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All content in this area was uploaded by Jianghai Xia on Feb 23, 2015
Content may be subject to copyright.
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 groups of earth
parameters: S-wave velocity, P-wave velocity, density, and thickness of layers. Analysis
of Jacobian matrix in a high frequency range (5- 30 Hz) provides a measure of sensitivity
of dispersion 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 inverted
results. The third example illustrates the combination of a standard CDP (common depth
point) roll-along acquisition format with inverting surface wave 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
v
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 direction 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 elastic
2
properties of the deeper layers (p. 30, Babuska and Cara, 1991). Shorter wavelengths are
sensitive to the physical properties of surficial layers. For this reason, a particular mode of
surface wave will possess a unique phase velocity for each unique wavelength, 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 sensitivity
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
velocity. Dispersion curves are inverted using least-squares techniques in SASW methods
(Stokoe and Nazarian, 1983; Nazarian et al., 1983).
The Kansas Geological Survey have conducted a three-phase research project to
estimate near-surface S-wave velocity from ground roll since 1995: acquisition of high
frequency (≥5 Hz) broad band ground roll, creation of efficient and accurate algorithms
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 1), 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
dramatically affects convergence of the inverse procedure (Xia, 1986). The practical way
to calculate the partial derivatives of 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
the inversion. Based on the
orthogonality between the partial
derivative vector with respect to
Table 1. An earth model parameters.
Layer
number
v
s
(m/s) v
p
(m/s)
ρ
ρρ
ρ
(g/cm
3
)
h (m)
1 194.0 650.0 1.82 2.0
2 270.0 750.0 1.86 2.3
3 367.0 1400.0 1.91 2.5
4 485.0 1800.0 1.96 2.8
5 603.0 2150.0 2.02 3.2
6 740.0 2800.0 2.09 infinite
3
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 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
contributes 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 (≥ 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 1), 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 1), however, numerical results from more than a hundred modeling testing
support these conclusions.
Free surface
______________________________
v
s1
v
p1
ρ
1
h
1
______________________________
v
s2
v
p2
ρ
2
h
2
______________________________
.
.
______________________________
v
si
v
pi
ρ
i
h
i
______________________________
.
.
______________________________
v
sn
v
pn
ρ
n
infinite
0
100
200
300
400
500
600
700
800
900
5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30 32.5
Frequency (Hz)
Phase velocity (m/s)
Model
Changes in density
Changes in P-wave v.
Changes in S-wave v.
Changes in thickness
Fig. 1. A layered earth model with parameters of shear-wave
velocity (v
s
), compressional wave velocity (v
p
), density
(
ρ
ρρ
ρ
), and thickness (h).
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 Rayleigh wave phase velocities after 25
percent changes in P-wave velocity, and so on.
4
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 sensitivity indicator
is greater for thickness variation than 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 unknowns in our inverse procedure.
We can reduce the number of unknowns from 4n -1 (n is number of layers) to n with these
assumptions. The fewer unknowns in an inverse procedure, the more efficient and stable
the process, and the more reliable the solutions.
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 n, x = [v
s1
, v
s2
, v
s3
, ..., v
sn
]
T
. 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
1
, b
2
, b
3
, ..., b
m
]
T
. After linearization of the dispersion function, an objective function
is defined as
Φ∆∆ ∆∆ ∆=− −+Jx bWJx b x
~~~
22
2
2
α
, (1)
where
∆
b[= b - c
R
(x
o
)] is the difference between measured data and model response to the
initial estimation, c
R
(x
o
) is the model response to the initial S-wave velocity estimates x
o
,
which are defined by phase velocities, see Xia et al. (in review) for details;
∆
x is a
modification of the initial estimation;
J
~
is the Jacobian matrix with m rows and n columns
(m > n) with the elements being the first order partial derivatives of
c
R
with respect to S-
wave velocities,
2
is the l
2
-norm length of a vector,
α
is the damping factor, and W
~
is a
weighting matrix, which can be determined by 1) differences in Rayleigh 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 minimum modification to
model parameters so the convergence procedure is stable for each iteration. This dose 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.
5
Iterative solutions of a weighted damping equation using Levenberg-Marquardt
method (L-M) (Marquardt, 1963) provide a stable and fast solution. Marquardt (1963)
pointed out the damping factor (
α
) controls the direction of
∆
x 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 (
α
) 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 IVI MiniVib was used as the energy source. Forty groups of 10 Hz geophones
were deployed on 1 m interval 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 filed data acquired by
the MASW method possess a strong ground roll
component (Figure 3). The dispersion curve (Figure
4a) of Rayleigh wave phase velocities have been
extracted from filed data (Figure 3), for frequencies
ranging from 15 to 80 Hz, using CCSAS processing
techniques (Park et al., in review).
Three-component borehole data were acquired
coincidentally to obtain P-wave and S-wave velocity
vertical profiles. A cross-correlation technique was
used to confidently determine S-wave arrivals on the
recorded three-component borehole data. Any error
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 borehole survey is
approximately 10%.
Inverting the Rayleigh wave
phase velocities to determine S-wave
velocities requires densities and P-
wave velocities be defined. Densities
were estimated and designated to
increase approximately linearly with
depth while P-wave velocities were
obtained from borehole data (Table 2).
The initial S-wave model (labeled
Fig. 3. Forty groups of 10 Hz geophones were
spread 1 m apart. An IVI MiniVib was used as
the energy source and located at 27 m away from
the
right side of the geophone spread. Two
linear events are velocities of dispersive ground
roll at
frequencies approximately 15 Hz and 50
Hz.
Table 2. The initial model of the real example.
Layer
number
v
s
(m/s) v
p
(m/s)
ρ
ρρ
ρ
(g/cm
3
)
h (m)
1 167.736 534.0 1.820 1.0
2 254.305 536.0 1.860 2.0
3 367.060 791.0 1.91 3.1
4 425.016 1212.0 1.96 3.1
5 472.324 1460.0 2.02 3.0
6 558.080 2400.0 2.09 4.6
7 672.877 2306.0 2.17 4.6
8 813.468 2226.0 2.26 6.0
9 813.468 2531.0 2.35 6.1
10 852.274 2410.0 2.4 infinite
6
“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 (66 m). Theoretically, considering this averaging
there should be only small differences between inverted velocity and 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
borehole values from two different directions (Figure 4b). Overall accuracy for both
inverted models are visually the same.
Vancouver, Canada
The Kansas Geological Survey and the Geological Survey of Canada conducted a
project of a surface wave technique testing in unconsolidated sediments of the Fraser
River Delta, Vancouver, Canada in Fall of 1998. 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 in 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.
Seismic source was a weight dropper built by the Exploration Services of the Kansas
Geological Survey. Three to ten impacts were vertical stacked at each offset. No
acquisition filter was applied during data acquisition. The record length is 2048
milliseconds with 1 millisecond sample interval. Overall difference between S-wave
0
100
200
300
400
500
600
700
800
900
1000
15 20 25 30 35 40 45 50 55 60 65 70 75 80
Frequency (Hz)
Phase velocity (m/s)
Measured
Initial A
Final A
Initial B
Final B
a
Fig. 4. Inverse results of Fig. 3. Labels on dispersion curves (a) and S-wave velocity profiles (b) have the same meaning as in
labels Figure 4 except that the dispersion curve labeled “measured” (a) is real data extracted from filed data (Figure 3) by CCSAS
techniques (Park et al., in review). “Initial B” model (b) was calculated from the “measured” 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).
0
200
400
600
800
1000
1200
1400
1600
0 5 10 15 20 25 30 35 40 45
Depth (m)
S-wave velocity (m/s)
Borehole
Initial A
Inverted A
Initial B
Inverted B
b
7
100
110
120
130
140
150
160
170
180
0 5 10 15 20
Frequency (Hz)
Phase velocity (m/s)
Measured
Final
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,
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
IVI 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. The total
about 180 shot gathers for each line were collected on 1.2 m spacing.
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). A 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 next station, 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 wave along the survey line is around 15 m. Inverted S-wave
velocities in the proximity of station 310 suggest a depth to the bedrock of more than 15 m
that doses not contradict the 21 m depth of the well data. Other three wells at stations 390,
15, and 65 confirmed inverted results if the bedrock corresponds the 500 m/s-contour line.
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.
0
50
100
150
200
250
0 5 10 15 20 25 30
Depth (m)
S-wave velocity (m/s)
Borehole FD97-2
Inverted
b
50
100
150
200
250
0 5 10 15 20 25 30
Frequency (Hz)
Phase velocity (m/s)
Measured
Final
0
50
100
150
200
250
300
350
0 5 10 15 20 25 30
Depth (m)
S-wave velocity (m/s)
Borehole FD92-4
Inverted
Fig. 6. Field shot gather (a) with 60 traces at location of borehole FD92-4, Rayleigh wave phase
velocities
(
b
)
extracted from
(
a
)
labeled Measured and from inverted Vs model
(
c
)
labeled Final.
a
b
c
c
a
b
8
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 Joe Anderson, David Laflen, and Brett Bennett
for their assistance during the field tests. The authors also appreciate the efforts of Marla
Adkins-Heljeson and Mary Brohammer in manuscript preparation.
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