Finite-difference Modeling of High-frequency Rayleigh waves Yixian Xu, China University of Geosciences, Wuhan, 430074, China; Jianghai Xia, and Richard D. Miller, Kansas Geological Survey, The University of Kansas, Lawrence, Kansas 66047-3726, USA. Summary The finite-difference (FD) method described here is specially developed to model high-frequency Rayleigh-wave propagation in near-surface mediums. Although many FD programs existed for earthquake research and oil exploration, none has been developed to model high-frequency Rayleigh-wave propagation in near-surface elastic mediums when the source and all receivers are on the free-surface. The scheme described here uses the second-order staggered grid, based on a set of first-order differential equations. Special implementations of combined elastic and acoustic free-surface conditions were developed, which were originally derived from surface wave propagation in a transversely isotropic medium. A combination of one-way sponge filtering and anisotropic filtering methods was used to minimize Rayleigh-wave reflections from artificial boundaries. Numerical stability was achieved by using a small spatial grid size and time step. This new FD method was tested and proved satisfactory using dispersion analysis of Rayleigh waves in a homogenous and layered half-space. The present study provides a practical tool for improving the confidence and uniqueness of deriving 2D interpretation of high-frequency Rayleigh-wave data, and makes successful in the first phase to 2D inversion. Introduction Because high-frequency (> 2 Hz) surface waves travel in a very shallow portion of the earth, spectral analysis of surface-wave (SASW) method was developed for site investigation over the last twenty years (e.g., Stokoe et al., 1994). A new technique incorporating multichannel analysis of surface waves (MASW) using active sources was developed (Park et al., 1999; Xia et al., 1999) to improve inherent difficulties in evaluating and distinguishing signal from noise with only a pair of receivers by the SASW method. The MASW method has become increasing popular for environmental and engineering applications in recent years (e.g., Xia et al., 2004). A couple reasons are likely responsible for increased use of this method. First is based on very characteristics of Rayleigh-wave energy. In the near-surface, Rayleigh waves represent the highest energy propagating from a source and therefore have a relatively high signal-to-noise ratio. Rayleigh waves are not limited by subsurface velocity properties, which sometime affect reflection and refraction methods. Rayleigh waves are predominately controlled by shear (S)-wave velocity properties of the subsurface which are key important for many geotechnical applications. The second comes from the advantages made possible with multichannel recording. MASW can use advanced array digital signal processing technologies and thus take advantage of the many benefits of time-space relationships of seismic wavefields. To our knowledge, ongoing interpretation methods are currently based on the plane-wave approximation (Xia et al., 1999). Few attempts have been made addressing Rayleigh-wave dispersion analysis of higher dimensional models. However, to utilize all Rayleigh-wave information, higher dimensional modeling codes must be developed. This paper aims to develop a 2D modeling method for high-frequency (>2Hz) Rayleigh-wave analysis. In the following sections, we will discuss the modeling method including implementations of the free-surface and artificial boundaries and numerical tests using Rayleigh-wave dispersion analysis. A 2D numerical example will demonstrate the effectiveness and accuracy of this 2D modeling scheme. Modeling method We choose the second-order staggered-grid finite-difference method (Virieux, 1986) because of its coding simplicity and accuracy for dense spatial sampling as demanded by Rayleigh-wave propagation. Wave propagation in a 2D heterogeneous elastic medium can be described by a first-order system of differential equations: tvx x xx xz tvz zz x xz t xx ( 2 ) xvx zv (1) z t zz xvx ( 2 ) zvz ( V V ) t xz x z z x where ( Vx, Vz) is the particle velocity vector, and ( xx, zz, xz ) is the stress tensor, x, is the density of a medium, and x, and x, are Lame coefficients. Discretization of the differential equations (1) for time-space coordinates at internal nodes using a staggered-grid scheme can be found in Virieux (1986). Selection of a time step and a spatial grid size for a specific velocity model and source energy frequency band is very important in Rayleigh-wave modeling. From numerical tests, a empirical rule, t 0.6 x/ Vmax, and x 0.125 min, where Vmax and min are the highest velocity and the shortest wavelength, SEG/Houston 2005 Annual Meeting 57
respectively, used in our modeling. Here the vertical grid size is assumed the same as the horizontal grid size. On a free-surface, zz and xz must be zero. We developed a practical implementation of the free-surface conditions by combining the image method (Levander, 1988) and Mittet s (2003) method. The procedure assumes that nodes at one-half of a spatial step length above the free-surface are filled by a medium with elastic properties symmetric about the free-surface; meanwhile, elastic properties at the free-surface were modified. Implementation in this fashion is physically reasonable, effective for Rayleigh-wave modeling. For absorbing artificial wave energy resulting from computing boundary, better results were achieved by joining the one-way sponge filtering (OSF) and the anisotropic filtering (Dai et al., 1994) methods. This implementation modified outgoing waves in a transition zone so that the waves impinge on the computing boundary at the normal angle. For all our numerical tests, we chose a source that change with time according to the first order derivative of Gaussian function. Because a sledgehammer or vibrator seem to be the most commonly used sources in high-frequency Rayleigh-wave surveys, the normal stress or velocity source can be implemented by simply defining the source function in terms of zz or V z, respectively, e.g., zz i,1 g( t), where (i, 1) represents the ith node at the free-surface, and g t is a source function. The initial condition is simply setting g t, when t 0. 0 Numerical tests of one-dimensional models Numerical tests by theoretical models were performed for a homogenous half-space and an isotropic layered half-space. Dispersive analysis provided a powerful tool for recognition and analysis of high-frequency Rayleigh-wave propagation and therefore was used for testing modeled vertical velocities. For our testing reported on here, P- and S-wave velocities of a homogenous half-space are 00 m/s and 00 / 3 m/s, respectively. These values define a Poisson s solid with a Rayleigh-wave velocity about 531m/s. Spatial grids consisted of 200 200 nodes with 30 nodes in the transition zones along the left and right sides of the model and 60 nodes in the transition zone along the bottom of the model. Such spatial grid size was 1 meter with time step of 0.1ms. The source is described by the first-order derivative of Gaussian function 2 t exp( t ) with controlling parameter 3000 and is located at node (80, 1). A shot gather (vertical particle velocity) and its dispersion curve are shown in Fig. 1. Small deviations can be identified, relative to the true Rayleigh-wave phase velocities at near source channels (Fig. 2a) and in lower frequency band (Fig. 2b). This phenomenon results from non-plane wave approximations and near-field effects. In the higher frequency band (> 70Hz), the dispersion curve appears to possess a slightly decreasing velocity trend as frequency increases. This is due to required spatial grid size exceeding the demands of the numerical dispersion. The non-dispersive phase velocity (531 m/s) obtained by the phase shift method agreed well with the Rayleigh-wave phase velocity of the homogeneous half-space in the frequency band commonly used for near-surface applications. The parameters of the two-layer model are as follows: the surface layer: V p = 800 m/s, V s = 200 m/s, = 2000 kg/m 3, and thickness = m; the half-space: V p = 1200 m/s, V s = 400 m/s, and = 2000 kg/m 3. The grid size, time step, and time function and location of sources were the same as the homogenous half-space. The snapshots of the vertical particle velocity of the two-layer model at 150 ms and 300 ms are shown in Fig. 2. The seismogram and dispersion curve of the vertical particle velocity are shown in Fig. 3. In the band of 15 to 30 Hz, the dispersion curve agrees very well with the analytical result (the maximum relative error within 2%), whereas in the band of 8~15 Hz, the relative error raises to about 5%. We realized the error is mainly caused by phase-shift preprocessing, near-field effects, (e.g., non-plane wave propagation and incorporated body wave energies), and finite-difference approximation, in which about 5% cannot quantitatively be separated into unique different contributions. Two-dimensional experiment The main object of this study is to support 2D interpretation of near-surface Rayleigh-wave investigations. 2D surface-wave modeling results for common geological features and anomalies would aid us greatly in understanding and detecting certain unique wave characteristics associated with Rayleigh waves. We designed a corner-edge model (Fig. 4) to simulate a vertical fault in a near-surface setting. The earth surface intersection point of the fault corner is at the center point of the 60-station-receiver spread (Fig. 4). O1 and O 2 are marked as the left source (on the left side of the corner) and right source (on the right side of the corner), respectively. All the modeling parameters (including the source time functions) are the same as the homogenous half-space. Scattering events that spread from the middle to the left quadrant can be clearly seen in vertical particle velocities excited by the left source (Fig. 5a). When the source is on the right quadrant, it is difficult to identify scattering events (Fig. 5b). When the frequencies approach the extremes, both SEG/Houston 2005 Annual Meeting 58
dispersion curves from sources on the left and right sides approach the same Rayleigh wave phase velocities for the top layer and the bottom half-space. However these two dispersion curves possess different phase velocities within the middle frequency band (9 to 18 Hz) illustrated clearly by Fig. 7. Interesting results can be obtained by examining amplitude spectra (Fig. 6). The amplitude spectra were calculated trace by trace. The main energies for the left source concentrate at about 9 Hz in the middle to the left portion of the profile, and 18 Hz in almost the whole profile for the right source. The fault edge was imaged by the maximum energy concentration observed when the left source was used. We can thus suggest the dispersive characterizations of Rayleigh waves excited by the left source are determined mainly by the left portion of subsurface structures; whereas the dispersive characterizations of Rayleigh waves excited by the right source are determined mainly by the right portion of subsurface structures. According to the traditional half wave-length method (Yang, 1993), one can estimate the phase velocity of Rayleigh waves about 321 m/s at 9 Hz from the record excited by the left source, which gives a depth to the top of the lower half-space at about 17.8 m. However, based on record excited by the right source one can estimate the phase velocity of Rayleigh waves about 230 m/s at 18 Hz, which gives a depth to the top of the lower half-space at about 6.4 m. These results possess a large deviation from the true depths on two sides of the fault. Conclusions We developed an effective FD method to model Rayleigh-wave propagation for near-surface applications. In this method, there is an important improvement: free-surface conditions combined the image method and a technique to modify medium properties just at the free-surface that is supported by the Rayleigh-wave propagation theory. We also achieved satisfactory implementations for artificial boundaries by combining one-way sponge filtering and anisotropic filtering methods. These make the present method feasible for near-surface applications. From the numerical perspective, the 2D model revealed some of the interesting properties of Rayleigh-wave propagation such as scattering events from the corner of the fault and error existed in the traditional half wave-length interpretation. These encourage the development of a 2D inversion method in the near future. Acknowledgments The first author thanks the Kansas Geological Survey for providing the surface analysis software used in the paper and the China Scholarship Council for the financial support. The first author appreciates Dr. Choon Park for the constructive discussion on examples of dispersion analysis used in the manuscript. References Dai, N.X., Vafidis A., and Kanasewich, E., 1994. Composite absorbing boundaries for the numerical simulation of seismic waves, Bull. Seis. Soc. Am., 84, 185-191. Ellefsen, K.J., 1993. Two-dimensional numerical simulation of elastic wave propagation for environmental and geotechnical studies, U.S. Geological Survey Open-File Report 93-714. Levander, A.R., 1988. Fourth-order finite-difference P-SV seismograms, Geophysics, 53, 1425 1436. Mittet, R., 2002. Free-surface boundary conditions for elastic staggered-grid modeling schemes, Geophysics, 67, 1616 1623. Park, C.B., Miller, R.D., and Xia, J., 1999. Multichannel analysis of surface waves, Geophysics, 64, 800-808. Stokoe II, K.H., Wright, G.W., James, A.B., and Jose, M.R., 1994. Characterization of geotechnical sites by SASW method, in Geophysical Characterization of Sites, ISSMFE Technical Committee #, edited by R.D. Woods, Oxford Publishers, New Delhi. Virieux, J., 1986. P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method, Geophysics, 51, 889 901. Xia, J., Miller, R.D., and Park, C.B., 1999. Estimation of near-surface shear-wave velocity by inversion of Rayleigh wave, Geophysics, 64, 691-700. Xia, J., Miller, R.D., Park, C.B., Ivanov, J., Tian, G., and Chen, C., 2004. Utilization of high-frequency Rayleigh waves in near-surface geophysics, The Leading Edge, 23, 753-759. Yang, C., 1993. Rayleigh wave exploration (in Chinese), Geological Pub. House, Beijing. Fig. 1. Vertical particle velocity and dispersion curve of a homogenous half-space. vertical particle velocity; dispersion curve. SEG/Houston 2005 Annual Meeting 59
1 80 50 140 1 80 50 40 40 70 70 0 0 130 130 Fig. 2. Snapshots of vertical particle velocities in a two-layered half-space at 150 ms (left panel) and 300 ms (right panel), respectively. Fig. 5. Vertical particle velocities and dispersion curves of the corner-edge model for different source layouts. vertical particle velocity with the left source; vertical particle velocity with the right source. Fig. 6. Amplitude spectra of the corner-edge model with different source layouts. the left source (left); the right source (right). 500 480 460 Phase velocity Corner-edge(left source) Corner-edge(right source) Layered half-space(5m thickness of upper layer) Layered half-space(m thickness of upper layer) 440 420 Fig. 3. Vertical particle velocity and dispersion curve of a 400 two-layered half-space. vertical particle velocity; 380 dispersion curve. O1 30m Spread 30m O2 Phase velocity (m/s) 140 360 340 320 300 280 260 5m 240 Vp=800m/s, Vs=200m/s. 5m 220 Vp=1200/s, 200 180 Vs=400m/s. Fig. 4. Illustration of a corner-edge model and layouts of sources and receivers. 8 12 14 16 18 20 Frequency (Hz) 22 24 26 28 30 Fig. 7. Comparison of phase velocities between corner-edge and layered half-space models. SEG/Houston 2005 Annual Meeting 60
EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2005 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. Finite-difference Modeling of High-frequency Rayleigh waves References Dai, N. X., A. Vafidis, and E. Kanasewich, 1994, Composite absorbing boundaries for the numerical simulation of seismic waves: Bulletin of the Seismology Society of America, 84, 185 191. Ellefsen, K. J., 1993. Two-dimensional numerical simulation of elastic wave propagation for environmental and geotechnical studies: U. S. Geological Survey Open-File Report 93 714. Levander, A. R., 1988. Fourth-order finite-difference P-SV seismograms: Geophysics, 53, 1425 1436. Mittet, R., 2002. Free-surface boundary conditions for elastic staggered-grid modeling schemes: Geophysics, 67, 1616 1623. Park, C. B., R. D. Miller, and J. Xia, 1999, Multichannel analysis of surface waves: Geophysics, 64, 800 808. Stokoe II, K. H., G. W. Wright, A. B. James, and M. R. Jose, 1994, Characterization of geotechnical sites by SASW method, in geophysical characterization of sites: Oxford Publishers. Virieux, J., 1986, P-SV wave propagation in heterogeneous media: Velocity-stress finitedifference method: Geophysics, 51, 889 901. Xia, J., R. D. Miller, and C. B. Park, 1999, Estimation of near-surface shear-wave velocity by inversion of Rayleigh wave: Geophysics, 64, 691 700. Xia, J., R. D. Miller, C. B. Park, J. Ivanov, G. Tian, and C. Chen, 2004, Utilization of high-frequency Rayleigh waves in near-surface geophysics: The Leading Edge, 23, 753 759. Yang, C., 1993, Rayleigh wave exploration: Geological Publishing House.