Study on the Free Surface Coupling EfTect of Seismic Waves

This article presents a simple method to calculate the angle of incidence and the slowness of Pand SV-waves from single 3-component set of data. The obtained slowness is used as the parameter to remove the free surface effect, such as phase distortion and amplification, and to recover the original incident wave. This method is derived from the calculation of ratios regarding the horizontal/vertical ·and the vertical/horizontal response for P-waves and SV-waves, respectively. From the comparison of the calculated ratios with the observed ones (estimated from data), the angle of incidence can be found. Synthetic data are firSt tested to prove their"accuracy, and then, the first '


INTRODUCTION
The slowness (or angle of incidence) is the major parameter used to both remove the free surface effect (Jepsen and Kennett, 1990;Kennett, 1991) and to study rupture propagation in the source region (Goldstein and Archuleta,199 la,b). Jepsen and Kennett (1990) estimate the slowne ss from array data to remove the free surface effect, and then, decompose observed data into new wavefields. Goldstein and Archuleta (1991b) use frequency-wavenumber techniques with dense array (SMART 1) to obtain more detailed measurements of rupture propagation . in the Taiwan area. Dankbaar (1985), using a group of geophones, calculates the slowness and designs P/S filters. In his analyses, an estimated slowness is used for all incident waves. Kennett (1991) roughly · estimates th · e range of the slowness for different wavetypes and isolates them according to the given slowness. In fact however, it is more likely that different body waves reveal with a different degree of slowness. In addition, the slowness is estimated from array data in his anaI · ysis (Dankbaar, 1985). For the small array cases, the estimation could be erroneous, thereby making large array data necessary. However, array data are not always available. Hence, this limits the application of his method.
The purpose of this study is to develop a method which can be applied to every single station to estimate the angle of incidence and the slowness. Moreover, the method must be able to be used for different incident waves independently.
Derived from calculating the ratio of horizontal/vertical and vertical/horizontal for all possible angles of incidence, the method developed is effective for finding the the best fi tting to observed ratio computed from peak spectra. It can be called the PEAK SPE METHOD (PSM). In this study, the PSM is first applied to synthetic data to calculate the angle of incidence. The results are then compared with those obtained from polarization analysis. Second, the PSM is applied to the first P-and SY-arrivals (other body waves are also applicable) in real strong motion data. Finally, the recovered incident waves with free surface effect removed are also displayed .

THEORY AND METHOD
If only single station 3-component data are available, yet there is some knowledge of the velocity structure regarding Pand S-waves in the neighbourhood of the receiver, the slowness or the angles of incidence of some particular body waves can be detem1ined. From the parameters of the velocity and slowness, the free surface response may be calculated and incident wave may be recovered by removing this response. Kennett's method (1983Kennett's method ( , 1991 is suitable for calculating near receiver response because the receiver is located at any depth. In the case of a receiver installed on free surface, it is better to use the equations expressed by Dankbaar (1985) or Aki and Richards (1980, Chapter 5). In this paper, the author adopts Dankbaar's (1985) expressions to show the near-receiver response in the slowness-frequency ( 1 ) where X Vs/Vp , ( ( x 2 -V } p2 ) i , 77 ( 1 -V8 2p2 ) t , and R 0 ( p ) ( l-2V8 2p2 ) 2+ 4p2V8 2(77, R with the subscript v or H, and the superscript p ors, respectively represents the response for the vertical or horizontal components for the Por S-waves. In this study, only Pand S-arrivals are considered so the vertical and the horizontal component records ( U v and UH ) can be expressed separately for incident P-waves as: U u ( p , f) Pin ( P , f)R';,( p , f), and for the incident S-waves as: where Pin and Sin represent incident P-and S-waves respectively. .

4 )
With all of the above expressions, the velocities of the P-and S� . waves at the free surface and the slowness must be used. The velocities can be obtained from other geophysical methods with the only unknown infor1nation · remaining, therefore, is the slowness ( p ). If pure P-or SY-waves, such as the first P or SV arrivals are selected, they can be considered plane waves with the particular slowness approximated. Hence, dividing (6) by (5), the ratio of horizontal/ vertical response for an incident P-wave is: This ratio is computed in the frequency domain for the windowed signals of P or SV where the horizontal and vertical component time series are both taken to the frequency domain.
Equation (15) is expressed for the entire frequency range with particular slowness. Although the P-wave amplitude spectrum is limited to frequency bands, the background noise covers a wide frequency range. The estimation may well be disturbed by existing noise meaning that the average ratio may be computed :from a frequency win1ow whose range covers the peak amplitude spectrum of the vertical component (it is rarely in .the peak of the horizontal component). The reason that this is done is that most P-wave energy is recorded in the vertical component and concentrated in the vicinity of the peak where SIN is high. The computed average ratio ( U u(Ptf)/Uv( -P�-f)) is . fixed for plane waves. Now, the problem is to fi nd a slowness (or angle of incidence) that can fit the observed ratio. The method is to search a ratio calculated from the right hand side of Equation ( 15) to fit the observed ratio.
Equations (1) and (2) can simply be calculated through 0 to 90 degree, then be substituted (15) to obtain many calculated ratios, and then compared to the observed ratio to find the best fitting for the angle. The best fitting angle, say Bp, is. the true angle of incidence, and the slowness is calculated from To remove the free sutface effect and to recover the original P-wave, P i is substituted into Equations (11), (12) and (9). By letting Vp=0.6, V8=0.14 km/s for the sutface layer, the PSM can be tested with modelling records. The synthetic P-wave incident at different angles is depicted in Figure   1. In this figure, the left side is an incident Ricker wavelet with the central frequency of 30 Hz, while the middle part is the surface record of the vertical (Z) and the horizontal (R) components with the angle of incidence marked in the beginning on the left side. Also, the right side is the recovered P-wave for each corresponding angle of incidence. Theoretically, after the PSM is applied, every recovered P-wave on the right side should be equal to the original Ricker wavelet on the left side, and this is illustrated in the figures. The free surface effect causes different amplification in different components for both P-and SY-waves, but there is no phase shift for the P-wave so the particle motion remains the P type. However, beyond the critical angle ( sin8 -.. . �, ) , phase shift happens to the SY-wave, and it becomes p a nondispersed surface wave (Ben-Menahan and Sign, 1981). Figure 2 displays the results of the incident SVwave at different angles of incidence under the same conditions as those in Figure 1. In Figure 2, a phase shift happens to those angles of incidence of a greater degree than that of the critical angle, 13.49°(sin-1(00�64)), thereby making the surface records nondispersed surface waves. To recover the incident SY-wave from the surface records, the ratio of vertical/horizontal is estimated by dividing Equation (7) by Equation (8): ( 1 7 ) The same procedure as for the P-wave is followed, but Equations (2) and (4) are calculated and substituted into Equation (17) to obtain the angles of incidence and the corre sponding slowness. This is substituted Equation (13), (14) and (10)  the PSM plays a useful role in calculating the angle of incidence and the slowness as well as in removing the free surface effect. A simple polarization analysis (Flinn, 1965) may be adopted to obtain the apparent angle of incidence, ¢, and then it may be corrected according to: However, this modifi cation can only be used for incident P-waves; there is no simple modifi cation for incident SY-wave. To explain . this in detail, the examples in Figures 1  and 2 are taken for illustration. With the polarization method (Flinn, 1965), the apparent angle (marked with symbol * in Figure 3) is obtained; after being modifi ed with Equation (18), the results (marked with symbol x in Figure 3a) are the same as those with the PSM under the noise-free situation. Unfortunately, for the SY-wave, the same result can not be obtained. Besides, SY-waves become nondispersed surface waves beyond a critical angle. Thus, after being processed by the polarization method, only two kinds of apparent angles, 0 and 90 degrees, showed up ( Figure 3b ). To obtain the real angle of incidence, therefore, the PSM must be employed. Moreover, the polarization method is seriously affected with noise, rendering the results unreliable unless other kinds of process are taken into consideration.

DATA ANALYSIS
The first P-and SY-arrivals of strong motion data are extracted for the PSM analysis. To increase the SIN ratio, the extracted P-and SY-waves were bandpassed in the frequency range of 2 -15 Hz and I -5 Hz respectively. The accelerograms used in this study are from the earthquake which occured on July 30, 1986 at 1131 UT, at epicenter 121°, 4 7 .650' E, 24°, 37.728'N at a depth 1.6 km, ML=6.2 and which was recorded by the SMART 1 array.
Twelve accelerograms recorded by inner array (station code 101 to 112) were used, and their relative station infonnation is listed in Ta ble 1. The velocities of the P-and S-waves for the top 15 m of this area are 0.6 and 0.14 km/s respectively (Wen and Yeh, 1984). Before being analyzed, all of the data used in this study were rotated into the vertical, radial and transverse directions by the calculation of the azimuthal angle (Table 1) and only the vertical (Z) and the radial (R)) components were used for the PSM analysis. Figure 4 shows the fi rst P arrival at each station, recorded in the vertical and the radial components. After the PSM analysis, the angle of incidence were obtained (marked on the top right), and the incident P wave is recovered as it was before hitting the free surface. Since the velocity of the P-wave on surface is very low, the P wave is supposed to be close to the vertical incidence, and the angle of incidence should also be very small. But the result is not as expected; one possible cause is the irregular boundary at the bottom of the Lanyang Basin (where the SMART 1 is located). From the data of Stations 102, 106, and I 11, it is found that the particle motion is not linear; on the contrary, it clearly appears to be nonlinear motion. Accordingly, it is not a pure P-wave. The factors, other than background noise, to have caused this phenomenon are very complicated and are therefore not discussed here. The results of the fi rst SV-arrival are listed in Figure 5. Comparing Figures 4 and 5, it is seen that the free surface effect has more influence on the SV-wave than on the P-wave. Figure 6 shows the angles of incidence of the P-and SV-waves, as calculated by the PSM and polarization method, and it is clear that among the data of the twelve stations, the calculated angles of the P-wave have a great variation, while those of the SV-wave show ve.:ry consi�tent results. Since the irregular layer boundary has a greater influence on the P-wave with a higher frequency than that on the S-wave with a lower frequency, this may be attributable to the deviation of the angles of the P-waves. · When the PSM applies to the real d · ata,, only the data around the peak amplitude spectra are used. Therefore, its availability may be suspected. To test this, data from one station (103) may be analyzed at all frequencies. Figure 7 shows the spectra of the P-wave in the     vertical and the radial components, while Figure 8 shows that of the SY-wave. Figure 7, shows that most of the P-wave energy lies in the vertical component, and its frequency range is limited in the window chosen by the PSM (2-· t5Hz); angles beyond this range are • all 90 degrees. Likewise, most of the SV-wave energy lies in the radial component, and its frequency range is limited to the window chosen by the PSM. This demonstrates the applicability of the PSM analysis.