A New Method in Polarization P / S Filters of Vector Wavefield

The two-dimensional finite element method is employed to simulateRayleigh waves in this study.Various dislocation sources located at variousdepths are considered.Through the analysis of amplitude variations,the relationshipsbetween the excitations of the Rayleigh waves and the faultingsare gradually understood.In order to determine the effects due to the dampingfactor involved in the calculations from the attenuation curves,a simpletechnique is proposed and applied in this paper.The results show that theamplitude is suppressed to a lower extent with lower frequency signals.Ingeneral,Rayleigh-wave excitation from any type of faulting decreases withsource depth.The steeper finite faultings at the shallower depth generatestronger Rayleigh waves.For 45°dip-slip sources,it is also easier to generatestronger Rayleigh waves for point sources than for finite ones.


INTRODUCTION
The elastic wave, as one kind of vector wavefield, encompasses the linear motion of P-and _S-waves, and the.elliptical motion of Rayleigh waves as well.Most exploration geophysi.sts are much interested in P-wa\1es, or in S-waves after the development of shear wave exploration.Yet, most of the data obtained from the outfield include these waves, so the signals become unclear and even mixed up due to interferences.Many solutions, such as bandpass filter, \telocity filter, ... etc. , have been used to partially overcome this problem at least.The polarization filter (Flinn, 1965), resulting fro1n the perspecting of particle motions has been employed in this field as one of the solutions.tAdditionally, the technique \\1hich reduces Rayleigh waves brought up by Shieh and Herrmann (1990) was considered a solution as well.In order to conduct a successful analysis, it is important to filter data so that it only contains one kind of wave (P-or S-).The main purpose of this study is to introduce a method to reach this goal.
Using the polarization method to analyze the characteristics of the elastic .wave has• already been widely discussed.Flinn ( 1965) first introduced the utilization of the covariance 160 TAO, Vol. 7, No.2, June 1996 matrix in the real time domain to manipulate earthquake problems.Later, studies conducted by Samson and Olson (1981) a data-adaptive polarization filter in their design to enhance waves of specific polarization.Ta ner et al. (1979) and Rene et al. (1986) used analytic   signals in the time domain for analysis, while Vidale (1986) employed the eigenvalue and eigenvector of the coherency matrix to determine the polarization parameters, which is consid ered to be more complete.Mott (1986) analyzed radar signals using the polarization method, and theoretically, although what he employed was still a analytic signal, it only involved 2-component data.The present study also deals with only 2-component data.Therefore, the data used here must be the signals received from the vertical and the radial components on the incident plane.

METHOD
Supposing the time series are recorded in the vertical (Z) and the radial (R) components, after the Hilbert transforrn, the signals can be expressed as: A time window is selected, and a coherency matrix is constructed: where </>=</> z-ef>r as the phase difference between the vertical and the radial components, the symbol * represents the complex conjugate and < > represents the average value in the time window.Matrix C is a non-negative Hermitian containing all the information needed to characterize the polarization parameters.Equation (2) was used by Vidale (1986) where he employed 3-component data in his study, but here, only 2-component data is used as Shieh and Her1•1nann did (1990).Equation ( 2) processed with eigenanalysis can be expressed as:

CU AU, (3)
where ;\ is a 2 x 2 diagonal matrix, representing eignevalues, and U is a 2 x 2 eigenvector matrix.It is found that the eigenvector corresponding to the maximum eigenvalue takes the following fo11n  It is clear that the phase difference ( efJ) between the vertical and the radial components can be calculated by (5) For the pure longitudinal signal (P-wave ), </> = 0°, and for the pure transverse signal (S-wave) ¢ = 18.0°.As for other nonlinear waves, this angle is located between 0° and 180°.From this perspective, two functions for passing P-and S-waves may be defined as: If the signal is pure P-wave, </> = 0, so Pc(t) = 1, and Sc(t) = 0.If the signal is pure S-wave, cjJ = 180, so Pc(t) = 0, and Sc(t) = 1.For other nonlinear waves, Pc(t) and Sc(t) are both smaller than I, so Equation ( 6) can be used to suppress the nonlinear waves.Note that for the pure incident S-wave with an angle of incidence beyond the critical angle, the particle motion becomes elliptically polarized.Equation ( 6) can't isolate this kind of S-wave successfully unless the ellipticity is very small.
As for defining the strength of linear polarization (Vi dale, 1986), it is stated as: (7) where Ai and .\2are the maximum and minimum eigenvalues, respectively.The main feature of this formula is to suppress the noise.When noise does not exist, A2 0 and PL 1; on the other hand, A2 #-0, and PL <I when noise exists.
In the final step, the nonlinear wave is reduced by ellipticity ( e ), and the function is defined as: Elliptic ity can be obtaine d by the rotation al angle as propose d by Vidale (1986).Under the 2-comp onent conditi on, the method is simplifi ed by finding an angle with the expone ntial tenn, cs exp(j B), then multip lying the eigenv ector (Equat ion 4) corresp onding to the maximu m eigenva lue and finally minimiz ing the real part by: The ellipticity is determined by: e-
The fi lter function of passing the transverse wave is defined as:

FREQUENCY AND TIME DECOMPOSITION
In practical applications, two procedures need to be foil owed before the polarization method can be used; they are frequency and time decomposition.One particular reason is that wave types with different frequencies may arrive at the same time, such that the polarization properties are mixed up.Jurkevics ( 1988) developed a rigorous algorithm to solve this problem.Subsequently, Shieh and Herrmann (1990) slightly modified the cosine window into the cosine square window.The following is a summary of the algorithm.
The first step is to narrow-bandpass the input data over the entire frequency range (fre quency decomposition).Second, a 50% overlap cosine square moving window is applied to the bandpassed data (time decomposition).The time-domain-moving window has a window width equal to 2/ fc, where �f c is the central frequency of the bandpass filter.Third, the polarization filter function (Equation ( 11) or ( 12)) is computed in the tapered time series.Fourth, this function is multiplied to the entire tapered time series.Fifth, the steps from the sec.and through the fourth are repeated until the entire time series is covered (the cosine square window effect is added).Finally, the steps from the first through the fi fth are repeated until the entire frequency range is C0\1ered (the bandpass effect is added).
The purpose of decomposing the time series as a function of both frequency and time is to separate .the contribution of signals with different arriv•al times and frequency responses.Obviously, simultaneous multiple arrivals with an identical frequency response can't be sep arated with such a method.

SYNTHETIC TEST
To start the synthetic test, five Ricker Wa\'elets with vertical and the radial (Z and R) components are simulated and listed in Figures 1-(a) and 1-(b ), respectively.The first one is a pure longitudinal wave, while the fifth is a pure transverse.wave.Others between the first and the fifth are nonlinear waves with phase diffe.rences ( ¢;) of 45°, 90° and 135° (Shieh and Herr1nann, 1990).Their particle motion can be found from the hodogram in Figure 1-( c ). Figures 2-( 11) and ( 12)).It is found that, in Figure 2-(a), only the longitudinal P-wave passes over, while the transverse S-wave and the nonlinear waves are all reduced or suppress to a lower level.Likewise, in Figure 2 11), (b) filtered S-waves using Equation (12. ).
. When noise exists� one may wonder if the function of the filter still works.To check its effectiveness, different levels of noise (from 10% to 60%) may be added to the original data in Figures 3-(a) and 3-(b).Then they can be made to pass over the polarization filter.Figures 4 and 5 show the results of passing P-waves and passing S-waves respectively.Although it is understood that the function of the filter decreases as the noise becomes larger, at the level of SIN =1.667 (60% noise), the function of the filter is still considered effective and acceptable in pratice.

CONCLUSION
It is very effective to use the polarization method to study the vector wavefield.Based on this viewpoint, this study presents a new fi ltering method.In addition to its function of reducing noise and the nonlinear waves, this method can be used to isolate linear longitudinal and transverse waves.Furthermore, it can also be applied to seismic waves, underwater acoustics, radar signals, ... etc., if the data are recorded by the 2-component receiver.5.00

•
where clz and a� are the nor1nalized directional projections.
a) and 2-(b) illustrate the signals being processed through the polarization Chiou-Fen Shieh 163 filter (as Equations ( -(b), only the S-wave passes over.The filtered waveforms are slightly distorted at the beginning and the end due to the imperfection of bandpass filtering and frequency decomposition.

Fig. 1 .Fig. 2 .
Fig. 1.Five 2-component Ricker wavelets with phase diferrence between (a) the vertical and (b) the radial components, 0°, 45°, 90°, 135° and 180° from the first to the fifth respectively.The wave types can be identified from the particle motion of (c), where the first and the fifth show linear motion while the other three show nonlinear motion.

Fig. 3 .
Fig. 3. Random noises added to the original Ricker wavelets.The noise level from the first (bottom) to the sixth (top )traces are 10, 20, 30, 40, 50 and 60% respectively: (a) the vertical component (b) the radial component.

Fig. 4 .
Fig. 4. Results of passing P-waves for different levels of noise added, (a) the vertical component (b) the radial component.

Fig. 5 .
Fig. 5. Results of passing S-waves for different levels of noise added, (a) the , vertical component (b) the radial component.