bFaculty of Geoscience, University of Potsdam, KarlâLiebknechtâStrasse 24â25, 14415 Potsdam, Germany. ABSTRACT. Determination of nonâstationary ...
Characterization of Rayleigh waves Polarization attributes using Continuous Wavelet Transform (CWT) M. S. D IALLO , a M. H OLSCHNEIDER ,a M. K ULESH ,a Q. X IE ,a F. S CHERBAUM b AND F. A DLER a a Applied and Industrial Mathematics, University of Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany bFaculty of Geoscience, University of Potsdam, Karl–Liebknecht–Strasse 24–25, 14415 Potsdam, Germany
ABSTRACT
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Determination of non–stationary polarization attributes and dispersion characteristics of Rayleigh wave using continuous wavelet transforms (CWT) and separation of surface and body waves.
Deformation in wavelet space
ϕ : H → H diffeomorphism of the upper halfplane H = R × R+ ⇒ deformation operator in wavelet space Φ[ϕ∗] : r 7→ Π ∗ (ϕ∗r),
1 Methodology
ϕ∗r = r ◦ ϕ.
Corresponds to “propagation”
Wg s(b, a) =
−∞
1 t−b g¯ s(t) dt, a a
a: scale parameter, b: location parameter, g(t): analysing wavelet. Here g progressive (i.e., only positive frequencies). Normalization of wavelet: Z∞ dω = 1. |ˆ g (ω)|2 ω 0
Inverse via adjoint +∞ Z∞ Z 0 −∞
Op[ϕ∗] reads in time space Op[ϕ∗] : s(t) 7→
t−b da 1 g r(b, a) db , a a a
Figure 5: Example of optimization showing the Fourier and wavelet propagated signals, their spectra an wavelet transform images.
+ − − + C +C C −C sign α 1+i H Dαs(t), 2 2
1
where H denotes the Hilbert transform, C ± = C ±(ϕ) := 2π
Mg Wg = 1.
• Wavelet deformation algebra approximates dispersive propagation ⇒ inverse problem allows dispersion estimate of Rayleigh waves. • Construction of non–stationary filters for wavefield separation.
Z∞
da ±iβa ˆ gˆ(±δa)e h(±αa) . a
2 Instantaneous Ellipse and Attribute Extraction Two orthogonal components of seismic data, X(t) and Y (t) define complex signal Z(t) = X(t) + iY (t).
−1
Dispersive propagation Z0(t) → Zd(t) modelled through semi–group of Fourier multipliers Z Zd(t) = dωA(ω)eiφ(ω)deiωtZˆ0(ω)
Taylor development around ω = ω0/a, A(ω) ' A(ω0/a), φ(ω) ' φ(ω0/a)+φ0(ω0/a)(ω− ω0/a) yields
−
Γb,a(t) = W +eiΩ t + W −e−iΩ t,
f (a) = dφ0(ω0/a)
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• minor half–axis r(b, a)
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X(t)
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Conclusions
Work in progress
• Future objectives are the analysis of non–linear diffeomorphisms via their associated vector fields. It is hoped that diffeomorphisms that behave like affine mappings “at small scale” generate algebras modulo smoothing operators. ⇒ for modelization of propagation phenomena in wavelet space deformation of phases must be different from deformation of modulus. • Construction of classes of diffeomorphisms with the following properties,
1. independence on the wavelets up to smoothing operators, 2. Op[φ∗1 ] Op[ϕ∗2 ] = Op[φ∗1 φ∗2 ], up to smoothing operators, 3. computability, i.e., explicit formulas for higher order terms should be obtainable.
[1] S.T. Ali, J.P. Antoine, J.P. Gazeau, and U.A. Mueller. Coherent states and their geeralizations: A mathematical overview. Rev. Math. Phys., 7(7):1013–1104, 1995.
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[2] I. Daubechies. Ten Lectures on Wavelets. CBMS-NSF Series in Applied Mathematics. SIAM, 1992. [3] I. Daubechies and Paul T. Time–frequency localisation operators – a geometric phase space approach: II The use of dilations. Inverse Problems, 4:661–680, 1988.
Wavelet spectrum after diffeomorphism
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Acknowlegments
References
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Wavelet modulus of r(b,a) and R(b,a) Frequency (Hz)
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This project is supported by a grant from the “DFG–Schwerpunktprogramm 1114”.
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dx=200 −5
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• Polarization analysis using CWT was developed. With the time and frequency localization, attirbutes of signals with different frequency content arriving at the same time can be distinguished.
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Spectrum of Source signal Spectrum from Wavelet diffeo.
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• Extension of the wavefiled separation → Application to multicomponent seismic data (e.g. 3C, 9C).
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• Derivation of the global dispersion law (phase and group velocity) and attenuation factor for surface waves in 2–D seismic data.
Source signal at dx=0 Propagated signal at dx=200 Calculated with Wavelet diffeo.
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Figure 4: Forward modelling of dispersion with Fourier and Wavelet diffeomorphism. For weak dispersion the comparison of the two methods shows good agreement.
Figure 2: Semi–major and semi–minor axis of a composite signal (mix of elliptical and linear polarization) determined from the method of (Rene et al. 1986) and the present study. 10
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• For reflection event (→ weak dispersion), the wavelet diffeomorphism can be effectively used for attenuation estimate.
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R(t), r(t) − This study
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• Diffeomorphism operator from optimization can be used for velocity dispersion and attenuation estimate.
Wavelet spectrum after diffeomorphism
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• The method can be effectively used to separate waves with different polarization attributes.
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• Linear diffeomorphism leaves the IW transformed signal invariant in time. → not suitable for dispersion investigation.
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Frequency spectrum of Morlet Wavelet − before diffeomorphism − after diffeomorphism
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Figure 1: Semi–major and semi–minor axis of elliptically polarized signal determined from the method of (Rene et al. 1986) and the present study. 10
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Real part of Morlet Wavelet − before diffeomorphism − after diffeomorphism
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This is an extension to time–frequency analysis of attributes in (Rene et al. 1986).
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• rotation speed of ellipses Σ(b, a) = (Ω+(b, a) − Ω−(b, a))/2 • major half–axis R(b, a)
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Figure 3: Linear diffeomorphism applied to a complex Morlet wavelet. The frequency content and position of the signal is unaltered as shown from the signals and spectra comparsion.
• average frequency within ellipse Ω(b, a) = (Ω+(b, a) + Ω−(b, a))/2
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ω p(a) = d(φ(ω0/a) − 0 φ0(ω0/a)) a
Interpretation: Γ describes rotating ellipses. Instantaneous parameters:
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arg Wg Z0(b, a) → arg Wg Z0(b + f (a), a) + p(a),
W ± = Wg Z ±(b, a)e±Ω (b,a)b.
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Figure 6: Estimated and calculated group velocities and the phase velocity. Very good agreement between the model and inveterted group velocity can be observed.
⇒ propagation of modulus
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(t) − optimized
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different from propagation of phase
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S
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0 0 Wg Z0 → Wg Zd ' Π ∗ Wg Z0(b + dφ0(ω0/a), a) A(ω0/a) eid(φ(ω0/a)− a φ (ω0/a)).
yields instantaneous complex curve
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opt
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Dispersive propagation in wavelet space
± Wg Z ±(t, a) ' Wg Z ±(b, a)e±iΩ (b,a)(t−b),
S (t) − after propagation of S (t) at D +dD 2
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Op[ϕ∗1 ◦ ϕ∗2 ] 6= Op[ϕ∗1 ] Op[ϕ∗2 ].
|Wg Z0(b, a)| → Wg Z0(b + f (a), a),
Local approximation t ' b
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Velocity [m/s]
Ω±(b, a) = ±∂b arg Wg Z ±(b, a).
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ω
Progressive and regressive wavelet transform of Z(t) yields Wg Z +(b, a) and Wg Z −(b, a). Instantaneous frequencies
Frequency spectra
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Only in the simplest case, α = δ and β = 0 does such an equality hold. Also, for a general wavelet g, only in this case is ran Wg (H) left invariant under ϕ∗.
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Op[ϕ∗1 ] Op[ϕ∗2 ] = Op[ϕ∗2 ] Op[ϕ∗1 ],
but in general
• Construction of ellipse–valued time–frequency picture from two components of seismic data ⇒ polarization attributes directly accessible.
S (t) − at D
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It follows
Π = Wg g.
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Reproducing kernel is orthogonal projection onto ran Wg , Wg Mg = Π∗,
A linear diffeomorphism ϕ : H → H has the form b α β b αb + βa ϕ: 7→ = , α, β ∈ R, α 6= 0, δ > 0. a 0 δ a δa
From dispersive signals Z0(t) and Z1(t) recorded at two stations, a distance d away from each other, minimize the following cost function with respect to f (a, Pi) in ϕ(b, a) = b + f (a), ZZ da min kWg (Z1) − Π ∗ (ϕ ◦ Wg (Z0))k2 db, a
Frequency (Hz)
+∞ Z
Inverse problem
Pi is the set of coeffcients to be fitted for. 1 ). ⇒ Group velocity can be calculated from ϕ (Vg (a) = f (a,P i) In applied seismic frequency range, attenuation factor A(ω) approxiamted by A(ω) = ¨ et al., 1978) with |Wg Z0(b, a)| and e−α ω ⇒ use spectral ratio method (Toksoz |Wg Z1(b, a)| (averaged over b) to estimate the attenuation fatcor α.
Op[ϕ∗] := Mg ϕ∗Wg : s 7→ Mg ϕ∗Wg s.
• Definition of the wavelet transform of a signal s(t)
Mg r(t) =
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[6] Y. Meyer. Wavelets and operators. Cambridge University Press, Cambridge, 1992.
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[7] Rene, R.M., Fitter J. L., Forsyth, P. M., Kim, K. Y., Murray, D., J., Walters, J. K., Westerman, J. D., 1986, Multicomponent seismic studies using complex trace analysis: Geophysics. 51, p. 1235–1251. Geophysics 51, 1 (1986).
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[4] M. Holschneider. Wavelets: an analysis tool. Oxford University Press, Oxford, 1995. [5] M. Holschneider. Wavelet analysis of partial differential operators. Akademie Verlag, 186:218–347, 1996.
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[8] M. N. Toksoz, ¨ D. H. Johston, and A. Timur. Attenuation of seismic waves in dry and saturated rocks:I laboratory measurements. Geophyscis, 44:681–690, 1978.
