autocorrelation and cross-correlation functions (in the time domain) or by averaging frequential sub-bands (in the frequential domain). For a better estimation of the multicomponent wideband spectral matrix, it is suitable to realize jointly spatial and frequential smoothing. For more details on averaging operators, we suggest to read .
2.3. Estimation of Signal Subspace
2.4. Filtering by Projection onto the Signal Subspace
3. Polarization Estimation
After presenting the separation processing part, we propose to find polarization parameters on each separated wavefield. State of polarization analysis is based on the computation of parameters describing the particle movement associated with wave propagation. That movement of the ground induced useful parameters which were first identified by Jolly in 1956 , whereas the first attempt to measure this movement was done by Shimshoni and Smith in 1964 . They introduced a successful method of polarization analysis for earthquake data. Many other algorithms were developed subsequently for seismic exploration applications [29–31]. One of the most effective and stable approaches in this regards is the algorithm developed by Flinn [32, 33] using the covariance matrix of the data. In the following part, we compare our proposed method (based on MWSM) with Flinn's algorithm.
3.2. Proposed Method
3.3. Comparison between Flinn's Method and Proposed Method Based on MWSM
MWSM's and Flinn's methods are two different approaches for polarization analysis. Flinn's method uses a covariance matrix and proposes a temporal approach on a single trace whereas our proposed method is a frequential approach which can either be on a single trace or on the whole array. In the case studied here, the waves' state of polarization can be considered as constant over distance. Consequently, for a given wave, we can estimate the amplitude ratio and phase shift between the components by carrying out an averaging of the parameters found on each sensor.
To compare Flinn's and our method and to illustrate polarization estimation, we consider the trivial case of a single wave with infinite velocity, received on a 2C-sensors array, whose phase shift (= 0.4 rad) and amplitude ratio (= 0.8) are constant over the array.
These figures show that for both methods, polarization analysis is very sensitive to noise and thus the estimates are better when SNR increases. We can notice that our MWSM-based method always gives better estimation than the Flinn's method for small SNR.
4. Direction-of-Arrival Estimation
4.1. Proposed Method
Just as we did for polarization state estimation, we propose a DOA estimation method based on the structure of Multicomponent Wideband Spectral Matrix. We call it MW-MUSIC for Multicomponent Wideband-MUSIC as it is an extension of the MUSIC (MUltiple SIgnal Classification) algorithm [34–37]. This method is an extension of the MUSIC algorithm for vector-sensor arrays called LV-MUSIC for Long-Vector MUSIC . The first extensions of MUSIC algorithm to polarized sources were made by Schmidt , Ferrara and Parks , Wong and Zoltowski [6, 8], and Wong et al. . Algorithms to estimate DOAs of polarized sources in electromagnetism were also proposed in [1, 39, 40]. Our proposed method has the advantage of being able to compute both DOA and offset.
The functional allows local maxima for a set of values and . We use the fact that two parameters have been already found by the method proposed on Section 3. After this stage, the functional only depends on two parameters, the direction of arrival and the offset, and it will give maximum for values of corresponding to the sources present in the signal. It is clear that processing will work with lot of efficiency in case of far field sources (waves can be considered as locally plane).
4.2. Study of Estimator Variance
5. Synthetic Examples
5.1. Wavefield Separation
5.2. Polarization Estimation
After separation of waves, the second step consists of making the polarization analysis of the two waves using the matrix elements of and (see Section 3). As previously presented, in Figure 8 (resp., Figure 9), blocks and correspond to wideband spectral matrices of wave 1 (resp., wave 2) on components and respectively. We denote them by and (resp., and ). Blocks and correspond to the cross-spectral matrices of wave 1 (resp., wave 2) containing the interactions between components and We denote them by and (resp., and ).
5.3. DOA Estimation
5.4. A More Realistic Example
A novel method providing wavefields separation along with an estimation of both the polarization parameters and the directions of arrival was presented. Taking into account the polarization and the widebandness of the signal leads to a better characterization of a greater number of waves ( ) as opposed to the monocomponent array case ( ). The performance and efficiency of the method was proven using several simulations. Comparison of the wideband matrix filtering method with those of the classic filtering technique has already been done  and gave encouraging result for wideband case. We also obtained promising results for DAO estimation using the proposed method which can be attributed to the fact that our method takes into account the entire frequency information and is therefore insensitive to frequency band selection.
The authors would like to thank reviewers for interesting remarks and A. A. Khan for improving the language.
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