Influence of Uneven Trace Spacing on Rayleigh Wave Dispersion

INTRODUCTION

The dispersion information of Rayleigh waves characterizes the wave propagation and the media properties. The dispersion curve can be inverted into an S-wave velocity profile depicting the underground structural property (Zhang et al, 2003; Park et al., 1999; Xia et al., 1999). However, the dispersion behaviors of Rayleigh waves may be affected by the field acquisition parameters. Principally, the nearest offset and spread length are realized the significant acquisition parameters (Chen et al., 2004; Zhang et al., 2004; Roesset et al., 1989). "Near-field effects" and "spreading length effects" are therefore the major effects concluded by some research works. For the near-field effects, the near-offset of the spread may affect the phase unwrapping at the extremely low frequencies of the dispersion curve (Bodet et al., 2009; Yoon and Rix, 2006, 2004; Lin and Chang, 2004). For the spreading length effects, the resolution of phase velocity is naturally determined by the spread length (Bodet et al., 2009, 2005; Forbriger, 2003; O'Neill, 2003); long spreading length is beneficial for obtaining high-resolution dispersion data of surface waves.

For a spatial sampling rate of signals, Nyquist sampling theorem is usually considered to avoid spatial aliasing. The criterion can be satisfied by selecting trace spacing shorter than λ_min/2, where λ_min is the minimum wavelength. However, previous studies ignored the case of uneven trace spacing. This study aims to investigate the influence of the uneven trace spacing on Rayleigh wave dispersion data in construction. A field trial is performed at the campus of Wuhan University for practical data. A 2.5D acquisition layout is specially designed with a linear array of geophones fixed and a mobile active source. The wave rays cover a fan shaped area. It emphasizes the trace spacing varying with observation directions, and the interferences from other acquisition parameters can be alleviated greatly. The observation angle θ (Fig. 1) changes from 0º to 90º. The distance between the shot point to the array center is fixed. Such geometry ensures the dependence of the trace spacing on the observation direction.

Figure  1.  Field testing layout.

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DATA ACQUISITION LAYOUT

Figure 1 shows the data acquisition spread in-situ. The soil field has been filled as playground for decades at the campus of Wuhan University. The linear array consists of 22 geophones of 2.5 Hz. The data acquisition system keeps the geophones fixed and allows the mobile source to rotate around the array center. As shown in Fig. 1, the radius L varies at 28, 33, and 38 m. The observation angle θ changes from 0° to 90° with an interval of 15°. When θ>0°, the trace spacing between the adjacent two receivers becomes uneven for a given array spread. This geometry ensures the spatial sampling uniformity to be dependent on the observation direction.

RAYLEIGH WAVE DISPERSION IMAGES

Trace Spacing

The geometry of the layout setting can be simplified into 2D space according to the source-receiver distances. Let the shot point be the coordinate origin, and r_i as the distance between the shot point and the ith receiver, which can be expressed in terms of observation angle. For , there is

and for , the source-receiver distance changes as

where N is the channel number; Δl is the distance between the adjacent two receivers in the receiving array. When θ=0°, the layout becomes a 2D space, trace spacing is equal to the receiver interval that is uniform. When θ>0°, the trace spacing becomes Δr_i=r_i+1–r_i≠Δl. Apparently, equations (1–2) demonstrate that the trace spacing changes with the observation angle. Define the Trace Spacing Distortion Rate (TSDR) σ to describe the distortion of the trace spacing for a given radius L, expressed by

where Δr_max and Δrmin are the maximum and minimum trace spacings, respectively; is the average channel interval of the array.

Figure 2 shows the relationship between the observation angle and the trace spacing distortion rate with three given radiuses. The trace spacing distortion rate increases with the increasing observation angle. When θ=0°, there is no distortion. When θ=30° and L=28 m, the maximum distortion approaches 1.0. When θ=45° and L=28 m, the maximum distortion approaches 1.5. On the other hand, such dependence proved the 2.5D data acquisition layout reasonable to reflect the uneven trace spacing.

Figure  2.  Trace spacing distortion rate versus observation angle.

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Dispersion Data Extracted by F-K Method

There are several methods dominating the construction of dispersion data. Spectral Analysis of Surface Waves (Sánchez-Salinero et al., 1987; Nazarian et al., 1983; Stokoe and Nazarian, 1983) is based on the data of two receivers to calculate the phase velocity of surface waves. It has high horizontal resolution but low ratio of S/N. Other methods are based on multichannel seismic data, such as the frequencywavenumber (F-K) method (Ashiya et al., 1999; Matsushima and Okada, 1990; Horike, 1985; Asten and Henstridge, 1984), Multichannel Analysis of Surface Waves (MASW, Miller et al., 1999; Park et al., 1999; Xia et al., 1999), the p-τ method (McMechan and Yedlin, 1981; McMechan and Ottolini, 1980), the recently developed slant staking (Xia et al., 2007) and linear radon transform (Luo et al., 2009). These methods have high S/N ratio, but low horizontal sensitivity due to the assumption of homogenous media within the spread. 2D geometry is widely used in data acquisition (Xia et al., 2006; Zhang et al., 2004). In the following study, F-K method is selected to discuss the uneven trace spacing issue for its convenience in mathematical terms.

F-K method

For a 2D Rayleigh wavefield f(t, x), its 2D Fourier transform can be expressed by (Xiong and Fang, 2002)

For a given time-domain discrete seismic signal f(mΔt, nΔx), the corresponding frequency-domain signal can be expressed in Fourier transform

where m, j=0, 1, … M–1; n=0, 1, … N–1, and M is the sampling point of each channel; N is the channel number; Δt and Δx are the time sampling interval and spatial sampling interval, respectively. Δf and Δk are the frequency sampling interval and wavenumber sampling interval, respectively. Based on the F-K transformation of multi-channel surface wave, the stacked strong energy bands represent the dispersion velocities of Rayleigh wave in the frequency-velocity space (Zhang et al., 2004; Park et al., 1999).

Raw data and dispersion features

The raw data were obtained (shown in Fig. 3) at sampling rate 1.0 ms and 2 048 sampling points for each trace. The horizontal coordination is the sourcereceiver distance of each trace. For different observation angle and radius, the same geophone has a different source-receiver distance respectively. Based on the seismic data, the Rayleigh wavefields can be used to extract the dispersion. The software system Geogiga Seismic Proc is used to process the data. The obtained Rayleigh wave dispersion images in the F-V space are shown correspondingly in Fig. 4.

Figure  3.  Raw data obtained at different L and θ. (a) L=38 m; (b) L=33 m; (c) L=28 m.

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Figure  4.  Dispersion images obtained at different L and θ. (a) L=38 m; (b) L=33 m; (c) L=28 m.

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The dispersion images have different patterns at different radius L and observation angle θ. At small observations, such as θ≤30°, the fundamental dispersion curves are clearly demonstrated in high energy bands. Some strong energy bands as pseudo hi-mode curves appeared because these bands deviated the true dispersion curves at big observation angles.

Figures 5–6 show the fundamental dispersion curves at different radiuses and different observation angles. As shown in Fig. 5, the fundamental dispersion curves deformed at non-zero observation angles. Figure 6 shows the dispersion curves obtained at three radiuses. For θ=0° and θ=15°, the three curves matched well at most frequencies except for the extremely low frequencies. Such mismatch at extremely low frequencies is mainly caused by near-field effects (Bodet et al., 2009; Yoon and Rix, 2006, 2004; Lin and Chang, 2004). Their composed dispersion curve can be understood as the ground truth. However, for larger observation angles, the high-energy bands behave "jumping", making it difficult to extract the dispersion curves from the dispersion images.

Figure  5.  Dispersion curves obtained with fixed L. (a) L=38 m; (b) L=33 m; (c) L=28 m.

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Figure  6.  Dispersion curves obtained with fixed azimuthal angle. (a) θ=0°; (b) θ=15°; (c) θ=30°; (d) θ=45°; (e) θ=60°; (f) θ=75°; (g) θ=90°.

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DISCUSSION

Referring to the reason why uneven trace spacing causes dispersion curve distortion, we can check the mechanism based on F-K method. For the multichannel Rayleigh wave, the stacking wavefield in frequency domain is given by

where M is the sampling number of each channel in the time domain; and N is the channel number of the layout. A uniform geophone array ensures that the wavefront channel distance is identical. In this case, equation (6) can be simplified further by deleting factor (Δre^ikr₁(θ))

where r_n(θ)=_r1(θ)+Δr_n(θ)=r₁(θ)+(n–1) Δlcosθ.

For the extreme case, when θ=0°, Δr=Δlcosθ= const and Δr_m becomes a constant, i.e., r_m=r₁+(m–1)Δl. When θ=90°, r_m=r₁+(m–1)Δl, the k_θ→0 and v_θ becomes unstable.

For the case of 0° < v_θ < 90°, Fig. 7 shows the near-field spread, the wavefront has a different angle to the spread from trace to trace. The surface wave of limited channels in the frequency-wavenumber space can be expressed as

Figure  7.  The angles of wavefront to the spread changes from trace to trace.

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where

Obviously, both r_i and Δr_i vary with channel number, the spatial sampling is nonuniform. The phase difference, Δϕ=k_θΔr_i, is variable, resulting in the wavefield stacking out-phase. Equation (8) is a Nonuniform Discrete Fourier Transform (NDFT). Δr_i is the critical factor varying in space. Let average wavefront channel distance in terms

where distance difference δr_j may be negative or positive values. The F-K integral equation (8) becomes

These strong stacking energy bands are no longer able to represent the real dispersion phase velocities, because such unevenly spaced data may cause the true wavefield to be deformed. When δr_i varies slightly, the strong stacking energy bands approach the real dispersion curves. How to overcome the effects of uneven trace spacing is out of the range of the present study, but is significant to the construction of dispersion data in nature.

In order to discern whether the distortion of the dispersion curve is caused by underground anisotropy, two perpendicular 2D seismic sections were acquainted. One is along the N-S direction; the other is along the W-E direction. Figures 8–10 demonstrate the raw data and dispersion data at the two perpendicular sections. The dispersion curves of the two sections are well matched. It means that the underground medium is homogeneous. Therefore, the deformations of the dispersion curves in Figs. 5–6 are mainly caused by uneven trace spacing.

Figure  8.  Raw data acquainted at two perpendicular sections.

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Figure  9.  Dispersion images.

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Figure  10.  Dispersion curves of the two perpendicular sections.

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CONCLUSIONS

Based on the discussions above, some conclusions can be given.

(1) The uneven trace spacing has significant influence on the Rayleigh wave dispersion curves. By using a 2.5D acquisition system, the observation angle determines the nonuniformity of the trace spacing.

(2) When the observation angle is less than 30°, the influence is slight and the dispersion data are accepted, but when the angle is larger than 45°, the dispersion data may be significantly deformed.

(3) For the observation angle θ=0°, the cutoff frequency of the fundamental mode dispersion curve seems dependent on the spreading length. At larger observation angles, the dispersion spectra may be contaminated with pseudo hi-mode dispersion curves.