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,

NSE 1.7

SEG/Houston 2005 Annual Meeting 1057

Dow

nloa

ded

07/0

3/14

to 1

29.2

37.1

43.1

6. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

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

NSE 1.7

SEG/Houston 2005 Annual Meeting 1058

Dow

nloa

ded

07/0

3/14

to 1

29.2

37.1

43.1

6. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

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)

NSE 1.7

SEG/Houston 2005 Annual Meeting 1059

Dow

nloa

ded

07/0

3/14

to 1

29.2

37.1

43.1

6. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

Finite-difference modeling of Rayleigh waves

5080110140

10

40

70

100

130

5080110140

10

40

70

100

130

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.

NSE 1.7

SEG/Houston 2005 Annual Meeting 1060

Dow

nloa

ded

07/0

3/14

to 1

29.2

37.1

43.1

6. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

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.

Dow

nloa

ded

07/0

3/14

to 1

29.2

37.1

43.1

6. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/