Analyses of the cross-hole method for determining shear wave velocitiesand damping ratios

L. Hall*, A. Bodare

Department of Civil and Environmental Engineering, Royal Institute of Technology, S-100 44 Stockholm, Sweden

Abstract

The cross-hole test is a common method used for determining shear wave velocities and, hence, to characterize the shear modulus. The

determination of the shear wave velocities can be greatly improved by signal processing techniques. With further signal processing the

damping ratios can also be determined. In this paper, the signal processing techniques to determine shear wave velocity and damping ratio is

explained. The techniques are then applied to a actual cross-hole test, and also veri®ed in a ®nite element analysis. q 2000 Elsevier Science

Ltd. All rights reserved.

Keywords: Cross-hole test; Shear wave velocity; Damping ratio; Signal processing; Field measurements; Finite element analysis

1. Introduction

Great progress has been made in recent years in the devel-

opment of analytical procedures, such as the ®nite element

method, for evaluating the response of soils under dynamic

loading. As these analytical methods are re®ned, procedures

to provide more realistic parameters for de®ning stress and

strain properties for use in these models must also be

improved. In cases where no permanent soil displacements

occur, the response to a speci®c input vibration is primarily

controlled by the shear modulus and the damping ratios of the

soil through which the vibrations travel. These properties are,

however, non-linear at larger strains with a shape and orienta-

tion that can only be determined in laboratory tests. At small

strains, the stress±strain characteristics are nearly linear elas-

tic, and better estimates of the low-strain soil properties are

made in situ by ®eld tests. Linear elastic stress±strain models

can be used for analyses of wave propagation and vibration

problems that usually are associated with small strains.

The easiest and most accurate way of determining the

low-strain shear modulus in situ is the cross-hole method.

The cross-hole method involves generating a seismic wave

in one bore-hole and measuring the vibration response by

geophones in two or more adjacent bore-holes at the same

depth. The shear wave velocity, which is closely related to

the shear modulus, can then be estimated by looking at the

average time for the wave to travel between two receivers.

However, using signal processing techniques such as the

Fourier transform algorithm, the determination of shear

wave velocity from cross-hole tests can be enhanced.

Now, the damping ratio can also relatively easy be esti-

mated. Fourier transforms can determine the phase velocity

and material damping by calculating the phase difference

between the measurements of two geophones. The phase

velocity is considered by many seismologists [8] to be a

better estimator of the shear wave velocity.

In this paper, analyses of measurements from in situ

cross-hole tests are presented together with analyses of

results from simulation of cross-hole tests from axisym-

metric ®nite element models. In these analyses, the shear

wave velocity (both phase and average time estimation) and

damping ratios were determined. The ®nite element simula-

tions showed that the shear wave velocity and material

damping could be accurately estimated. The in situ

measurements were, however, very sensitive to noise and

other disturbances. The results can be greatly improved by

®rst re®ning the measured signals by using signal proces-

sing techniques such as ®ltering and windowing.

2. Digital signal analysis

The development of digital data acquisition system has

increased the possibility of using signal processing techni-

ques on the collected data. The most important improve-

ment is the possibility of signal analyses in the frequency

domain. The transformation of data from the time domain

into the frequency domain is based on the Fourier transform.

Soil Dynamics and Earthquake Engineering 20 (2000) 167±175

0267-7261/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved.

PII: S0267-7261(00)00048-8

www.elsevier.com/locate/soildyn

* Corresponding author. Present address: FB Engineering AB, S-402 41

Gothenburg, Sweden.

E-mail address: lhl@fbe.se (L. Hall).

In the frequency domain, many useful parameters describ-

ing the measured signals as well as soil properties can be

determined.

A periodic sampled signal (e.g. velocity time history) can

be transformed into the frequency domain by the discrete

Fourier transform (DFT). For a variable x(tk), k � 0; N 2 1,

the discrete Fourier transform is given by:

X� fn� � DtSXN 2 1

k�0

�x�tk� cos �2pfntk�2 ix�tk� sin �2pfntk��

� DtSXN 2 1

k�0

x�tk�´e2i2pfntk for n � 0¼N 2 1 (2.1)

where tk � kDtS and fn � nDf � n=NDtS:

In general, the transform into the frequency domain will

be a complex valued function, which therefore is commonly

described with Fourier magnitude and phase angle. The

Fourier magnitude and phase angle from the Fourier trans-

form are determined as follows:

iX� fn�i �XN 2 1

k�0

�X�k�real´X�k�real 1 X�k�imag´X�k�imag�0:5; respectively;

�2:2�

fX� fn� �XN 2 1

k�0

tan21 X�k�imag

X�k�real

� �: �2:3�

A plot of the magnitude versus frequency is known as the

Fourier amplitude spectrum; and a plot of the phase angle

against frequency is called the Fourier phase spectrum. The

frequency resolution (Df ) in the spectra is obtained from the

number of samples N and the sampling time step DtS as

follow: Df � 1=T where T � NDtS:

The Fourier amplitude spectrum illustrates how a quan-

tity varies with the frequency. This information can also be

expressed in terms of power. The power spectrum also

called power spectral density function (PSD) is de®ned as:

SXX� fn� � Xp� fn�´X� fn�; �2:4�where Xp( f ) is the complex conjugate of X( f ).

Frequency analyses can also be used as an indication of

mutual frequencies from two records obtained simulta-

neously. This is done with the cross power spectrum (SXY)

that is the product of the linear spectrum of record 1 (i.e. X)

and the complex conjugate of the linear spectrum of record

2 (i.e. Yp) as:

SXY � fn� � Yp� fn�´X� fn�: �2:5�The cross power spectrum is a useful tool to determine

relative phase difference Df between two signals caused

by time delay, propagation delays and varying wave paths

between the receivers. The phase difference Df from the

cross power spectrum can be calculated as follows:

DfXY � fn� �XN 2 1

k�0

tan21 SXY �k�imag

SXY �k�real

� �; �2:6�

or just as the difference between phases of the two signals:

DfXY � fn� � fX� fn�2 fY � fn�k k: �2:7�The use of the phase difference is explained further in

Section 3.

3. Determination of material properties

When measuring two or more signals at different

distances, the dynamic material properties such as propaga-

tion velocities and material damping can be determined.

This can be done both in the time domain or in the frequency

domain. The most straightforward method to determine soil

properties is by analyzing the signals in the time domain.

However, if the signals are digitized it might be easier to

determine the signals in the frequency domain.

3.1. Time domain analyses

In time domain, the propagation velocity is determined

by looking at the arrival time of the waves and the known

distance between the receivers. Material damping can be

estimated by examining the amplitude decay between the

signals. The amplitude decay is, however, very dif®cult to

determine as the signals are corrupted by inhomogeneous

effects in the soils through which the ground motions travel.

3.2. Frequency domain analyses

In the frequency domain, the phase difference between

two signals from the cross power spectrum is used in order

to determine the propagation velocity and material damp-

ing. The phase difference is determined according to Eqs.

(2.6) or (2.7). A phase shift equal to 360 degrees (2 rad) is

equivalent to one period. As such, the travel time of the

waves with different frequencies can be determined by

tph� fn� � 1

fn

DfXY � fn�2p

� �: �3:1�

Now the propagation velocity can be determined by dividing

the distance between the receivers by the travel time obtained

from the phase difference of the two signals. As the deter-

mined propagation velocity depends on the frequency, it is

more commonly called the phase velocity. The phase velo-

city, Cph, for a given frequency can be calculated as

Cph� fn� � Dr

tph� fn� ; �3:2�

where Dr is the distance between the receivers �Dr �r2 2 r1�:

The coef®cient of attenuation, a , can be determined by

the ratio of the magnitude spectra between two signals and

L. Hall, A. Bodare / Soil Dynamics and Earthquake Engineering 20 (2000) 167±175168

the geometrical damping as follows [2]:

aph� fn� �ln

X� fn�´r1

Y� fn�´r2

� ��Dr� : �3:3�

The coef®cient of attenuation varies depending on soil type

and vibration frequency according to the following

equation:

a � 2pf´j

C; �3:4�

where f is the frequency, j the damping ratio and C the wave

propagation velocity

A more common way is to express material damping in

terms of the damping ratio. By using the phase velocity and

Eq. (3.4), the damping ratio becomes:

jph� fn� �ln

X� fn�´r1

Y� fn�´r2

� �´Cph� fn�

2pfn´Dr�

lnX� fn�´r1

Y� f �´r2

� �2pf´tph� fn� : �3:5�

It should, however, be noted that the phase velocity and

material damping are determined with the following

assumptions [7]: (1) the damping ratio is independent of

frequency and strain amplitude; (2) measured wave ampli-

tudes are not affected by re¯ected or refracted waves; (3)

any additional near ®eld effects on amplitude are negligible;

and (4) particle motions are accurately tracked at both

measurement points.

When analyzing the soil properties obtained (phase velo-

city and material damping) from the phase difference, the

power spectra are used to determine the frequencies in the

signals that are of poor quality. A schematic description of

determination of the phase velocity and material damping is

shown in Fig. 1.

4. Cross-hole measurements

The cross-hole measurements took place in LedsgaÊrd just

outside Kungsbacka, Sweden, in conjunction with a project

supervised by the Swedish Rail Administration (Banverket)

and with participants from Swedish Geotechnical Institute

(SGI), Norwegian Geotechnical Institute (NGI) and others.

The project was initiated because very large vibrations had

L. Hall, A. Bodare / Soil Dynamics and Earthquake Engineering 20 (2000) 167±175 169

Fig. 1. Schematic description for determining phase velocity and damping ratio by analysis of the signals in the frequency domain.

been measured in the embankment for trains passing at high

speeds. The soil conditions in LedsgaÊrd consisted of a very

deep (,75 m) and soft clay layer. In order to investigate the

cause of these large ground vibrations, Banverket performed

extensive measurements of the embankment's response due

to passing high-speed trains. Soil investigations of the clay

layer were also performed. For this reason, seismic tests (i.e.

cross-hole tests) were performed by KTH (Royal Institute of

Technology) in order to determine the dynamic soil proper-

ties. KTH also performed measurements on train-induced

ground vibrations on the ground surface just outside the

embankment, which were used for comparing results from

®nite element analyses of train-induced ground vibrations

[4].

Standard geotechnical soil investigations were also

performed [1,6] and soil samples were taken and routinely

investigated for soil type, density, water content, plastic

index, sensitivity, and shear strength as well as odometer

modulus. The seismic in situ tests by KTH were performed

the same week as the measurements of the train-induced

ground vibrations in the fall of 1997. The focus on these

measurements was on cross-hole tests, though down-hole

tests and fall weight tests also were performed. The cross-

hole method applied in LedsgaÊrd is described in the follow-

ing section.

4.1. Description of the cross-hole tests

The instruments used for the cross-hole and down-hole

measurements were developed at KTH [5]. This instrument,

shown in Fig. 2, consisted of a cone of solid steel containing

a geophone with a resonance frequency of 10 Hz.

In the cross-hole tests the cones (two or more) were

forced down by hand to the desired depth with aid of the

connecting steel tubes. This was possible as the ground

consisted of soft clay. The cones were then released from

the steel tubes by lifting up the steel tubes about 10 cm. In

addition, the cones were kept vertical by steel wires, which

were also used to retain the cones when all the measure-

ments were complete.

In order to induce a ground motion at the desired depth,

the same excitation equipment as used in Bodare [3] was

applied. This equipment consisted of a screw-plate

connected to a steel tube (see Fig. 2a). By rotating the

tube, the plate was lowered to the same depth as the cones

containing the geophones. Then, by releasing an inner rod

on the screw-plate it was the possible to induce the waves at

the desired depth.

L. Hall, A. Bodare / Soil Dynamics and Earthquake Engineering 20 (2000) 167±175170

Fig. 2. Cross section of the instruments used in the cross-hole measure-

ments in LedsgaÊrd. (a) Excitator±Screwplate on which an inner rod was

released. (b) Receiver±cone containing a geophone.

Fig. 3. Cross-hole setup used in LedsgaÊrd.

The signals measured by the geophones were sampled

digitally with a sampling frequency of 1000 Hz to a compu-

ter, where they were stored for future analyses. This enabled

the results also to be viewed immediately after the measure-

ments were performed. If the results were satisfactory, the

cones and the screw-plate were lowered to the next depth to

be evaluated. The applied cross-hole setup is shown in

Fig. 3.

L. Hall, A. Bodare / Soil Dynamics and Earthquake Engineering 20 (2000) 167±175 171

Fig. 4. Results from a cross-hole test at a depth of 3 m below the ground surface (v2� vertical particle velocity). #1, #2 and #3 refer to location of geophones

as numbered in Fig. 3. The estimated arrival times for the P-wave (Cp) and S-wave (Cs) are also shown in the ®gure.

Fig. 5. Results from signal analyses of windowed signals as shown in Fig. 4. (a) Normalized power spectra (PSD) of signal #1, #2 and #3. (b) Phase velocity

(Cph). (c) Damping ratio (j) obtained from the phase difference between signals #1±#2, #1±#3 and #2±#3.

4.2. Analyses of cross-hole measurements

Results from a cross-hole test are shown in Fig. 4. As seen

in Fig. 5, three cones with geophones were used in the tests.

Totally, cross-hole tests were performed at ®ve different

elevations from 2 to 6 m below the ground surface.

When analyzing the cross-hole tests, the signals were ®rst

improved by baseline correction and noise removals by a

lowpass ®lter (100 Hz). The S-wave propagation velocity

was then estimated by studying the time of arrival of the

geophones and using the known spacing of the cones. This

was, however, a very subjective and cumbersome process.

The S-wave propagation velocity was therefore also esti-

mated from the phase velocity in the frequency domain by

®rst determining the phase difference between the signals.

When determining the phase velocity, it was found that the

result could be improved if the signals were windowed (see

windows in Fig. 4) before the Fourier transformation into

the frequency domain. After windowing had been

performed, it was also possible to estimate the damping

ratio. Fig. 5 shows the normalized power spectrum, phase

velocity and damping ratio estimated from windowed

signals of a cross-hole test. The theory for estimating mate-

rial properties in the frequency domain is given in Section 5.

In Fig. 6, the results from all the cross-hole measurements

and other seismic ®eld and laboratory tests performed in

LedsgaÊrd are summarized. Cross-hole tests were available

from 2 to 6 m. At 6 m below the ground surface, the shear

wave velocity was correlated with the shear strength

obtained from laboratory tests (see Fig. 6(a)). The ratio

between the maximum shear modulus and the undrained

shear strength was estimated to be 335. This also agrees

with the values obtained by others [9] considering that the

plasticity index PI for the clay in LedsgaÊrd was very high

(PI < 139, [6]). The shear strength was measured down to

15 m below the ground surface. The shear wave velocity in

the crust layer was estimated from fall weight tests. In the

mud layer, the results from the cross-hole measurements

were con®rmed with the results from down-holes tests and

shear wave velocity measurements with bender elements.

The damping ratios were obtained from cross-hole tests

and dynamic triaxial tests. As seen in Fig. 6b, the low strain

measurements with the dynamic triaxial tests con®rms the

results from the cross-hole measurements. The obtained

damping ratio seemed to be around 4.8 in the layers of

mud and around 2.5% in the layers of clay.

5. Simulation of a cross-hole test in a ®nite elementmodel

In order to validate the appropriateness of the determina-

tion of the soil properties, and especially the damping ratio,

from cross-hole tests, a cross-hole test was simulated in a

®nite element analysis. The dynamic ®nite element program

ABAQUS was used in the analysis using direct time inte-

gration in the time domain.

5.1. Finite element model and analysis

The ®nite element model used was 16 £ 15 m2 and

consisted of axisymmetric elements. The elements had the

dimensions 0.1±0.3 m, with a smaller element closer to the

source. The left boundary was locked in the horizontal

direction simulating the symmetry line. The right and

bottom boundaries consisted of in®nite elements (absorbing

boundaries) simulating in®nite regions. The ®nite element

model is shown in Fig. 7(a).

In this analysis, a cross-hole test at 6 m below the ground

L. Hall, A. Bodare / Soil Dynamics and Earthquake Engineering 20 (2000) 167±175172

Fig. 6. Summary of results from soil investigation in LedsgaÊrd concerning: (a) shear wave velocity and (b) damping ratio.

surface was simulated. The model was given material prop-

erties that would give S-wave and P-wave propagation velo-

cities of 56 and 1050 m/s, respectively. The material models

were linear elastic and de®ned by the Young's modulus and

Poisson's ratio. The density was set to 1450 kg/m3 and the

damping ratio of approximately 3% was chosen. In direct

time integration analyses, the material damping is modeled

using the Rayleigh damping model. The damping ratio for

the Rayleigh damping model is, however, not frequency

independent as the actual material damping. The parameters

L. Hall, A. Bodare / Soil Dynamics and Earthquake Engineering 20 (2000) 167±175 173

Fig. 7. (a) Axisymmetric ®nite element model used in the cross-hole test simulation. The ®gure shows the deformed mesh (magni®ed with a factor of 20 000)

at time 60 ms after initiation of load. Numbers in mesh (#1, #2 and #3) indicate location of nodes simulating geophones. (b) Vertical load applied in the ®nite

element analysis. Arrow shows location of load. (c) The damping ratio used in the ®nite element analysis.

Fig. 8. Results from ®nite element simulation of cross-hole tests of uniform soil (v2� vertical particle velocity), #1, #2 and #3 refer to location in ®nite

element mesh as shown in Fig. 7. The arrival times with corresponding P-wave (Cp) and S-wave (Cs) are also shown in the ®gure.

of the Rayleigh damping model were therefore chosen so

that the damping ratios would be relatively constant over the

frequency span of interest. The applied Rayleigh damping

model used in the analysis is shown in Fig. 7(c).

The source, modeled as half-sinus pulse force with a

duration of 20 ms in the vertical direction was applied in

the symmetry line (see Fig. 7(b)). This was found to agree

with the source used in the cross-hole tests in LedsgaÊrd. The

®nite element analysis was performed with a time step of

0.001 s. The vertical particle velocity was then obtained: 3,

6 and 9 m from the source (see numbering #1, #2 and #3 in

Fig. 7(a)). The results from the ®nite element analysis were

analyzed in the same manner as the actual cross-hole

measurements.

5.2. Analyses of results from ®nite element analysis

The particle velocities obtained from the ®nite element

analysis simulating the cross-hole test are shown in Fig. 8.

Windows were also applied here to minimize the effect of

the P-wave and re¯ection of waves from the ground surface

(see windows in Fig. 8). From the windowed signals, the

phase velocity and damping ratio were calculated according

to theory described in Section 3. The results, shown in Fig.

9, were also compared with the true values of the shear wave

velocity and damping ratio as given in the ®nite element

model. This indicated that the phase velocity and damping

ratio could be estimated using the phase difference from the

cross-power spectrum. The phase velocity seemed to give a

slightly dispersive (the velocity varies with frequency)

behavior despite the homogenous ªsoilº (see Fig. 9(b)).

The dispersive behavior is probably caused by some numer-

ical errors when calculating the phase difference. Neverthe-

less, the agreement with the true shear wave velocity was

very good. The damping ratio estimation also agreed very

well with the true damping ratio (see Fig. 9(c)).

A ®nite element analysis was also performed with layered

ªsoilº with an increasing shear wave velocity pro®le with

depth. In the result from this analysis, the damping ratio was

shown to be very sensitive to re¯ection and refraction of

waves. Small windows were required in order to obtain

reasonably good results. The phase velocity did not seem

as sensitive to re¯ection and refraction of waves. Here

again, the results improved if windows were applied in the

time domain before the phase differences were calculated.

6. Conclusions

The cross-hole method using cones of solid steel contain-

ing geophones was shown to be a simple in situ technique to

determine low strain soil properties such as the shear wave

velocity and the damping ratio in soft soils. The determina-

tion of the soil properties was improved by calculations of

the phase differences between the measured signals. From

the phase difference, shear wave velocity (phase velocity)

and damping ratio could be estimated. The method of deter-

mining soil properties from the phase difference of the

L. Hall, A. Bodare / Soil Dynamics and Earthquake Engineering 20 (2000) 167±175174

Fig. 9. Results from signal analyses of the windowed signals as shown in Fig. 8. (a) Normalized power spectra (PSD) of signals #1, #2 and #3. (b) Phase

velocity (Cph), (c) Damping ratio (j) obtained from the phase difference between signals #1±#2, #1±#3 and #2±#3.

signals was veri®ed in a ®nite element analysis. The ®nite

element analyses showed that the true shear wave velocity

and damping ratio could be obtained with this method. The

analyses showed that the results could be improved, espe-

cially for the damping ratio, if windows were applied to the

signals in the time domain before the phase differences were

calculated. The results from actual ®eld tests with the cross-

hole method were also con®rmed with other ®eld and

laboratory tests.

Acknowledgements

This work was supported by the Swedish Railway

Administration (Banverket) and the Swedish Council for

Building Research (ByggfoskningraÊdet) under Grant #93-

0768-9.

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