ABSTRACT
CONTENTS 5.1 Introduction ......................................................................................................................... 76 5.2 Matrix Data Sets.................................................................................................................. 76
5.2.1 Acquisition............................................................................................................... 77 5.2.2 Matrix Model........................................................................................................... 77
5.3 Matrix Processing ............................................................................................................... 78 5.3.1 SVD ........................................................................................................................... 78
5.3.1.1 Definition.................................................................................................. 78 5.3.1.2 Subspace Method.................................................................................... 78
5.3.2 SVD and ICA........................................................................................................... 79 5.3.2.1 Motivation................................................................................................ 79 5.3.2.2 Independent Component Analysis ...................................................... 79 5.3.2.3 Subspace Method Using SVD-ICA...................................................... 81
5.3.3 Application .............................................................................................................. 82 5.4 Multi-Way Array Data Sets............................................................................................... 85
5.4.1 Multi-Way Acquisition .......................................................................................... 86 5.4.2 Multi-Way Model ................................................................................................... 86
5.5 Multi-Way Array Processing ............................................................................................ 87 5.5.1 HOSVD..................................................................................................................... 87
5.5.1.1 HOSVD Definition.................................................................................. 87 5.5.1.2 Computation of the HOSVD................................................................. 88 5.5.1.3 The (rc, rx, rt)-rank................................................................................... 89 5.5.1.4 Three-Mode Subspace Method............................................................. 90
5.5.2 HOSVD and Unimodal ICA ................................................................................. 90 5.5.2.1 HOSVD and ICA..................................................................................... 91 5.5.2.2 Subspace Method Using HOSVD-Unimodal ICA............................ 91
5.5.3 Application to Simulated Data............................................................................. 92 5.5.4 Application to Real Data ....................................................................................... 97
5.6 Conclusions........................................................................................................................ 100 References ................................................................................................................................... 100
76 Signal and Image Processing for Remote Sensing
This chapter describes multi-dimensional seismic data processing using the higher order singular value decomposition (HOSVD) and partial (unimodal) independent component analysis (ICA). These techniques are used for wavefield separation and enhancement of the signal-to-noise ratio (SNR) in the data set. The use of multi-linear methods such as the HOSVD is motivated by the natural modeling of a multi-dimensional data set using multiway arrays. In particular, we present a multi-way model for signals recorded on arrays of vector-sensors acquiring seismic vibrations in different directions of the 3D space. Such acquisition schemes allow the recording of the polarization of waves and the proposed multi-way model ensures the effective use of polarization information in the processing. This leads to a substantial increase in the performances of the separation algorithms. Befo re in troducing the mu lti-way mo del and process ing, we first describe the classic al
subsp ace method based on the SVD and ICA techn iques for 2D (mat rix) seismic data sets. Using a matrix model for these data sets, the SV D-bas ed subsp ace me thod is pres ented and it is shown how an extra ICA step in the pr ocessin g allows bette r wave field separation. Then, conside ring sign als recorded on vector-sensor arrays , the multi-wa y mode l is define d and discusse d. The HOSVD is pre sented and som e proper ties det ailed. Bas ed on this multi-linear decomp osition, we propose a subspace method that allows separ ation of polarize d wave s unde r orthogo nality co nstrai nts. We then introduce an ICA step in the pro cess that is perform ed here uni quely on the temp oral mode of the data set, leading to the so-call ed HOSV D-unim odal ICA subsp ace algorit hm. Resul ts on sim ulated and real polarize d data sets sho w the ability of this algorit hm to surpas s a matr ix-based algorithm and subspace method usin g only the HOSVD. Sectio n 5.2 pre sents matr ix da ta sets and their associa ted mod el. In Section 5.3, the well-
known SVD is detailed, as well as the matrix-base d subspace method. The n, we pr esent the ICA co ncept and its contrib ution to subspace formulat ion in Section 5.3.2. App lications of SVD-ICA to seismic wave field separatio n are discussed by way of illu stration s. Sectio n 5.4 exp oses how sign al mixtur es recorded on vecto r-sens or array s can be desc ribed by a mult i-way mod el. Then, in Se ction 5.5, we introdu ce the HO SVD and the associa ted subspace me thod for multi-wa y data proces sing. As in the matrix data set case , an extra ICA st ep is proposed leading to a HO SVD-un imoda l ICA subspace method in Section 5.5.2. Final ly, in Sectio n 5.5.3 and Section 5.5 .4, we illustrat e the propose d algorithm on simulated and real multi-way polarized data sets. These examples emphasize the potential of using both HOSVD and ICA in multi-way data set processing.
