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Misidentification caused by leaky surface wave in high-frequency surface wave method


Geophysical Journal International
Geophys. J. Int. (2014) 199, 1452–1462 GJI Seismology doi: 10.1093/gji/ggu337

Misidenti?cation caused by leaky surface wave in high-frequency surface wave method
Lingli Gao,1 Jianghai Xia1,2 and Yudi Pan1
1 Subsurface

Imaging and Sensing Laboratory, Institute of Geophysics and Geomatics, The China University of Geosciences, 388 Rumo Rd., Wuhan, Hubei 430074, China 2 Hubei Subsurface Multi-scale Imaging Laboratory, Institute of Geophysics and Geomatics, The China University of Geosciences, 388 Rumo Rd., Wuhan, Hubei 430074, China. E-mail: jxia@cug.edu.cn

Accepted 2014 September 1. Received 2014 August 22; in original form 2013 November 27

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SUMMARY Multichannel analysis of surface waves (MASW) method analyses high-frequency surface waves to determine shear (S)-wave velocities of near-surface materials, which are usually unconsolidated and possess higher Poisson’s ratios. One of key steps using the MASW method to obtain the near-surface S-wave velocities is to pick correct phase velocities in dispersive images. A high-frequency seismic survey conducted over near-surface materials with a higher Poisson’s ratio will often result in data that contains non-geometric wave, which will raise an additional energy in the dispersion image. Failure to identify it may result in misidenti?cation. In this paper, we have presented a description about leaky surface wave and the in?uence caused by the existence of leaky waves in a high-frequency seismic record. We ?rst introduce leaky wave and non-geometric wave. Next, we use two synthetic tests to demonstrate that non-geometric wave is leaky wave and show the properties about leaky surface wave by eigenfunctions using Chen’s algorithm. We show that misidenti?cation may occur in picking the dispersion curves of normal Rayleigh wave modes because the leaky-wave energy normally connects energy of fundamental and/or higher modes. Meanwhile, we use a real-world example to demonstrate the in?uence of leaky wave. We also propose that muting and ?ltering should been applied to raw seismic records prior to generating dispersive images to prevent misidentifying leaky surface waves as modal surface waves by a real-world example. Finally, we use a three-layer model with a low-velocity half-space to illustrate that leaky surface waves appear on condition that the phase velocities are higher than maximum S-wave velocity of the earth model when solving the Rayleigh equation. Key words: Surface waves and free oscillations; Guided waves; Wave propagation.
ysis of surface waves (MASW) method (e.g. Song et al. 1989) greatly improved the accuracy of estimated S-wave velocities (Xia et al. 1999). The dispersion curves of the surface waves are used to obtain the near-surface S-wave velocities. The difference between inverted S-wave velocities using the MASW method and borehole measurements is about 15 per cent or less if high-mode data are available (Xia et al. 2003). Hence, picking accurate phase velocity information is a primary task of any high-frequency surface wave survey. Shallow subsurface typically consists of unconsolidated sediments characterized by very high ratios of P- to S-wave velocities (St¨ umpel et al. 1984). Seismic data from the unconsolidated sediments with high Poisson’s ratio is complex. For example, Roth et al. (1999) noticed that a seismic surface wave in an environment with a very high Poisson’s ratio had a phase velocity lower than P-wave velocity, but higher than S-wave velocity. Later, Roth & Holliger (2000) identi?ed the signal as non-geometric wave based on analytic and numerical analyses. Xia et al. (2002) found the similar

I N T RO D U C T I O N Surface waves are guided and dispersive in all earth models except for the case of the elastic half-space. Rayleigh (1885) waves are surface waves that travel along a free surface, such as the Earth–air interface, or along the Earth–water interface. Rayleigh waves are the result of interfering P and SV wave. Particle motion of the fundamental mode of Rayleigh waves in a homogeneous medium moving from left to right is elliptical in a counter-clockwise (retrograde) direction along the free surface. As depth increases, the particle motion becomes prograded and is still elliptical when reaching suf?cient depth. Rayleigh waves are usually characterized by relatively low velocity, low frequency and high amplitude (Sheriff 1991). Shear (S)-wave velocity is a fundamental physical parameter for many near-surface geophysical studies. Spectral analysis of surface waves (SASW) method (Stokoe & Nazarian 1983) was introduced to analyse dispersive curves of ground roll to generate near-surface S-wave velocity pro?les. The latter developed multichannel anal-

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The Authors 2014. Published by Oxford University Press on behalf of The Royal Astronomical Society.

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Figure 1. Snapshots of the vertical particle velocities for two high Poisson’s ratio earth models with the M-PML technique at t = 60 ms, t = 120 ms, t = 210 ms. (a) Snapshots for a homogeneous earth model. (b) Snapshots for a two-layer earth model. Letters ‘L’, ‘P’, ‘R’, ‘S’, ‘SS ’ and ‘SS’ represent for leaky wave, P wave, Rayleigh wave, S wave, re?ected and refracted S wave, respectively. Table 1. Roots of Rayleigh equation vary with different Poisson’s ratios. Vs stands for S-wave velocity. Poisson’s ratio 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 ... 0.49 Real root (s) 0.92Vs 1.85Vs 1.93Vs 0.92Vs 0.92Vs 0.93Vs 0.93Vs 0.93Vs 0.93Vs 0.93Vs 0.93Vs 0.94Vs ... 0.95Vs Complex roots – (1.89 ± 0.08i)Vs (1.89 ± 0.13i)Vs (1.90 ± 0.17i)Vs (1.90 ± 0.19i)Vs (1.90 ± 0.22i)Vs (1.90 ± 0.24i)Vs (1.91 ± 0.27i)Vs (1.91 ± 0.29i)Vs (1.91 ± 0.31i)Vs ... (1.96 ± 0.55i)Vs

signal in the data acquired in Wyoming. Xu et al. (2007) showed an unexpectedly small jump resulting from the non-geometric wave arrived in the seismogram calculated from the stress image method (SIM). Roth & Holliger (2000) summarized that the non-geometric wave will appear when the Poisson’s ratio is high and the velocity of non-geometric wave is nearly twice the shear wave velocity of the surface layer. However, it is still unknown what non-geometric wave really is and what in?uence will be caused by non-geometric wave. A point source placed on the surface of an earth model excites different waves, such as P wave, S wave and Rayleigh/Love wave. As Figs 1(a) and (b) show (details in the following paragraph), there are leaky surface waves (letter ‘L’) that travel along the surface. The leaky surface wave has been observed by various authors in experimental and numerical studies. For example, Oliver & Major (1960) noticed a decaying oscillatory long-period motion following P wave, called PL phase, in the data obtained through the operation of a network of long period seismographs from the International

Geophysical Year. They interpreted the PL phases are likely corresponding to leaky modes. Rosenbaum (1960) summarized a complete representation of the total wave ?eld as a sum of normal and leaky modes. Phinney (1961) conducted a theoretical study of leaky waves, which were called Pseudo-P mode in his research. Aki & Richards (1980) provides a theoretical study of leaky surface waves for the simplest layered medium-liquid layer overlying a liquid halfspace. Schr¨ oder & Scott (2001) described the theoretical derivation of leaky surface waves in some details. They demonstrated that in materials of high Poisson’s ratio (>0.26), not only modal surface waves, but also leaky surface waves will generate. Leaky wave is an inhomogeneous wave that propagates along the ground surface with a phase velocity higher than S-wave velocity, lower than Pwave velocity. It showed that for materials with high Poisson’s ratio a leaky wave exists, velocities of which can determined by solving the Rayleigh equation. In this paper, we ?rst show what leaky surface wave and nongeometric wave are. Then we demonstrate that the non-geometric wave acquired in the shot gather is leaky surface wave by comparing the theoretical values of phase velocities calculated by the generalized R/T coef?cients algorithm (Chen 1993) with the dispersive energy of non-geometric wave. Synthetic tests and a real-world example indicate in a high Poisson’s ratio multilayer model, the leaky waves exist and their dispersion energy in the frequency–velocity (f–v ) domain may connect to modal surface wave energy, which could result in recognizing the dispersion image of leaky-wave energy as one part of dispersive energy of modal surface waves from the dispersion image. Such misidenti?cation will produce incorrect dispersion curves and eventually bring wrong inversion results. We propose some processing techniques such as muting and ?ltering to reduce the in?uence caused by leaky surface wave and provide suggestions to avoid misidenti?cation of leaky surface waves. Finally, we discuss complex roots of Rayleigh equation caused by an earth model with a low-velocity half-space.

Table 2. Parameters of a homogeneous earth model. Layer number 1 vs (m s–1 ) 220 v p (m s–1 ) 1000 ρ (kg m–3 ) 2000 h (m) In?nite Poisson’s ratio 0.475

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Eq. (1) is commonly called Rayleigh equation (Schr¨ oder & Scott 2001). By rationalizing, eq. (1) can be expressed as v vs
2

v vs

6

?8

v vs

4

+ 24 ? 16 vs /v p

2

v vs
2

2

? 16 1 ? vs /v p

= 0,

(2)

where v p , vs and v represent the P-wave velocity, S-wave velocity and the phase velocity of Rayleigh wave, respectively. Depending on the Poisson’s ratio σ of a material, different kinds of roots of eq. (2) exist. We calculated the roots with various Poisson’s ratios and obtained Table 1 (omit the negative roots). For σ ≤ 0.26, eq. (2) has three real roots. For σ > 0.26, eq. (2) has one real root and two complex conjugate roots (omit the negative roots). The real root that is the smallest in magnitude gives rise to the wellknown Rayleigh wave phase velocity, and two complex conjugate roots used to be supposed as erroneous roots of Rayleigh equation (Achenbach 1973; Graff 1975). However, Schr¨ oder & Scott (2001) demonstrated that the complex conjugate roots of Rayleigh equation in fact give rise to a leaky surface wave. It is an inhomogeneous wave that propagates along the surface with a phase velocity higher than the S-wave velocity, lower than the P-wave velocity. It couples into a plane wave that propagates in the medium. For the complex conjugate roots, real parts of complex roots represent propagation velocities of leaky surface waves and the imaginary parts represent attenuation. Table 1 indicates that the leaky-wave velocities are nearly twice of S-wave velocities, same as the conclusion summarized by Roth & Holliger (2000). For an isotropic elastic model that has n parallel layers, Haskell (1953) present the transfer matrix method for the ?rst time to compute the dispersion curves, and then many improved methods were developed (Knopoff 1964; Dunkin 1965; Thrower 1965; Schwab & Knopoff 1970, 1972; Kennett 1974; Abo-Zena 1979; Menke 1979; Chen 1993; Zhang et al. 1997). Here, we use a generalized re?ection and transmission coef?cients algorithm (Chen 1993) in the calculation of theoretical phase velocities because Chen’s algorithm is not only ef?cient and accurate, but also can be extended to determine the corresponding eigenfuncitons. The dispersion equation (Chen 1993) can be written as ?1 ?0 det( I ? R ud R du ) = 0,
0 1

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(3)

Figure 2. (a) A shot gather of a homogeneous earth model (Table 2). (b) Its dispersive image generated by the high-resolution linear Radon transform (all following dispersive images are generated by the high-resolution linear Radon transform). The green solid dots and yellow triangles represent the theoretical values of Rayleigh-wave phase velocities and leaky surface wave phase velocities, respectively.

L E A K Y S U R FA C E WAV E For a homogeneous half-space, the phase velocity of Rayleigh wave can be determined by 2? v2 2 vs
2

+4

v2 v2 ? 1 2 ? 1 = 0. 2 vp vs

(1)

? ud represent the gener? ud and R where I is a 2 × 2 identity matrix, R alized R/T coef?cients for P–SV waves (details about the derivation in Chen 1993). For a given frequency f we search for the roots of the dispersion equation, which gives the phase velocities of Rayleigh waves and leaky surface waves. For Rayleigh-wave phase velocities, we set the search range from 0.8 × VSmin to VSmax (VSmin and VSmax represent for the minimum and the maximum S-wave velocities for a given model, respectively); for leaky-wave phase velocities, we set the search range from VSmax to 2 × VSmax . To show the propagating characters of leaky surface wave, we set a homogeneous half-space model with a P-wave velocity of 1500 m s?1 , S-wave velocity of 110 m s?1 and density of 2.0 g cm–3 and a two-layer earth model with a P-wave velocity of 800 m s?1 , S-wave velocity of 200 m s?1 , thickness of 8 m and density of

Table 3. Parameters of the ?rst two-layer earth model. Layer number 1 2 vs (m s–1 ) 72 140 v p (m s–1 ) 520 1050 ρ (kg m–3 ) 1860 1860 h (m) 4 In?nite Poisson’s ratio 0.490 0.491

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Figure 3. (a) A shot gather of a two-layer earth model (Table 3). (b) Its dispersive image. The green solid dots and yellow triangles represent the theoretical values of Rayleigh-wave phase velocities and leaky surface wave phase velocities, respectively. (c) The eigendisplacements; the solid and dashed lines are the vertical and horizontal components, respectively, and each trace has been normalized by the maximum value of corresponding vertical displacement. (d) The eigentractions; the solid and dashed lines are the vertical and horizontal components, respectively, and each trace has been normalized by the maximum value of corresponding horizontal traction. ‘L0, S0’, ‘L1, S1’, ‘L2, S2’, and ‘L3, S3’ represent fundamental mode, ?rst higher mode, second higher mode and third higher mode of leaky wave and Rayleigh wave at f = 5, 15, 25 and 35 Hz, respectively. Table 4. Parameters of a six-layer earth model (Xia et al. 1999). Layer number 1 2 3 4 5 6 vs (m s–1 ) 194 270 367 485 603 740 v p (m s–1 ) 650 750 1400 1800 2150 2800 ρ (kg m–3 ) 1820 1860 1910 1960 2020 2090 h (m) 2.4 2.4 2.4 2.4 3.6 In?nite Poisson’s ratio 0.451 0.426 0.463 0.461 0.457 0.462

1.92 g cm–3 for the surface layer, P-wave velocity of 1600 m s?1 , S-wave velocity of 400 m s?1 and density of 2.0 g cm–3 for the halfspace. We use a 30 Hz Ricker wavelet for all the examples provided in this paper unless otherwise stated. Figs 1(a) and (b) show the snapshots (t = 60, 120 and 210 ms) of the seismic waves (letters ‘L’, ‘P’, ‘R’, ‘S’, ‘SS ’ and ‘SS’ represent for leaky wave, P wave, Rayleigh wave, S wave, re?ected and refracted S wave, respectively) propagating in the homogeneous half-space and the two-layer model, respectively. It is obvious that the leaky surface wave propagates faster than the Rayleigh wave, and attenuates along the direction of

propagation. Fig. 1(b) indicates that the leaky surface wave can also be observed in a multilayer earth model. N O N - G E O M E T R I C WAV E If an explosive seismic source is placed near a seismic discontinuity such as the free surface in a homogeneous full space, it radiates homogeneous (non-decaying) and inhomogeneous (evanescent) P waves and the inhomogeneous P waves will be transformed into homogeneous S waves by the re?ection (Roth & Holliger 2000).

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velocities of leaky surface waves are right on the peaks of the nongeometric wave energy, which veri?es that non-geometric wave is leaky surface wave. The second data set used to verify the ?tness of leaky surface wave phase velocities to non-geometric energy is due to a two-layer earth model (Table 3). Rayleigh-wave energy is clearly modelled in a synthetic 48-channels shot gather (Fig. 3a) with a simpli?ed M-PML technique (Zeng et al. 2011). In the shot gather, there is a signal with high amplitude arriving before Rayleigh wave, which appears at the offset of 2 m. The distance that the diffraction wave appeared is on condition that 2h tan θ > 2 m (h and θ represent the thickness of the ?rst layer and the incident angle, respectively), which means that this signal cannot be diffraction wave. As the speed of this wave is nearly twice of S-wave velocity of the topmost layer, this signal represents non-geometric wave. In the dispersive image (Fig. 3b), the phase velocities of dispersive energy from 5 to 8 Hz are higher than 140 m s?1 , which exceed the maximum S-wave velocity of the earth model. This phenomenon is arisen from the non-geometric wave. The green solid dots and yellow triangles present the phase velocities of Rayleigh wave and leaky surface wave, respectively. The fundamental and higher modes of leaky surface waves correspond to the yellow triangles very well, which veri?es that non-geometric wave is leaky surface wave. From the numerical tests, we demonstrate that the non-geometric wave acquired in seismic records is leaky surface wave and the distinction of phase velocities of non-geometric wave and the theoretical values of leaky surface wave is negligible. Figs 3(c) and (d) show the eigendisplacements and eigentractions of leaky modes and Rayleigh wave modes for the same model shown in Table 3. The higher leaky modes penetrate deeper than fundamental leaky mode. The eigenfunction indicates that the leaky modes and Rayleighwave modes have a similar physical mechanism: P–S or S–P conversions at the free surface and the re?ections at subsurface. Rayleigh waves are dispersive in a layered-earth model and possess the properties that asymptotes at the low and high frequencies of the fundamental and high modes are associated with the S-wave velocities of the half-space and the surface layer, respectively. In the two dispersion images, we notice that the leaky surface wave will generate additional energy in dispersive images in a lower frequency range, which exceed the asymptotes of the fundamental mode or higher modes at low frequencies. Furthermore, it may cause misidenti?cation when picking phase velocities of leaky surface waves as Rayleigh-wave phase velocities for inversion. In the following, we will present the in?uence of leaky surface wave by some numerical tests and real world example.

Figure 4. A dispersive image is generated based on an earth model described in Table 4. Misidenti?cation of the fundamental mode could happen due to the existence of leaky surface wave. The green solid dots and yellow triangles represent the theoretical values of Rayleigh wave and leaky surface wave, respectively.

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The transformed waves are linked to a complex incidence angle and cannot be associated with a geometrical ray path, so such waves are usually referred as non-geometric waves (Kuhn 1985). Nongeometric waves have been mentioned in many literatures (e.g. Brekhovskikh 1980; Tsvankin 1995). One best-known type of nongeometric waves is the S? phase (Hron & Mikhailenko 1981). However, this kind of non-geometric S? wave cannot be observed in the seismic record because it propagates with Rayleigh wave and its amplitude is close to zero. Roth & Holliger (2000) showed that for materials with high Poisson’s ratio, an additional non-geometric wave with high amplitudes appears at near offsets and can be observed in a shot gather. They analysed the signal collected in Switzerland (Roth et al. 1999) by following the approach taken by Hron & Mikhailenko (1981) in their study of S? phase. They showed the results obtained with the Cagniard-de Hoop method (Cagniard 1939; De Hoop 1960) for homogeneous half-space models with different Poisson’s ratios, and then inferred that the velocity of non-geometric wave is nearly twice as much as the S-wave velocity. In order to illustrate the relationship between non-geometric wave and leaky surface wave, we compare the theoretical phase velocities of leaky surface wave with the dispersion energy of non-geometric wave. We use a homogeneous earth model and a two-layer earth model to demonstrate that the non-geometric wave is leaky surface wave. The homogeneous earth model’s Poisson’s ratio is 0.475, with all the other model parameters showed in Table 2. Surface wave is not dispersive in a homogeneous earth model. Rayleigh-wave energy is clearly modelled in a synthetic 48-channel shot gather (Fig. 2a) with a simpli?ed M-PML technique (Zeng et al. 2011). In the shot gather, there is a signal arriving before Rayleigh wave, with a phase velocity between the P- and S-wave velocity. In the dispersive image (Fig. 2b) obtained by high-resolution linear Radon transform (Luo et al. 2009), the non-geometric wave leads to additional energy at the frequencies between 5 and 10 Hz, and its phase velocity is nearly 430 m s?1 , almost twice as the S-wave velocity. The green solid dots and yellow triangles represent Rayleigh-wave phase velocities and leaky-wave phase velocities, respectively. The theoretical phase

M I S I D E N T I F I C AT I O N Leaky surface wave exists commonly in unconsolidated nearsurface sediments. Theoretically, it exists in materials with Poisson’s ratio being bigger than 0.26 (Schr¨ oder & Scott 2001). Due to the uncertainties of the Earth medium of the real world, leaky surface wave may cause misidenti?cation by regarding the leakywave energy as part of fundament- or higher-mode Rayleigh waves. For example, for a shot gather (Fig. 2a) due to a homogeneous

Table 5. Parameters of the second two-layer earth models for comparison. Layer number 1 2 vs (m s–1 ) 200 400 v p (m s–1 ) 470 748 320 640 ρ (kg m–3 ) 1820 1860 h (m) 3 In?nite Poisson’s ratio 0.389 0.300 0.18 0.18

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Figure 5. Four dispersive images generated based on four earth models with parameters in Table 5. Misidenti?cation of the ?rst higher mode could happen due to the existence of leaky surface wave. The green solid dots and yellow triangles represent the theoretical values of Rayleigh waves and leaky surface waves, respectively. (a) Dispersive image of Model I (the P-wave velocities of the ?rst layer and second layer are 470 and 748 m s?1 , respectively). (b) Dispersive image of Model II (the P-wave velocities of the ?rst layer and second layer are 320 and 640 m s?1 , respectively). (c) Dispersive image of Model III (the P-wave velocities of the ?rst layer and second layer are 470 and 640 m s?1 , respectively). (d) Dispersive image of Model IV (the P-wave velocities of the ?rst layer and second layer are 320 and 748 m s?1 , respectively).

earth model, the leaky surface wave will generate an additional energy at low frequencies (Fig. 2b), which could be identi?ed as part of Rayleigh-wave dispersive energy. As for multilayer earth models, the situation is even more complicated. We will use two numerical tests of multilayer earth models to illustrate the misidenti?cation caused by leaky surface wave. In the following discussion, all dispersive images are generated by high-resolution linear Radon transform (Luo et al. 2009). In the dispersive images, the green solid dots and the yellow triangles represent the theoretical phase velocities of Rayleigh waves and leaky surface waves, respectively. A six-layer earth model same as Xia’s model (Xia et al. 1999) is used to present the in?uence of leaky surface wave on identifying fundamental mode (Table 4). According to the character of the Rayleigh wave, asymptotes at the low frequencies of the fundamental mode would approach about 0.91 times to the velocity of the half-space. Compare the dispersion curve to the theoretical values (green dots and yellow triangles) in Fig. 4, the phase velocities of dispersive energy at 5–12 Hz do not coincide to any mode of Rayleigh-wave phase velocities, however, the energy matches with phase velocities of leaky surface wave (the yellow triangles). When picking phase velocities for inversion, it is easy to mistake leakywave phase velocities as part of fundamental mode Rayleigh-wave phase velocities, which will result in wrong inversion results that possess higher S-wave velocities than the true model.

Other two-layer earth models (Table 5) are used to show the in?uence on identifying phase velocities of leaky surface wave as the high-mode Rayleigh-wave dispersion curve. Asymptotes at the low frequencies of ?rst higher mode approach the correct phase velocities for the half-space (?400 m s?1 ). Fig. 5(a) is the dispersive image of Model I (the P-wave velocities of the ?rst layer and second layer are 470 and 748 m s?1 , respectively) whose Poisson’s ratios of both layers are bigger than 0.26. In the dispersive image, the ?rst higher mode is unavailable, and an additional energy appears at frequencies between 10 and 20 Hz whose phase velocities exceed the maximum S-wave velocity of the model (?400 m s?1 ). The ‘exceed’ phase velocities that represent leaky surface waves (yellow triangles) are easily regarded as ?rst high-mode dispersion curve due to uncertainness of the earth model. For Model II (the P-wave velocities of the ?rst layer and second layer are 320 and 640 m s?1 , respectively), the Poisson’s ratio of both layers are smaller than 0.26, the leaky surface wave will not generate. It is easy to pick the Rayleigh-wave phase velocities (Fig. 5b). Fig. 5(c) is the dispersion image of Model III (the P-wave velocities of the ?rst layer and second layer are 470 and 640 m s?1 , respectively), the Poisson’s ratio of the second layer is smaller than 0.26, the leaky surface wave energy appears in the low frequency range and could also cause misidenti?cation. For Model IV (the P-wave velocities of the ?rst layer and second layer are 320 and

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Figure 6. (a) Raw data acquired from Wyoming in 1998 using 48 8 Hz vertical-component geophones on an interval (0.9 m) with 1.8 m from the source to the nearest geophone. The source was a 6.3 kg hammer vertically impacting a metal plate. (b) Processed data by muting leaky wave (non-geometric wave). (c) A dispersive image of raw ?eld data. (d) A dispersion image of processed data. The green solid dots and yellow triangles represent the theoretical values of Rayleigh wave and leaky surface wave, respectively.

748 m s?1 , respectively), the Poisson’s ratio of the ?rst layer is smaller than 0.26, there is no any energy of leaky surface wave in the dispersion image (Fig. 5d). These dispersion images indicate that leaky surface wave will generate when the surface layer possesses high Poisson’s ratio. We demonstrated that the existence of leaky wave will generate unexpected additional energy, which might be misidenti?ed as part of the fundamental- or high-mode Rayleigh-wave energy. It is interesting to notice that mode transformation happens at the cut-off frequency, which suggests the higher modes may turn to leaky mode at the cut-off frequency.

F I E L D D ATA E X A M P L E A surface wave survey was conducted in Wyoming in 1998 to determine S-wave velocities of the near surface materials (Xia

et al. 2002). Forty-eight 8 Hz vertical-component geophones were placed along a line at an interval (0.9 m) with 1.8 m from the source to the nearest geophone. The source was a 6.3 kg hammer vertically impacting a metal plate. The multichannel raw ?eld data (Fig. 6a), which contains enhanced Rayleigh-wave signals and leaky surface wave (non-geometric wave in Xia et al. 2002, red circle in Fig. 6a), are acquired. Leaky surface wave propagates faster than Rayleigh wave, and attenuates along the direction of propagation. The processed ?eld data (Fig. 6b) is obtained by muting leaky waves before Rayleigh-wave arrivals from the raw data (details in Yilmaz 1987). Two dispersive images (Figs 6c and d) of the raw ?eld data and the processed data, respectively, are obtained by high-resolution linear Radon transform (Luo et al. 2009). The fundamental mode of Rayleigh waves is distinct at 10–50 Hz and higher modes of Rayleigh waves appear at 42–50 Hz in Figs 6(c) and (d). Additional energy appears with phase velocities between 500 and 800 m s?1 at low frequencies in Fig. 6(c). However, no such energy appears

Misidenti?cation caused by leaky surface wave
Table 6. Wyoming example. Inversion results of S-wave velocities by the L-M method. Layer number 1 2 3 4 5 6 7 8 9 10 vs (m s–1 ) 145 162 207 226 235 242 255 269 284 474 v p (m s–1 ) 270 340 510 510 900 900 900 900 900 900 ρ (kg m–3 ) 1820 1840 1860 1910 1930 1960 1980 2010 2020 2050 h (m) 1 1 1 1 1 1 1 1 1 In?nite Poisson’s ratio 0.297 0.353 0.401 0.378 0.463 0.461 0.456 0.451 0.445 0.308

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Table 7. Parameters of a three-layer earth model which contains a low-velocity halfspace. Layer number 1 2 3 vs (m s–1 ) 200 300 100 v p (m s–1 ) 1000 1800 500 ρ (kg m–3 ) 1960 2030 2060 h (m) 5 5 In?nite Poisson’s ratio 0.479 0.486 0.479

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in Fig. 6(d), which suggests the additional energy results from the leaky wave. Meanwhile, an inversion test performed by the L-M method is used to verify the characteristics of the additional energy. Rayleigh-wave phase velocity is extracted from the dispersive image in the f–v domain. A 10-layer model is used when inverting the dispersion curve (Fig. 6c) with layer thickness of 1 m. P-wave velocities of each layer are estimated using the ?rst arrivals (Xia et al. 2002) with the density of each layer is chosen at 2.0 g cm–3 . Inverted S-wave velocities for the dispersion curve (Fig. 6c) are shown in Table 6. Then, we calculate the theoretical values of the inverted S-wave velocities model to verify if the additional energy will correspond to the theoretical values. In Figs 6(c) and (d), the green solid dots and the yellow triangles represent Rayleigh-wave and leaky surface-wave phase velocities, respectively. As Fig. 6(c) shows, the yellow triangles match with the additional energy well, which con?rms that the leaky wave generate the additional energy in the dispersive image (Fig. 6c) and could cause misidenti?cation. Such misidenti?cation will result in wrong inversion results. The real-world example shows that leaky surface wave can be detected in exploration seismic data, and commonly referred as non-geometric wave. In seismic records, the leaky surface wave has high amplitude, and arrives before Rayleigh waves. To avoid misidenti?cation caused by leaky surface waves, muting and (or) ?ltering should be applied to raw ?eld data to remove the leaky waves before we extract Rayleigh-wave phase velocities from the dispersive curve in the f–v domain.

DISCUSSION Phase velocities can be determined by solving the real parts of Rayleigh-wave dispersion equation in the situation that Rayleighwave dispersion equation becomes complex numbers at some frequencies when S-wave velocities of the half-space is lower than some of the layers above (Pan et al. 2013). We showed in this paper that roots higher than the maximum S-wave velocity of the earth model correspond to leaky surface waves and the others correspond to Rayleigh waves. A three-layer model (Table 7) with a low-velocity half-space is used to illustrate that leaky surface waves appear on condition that the phase velocities are higher than maximum S-wave velocity of the earth model when solving the Rayleigh equation.

Figs 7(a) and (b) are shot gathers of the vertical component and horizontal component of a three-layer earth model which contains a low-velocity half-space (Table 7). The raw shot gathers are analysed to determine the general velocities of Rayleigh wave and non-geometric wave as Figs 7(a) and (b) show. To obtain a better understanding of the characteristics of the synthetic seismic signals in Figs 7(a) and (b), the particle motions of the signals are displayed in Figs 7(c) and (d). In the diagrams, each hodogram is normalized to the maximum displacement at the given receiver positions (trace 4 and trace 60). For trace 4, time increases from 1 to 500 ms. Compare to the shot gathers (Figs 7a and b), leaky waves appear at 1–150 ms, and the particle motion is elliptical due to the leaky wave. Then, the particle motion is retrograde elliptical respect to the vertical direction. As time increases, the particle motion is prograde elliptical respect to the vertical direction. This is corresponded with the propagation characteristics of Rayleigh waves. For trace 60, time increases from 1 to 1000 ms. The displacements of leaky surface waves (t = 1–300 ms) are smaller in Fig. 7(d) than the displacements in Fig. 7(c). Clearly, leaky waves decay as they travel along the surface (Fig. 7a). Then, the particle motion is retrograde elliptical respect to the vertical direction. As time increases, the particle motion is prograde elliptical respect to the vertical direction. In a word, the propagation characteristics of the signals in the shot gathers indicate that the shot gathers contain leaky waves and Rayleigh waves. Dispersive image (Fig. 7e) indicates the characteristics of the energy distribution. The fundamental-mode energy and a leaky-wave energy which exceed the maximum S-wave velocity are distinct in the dispersive image (Fig. 7e). When the phase velocities are higher than the S-wave velocity of the half-space, the dispersion equation will become complex number (Pan et al. 2013). The green solid dots are determined by solving the real parts of the dispersion equation, the phase velocities are lower than the maximum S-wave velocity, which are Rayleigh-wave phase velocities and correspond to the energy trend of the fundamental mode very well. The yellow triangles are also determined by the dispersion equation, whose phase velocities are higher than the maximum S-wave velocity, represent the theoretical values of leaky surface wave. They correspond to the leaky-wave energy exceed the maximum S-wave velocity well. To verify that the phase velocities exceed the maximum Swave velocity of the earth model are leaky mode, a processed shot

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Figure 7. (a) A shot gather of the vertical component of a three-layer earth model which contains a low-velocity half-space (Table 7). (b) A shot gather of the horizontal component of a three-layer earth model which contains a low-velocity half-space (Table 7). (c) Particle motion of trace 4 from t = 1 to 500 ms. (d) Particle motion of trace 60 from t = 1 to 1000 ms. (e) A dispersive image of raw ?eld data (a). (f) A shot gather after velocity-band-reject ?ltering (300–1000 m s?1 ) applied to the synthetic seismic record (a). (g) A dispersive image of processed data (f). The green solid dots and yellow triangles represent the theoretical values of Rayleigh wave and leaky surface wave, respectively.

Misidenti?cation caused by leaky surface wave
gather (Fig. 7f) is generated by velocity-band-reject ?ltering (300– 1000 m s?1 ) applied to the synthetic seismic record (Fig. 7a). Fig. 7(g) is the dispersion image of processed data. Compare Figs 7(e) and (g), the leaky wave energy disappears. REFERENCES

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C O N C LU S I O N S We have presented a detail description about the calculation and properties of leaky surface waves and the in?uence caused by the existence of leaky waves in a high-frequency seismic record. Velocities of leaky surface waves can be determined by solving dispersion equation (Poisson’s ratio >0.26). We recommend Chen’s algorithm for calculation because it is a stable and effective approach for attaining the phase velocities and corresponding eigenfunctions for both leaky surface waves and Rayleigh waves. Eigenfunctions indicate that leaky-wave modes and Rayleigh-wave modes have similar physical mechanism. Leaky surface waves are dispersive in a multilayer earth model whose surface layer possesses a high Poisson’s ratio. It is well known that Rayleigh-wave phase velocities are lower than the maximum S-wave velocity for a given velocity model and attenuate as depth increases. Leaky-wave phase velocities, however, are between the maximum Rayleigh-wave phase velocity and P-wave velocity and attenuate along the propagation direction. In high-frequency seismic surveys, for a high Poisson’s ratio material, an additional non-geometric wave will be generated, which can emerge at near offsets of a shot gather with high amplitudes. The non-geometric wave is particularly common in shallow highfrequency seismic surveys because high Poisson’s ratio materials are common in near surface. We have demonstrated that nongeometric waves are leaky waves by comparing theoretical values of leaky surface waves and dispersion energy curves of non-geometric waves. Leaky waves increase the complexity of a high-frequency seismic record. Two synthetic tests have demonstrated that leaky surface waves could be easily treated as part of fundament- or highmode Rayleigh-wave energy because the leaky-wave energy normally connects energy of fundamental and/or higher modes. This misidenti?cation could result in an inverted S-wave velocity model possessing higher values than the true model. The ?eld example has demonstrated that effects of the misidenti?cation caused by leaky surface waves can be reduced by muting and/or ?ltering prior to transferring ?eld data from the time-space domain into the frequency-velocity domain. Our research is a reminder that misidenti?cation will be caused by the existence of leaky waves (non-geometric waves) in a high-frequency seismic record if no processing is taken, especially surveys at high Poisson’s ratio materials. Muting and/or ?ltering are (is) needed and ef?cient in processing of high-frequency Rayleigh-wave data.

AC K N OW L E D G E M E N T S We appreciate two anonymous reviewers and the Editor Xiaofei Chen for their constructive and detailed comments and suggestions. This research is supported by the National Natural Science Foundation of China (NSFC, Grant No. 41274142) and the Fundamental Research Funds for Institute for Geophysical and Geochemical Exploration, Chinese Academy of Geological Sciences, Grant Nos. WHS201201 and WHS201203.

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