Finite‐difference modeling of high‐frequency …...Finite-difference modeling of Rayleigh waves...
Transcript of Finite‐difference modeling of high‐frequency …...Finite-difference modeling of Rayleigh waves...
Finite-difference Modeling of High-frequency Rayleigh waves
Yixian Xu, China University of Geosciences, Wuhan, 430074, China; Jianghai Xia, and Richard D. Miller,
Kansas Geological Survey, The University of Kansas, Lawrence, Kansas 66047-3726, USA.
Summary
The finite-difference (FD) method described here is
specially developed to model high-frequency Rayleigh-wave
propagation in near-surface mediums. Although many FD
programs existed for earthquake research and oil exploration,
none has been developed to model high-frequency
Rayleigh-wave propagation in near-surface elastic mediums
when the source and all receivers are on the free-surface. The
scheme described here uses the second-order staggered grid,
based on a set of first-order differential equations. Special
implementations of combined elastic and acoustic free-surface
conditions were developed, which were originally derived
from surface wave propagation in a transversely isotropic
medium. A combination of one-way sponge filtering and
anisotropic filtering methods was used to minimize
Rayleigh-wave reflections from artificial boundaries.
Numerical stability was achieved by using a small spatial grid
size and time step. This new FD method was tested and proved
satisfactory using dispersion analysis of Rayleigh waves in a
homogenous and layered half-space. The present study
provides a practical tool for improving the confidence and
uniqueness of deriving 2D interpretation of high-frequency
Rayleigh-wave data, and makes successful in the first phase to
2D inversion.
Introduction
Because high-frequency (> 2 Hz) surface waves travel in
a very shallow portion of the earth, spectral analysis of
surface-wave (SASW) method was developed for site
investigation over the last twenty years (e.g., Stokoe et al.,
1994). A new technique incorporating multichannel analysis of
surface waves (MASW) using active sources was developed
(Park et al., 1999; Xia et al., 1999) to improve inherent
difficulties in evaluating and distinguishing signal from noise
with only a pair of receivers by the SASW method. The
MASW method has become increasing popular for
environmental and engineering applications in recent years
(e.g., Xia et al., 2004). A couple reasons are likely responsible
for increased use of this method. First is based on very
characteristics of Rayleigh-wave energy. In the near-surface,
Rayleigh waves represent the highest energy propagating from
a source and therefore have a relatively high signal-to-noise
ratio. Rayleigh waves are not limited by subsurface velocity
properties, which sometime affect reflection and refraction
methods. Rayleigh waves are predominately controlled by
shear (S)-wave velocity properties of the subsurface which are
key important for many geotechnical applications. The second
comes from the advantages made possible with multichannel
recording. MASW can use advanced array digital signal
processing technologies and thus take advantage of the many
benefits of time-space relationships of seismic wavefields. To
our knowledge, ongoing interpretation methods are currently
based on the plane-wave approximation (Xia et al., 1999).
Few attempts have been made addressing Rayleigh-wave
dispersion analysis of higher dimensional models. However, to
utilize all Rayleigh-wave information, higher dimensional
modeling codes must be developed.
This paper aims to develop a 2D modeling method for
high-frequency (>2Hz) Rayleigh-wave analysis. In the
following sections, we will discuss the modeling method
including implementations of the free-surface and artificial
boundaries and numerical tests using Rayleigh-wave
dispersion analysis. A 2D numerical example will demonstrate
the effectiveness and accuracy of this 2D modeling scheme.
Modeling method
We choose the second-order staggered-grid
finite-difference method (Virieux, 1986) because of its coding
simplicity and accuracy for dense spatial sampling as
demanded by Rayleigh-wave propagation. Wave propagation
in a 2D heterogeneous elastic medium can be described by a
first-order system of differential equations:
( 2 )
( 2 )
( )
t x x xx z xz
t z z zz x xz
t xx x x z z
t zz x x z z
t xz x z z x
V
V
V V
V V
V V
(1)
where is the particle velocity vector, and( , )x zV V
, )xz( ,xx zz is the stress tensor, ,x z is the density of a
medium, and ,x z and ,x z are Lame coefficients.
Discretization of the differential equations (1) for
time-space coordinates at internal nodes using a staggered-grid
scheme can be found in Virieux (1986). Selection of a time
step and a spatial grid size for a specific velocity model and
source energy frequency band is very important in
Rayleigh-wave modeling. From numerical tests, a empirical
rule, , and max0.6 /t x V min0.125x , where andmaxV min
are the highest velocity and the shortest wavelength,
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Finite-difference modeling of Rayleigh waves
respectively, used in our modeling. Here the vertical grid size
is assumed the same as the horizontal grid size.
On a free-surface, zz and xz must be zero. We
developed a practical implementation of the free-surface
conditions by combining the image method (Levander, 1988)
and Mittet’s (2003) method. The procedure assumes that nodes
at one-half of a spatial step length above the free-surface are
filled by a medium with elastic properties symmetric about the
free-surface; meanwhile, elastic properties at the free-surface
were modified. Implementation in this fashion is physically
reasonable, effective for Rayleigh-wave modeling.
For absorbing artificial wave energy resulting from
computing boundary, better results were achieved by joining
the one-way sponge filtering (OSF) and the anisotropic
filtering (Dai et al., 1994) methods. This implementation
modified outgoing waves in a transition zone so that the waves
impinge on the computing boundary at the normal angle.
For all our numerical tests, we chose a source that change
with time according to the first order derivative of Gaussian
function. Because a sledgehammer or vibrator seem to be the
most commonly used sources in high-frequency
Rayleigh-wave surveys, the normal stress or velocity source
can be implemented by simply defining the source function in
terms of zz or zV , respectively, e.g., , where
(i, 1) represents the ith node at the free-surface, and
,1 ( )zz i g t
g t is
a source function. The initial condition is simply setting
0g t , when .0t
Numerical tests of one-dimensional models
Numerical tests by theoretical models were performed for
a homogenous half-space and an isotropic layered half-space.
Dispersive analysis provided a powerful tool for recognition
and analysis of high-frequency Rayleigh-wave propagation
and therefore was used for testing modeled vertical velocities.
For our testing reported on here, P- and S-wave velocities
of a homogenous half-space are 1000 m/s and 3/
3000
1000
m/s, respectively. These values define a Poisson’s solid with a
Rayleigh-wave velocity about 531m/s. Spatial grids consisted
of 200 200 nodes with 30 nodes in the transition zones along
the left and right sides of the model and 60 nodes in the
transition zone along the bottom of the model. Such spatial
grid size was 1 meter with time step of 0.1ms. The source is
described by the first-order derivative of Gaussian function
with controlling parameter)exp( 2tt and is
located at node (80, 1). A shot gather (vertical particle velocity)
and its dispersion curve are shown in Fig. 1. Small deviations
can be identified, relative to the true Rayleigh-wave phase
velocities at near source channels (Fig. 2a) and in lower
frequency band (Fig. 2b). This phenomenon results from
non-plane wave approximations and near-field effects. In the
higher frequency band (> 70Hz), the dispersion curve appears
to possess a slightly decreasing velocity trend as frequency
increases. This is due to required spatial grid size exceeding
the demands of the numerical dispersion. The non-dispersive
phase velocity (531 m/s) obtained by the phase shift method
agreed well with the Rayleigh-wave phase velocity of the
homogeneous half-space in the frequency band commonly
used for near-surface applications.
The parameters of the two-layer model are as follows: the
surface layer: Vp = 800 m/s, Vs = 200 m/s, = 2000 kg/m3,
and thickness = 10 m; the half-space: Vp = 1200 m/s, Vs = 400
m/s, and = 2000 kg/m3. The grid size, time step, and time
function and location of sources were the same as the
homogenous half-space. The snapshots of the vertical particle
velocity of the two-layer model at 150 ms and 300 ms are
shown in Fig. 2. The seismogram and dispersion curve of the
vertical particle velocity are shown in Fig. 3. In the band of 15
to 30 Hz, the dispersion curve agrees very well with the
analytical result (the maximum relative error within 2%),
whereas in the band of 8~15 Hz, the relative error raises to
about 5%. We realized the error is mainly caused by
phase-shift preprocessing, near-field effects, (e.g., non-plane
wave propagation and incorporated body wave energies), and
finite-difference approximation, in which about 5% cannot
quantitatively be separated into unique different contributions.
Two-dimensional experiment
The main object of this study is to support 2D
interpretation of near-surface Rayleigh-wave investigations.
2D surface-wave modeling results for common geological
features and anomalies would aid us greatly in understanding
and detecting certain unique wave characteristics associated
with Rayleigh waves. We designed a corner-edge model (Fig.
4) to simulate a vertical fault in a near-surface setting. The
earth surface intersection point of the fault corner is at the
center point of the 60-station-receiver spread (Fig. 4). O
and are marked as the “left” source (on the left side of the
corner) and “right” source (on the right side of the corner),
respectively. All the modeling parameters (including the
source time functions) are the same as the homogenous
half-space.
1
2O
Scattering events that spread from the middle to the left
quadrant can be clearly seen in vertical particle velocities
excited by the left source (Fig. 5a). When the source is on the
right quadrant, it is difficult to identify scattering events (Fig.
5b). When the frequencies approach the extremes, both
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Finite-difference modeling of Rayleigh waves
dispersion curves from sources on the left and right sides
approach the same Rayleigh wave phase velocities for the top
layer and the bottom half-space. However these two dispersion
curves possess different phase velocities within the middle
frequency band (9 to 18 Hz) illustrated clearly by Fig. 7.
Interesting results can be obtained by examining
amplitude spectra (Fig. 6). The amplitude spectra were
calculated trace by trace. The main energies for the “left”
source concentrate at about 9 Hz in the middle to the left
portion of the profile, and 18 Hz in almost the whole profile
for the right source. The fault edge was imaged by the
maximum energy concentration observed when the “left”
source was used. We can thus suggest the dispersive
characterizations of Rayleigh waves excited by the “left”
source are determined mainly by the left portion of subsurface
structures; whereas the dispersive characterizations of
Rayleigh waves excited by the “right” source are determined
mainly by the right portion of subsurface structures.
According to the traditional half wave-length method
(Yang, 1993), one can estimate the phase velocity of Rayleigh
waves about 321 m/s at 9 Hz from the record excited by the
“left” source , which gives a depth to the top of the lower
half-space at about 17.8 m. However, based on record excited
by the “right” source one can estimate the phase velocity of
Rayleigh waves about 230 m/s at 18 Hz, which gives a depth
to the top of the lower half-space at about 6.4 m. These results
possess a large deviation from the true depths on two sides of
the fault.
Conclusions
We developed an effective FD method to model
Rayleigh-wave propagation for near-surface applications. In
this method, there is an important improvement: free-surface
conditions combined the image method and a technique to
modify medium properties just at the free-surface that is
supported by the Rayleigh-wave propagation theory. We also
achieved satisfactory implementations for artificial boundaries
by combining one-way sponge filtering and anisotropic
filtering methods. These make the present method feasible for
near-surface applications.
From the numerical perspective, the 2D model revealed
some of the interesting properties of Rayleigh-wave
propagation such as scattering events from the corner of the
fault and error existed in the traditional half wave-length
interpretation. These encourage the development of a 2D
inversion method in the near future.
Acknowledgments The first author thanks the Kansas
Geological Survey for providing the surface analysis software
used in the paper and the China Scholarship Council for the
financial support. The first author appreciates Dr. Choon Park
for the constructive discussion on examples of dispersion
analysis used in the manuscript.
ReferencesDai, N.X., Vafidis A., and Kanasewich, E., 1994. Composite
absorbing boundaries for the numerical simulation of seismic waves, Bull. Seis. Soc. Am., 84, 185-191.
Ellefsen, K.J., 1993. Two-dimensional numerical simulation of elastic wave propagation for environmental and geotechnicalstudies, U.S. Geological Survey Open-File Report 93-714.
Levander, A.R., 1988. Fourth-order finite-difference P-SVseismograms, Geophysics, 53, 1425–1436.
Mittet, R., 2002. Free-surface boundary conditions for elasticstaggered-grid modeling schemes, Geophysics, 67,1616–1623.
Park, C.B., Miller, R.D., and Xia, J., 1999. Multichannel analysis of surface waves, Geophysics, 64, 800-808.
Stokoe II, K.H., Wright, G.W., James, A.B., and Jose, M.R., 1994.Characterization of geotechnical sites by SASW method, inGeophysical Characterization of Sites, ISSMFE TechnicalCommittee #10, edited by R.D. Woods, Oxford Publishers,New Delhi.
Virieux, J., 1986. P-SV wave propagation in heterogeneous media:Velocity-stress finite-difference method, Geophysics, 51,889–901.
Xia, J., Miller, R.D., and Park, C.B., 1999. Estimation of near-surface shear-wave velocity by inversion of Rayleigh wave, Geophysics, 64, 691-700.
Xia, J., Miller, R.D., Park, C.B., Ivanov, J., Tian, G., and Chen, C.,2004. Utilization of high-frequency Rayleigh waves innear-surface geophysics, The Leading Edge, 23, 753-759.
Yang, C., 1993. Rayleigh wave exploration (in Chinese),Geological Pub. House, Beijing.
(b)
Fig. 1. Vertical particle velocity and dispersion curve of a homo-
genous half-space. (a) vertical particle velocity; (b) dispersion curve. (a)
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Finite-difference modeling of Rayleigh waves
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Fig. 2. Snapshots of vertical particle velocities in a two-layered
half-space at 150 ms (left panel) and 300 ms (right panel),
respectively.
Fig. 3. Vertical particle velocity and dispersion curve of a
two-layered half-space. (a) vertical particle velocity; (b)
dispersion curve.
Fig. 4. Illustration of a corner-edge model and layouts of sources
and receivers.
(a)
(b)
Fig. 5. Vertical particle velocities and dispersion curves of the
corner-edge model for different source layouts. (a) vertical particle
velocity with the “left” source; (b) vertical particle velocity with the
“right” source.
(a)
(b)
Fig. 6. Amplitude spectra of the corner-edge model with different
source layouts. the “left” source (left); the “right” source (right).
8 10 12 14 16 18 20 22 24 26 28Frequency (Hz)
30
180
200
220
240
260
280
300
320
340
360
380
400
420
440
460
480
500
Ph
ase
velo
city
(m
/s)
Phase velocity
Corner-edge(left source)
Corner-edge(right source)
Layered half-space(5m thickness of upper layer)
Layered half-space(10m thickness of upper layer)
Spread30m 30m
O2O1
5mVp=800m/s, Vs=200m/s.
5mVp=1200/s,
Vs=400m/s.
Fig. 7. Comparison of phase velocities between corner-edge and
layered half-space models.
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EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2005 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web.
Finite-difference Modeling of High-frequency Rayleigh waves
References Dai, N. X., A. Vafidis, and E. Kanasewich, 1994, Composite absorbing boundaries for the
numerical simulation of seismic waves: Bulletin of the Seismology Society of America, 84, 185–191.
Ellefsen, K. J., 1993. Two-dimensional numerical simulation of elastic wave propagation for environmental and geotechnical studies: U. S. Geological Survey Open-File Report 93–714.
Levander, A. R., 1988. Fourth-order finite-difference P-SV seismograms: Geophysics, 53, 1425–1436.
Mittet, R., 2002. Free-surface boundary conditions for elastic staggered-grid modeling schemes: Geophysics, 67, 1616–1623.
Park, C. B., R. D. Miller, and J. Xia, 1999, Multichannel analysis of surface waves: Geophysics, 64, 800–808.
Stokoe II, K. H., G. W. Wright, A. B. James, and M. R. Jose, 1994, Characterization of geotechnical sites by SASW method, in geophysical characterization of sites: Oxford Publishers.
Virieux, J., 1986, P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method: Geophysics, 51, 889–901.
Xia, J., R. D. Miller, and C. B. Park, 1999, Estimation of near-surface shear-wave velocity by inversion of Rayleigh wave: Geophysics, 64, 691–700.
Xia, J., R. D. Miller, C. B. Park, J. Ivanov, G. Tian, and C. Chen, 2004, Utilization of high-frequency Rayleigh waves in near-surface geophysics: The Leading Edge, 23, 753–759.
Yang, C., 1993, Rayleigh wave exploration: Geological Publishing House.
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