Approximation to Cutoffs of Higher Modes of Rayleigh Waves for a Layered Earth Model

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1 Pure appl. geophys. 166 (2009) Ó Birkhäuser Verlag, Basel, /09/ DOI /s Pure and Applied Geophysics Approximation to Cutoffs of Higher Modes of Rayleigh Waves for a Layered Earth Model YIXIAN XU, 1 JIANGHAI XIA, 2 and RICHARD D. MILLER 2 Abstract A cutoff defines the long-period termination of a Rayleigh-wave higher mode and, therefore is a key characteristic of higher mode energy relationship to several material properties of the subsurface. Cutoffs have been used to estimate the shear-wave velocity of an underlying half space of a layered earth model. In this study, we describe a method that replaces the multilayer earth model with a single surface layer overlying the half-space model, accomplished by harmonic averaging of velocities and arithmetic averaging of densities. Using numerical comparisons with theoretical models validates the single-layer approximation. Accuracy of this single-layer approximation is best defined by values of the calculated error in the frequency and phase velocity estimate at a cutoff. Our proposed method is intuitively explained using ray theory. Numerical results indicate that a cutoffs frequency is controlled by the averaged elastic properties within the passing depth of Rayleigh waves and the shear-wave velocity of the underlying half space. Key words: Layered half-space model, cutoffs, higher modes, Rayleigh waves. 1. Introduction Cutoff is the long-period termination of a surface-wave higher mode and, therefore, can be defined by a discrete point in the phase velocity vs. frequency plot. For a layered earth model, phase velocities of Rayleigh waves higher-mode approach the shear (S)-wave velocity of the underlying half space at the cutoffs (e.g., LAY and WALLACE, 1995; UDIAS, 2000). It is therefore reasonable to intuitively think of cutoffs as the point at which energy begins transforming into S-wave radiation in the bounding half space (OLIVER and EWING, 1957). From numerical investigation of the first three Rayleigh modes for a layered crustalmantle model, MOONEY and BOLT (1966) concluded that the ratio of S-wave velocities in the surface layer to the lower half space is the main factor that controls the cutoffs. Although cutoffs were used in guided-wave studies, analytical study and the potential use of surface-wave cutoffs for exploration applications is limited to a single case in the literature (XIA et al., 2006). In that study, S-wave velocity and layer thickness were 1 State Key Laboratory of Geological Processes and Mineral Resources, Institute of Geophysics & Geomatics, China University of Geosciences, Wuhan, Hubei , P.R. China. xyxian@cug.edu.cn 2 Kansas Geological Survey, The University of Kansas, 1930 Constant Avenue, Lawrence, Kansas , U.S.A.

2 340 Y. Xu et al. Pure appl. geophys., estimated from Love-wave energy. Higher modes of Rayleigh waves have proven very important in investigations of subsurface rigidity, from global or regional scale (e.g., PILIDOU et al., 2005; LEBEDEV et al., 2005) to near-surface applications (e.g., XIA et al., 2003; BEATY and SCHMITT, 2003). Use of the higher modes in velocity-model inversions increases exploration depths and improves resolution of inverted S-wave velocity models from Rayleigh waves. Here we are referring to exploration depth as the depth from the ground surface to the lower half-space contact for a layered earth model. Exploration depth is a dynamic parameter that conventionally can be adjusted from wavelength estimation at long-period termination made from analysis of the extracted dispersion curve. The value of increased exploration depths and improved resolution has lead to development and now common use of a procedure for joint inversion of multimode Rayleigh waves in practice (XIA et al., 2003 and 2006; DAL MORO and PIPAN, 2007). A good initial model is key to reducing the risk of errant results and speeding the inversion of multimode Rayleigh-wave dispersion curves (e.g., XIA et al., 1999; RYDEN and PARK, 2006). The exploration depth and the S-wave velocity of the lower half space are initial model parameters invariably difficult to estimate accurately but necessary for accurate inversion of Rayleigh waves. Additionally, imaging the subsurface structure using converted waves (e.g., PS waves) has been a topic of interest in the petroleum industry for more than a decade (e.g., SERIFF and SRIRAM, 1991; GRECHKA and TSVANKIN, 2002). Imaging the deep subsurface structures via converted waves requires knowledge of the near-surface S-wave velocity field that is difficult to estimate by conventional oil exploration seismic reflection technologies. The utility of higher-mode Rayleigh waves for estimating S-wave velocities can be effectively exploited at no additional cost in data acquisition. Therefore, the use of Rayleigh-wave higher-mode cutoffs may have great potential for petroleum application compared to conventional fundamental mode approaches if the exploration depth that defines near surface can be extended, and S-wave velocity estimate of the underlying half space can be obtained and used in the construction of an initial model for inversion. Improved imaging accuracy from higher-mode, high-frequency Rayleigh-wave energy is sufficient justification to utilize higher-mode cutoffs. Although from classical dispersion curve analysis it is not possible to obtain cutoffs accurately for real world data (e.g., XIA et al., 2007), a high-resolution linear Radon transform method (LUO et al., 2008) appears feasible for extracting high-quality dispersion curves from higher modes. The study discussed here discloses an approximation method for estimating the cutoffs in a layered earth model. Explicit formulations used to calculate cutoffs can only be developed for the simplest layered earth model with a single surface layer (NEWLANDS, 1952; KUO and NAFE, 1962). In the next section we develop a modified formula for calculating the cutoffs for one surface layer earth model that is fundamentally different from high wavenumber approximation (NEWLANDS, 1952). We also introduce an approximation method that is based on harmonic averaging of velocities and arithmetic averaging of densities to identify a single layer that reasonably approximates multiple layers. Estimating cutoffs using this approach is numerically investigated to establish its accuracy. The presented numerical testing results are in good agreement with cutoffs

3 Vol. 166, 2009 Higher Modes of Rayleigh Waves 341 defined as true versus those selected using approximation models. Using the method described here, the exploration depth and the averaged P- and S-waves velocities of the approximated surface layer can be estimated. 2. Estimation of Cutoffs Based on a Layered Half-space Model with a Single-surface Layer For high wavenumber approximation, i.e., k??, where k is the wavenumber, of a layered half-space model with a single-surface layer and thickness H, NEWLANDS (1952) formulated the frequency at cutoffs as fc h ¼ h c þ np 2pH b 2 2 =b =2 b 2 ; n ¼ 1; 2; 3;...; ð1þ h c ¼ tan 1 2 b =b2 1 4 b 2 2 =b =2 1 b 2 1=2 ; ð2þ 2 =a2 1 in the case of b 1 < c R < b 2 ; where a and b are the velocities of P and S waves; subscripts 1 and 2 denote the surface layer and the lower half space, respectively; n denotes the mode number. A modified approximation of the cutoffs, which eliminates the dependence on the high wavenumber approximation, can be written as 2 n tan kh b 2 2 =b =2 o 1 2 b b 2 2 =b =b2 1 ð T1 A U 1 BÞ 1=2 1 b 2 1=2 Y =a2 1 ð T1 A þ U 1 BÞ ð3þ ¼ tan h a c ; p 2 \h a c \ p ; 2 where T 1 ¼ 1 b 2 1=2 2 =a2 1 ð l2 =l 1 2Þ 2 þ 1 b 2 b 2 2 =a2 2 l 2 2 b 2 ; l 1 1 U 1 ¼ 1 b 2 1=2 2 =a2 1 ð l2 =l 1 2Þ 2 1 b 2 b 2 2 =a2 2 l 2 2 ; l 1 "!!# l 2 þ 2 b2 2 l 1 b þ 4 2 b2 2 l 1 1 b 2 ; 1 h A ¼ cos H 1 b 2 1=2 i h 2 =a2 1 ; B ¼ sin H 1 b 2 1=2 i 2 =a2 1 ; Y 1 ¼ 1 b 2 1=2 l 2 2 =a2 2 1 l 1 ¼ q 1 b 2 1 ; l 2 ¼ q 2 b 2 2 : b 2 1

4 342 Y. Xu et al. Pure appl. geophys., Equation (3) is derived from equations (14.8) and (14.9) in NEWLANDS (1952) after a tedious mathematical process. The frequencies at cutoffs from equation (3) are fc a h a c þ np 2pH b 2 2 =b =2 b 2 ; n ¼ 1; 2; 3;... ð4þ where the superscript denote the cutoff approximations. We evaluate the accuracy of equations (1) and (4) against numerical results using the generalized R/T coefficients method (e.g., HISADA, 1994; LAI, 1998). The earth model used in the numerical simulation is composed of a half space underlying a surface layer with the properties: Surface layer : v p1 ¼ 800 m=s, v s1 ¼ 200 m=s, q 1 ¼ 2000 kg=m 3 ; Half space: v p2 ¼ 1200 m=s, v s2 ¼ 400 m=s; q 2 ¼ 2000 kg=m 3 : The thickness of the surface layer is allowed to change from 5 m to 40 m. The frequency increment for this numerical calculation is 1 Hz. The results from equation (1) or (4) and the R/T coefficients method should be considered the same if the frequency deviation is within ± 0.5 Hz. (Fig. 1). The frequencies at cutoffs calculated from Eq. (4) are more accurate than those from Eq. (1) for the first three higher modes of the study model, especially where the cutoffs possess higher frequencies for a thinner surface layer. The accuracy of the high wavenumber approximation, however, is also sufficient for near-surface applications. 3. Approximation of Cutoffs Based on Spatial Averaging of Layers Parameters In order to use Eq. (4) to calculate cutoff frequencies for a model consisting of m (m > 1) isotropic layers, we approximate the multilayer model with a single-layer model through spatial harmonic averaging of P- and S-wave velocities and spatial arithmetic averaging of density: Figure 1 Comparison of cutoff frequencies of the first three higher modes for the model listed in the text calculated by the high wavenumber approximation (dashed line), the modified approximation (dashed-dot line) developed in this study, and the analytical method (solid line). a) The first higher mode, b) the second higher mode, and c) the third higher mode.

5 Vol. 166, 2009 Higher Modes of Rayleigh Waves 343 v pors ¼ * + 1 Xm r i v p or s ; ð5þ i¼1 i * + q ¼ Xm r i q i ; ð6þ i¼1 where r i (i = 1,... m) is the weighting and equals h i /H, where h i and H are the i-layer and the total thickness, respectively; v i P or S represents the velocity of P or S wave of the i-layer (the subscript denotes the layer number); and q i denotes the density of the i-layer. In order to evaluate the validity and accuracy of our averaging procedure (Eqs. 5 and 6), we conducted a series of numerical simulations using the R/T coefficients method (HISADA, 1994; LAI, 1998) and several earth models. A frequency increment of 0.49 Hz is used in all calculations. Using the averaging procedure (Eqs. 5 and 6), we replaced the second and third layers of a four-layer model (Model 1 in Table 1) with an intermediate layer. The dispersion curves were calculated for the fundamental and the first three higher modes of the original and the approximated models and displayed in Figure 2. For each higher mode, the dispersion curve of a single layer that is an approximation to two intermediate layers closely resembles the dispersion curve of the true model, which consists of two intermediate layers, even with the slight distortion in the intermediate frequency range. This observation is not surprising, considering the fact that parameters for the top layer and the underlying half space are the same for both models. Table 1 Parameters of Models 1, 2, 3, 4, and 5 Model Parameters Vp (m/s) Vs (m/s) q (kg/m 3 ) Thickness (m) 1 Top layer Second layer Third layer Half space ? 2 Top layer /10/20 Intermediate layer Half space ? 3 Top layer The second layer The third layer Half space ? 4 Top layer /10/20 Intermediate layer Half space ? 5 Top layer The second layer The third layer Half space ?

6 344 Y. Xu et al. Pure appl. geophys., Figure 2 Comparison of the dispersion curves for single intermediate layer approximation to two intermediate layers. Solid and dashed lines indicate dispersion curves of the true model (Model 1 in Table 1) and the approximated model, respectively. Now, we examine a model consisting of two layers overlying the half space (Model 2 in Table 1). Using the same averaging procedure (Eqs. 5 and 6), an approximated surface layer can be defined using the top two layers. The dispersion curves were calculated for different first and second layer thickness ratios (Figs. 3a 3c). For the first three higher modes, the maximum deviation of frequency at cutoffs is two frequency increments (0.98 Hz) and the maximum phase velocity error is 8.1% (Figs. 4a 4c). Based on the accuracy requirements of most current near-surface applications, both frequency and phase velocity estimates are considered in good agreement for semi-quantitative interpretations of dispersion curves and formulation of an initial model for inversion. To test the accuracy of the proposed method even further, a comparison similar to the previous examples was made for a three-surface layer model (Model 3 in Table 1). Similar to the two-layer model result, a three-layer numerical comparison reveals differences at cutoffs being within two frequency increments and 8.5% in the phase velocities (Figs. 5 and 6). To validate the breadth of applicability in real-world stratigraphy, two special models are tested. The first one consists of two layers overlying the half space (Model 4 in Table 1) with a low-velocity layer sandwiched between a surface layer of varying thickness and the underlying half space. The second one consists of three layers overlying the half space (Model 5 in Table 1), where the P-wave velocity of the second layer is equal to the S-wave velocity of the third layer so that the singularity arises. The results also show excellent agreement of cutoffs (Figs. 7 and 8 for Model 4, and Fig. 9 for Model 5). For most near-surface applications, our modeling results (including some results not shown here) possess errors of cutoffs and corresponding phase velocities that are within 15% (based on the proposed approximation method when higher modes are present). 4. Explanation by Ray Theory Next we will provide a physically reasonable explanation of the proposed method. The higher modes of Rayleigh waves result from constructive interference of reflected

7 Vol. 166, 2009 Higher Modes of Rayleigh Waves 345 Figure 3 Comparison of the dispersion curves for single-surface layer approximation to two surface layers. Solid and dashed lines indicate dispersion curves of the true model and the approximated model, respectively. The thicknesses of the top two layers are a) 5 m and 5 m, b) 10 m and 5 m, and c) 20 m and 5 m, respectively. Figure 4 Misfit illustrations of phase velocity (top panel) and frequency (bottom panel) at cutoffs with different ratios of the top two layers for calculations in Figure 3. Pluses and triangles represent the results of the true model and the approximated model, respectively. a) The first higher mode, b) the second higher mode, and c) the third higher mode. and refracted P and S waves within a surface layer acting as a waveguide (TOLSTOY and USDIN, 1953; XU et al., 2006). LEVSHIN et al. (2005) provided a good analogy by comparing them to crustal higher modes. If one model can be considered an

8 346 Y. Xu et al. Pure appl. geophys., Figure 5 Comparison of the dispersion curves for single-surface layer approximation to three surface layers. Solid and dashed lines indicate dispersion curves of the true model and the approximated model, respectively. Figure 6 Misfits of frequency (a) and phase velocity (b) at cutoffs associated with different modes in Figure 5. Pluses and triangles represent the results of the true model and the approximated model, respectively. approximation of another, then from an application viewpoint the observed wave arrivals for the two models must be the same or very similar regardless of whether the travel paths are the same or not. Assuming a model with two surface layers over a half space (Fig. 10), the travel time of a wave along the true path is t ¼ 2h 1=cos h 1 þ 2h 2=cos h 2 : ð7þ v 1 v 2 As previously discussed, the travel time for an approximated model constructed by averaging is t 0 ¼ 2H=cos h v ¼ 2H h 1=H þ h 2=H cos h ¼ 2 h 1 þ h 2 cos h: v 1 v 2 v 1 v 2 ð8þ

9 Vol. 166, 2009 Higher Modes of Rayleigh Waves 347 Figure 7 Comparison of the dispersion curves for single-surface layer approximation to two-surface layers when a low velocity layer is present. Solid and dashed lines indicate dispersion curves of the true model and the approximated model, respectively. The thicknesses of the top two layers are a) 5 m and 5 m, b) 10 m and 5 m, and c) 20 m and 5 m, respectively. Figure 8 Misfit illustrations of phase velocity (top panel) and frequency (bottom panel) at cutoffs with different ratios of the top two layers for calculations in Figure 7; pluses and triangles represent the results of the true model and the approximated model, respectively. (a) The first higher mode, b) the second higher mode, and c) the third higher mode.

10 348 Y. Xu et al. Pure appl. geophys., Figure 9 Comparison of the first three higher modes for Model 5 in Table 1. a) Dispersion curves, solid and dashed lines indicate the results of the true model and the approximated model, respectively. and b) cutoffs, pluses and triangles represent the results of the true model and the approximated model, respectively. Figure 10 Explanation of the proposed approximation method by the ray theory. After a simple rearrangement, the difference of t and t 0 is given Dt ¼ 2h þ 2h : ð9þ v 1 cos h 1 cos h v 2 cos h 2 cos h It is obvious that the difference in travel times is the same when h = h 1 = h 2, which can occur when the incident wave is at an angle of zero. It should be noted that Dt isin general small, because h 1 < h < h 2 in the studied model (where v 1 < v 2 is required for a waveguide). A major portion of the difference in the first term cancels due to the difference in the second term (Eq. 9). Looking closer at Equation (9), we notice that the travel time difference increases with incident angle (and therefore increases with offset). This characteristic seems problematic for utilizing the cutoffs of higher-mode Rayleigh waves because higher modes are better developed in the far field (XIA et al., 2006). Conversely, the lower the mode number, the steeper the dispersion curve becomes near

11 Vol. 166, 2009 Higher Modes of Rayleigh Waves 349 Figure 11 A layered earth model with borehole information for which the proposed method can be used to determine other parameters. the cutoffs. Therefore, the most accurate approximation will generally be within an intermediate offset range. 5. Discussion and Conclusions In this study we developed a method for replacing a multilayer earth model with a single-surface layer over the half-space model using harmonic averaging of the velocities and arithmetic averaging of densities. From numerical tests, the accuracy of estimated frequencies and phase velocities at cutoffs is less than 1 Hz and 9%, respectively. According to Eqs. (4), (5) and (6), the frequency at cutoffs is controlled by the average elastic properties of the surface layers and the S-wave velocity of the half space. The effectiveness and accuracy of the proposed method can be further extended if additional data constraints are available from other techniques (e.g., borehole and logging). For example, if the thickness and velocities for the top two layers are available from borehole logging data for a three layer over a half-space model (Fig. 11), one can then easily obtain the S-wave velocity for the third layer using the following relations: b a 1 ¼ h r 1=b 1 þ r 2 =b 2 þ r 3 =b 3 i 1 ; r 1 ¼ h 1 =H; r 2 ¼ h 2 =H; r 3 ¼ðHh 1 h 2 Þ=H where b i (i = 1, 2, 3) denotes S-wave velocities of surface layers, and h i (i = 1, 2) are the corresponding thicknesses. For this example, b i (i = 1, 2) and h i (i = 1, 2) are obtained from borehole data. The S-wave velocity of the approximated surface layer, b a, and the depth to the top of the half space, H, can be estimated by Eq. (4)or(1) if cutoffs of higher modes are available. There are similar approaches for estimating the unknown P-wave velocity. This situation may occur for either exploration applications or investigation of the earth s interior.

12 350 Y. Xu et al. Pure appl. geophys., Acknowledgments The first author is grateful to the China Scholarship Council, and the Kansas Geological Survey, the University of Kansas, for their financial support of this study. The authors thank Dr. Baker and another anonymous reviewer and the Editor Howard Patton for their constructive and detailed reviews. REFERENCES BEATY, K.S. and SCHMITT, D.R. (2003), Repeatability of multimode Rayleigh-wave dispersion studies, Geophysics 68, DAL MORO, G. and PIPAN, M. (2007), Joint inversion of surface wave dispersion curves and reflection travel times via multi-objective evolutionary algorithms, J. Appl. Geophys 61(1), GRECHKA, V. and TSVANKIN, I. (2002), PP? PS = SS, Geophys 67, HISADA, Y. (1994), An efficient method for computing Green s functions for a layered half-space with sources and receivers at close depths, Bull. Seismol. Soc. Am 84, KUO, J. T. and NAFE, J. E. (1962), Period equation for Rayleigh waves in a layer overlying a half space with a sinusoidal interface, Bull. Seismol. Soc. Am 52, LAI, C.G. (1998), Simultaneous inversion of Rayleigh phase velocity and attenuation for near-surface site characterization, Ph.D. Dissertation, Georgia Inst. of Tech., Atlanta, U.S.A. LAY, T. and WALLACE, T.C. Modern Global Seismology (Academic Press 1995). LEBEDEV, S., NOLET, G., MEIER, T., and VAN DER HILST, R.D. (2005), Automated multimode inversion of surface and S waveforms, Geophys. J. Int. 162, LEVSHIN, A.L., ROTZWOLLER, M.H., and SHAPIRO, N.M. (2005), The use of crustal higher modes to constrain crustal structure across Central Asia, Geophys. J. Int. 160, LUO, Y., XIA, J., MILLER, R.D., XU, Y., LIU, J., and LIU, Q. (2008), Rayleigh-wave dispersive energy imaging by high-resolution linear Radon transform, Pure Appl. Geophys. 165, MOONEY, H. M. and BOLT, B. A. (1966), Dispersive characteristics of the first three Rayleigh modes for a single surface layer, Bull. Seismol. Soc. Am. 56, NEWLANDS, M. (1952), The disturbance due to a line source in a semi-infinite elastic medium with a single surface layer, Phil. Trans. Roy. Soc. London, A 245, OLIVER, J. and EWING, M. (1957), Higher modes of continental Rayleigh waves, Bull. Seismol. Soc. Am. 47, PILIDOU, S., PRIESTLEY, K., DEBAYLE, E., and GUDMUNDSSON, O. (2005), Rayleigh wave tomography in the North Atlantic: high resolution images of the Iceland, Azores and Eifel mantle plumes, Lithos. 79, RYDEN, N. and PARK, C.B. (2006), Fast simulated annealing inversion of surface waves on pavement using phase-velocity spectra, Geophysics 71(4), R49 R58. SERIFF, A.J. and SRIRAM, K.P. (1991), P-SV reflection moveouts for transversely isotropic media with a vertical symmetry axis, Geophysics 56, TOLSTOY, I. and USDIN, E. (1953), Dispersion properties of stratified elastic and liquid media: A ray theory, Geophysics 18, UDIAS, A.Principles of Seismology (Cambridge University Press 2000). XIA, J., MILLER, R.D., and PARK, C.B. (1999), Estimation of near-surface shear-wave velocity by inversion of Rayleigh wave, Geophysics 64, XIA, J., MILLER, R. D., PARK, C. B., and TIAN, G. (2003), Inversion of high frequency surface waves with fundamental and higher modes, J. Appl. Geophys. 52, XIA, J., XU, Y., CHEN, C., KAUFMANN, R.D., and LUO, Y. (2006), Simple equations guide high-frequency surfacewave investigation techniques, Soil Dyna. Earthq. Engin. 26(5), XIA, J., XU, Y., and MILLER, R.D. (2007), Generating image of dispersive energy by frequency decomposition and slant stacking, Pure Appl. Geophys. 164(5),

13 Vol. 166, 2009 Higher Modes of Rayleigh Waves 351 XU, Y., XIA, J., MILLER, R.D. (2006), Quantitative estimation of minimum offset for multichannel surface-wave survey with actively exciting source, J. Appli. Geophys. 59(2), (Received October 18, 2007, accepted October 14, 2008) Published Online First: February 20, 2009 To access this journal online: