We want to develop theoretical and numerical tools that allows us to extract dynamical behavior from wavelet transforms corresponding to multivariate (multichannel) signals. Mathematically this amounts to the construction of an algebra of pseudo differential operators that act in wavelet space via a diffeomorphism deforming the wavelet half-plane. This technique will be applied to the problem of separating seismic signals into components, corresponding to different propagative behavior. In particular, we want to separate surface waves from body waves in seismic records. Our approach is based on the combination of (i) a recently developed polarization filter that operate entirely in the wavelet domain and (ii) an approximative description of the dispersive propagation in wavelet space.

As a rule, the waves in solids and geological media are observed only indirectly, namely, through interpreting the data obtained from vibration measurements in the form of seismograms. However, these seismograms contain usually a mix of different wave types, which makes it necessary to apply high-quality filtering procedures. One of the possible methods for the waves filtering relies on the polarization analysis serving to derive the parameters that characterize the ellipticity of the signal.

For further interpretation, one should be able to retrieve the dispersion and dissipation parameters of the medium of propagation from source seismograms. This data can be obtained by inversion of the waves with dispersion in an analysis of one-component or multicomponent signals.

Most of the experimental studies of dispersive waves rely on Fourier analysis. However, these waves are relatively difficult to study; their characteristic feature, namely, dispersion, is described by a function of frequency rather than by a single variable. In view of this, the time-frequency approach seems to be rather promising. This approach is particularly remarkable because of its ability to discriminate between signals according to their dispersion curves. The time-frequency approach known today as wavelet analysis has vigorously advanced in the last decades and has become one of the most important methods of digital signal processing.

Thus, the purpose of this work is to develop methods for processing multicomponent seismograms that are capable of retrieving the parameters of ellipticity (ellipticity, phase shift, and dip angle of the polarization ellipse), the dispersion (group and phase velocity), and the dissipation (attenuation function) of the waves. As the basis for these methods, we choose wavelet analysis, in particular, the direct and inverse continuous wavelet transforms.

1) We have successfully developed and applied a technique for wavefiled separation based on the analysis of instantaneous polarization attributes in the wavelet domain. The methodology was initially developed for two component data set [2,8].

2) Recently we extended it to the higher component data set [4,9,10] using the covariance method and an approximate approach to compute the elements of the covariance matrix for a time window which is derived from an averaged instantaneous frequency of the multicomponent record. The technique has been successfully applied to both synthetic and field data.

The adaptive aspect of this method comes from the fact that we locally adjust the window size for the covariance matrix computation to the period of the signal. Obviously, our method can be applied for any number of components. Time-frequency representation allows the detection of dispersion in the polarization attributes which can be further exploited to infer some physical characteristics of the medium under investigation. Moreover, this representation offers the ability to distinguish between attributes belonging to different coherent events that may overlap in time but with different frequency content separated by time-dependent frequency cut-offs.

The novelty in the present polarization method resides in the combined use of time-frequency representation along with an adaptive method of estimating the window for the covariance matrix. Some advantages of our method are:

3) We have developed the wavelet propagator that enables us to model the signal dynamics of specific waves type that can be identified as coherent event in the wavelet domain [2,3]. The idea behind the wavelet propagation consisted of the deforming the wavelet modulus and the wavelet phase differently but in consistent manner and then recombining them to reproduce the full wavelet transform of the propagated waveform at a given distance.

Proposed wavelet-propagator has the following interpretation (Figure 3). The group velocity is a function that "deforms" the image of the absolute value of the source signal's wavelet spectrum, the phase velocity "deforms" the matrix of the wavelet spectrum phase, and the attenuation function determines the frequencydependent real coefficient by which the spectrum is multiplied.

4) We have used this model to construct inversion scheme for obtaining the dispersive and dissipative characteristics of the analyzed wave type through parametric modeling of these properties [5]. The method has been applied to synthetic and real data set. The results from the real data set were compared with those from independent methods for ground-truthing.

5) Finally, we proposed a software package GWL based on the continuous wavelet transform that allows to perform the direct and inverse continuous wavelet transform, 2C and 3C polarization analysis and filtering, modeling the dispersed and attenuated wave propagation in the time-frequency domain and optimization in signal and wavelet domains with the aim to extract velocities and attenuation parameters from a seismogram. 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 and can be particularly suitable in geophysical problems that we illustrate with the analysis of synthetic, geomagnetic or real seismic data.

[1]	M.Kulesh, M.Holschneider, M.S.Diallo, F.Scherbaum, M.Ohrnberger, E.Lueck Estimating attenuation, phase and group velocity of surface waves observed on 2D shallow seismic line using continuous wavelet transform // Proceedings of the XXXII International Summer School "Advanced Problems in Mechanics" (Russia, St. Petersburg, June 24-July 1, 2004). P. 257-262 (2004). Full text (pdf, eng, 3773Kb); About the conference
[2]	M.A. Kulesh, M.S. Diallo and M. Holschneider Wavelet analysis of ellipticity, dispersion, and dissipation properties of Rayleigh waves // Acoustical Physics. V. 51. No. 4. P. 421-434 (2005). Full text (pdf, eng, 328Kb); Abstract (html, eng); About the journal
[3]	M. Kulesh, M. Holschneider, M.S. Diallo, Q. Xie and F. Scherbaum Modeling of wave dispersion using continuous wavelet transforms // Pure and Applied Geophysics. V. 162. No. 5. P. 843-855 (2005). Full text (pdf, eng, 2725Kb); Abstract (html, eng); About the journal
[4]	M.S. Diallo, M. Kulesh, M. Holschneider and F. Scherbaum Instantaneous polarization attributes in the time-frequency domain and wave field separation // Geophysical Prospecting. V. 53. No. 5. P. 723-731 (2005). Full text (pdf, eng, 939Kb); About the journal
[5]	M. Holschneider, M.S. Diallo, M. Kulesh, M. Ohrnberger, E. Lueck and F. Scherbaum Characterization of dispersive surface waves using continuous wavelet transforms // Geophysical Journal International. V. 163. No. 2. P. 463-478 (2005). Full text (pdf, eng, 4364Kb); Abstract (html, eng); About the journal
[6]	Michail A. Kulesh, Matthias Holschneider, Mamadou S. Diallo, Kristina I. Kurennaya Elliptic properties of elastic surface waves in wavelet domain // Proceedings of the International Summer School "Advanced Problems in Mechanics" (Russia, St. Petersburg, June 28-July 5, 2005). P. 361-366 (2005). Full text (pdf, eng, 826Kb); Presentation (pdf, eng, 3187Kb); About the conference
[7]	Mamadou S. Diallo, Michail Kulesh, Matthias Holschneider, Kristina Kurrenaya and Frank Scherbaum Estimating polarization attributes with an adaptive covariance method in the wavelet domain // SEG Technical Program Expanded Abstracts. V. 24. P. 1014-1017 (2005). Full text (pdf, eng, 2090Kb); Abstract; Presentation (pdf, eng, 2998Kb); About the conference
[8]	M.S. Diallo, M. Kulesh, M. Holschneider, F. Scherbaum and F. Adler Characterization of polarization attributes of seismic waves using continuous wavelet transforms // Geophysics. V. 71. No. 3. P. V67-V77 (2006). Full text (pdf, eng, 1510Kb); Abstract (html, eng); About the journal
[9]	M.S. Diallo, M. Kulesh, M. Holschneider, K. Kurennaya and F. Scherbaum Instantaneous polarization attributes based on an adaptive approximate covariance method // Geophysics. V. 71. No. 5. P. V99-V104 (2006). Full text (pdf, eng, 1294Kb); Abstract (html, eng); About the journal
[10]	M. Kulesh, M.S. Diallo, M. Holschneider, K. Kurennaya, F. Kruger, M. Ohrnberger and F. Scherbaum Polarization analysis in the wavelet domain based on the adaptive covariance method // Geophysical Journal International (2007, in press). Full text (pdf, eng, 3137Kb); About the journal

[1]	M.S.Diallo, M.Holschneider, F.Scherbaum and M.Kulesh Characterization of dispersive Rayleigh waves using wavelet transform // Eos. Trans. AGU, 84(46), Fall Meet. Suppl., Abstract S22B-0442 (2003). Full text (pdf, eng, 199Kb); Presentation (pdf, eng, 1662Kb); About the conference
[2]	M.S.Diallo, M.Holschneider, M.Kulesh, F.Scherbaum and F.Adler Characterization of the Rayleigh wave polarization attributes with continuous wavelet transform // Geophysical Research Abstracts, Vol. 5, 11237 (2003). Full text (pdf, eng, 12Kb); Presentation (pdf, eng, 642Kb); About the conference
[3]	M.Kulesh, M.Holschneider, M.S.Diallo, F.Scherbaum and M.Ohrnberger Estimating attenuation, phase and group velocity of surface waves observed on a 2D shallow seismic line using continuous wavelet transform // Book of abstracts of XXXII International Summer School - Conference "Advanced Problems in Mechanics" (June 24 - July 1, 2004, St.Petersburg, Russia). P. 65-66. Full text (pdf, eng, 1242Kb); Presentation (pdf, eng, 3250Kb); About the conference
[4]	M.S.Diallo, M.Holschneider, M.Kulesh, F.Scherbaum, M.Ohrnberger and E.Lueck Toward improved methods of estimating attenuation, phase and group velocity of surface waves observed on shallow seismic records // Eos Trans. AGU, 85(17), Jt. Assem. Suppl., Abstract S51A-02 (2004). Full text (pdf, eng, 102Kb); Presentation (pdf, eng, 3890Kb); About the conference
[5]	M.Holschneider, M.S.Diallo, M.Kulesh, F.Scherbaum and M.Ohrnberger Estimating attenuation, phase and group velocity of surface waves observed on a 2D shallow seismic line using continuous wavelet transform // Geophysical Research Abstracts, Vol. 6, 03129 (2004). Full text (pdf, eng, 35Kb); Presentation (pdf, eng, 3362Kb); About the conference
[6]	Diallo M.S., Kulesh M., Holschneider M. and Scherbaum F Instantaneous polarization attributes in the time-frequency domain: application to wave field separation // Eos Trans. AGU, 85(47), Fall Meet. Suppl., Abstract S31B-1063 (2004). Full text (html, eng); Presentation (pdf, eng, 4686Kb); About the conference
[7]	Diallo M.S., Kurennaya K., Kulesh M. and Holschneider M Elliptic properties of surface elastic waves in wavelet domain (in Russian) // Book of abstracts of XIV Winter School on Continuous Media Mechanics (28 February - 3 March 2005, Perm). P. 100. Full text (pdf, rus, 257Kb); Presentation (pdf, eng, 3855Kb); About the conference
[9]	M.A.Kulesh, M.Holschneider, M.S.Diallo, K.Kurennaya and F.Scherbaum Modeling of wave dispersion using continuous wavelet transforms: incorporating causality constraint with non-linear frequency-dependent attenuation // Eos Trans. AGU, 86(52), Fall Meet. Suppl., Abstract S33A-0289 (2005). Full text (html, eng); Presentation (pdf, eng, 630Kb); About the conference
[10]	M. Kulesh, M. Nose and M. Holschneider Polarization Analysis of Pi2 Pulsations Using Continuous Wavelet Transform // Eos Trans. AGU, 87(52), Fall Meet. Suppl., Abstract SM43D-02 (2006). Full text (html, eng); Presentation (pdf, eng, 870Kb); About the conference

[1]	Q.Xie, M.Holschneider, M.Kulesh Some remarks on linear diffeomorphisms in wavelet space // Preprint Series DFG SPP 1114, University of Bremen. Preprint 37 (2003). Full text (pdf, eng, 1548Kb)
[2]	M.S.Diallo, M.Holschneider, M.Kulesh, F.Scherbaum and F.Adler Characterization of seismic waves polarization attributes using continuous wavelet transforms // Preprint Series DFG SPP 1114, University of Bremen. Preprint 38 (2003). Full text (pdf, eng, 1104Kb)
[3]	M.Kulesh, M.Holschneider, M.S.Diallo, Q.Xie and F.Scherbaum Modeling of wave dispersion using continuous wavelet transforms // Preprint Series DFG SPP 1114, University of Bremen. Preprint 40 (2003). Full text (pdf, eng, 1645Kb)
[4]	M.Holschneider, M.S.Diallo, M.Kulesh, F.Scherbaum, M.Ohrnberger and E.Lueck Characterization of dispersive surface wave using continuous wavelet transforms // Preprint Series DFG SPP 1114, University of Bremen. Preprint 56 (2004). Full text (pdf, eng, 13218Kb)
[5]	M.S.Diallo, M.Kulesh, M.Holschneider and F.Scherbaum Instantaneous polarization attributes in the time-frequency domain and wave field separation // Preprint Series DFG SPP 1114, University of Bremen. Preprint 57 (2004). Full text (pdf, eng, 1085Kb)
[6]	M.S.Diallo, M.Kulesh, M.Holschneider, K.Kurennaya and F.Scherbaum Instantaneous polarization attributes based on adaptive covariance method // Preprint Series DFG SPP 1114, University of Bremen. Preprint 87 (2005). Full text (pdf, eng, 396Kb)
[7]	M. Kulesh, M. S. Diallo, M. Holschneider, K. Kurennaya, F. Krueger, M. Ohrnberger, F. Scherbaum Polarization analysis in wavelet domain based on the adaptive covariance method // Preprint Series DFG SPP 1114, University of Bremen. Preprint 137 (2005). Full text (pdf, eng, 5290Kb)
[8]	M. Kulesh, M. Nose, M. Holschneider, K. Yumoto Polarization analysis of a Pi2 pulsation using continuous wavelet transform // Preprint Series DFG SPP 1114, University of Bremen. Preprint 153 (2007). Full text (pdf, eng, 1418Kb)
[9]	M. Kulesh, M. Holschneider, M. Ohrnberger, E. Lueck Modeling of wave dispersion using continuous wavelet transforms II: wavelet based frequency-velocity analysis // Preprint Series DFG SPP 1114, University of Bremen. Preprint 154 (2007). Full text (pdf, eng, 2602Kb)