FEASIBILITY OF DETECTING VOIDS WITH RAYLEIGH-WAVE DIFFRACTION

Jianghai Xia, Kansas Geological Survey, The University of Kansas, Lawrence, Kansas

Jonathan E. Nyquist, Temple University, Philadelphia, Pennsylvania Yixian Xu, Open Laboratory of Engineering Geophysics of China Department of Land and

Resources, China University of Geosciences, Wuhan, China Mary J. S. Roth, Lafayette College, Easton, Pennsylvania

Abstract

Void detection is challenging due to the complexity of near-surface materials and the limited resolution of geophysical methods. Although multichannel, high-frequency, surface-wave techniques can provide reliable S-wave velocities in different geological settings, they are not suitable for detecting voids directly based on anomalies of the S-wave velocity because of limitations on the resolution of S-wave velocity profiles inverted from surface-wave phase velocities. Therefore, we studied the feasibility of directly detecting voids with surface-wave dif-fractions. Based on the properties of surface waves, we have derived a Rayleigh-wave diffraction traveltime equation. We also have solved the equation for the depth to the top of a void and an average velocity of Rayleigh waves. Using these equations, the depth to the top of a void can be determined based on traveltime data from a diffraction curve. In practice, only two diffraction times are necessary to define the depth to the top of a void and the average Rayleigh-wave velocity that generates the diffraction curve. We used four two-dimensional square voids to demonstrate the feasibility of detecting a void with Rayleigh-wave diffractions: a 2 m by 2 m with a depth to the top of void of 2 m, 4 m by 4 m with a depth to the top of the void of 7 m, and 6 m by 6 m with depths to the top of the void 12 m and 17 m. Rayleigh-wave diffractions were recognizable for all these models after FK filtering was applied to the synthetic data. The Rayleigh-wave diffraction traveltime equation was verified by the modeled data. A real-world example is presented to show how to utilize the derived equation of surface-wave diffractions.

Introduction

Elastic properties of near-surface materials and their effects on seismic wave propagation

are of fundamental interest in ground-water, engineering, and environmental studies. Shear (S)-wave velocities can be derived from inverting dispersive phase velocities of the surface (Ray-leigh and/or Love) waves (e.g., Dorman and Ewing, 1962; Aki and Richards, 1980, p. 664). Spectral Analysis of Surface Waves (SASW) (Stokoe and Nazarian, 1983; Stokoe et al., 1989) analyzes the dispersion curve of the ground roll to produce near-surface S-wave velocity profiles. The other method developed in the last 10 years utilizes a multichannel recording system to estimate near-surface S-wave velocity from high-frequency (≥ 2 Hz) Rayleigh waves (Multi-channel Analysis of Surface Waves—MASW, Park et al., 1999; Xia et al., 1999). The MASW method possesses the advantage of easily recognizing surface waves, effectively eliminating body-wave energy, and accurately defining dispersion energy. Errors associated with S-wave velocities obtained by MASW method are 15% or less and random after comparison with borehole direct measurements (Xia et al., 2000a and 2002a). If higher mode data are available,

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the accuracy of inverted S-wave velocity can be significantly improved (Xia et al., 2000b and 2003; Beaty et al., 2002). A summery of the MASW method can be found in Xia et al. (2004a).

More and more publications have appeared on utilizing surface waves in defining near-

surface S-wave velocities (e.g., Rix and Leipski, 1991; Luke and Stokoe, 1998; Xia et al., 2002b, in press (a), and in review; Beaty and Schmitt, 2003; Yilmaz and Eser, 2002; Forbriger, 2003a and 2003b; Hayashi and Hikima, 2003; Lin and Chang, 2004; Lin et al., 2004; Ivanov et al., 2005; Xu et al., in press), attenuation properties (Xia et al., 2002c), and combined analysis of active and passive surface waves (Park et al., 2005). In environmental and geotechnical studies, the MASW method was employed to generate 2-D shear-wave velocity fields calculated from inversion of Rayleigh-wave phase velocities and was successfully used to define bedrock interfaces and near-surface geological structures from 2 to 50 meters (Xia et al., 1998; Miller et al., 1999; Ryden et al., 2004; Tian et al., 2003a and 2003b; Kaufmann et al., 2005) and to determine a collapse feature in an extremely noisy environment (Xia et al., 2004b). Research on surface-wave techniques was extended to a non-layered Earth model (Gibson-half space) by Xia et al. (in press (b)). This model is useful for assessing man-made structures, especially in case of few data points available.

Xia et al. (2005) discussed the resolving power of MASW techniques and improving

resolution of S-wave velocity results. Based on their work, we realized that inverted S-wave velocity profiles might not possess high enough resolution to locate voids even though the velocity contrast is high. This is mainly because voids are small relative to the layered half space model, which is the model currently used in surface-wave analysis. The inverted S-wave profiles are averaged S-wave velocities within a geophone spread. The unblurring process (Xia et al., 2005) is still not able to increase the resolution of S-wave velocities to the level where we can confidently identify voids on an S-wave velocity section.

Several researchers published results on void detection with surface waves. Inversion of

scattered surface waves was discussed by Riyanti et al. (2005). They concluded that scattered surface waves can be used for near-surface characterization using inversion methods. Analysis attenuation of Rayleigh waves (AARW) for detecting voids was presented (Nasseri-Moghaddam et al., 2005). The numerical results show the promise of AARW for detecting voids in the real world. Gelis et al. (2005) reported numerical modeling results. A cavity with a rectangular section generates more diffraction than a cavity with a circular section. They also found that a low-velocity zone around and above a cavity might increase Rayleigh-wave attenuation and could possibly mask the cavity signature.

In this paper, we present a simple method in the time-space domain based a traveltime

equation of surface-wave diffractions. It can be used to detect a void directly from a shot gather. First, we derived the traveltime equation of Rayleigh-wave diffractions based on the properties of surface waves. Second, we solved the equation for the depth to the top of a void and an average velocity of Rayleigh waves. With these equations, a depth to the top of a void can be determined based on traveltime data from a diffraction curve. In practice, only two diffraction times are necessary to define the depth to the top of a void and the average Rayleigh-wave velocity that generates a diffraction curve. To increase inversion robustness and improve the accuracy of calculated parameters, a least-square method of fitting multi-points on the traveltime curve can be used.

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The Diffraction Equation of Surface Waves

Suppose surface waves with a dominant phase velocity v travel along the ground surface (segment 1 in Figure 1); then a diffraction of surface waves with the same phase velocity due to a void can occur at the top left corner of the void (segment 2 in Figure 1). We can write the traveltime equation for the diffraction as

( )[ ]2/1221hxd

vtx ++= , (1)

where tx is the diffraction arriving time at the offset of x (the horizontal distance between the apex of the hyperbola corresponding to the edge of the void and a receiver), v is a phase velocity of the diffraction, d is the horizontal distance from the source to the apex of the hyperbola, and h is the depth to the top of the void (the diffraction point).

Figure 1. Geometry of surface-wave diffractions.

When x = 0, we obtain the arrival time t0 at the apex of the hyperbola:

( )hdv

t +=1

0 . (2)

Solving the diffraction equation for h and v, we substitute Equation (2) into Equation (1) to eliminate v and obtain a quadratic equation ah2 + bh + c = 0, (3)

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where

a = 12

0

−⎟⎟⎠

⎞⎜⎜⎝

⎛tt x , b = d

tt

tt xx

⎟⎟⎠

⎞⎜⎜⎝

⎛−12

00

, and c = 22

0

2

0

12 xdtt

tt xx −

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛.

The solution for the quadratic equation (2) is

h = a

acbb2

42 −+− , and (4)

the phase velocity can be found by

( )hdt

v +=0

1 . (5)

The solutions are based on only two traveltimes extracted from the diffraction: one at the

apex of the hyperbola and the other at any other place along the hyperbola. In the following discussion, we will apply Equations (4) and (5) to synthetic data and real-world data to show the feasibility of using surface-wave diffractions to detect a void. Obviously, we can improve accuracy of calculated h and v and robustness by least-square fitting additional traveltimes taken from the hyperbola.

Modeling Results

We used four two-dimensional square voids: a 2 m × 2 m with a depth to the top of the void of 2 m, a 4 m × 4 m with a depth to the top of the void of 7 m, and 6 m × 6 m with a depth to the top of the void of 12 m and 17 m, to demonstrate feasibility of directly detecting a void with Rayleigh waves and to verify the traveltime equation (1). The voids are within a homogenous half space. We used an algorithm developed by Xu et al. (2005) to generate synthetic seismographs.

P- and S-wave velocities of the homogenous half-space are 1000 m/s and 200 m/s, respectively. These values define a Rayleigh-wave velocity about 184 m/s (Sheriff and Geldart, 1985, p. 49). To avoid numerical difficulties, we used P- and S- wave velocities of the void at 340 m/s and 17 m/s, respectively. Densities of the half space and the void are 2000 kg/m3 and 10 kg/m3, respectively. With a cell size of 1 m by 1 m, a 200 m × 200 m subspace was divided into 200 × 200 nodes with 30 nodes in the transition zones along the left and right sides of the model and 30 nodes in the transition zone along the bottom of the model.

Synthetic seismographs simulated the following field layout. Sixty vertical component

geophones were located in the middle of the subspace from 71 to 130 m with an interval of 1 m. The void was at the center of the geophone spread with different depths. A source was at 70 m.

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Figure 2 shows a synthetic shot gather for void Model 1 with a source to diffraction point dis-tance (d) of 29 m. Diffractions (Figure 2a) can be identified. Notice that the phase of diffract-tions possesses a 180-degree shift. Diffractions were relatively strong (Figure 2b) after FK filtering was applied to the synthetic data to remove linear events with veloci-ties of 180 m/s to 200 m/s that mainly filtered surface waves with a phase velocity of 184 m/s. Based on their frequency and velocity, we are certain that diffractions are surface-wave diffractions due to the void at a depth of 2 m. We used Equation (1) with the phase velocity of 184 m/s and h = 2 m to model a traveltime curve (a solid hyperbolic line in Figure 2b). The traveltime curve matches diffrac-tions perfectly. We picked t0 and tx (x = 18 m) as 168.5 ms and 256.0 ms from Figure 2b, respectively. Using Equations (4) and (5), we obtained estimates of the depth to the void (2.01 m) and the phase velocity of the half space (184.0 m/s) that are very close to our model parameters (h = 2 m and v = 184 m/s).

Figure 2. a) A synthetic shot gather of a 2 m × 2 m voidat the depth of 2 m in a homogeneous half space. b) FKfiltered data with a diffraction traveltime curve.

Investigation depth using

surface-wave techniques is limited by wavelengths of surface waves. The longest wavelength of the surface waves generated by our source wavelet was around 35 m. Figure 3a shows a shot gather for the void at a depth of 7 m. As expected, when the depth of a void increases the energy of diffractions decreases signifi-cantly. Certain processing techniques must be applied to enhance the relative amplitude of diffractions before using Equation (1) to fit diffractions. We applied the same FK filter used in the case shown in Figure 2 to the synthetic data (Figure 3a). The FK filtered shot gather is shown in Figure 3b. Diffraction energy highlighted by a hyperbola was much stronger than that in raw data (Figure 3a). As depth of a void increases from 2 m to 7 m, the apex of the hyperbola (Figure

a

b

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3b) moves down from 168.5 ms to 190.2 ms. The horizontal distance from a source to the edge of the void (d) was 28 m in Figure 3a. 21.7 ms (= 190.2 ms – 168.5 ms) is exactly the time for surface waves traveling 4 m with a phase velocity of 184 m/s. We picked t0 and tx (x = 18 m) as 190.2 ms and 257.0 ms from Figure 3b, respec-tively. Using Equations (4) and (5), we obtain estimations of the depth to the void (7.02 m) and the phase velocity of the half space (184.1 m/s).

Figure 3. a) A synthetic shot gather of a 4 m × 4 m voidat the depth of 7 m in a homogeneous half space. b) FKfiltered data with a diffraction traveltime curve.

We modeled deeper voids,

12 m and 17 m, and increased the size of the void to 6 m × 6m. The horizontal distance from the source to the top left corner is 27 m. Figures 4 and 5 show our results. Synthetic shot gathers were shown in Figures 4a and 5a for the void at the depth of 12 m and 17 m, respectively. After applying the FK filter to the synthetic data (Figures 4a and 5a), surface-wave diffrac-tions can be identified without any difficulty. We picked t0 and tx (x = 18 m) as 212.0 ms and 264.3 ms from Figure 4b and t0 and tx (x = 18 m) as 239.1 ms and 281.3 ms from Figure 5b, respectively. Using Equations (4) and (5), we obtain accurate estimations of the depth to the void and the phase velocity of the half space in both cases.

b

a

Energy of diffractions on

the right side of a void (a source on the left side of a void) is significantly weaker than direct surface waves, so the direct surface waves completely mask the diffractions. It is difficult to enhance surface-wave diffractions on that side because they are in the “same slope.” Modeling results suggest that in practice backscattering (diffractions on the same side as the source) is the signal to use for void detection.

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Our modeling results are limited to a two-dimensional space and the specific case of a void that possesses a square section in a homogeneous half space. The real world situations are more complex than this setting. As in numerical modeling results shown by Gelis et al. (2005), a cavity with a circular section may produce much weaker surface-wave diffraction energy, which makes detecting a void difficult with surface-wave diffractions. Nevertheless, the surface-wave traveltime equation (1) and our modeling results provide hope of directly detecting a void simply with a shot gather and some filtering.

a

b a

Figure 5. a) A synthetic shot gather of a 6 m × 6 m void at the depth of 17 m in a homogeneoushalf space. b) FK filtered data with a diffraction traveltime curve.

Figure 4. a) A synthetic shot gather of a 6 m × 6 m void at the depth of 12 m in a homogeneoushalf space. b) FK filtered data with a diffraction traveltime curve.

b

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A Real-world Example

An ongoing research program at Metzgar Fields, an athletic facility owned by Lafayette College in Easton, Pennsylvania has the primary objective of improving site investigation methods for karst features using geophysical methods (Jenkins and Nyquist, 1999; Mackey et al., 1999; Maule et al., 2000; Roth et al., 2000, Roth et al., 2002, Nyquist and Roth, 2003; Manney et al., 2005; Nyquist et al., 2005; Nyquist and Roth, 2005). Meztgar Field is underlain by the Epler formation, an Ordovician age limestone prone to dissolution. Multi-electrode resistivity testing completed in 1999 located a significant anomaly at the test site and subsequent drilling confirmed the existence of an air-filled void under approximately seven meters of bedrock. During the summer of 2003, we triangulated the dimensions of this cave using laser pointers and a downhole camera originally designed for search and rescue operations (Roth et al., 2004), determining that the cave is roughly 4 m long, 1.5 m wide, and 2 m high.

Figure 6 shows electrical resistivity inversion results for a line collected perpendicular to

geologic strike and centered over the void. The data were acquired using an AGI Supersting, and a dipole-dipole array comprised of 28 electrodes spaced 3 m apart. The limestone bedrock surface under Metzgar Field is highly irregular, with narrow bedrock ridges running parallel to strike separated by soil-filled depressions. The white line in Figure 6 shows the bedrock surface based on auger refusal depths. The resistivity data clearly respond to the strong soil/limestone resistivity contrast, but the cavity itself is hard to detect, appearing only as an anomaly when lines perpendicular and parallel to strike are compared for anisotropy (Manney et al, 2005).

Figure 6. Resistivity results for a line perpendicular to geologic strike centered over the cavity.The white line shows auger refusal depth.

Surface–wave data were acquired with a 48-channel system and 4.5 Hz geophones spaced 0.5 m apart. The geophones were deployed on the SE and NW line with Channel 24 over the top of the void (Figure 6). The data from three blows of a 6-kg sledgehammer on a 15-cm square aluminum striker plate were vertically stacked. The nearest source-receiver offset was 1 m. To enhance surface-wave diffraction, we applied an FK filter from –800 m/s to -150 m/s to data (Figure 7a). FK filtered data (Figure 7b) possess no direct body and surface waves, and backscattering due to surface waves was booted notably.

The apex of the hyperbolic event (Figure 7b) is at x = 0 (trace 24) and t0 = 0.0547 s. We

also picked tx = 0.0735 s at x = 11 m (trace 46) on the hyperbola. The horizontal distance (d)

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from the source to the apex of the hyperbola is 13 m. Based on Equations (4) and (5), the depth to the top of void and the velocity of this hyperbola are 6.0 m and 347 m/s, respectively.

b a

Figure 7. a) A shot gather on the top of a known void at trace 24 with a diffraction traveltimecurve. b) FK filtered data with a diffraction traveltime curve.

Conclusions We studied feasibility of detecting a void with surface-wave diffractions by 2-D surface-

wave modeling. We derived a traveltime equation of surface-wave diffractions based on properties of surface waves. We solved this equation for a phase velocity and depth to a void. In practice, only two diffraction times were necessary to define the depth to the depth and the phase velocity. Both synthetic and real-world examples demonstrated that it is feasible to directly detect a void in a shot gather. Even though our studies were restricted to a 2-D model, and do not consider attenuation of surface waves, the results of this paper are directly applicable to the detection of 2-D structures such as tunnels, and, as our case study shows, may be applicable to 3-D void detection as well. Because surface-wave diffractions are relatively weak, especially in a high attenuation medium, a real challenge in detecting a deeper void is to generate sufficient surface-wave energy with a long wavelength and a high signal-to-noise ratio.

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Tian, G., Steeples, D.W., Xia, J., and Spikes, K.T., 2003a, Useful resorting in surface wave method with the autojuggie: Geophysics, 68(6), 1906-1908.

Tian, G., Steeples, D.W., Xia, J., Miller, R.D., Spikes, K.T., and Ralston, M.D., 2003b, Multichannel analysis of surface wave method with the autojuggie: Soil Dynamics and Earthquake Engineering, 23(3), 243-247.

Xia, J., Miller, R.D., and Park, C.B., 1998, Construction of vertical section of near-surface shear-wave velocity from ground roll: Technical Program, The Society of Exploration Geophysicists and The Chinese Petroleum Society Beijing 98 International Conference, 29-33.

Xia, J., Miller, R.D., and Park, C.B., 1999, Estimation of near-surface shear-wave velocity by inversion of Rayleigh wave: Geophysics, 64(3), 691-700.

Xia, J., Miller, R.D., Park, C.B., Hunter, J.A., and Harris, J.B., 2000a, Comparing shear-wave velocity profiles from MASW with borehole measurements in unconsolidated sediments, Fraser River Delta, B.C., Canada: Journal of Environmental and Engineering Geophysics, 5(3), 1-13.

Xia, J., Miller, R.D., and Park, C.B., 2000b, Advantage of calculating shear-wave velocity from surface waves with higher modes: Technical Program with Biographies, SEG, 70th Annual Meeting, Calgary, Canada, 1295-1298.

Xia, J., Miller, R.D., Park, C.B., Hunter, J.A., Harris, J.B., and Ivanov, J., 2002a, Comparing shear-wave velocity profiles from multichannel analysis of surface wave with borehole measurements: Soil Dynamics and Earthquake Engineering, 22(3), 181-190.

Xia, J., Miller, R.D., Park, C.B., Wightman, E., and Nigbor, R., 2002b, A pitfall in shallow shear-wave refraction surveying: Journal of Applied Geophysics, 51(1), 1-9.

Xia, J., Miller, R.D., Park, C.B., and Tian, G., 2002c, Determining Q of near-surface materials from Rayleigh waves: Journal of Applied Geophysics, 51(2-4), 121-129.

Xia, J., Miller, R.D., Park, C.B., and Tian, G., 2003, Inversion of high frequency surface waves with fundamental and higher modes: Journal of Applied Geophysics, 52(1), 45-57.

Xia, J., Miller, R.D., Park, C.B., Ivanov, J., Tian, G., and Chen, C., 2004a, Utilization of high-frequency Rayleigh waves in near-surface geophysics: The Leading Edge, 23(8), 753-759

Xia, J., Chen, C., Li, P.H., and Lewis, M.J., 2004b, Delineation of a collapse feature in a noisy environment using a multichannel surface wave technique: Geotechnique, 54(1), 17-27.

Xia, J., Chen, C., Tian, G., Miller, R.D., and Ivanov, J., 2005, Resolution of high-frequency Rayleigh wave data: Journal of Environmental and Engineering Geophysics, Special issue on Seismic Surface Waves, 10(2), 99-110.

Xia, J., Xu, Y., Chen, C., Kaufmann, R.D., and Luo, Y., in press (a), Simple equations guide high-frequency surface-wave investigation techniques: Soil Dynamics and Earthquake Engineering.

Xia, J., Xu, Y., Miller, R.D., and Chen, C., in press (b), Estimation of elastic moduli in a compressible Gibson half-space by inverting Rayleigh wave phase velocity: Surveys in Geophysics.

Xia, J., Xu, Y., and Miller, R.D, in review, Generating image of dispersive energy by slant stacking: Pure and Applied Geophysics.

Xu, Y., Xia, J., and Miller, R.D., 2005, Finite-difference modeling of high-frequency Rayleigh waves: Technical Program with Biographies, SEG, 75th Annual Meeting, Houston, TX, 1057-1060.

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Xu, Y., Xia, J., and Miller, R.D., in press, Quantitative estimation of minimum offset for multichannel surface-wave survey with actively exciting source: Journal of Applied Geophysics.

Yilmaz, O., and Eser, M., 2002, A unified workflow for engineering seismology: Technical Program with Biographies, SEG, 72nd Annual Meeting, Salt Lake City, UT, 1496-1499.

Acknowledgments

The authors appreciate efforts of geophysics students at Temple University and Temple

Grant-in-Aid support in acquiring seismic data used in the paper. The authors thank Marla Adkins-Heljeson and Mary Brohammer of the Kansas Geological Survey for editing the manuscript.

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