CONFIGURATION OF NEAR-SURFACE SHEAR-WAVE VELOCITY BY INVERTING SURFACE WAVE

Jianghai Xia, Richard D. Miller and Choon B. Park Kansas Geological Survey, The University of Kansas

1930 Constant Ave., Lawrence, Kansas 66047

ABSTRACT

The shear (S)-wave velocity of near-surface materials (such as soil, rocks, and pavement) and its effect on seismic wave propagation are of fundamental interest in many groundwater, engineering, and environmental studies. Ground roll is a Rayleigh-type surface wave that travels along or near the surface of the ground. Rayleigh wave phase velocity of a layered earth model is a function of frequency and four earth parameters: S- wave velocity, P-wave velocity, density, and thickness of layers. Analysis of the Jacobian matrix in a high frequency range (5-30 Hz) provides a measure of sensitivity of disper- sion curves to earth model parameters. S-wave velocities are the dominant influence of the four earth model parameters. With the lack of sensitivity of the Rayleigh wave to P- wave velocities and densities, estimations of near-surface S-wave velocities can be made from high frequency Rayleigh wave for a layered earth model. An iterative technique applied to a weighted equation proved very effective when using the Levenberg- Marquardt method and singular value decomposition techniques. The convergence of the weighted damping solution is guaranteed through selection of the damping factor of the Levenberg-Marquardt method. Three real world examples are presented in this paper. The first and second examples demonstrate the sensitivity of inverted S-wave velocities to their initial values, the stability of the inversion procedure, and/or accuracy of the in- verted results. The third example illustrates the combination of a standard CDP (common depth point) roll-along acquisition format with inverting surface waves one shot gather by one shot gather to generate a cross section of S-wave velocity. The inverted S-wave velocities are confirmed by borehole data.

INTRODUCTION

Elastic properties of near-surface materials and their effects on seismic wave propagation are of fundamental interest in groundwater, engineering, and environmental studies. S-wave velocity is one of the key parameters in construction engineering. As an example, Imai and Tonouchi (1982) studied P- and S-wave velocities in an embankment, and also in alluvial, diluvial, and Tertiary layers, showing that S-wave velocities in such deposits correspond to the N-value (Craig, 1992), an index value of formation hardness used in soil mechanics and foundation engineering.

Surface waves are guided and dispersive. Rayleigh (1885) waves are surface waves that travel along a “free” surface, such as the earth-air interface. Rayleigh waves are the result of interfering P and S, waves. Particle motion of the fundamental mode of Rayleigh waves moving from left to right is elliptical in a counter-clockwise (retrograde) direction. The motion is constrained to the vertical plane that is consistent with the direc- tion of wave propagation. Longer wavelengths penetrate deeper than shorter wavelengths for a given mode, generally exhibit greater phase velocities, and are more sensitive to the

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elastic properties of the deeper layers (p. 30, Babuska and Cara, 1991). Shorter wave- lengths are sensitive to the physical properties of surficial layers. For this reason, a par- ticular mode of surface wave will possess a unique phase velocity for each unique wave- length, hence, leading to the dispersion of the seismic signal.

Ground roll is a particular type of Rayleigh wave that travels along or near the ground surface and is usually characterized by relatively low velocity, low frequency, and high amplitude (p. 143, Sheriff, 1991). Stokoe and Nazarian (1983) and Nazarian et al. (1983) presented a surface-wave method, called Spectral Analysis of Surface Waves (SASW), that analyzes the dispersion curve of ground roll to produce near-surface S- wave velocity profiles. SASW has been widely applied to many engineering projects (e.g., Sanchez-Salinero et al., 1987; Sheu et al., 1988; Stokoe et al., 1989; Gucunski and Woods, 1991; Hiltunen, 1991; Stokoe et al., 1994).

Inversion of dispersion curves to estimate S-wave velocities deep within the Earth was first attempted by Dorman and Ewing (1962). Song et al. (1989) related the sensi- tivity of model parameters to several key earth properties by modeling and presented two real examples using surface waves to obtain S-wave velocities. Turner (1990) examined the feasibility of inverting surface waves (Rayleigh and Love) to estimate S-wave and P- wave velocities. Dispersion curves are inverted using least-squares techniques in SASW methods (Stokoe and Nazarian, 1983; Nazarian et al., 1983).

Since 1995, the Kansas Geological Survey have been conducting a three-phase research project to estimate near-surface S-wave velocity from ground roll: acquisition of high frequency (2 5 Hz) broad band ground roll, creation of efficient and accurate algo- rithms organized in a basic data processing sequence designed to extract Rayleigh wave dispersion curves from ground roll, and development of stable and efficient inversion algorithms to obtain near-surface S-wave velocity profiles. Research results of the first two phases can be found in Park et al. (1996) and Park et al. (in review). This paper will discuss research results related to phase three (Xia et al., in review).

METHOD

Consideration on Numerical Calculations

For a layered earth model (Figure l), Rayleigh wave dispersion curves can be calculated by Knopoff s method (Schwab and Knopoff, 1972). Accuracy of the partial derivatives is key in determining modifications to the earth model parameters and dra- matically affects convergence of the inverse procedure (Xia, 1986). The practical way to calculate the partial derivatives of the Rayleigh wave dispersion function is by evaluating finite-difference values because it is in an implicit form. In this study, Ridder’s method of polynomial extrapolation (p. 186, Press et al., 1992) is used to calculate the partial derivative or Jacobian matrix during Table 1. An earth model parameters.

R 1 h fm‘l the inversion. Based on the orthog- onality between the partial derivative vector with respect to density and the vector of density, the accuracy of numerical derivatives can be evaluated using an earth model (Table 1). Numerical results indicate that the

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average relative error in the estimated elements of the Jacobian matrix is approximately 0.1 percent with at least three significant figures. Our experiences show that estimating the Jacobian matrix in a high frequency range (> 5 Hz) by Ridder’s method is stable. Most importantly, Ridder’s method provides an efficient means to calculate the Jacobian matrix of the Rayleigh wave phase velocity for the layered earth model.

Analysis of Sensitivity of Earth Model Parameters

Rayleigh wave phase velocity (dispersion data) is the function of four parameters: S-wave velocity, P-wave velocity, density, and layer thickness. Each parameter contrib- utes to the dispersion curve in a unique way. A parameter can be negated from the inverse procedure if contributions to the dispersion curve from that parameter are relatively small in a certain frequency range. Contributions to the Rayleigh wave phase velocity in the high frequency range (2 5 Hz) from each parameter are evaluated to determine which parameter can be inverted with reasonable accuracy (Figure 2).

Based on the analysis of the Jacobian matrix of the earth model (Table l), we may conclude that the ratio of percentage change in the phase velocities to percentage change in the S-wave velocity, thickness of layer, density, or P-wave velocity are 1.56, 0.64, 0.4, or 0.13, respectively. The S-wave velocity is the dominant parameter influencing changes in Rayleigh wave phase velocity for this particular model in the high frequency range (> 5 Hz), which is therefore the fundamental basis for the inversion of S-wave velocity from Rayleigh wave phase velocity. Analysis presented in this section is based on a single model (Table l), however, numerical results from more than a hundred modeling testing support these conclusions.

Free surface 900 T

VSl vPl Pl h,

V s2 % P2 h2

vsi vpi Pi hi

V sn V Pn Pll infinite

Fig. 1. A layered earth model with parameters of shear- wave velocity (v,), compressional wave velocity (vr), density ( p), and thickness (h).

; 800

t

l l . . l

. Changes in S-wave v

‘00 -- A Changes in lhickness

0, I

5 7.5 10 12.5 15 175 20 22.5 25 27.5 30 325

Frequency (Hz)

Fig. 2. Contributions to Rayleigh wave phase velocity by 25 percent changes in each earth model parameter (Table 1). The solid line is Rayleigh wave phase velocity due to the earth model listed in Table 1. Squares represent Rayleigh wave phase velocities after 25 percent changes in density, diamonds represent Rayleigh wave phase velocities after 25 percent changes in P-wave velocity, and so on.

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In summary, a 25 percent error in estimated P-wave velocity or rock density results in less than a 10 percent difference between the modeled and actual dispersion curves. Since in the real world, it is relatively easy to obtain density information with accuracy greater than 25 percent (p. 173, Carmichael, 1989), densities can be assumed known in our inverse procedure. It is also reasonable to suggest relative variations in P- wave velocities can be estimated within 25 percent of actual, and therefore P-wave velocities will be also be assumed known. Inverting Rayleigh wave phase velocity for layer thickness is more feasible than for P-wave velocity or density because the sensi- tivity indicator is greater for thickness variation than for P-wave velocity or density. However, because the subsurface can always be subdivided into a reasonable number of layers, each possessing an approximate constant S-wave velocity, thickness can be eliminated as a variable in our inverse procedure. Only S-wave velocities are left as un- knowns in our inverse procedure. We can reduce the number of unknowns from 4pt -1 (PI is number of layers) to 12 with these assumptions. The fewer unknowns in an inverse pro- cedure, the more efficient and stable the process, and the more reliable the results.

Inversion Algorithm

The basis was developed for suggesting S-wave velocities fundamentally control changes in Rayleigh wave phase velocities for the layered earth model in the previous section. Therefore, S-wave velocities can be inverted adequately from Rayleigh wave phase velocities.

Let S-wave velocities (earth model parameters) be the elements of a vector x of length yt, x = [vsl, vs2, vs3, . . . . v~,]‘. Similarly, let the measurements (data) of Rayleigh wave phase velocities at m different frequencies be the elements of a vector b of length m, b = [b,, b,, b,, . . . . b,lT. After linearization of the dispersion function, an objective function is defined as

where Ab [= b - cR(x,)] is the difference between measured data and model response to the initial estimation, cR(x,) is the model response to the initial S-wave velocity estimates x,, which are defined by phase velocities, see Xia et al. (in review) for details; Ax is a modification of the initial estimation; J is the Jacobian matrix with m rows and y1

columns (m > n) with the elements being the first order partial derivatives of cR with respect to S-wave velocities, II II 2 is the &norm length of a vector, oz is the damping

factor, and W is a weighting matrix, which can be determined by 1) differences in Ray-

leigh wave phase velocities with respect to frequency, 2) signal to noise (surface wave signal to body wave signal) ratio, or 3) users. We are searching for a solution with mini- mum modification to model parameters so the convergence procedure is stable for each iteration. This does not mean the final model will be closer to the initial model than other optimization techniques such as the Newton method. After several iterations, the sum of the modifications is added to the initial model, making a final model that can be significantly different from the initial model.

Iterative solutions of a weighted damping equation using Levenberg-Marquardt method (L-M) (Marquardt, 1963) provide a stable and fast solution. Marquardt (1963)

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pointed out that the damping factor (a) controls the direction of Ax and the speed of convergence. By adjusting the damping factor, we can improve processing speed and guarantee the stable convergence of the inversion, Employing the SVD technique (Golub and Reinsch, 1970) to minimize the objective function (1) allows us to change the damping factor (a ) without recalculating the inverse of the normal matrix.

REAL WORLD EXAMPLES

Lawrence, Kansas

Surface wave data were acquired during the Winter of 1995 near the Kansas Geological Survey in Lawrence, Kansas, using the MASW acquisition method (Park et al., 1996). An IV1 MiniVib was used as the energy source. Forty groups of 10 Hz geophones were deployed on 1 m intervals with the first group of geophones two meters from a test well. The source was located adjacent to the geophone line relative to the test well with a nearest source offset of 27 m. A 10 second linear up-sweep with frequencies ranging from 10 to 200 Hz was generated for each shot station. The raw field data acquired by the MASW method possess a strong ground roll component (Figure 3). The dispersion curve (Figure 4a) of Rayleigh wave phase mm3 (ml velocities has been extracted from field data (Figure

I 27 0

3) for frequencies ranging from 15 to 80 Hz, using CCSAS processing techniques (Park et al., in review). PI

Three-component borehole data were acquired coincidentally to obtain P-wave and S-wave velocity ua vertical profiles. A cross-correlation technique was m

: used to confidently determine S-wave arrivals on the c

recorded three-component borehole data. Any error Em

on the S-wave velocity profile (the solid line in Figure 4b) is mainly due to the 0.5 ms sampling interval. The overall error in S-wave velocity of the borehole survey is approximately 10%. Fig. 3. Forty groups of 10 Hz geophones were

Inverting the Rayleigh wave phase velocities spread 1 m apart. An IV1 MiniVib was used as

to determine S-wave velocities requires densities the energy source and located at 27 m away from the right side of the geophone spread. Two

and P-wave velocities be defined. Densities were linear events are velocities of dispersive ground

estimated and designated to increase approximately HZ. roll at frequencies approximately 15 Hz and 50

linearly with depth while P-wave velocities were obtained from bore- hole data (Table 2). The initial S-wave model (labeled “initial B” on Figure 4) was created by the inverse program based on equation (2). The rms error between measured data and modeled data dropped from 70 m/s to 30 m/s with two iterations. The inverted S-wave velocity profile is horizontally averaged across the length of the source-geophone spread 10 1 852.274 1 2410.0 1 2.4 1 infinite

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(66 m). Theoretically, considering the averaging, there should be only small differences between the inverted velocity and the borehole-measured velocity. The average relative difference between inverted S-wave velocities and borehole-measured S-wave velocities is 18 percent. If the first layer is excluded, the difference is only 9%.

To analyze the sensitivity of the inverted model to initial values, we manually select initial values for Vs that are uniformly greater than borehole values (Figure 4). “Initial A” and “Initial B” are symmetrical to the borehole values and converge to bore- hole values from two different directions (Figure 4b). Overall accuracy for both inverted models are visually the same.

Fig. 4. Inverse results of Fig. 3. The dispersion curve labeled “measured” (a) is real data extracted from data (Figure 3) by CCSAS techniques (park et al., in review). “Initial B” model (b) was calculated from the “measured” field data in (a). “Borehole” (b) was S-wave velocities derived from the 3-component seismic borehole survey. “Initial A” and “initial B” models (b) are symmetrical to the borehole values. Both initial models converge to the model determined by borehole data. One of every two phase velocities due to the inverted models is shown by diamonds and dots (a).

Vancouver, Canada

The Kansas Geological Survey and the Geological Survey of Canada conducted surface wave technique testing in unconsolidated sediments of the Fraser River Delta, near Vancouver, Canada in Fall of 1998. A thorough study of S-wave velocity in this area has been done (Hunter et al., 1998). Vertical profiles of S-wave velocity based on borehole measurements are available for more than 30 locations. These S-wave velocity profiles provide the ground truth of S-wave velocity in this area. Eight sites were selected based on geographic location, accessibility and availability of boreholes, and the pattern of S-wave velocity from borehole measurements. Multi-channel surface wave data were acquired by 60 (or 48) 4.5 Hz vertical component geophones at eight borehole locations. The seismic source was a weight dropper built by the Exploration Services of the Kansas Geological Survey. Three to ten impacts were vertically stacked at each offset. No acqui- sition filter was applied during data acquisition. The record length is 2048 milliseconds with 1 millisecond sample interval. Overall difference between S-wave velocities from MASW and borehole measurements is about 15%. Figure 5 and Figure 6 show results from two borehole locations.

Joplin, Missouri

A test conducted during the Summer of 1997 included collection of surface wave data in a standard CDP (common depth point) roll-along acquisition format (Mayne,

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1962) similar to conventional petroleum exploration data acquisition. Thirty groups of 10 Hz geophones were spaced 1.2 m apart. The nearest source-receiver offset was 12 m. An IV1 MiniVib was used as the energy source. A linear up-sweep with frequencies ranging from 10 to 200 Hz and lasting 10 seconds was generated for each shot station. About 180 shot gathers were collected on 1.2 m spacing for each line.

Frcqucnc~ (W 0 5 10 15 20 25 30

Fig. 5. Field shot gather (a) with 60 traces at location of borehole FD97-2, Rayleigh wave phase velocities (b) extracted from (a) labeled Measured and from inverted Vs model (c) labeled Final.

Fig. 6. Field shot gather (a) with 60 traces at location of borehole FD92-4, Rayleigh wave phase velocities (II) extracted from (a) labeled Measured and Erom inverted Vs model (c) labeled Final.

The inverse results provided a vertical profile of S-wave velocity vs. depth for each source station. The inverted S-wave velocity profile for each shot gather is the result of horizontally averaging across the length of the source-geophone spread (48 m). Contour drawing software was used to generate two 2-D S-wave velocity maps (Figure 7). Figure 7 shows that the S-wave velocity changes smoothly from one station to the next, suggesting stability in the inversion algorithm and reliability of the inverted results. A landfill area associated with lower S-wave velocity (275 m/s) is located around station 325. A gravel road with a relative higher S-wave velocity (425 m/s) is located at station 340. Depth to the bedrock at the two well locations along the line is consistent with the high gradient portion of the contour plot. Because the lowest frequency used in the test is 10 Hz, the average penetration depth of Rayleigh waves along the survey line is around 15 m. Inverted S-wave velocities in the proximity of station 3 10 suggest a depth to the bedrock of more than 15 m that does not contradict the 21 m depth of the well data. Three other wells at stations 390, 15, and 65 confirmed inverted results if the bedrock corresponds the 480 m/s-contour line.

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S-wave velocity of line 1, Joplin, MO

Well, 21 m to bedrock Well, 12 m to bedrock

3+0 360 360 400 Station number

S-wave velocity of line 2, Joplin, MO

Well, 10.8: to bedrock Well, 15.3 m to bedrock 0

-1 .!3

12

Station number

I 0 15 30 45 60 Meters

m/S

1080

960

720

600

360

240

Fig. 7. S-wave velocity maps inverted from surface wave dispersion data. The vertical scale is exaggerated 1.3 times compared to the horizontal scale. Contouring interval is 60 m/s.

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CONCLUSIONS

Inverting high frequency Rayleigh wave dispersion data can provide reliable near- surface S-wave velocities. Through analysis of the Jacobian matrix, we can begin to quantitatively sort out some answers to questions about the sensitivity of Rayleigh wave dispersion data to earth properties. For a layered earth model defined by S-wave velocity, P-wave velocity, density, and thickness, S-wave velocity is the dominant property for the fundamental mode of high frequency Rayleigh wave dispersion data. In practice, it is reasonable to assign P-wave velocities and densities as known constants with a relative error of 25 percent or less. It is impossible to invert Rayleigh wave dispersion data for P- wave velocity and density based on analysis of the Jacobian matrix for the model (Table 1). We have presented iterative solutions to the weighted equation by the L-M method and the SVD techniques. Synthetic and real examples demonstrated calculation efficiency and stability of the inverse procedure. The inverse results of our real example are verified by borehole S-wave velocity measurements.

ACKNOWLEDGEMENTS

The authors would like to thank Jim Hunter and Ron Good of Geological Survey of Canada, Joe Anderson, David Laflen, and Brett Bennett for their assistance during the field tests. The authors also appreciate the efforts of Marla Adkins-Heljeson, Mary Brohammer and Amy Stillwell in manuscript preparation.

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