Pure appl. geophys. 162 (2005) 843–855 0033 – 4553/05/050843 – 13 DOI 10.1007/s00024-004-2644-9
Ó Birkha¨user Verlag, Basel, 2005
Pure and Applied Geophysics
Modeling of Wave Dispersion Using Continuous Wavelet Transforms M. Kulesh,1 M. Holschneider,1 M. S. Diallo,1 Q. Xie,1 and F. Scherbaum2
Abstract—In the estimate of dispersion with the help of wavelet analysis considerable emphasis has been put on the extraction of the group velocity using the modulus of the wavelet transform. In this paper we give an asymptotic expression of the full propagator in wavelet space that comprises the phase velocity as well. This operator establishes a relationship between the observed signals at two different stations during wave propagation in a dispersive and attenuating medium. Numerical and experimental examples are presented to show that the method accurately models seismic wave dispersion and attenuation. Key words: Continuous wavelet transform, dispersion, modeling, propagator in wavelet space, phase deformation, attenuation.
1. Introduction Wave dispersion expresses the phenomenon by which the phase and group velocities are functions of the frequency. The cause of dispersion may be either geometric or intrinsic. For seismic surface waves the cause of dispersion is of a geometrical nature. Geometric dispersion results from the constructive interferences of waves in bounded or heterogeneous media. Intrinsic dispersion arises from the causality constraint imposed by the Kramers-Kro¨nig relation. For these waves, the determination of the phase and group velocity as well as of the quality factor Q can lead to a model of the physical properties of the sub-surface structure (BUTTKUS, 2000). The analysis of surface waves are very often used as an indirect characterization and imaging tool for various geotechnical projects. Longperiod surface waves such as those generated by large earthquakes are extensively exploited for the investigation of the earth’s deep interior (KEILIS-BOROK, 1989; LAY and WALLACE, 1995). One problem in the context of surface wave analysis (especially with high frequency signals) is the robust determination of dispersion curves.
1 Institute of Mathematics, University of Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany. E-mail:
[email protected] 2 Institute of Geoscience, University of Potsdam, Karl-Liebknecht-Str. 24-25, 14415 Potsdam, Germany.
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Several methods exist for the estimation of group and phase velocities. The choice of methods depends on the signal-to-noise ratio, the assumed or known degree of interference between modes and/or between surface and body waves, as well as on the number of stations available. For instance, the group velocity can be determined from a single station record of a well dispersed surface wave (HERMANN, 1973) using the multiple filter technique originally developed by DZIEWONSKI et al. (1969). However, for the phase velocity this method is not easy to implement as it requires correcting the effect of the initial phase of the outgoing pulse and the instrumental phase shift. An alternative method uses surface wave trains observed at two closely spaced stations. This method exploits the frequency-dependent phase differences between the two seismograms to derive the phase velocity. One can subsequently obtain the group velocity from the phase velocity by numerical derivation. In this context, however, small measurement errors in the phase velocity can explode into significant errors in the computed group velocity, making the interpretation of the latter less reliable. An estimate of the group velocity less prone to noise instabilities can be derived from the Fourier transform of the cross-correlated seismograms. Even if noise filtering can be properly performed, the problems of phase unwrapping, multiple modes and interference make up the next hurdle to be tackled. Most of the problems mentioned above, however, can be overcome by using timefrequency representation (TFR) of the seismic signals of interest. Time-frequency analysis consists of examining the variation of the frequency content of a signal with time. TFR methods such as the Gabor transform, the continuous wavelet transform (CWT), and the Wigner-Ville distribution (WVD) are widely used in the analysis of nonstationary signals to determine the group velocity dispersion (LEVSHIN et al., 1972; PROSSER et al., 1999; PEDERSEN et al., 2003). With these techniques, the limitations related to the superposition of multiple modes or temporal interference with reflections may be overcome. The wavelet and the Gabor transform are computed by correlating the signal with families of time-frequency atoms. Therefore, the time-frequency resolution of these methods is limited by the uncertainty relation bound to the simultaneous time-frequency resolution of the analyzing atoms. Since the WVD is a correlation of the signal with a time and frequency translation of itself, there is no limitation in the time-frequency resolution. However its distribution may show negative energy levels and exhibit interferences that have no physical meaning. To minimize the effect of these interferences, the smoothed WVD, usually referred to as pseudo-Wigner-Ville distribution (PWVD) is considered. Unfortunately, the use of the smoothing operator introduces time-frequency smearing in the distribution and therefore downgrades its very advantage over the Gabor or wavelet transform, namely the exact localization over time and frequency that would make it more attractive for specific problems. In general, it is possible to improve the timefrequency resolution of a TFR with the reassignment method proposed originally by
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KODERA et al. (1976) and reformulated by AUGER and FLANDRIN (1995). The reassignment method consists of moving the energy of a given time-frequency domain to the location of its energy center of gravity. Recent applications of this technique in dispersion analysis can be found in NIETHAMMER et al. (2001), PEDERSEN et al. (2003). Very often, despite the availability of phase and amplitude information from a TFR, it is usually the modulus of the transform (energy density) that is manipulated and hence only the group velocity dispersion curve can be extracted. Even if the energy distribution is represented as accurately as possible in the time-frequency plane, there is no way of reconstructing the original signal from this distribution if the phase information is not considered. In ignoring the phase information one loses a valuable piece of information that, if correctly interpreted, could provide a means of estimating the phase velocity as well. An alternative approach would be to find a propagation operator that links the time-frequency transforms of observed signals at different stations. The operator thus defined would characterize the physics of the medium and could be used to extract its full dispersion and attenuation characteristics. The idea of a propagation operator is similar to that developed by SONG and INNANEN (2002) for multi-resolution wavefield reconstruction using the discrete wavelet transform (DWT) in which a relationship between propagated and unpropagated wavefields is defined through a convolution with a viscoelastic kernel function that characterizes both the attenuation and dispersion properties of the medium. For the purpose of analyzing dispersive waves, a forward modeling scheme that takes the parameterized phase and group velocity as input is more desirable. With such a scheme, estimates of the velocity dispersion curves could subsequently be used with appropriate inversion techniques to obtain optimal earth structure models matching the observed data. With such a goal in mind, we propose a model to establish a formal link between the wavelet transforms of the propagated signals through a transformation operator that explicitly incorporates the phase and group delay as well as the attenuation factor of the medium. Subsequently, one can use the full information provided by the complex wavelet transform to simultaneously extract the attenuation- and dispersion characteristics in terms of phase and group velocities of the medium. The paper is organized as follows. First, we provide a brief introduction to the continuous wavelet transform (CWT) method, followed by the development of the mathematical model establishing the relationship between the full wavelet transforms of propagated dispersive signals observed at closely spaced stations. Next, applications of the method to numerical and experimental data are presented and discussed. In an appendix, the geometric aspects of the dynamics in wavelet space are discussed.
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2. Introduction to the Continuous Wavelet Transform The wavelet transform of a signal s 2 L2 ðRÞ with respect to a mother wavelet gðtÞ is defined as follows, W g sðb; aÞ ¼
Zþ1
1 t b g sðtÞdt ¼ hgb;a ; si; a a
ð1Þ
1
where h; i is the L2 –scalar product and gb;a ðtÞ ¼ 1a gððt bÞ=aÞ is generated from g through dilation (a > 0) and translation (b 2 R). The symbol denotes the usual complex conjugate. The wavelet g is assumed to be a function well localized in the time and frequency domains and obeying the oscillation condition Zþ1
gðtÞdt ¼ 0:
ð2Þ
1
The wavelet transform can be expressed in terms of the Fourier transform ^s of s,
W g sðb; aÞ ¼
Zþ1
g^ ðafÞe2pibf ^sðfÞdf:
ð3Þ
1
From this we see that the inverse 1=a of the scale may be associated with a frequency measured in units of the central frequency of g. If the central frequency of the wavelet is assumed to be f0 , then each scale a can be related to the physical frequency f by a ¼ f0 =f . Therefore, if we select a wavelet with a unit central frequency, we can directly obtain the physical frequency by taking the inverse of the scale. For the sake of clarity, we will consider a wavelet with unit central frequency and proceed with the physical frequency instead of the scale in the next sections. The signal sðtÞ can be recovered from its wavelet transform as follows, sðtÞ ¼
1 Cg
Z1 Zþ1 0
1 tb dbda g ; W g sðb; aÞ a a a
ð4Þ
1
where the constant Cg is defined below. In this paper we limit ourselves to real-valued signals and admissible, progressive analyzing wavelets, i.e., we assume g^ðf Þ ¼ 0 for f 0, and that the constant R1 Cg ¼ 0 j^ gðf Þj2 dff is finite.
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3. Estimation of the Propagator in Wavelet Space In this section we construct a link between the wavelet transform of the signal s1 ðtÞ and the wavelet transform of s2 ðtÞ through an operator that explicitly entails the wavenumber, its derivative and the attenuation function. For a linear and stationary medium, the deformation of the signal s1 ðtÞ that has traveled a distance d through a dispersive, dissipative medium (yielding the observed signal s2 ðtÞ) can be represented by s2 ðtÞ ¼
Zþ1
edAðf Þ e2pi½ftdkðf Þ ^s1 ðf Þ df ;
ð5Þ
1
or ^s2 ðf Þ ¼ ^s1 ðf Þe2pidkðf Þ edAðf Þ :
ð6Þ
The wavenumber kðf Þ and the attenuation coefficient Aðf Þ contain all information about the physical properties of the medium. The wavenumber kðf Þ describes the medium’s dispersive characteristics and can be used to derive the phase and group velocities, f dkðf Þ 1 1 vphase ðf Þ ¼ ; vgroup ðf Þ ¼ : ð7Þ ¼ 0 kðf Þ df k ðf Þ It follows that the group velocity can be easily calculated from the phase velocity, but not vice versa (KEILIS-BOROK, 1989), unless the phase velocity is known for some reference frequency used for the determination of the constant brought in by the integration. From the spectra of ^s1 ðf Þ and ^s2 ðf Þ in equation (6), the phase velocity can be estimated using the phase difference or cross-correlation methods. As observed, seismograms are generally a superposition of different types of waves that may overlap in arrival time and frequency content, application of Fourier based methods would not provide an accurate estimate for the dispersion and attenuation characteristics. An efficient way to circumvent the limitation of the Fourier based methods is to use time-frequency analysis such as the continuous wavelet transform. As mentioned earlier, our strategy is to construct a propagation operator that would link the wavelet transforms of the signals s1 ðtÞ and s2 ðtÞ in a way very similar to the relation between the Fourier spectra of ^s2 ðf Þ and ^s1 ðf Þ in equation (6), but also involving the time variable. For the sake of clarity, we will consider a wavelet with unit central frequency for the wavelet transforms so that we easily obtain the correspondence between the scale a and the physical frequency f and write the wavelet transforms of the signal sðtÞ as a function of the time t and frequency f instead of b and a, respectively.
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The challenge is to express the spectral relation of equation (6) in terms of the wavelet transform of the source signal, W g s1 ðt; f Þ and that of the propagated signal, W g s2 ðt; f Þ. From (3) the wavelet transform of s2 ðtÞ expressed in terms of the Fourier transform of s1 ðtÞ yields Z þ1 W g s2 ðt; f Þ ¼ ð8Þ g^ ðf=f Þe2pitf ^s1 ðfÞe2pidkðfÞ edAðfÞ df: 1
If we assume that the attenuation function AðfÞ and the phase function kðfÞ are slowly varying with respect to the effective size of the spectrum of the wavelet transform, we can make useful approximations. For instance, for fixed point ðt; f Þ on the time frequency half plane, we may develop A and k around the central frequency f . For moderate dispersion, the wavenumber term kðf Þ on the right-hand side of equation (8) can be approximated by the first two terms of its Taylor series around f , i.e., AðfÞ ¼Aðf Þ þ Oðjf f jÞ;
ð9Þ 2
0
kðfÞ ¼kðf Þ þ ðf f Þk ðf Þ þ Oðjf f j Þ:
ð10Þ
Upon inserting the above approximations into the integral (8) we obtain W g s2 ðt; f Þ ¼
Zþ1
0
0
g^ ðf=f Þe2pi½tdk ðf Þf ^s1 ðfÞedAðf Þ e2pidðkðf Þfk ðf ÞÞ df
1
¼e
dAðf Þ 2pid½kðf Þf k 0 ðf Þ
¼e
dAðf Þ 2pid½kðf Þfk 0 ðf Þ
Zþ1
e
0
g^ ðf=f Þe2piðtdk ðfÞÞf ^s1 ðfÞdf
ð11Þ
1
e
W g s1 ðt d k 0 ðf Þ; f Þ:
The above equation (corresponding to equation (17) in the Appendix) expresses the relationship between the wavelet transforms of s1 ðtÞ and s2 ðtÞ. With some assumptions about the signal support and the phase of the wavelet, another wavelet propagator (equation (18) in the Appendix) that yields similar results as the previous one, can be defined. A rigorous mathematical development that discusses the derivation of these operators is given in the Appendix. The derived relationships between the wavelet transforms provide an advantage over the traditional use of other time-frequency analysis methods such as the WVD, which is a real distribution with an implicitly encoded phase information. With such a transform it would not be straightforward to have the phase and group delay and attenuation. In equation (11) all parameters that characterize the dispersion and attenuation are expressed explicitly and could be therefore easily extracted. In essence, the so defined wavelet propagator is an extension of the Fourier propagator (3) to the time-frequency half-space. This provides a flexible way of
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determining the dispersion and attenuation characteristics of coherent energy arrivals identifiable in the wavelet image.
4. Application 4.1 Synthetic Example To test whether our propagation operator can properly handle dispersion and dissipation, we simulate the propagation of a Ricker (RICKER, 1935) waveform (central frequency around 30 Hz) in a dispersive and attenuating medium. In the present example, the dispersion of the propagating wave is described by the frequency-dependent phase velocity vðf Þ given by vðf Þ ¼ v0 þ Dvef
2
=r2
ð12Þ
;
where v0 is the velocity at f ¼ 1 and Dv describes the extent of the dispersion. The velocity parameters used for the simulation of propagating waveforms are as follows, v0 ¼ 1300 m/s, Dv ¼ 200 m/s and r ¼ 30. Figure 1b shows the phase and group
0.1
Velocity (m/s)
0.08
k(f)
x 10
(b) 1600
0.06 0.04 0.02
(c)
Phase velocity Group velocity
5
1500 4
A(f)
(a)
1400
3 2
1300 1
0
1200 0
20
40
60
80
100
0 0
Frequency (Hz)
20
40
60
80
100
0
20
Frequency (Hz)
40
60
80
100
Frequency (Hz)
(d)
Trace Number
4
3
2
1
0
0.2
0.4
0.6
0.8
1 Time (s)
1.2
1.4
1.6
1.8
2
Figure 1 (a) Wavenumber curves, (b) phase and group velocity, (c) attenuation coefficient, (d) propagated Ricker wavelet using Fourier method (grey) and the wavelet operator (black line).
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velocities used to model the simulation of dispersion. Figure 1a shows the frequencydependent wavenumber and Figure 1c the attenuation coefficient. The signals propagated at four successive stations using the Fourier integral method and wavelet diffeomorphism are shown in Figure 1d for comparison. Note the excellent agreement between the results from Fourier and wavelet propagation. Figures 2a,b show the wavelet modulus and the phase images for the propagated waveforms. Considering the deformation pattern of modulus and phase images, we see that the propagation operator manipulates the phase and the modulus in different ways, as elucidated in the mathematical treatment. By definition, the wavenumber is given by kðf Þ ¼
f : vðf Þ
ð13Þ
The approximation in equation (10) is accurate only if the second derivative of kðf Þ and successive higher-order derivatives are very small compared to the first two terms or identically zero.
(a)
100 90
Frequency (Hz)
80
Wgs1(t,f)
Wgs2(t,f)
Wgs3(t,f)
Wgs4(t,f)
Wgs1(t,f)
Wgs2(t,f)
Wgs3(t,f)
Wgs4(t,f)
70 60 50 40 30 20 10 0
(b)
100 90
Frequency (Hz)
80 70 60 50 40 30 20 10 0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time (s)
Figure 2 (a) Modulus of the wavelet transforms of the different waveforms in (Figure 1d) obtained with wavelet method and (b) the corresponding phase pictures. Note the difference in the deformation of the phase and the modulus of the transforms.
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At first glance this may mean that velocity inversion would make sense only for wave propagation in nondispersive or weakly dispersive media which mimic a linear phase process. In practice however, velocity inversion for highly dispersive media can still be performed if one selects highly localized (rapidly decaying) frequency spectra for the wavelet. 4.2 Experimental Data We apply the propagation operator to a real data consisting of seismic data obtained from a shallow seismic experiment. From the recorded seismograms we selected some traces on successive stations for our test. The Hammer strike is used as seismic source with equidistant geophones (distance between stations 2 m) as receiver. First we determine the phase velocity, the group velocity and the attenuation coefficient using Fourier-based methods. Instead of using an approach based on two stations, we considered the parameterization of the wavenumber and
(b) 300 250
0.6 0.4 0.2
(c)
Phase velocity Group velocity
2 1.5
200
A(f)
k(f)
Velocity (m/s)
1
0.8
(a)
150
1
100 0.5 50
0 20
30
40
50
60
0 20
Frequency (Hz)
30
40
50
60
0 20
30
Frequency (Hz)
40
50
60
Frequency (Hz)
(d) 6
Trace Number
5
4
3
2
1
0
0.05
0.1
0.15
0.2
0.25 Time (s)
0.3
0.35
0.4
0.45
0.5
Figure 3 Test of the wavelet propagator on experimental data. (a) The wavenumber curve, (b) the phase and group velocity and (c) the attenuation coefficient are determined within the Fourier domain by a minimization process that seeks to find the parameterized wavenumber and attenuation which minimize a properly chosen objective function. (d) Comparison between the original traces (grey) with those reproduced by propagating the reference trace (bottom trace on the panel) to the successive receiver stations (black lines). Note the fairly good agreement between the original and reproduced signal especially for the strong arrival associated with the surface waves.
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attenuation functions and estimated them via a properly defined cost function that involves all the preselected traces. The advantage of such an approach is that phase unwrapping and the search for the number of 2p-cycle skips to be added to the phase is no longer. Taking the traces closest to the source for reference, we then try to reproduce the observed signal at further stations, using the propagation operator with the attenuation and the wavenumber (and its derivative) estimated above. Figures 3a,b,c show the estimated wavenumber, phase and group velocities and the attenuation curves. In Figure 3d, the original traces and those obtained with the wavelet as described above are plotted together. Note that the most prominent arrival that corresponds to the surface waves is fairly well reproduced at all stations. Figures 4a,b show the modulus and phase picture of the reference trace for the frequency range 20 to 60 Hz. A similar plot (Figs. 4c,d) is produced for the furthest traces, too.
Frequency (Hz)
(a)
60
Wgs1(t,f)
50 40 30 20
Frequency (Hz)
(b) 60 Wgs1(t,f)
50 40 30 20
Frequency (Hz)
(c)
60 50
Wgs5(t,f) 40 30 20
Frequency (Hz)
(d) 60 50
Wgs5(t,f) 40 30 20 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Time (s)
Figure 4 (a) The modulus of the wavelet transforms of reference waveforms (bottom Fig. 3d and (b) its phase image. (c) The modulus of the wavelet transforms of last waveforms (fifth waveform in Figures 3d and (d) its phase image.
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This is an important result as, to our knowledge, a method that offers the possibility of propagating signals using the continuous wavelet transform has not yet been proposed in the literature. We want to emphasize strongly at this point that the importance of using the wavelet propagator is not only in the simultaneous estimate of phase and group velocities and attenuation, but also in the possibility of separating different coherent energy arrivals. For this, one would first determine the dispersion and attenuation of each coherent arrival independently by applying the propagation operator to the corresponding time and frequency window in the wavelet domain, and then taking the inverse wavelet transform to obtain the signal in the time domain. This constitutes the principal advantage of the proposed propagator in the wavelet domain over other methods that involve only the modulus of the time-frequency transform and therefore cannot be used to reconstruct the signal. Comparing the reconstructed signal with the observed signal provides an additional means of assessing the accuracy of the estimated dispersion curves.
5. Conclusions In this study we propose a mathematical model to establish a link between the continuous wavelet transform of a signal and its propagated counterpart in a dispersive and attenuating medium. We show that in dispersive and attenuating media it is possible to predict the propagated signals with great accuracy, thus providing a means to estimate both the phase and group velocity, as well as the attenuation coefficient. The advantage of using the proposed propagator over traditional methods such as the Wigner-Ville or time frequency reassignment for dispersion curves estimates is that the full dispersion and dissipation characteristics are explicitly expressed and therefore can be easily extracted. The recovered parameters can be used to reconstruct the signal in the time domain. Comparing the actual signal with the reconstructed one provides a way of checking the accuracy of the derived dispersion curves in our examples.
Acknowledgment This project is supported by a grant from the Deutsche Forschungsgemeinschaft (DFG) within the framework of the priority program SPP 1114,‘‘Mathematical methods for time series analysis and digital image processing.’’ We thank Matthias Ohrnberger and Erika Lu¨ck from the Institute of Geoscience, University of Potsdam for providing the experimental data.
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Appendix Asymptotic Propagator in Wavelet Space We analyze the propagation of the wavelet transforms of the signals s1 and s2 given by (5) via P½A;k;d :¼ F 1 edðAðÞþ2pikðÞÞ F , where F denotes the usual Fourier transform. Making use of the inverse wavelet transform, we describe the relationship between r1 ¼ W g s1 and r2 ¼ W g s2 by r2 ðb; aÞ ¼
W ðP½A;k;d r1 Þðb; aÞ
Z1 Zþ1 :¼ 0
Lðb; a; b0 ; a0 Þr1 ðb0 ; a0 Þ
db0 da0 ; a0
ð14Þ
1
where the kernel L is given by 1 1 Lðb; a; b ; a Þ ¼ hgb;a ; P½A;k;d gb0 ;a0 i ¼ Cg Cg 0
0
Z1
0
gða0 fÞe2piðbb Þf edAðfÞ e2pidkðfÞ df: g^ ðafÞ^
1
ð15Þ Under the linear approximations (9) and (10) for the attenuation function AðfÞ and the wavenumber function kðfÞ, and introducing a diffeomorphism of the scale-timespace H ¼ R Rþ 2 ðb; aÞ via Uðb; aÞ :¼ ðb þ d=vgroup ðaÞ; aÞ;
ð16Þ
we obtain a first approximation of the dispersive propagation, 1 0
W ðP½A;k;d r1 Þðb; aÞ ’ edAðaÞ e2pidðkðaÞa
k ðaÞÞ
ðr1 U1 Þðb; aÞ:
ð17Þ
Assuming that the signal s is supported in some compact time interval ½T ; T , and that the analyzing wavelet has linear phase (with time-derivative approximately equal to 2p, as is the case for the Morlet wavelet), we have arg r1 ðb; aÞ ’ 2pb=a for a T and (17) can be further refined to yield W ðP½A;k;d r1 Þðb; aÞ ’ edAðaÞ ðjr1 j U1 Þðb; aÞei argðr1 W
1
Þðb;aÞ
:
ð18Þ
Here we have introduced a new diffeomorphism of the scale-time-space via Wðb; aÞ :¼ ðb þ d=vphase ðaÞ; aÞ:
ð19Þ
It is clear that with respect to the modulus the dispersion k acts solely via a deformation U of the wavelet plane. The above discussion demonstrates that the whole dynamic in wavelet space can be understood as comprising three contributions, an overall attenuation given by edAðaÞ , propagation of the modulus with the group velocity (composition with U1 ), propagation of the phase with the phase velocity (composition with W1 ).
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