estimation of polarization parameters using time–frequency...

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Signal Processing 86 (2006) 3714–3731

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Estimation of polarization parameters using time–frequencyrepresentations and its application to waves separation

Antoine Roueffa, Jocelyn Chanussotb,�, Jerome I. Marsb

aInstitut Fresnel, Groupe Physique et Traitement de l’Image, DU Saint Jerome 13397 Marseille cedex 20, FrancebSignals and Images Laboratory, LIS, Grenoble, LIS/ENSIEG, Domaine Universitaire, BP 46, 38402 Saint-Martin-d’Heres Cedex, France

Received 20 September 2005; received in revised form 3 January 2006; accepted 8 March 2006

Available online 9 May 2006

Abstract

This paper deals with the detection of polarized seismic waves, the estimation of their polarization parameters, and the

use of these parameters to apply waves separation. The data, containing several polarized waves together with some noise,

are recorded by two-component sensors. After a review presenting the tools classically used to estimate the polarization

parameters of a wave in the time domain and in the time–frequency domain, respectively, we present a new methodology to

detect polarized waves, and estimate their polarization parameters automatically. The proposed method is based on the

segmentation of a time–frequency representation of the data. In addition, after describing the proposed polarization

estimation method, we present the oblique polarization filter (OPF) that enables the separation of two polarized waves

using their polarization parameters, even if the corresponding patterns partially overlap in the time–frequency plane. The

OPF consists in applying phase shifts, rotations, and amplifications in order to project one wave on one single component

and the other wave on the other component. Being more efficient than classical polarization estimation methods, our

approach greatly increases the separation performances of the OPF. Results are presented both on synthetic and real

seismic data.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Time–frequency analysis; Polarization estimation; Waves separation; Singular value decomposition

1. Introduction

In the geophysical community, sensors recordingvibrations in several directions are more and morewidely used [1]. The main advantage provided bythese vectors (or multi-component) lies in the access

e front matter r 2006 Elsevier B.V. All rights reserved

gpro.2006.03.019

ing author. Tel.: +33 4 76 82 62 53;

2 63 84.

resses: [email protected] (A. Roueff),

[email protected] (J. Chanussot),

is.inpg.fr (J.I. Mars).

to polarization information they provide. Polariza-tion and related signal processing tools are veryuseful to differentiate and characterize polarizedwaves when traditional algorithms such as timefiltering, frequency filtering or time–frequencyfiltering fail (i.e. when the patterns correspondingto the different waves overlap in the time–frequencyplane) [2,3].

In particular, polarization can be used to separatepolarized seismic waves. It actually enables thecombination of the information provided by, e.g.,the vertical component with the information

.

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ARTICLE IN PRESSA. Roueff et al. / Signal Processing 86 (2006) 3714–3731 3715

provided by the horizontal one. In the sourceseparation community, some blind statistical meth-ods have been used to separate waves with a two-component sensor model [4,5]. However, thesemethods are designed for random processes andrequire specific statistical signal properties. Forexample, JADE algorithm [6] assumes that thedistributions are independent and non-Gaussian.Consequently, these methods often fail when deal-ing with deterministic non-stationary seismic waves.Furthermore, blind methods usually prevent the useof a priori information.

In practice, waves separation is traditionallyperformed using filtering algorithms: a transformis mapping the data in a domain where waves haveseparated supports is applied. Dealing with datarecorded by a sensor array, the classical transformsare the 2D Fourier transform [7] and the Radontransform [8]. When one wants to process each traceseparately, time–frequency filtering is another inter-esting alternative [9]. The main limitation of suchmethods is that the separation is only partial whenthe patterns of the different waves overlap in theconsidered domain. When vector data are available,these filtering procedures are usually applied oneeach component separately.

In the case of vector data, it is theoreticallypossible to perfectly separate two polarized waves,even if their pattern overlap, by using the obliquepolarization filter (OPF) [10]. However, this filterrequires a precise estimation of each wave’s

two componentsensor

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section II.A

section I

section I

Fig. 1. Separation of two polarized w

polarization parameters. Since the estimation ofthese parameters is a difficult task, the obliquepolarization filter is seldom used in practice. As aconsequence, the focus of this paper is on thiscrucial issue: the automatic estimation of thepolarization parameters and how they are used bythe oblique polarization filter to perform wavesseparation.

Nehorai et al. presented an important pioneeringwork on the modeling and statistical evaluation ofvector-sensor array processing [11]. In this paper,the description and the processing of the data forestimating the polarization parameters of seismicwaves from two-component sensors using time–frequency representations are addressed. The aim isto overcome the limitation of the OPF by providinga robust and reliable estimation and thus increase itsseparation performances. We show that combiningthe oblique polarization filter with the proposedpolarization parameter estimation algorithm allowsthe separation of two polarized waves, even whenthey partially overlap in the time–frequency plane.

The paper is organized as illustrated in Fig. 1.Section 2 covers the context of the study. It providesthe model used for two-component data (Section2.1) and presents the OPF (Section 2.2). Section 3presents the proposed polarization estimation algo-rithm. It consists in segmenting the time–frequencyplane in several regions (one for each wave). Thenthe polarization parameters are estimated in eachregion individually. Finally, Section 4 presents the

polarizationparameterestimation

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Oblique

Filter (OPF)Polarization

II

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section IV

aves. Organization of the paper.

ARTICLE IN PRESSA. Roueff et al. / Signal Processing 86 (2006) 3714–37313716

application of the oblique polarization filterto waves separation, both on synthetic and realdata.

2. Context of the study

2.1. A model for polarized waves

Let shðtÞ and svðtÞ be the two row vectorscontaining the sampled signals recorded on a two-component single sensor (h stands for horizontaland v for vertical) in a time window. In presence ofone wave, we use the following model:

sahðtÞ

savðtÞ

" #¼

1

reiy

" #waðtÞ, (1)

where wðtÞ is the row vector containing the samplesof the polarized wave, and ðr; yÞ are the polarizationparameters of the wave wðtÞ. Note that sa denotesthe analytic signal of s [12]. A physical interpreta-tion of the polarization parameters of a moregeneral model is given by Anderson et al. [13].

Our model assumes that the signals recorded oneach component are identical, close to one phaseshift y and one amplification r. There is no temporal

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sh(a) (b)

s s

Fig. 2. Example of three kinds of polarization: (a), (b), (c) Plots on the

solid line and vertical component svðtÞ in dashed line. Plots on the

polarization and (c) r ¼ 0.

delay since waves reach each component at the sametime. The use of analytic forms allows us to modelthe phase shift y as a simple multiplication by thecomplex exponential eiy. This model implies that thepolarization of the waves depends neither on timenor on frequency, which is fair when dealing withseismic waves.

In presence of Nw waves and a random whitenoise, the model becomes

sahðtÞ

savðtÞ

" #¼XNw

n¼1

1

rneiyn

" #wa

nðtÞ þnahðtÞ

navðtÞ

" #, (2)

where nhðtÞ and nvðtÞ are the row vectors containingthe noise on the h and the v components,respectively. In this paper, we consider that thisnoise is random and uncorrelated from onecomponent to the other.

Fig. 2 presents three examples of polarizationwhen only one wave is present:

(a)

0 8

time

0sh

top a

bott

corresponds to the most general case where ra0and ya0. Signals are not in phase: thepolarization is called elliptic.

(b)
corresponds to the linear case where ra0 andy ¼ 0. Signals are in phase.
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sh(c)

s

re the time representations: horizontal component shðtÞ is in

om are Lissajous plots: (a) elliptic polarization, (b) linear


(c)

p

Fig.

arou

(the

corresponds to the case r ¼ 0. The wave isprojected on one single component.

The Lissajous plots t 7!ðshðtÞ; svðtÞÞ presented inFig. 2 (bottom) enable to visually characterize thepolarization.

2.2. The oblique polarization filter

The previous illustration in Fig. 2 shows thatleaving from the general case (ra0, ya0), it ispossible to project the wave on one single compo-nent (case r ¼ 0) by applying two simple operatorscompensating the phase shift and rotation para-meters. This operation can be written as follows:

cosðgÞ sinðgÞ

� sinðgÞ cosðgÞ

" #1 0

0 e�iy

� �sahðtÞ

savðtÞ

" #, (3)

where g ¼ arctanðrÞ.Similarly, in presence of two waves on the two

components, a perfect separation is possible byprojecting the first wave on one single com-ponent and the second wave on the other one.This operation can be done using the OPF proposedin [10]. To ensure a correct separation, this

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amplificrotation

Lissajous sh, sv

phase sh

sh in solidand sv in dash

hase shift

3. Flow graph presenting the oblique polarization filter. Initially, th

nd time sample 60, and the second one is around sample 210 (first

re is one single wave on each component).

algorithm assumes that the model recalled in Eq.(1) holds for both waves and that the polarizationparameters of both waves are known. It does notrequire the waves to be separated in the time–frequency plane, or statistically independent, ornon-Gaussian.

In order to present the OPF, we consider a simpleexample where waves have separated supports forillustrative purpose. This is presented in Fig. 3.Initially, both waves are elliptically polarized (twoellipses are visible on the Lissajous plots). As shownin Fig. 3, the OPF is performed using a simplecombination of operators:

Step 1: A phase shift is applied on the secondcomponent. The polarization of the left wave turnslinear. Note that the ellipse of the second wave isalso changed because the whole signal is phaseshifted.

Step 2: A rotation is applied in order to projectthis wave on the first component.

Step 3: A phase shift is applied on the secondcomponent. The polarization of the right wave turnslinear.

Step 4: A second rotation is applied in order tohave the vertical axis as an axis of symmetry (in theLissajous plot).

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ation rotation of π / 4

rotationift

e two waves have an elliptic polarization. The first wave is located

plot of Fig. 3). At the end, the two waves are perfectly separated


Step 5: An amplification leads to orthogonalwaves (in the Lissajous plot).

Step 6: A last rotation leads to the perfectseparation of the two waves (there is one singlewave on each component).

The details of the mathematical operationscorresponding to the different steps of the OPFare presented in Appendix A.

The key-point of this method is the requiredknowledge of the polarization parameters of eachwave. If the waves have different time supports(non-overlapping signals), these parameters caneasily be found by cross-correlation in the timewindow corresponding to each wave separately. Itcan also be estimated from the Lissajous plots.However, when the temporal support of the wavesare identical, one cannot estimate the polarizationparameters in this way and the oblique polarizationfilter cannot be used directly. To overcome thislimitation and increase the applicability of the OPF,we propose a new algorithm to detect polarizedwaves, and estimate their polarization parametersbased on time–frequency representations. This isdeveloped in the following section.

3. Detection of polarized waves and estimation of

their polarization parameters

3.1. Introduction

In this section, the classical tools used to estimatethe polarization in the time domain and in the time–frequency domain are reviewed first. Starting fromone example, we then propose an algorithm todetect polarized waves and estimate their polariza-tion parameters. It consists in segmenting the time–frequency plane in several regions (one for eachwave). The novelty in this partitioning process isthat it is not based on the energy of the waves as in[9]. This segmentation enables the detection of thedifferent polarized waves, and also the design, foreach wave, of an optimized time–frequency regionwhere the polarization can be estimated with a goodsignal-to-noise ratio (SNR).

3.2. Estimation in the time domain

3.2.1. Estimation using the cross-product

We address the estimation of the polarizationparameters r and y of a wave w recorded by a two-component sensor ðsh; svÞ. We assume that the wavew is corrupted by an additive noise n ¼ ðnh; nvÞ. This

corresponds to the following model:

sahðtÞ

savðtÞ

" #¼

1

reiy

" #waðtÞ þ

nahðtÞ

navðtÞ

" #. (4)

nh and nv are either uncorrelated, if it correspondsto a random noise of the model (Eq. (2)), orcorrelated if it includes other polarized wavespresent in the data.

The simplest method to perform the estimationconsists in finding a time window I where the wavew is dominant compared to the noise n, and incomputing the cross-product C defined by

C ¼

RI

savðtÞsa�h ðtÞdtR

IjsahðtÞj

2 dt, (5)

where * stands for the complex conjugate operator.This complex number leads to reiy when the noise

is negligible. However, if the energy of the noise nh

is bigger than the energy of w, or if the energy of nv

is bigger than the energy of rw, the estimation is notaccurate. In order to take this asymmetry intoaccount, the SNR is defined as the minimumbetween the SNR of each component:

SNR ¼ minEw

Enh

;r2Ew

Env

� �, (6)

where Ew (respectively Enh and Env ) is the energy ofthe signal w (respectively nh and nv).

This SNR will be used to quantify the robustnessof the polarization estimation in Section 3.5.

3.2.2. Estimation using the singular value

decomposition

Another polarization estimation method, studiedin [14], consists in estimating the singular valuedecomposition (SVD) [15] of the matrix defined asthe stacking of the two row vectors sahðtÞ and savðtÞ:

sahðtÞ

savðtÞ

" #¼ UDV�, (7)

where D is the diagonal matrix containing thesingular values l1, l2 in decreasing order, and U ¼

ðui;jÞ and V ¼ ðvi;jÞ contain the left and the rightsingular vectors, respectively.

Assuming that the energy of the wave is higherthan the energy of the noise, the polarizationparameters can be estimated from the followingratio:

uð2; 1Þ

uð1; 1Þ¼ reiy. (8)


In addition, the ratio between the first and thesecond singular value gives an evaluation of analternative SNR defined as

dSNR ¼l1l2

. (9)

When the data sh and sv contain random noise, thisratio is close to one, but when the data contain onlyone polarized wave, this ratio goes to infinity. dSNRcan be computed from sh and sv directly whereas theprevious SNR, as defined by Eq. (6) can only becomputed when w and r are known. As aconsequence, this second SNR can be used foradaptive filtering [16]. In this paper, we will use thissingular value ratio to detect polarized waves.

In addition, a second advantage of this SVDestimator lies in the symmetric role played by sh andsv.

3.3. Extension of the classical methods in the

time– frequency plane

3.3.1. Estimation using the cross-spectrogram

In this section, we study the possible time–frequency extensions of the classical methods. Inthe fifth chapter of Keilis-Borok’s book [17],formula (5) is generalized as follows:

Cðt; nÞ ¼

RIt;n

TFRvðt; f ÞTFR�hðt; f Þdt dfRI t;njTFRhðt; f Þj2 dtdf

, (10)

where TFRh and TFRv are the linear time–frequency representations of sh and sv, respectively,and I t;n is a time–frequency region of fixed dimen-sion and centered on the time–frequency locationðt; nÞ.

When using the wavelet transform, one shouldadapt the size of the region I t;n to the scale. In thispaper, we only used the short-time Fourier trans-form for the sake of simplicity. In addition, we useda Gaussian window.

In a similar way as with Eq. (5), in the case of ahigh SNR (meaning that the energy of the noise isnegligible compared to the energy of the signalwithin the time–frequency area defined by I t;n), thisresults in an estimation of r and y:

Cðt; nÞ ¼ reiy. (11)

In [18], the time–frequency region I t;n is chosen asthe singleton ft; ng, ensuring a good time–frequencyresolution. In this case, Eq. (10) turns to a simplecross-spectrogram (or cross scalogram, depending

on whether the TFR is the Short Term FourierTransform (STFT) or the continuous wavelettransform (CWT)):

Cðt; nÞ ¼TFRvðt; nÞTFR�hðt; nÞ�þ jTFRhðt; nÞj2

, (12)

where � is a constant aiming at avoiding nulldenominators.

3.3.2. Estimation using the singular value

decomposition

As proposed in [19], the extension of the SVDmethod to the time–frequency plane can be achievedby computing, for each time–frequency locationðt; nÞ, the SVD on the matrix formed by the twovectors TFRhðt; nÞ and TFRvðt; nÞ, where t variesfrom t� nbs to tþ nbs (nbs is a constant definingthe number of considered samples). The matrix isthus of dimension 2 by nbs � 2þ 1. Here, thedimensions of the time–frequency region I t;n arecharacterized by the bandwidth of the window usedin the STFT and the number of samples 2 � nbsþ 1.

As in the time domain, an estimation of the SNRis given by the ratio between the two singular valuesat each time–frequency location, in addition to theestimation of the complex number reiy. This time–frequency representation enables some interpreta-tion of the data, but it is usually not used to estimatethe polarization parameters.

3.4. Proposed detection and estimation method

3.4.1. Introduction

In this paper, we propose to use the previoustime–frequency representation generated using theSVD in order

�
to detect polarized waves, � to design a time–frequency region for eachdetected wave, enabling an accurate estimationof the polarization parameters.
An example is presented in Fig. 4. The syntheticdata correspond to the sum of two waves: oneRicker wave and one Morlet wave, together withsome Gaussian white noise. The time–frequencypatterns of the two waves overlap. The theoreticalSNR (see Eq. (6)) between the Ricker wave and therandom noise is 3.4 dB, and 6 dB between theMorlet wavelet and the random noise.

The polarization parameters of the Ricker waveare: y ¼ 1:57 and r ¼ 1:3, and y ¼ �0:5 and r ¼ 1

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(e) (f)

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Fig. 4. Representations of a two component synthetic signal, which is the sum of a Ricker wave and a Morlet wave, together with some

Gaussian white noise. (f) Presents the result of the automatic detection of polarized waves. Each detected wave is represented by a contour

and the polarization parameters are represented by the corresponding ellipses. (a) shðtÞ. (b) svðtÞ. (c) Spectrogram of sh. (d) Spectrogram of

sv. (e) Image of the ratio l1=l2 and (f) Result of the detection algorithm.

A. Roueff et al. / Signal Processing 86 (2006) 3714–37313720

for the Morlet wave. For this example, thebandwidth of the Gaussian window is 0:1Hz, andnbs ¼ 10 samples. The white areas on the left and on

the right of Fig. 4(e) are due to side effects. For thespectrogram images, the frequency axis correspondsto the normalized frequency.


When one of the waves is dominant compared tothe other wave and the random noise, the singularvalue ratio is significantly bigger than one (see Fig.4(e)). On the contrary, in the regions where therandom noise is dominant, this ratio is close to one.Finally, when the two waves overlap, this ratio isalso fairly low, even though the energy present atthe corresponding locations is important.

Another interesting property of the image pre-sented in Fig. 4(e) is that each polarized wave can beassociated to a dome. As a consequence, we proposeto automatically detect the presence of polarizedwaves by studying this ratio image. The segmenta-tion of this image should provide one time–frequency region for each wave where the corre-sponding polarization can be estimated. Thisprocess is described in the flow graph of Fig. 5and explained in the next paragraph. When thepolarized waves have been detected and theirrespective polarization parameters estimated, wepresent as a result a time–frequency image where the

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step 1

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Fig. 5. Flow graph presenting the polarization estim

contours of each detected wave are shown. Thepolarization parameters are represented by thecorresponding ellipse inside each contour. Thisprovides a complete visual interpretation of thedata. An example is presented in Fig. 4(f).

3.4.2. Segmentation of the time– frequency plane

In order to design a region for each wave wherethe estimation of the polarization can be achievedaccurately, we first need to separate the differentwaves. Since these regions correspond to differentdomes in the ratio image, we apply the segmentationtool called the watershed algorithm [20]. Thissegmentation consists in grouping all pixels con-nected to the same local maximum. The algorithmprovides one region for each local maximum of theimage.

After this segmentation, all the different domesare separated. However, to decide whether eachdome should be associated to a polarized wave ornot, we apply a threshold on the singular value ratio

time frequencyfiltering

same process for all regionsas long as the local maximumis bigger than the threshold

step 6

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ρ, θ estimation of one wave

aximum is the biggest

step 4

step 5

election of the region whose local

ation method. See Section 3.4.2 for the details.


of each local maximum. The estimation of thisthreshold is described in the next paragraph. Thenumber of remaining regions corresponds to thenumber of detected polarized waves. Note on theexample presented in Fig. 5, that local maxima thatare below the chosen threshold are not consideredas representing a polarized wave and are thereforenot selected by the algorithm.

At this step, we have one region for each wave.Now, in order to have a robust estimation,information actually related to the polarized waveis separated from the information related to thenoise in each segmented region using a secondthreshold. The setting of this threshold is alsodetailed in the next paragraph. This second step inthe segmentation process leads to the contour imageof Fig. 5.

Finally, the polarization parameters of each waveare estimated in the corresponding time–frequencyarea (see Fig. 5, steps 4 and 5). The denoisingprocess is performed by the classical time–frequencyfiltering [21] in the region defined by the segmenta-tion (see Fig. 5 step 4). The output SNR increases,assessing the accuracy of the estimation based onthe SVD method. The same procedure is applied oneach segmented region, to estimate the polarizationof each detected wave.

3.4.3. Setting of the thresholds

In this paragraph, we detail the setting of the twothresholds applied in the segmentation algorithmpreviously described. The first threshold, applied onthe local maxima of the ratio image, enables the

cum

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df (

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9099

100

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singular value ratio (dB)(a) (

Fig. 6. Cumulated probability density functions of the singular value ra

are considered, on (b) only locations corresponding to local maxima a

selection of the most significant domes, i.e. thedetection of the polarized waves. In each segmentedregion, the second threshold allows the separationof the wave from the random noise. The optimalsetting of these thresholds could be derived from theprobability density function (pdf) of the singularvalue ratio in presence of a polarized wave, and thepdf of the singular value ratio in presence ofuncorrelated noise by minimizing the falsealarm probability and maximizing the detectionprobability. Unfortunately, the pdf of thesingular value ratio in presence of a polarized waveremains unknown and cannot be estimated. As aconsequence, we simply estimate the pdf of thesingular value ratio in presence of noise usingsimulations, and we estimate the upper boundensuring a probability of having some noise smallerthan 1%, thus minimizing the false alarm prob-ability.

The simulations are done by generating uncorre-lated white noise on both components one thousandtimes. No signal is present. Note that when the noisehas the same variance on both component, itsamplitude has no impact on the singular value ratio.Fig. 6 presents the estimated cumulated pdfs of thecorresponding singular value ratio. On the left, weconsider all locations in the time–frequency images,whereas on the right we only consider locationscorresponding to local maxima. In addition, in bothcases, we consider different sizes of time–frequencyregion I t;n. Actually, the choice of I t;n is animportant parameter: the bigger I t;n is, the whiterthe noise is.

cum

ulat

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df (

%)

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nbs = 10, B = 0.1 Hz

nbs = 20, B = 0.1 Hz

singular value ratio at the local maxima (dB)b)

tio in presence of white noise. On (a) all time–frequency location

re considered.


From the first Fig. 6 (left), the first threshold is setsuch that the false alarm probability remains below1% (i.e. the probability of considering the noise as apolarized wave). For example, for a window with a0.1Hz bandwidth and nbs ¼ 10, if, for one givenpixel of Fig. 4(e), the singular value ratio is biggerthan the threshold value set to 16 dB, the prob-ability of actually being in presence of noise issmaller than 1%. Similarly, the second threshold isset from the second Fig. 6 (right), where the curvesare plot by only considering time–frequency loca-tions corresponding to local maxima.

As a conclusion, the proposed method leads to atime–frequency representation where the contoursassociated to the different waves are plot with theirassociated polarization parameters. The detection ofthe waves and the estimation of the correspondingpolarization are done automatically. Only twoparameters should be set, namely the bandwidthof the window and nbs (the length of the vectorwhich is used to compute the SVD). In order to setthese parameters, a time–frequency region is de-signed such that for each wave, one can find alocation ðt; nÞ with a good SNR. For the examplepresented in Fig. 4, we chose a bandwidth of 0.1Hzand nbs ¼ 10 samples.

3.5. Quantitative evaluation of the results on the

synthetic data set

In the previous paragraph, the proposed detec-tion and estimation algorithm has been presented.To assess the robustness of the proposed polariza-

sing

ular

val

ue r

atio

(dB

)

-10 -5 0 5 100

5

10

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20

25

30

35

40

45

50T = 6, B = 0.1 Hz

T = 10, B = 0.1 Hz

T = 20, B = 0.1 Hz

signal to noise ratio (dB)(a)

Fig. 7. Influence of the signal-to-noise ratio on (a) the singul

tion estimation method, different powers of noiseare considered. For this simulation, some whitenoise with varying energy is added to the Rickerwave previously presented. The different resultsobtained from this analysis are plotted in Fig. 7.The SNR is computed as described in Eq. (6). Thepolarization estimation error is given by themodulus of the difference between the actual reiy

and the resulting estimation.From the singular value ratio plots, we observe

that the choice of T and B has a major impact onthe singular value ratio, especially when the SNR isgreater than 2 dB. However, from the polarizationestimation error plots, we observe that the estima-tion results are close when the SNR is greater than2 dB: there is no dependence on the chosen time–frequency region. This means that the used thresh-old based on the singular value ratio will bedifferent each time one change T or B, whereasthe estimation error may not be changed.

In the next section, the proposed estimationmethod is used together with the oblique polariza-tion filter to perform waves separation. The wholeseparation process is tested on the previouslydescribed synthetic data set, and also to some realdata sets.

4. Results of the oblique polarization filter

4.1. Application on the synthetic data set

To illustrate the algorithm, we apply the OPF tothe previously described example with two synthetic

pola

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estim

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)

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ar value ratio and (b) the polarization estimation error.

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waves. Fig. 4(f) shows that the two waves have beencorrectly detected, with no false alarm, and theirpolarization parameters have been estimated. Usingthese parameters, the OPF is applied. The outputsignals (noted s1 and s2) are presented in Fig. 8: thetwo waves are correctly separated. Note that thefilter did not remove the random noise, because it isnot polarized. However, since it is negligible, it didnot prevent the algorithm from separating the twowaves.

It is clear on this example that a separation by aclassical time–frequency filtering of each componentseparately could not have performed such a resultsince the patterns of the different waves overlap inthe time–frequency plane.

4.2. Application on real data

The first considered data set was recorded by theLaboratoire de Geophysique Interne et Tectono-physique (LGIT). It was obtained from an explosive

source and a line of 47 receivers (two componentsgeophones). The distance between two geophones is10m. The horizontal component is a horizontalgeophone which vibration axis is in the plane goingthrough the source point and the seismic line (inline). The second component is a geophone with avertical axis. The offset between the source and thefirst receiver is 50m. Data are sampled every 16msand the recording’s duration is limited to 4 s. Fig. 9shows the initial data recorded on the horizontal(left) and the vertical (right) component,respectively. On this recording, we can notice arefracted wave, a reflected arrival, and two dis-persive Rayleigh waves characterized by lowerapparent velocities and by low frequency contents,especially on the h component. The slower Rayleighwave will be called the slow Rayleigh wave, and thefaster one wave will be called the fast Rayleighwave.

The two initial profiles are presented in Fig. 9(a)and (b), respectively. We focus on the 47th traces

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0 50 100 150 200 250time time

(a) (b)

(c) (d)

Fig. 9. Illustration of the algorithm on a real Seismic profile. (a) and (b) Initial profiles recorded by the two-component sensor. (c) and (d)

After the OPF. At the output, the slow Rayleigh wave is projected on output 1, and the fast Rayleigh wave is projected on output 2. The

detection of polarized waves and the estimation of polarization parameters is performed on the last trace (see Fig. 10).

A. Roueff et al. / Signal Processing 86 (2006) 3714–3731 3725

(the last ones) noted sh and sv. The spectrograms ofsh and sv are presented in Fig. 10(a) and (b),respectively. On these images, the different wavesare visible. On the contour image presented in Fig.10(d), the two Rayleigh waves are detected. Thereflected and refracted waves are mixed in oneregion. From the polarization ellipse, one can seethat these waves are mainly present on the verticalcomponent. In addition of these waves, severalother waves are also detected, but they are lessenergetic.

We decide to separate the slow Rayleigh wavefrom the fast one. The result is plot in Fig. 10. Theslow Rayleigh wave has been efficiently projected onthe first component, but the fast Rayleigh wave hasbeen separated in two parts. This means that the

signal contain one (or more) other wave(s) in thistime–frequency area.

Assuming that the polarization parameters areconstant over the whole profile (which is fair ingeophysical prospecting), the estimation obtainedfrom the last trace is used for all the traces of theprofile. The same oblique polarization filter is thusapplied to each trace. The resulting profiles arepresented in Fig. 9(c) and (d). The slow Rayleighwave has been efficiently projected on one compo-nent, and the fast one has been separated in twoparts.

The second data set is a marine seismic campaignrecorded by the Compagnie Generale de Geophy-sique (CGG). The sensor array was laid on the seafloor (14m deep) and each receiver recorded two

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some hints on the presence of polarized waves. (d) Detected waves and corresponding polarization ellipses. Choosing to separate the slow

Rayleigh wave (around sample 180) from the fast Rayleigh wave (around sample 100), corresponding contours are selected. The

spectrograms of the output separated waves are presented on (e) (slow Rayleigh wave on s1) and (f) (fast Rayleigh wave on s2).


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0 20 40 60 80 100 120

time time

(a)

(c) (d)

(b)

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of polarized waves and the estimation of polarization parameters is done on the 10th trace (see Fig. 12).

A. Roueff et al. / Signal Processing 86 (2006) 3714–3731 3727

components. The maximum offset between thesource and the receivers is 900m, time sampling is2ms with a listening time of 1 s. The spatialsampling is 50m.

The two initial profiles are presented in Fig. 11(a)and (b), respectively. We focus on the 10th traces,noted sh and sv. The spectrograms of sh and sv arepresented in Fig. 12(a) and (b), respectively.These images present an energetic pattern in thecenter of the image (around coordinates ð50; 0:15Þ).In addition, in the spectrogram of sv, anotherpattern is visible on the right (around coordinates(110,0.08)).

It is hard to drive some precise interpretationsfrom the spectrogram images only. On the contrary,looking at the contour image (see Fig. 12(d)),several regions are found with the same polarization

ellipse, but on the right part of the image, anotherinteresting region with a significantly differentpolarization has been detected. Thus, the contourimage enable us to confirm the presence of adifferent wave on the right. We decide to separatethese two waves. After the OPF, the two waves(whose spectrograms are presented in Fig. 12(e) and(f), respectively) are separated.

Again, assuming the polarization parameters areconstant over the whole profile, the estimationobtained from one trace is used on all the traces andthe same oblique polarization filter is applied toeach trace. Resulting profiles are presented in Fig.11(c) and (d): the energetic wave is projected on thefirst component, while the second wave is projectedon the second one. This processing allows us toreveal the presence of a weak wave (on the right)

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giving some hints on the presence of polarized waves. (d) Detected waves and corresponding polarization ellipses. Choosing to separate the

energetic wave (around sample 40) from the ‘‘small’’ wave (around sample 110), corresponding contours are selected. The spectrograms of

the output separated waves are presented on (e) (energetic wave) and (f) (‘‘small’’ wave).



that was not visible before. This has been validatedby geophysicists.

Obtaining similar result using standard methodssuch as 2D Fourier transform would be impossiblesince the patterns of the different waves overlap inthe frequency images: the two waves have the samefrequency and the same velocity. In addition, theamplitude difference between the two waves tendsto hide the ‘‘small’’ wave on the right in the time-distance images presented in Fig. 11(c) and (d). As aconclusion, the proposed polarization methodprovided complementary information to spectro-gram images, enabling the detection and separationof different waves.

5. Conclusion

In the general frame of waves separation,polarization can be a discriminant parameter whentraditional methods fail (for instance when thedifferent waves have overlapping time–frequencysupports). Two-component sensors record vibra-tions in two orthogonal directions and actuallyprovide an access to a polarimetric characterizationof the signals, assuming the vector nature of thedata is correctly handled.

In this paper, we presented an algorithm thatautomatically detects the polarized waves andestimates the corresponding polarization para-meters. This is achieved by the watershed segmenta-tion of one time–frequency representation of thepolarization information (namely the singular va-lues ratio image). The setting of the segmentationparameters is derived from statistical simulationsand keeps the false alarm rate below 1%. In asecond step, the estimated polarization parametersare used to actually separate the waves using theoblique polarization filter. The proposed algorithmassumes that the polarization of one given wave isconstant over the time–frequency plane. It alsoassumes that for each wave, there exists one time–frequency region where the wave is dominantcompared to the other waves and to the noise.

Compared with classical approaches operating oneach component separately, we improved the separa-tion performances since the proposed algorithm isable to separate two waves whose time–frequencysupports overlap. When more than two waves arepresent, the proposed algorithm needs to be applied incombination with other time/frequency/time–fre-quency filtering techniques. But when three or morewaves actually overlap at the same time–frequency

location, none of the existing methods will be able toperform a correct separation.

We brought in this paper a methodology toconstruct time–frequency images describing two-component data. The generalization of the methodto n-component data is trivial by using the SVDmethod. It only requires to model the Lissajous plotswith a third parameter (spherical parametrization). Inaddition, as a prolongation of this work, one may tryto adapt this study to the more complex case wherethe polarization depends on frequency. In particular,this requires the estimation of the dispersion from onecomponent to the other as in [22].

Appendix A. Stepwise description of the oblique

polarization filter

We detail in this Appendix the different mathe-matical steps involved in the oblique polarizationfilter leading to the perfect separation of two waveswith known polarization parameters.

Two polarized waves, w1 and w2, are considered,with known polarization parameters: reiy and leic,respectively. This corresponds to the followingsituation:

sahðtÞ

savðtÞ

" #¼

1

reiy

" #wa1ðtÞ þ

1

leic

� �wa2ðtÞ. (13)

A.1. Step 1: Phase shift

The first phase shift corresponds to the multi-plication by the following matrix:

M1 ¼1 0

0 e�iy

� �which leads to

sahðtÞ

savðtÞ

" #¼

1

r

" #wa1ðtÞ þ

1

leic1

� �wa2ðtÞ, (14)

where c1 ¼ c� y.

A.2. Step 2: Rotation

The first rotation corresponds to the followingmatrix:

M2 ¼cosðgÞ sinðgÞ

� sinðgÞ cosðgÞ

" #


where g ¼ arctanðrÞ. This leads to

sahðtÞ

savðtÞ

" #¼

r20

� �wa1ðtÞ þ

l2eic2

l3eic3

" #wa2ðtÞ, (15)

where r2 ¼ cosðgÞ þ r sinðgÞ, l2eic2 ¼ cosðgÞ þ lsinðgÞeic1 , and l3eic3 ¼ � sinðgÞ þ l cosðgÞeic1 .

A.3. Step 3: Phase shift

The second phase shift corresponds to the multi-plication by the following matrix:

M3 ¼1 0

0 eiðc2�c3Þ

� �.

This leads to

sahðtÞ

savðtÞ

" #¼

r20

� �wa1ðtÞ þ

l2eic2

l3eic2

" #wa2ðtÞ. (16)


The second rotation corresponds to the followingmatrix:

M4 ¼cosðg2Þ sinðg2Þ

� sinðg2Þ cosðg2Þ

" #,

where g2 ¼12arctanðl3=l2Þ.

This leads after a tedious but direct calculus to

sahðtÞ

savðtÞ

" #¼

cosðg2Þr2

� sinðg2Þr2

" #wa1ðtÞ

þl3eic2=ð2 sinðg2ÞÞ

l3eic2=ð2 cosðg2ÞÞ

" #wa2ðtÞ. ð17Þ

A.5. Step 5: Amplification

The amplification corresponds to the followingmatrix:

M5 ¼tanðg2Þ 0

0 1

� �

This leads to

sahðtÞ

savðtÞ

" #¼

sinðg2Þr2

� sinðg2Þr2

" #wa1ðtÞ

þl3eic2=ð2 cosðg2ÞÞ

l3eic2=ð2 cosðg2ÞÞ

" #wa2ðtÞ. ð18Þ


The final rotation corresponds to the followingmatrix:

M6 ¼1 �1

1 1

� �This leads to

sahðtÞ

savðtÞ

" #¼

2 sinðg2Þr2

0

" #wa1ðtÞ

þ0

l3eic2=ðcosðg2ÞÞ

" #wa2ðtÞ, ð19Þ

where the two waves are perfectly separated (w1 ison the h component, while w2 is on the v

component).

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