geophysical wavelet library: applications of the continuous wavelet transform to the polarization...

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Geophysical wavelet library: Applications of the continuous wavelettransform to the polarization and dispersion analysis of signals$

M. Kulesh �, M. Holschneider, M.S. Diallo 1

Institute for Mathematics, University of Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany

a r t i c l e i n f o

Article history:

Received 2 March 2007

Received in revised form

19 March 2008

Accepted 25 March 2008

Keywords:

Continuous wavelet transform

Signal processing

Dispersion

Polarization

MATLAB

a b s t r a c t

In the present paper, we consider and summarize applications of the continuous wavelet

transform to 2C and 3C polarization analysis and filtering, modeling the dispersed and

attenuated wave propagation in the time–frequency domain, and estimation of the phase

and group velocity and the attenuation from a seismogram. Along with a mathematical

overview of each of the presented methods, we show that all these algorithms are

logically combined into one software package ‘‘Geophysical Wavelet Library’’ developed

by the authors. The novelty of this package is that we incorporate the continuous wavelet

transform into the library, where the kernel is the time–frequency polarization and

dispersion analysis. This library has a wide range of potential applications in the field of

signal analysis and may be particularly suitable in geophysical problems that we illustrate

by analyzing synthetic, geomagnetic and real seismic data.

& 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Frequency-dependent measurements or time–frequencyanalysis (TFA) offers additional insight and performance toany applications where Fourier techniques have been used.This analysis consists of examining the variation of thefrequency content of a signal with time, and it isparticularly suitable in geophysical applications. We canemphasize three directions to use TFA to analyze geophy-sical data. First of all, time–frequency representations canbe incorporated in polarization analysis (Soma et al., 2002;Schimmel and Gallart, 2005; Diallo et al., 2005, 2006b;Pinnegar, 2006; Pacor et al., 2007; Kulesh et al., 2007a). It isalso possible to model dispersive and dissipative wavepropagation in the time–frequency domain (Kulesh et al.,2005a, b). Finally, TFA is suitable to estimate phase and

group velocities and the attenuation coefficient (Levshinet al., 1972; Prosser et al., 1999; Pedersen et al., 2003;Holschneider et al., 2005; Kulesh et al., 2008).

The continuous wavelet transform (CWT) gives asuitable general framework for solving these types ofproblems; this approach is powerful and elegant, but it isnot the only method available for practical applications.Other TFA methods, such as the Gabor transform, theS-transform (Schimmel and Gallart, 2005), or bilineartransforms, like the Wigner–Ville (Pedersen et al., 2003)or smoothed Wigner–Ville transform, can be used as well.The performance of TFA relative to different TFA approachesis primarily controlled by the frequency resolution cap-ability that motivated the use of CWT in the present work.

This article summarizes our previous works aimed atpolarization and dispersion analysis of signals in thewavelet domain and offers the geophysical wavelet library(GWL)—a new free software package based on CWT withthe following key features:

(1) object-based implementation of main data types likea vector, an axis, a matrix, a multi-channel signal, anda multi-channel wavelet spectrum;

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/cageo

Computers & Geosciences

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0098-3004/$ - see front matter & 2008 Elsevier Ltd. All rights reserved.

doi:10.1016/j.cageo.2008.03.004

$ Code on server at http://users.math.uni-potsdam.de/�gwl and

http://www.iamg.org/CGEditor/index.htm� Corresponding author. Tel.: +49 331977 1823.

E-mail address: [email protected] (M. Kulesh).1 Now at ExxonMobil Upstream Research Company, Houston, TX,

USA.

Computers & Geosciences 34 (2008) 1732–1752


(2) object-based implementation of main mathematicalobjects like Morlet and Cauchy wavelets, some functionapproximations, 2C and 3C polarization parameters,and dispersion parameters;

(3) command line and MATLAB interface for transforma-tions like the Fourier transform, the direct and inverseCWT, 2C and 3C polarization transforms (Diallo et al.,2005, 2006b; Kulesh et al., 2007a), and linear andnon-linear deformations of a wavelet spectrum intro-duced by Xie et al. (2003) and Kulesh et al. (2005a,2008);

(4) command line interface to optimize in signal andwavelet domains using the algorithm of Levenberg–Marquardt optimization with the aim of extractingvelocities and attenuating parameters from a seismo-gram (Holschneider et al., 2005);

(5) data import from tabulated and plain ASCII files anddata export into ASCII files.

One can find and download a great number of other freeor commercial wavelet-based software, for example:

(1) ImageLib is a Cþþ class library providing imageprocessing and related facilities. The main set ofclasses provides a variety of image and vector types,with additional modules that support scalar andvector quantization, discrete wavelet transforms, andsimple histogram operations.

(2) LIFTPACK is a software package written in C for fastcalculation of 1D and 2D Haar and biorthogonalwavelet transforms using the lifting scheme.

(3) The Rice Wavelet Toolbox is a collection ofMATLAB M-files and C MEX-files for 1D and 2Dwavelet and filter bank design, analysis, and proces-sing. The toolbox provides tools for denoising andinterfaces directly with MATLAB code for hiddenMarkov models in the wavelet domain and waveletregularized deconvolution.

(4) MathWorks’ Wavelet Toolbox for MATLAB imple-ments standard wavelet families, including Daube-chies wavelet filters, complex Morlet and Gaussian,real reverse biorthogonal, and discrete Meyer. It hasinteractive tools for continuous and discrete waveletanalysis and methods for adding wavelet families.

(5) MR/1 is a large package written in Cþþ and IDL(Interactive Data Language, Research Systems Inc.) forfiltering, deconvolution, object detection and analysis,vision modeling, compression, registering, etc. A widerange of wavelet and other multi-scale transforms issupported.

(6) Wavelet Explorer is a package for wavelets inMathematica. It includes common filters, such asDaubechies’ extremal phase and least asymmetricfilters, spline filters, and contains transforms towavelet bases, wavelet packet bases, or local trigono-metric bases in one and two dimensions, as well asdata compression and denoising.

However, none of these software packages have anyfeature tailored for geophysical problems. With GWL, we

try to address these limitations and incorporate CWT intothe library where the kernel is the time–frequency pola-rization and dispersion analysis. The main purpose of thisarticle is to show not only mathematical aspects of thisproblem but also some peculiarities of implementation.

This paper is organized as follows. After a short over-view of GWL structure and implementation technology,we briefly introduce the direct and inverse wavelettransforms and some of their properties that we will needafterwards. Then, we describe three different wavelet-based polarization analysis methods for two, three, ormore components. Next, we introduce wavelet deforma-tion algebra and describe three approximate methods forwave propagation modeling in the time–frequency do-main. In the last section, we demonstrate the applicationsof these propagation models in two inversion algorithmsto characterize dispersion and attenuation properties ofsurface waves. In Appendix A, we summarize a list of usedmathematical notations.

Sections 3–7 have the following logical structure. First,we denote the modules of GWL, related to the correspond-ing section. After the base equations and a short mathe-matical description of each method, we show how each isimplemented in GWL. Finally, we illustrate the discussedmethods using synthetic or real data and compare it withother methods.

2. GWL structure and implementation technology

GWL includes three logical levels: the library level, thelevel of command line tools, and the interface level asshown in Fig. 1. The aim of such separation is to structurethe mathematical algorithms, the calculation interface,and the post-processing graphical interface.

The main part of the library level is a Cþþ hierarchicalobject library called PPP (parametric processing of pulsa-tions). This library contains an object-based implementa-tion of main data types and mathematical objects andimplements all used algorithms. We designed PPP withthe assumption that it could be potentially used outside ofGWL in any other project related to the TFA of the signals.PPP uses the Cþþ Standard Template Library, an ANSI Ccommand line parser (argtable2 version 2.6), a C sub-routine library for computing the discrete Fourier trans-form (FFTW version 3.0.1, Frigo and Johnson, 1998),and Linux GUI components and utility classes (Qwtversion 5.0.1).

The GWL command line level is a set of independentCþþ modules. These modules are based on the PPPlibrary and provide a command line interface for allmethods implemented in this library. After the compila-tion, we obtain a set of executable modules placed in theGWL/bin directory. To perform a calculation using GWL,we run certain modules from this directory in theappointed order. The calculation parameters must begiven by command line. The data exchange betweendifferent modules is implemented by the data files withbinary stream formats. After the calculation process isfinished, we collect ASCII or binary files with thecalculation results.

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M. Kulesh et al. / Computers & Geosciences 34 (2008) 1732–1752 1733


In general, we can plot these calculation results, savedin ASCII format, using any plotting software. To reduce therequired hard drive space and improve plotting perfor-mance, we can also store the results as binary files.Toward this end, we developed a special MATLAB packageplaced on the interface level of GWL in the directoryGWL/mshell. This package allows us to read all binaryformats supported in GWL and plot GWL objects, such as amulti-channel signal or a multi-channel wavelet spec-trum, using high-level subroutines based on the standardMATLAB plotting commands. This package is developed asan M-file library and does not need any installation beforeuse. The second possible use of the GWL/mshell library isan integration of the calculation process with plottingsubroutines using an M-file program. In this way, weadded some procedures into GWL/mshell to directlyexecute modules from the GWL command line level.

To demonstrate the calculation technique using GWL,we stored many examples into the GWL/solutions direc-tory. Together with GWL MATLAB tools, this folderconstitutes a part of the GWL interface level. Diallo et al.(2005, 2006a, b), Holschneider et al. (2005), and Kulishet al. (2005a, b, 2007a, b, 2008) previously used all ofthese examples to show the application possibilities of theCWT for dispersion and polarization analysis of synthetic,geomagnetic, and real seismic data.

3. The CWT

In this section, we introduce two modules, gwlCwt andgwlIwt, from the command line level. These modulesimplement direct and inverse CWT, respectively.

3.1. The direct wavelet transform

The wavelet transform of a real or complex signal sðtÞ 2

L2ðRÞ with respect to a real or complex mother wavelet

gðtÞ is the set of L2-scalar products of all dilated andtranslated wavelets with an arbitrary signal to be analyzed(Holschneider, 1995):

Wgsðt; aÞ ¼ hgt;a; si ¼

Z þ1�1

1

agn t� t

a

� �sðtÞdt,

a 2 R; t 2 R, (1)

where gt;a ¼ ð1=aÞgððt� tÞ=aÞ is generated from gðtÞ

through dilation a and translation t. The symbol ð�Þn

denotes the complex conjugate. There are differentpossibilities to define the prefactor in the wavelet. Forexample, the factor 1=

ffiffiffiap

is widely used. We used here thefactor 1=a instead because the amplitude of the waveletspectrum is proportional to the amplitude of the signal forpure frequencies in this case. For other normalizations,this relation could be frequency (scale) dependent.

The inverse of the scale a may be associated with afrequency measured in units of the wavelet centralfrequency. If the central frequency of the wavelet isassumed to be f 0, each scale, a, can be related to thephysical frequency, f, by a ¼ f 0=f . Therefore, if we select awavelet with a unit central frequency, it is possible toobtain the physical frequency directly by taking theinverse of the scale.

Procedure (1) is very slow and, therefore, cannot beused for long signals. We can construct a more effectivealgorithm if we use the convolution theorem and express

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Library level

argtable21)

1)Argtable2 is an ANSI C command line parser (version 2.6)2)Fast Fourier Transforms by using FFTW (version 3.0.1)3)The Qwt library contains GUI Components and utility classes (version 5.0.1)

FFTW32)PPP

Command line level

GWL command line tools

Interface level

Matlab interfacelibrary

Matlab exampleslibrary

/argtable2/fftw3/qwt/PPP

/source/commshell

/bin

/mshell/solutions

GWL

QWT3)

Fig. 1. Structure of GWL.

M. Kulesh et al. / Computers & Geosciences 34 (2008) 1732–17521734


the wavelet transform in terms of the Fourier transformsðzÞ of sðtÞ as

Wgsðt; f Þ ¼

Z þ1�1

gnðz=f Þ expð2pitzÞsðzÞdz. (2)

The direct wavelet transform (1) is implemented in themodule gwlCwt with the parameter wttype ¼ 0, whichcalculates the wavelet spectrum related to the physicalfrequency f and time t or the time–frequency domain. Thefast approach (2) is implemented in the module gwlCwt

as well but with the parameter wttype ¼ 1, where the FFT(Press et al., 1992) or FFTW3 (Frigo and Johnson, 1998)procedure is used for the convolution calculation. Incomparison with approach (1), this option requires asignal with 2n points’ length.

3.2. Wavelets

The wavelet gðtÞ is assumed to be a function that is welllocalized in time and frequency. This function does notcontain the zero frequency or, equivalently, its average isZ þ1�1

gðtÞdt ¼ 0.

The choice of the analyzing wavelet depends on whattypical oscillation modes are present in the signal. Forexample, to analyze rectangular oscillations, the realHAAR wavelet or a similar but symmetric FHAT waveletis most suitable. If only one component of seismic orgeomagnetic signals is analyzed, any real wavelet basedon different-order derivatives of the Gaussian is conve-nient. However, the method of polarization analysisproposed in the next sections is based on a progressivewavelet.

We can choose the analyzing wavelet in the modulegwlCwt using the parameter wavelet. Below, we detailtwo progressive wavelets implemented in GWL:

(1) wavelet ¼ morlet corresponds to the complex Mor-let wavelet (Holschneider, 1995). This wavelet isshown in Fig. 2 and can be written with its Fouriertransform gðoÞ as

gðtÞ ¼ expð2pitÞ expð�t2=ð2s2ÞÞ,

gðoÞ ¼ sffiffiffiffiffiffi2pp

expð�ðo� 2pÞ2s2=2Þ, (3)

where o is the angular frequency and parameter sdescribes the variance of the wavelet.

(2) The second wavelet implemented in GWL is thecomplex Cauchy wavelet ðwavelet ¼ cauchyÞ

gðtÞ ¼ 1�2pit

p� 1

� ��p

,

gðoÞ ¼ ðp� 1Þp

ðp� 1Þ!

o2p

� �p�1

exp �ðp� 1Þo

2p

� �. (4)

Both Morlet and Cauchy wavelets are progressive, i.e.,their Fourier coefficients for negative frequencies are zero.This feature allows us to separate the wavelet spec-trum into prograde Wþ

g sðt; f Þ and retrograde W�

g sðt; f Þ

components,

Wgsðt; f Þ ¼Wþ

g sðt; f Þ þW�

g sðt; f Þ,

where

Wþ

g sðt; f Þ ¼Wgsðt; f Þ; fX0;

0; fo0;

(

W�

g sðt; f Þ ¼0; fX0;

Wgsðt; f Þ; fo0:

((5)

Strictly speaking, Morlet wavelet is not a progressivewavelet; it is not even a wavelet because it is not of zeromean. However, the negative frequency components of g

are small compared to the progressive component.

3.3. The inverse wavelet transform

We implemented the inverse CWT in the modulegwlIwt:

sðtÞ ¼MhWgsðt; f Þ

¼1

Cg;h

Z þ1�1

Z þ1�1

hððt � tÞf ÞWgsðt; f Þdtdf , (6)

where hðtÞ is the wavelet used for the inverse wavelettransform Mh. In the general case, this wavelet can bedifferent from the analyzing wavelet gðtÞ.

As in the case of the direct integration (1), proce-dure (6) is very slow. However, one can choose thed-function as the wavelet hðtÞ for the inverse CWTðwavelet ¼ deltaÞ, which gives us a rather simple and

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−3 −2 −1 0 1 2 3−1

−0.5

0

0.5

1

Time

real partimaginary part

−5 0 5Frequency

Fig. 2. Representation of Morlet wavelet in (a) time and (b) frequency domain, s ¼ 0:7.



fast reconstruction formula

sðtÞ ¼MdWgsðt; f Þ ¼1

Cg;d

Z þ1�1

Wgsðt; f Þdf

f. (7)

Cg;h in Eq. (6) and Cg;d in Eq. (7) are the normalizationcoefficients related to the direct and inverse motherwavelets, respectively, and

Cg;h ¼

Z þ10ðg

nðoÞhðoÞ þ g

nð�oÞhð�oÞÞ do

o.

For a successful inverse transform, this constant has tosatisfy the admissibility condition: 0oCg;ho1. We do notneed to calculate this coefficient for all applications;therefore, we implemented the parameter ampl in themodule gwlIwt, which defines the normalization mode ofthe inverse signal.

3.4. Wavelet transform of a complex signal

As we mentioned above, the signal sðtÞ can be complex.Using a complex progressive wavelet and Eq. (5), we canseparate the wavelet spectrum of such signal intoprograde and retrograde components. An example of theprograde and retrograde wavelet spectrum is shown inFig. 3, where we consider a two-component syntheticseismogram related to a Rayleigh wave. From two signals,uxðtÞ and uzðtÞ, that correspond to the orthogonal compo-nents of the record, we construct a complex signal

zðtÞ ¼ uxðtÞ þ iuzðtÞ. (8)

Next, we perform CWT (2) for this complex signal usingCauchy wavelet (p ¼ 5) and plot only the absolute valuesof the complex wavelet coefficients as gray-scaled imagesseparately for the prograde and the retrograde compo-nents (5) of the wavelet spectrum. The analysis of thedifference between these two spectra is exactly the idea tofind out polarization properties of a signal as is introducedin the next section.

4. Polarization properties of two-component data andpolarization filtering

GWL contains several modules for polarization analysisand filtering of two-component signals. Polarizationanalysis can be carried out using the module gwlET2D,while polarization filtering is performed using the othermodule, gwlET2DFilter.

4.1. Polarization analysis in the time domain

Given a signal from a three-component record, withuxðtÞ, uyðtÞ, and uzðtÞ representing the seismic tracesrecorded in three orthogonal directions (radial–transversal–vertical), any combination of two orthogonal componentscan be selected for polarization analysis. In its fullgenerality, an elliptically polarized rotating signal (8) isdescribed by the following geometric parameters (Fig. 4):

(1) R: the semi-major axis RX0,(2) r: the semi-minor axis RXrX0,

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−2500

250

u x (t

)

−2500

250

u z (t

)

Time (s)

Freq

uenc

y (H

z)

Wgz+(t,f)

Wgz−(t,f)

0 1 2 3 4 5 6 7 8 9 10−4

−2

0

2

4

−500 0 500−500

0

500

u z (t

)

ux (t)

Fig. 3. Wavelet transform of Rayleigh wave arrival. (a) Radial uxðtÞ and vertical uzðtÞ components of synthetic 2C seismograms. (b) Its prograde and

retrograde wavelet transform. (c) Hodogram showing particle motions over entire time window. Solid lines in panel (b) indicate maximum wavelet

spectrum modulus.



(3) y: the tilt angle, which is the angle of the semi-majoraxis with the horizontal axis, y 2 ð�p=2;p=2�, and

(4) Df: the phase difference between two components.

There are several methods to obtain these polarizationattributes from a signal; some of them we implemented inthe module gwlET2D using the parameter type:

(1) type ¼ rene performs the complex trace analysis(CTA) proposed by Rene et al. (1986),

(2) type ¼ morozov corresponds to the method pro-posed by Morozov and Smithson (1996) and isadapted to the two-component case, and

(3) type ¼ scovar allows us to calculate polarizationproperties by the eigenanalysis of the cross-energymatrix of a two-component record (Flinn, 1965;Kanasewich, 1981; Jurkevics, 1988).

4.2. Complex trace method in the wavelet domain

The above-listed methods operate only in the timedomain, and the estimated attributes represent an averageover all frequencies and, therefore, do not provideinformation about their frequency dependency. To copewith these limitations inherent to time–frequency resolu-tion, we previously proposed a method based on CWT(Diallo et al., 2006b).

When a complex progressive wavelet is used, CWTproperty (5) allows us to represent the wavelet spectrumas a superposition of its prograde and retrograde compo-nents. Let us consider the instantaneous angular fre-quency defined as the derivative of the complexspectrum’s phase

O�ðt; f Þ ¼ q arg W�

g zðt; f Þ=qt.

As long as jWgzðt; f Þj varies in time more slowly thanarg Wgzðt; f Þ, the Taylor expansion of the wavelet coeffi-cients reads as Wgzðt þ t; f Þ ’Wgzðt; f Þ expðiOðt; f ÞtÞ.Then, near time instant t, each component can be

represented as follows:

Wgzðt þ t; f Þ ’Wþ

g zðt; f Þ expðiOþðt; f ÞtÞþW�

g zðt; f Þ expð�iO�ðt; f ÞtÞ,

which yields the time–frequency spectrum for each of theparameters

Rðt; f Þ ¼ jWþ

g zðt; f Þj þ jW�

g zðt; f Þj,

rðt; f Þ ¼ jjWþ

g zðt; f Þj � jW�

g zðt; f Þjj,

yðt; f Þ ¼ arg½Wþ

g zðt; f ÞW�

g zðt; f Þ�=2,

Dfðt; f Þ ¼ argWþ

g zðt; f Þ þW�

g zðt; f Þn

Wþ

g zðt; f Þ �W�

g zðt; f Þn

!þ p=2. (9)

The module gwlET2D with the input parameter type ¼complex performs the calculation of polarization attri-butes in the time domain if the input object is a signal andin the time–frequency domain using the procedure (9) if aspectrum object is given as input.

4.3. Two-component polarization filter

If we analyze seismic data, an advantage of method (9)is the possibility to perform the complete wave-modeseparation/filtering process in the wavelet domain andprovide the frequency dependence of ellipticity, whichcontains important information about the subsurfacestructure. Using 2C synthetic and real seismic shotgathers, we previously showed how to use the methodto separate different wave types and identify zones ofinterfering wave modes (Diallo et al., 2006b).

With the extension of the polarization analysis tothe wavelet domain, we can construct filtering algo-rithms to separate different wave types based on theinstantaneous attributes by a combination of constraintsposed on the range of the ellipticity, rðt; f Þ ¼ rðt; f Þ=Rðt; f Þ

and the tilt angle, yðt; f Þ. Formally the algorithm can be

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R

rθ

x

z

Time

Δφ

ux(t)uz(t)

Fig. 4. Schematic representation of an ellipse with its geometric parameters: semi-major axis R, semi-minor axis r, tilt angle y, and phase difference Dfbetween uxðtÞ and uzðtÞ components.



represented as

zf ðtÞ ¼MhEryWgzðt; f Þ,

Eryðt; f Þ ¼

Wgzðt; f Þ for rðt; f Þ 2 Pr and

yðt; f Þ 2 Py;

0 otherwise;

8>><>>: (10)

where Ery is the filter operator of the wavelet-spectrum.Sets Pr and Py define the range of r and y, which are keptin the filtered signal. In Table 1, we summarize filtersettings that can be used to detect signals with specificpolarization.

According to the filter settings defined in Table 1,the filtering approach (10) is implemented in themodule gwlET2DFilter with the input parameter

type ¼ complex. To choose the filter type, the inputparameter filter is used; the value filter ¼ linhor

corresponds to ELH , filter ¼ linvert to ELV , filter ¼ellihor to EEH , and filter ¼ ellivert to EEV filter.However, the object structure of the PPP library allows usto extend the set of the filter types.

4.4. Application to the analysis of geomagnetic data

The above described method can be applied not onlyfor seismic data. We also used this polarization analysis toexamine characteristics of geomagnetic Pi2 pulsations.These pulsations are defined as damped oscillations of thegeomagnetic field with the period range of 40–150 s.These kind of pulsations can be observed clearly at low-and mid-latitude ground stations at substorm onset. Inour previous work (Kulesh et al., 2007b), we analyzed indetail the time dependence of the phase differencebetween two components of geomagnetic record arounda Pi2 pulsation. An important application of suchinformation is to determine oscillation properties of Pi2pulsation during the event because it provides informa-tion to reveal the excitation mechanism of these pulsa-tions. Here, we consider some additional examples thatwere not included in that paper.

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Table 1Classification of wave types using polarization attributes

Filter name Notation Pr Py

Linear and horizontal ELH ½0;rf � ½�yf ; yf �

Linear and vertical ELV ½0;rf � ½�p=2;�yf � [ ½yf ;p=2�

Elliptical and horizontal EEH ½rf ;1� ½�yf ;yf �

Elliptical and vertical EEV ½rf ;1� ½�p=2;�yf � [ ½yf ;p=2�

rf and yf are two filter parameters.

−1.5−1

−0.50

0.5 HER, North

1996−12−07

0

0.5

1ρ (t,f)

17.4 17.5 17.6 17.7 17.8 17.9

1

2 HER, East

1996−12−07

−50

0

50Θ (t,f)

17.4 17.5 17.6 17.7 17.8 17.9

−0.50

0.51 HER, North

1997−01−04

Fre

quen

cy (

Hz)ρ (t,f)

15.7 15.8 15.9 16 16.10.01

0.015

0.02

0.0252.22.42.62.8 HER, East

1997−01−04

Fre

quen

cy (

Hz)Θ (t,f)

15.7 15.8 15.9 16 16.10.01

0.015

0.02

0.025

00.5

1 HER, North

1997−01−14

Time (hours)

Fre

quen

cy (

Hz)ρ (t,f)

11.8 11.9 12 12.1 12.2 12.30.01

0.015

0.02

0.0250.5

1 HER, East

1997−01−14

Time (hours)

Fre

quen

cy (

Hz)Θ (t,f)

11.8 11.9 12 12.1 12.2 12.30.01

0.015

0.02

0.025

Fig. 5. Polarization properties of three geomagnetic records. Vertical lines bound interval of Pi2 pulsation in corresponding record.



To demonstrate the time–frequency polarization ana-lysis, we examine some records from a magnetic groundstation (HER station, 124:425� co-latitude, 19:225� long-itude) around a Pi2 pulsation for three events thatoccurred on 7 December 1996, 4 January 1997, and14 January 1997. The north and east components of therecords, as well as the computed ratio and the tilt angle inthe time–frequency domain, are shown in Fig. 5. Duringthe event, we found that the ellipticity ratio bounded byvertical lines is small, which indicates that oscillations are

dominant in the direction defined by the tilt angle. Tiltangles close to 0� indicate that the semi-major polariza-tion axis is directed almost along the north–southdirection.

In Fig. 6, we show an example of a polarization filtercalculated using the module gwlET2DFilter - -type=complex. We consider here three different eventsshown above in Fig. 5, but now we analyze the wholesignal in the context of an entire day that contains a Pi2pulsation. With the aim of extracting the Pi2 pulsation

ARTICLE IN PRESS

−101 1996−12−07, HER, North

Spectral bandpass filterPi 2

4 6 8 10 12 14 16 18 20

−101 HER, East Pi 2

−101 1996−12−07, HER, North

Total horizontal filterPi 2

4 6 8 10 12 14 16 18 20


−101 1997−01−04, HER, North Pi 2

4 6 8 10 12 14 16 18 20


−101 1997−01−04, HER, North Pi 2

4 6 8 10 12 14 16 18 20


−101 1997−01−14, HER, North Pi 2

4 6 8 10 12 14 16 18 20

−101

Time (hours)

HER, East Pi 2

−101 1997−01−14, HER, North Pi 2

4 6 8 10 12 14 16 18 20

−101

Time (hours)

HER, East Pi 2

−101 1996−12−07, HER, North

Linear horizontal filterPi 2

4 6 8 10 12 14 16 18 20


−101 1996−12−07, HER, North

Elliptical horizontal filterPi 2

4 6 8 10 12 14 16 18 20


−101 1997−01−04, HER, North Pi 2

4 6 8 10 12 14 16 18 20


−101 1997−01−04, HER, North Pi 2

4 6 8 10 12 14 16 18 20


−101 1997−01−14, HER, North Pi 2

4 6 8 10 12 14 16 18 20

−101

Time (hours)

HER, East Pi 2

−101 1997−01−14, HER, North Pi 2

4 6 8 10 12 14 16 18 20

−101

Time (hours)

HER, East Pi 2

Fig. 6. An example of polarization filtering as applied to geomagnetic records.



from all these signals, we use the Fourier band-pass filterand the so-called ‘‘total horizontal filter’’ ELH þEEH , whichis defined in the same frequency band as the Fourier filterbut with an additional restriction on the tilt angle ofjyjo35�. We can see that the polarization filter picks outthe Pi2 pulsation clearer than the Fourier band-pass filter.

The next example is a division of previous horizontalpolarized signals into the linear ELH and elliptical EEH

parts, where rf ¼ 0:3. Yumoto et al. (1994) reported thatthe ellipticity ratio of a pulsation recorded on the night-side is small, which indicates that oscillations aredominant in the direction defined by the tilt angle; wecan see this behavior in Fig. 5. Filtering procedure ELH isconsistent with this result; all of Pi2 pulsations in Fig. 6are found only on the left-bottom panels related tolinearly polarized signals.

5. Polarization analysis and filtering of 3C data

In this section, we briefly introduce two wavelet-basedpolarization methods suitable to analyze three-compo-nent records. This analysis and the corresponding polar-ization filtering are implemented in the modules gwlET3Dand gwlET3DFilter, respectively.

Unfortunately, there is no mathematically exact apriori definition for the instantaneous polarization attri-butes of a multi-component signal. Therefore, any attemptto produce one is usually arbitrary. Morozov and Smithson(1996) proposed a method based on a variational principlethat allows one to analyze any number of components,and they briefly addressed the possibility of usinginstantaneous polarization attributes for wavefield se-paration and shear-wave splitting identification. UsingGWL, we can execute this algorithm if we run the modulegwlET3D - -type=morozov for an input object having asignal format.

The above-mentioned method is, by design, restrictedto the characterization of an ellipse. In more generalterms, particle motions captured with three-componentrecordings can be characterized by a polarization ellip-soid. Several methods are proposed in the literature tointroduce such an approximation. They are based on theanalysis of the covariance matrix of multi-componentrecordings and principal components analysis usingsingular value decomposition (Kanasewich, 1981; Vidale,1986; Park et al., 1987; Jurkevics, 1988; Jackson et al.,1991). The module gwlET3D - -type ¼ scovar fulfills acovariance analysis in the case where an input object has asignal format.

5.1. CWT-based polarization properties of an ellipse

We extended the method of Morozov and Smithson(1996) to the wavelet domain in order to use theinstantaneous attributes for filtering and wavefield se-paration (Diallo et al., 2005). To introduce this method, wedefine Wguðt; f Þ as the component-wise calculated pro-grade wavelet spectra (f40, wavelet is progressive) of themulti-component signal uðtÞ ¼ ½uxðtÞ;uyðtÞ;uzðtÞ�. The re-sulting semi-major axis and semi-minor axis in this

approach are

Rðt; f Þ ¼ R½Wguðt; f Þ expð�ic0ðt; f ÞÞ�,

rðt; f Þ ¼ R½Wguðt; f Þ expð�iðc0ðt; f Þ þ p=2ÞÞ�,

c0ðt; f Þ ¼1

2arg½Aðt; f Þ þ eBðt; f Þ� þ pn; n 2 N,

Aðt; f Þ ¼1

2

Xj

Wgujðt; f Þ2,

Bðt; f Þ ¼1

2

Xj

Wgujðt; f Þ

0@

1A

2

, (11)

where e51 is a regularization parameter to stabilize thecalculation when the term Aðt; f Þ is close to zero. Torun this method, we use the module gwlET3D - -type=morozov with an input object in the spectrumformat.

As an example, Pacor et al. (2007) used method (11) forspectral analysis and multi-component polarization ana-lyses on the Gubbio Piana (central Italy) earthquakerecordings to identify the frequency content of thedifferent phases that composes the recorded wavefieldand highlight the importance of basin-induced surfacewaves in modifying the main strong ground-motionparameters.

5.2. CWT-based polarization properties of an ellipsoid

We also extended the covariance method to thetime–frequency domain (Kulesh et al., 2007a). Followingthis technique, we use an approximate analytical formulato compute the elements of the covariance matrix for atime window that is derived from an averaged instanta-neous frequency of the multi-component record (Dialloet al., 2006a). To calculate the covariance matrix, we needonly the real part of the complex wavelet coefficients.Therefore, we approximate each voice of the wavelettransform with frequency f around time t using themodulus and phase of the wavelet coefficients by

R½Wgujðt þ t; f Þ� ’ jWgujðt; f Þj

� cos½Ojðt; f Þtþ argWgujðt; f Þ�,

where the instantaneous frequency is calculated using theprograde wavelet spectrum

Ojðt; f Þ ¼ q argWþ

g ujðt; f Þ=qt.

Using this approximation, the entries of the cross-energymatrix Mðt; f Þ ¼ ½Mjmðt; f Þ� can be calculated as

Mjmðt; f Þ ¼ jWgujðt; f ÞjjWgumðt; f Þj

� sincOjðt; f Þ �Omðt; f Þ

2Dtjmðt; f Þ

� �� cosðA�jmðt; f ÞÞ

þ sincOjðt; f Þ þOmðt; f Þ

2Dtjmðt; f Þ

� �

�cosðAþjmðt; f ÞÞ

� mjmmmj,

A�jmðt; f Þ ¼ argWgujðt; f Þ � argWgumðt; f Þ,

j;m ¼ x; y; z, (12)

ARTICLE IN PRESS



where sincðxÞ ¼ sinðxÞ=x indicates the sine cardinal func-tion, and the mean values mjm are defined as

mjm ¼ R½Wgujðt; f Þ�sinc½Dtjmðt; f ÞOjðt; f Þ=2�.

For example, we can define Dtjmðt; f Þ in a specific mannerfor each entry of the cross-energy matrix Mðt; f Þ as

Dtjmðt; f Þ ¼4pn

Ojðt; f Þ þOmðt; f Þ; n 2 N. (13)

The eigenanalysis performed on Mðt; f Þ yields theprincipal component decomposition of the energy. Sucha decomposition produces three eigenvalues, l1ðt; f ÞX

l2ðt; f ÞXl3ðt; f Þ and three corresponding eigenvectors,vjðt; f Þ, that fully characterize the magnitudes and direc-tions of the principal components of the ellipsoid thatapproximates the particle motion in the considered timewindow Dtjmðt; f Þ:

the semi-major axis Rðt; f Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil1ðt; f Þ

pv1ðt; f Þ=kv1ðt; f Þk;

the semi-minor axis rðt; f Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil3ðt; f Þ

pv3ðt; f Þ=kv3ðt; f Þk;

the intermediate axis rsðt; f Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2ðt; f Þ

pv2ðt; f Þ= jv2ðt; f Þk;

the ellipticity rðt; f Þ ¼ krsðt; f Þk=kRðt; f Þk; the minor ellipticity r1ðt; f Þ ¼ krðt; f Þk=krsðt; f Þk;

the dip angle bðt; f Þ ¼ arc tanðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv1;xðt; f Þ

2þ v1;yðt; f Þ

2q

=

v1;zðt; f ÞÞ;

the azimuth gðt; f Þ ¼ arc tanðv1;yðt; f Þ=v1;xðt; f ÞÞ.

We implemented methods (12) and (13) in the modulegwlET3D - -type=acovar, both for a signal and for awavelet spectrum, depending on the input object format.When the instantaneous frequencies are the same for allcomponents, this method produces the same results asthose by Morozov and Smithson (1996) in terms ofpolarization parameters.

5.3. Application to the analysis and filtering of seismic

signals

First, we show an example of polarization filteringbased on approach (11). The real seismograms (Fig. 7)used in this example are three-component recordingsfrom an explosive-source experiment aimed at ima-ging the Dead Sea transform in the Middle East

(DESERT-Group, 2000). Strong surface wave arrivals canbe observed between 3 and 5 s.

We perform two different polarization filters (see, e.g.,Diallo et al., 2005 for more information):

(1) gwlET3DFilter - -type=morozov - -filter¼ ellixy extracts a seismic arrival with a polariza-tion in the horizontal ðx� yÞ plane. It selects regions inthe time–frequency plane where yzðt; f Þo40� and thensets the corresponding Wguðt; f Þ to zero, followed byan inverse CWT, where

yjðt; f Þ ¼ arccosjpjðt; f Þj

kpðt; f Þk

� �2 0;

p2

h i; j ¼ x; y; z

are the different angles between planarity vectorpðt; f Þ ¼ Rðt; f Þ � rðt; f Þ and orthogonal axes x, y, and z

at each point ðt; f Þ. Fig. 8a corresponds to this filter. Itis important to note that the energy arrival from thefiltered vertical component is very weak. It suggeststhat this filter is capable of detecting the desiredsignal which is expected to consist mainly of thex-and y-components.

(2) gwlET3DFilter - -type=morozov - -filter=normperforms another application of polarization analysis,which is important in geologically complex areas. Thisis the removal of out-of-plane energy. This is achievedby normalizing the different cosine directions yjðt; f Þ,so that the range of each direction is between 0 and 1.Then, each normalized yjðt; f Þ is multiplied by thewavelet transform of the signal component along thecorresponding j-axis ðj ¼ x; y; zÞ, followed by an in-verse CWT. Fig. 8b shows the result of this procedure.This filtering has the effect of reducing the influenceof energy arrivals that come from directions notparallel to the direction where each component hasits highest sensitivity to particle motion.

Now let us demonstrate adaptive covariance methods(12) and (13) as applied to earthquake data analysis.Fig. 9a shows the three-component seismograms of theMw ¼ 7:8 earthquake of June 13, 2005, at the Chilean–Bolivian border, recorded at the GRA1 station of the GRFarray in northern Bavaria (Germany). The hypocentraldepth was estimated as 114 km.

ARTICLE IN PRESS

0 1 2 3 4 5 6 7 8 9 10Time (s)

uv (t)

ut (t)

ur (t)

Fig. 7. Three-component real seismograms showing radial urðtÞ, transversal utðtÞ, and vertical uvðtÞ components, respectively.



In Fig. 10, we compare the semi-major and semi-minoraxes of polarization computed from the adaptive covar-iance method and the standard covariance method (see,e.g., Diallo et al., 2006a, for more information). As one cansee by comparing the black and gray curves, the standardmethod represents a smoothed version of the instanta-neous attributes from the adaptive method. Furthermore,because the time window is fixed for the standardmethod, it is not possible to characterize polarizationattributes of a seismic event with a period lower than thatof the time window used for the analysis. We circumventthis problem with the adaptive covariance methodthrough the adaptive selection of the time window (13).

Let us consider a new example that is not included inour earlier publications. An interesting feature in the

seismograms (Fig. 9a) is the presence of SKS arrivals,which is one of the most important examples of shear-wave splitting in the Earth. This observation indicates akind of anisotropy of the upper mantle. The degree ofanisotropy is generally quantified by the estimate of theamount and direction of this shear-wave splitting. Here,we are not trying to perform a state-of-the-art analysis ofthe observed SKS phase for continental shear-wavesplitting; our aim is to show how the wavelet-basedpolarization method can improve shear-wave splittinganalysis in general. Therefore, we will focus our attentionon the time window for the SKS arrival shown in Fig. 11a.

Fig. 11b corresponds to the wavelet transform of theradial component, which entails most of the energy fromthe SKS arrivals. The image of rðt; f Þ for this phase reveals

ARTICLE IN PRESS

0 1 2 3 4 5 6 7 8 9 10

urv (t)

urt (t)

urr (t)

0 1 2 3 4 5 6 7 8 9 10Time (s)

uθv (t)

uθt (t)

uθr (t)

Fig. 8. Example of two different polarization filters applied to three-component seismograms presented in Fig. 7.

0 500 1000 1500 2000 2500 3000

Time (s)

uv (t)

ut (t)

ur (t)

Pdiff

SKS Sdiff

Fig. 9. Three-component real seismograms showing radial urðtÞ, transversal utðtÞ, and vertical uvðtÞ components, respectively.



some degree of ellipticity, which varies between 0.2 and0.4 as indicated by the color code. This is a weakellipticity, but it is consistent with the amount of SKSsplitting for typical teleseismic observations. It is alsoimportant to note that the average ellipticity ratio for theSKS is about three times higher than its counterpartfor the Pdiff (diffracted P-wave), which we used asreference for a linear polarization. For instance, in anillustration such as Fig. 11d, any noticeable frequencydependencies of the splitting parameters with frequency

or a comparatively large r1ðt; f Þ with respect to rðt; f Þshould be a hint towards a shear-wave splitting occurringat different scales and different directions.

6. Modeling of wave propagation using adiffeomorphism in wavelet space

To analyze the dynamic behavior of multi-variatesignals using CWT, one may be interested in investigating

ARTICLE IN PRESS

00.20.40.60.8

1

R

ACMSCM

00.20.40.60.8

1

r s

400 600 800 1000 1200 1400 1600 1800 20000

0.20.40.60.8

1

Time (s)

r

Fig. 10. Comparison of polarization attributes obtained from adaptive covariance method (ACM) with those computed using standard covariance method

(SCM). (a) Semi-major polarization axis, (b) intermediate polarization axis, and (c) semi-minor polarization axis.

uv (t)

ut (t)

ur (t)

1

2

3

x 104

Fre

quen

cy (

Hz)

Wgur (t,f)

0.05

0.1

0.15

0.2

0.20.40.60.8

Fre

quen

cy (

Hz)

Wgρ (t,f)

0.05

0.1

0.15

0.2

0.20.40.60.8

Time (s)

Fre

quen

cy (

Hz)

Wgρ1 (t,f)

1100 1105 1110 1115 1120 1125 1130 1135 1140 1145 1150

0.05

0.1

0.15

0.2

Fig. 11. Polarization analysis restricted to time window corresponding to SKS phase arrival: (a) three-component seismogram for considered time

window, (b) wavelet transform of radial component, (c) ellipticity rðt; f Þ, and (d) minor ellipticity r1ðt; f Þ.



a diffeomorphic deformation of the wavelet space. Thesedeformations establish an algebra of wavelet pseudodif-ferential operators acting on signals. We developed twomodules related to these operators—gwlDiffeoLin andgwlDiffeoDisp.

In the most general case, a wavelet deformationoperator D can be defined as

O½D�: sðtÞ7!MhDWgsðt; f Þ; D: H!H,

H:¼fðt; f Þ: t 2 R; f40g.

6.1. Linear diffeomorphism of the wavelet spectrum

The simplest deformation is a set of all linear maps; weimplemented it in the module gwlDiffeoLin as follows:

DL: ðt; f Þ7!ðb1t þ b2=f ; b3f Þ. (14)

In this particular case, if the analyzing wavelet gðtÞ and asource signal sðtÞ are both progressive, this action is givenin the time domain by the following expression (Xie et al.,2003):

O½DL�: sðtÞ7!Cþsðb1tÞ,

C� ¼ 2pZ 1

0gnð�b3oÞhð�b1oÞ expð�ib2oÞ

doo .

If we select b1 ¼ 1, the propagated signal can besimply obtained by multiplying the source signal withconstant Cþ. This interesting mathematical result isdemonstrated in Fig. 12, where we consider a complexMorlet wavelet as source signal s1ðtÞ. We apply the CWTprocedure using the Cauchy wavelet and the linearpropagator DL : ðt; f Þ7!ðt þ 3:3=f ; f Þ, and, finally, we cal-culate IWT, followed by multiplying by constant Cþ. Afterthis procedure, we obtain the same signal s2ðtÞ ¼ s1ðtÞ, butits wavelet spectrum is different from the source waveletspectrum.

6.2. Describing wave dispersion and attenuation with non-

linear diffeomorphism

The linear propagator (14) is not relevant for practicalgeophysical applications, but it demonstrates an idea ofhow to model a signal propagation using some diffeo-morphic operators in the time–frequency domain. In thiscontext, we have shown previously (Kulesh et al., 2005a;Holschneider et al., 2005) how the wavelet transform ofthe source and the propagated signals are related througha transformation operator that explicitly incorporates thephase and group velocities as well as the attenuationfactor of the medium.

Assume that sjðtÞ and smðtÞ represent two signalsobserved at two stations that are distance Dmj ¼ Dm � Dj

apart. If the dispersive and dissipative characteristics ofthe medium are represented by the frequency-dependentwavenumber kðf Þ and attenuation coefficient aðf Þ, therelation between the Fourier transforms of these signalsreads

smðf Þ ¼ DFsjðf Þ ¼ expð�iKðf ÞDmj � 2pinÞsjðf Þ or

Tmjðf Þ ¼ DCTrrðf Þ ¼ smðf Þsn

j ðf Þ

¼ expð�aðf ÞSmjÞ expð�2pikðf Þ

�Dmj � 2pinÞTrrðf Þ, (15)

where n 2 N is any integer number and Kðf Þ is thecomplex wavenumber, which can be defined by realfunctions kðf Þ and aðf Þ as Kðf Þ ¼ 2pkðf Þ � iaðf Þ. Tmjðf Þ isthe cross-correlation in the Fourier domain when we takea trace srðtÞ as reference and two other traces, smðtÞ atdistance Dm and sjðtÞ at distance Dj (distance with respectto the position of the reference when all three stations arealigned): Trrðf Þ ¼ jsrðf Þj

2, Smj ¼ Dm þ Dj.Let us assume that the frequency-dependent wave-

number and attenuation vary slowly with respect tothe frequency range of the mother wavelet. For mode-rate dispersion, the complex wavenumber can be

ARTICLE IN PRESS

s1 (t)

Fre

quen

cy (

Hz)

|Wg s1 (t,f)|

2468

101214

Time (s)

Fre

quen

cy (

Hz)

arg Wg s1 (t,f)

0 1 2 3 4 5

2468

101214

s2 (t)

|Wg s2 (t,f)|

2468

101214

Time (s)

arg Wg s2 (t,f)

0 1 2 3 4 5

2468

101214

Fig. 12. An example of linear propagator applied to progressive Morlet wavelet as source signal.



approximated by the first two terms of its Taylor seriesaround the frequency f. Next, we assume that theattenuation shows nearly linear frequency dependence(‘‘constant-Q attenuation’’). In such a case, a0ðf Þ�0 and theasymptotic propagator in the wavelet space has the formas defined in Kulesh et al. (2005a):

Wgsmðt; f Þ ¼ DWWgsjðt; f Þ

¼ expð�aðf ÞDmjÞ expð�ic1ðf ÞÞ

�Wgsjðt � k0ðf ÞDmj; f Þ or

WgTmjðt; f Þ ¼ DCWWgTrrðt; f Þ

¼ expð�aðf ÞSmjÞ expð�ic1ðf ÞÞ

�WgTrrðt � k0ðf ÞDmj; f Þ, (16)

where c1ðf Þ ¼ 2p½kðf Þ � fk0ðf Þ�Dmj þ 2pn.In a special case, with the assumption that the

analyzing wavelet has a linear phase (with time derivativeapproximately equal to 2p, as is the case for the Morletwavelet (3)), approximation (16.1) can be written in termsof the phase, Cpðf Þ, and group, Cgðf Þ, velocities as

Wgsmðt; f Þ ¼ expð�aðf ÞDmjÞ � Wgsj t �Dmj

Cgðf Þ; f

� ��

� exp i arg Wgsj t �Dmj

Cpðf Þ�

n

f; f

� �� ,

where Cpðf Þ ¼f

kðf Þ; Cgðf Þ ¼

1

k0ðf Þ

¼C2

pðf Þ

Cpðf Þ � fC0pðf Þ. (17)

Procedure (16) is implemented in the module gwlDif-

feoDisp with the parameter prop ¼ 1, while procedure(17) can be executed with the parameter prop ¼ 2. Thismodule has an additional parameter, acorr, which can beused when the source signal is given by cross-correlationTrr , and we obtain the cross-correlation after the propaga-tion as well. The second feature implemented in themodule gwlDiffeoDisp is the possibility of using a

two-component signal in the complex form (8) as input. Adetailed description of this calculation potentiality isgiven by Kulesh et al. (2005b) as applied to the dispersionanalysis of Rayleigh waves.

Relationship (17) has the following interpretation. Thegroup velocity is a function that ‘‘deforms’’ the image ofthe absolute value of the source signal’s wavelet spectrum,the phase velocity ‘‘deforms’’ the image of the waveletspectrum phase, and the attenuation function determinesthe frequency-dependent real coefficient by which thespectrum is multiplied. This behavior is demonstrated inFig. 13, where we consider a synthetic signal s1ðtÞ. In thisexample, we use a propagation phase and group velocitiesthat are not based on a physical model. These frequency-dependent velocities are shown in Figs. 13c and d as solidcurves. We perform the propagation of signal s1ðtÞ usingprocedure (15.1) and obtain a propagated counterparts2ðtÞ. The gray-scaled images in Fig. 13 show the absolutevalues and phases of wavelet spectra Wgs1ðt; f Þ andWgs2ðt; f Þ. We see that the deformations of the imageslabeled as jWgs2ðt; f Þj and arg Wgs2ðt; f Þ agree in generalwith velocities curves Cgðf Þ and Cpðf Þ, respectively, buthave small distinctions that demonstrate the asymptoticproperties of propagator (17).

6.3. The case for non-linear frequency-dependent

attenuation

Below, we consider two simple extensions of theproposed wavelet-based propagators. We have not pub-lished these new propagation models before; therefore,we only show the mathematical formulations without anyinterpretations.

In propagation models (16) and (17), the frequency-dependent wavenumber and attenuation are independentand, therefore, do not satisfy the causality constraint. Tosatisfy it, we approximate the complex wavenumber as inthe previous section by the first two terms of its Taylor

ARTICLE IN PRESS

s1 (t)

Fre

quen

cy (

Hz)

|Wg s1 (t,f)|

102030405060708090

Time (s)

Fre

quen

cy (

Hz) arg Wg s1 (t,f)

0 0.5 1 1.5 2

102030405060708090

s2 (t)

|Wg s2 (t,f)|

Cg (f)

102030405060708090

Time (s)

arg Wg s2 (t,f)Cp (f)

0 0.5 1 1.5 2

102030405060708090

Fig. 13. Propagated synthetic signal and its wavelet transform: (a), (c) are powers (absolute value squared) of wavelet coefficients and (b), (d) are

corresponding phase images. Lines in (c) and (d) show frequency-dependent group and phase velocities used in propagation model.



series around the frequency f. However, instead of theassumption a0ðf Þ�0, a special wavelet, like Cauchy wavelet(4), can be used; this allows us to derive a relationshipbetween the wavelet transforms of signals observedat two different stations in terms of the complexwavenumber:

Wgsmðt; f Þ ¼ DCCWgsjðt; f Þ

¼expð�i½Kðf Þ � fK0ðf Þ�DmjÞ

f p�1a ðf Þ

Wgsj

� t �Dmj

2p RK0ðf Þ;f

f aðf Þ

� �, (18)

where f aðf Þ ¼ 1�fDmj

p� 1IK0ðf Þ.

This propagator is implemented in the module gwl

DiffeoDisp with the parameter prop ¼ 3.

6.4. Modeling wave propagation in the space of polarization

parameters

A special feature of propagator (16) is the possibility ofjoining it with the polarization properties. If a two-component input signal is given in the complex form (8)and the frequency-dependent wavenumber and attenua-tion are independent, we can then obtain the dispersivepropagator in the space of polarization properties (9) asfollows:

Rmðt; f Þ ¼ expð�aðf ÞDmjÞRjðt � k0ðf ÞDmj; f Þ,

rmðt; f Þ ¼ expð�aðf ÞDmjÞrjðt � k0ðf ÞDmj; f Þ,

ymðt; f Þ ¼ yjðt � k0ðf ÞDmj; f Þ,

Fmðt; f Þ ¼ Fjðt � k0ðf ÞDmj; f Þ � c1ðf Þ, (19)

where we use the signed semi-minor axis rðt; f Þ ¼

jWþ

g zðt; f Þj � jW�

g zðt; f Þj and the additional phase para-meter Fðt; f Þ ¼ ðarg Wþ

g zðt; f Þ � arg W�

g zðt; f ÞÞ=2.In the special case of the Morlet wavelet, this

propagator can be rewritten in terms of the phase andgroup velocities as

Rmðt; f Þ ¼ expð�aðf ÞDmjÞRjðt � Dmj=Cgðf Þ; f Þ,

rmðt; f Þ ¼ expð�aðf ÞDmjÞrjðt � Dmj=Cgðf Þ; f Þ,

ymðt; f Þ ¼ yjðt � Dmj=Cpðf Þ; f Þ,

Fmðt; f Þ ¼ Fjðt � Dmj=Cpðf Þ; f Þ. (20)

These relationships between the polarization para-meters observed at stations m and j allow us to constructthe following propagation procedure (polarization propa-gator):

W�

g zmðt; f Þ ¼ ðRmðt; f Þ � rmðt; f ÞÞ

� expðiðymðt; f Þ �Fmðt; f ÞÞÞ,

zmðtÞ ¼Mh Wþ

g zmðt; f Þ þW�

g zmðt; f Þ� �

. (21)

We implemented this propagator in the modulegwlDiffeoDisp with the parameter prop ¼ 4 orprop ¼ 5, depending on which relationship, (19) or (20),is used for the propagation procedure (21).

6.5. Parametrization of dispersion and attenuation

To perform each of the above-mentioned propagatorsin the module gwlDiffeoDisp, we need to define first adispersion model. To prepare a dispersion model, themodule gwlDispModel can be used. We predefine somemodels in this module:

(1) gwlDispModel - -wn ¼ gauss - -atn ¼ gauss cor-responds to the three-parameter exponential ap-proximation for the wavenumber kðf Þ and theattenuation function aðf Þ:

Fðf Þ ¼ p1f þ p2f expð�f 2=ð2p23ÞÞ. (22)

(2) In this module we can also parameterize aðf Þ andkðf Þ using polynomials functions of power N

ðwn=polin atn=polinÞ:

Fðf Þ ¼ p0 þXN

j¼1

pj

jf j. (23)

(3) To parameterize B-spline functions of fourth orderðwn ¼ bspline atn ¼ bsplineÞ, we define aðf Þ andkðf Þ as

Fðf Þ ¼XN�1

j¼0

pjBf � f 1 � jD

Dþ 3

� �,

D ¼f 2 � f 1

N � 3, (24)

where ½f 1; f 2� is the frequency interval in which wewant to define this function.

(4) For model (18), it is convenient to use an isotropiclinear viscoelastic model (Cole–Cole model intro-duced by Lu and Hanyga, 2004), where the fre-quency-dependent wavenumber and attenuation arerelated ðwn ¼ colecole atn ¼ colecoleÞ:

kðf Þ ¼ RKð2pf Þ=ð2pÞ; aðf Þ ¼ �IKð2pf Þ,

KðoÞ ¼ offiffiffiffiffiffiffiffiffiffiffisðoÞ

p ,

sðoÞ ¼ Mr1þ ðioteÞg

1þ ðiotsÞg. (25)

Note that, in the case of independent wavenumber andattenuation such as for exponential, polynomial and B-spline approximations, we can combine different types ofparametrization for kðf Þ and aðf Þ.

7. An estimate of phase and group velocitiesand attenuation

Eq. (17) and Fig. 13 allow us to formulate how thefrequency-dependent phase velocity can be obtainedusing the wavelet spectra phases of source and propa-gated signals. On one hand, we can calculate thecorrelations between two wavelet spectra to obtain theapproximate phase velocities of each mode in a signal(Kulesh et al., 2008). On the other hand, one can alsoformulate an optimization problem and solve it through

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the minimization of an appropriately defined cost func-tion to precisely define the parameters of one mode(Holschneider et al., 2005). Both of these possibilities areimplemented in modules gwlTransFK, gwlOptiSI, andgwlOptiSP.

7.1. Wavelet-based frequency–velocity analysis

Using the correlations between two spectra, we canperform ‘‘frequency–velocity’’ analysis on the analogy ofthe frequency–wavenumber method (Capon, 1969) for aseismogram sjðtÞ; j ¼ 1; . . . ;N. The main part of thisanalysis consists of calculating the correlation spectrumMðf ; cÞ as follows (Kulesh et al., 2008):

Mðf ; cÞ ¼

Z tmax

tmin

Xj;m

Ajðt; f ÞAm t�Dmj

c; f

� ��dt

¼

Z tmax

tmin

Xj;m

expðiBjðt; f ÞÞ

�� exp �iBm t�

Dmj

c; f

� �� dt, (26)

Ajðt; f Þ ¼Wgsjðt; f Þ=jWgsjðt; f Þj,Bjðt; f Þ ¼ arg Wgsjðt; f Þ,

where ½tmin; tmax� indicates the total time range for whichthe wavelet spectrum was calculated, c 2 ½Cmin

p ;Cmaxp � is an

unbounded variable corresponding to the phase velocity,Aj is a complex-valued wavelet phase and Bj is a real-valued wavelet phase.

The correlation spectrum Mðf ; cÞ is calculated usingonly the correlations between phases of wavelet spectra;these phases do not contain any amplitude information.Using this fact, we can use an alternative definition of amulti-trace correlation expression:

Mðf ; cÞ ¼Z tmax

tmin

YNj¼1

Bj t�D1j

c; f

� ��dt. (27)

We can perform the ‘‘frequency–velocity’’ analysis of amulti-channel wavelet spectrum using both Eqs. (26) and(27) in the module gwlTransFK. These two methods arerelated with the input parameters corr ¼ cphase andcorr ¼ arg, respectively.

To demonstrate this concept, we re-analyzed a syn-thetic seismogram having two propagated modes withdifferent wavenumbers, but with the same frequencycontent (Kulesh et al., 2008). We consider the situationwithout attenuation: aðf Þ ¼ 0:

smðf Þ ¼ exp�2piDmjf

C1ðf Þ

� �sjðf Þ

þ exp�2piDmjf

C2ðf Þ

� �sjðf Þ.

This seismogram is shown in Fig. 14a. The phase velocities,C1ðf Þ and C2ðf Þ, used for this seismogram generation areplotted in Figs. 14b and c as solid curves; fundamentally,this situation describes first symmetric and asymmetricmodes of a Lamb wave.

The gray-scaled background image in Fig. 14b showsthe power function of normalized correlation coefficientsfor real-valued wavelet phases using method (27), andFig. 14c shows the correlation result where complex-valued phases and method (26) are used. The agreementof the correlation coefficients’ maximum lines withtheoretical phase velocities is very good in both methods.

The four extra ‘‘pseudo-modes’’ presented in Figs. 14band c are justified on the basis of 2p-cycle skips betweenthe stations introduced in relationship (15) and remainingin the wavelet propagator (17) as the n=f term. These‘‘pseudo-modes’’ can be filtered by analyzing the groupvelocities.

7.2. Inversion for the dispersion and attenuation

characteristics of the medium

Given the recorded signals at two or more stations, wewill present an approach to simultaneously estimate kðf Þ

and the attenuation coefficient aðf Þ.For a given parametrization of dispersion and attenua-

tion functions, finding an acceptable set of parameters canbe thought of as an optimization problem that seeks tominimize a cost function w2 and can be formulated asfollows:

w2ðaðf ; aÞ; kðf ;kÞÞ !min; a 2 RNa ; k 2 RNk ,

where Na is the number of parameters used to model theattenuation aðf Þ and Nk is the number of parameters usedto model the wavenumber kðf Þ. Vectors a and k representthe parameters describing approximations (22)–(25) offunctions aðf Þ and kðf Þ, respectively.

The cost function w2ðaðf ; aÞ; kðf ;kÞÞ involves a propa-gator described in the previous section depending on thenature of the signal to be analyzed. In the following, wewill discuss the different steps involved in our inversionalgorithm as depicted in the chart of Fig. 15 (Holschneideret al., 2005).

(1) The first step consists of seeking a good initialcondition by performing an image matching usingthe modulus of the wavelet transforms of a pair oftraces (the module gwlOptiSP - -cmpl ¼ 3). In thiscase the cost function w2 is defined as

w2ða;kÞ ¼

Z ZjWgs2ðt; f Þj��

�jDWða;kÞWgs1ðt; f Þj��2 dt df . (28)

Optimization is carried out over the whole frequencyrange of the signal. At the end, we get the frequency-dependent derivative of the wavenumber k0ðf Þ andattenuation aðf Þ. At this stage, we need to distinguishbetween the case where the analyzed signal consistsof only one coherent arrival and the case where itconsists of several coherent arrivals. In the former, thederived functions are meaningful and characterizethose analyzed event. However, these functionscannot be easily interpreted in the latter because, onone hand, they were obtained from an optimizationthat only takes the modulus of the transform into

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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ce N

umbe

r

Frequency (Hz)

Pha

se v

eloc

ity

(m/s

)

50 100 150 200 250

1200

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1300

1350

1400

1450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

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50 100 150 200 250

Fig. 14. Frequency–velocity analysis of a noisy seismic arrival with two interfering pulses. Noise level is about 5% of peak-to-peak amplitude.

Wavelet-optimization (modulus)Frequency: fmin-fmax Hz

Signal-optimizationFrequency: fmin-fmax HzStep 2: more than

two signals

global phaseand group

velocity

globalattenuationparameters

phase and groupvelocity for

single mode

attenuationparameters for

single mode

Step 1: two signals

Wavelet optimization(argument)T-F band: f ∈F1, t ∈T1

T-F band: f ∈F2, t ∈T2





Wavelet optimization(modulus)

Step 3: morethan twosignals

first set ofparameters

verification

verification

Fig. 15. Flowchart showing optimization sequence to be performed in order to extract dispersive and dissipative characteristics of individual modes (that

can be identified with CWT) from multimode surface wave records.



account and the phase information is lost, and on theother hand, the signals involved consist of manyoverlapping arrivals.

(2) If only one single phase is observed in all traces sjðtÞ, itis enough to minimize a cost function that involvessome selected seismic traces in order to estimate theattenuation and phase velocity using propagator (15)in the module gwlOptiSI. A possible definition ofthis cost function may be

w2ða;kÞ ¼Xm;j

Zj smðtÞ �F�1DFða;kÞsjðf Þj

2 dt or

w2ða;kÞ ¼Xm;j

ZjTmjðtÞ �F�1DCða;kÞTrrðf Þj

2 dt, (29)

where F�1 indicates the inverse Fourier transform.Minimizing the cost function based on the cross-correlations is more advantageous for two reasons.On one hand, the effect of random noise is canceled;on the other, with geophones laid out symmetri-cally around the source in a seismic survey, cross-correlations of traces from seismic waves propagatingin opposite directions can be combined in theoptimization. For single phase arrival in all traces,the output at the end of this step yields the desiredresult.

(3) In the case where the observed signals consistof a mixture of different wave types and modes, acascade of optimizations in the wavelet domain isnecessary in order to fully determine the dispersionand attenuation characteristics specific to each co-herent arrival. First, we perform the optimization onthe modulus of the transforms, in which case theattenuation for the specified event is derivedðgwlOptiSP - -cmpl=3Þ:

w2ða;kÞ ¼Xm;j

Z ZjjWgTmjðt; f Þj

� jDCWða;kÞWgTrrðt; f Þjj2 dt df . (30)

Next, we perform an optimization involving theargument of the wavelet transforms, which finallyprovides the phase and group velocity curves of theanalyzed coherent arrival ðgwlOptiSP - -cmpl=4Þ:

w2ða;kÞ ¼Xm;j

Z Zj argWgTmjðt; f Þ

� argDCWða;kÞWgTrrðt; f Þj2 dt df . (31)

This optimization can be repeated to characterizeeach coherent arrival separately.

Because the dependence of the cost functions (28)–(31)on the parameters a and k is highly non-linear, eachfunction may have several local minima. To obtain theglobal minimum that corresponds to the true parameters,a non-linear least-squares minimization method thatproceeds iteratively from a reasonable set of initialparameters is required. In the present contribution,we use the Levenberg–Marquardt algorithm (Presset al., 1992).

7.3. Application to experimental data

As an example, we previously analyzed the field data inHolschneider et al. (2005), where we successfully esti-mated the phase and group velocities as well as attenua-tion using this minimization procedure. The experimentaldata shown in Fig. 16 consist of a shallow seismic survey(stations along a line) at Kerpen, a particular site in theLower Rhine embayment where the buried scarp of ahistorically active fault is presumed. Several profiles of 48channels with 2 m inter-receiver spacing were collectedusing hammer blows as seismic source. We selected aseismogram profile with prominent low frequency, highamplitude arrivals that correspond to the surface wavearrivals we intend to characterize.

To check the quality of such an inversion, we evaluatedslowness using two alternative methods as described byHolschneider et al. (2005): CAPON high resolution method(Capon, 1969) and MUSIC high resolution method(Schmidt, 1986).

We selected the complete waveform windows ofsubarrays corresponding to subsections A and B alongthe shot profile. The wavenumber k is sampled equidis-tantly in one dimension, and a set of 200 discrete frequen-cies is spaced equidistantly on a logarithmic frequencyscale from 10 to 50 Hz. In Fig. 17, we show the resultsof this analysis for the two subarrays. Each subfigurepictures the results of the MUSIC approach as a gray-scaled background image. For better visibility, we normal-ized the maxima to one for each individual analysisfrequency. The superimposed contour lines give the resultfrom Capon’s analysis method. In order to compare theseresults with the phase velocity estimates from the CWT-based method, we finally superimposed the correspond-ing estimates as solid curves.

For subsection A, all three methods show a consistentestimate of the phase velocities from the seismic records.For subsection B, we recognize less consistency betweenthe different estimates of the phase velocities. It is inte-resting to note that the contour plot of Capon’s analysisresults does not reach the frequency range where thesurface wave is highly attenuated. In addition, the gray-scale background depicting the result from the MUSICapproach shows a decreased resolution of the velocity inthis frequency range, indicated by the peak broadeningalong the vertical scale. The region where the phasevelocity estimates from all methods are similar corre-sponds to those where the energy of the surface arrival issignificantly high (around 25–35 Hz).

8. Conclusion

We propose a software package that implements somemethods of polarization and dispersion analysis in thewavelet domain using the continuous wavelet transform:

(1) A method for computing instantaneous attributes of2C signals in the time–frequency domain. The advan-tage of this method over previous techniques (Reneet al., 1986) is that both the time and frequency

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dependence of the attributes can be obtained andused for wave-mode separation and filtering.

(2) An extension of the polarization analysis technique formulti-component data initially proposed by Morozovand Smithson (1996) into the time–frequency domain.

(3) A method to estimate the instantaneous polarizationattributes based on an approximation to the covar-iance matrix and its extension to the time–frequencydomain. The advantage of the proposed method overthe standard method is that the length of the window

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20 25 30 35 40 45

4

5

6

7

8

9

10x 10−3

Slow

ness

(s/

km)

Subsec. A

20 25 30 35 40 45

4

5

6

7

8

9

10x 10−3

Slow

ness

(s/

km)

Subsec. B

Fig. 17. Comparison of slowness estimates obtained for (a) subsection A and (b) subsection B from CWT method (solid curve), Capon’s high resolution f-k

method (contour lines), and MUSIC algorithm (gray-scaled background image).

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

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Subs

ec. A

Subs

ec. B

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eive

r st

atio

n

Fig. 16. Observed seismograms obtained from shallow seismic experiment using sledgehammer as a source.



size for the covariance computation is adaptivelyadjusted with the help of the instantaneous frequen-cies from the different components.

(4) Some methods to establish a link between thecontinuous wavelet transforms of a signal and itspropagated counterpart in a dispersive and attenuat-ing medium. The advantage of using the proposedpropagator over traditional methods, such as theWigner–Ville or time–frequency reassignment fordispersion curves estimates, is that the full dispersionand dissipation characteristics are explicitly expressedand, therefore, can be easily extracted.

(5) An approach to use this wavelet propagator in themethod of ‘‘frequency–velocity’’ analysis that isanalogous to the classical frequency–wavenumber(f–k) analysis methods. Using this method, thedetermination of several mode branches is feasible.

(6) A method of simultaneous computations of bothphase and group velocities in the wavelet domain.The method owes its robustness to the fact thatthe minimization process involves not only themodulus but also the phase of the wavelet transform,thus making it possible, in principle, to reconstructthe dispersed signal from the manipulated waveletcoefficients.

The mathematical aspects of all these methods havebeen separately published by Kulesh et al. (2005a, b,2007a), Holschneider et al. (2005), and Diallo et al. (2005,2006a, b). In this paper, we showed that all of thesemethods could be logically combined into one library. Allexamples presented in this paper are generated using thislibrary and are included in the installation of our software.

Acknowledgments

This project is supported by a grant from the DeutscheForschungsgemeinschaft (DFG) within the framework ofthe priority program SPP 1114 ‘‘Mathematical methods fortime series analysis and digital image processing’’.

We are especially grateful to Peter R. Sutcliffe (Herma-nus Magnetic Observatory) for providing the experimentaldata set for HER station used in Section 4.4. We wish tothank Masahito Nose (Data Analysis Center for Geomag-netism and Space Magnetism, Kyoto University) for hisvaluable comments and suggestions related to theanalysis of this geomagnetic data.

In Section 5.3, we used data of the controlled sourcearray (CSA) experiment as part of the Dead Sea ResearchProject (DESERT-Group, 2000). DESERT was financed bythe Deutsche Forschungsgemeinschaft (DFG), the Geo-ForschungsZentrum Potsdam (GFZ), and the MinervaFoundation. Instruments were provided by the Geophysi-cal Instrument Pool Potsdam (GFZ) and the Free Universityof Berlin.

We thank Matthias Ohrnberger, Frank Scherbaum,Frank Kruger, and Erika Luck from the Institute ofGeoscience, University of Potsdam for providing theexperimental data demonstrated in Sections 5.3 and 7.3.

We also thank the two anonymous reviewers for theirhelpful and constructive comments.

Appendix A. List of symbols

t time

f frequency

a ¼ f 0=f scale parameter of the CWT

o angular frequency

ð�Þn complex conjugate

sðtÞ one-channel signal

uðtÞ ¼ ½uxðtÞ;uyðtÞ;uzðtÞ� three component recording, where

ux—radial, uy—transversal and

uz—vertical components

zðtÞ ¼ uxðtÞ þ iuzðtÞ a complex signal

sjðtÞ; j ¼ 1; . . . ;N a seismic section (N stations aligned

along a line)

Dmj ¼ Dm � Dj distance between two stations

ð:Þ Fourier transform

Wgsðt; f Þ the continuous wavelet transform of

sðtÞ (CWT)

Wþ

g sðt; f Þ, W�

g sðt; f Þ prograde and retrograde wavelet

spectrum

Wguðt; f Þ the component-wise calculated

wavelet spectrum of a multi-

component signal

MhWgsðt; f Þ the inverse wavelet transform (IWT)

gðtÞ, hðtÞ wavelets used for CWT and IWT

Oðt; f Þ ¼ q argWgsðt; f Þ=qt instantaneous frequency

R, Rðt; f Þ, Rðt; f Þ the semi-major axis as scalar, a

function of time and frequency or a

vector

r, rðt; f Þ, rðt; f Þ the semi-minor axis

rs, rsðt; f Þ, rsðt; f Þ the intermediate axis

pðt; f Þ ¼ Rðt; f Þ � rðt; f Þ the planarity vector

rðt; f Þ ¼ krðt; f Þk=kRðt; f Þk the ellipticity

y, yðt; f Þ the tilt angle

Df, Dfðt; f Þ the phase difference between two

components

D a wavelet deformation operator

kðf Þ frequency-dependent wavenumber

aðf Þ frequency-dependent attenuation

coefficient

Kðf Þ ¼ 2pkðf Þ � iaðf Þ complex wavenumber

TmjðtÞ the cross-correlation of two signals

sjðtÞ and smðtÞ

Cpðf Þ, Cg ðf Þ the phase and group velocities

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