Seismic surface wave profiling is gaining popularity in engineering practice
for determining shear-wave velocity profile. The main advantage of surface wave
testing is essentially related to its non-destructive and non-invasive nature
that allows the characterization of hard-to-sample soils without the need for
boreholes that makes the subsurface seismic methods (such as down-hole and cross-hole
methods) expensive and time consuming. One of the familiar seismic method for
determining the shear wave velocity is the Spectral Analysis of Surface Waves
(SASW) method, based on dispersive characteristics of surface waves. The SASW
method is widely established as a subsurface investigative tool and is implemented
in a wide variety of geotechnical environments, including pavements, solid waste
landfills and sea beds (Roesset, 1998).
A unique method of collecting and calculating data to produce phase plots was utilized for the SASW testing based on two receivers. Phase plots were calculated and shown in the field by the dynamic signal analyzer, using information from the receivers as well as the source. Using the source function verifies that the phase shift between receivers is correlated to the source signal. The testing procedure of SASW method is revealed in Fig. 1.
The wrapped phase angle and unwrapped phase angle corresponding to SASW method
are plotted in Fig. 2.
||General SASW field testing source-receiver geometry
|| (a) Wrapped phase angle in deg, (b) Unwrapped phase angle
The surface wave phase velocity is calculated from phase angle as:
where, D is the distance between two receivers and φ12 is the
phase angle between two receivers. These calculations yield an experimental
dispersion curve from SASW method. The evaluation method of the dispersive phase
and group velocities for SASW technique was also proposed using the harmonic
wavelet transformation (Park and Kim, 2001; Kim
and Park, 2002).
During the field testing various kinds of noises are in existence. The conventional phase unwrapping method with filter criteria usually leads to the correct dispersion curve. However, under some irregular profiles where the more modes dominate the surface motion, the conventional phase unwrapping method can lead to incorrect dispersion curve. For the phase unwrapping method, it is difficult to evaluate correct number of 3600 cycles in the phase spectrum with noises. Furthermore, due to the short distance coverage of SASW method, the spatial resolution is limited.
Park et al. (1999a, b,
2000) has shown that Multichannel Analysis of Surface
Waves (MASW) method represents an improvement over SASW, overcoming the few
but significant weaknesses of the SASW method. MASW is also a fast method of
evaluating near-surface vs profile because the entire range of investigation
depth is covered by one or a few generation of ground roll without changing
receiver configuration. The main advantage of the MASW method is its ability
to take a full account of the complicated nature of seismic waves that always
contain higher modes (Strobbia and Foti, 2006;
Penumadu and Park, 2005) of surface waves, body waves, scattered waves,
traffic waves. Incorporating multi-station receivers and 2-D wavefield transformation
(Lin and Chang, 2004; Park et
al., 1999a) improves inherent difficulties in evaluating signal from
noise with only a pair of receivers. The MASW technique has proven to mute the
interfering seismic waves in the shot records and filter noisy surface-wave
modes and thus significantly improve the range and resolution of multimodal
dispersion curves in the phase-velocity-frequency domain (Ivanov
et al., 2005; Penumadu and Park, 2005). Observation
and detection of higher mode surface wave is the pioneering success of the MASW
method (Park et al., 2000, 2001a,
b). MASW method is also convenient to apply in a shallow
marine environment (Kaufmann et al., 2005), to
evaluate stiffness of water-bottom sediments (Park et
al., 2005), to determine a sinkhole impact area as well as collapse
and subsidence feature (Xia et al., 2004), to
represent near surface anomalies (Park et al., 1999b)
and to map bedrock surface (Miller et al., 2000,
The MASW method is arranged as data acquisition, data processing for dispersion,
finally inversion analysis shown in Fig. 3. This method simplifies
the testing procedure and automates the data analysis. The determination of
the dispersion spectrum S (w, v) from multichannel surface wave data is based
on the Eq. 2 in which A (x, ω) is the normalized energy
spectrum for each receiver, v is the assumed phase velocity and V is the phase
velocity for a given frequency.
|| Scheme of the three steps of the MASW method
S(ω,v) is maximized when the two velocities are equal.
However, the usual MASW method is constraint to adjust irregularities and presence of noises for phase velocity extraction using Fourier transform. However, because of characteristic of Fourier transform used in the conventional phase extraction method, dispersion curve is sensitive to background noise and body waves in the low frequency range. Furthermore, under some field conditions such as pavement site, the phase extraction method can lead to erroneous dispersion curve. To overcome these problems, a new method of determining the group velocities and phase velocities using time frequency analysis based on Complex Morlet wavelet transform is highlighted with the algorithm.
In this study, the new evaluation method of phase and group velocities is proposed
based on complex Morlet wavelet transform. Wavelet decompositions and windowed
Fourier decompositions have emerged in the last few years as tools of great
importance for the analysis of signals in which local frequencies can be extracted
(Torresani, 1993). Wavelet theory can be viewed as a
modern improvement and extension of the Fourier theory and hence allow a flexible
alternative to the Fourier method in non-stationary signal analysis (Cho
and Chon, 2006). Wavelet transform has recently been through considerable
development adding new instruments and special functions and considered more
robust with noise deduction performance. The aim of this study is to develop
the consistent surface wave analysis technique with noise lessening performance
using complex Morlet wavelet transform.
MATERIALS AND METHODS
Location of study: The study of the proposed method is conducted at University Kebangsaan Malaysia since January, 2008 to March, 2009. The seismic signal for soil profile collected by MASW testing at University Kebangsaan Malaysia (UKM) in Bangi, Selangor, Malaysia is used in this study. The schematics arrangement of the test is shown in Fig. 4. The propagation of the waves were detected using multiple receiving geophones transferred digitally a notebook computer. The recorded time domain signals were saved in notebook computer and were then converted to frequency domain where noise contaminations were analyzed by using MATLAB. The data acquisition and analysis are performed since April, 2008 to January, 2009 at University Kebangsaan Malaysia (UKM) in Bangi, Selangor, Malaysia.
Proposed method using wavelet transform: Wavelet Transformation (WT)
is equivalent to the convolution of the wavelet function and the signal under
investigation. The WT is performed by projecting a signal s(t) onto a family
of zero mean functions deduced from an elementary function ψ by translations
and dilations (Yves et al., 1992).
The variables a and b control the scale and position of the wavelet, respectively.
For Seismic test, we are required to work with complex signals. The Fourier
Transformation (FT) coefficients of such signals are no longer symmetric.
|| Data acquisition using MASW method
Likewise it is convenient to work with analytic wavelets which can separate
amplitude and phase components and allow the measurement of the time evolution
of frequency transients. The WT for processing of surface waves can be implemented
in such a way that only the coefficients resulting from the forward flow components
are obtained when the scale is positive and only the coefficients resulting
from the reverse flow components are obtained when the scale is negative. Ignoring
the translational parameter b, the scale dependent complex Morlet wavelet is
given by equation:
where, ω0 is the nondimensional frequency and usually assumed
to 5 to 6 satisfy the admissibility condition. The FT is given by equation:
where, H stands for the Heaviside step function. From Eq. 5, one can observe that a frequency spectrum of an upper analytic signal is obtained for a>0 and a frequency spectrum of a lower analytic signal is obtained for a<0.
The WT of a signal s(t) with the Morlet wavelet is given by Eq.
If the number of scales is J, a complete set of directional wavelet coefficients
can be mapped over the scales from a = -J to b = J, excluding a = 0. In order
to see how negative scales have been utilized to obtain directional wavelet
coefficients, let us evaluate Eq. 6 for one positive and one
negative scale. Assuming b = 0 for simplicity, the WT of the signal for a =
1 and b = -1 are given, respectively, by equation:
||Schematic diagram for the wavelet analysis in MASW method
where, the constant C = π-1/4|a|G1/2.
Taking the FT of Eq. 7 and 8 yield, respectively
where, S (ω) is the FT of the signal and C is a constant. The inverse
FT of Eq. 9 and 10 yields complex wavelet
coefficients formed only by the forward and the reverse flow components respectively.
The wavelet coefficients are used to determine the phase delay and group delay
for MASW method. The crucial attention of the proposed method is dispersion
curve analysis for the representation of phase velocity versus frequency curve.
The design of the new method is comparable to the phase velocity extraction
using Fourier transform for MASW method (Park et al.,
2000, 2001a, b). The implementation
of wavelet transform to determine phase velocity and group velocity for multi-channels
surface wave method is revealed in the Flow-chart of wavelet analysis in MASW
method shown in Fig. 5.
Determination of phase and group velocity: The wavelet transform is used to decompose a signal into wavelets, small oscillations that are highly localized in time. Whereas the Fourier transform decomposes a signal into infinite length of sines and cosines and effectively losing all time-localization information, to which the WT's basis functions are scaled and hence shifted versions of the time-localized mother wavelet. The WT is used to construct a time-frequency representation of a signal that offers very good time and frequency localization.
The WT is an excellent tool for mapping the changing properties of non-stationary signals. The WT is also an ideal tool for determining whether or not a signal is stationary in a global sense. When a signal is judged non-stationary, the WT can be used to identify stationary sections of the data stream.
The definition of the continuous WT is given by Eq. 11 and
The WT is a convolution of the data sequence with a scaled and translated version
of the mother wavelet, the psi function. This convolution can be accomplished
directly, as in the first equation, or via the Fast Fourier Transformation (FFT)-based
fast convolution in the second equation. Note that the WT is a continuous function
except for the discrete data series x and its discrete Fourier transform. In
these equations the *symbolizes complex conjugation, N is the data series length,
s is the wavelet scale, dt is the sampling interval, n is the localized time
index and omega is the angular frequency. Each of the equations contains a normalization
so that the wavelet function contains unit energy at every scale.
In the WT, for each value of the scale used, the correlation between the scaled
wavelet and successive segments of the data stream is computed. Unless reconstruction
is needed, there are no restrictions in the WT as to the number of scales are
to be used, nor of the spacing between the scales. A WT spectrum can use linear
or logarithmic scales of any density desired. If needed, a high resolution spectrum
can be generated for a narrow range of frequencies. The convolutions can be
done up to N times at each scale and must be done all N times if the FFT is
used. The WT consists of N spectral values for each scale used, each of these
requiring an inverse FFT. The computational load of the WT and its memory requirements
are thus considerable. The benefit from this high measure of redundancy in the
WT is an accurate time-frequency spectrum. The most commonly used WT wavelet
is the Gaussian wavelet is defined as:
by taking the pth derivative of f. The integer p is the parameter
of this formula, Cp is the such that:
where f (p) is the pth derivative of f.
The Morlet wavelet depicted as Gaussian-windowed complex sinusoid is also commonly
used WT wavelet that is defined as following in the time and frequency domains:
In the Eq. 17 and 18, ψ, eta is
a non-dimensional time parameter, m is the wavenumber and H is the Heaviside
function. In the time domain plot that follows, the complex Morlet wavelet
is shown in Fig.6 with an adjustable parameter m (wavenumber) of 8. In addition,
the complex Gaussian wavelet is defined as:
|| Representation of the complex morlet wavelet
So, the complex Morlet wavelet is defined by Eq. 20.
where, fb is a bandwidth parameter.
fc is the wavelet centre frequency.
The complex Gaussian wavelet and complex Morlet wavelet is expedient for the decomposition of the complex wave like as the seismic wave.
We can use Eulers formula to define the analytical signal as,
where, i is the imaginary unit to give a more concise formula.
The Fourier coefficients are then given by equation:
The Fourier coefficients an, bn, cn are related
However, after decomposition of the seismic signal basis on the complex Morlet
wavelet, the wavelet coefficient can be represented as the decomposed signal
by Eq. 26:
where, H represents the Hilbert transform; the magnitude of ψ (t) is
and the phase of ψ (t) is:
The Eq. 29 and 30 show the magnitude
and phase extraction through the complex Morlet wavelet transform. The phase
spectrum using complex Morlet wavelet transform is revealed in the Fig.
7. This phase extraction using complex Morlet wavelet is the significant
step to determine phase velocity.
The determination of group velocity and phase velocity is correlated. The group
velocity is obtained from the group delay which is the guiding principle to
obtain phase velocity. Figure 8 shows the group delay corresponding
to the time domain signal between two receivers.
||Determination of phase spectrum through complex morlet wavelet
||Determination of group delay tg1 and
tg2 at receiver No. 1 and 2
||Determination of Phase delay tph1 and
tph2 at receiver No. 1 and 2
|| Multi-channels surface wave signal
The phase delay determination
correlated with group delay is the key step to determine phase velocity. The phase delay is also shown by the Fig. 9 which is used
to extract phase velocity. Extraction of phase velocity through MASW method
is the key concentration of this study. The concept of MASW method is developed
on using multiple receivers for acquisition (Park et
al., 1999a, b). The acquired multiple receiver
corresponding signal is processed to determine phase velocity as well as group
velocity with robust performance. The concept of the extraction of phase velocity
and group velocity is implemented corresponding to multiple-channels surface
wave signal shown in Fig. 10. Every adjacent pair is regarded
to attain phase velocity corresponding to multiple receivers. The phase velocity
versus frequency curve known as dispersion curve is an attempt to deliver the
information of soil characteristics through inversion analysis. The multiple
dispersion curve-phase velocity versus frequency curve is compared to reduce
the presence of noise and erroneous outcome.
The procedure to determine the group and phase velocities with frequency is
||Compute Complex Morlet wavelet transform of signals obtained
at receiver 1 and 2.
||Determine phase and group delays in frequency at fc frequency which is the centre frequency of complex Morlet wavelet.
||The group delays at receivers 1 and 2 which are tg1 and tg2 are obtained. The group delay is a time corresponding
to the maximum of magnitude of ψ1 and ψ2
||From phase information of, ψ1 (t) is taken
as a phase corresponding to tg1
||The tL and tR are obtained from phase
information of ψ2. The tL is the time corresponding
to θ1 which is the most close to tg2 on the left side of tg2 and tR is the time
corresponding to θ1 which is the most close to tg2 on the right side of tg2
||tph1 is defined as tg1 and tg2 is either tL or tR depending
upon which is closer to tg2
||To determine the phase and group delays at whole frequencies,
the procedure (2) has to be repeated for all complex Morlet wavelet coefficients.
||If the distance between receivers 1 and 2 is D, then the group
velocity Vgr and phase velocity Vph at each frequency
are obtained as follow:
||The procedure is continued until all receivers corresponding
signal is regarded iteratively for processing.
The new method is more robust and persistent than conventional phase extraction
method because the reduction of noises during analysis as well as easier and
faster analysis. The dispersion curve extracted using proposed method is shown
by Fig. 11 to estimate the performance. At frequencies larger
than about 150 Hz, it can be noticed that the dispersion curve obtained by the
conventional MASW method in Fig. 12a coincides well with
the dispersion curve obtained by the proposed method in Fig.
11. With comparing the result of new method with previous finding relates
through the high frequency corresponding low phase velocity. The good agreement
of both finding is also supportive by the dispersive characteristic of soil
because high frequency penetrates near surface layer consisting low shear wave
||Phase velocity versus frequency curve depicted from phase
However, at low frequencies below 150 Hz, both dispersion curves do not match.
This disagreement may be owing to the near field effect which comes from the
coupling between surface and body waves as noises. The disagreement at very
low frequency like below 10 Hz is not so concentric for the measurement of near
surface layer profile. Among the frequencies about 50 to 100 Hz the irregularities
is found regarding the dispersive trend in the conventional extracted phase
velocity in Fig. 12a and this portion of previous findings
contradicts with the phase velocity of those ranges in new findings of Fig.
11. According to conventional result in Fig. 12a, low
phase velocity 200 m sec-1 is shown at low frequency 50Hz and phase
velocity about 500 m sec-1 is pointed at 100 Hz. This outcome means
deep layer consisting soft soil surrounded by hard soil which is not supportive
by dispersive characteristics of soil. According to this usual MASW method,
the irregularities at frequency range 50-100 Hz in Fig. 12a
is amended by curve fitting in Fig. 12b. The adjusted phase
velocity by curve fitting criteria in Fig. 12b is well suited
with the extracted phase velocity by proposed method in Fig.
11 except the very low frequencies below 10 Hz. For the near surface layer
profile, the phase velocity at very low frequencies is regardless for the half-space
The significant outcome for the new method is representation of consistent
dispersion curve avoiding irregularities and noise impressions. The reliable
dispersion curve is the key success of this study because this stage is ineffort
to characterize soil profile through inversion method. The dispersive outcome
of the proposed method is sustained with outcome of MASW method regarding noise
filter criteria. The presence of noises during signal processing is kept away
from the extracted phase velocity based on new method.
||(a) Extracted phase velocity using conventional surface wave
method, (b) adjusted dispersion curve with erroneous
Furthermore, the analysis of the proposed method is robust and persistent for using the wavelet transformation. The outcome of this new method is obtained easily and rapidly avoiding Fourier transformation. The main benefit from this high measure of redundancy in the WT is an accurate time-frequency spectrum.
The performance of new method is estimated by comparing the dispersive output
with that of conventional MASW method. For the accuracy and pioneering performance,
the new MASW method possesses several advantages for different Geotechnical
application. Anomalies that include fracture zones within bedrock, dissolution/potential
subsidence features, voids associated with old mine works and erosional channel
eched into the bedrock surface have been effectively identified by proposed
MASW method. Advantages of using wavelet transform in MASW method are to detect,
delineate, or map anomalous subsurface materials including insensitivity of
cultural noise, ease of generating and propagating surface wave energy and its
sensitivity to change in velocity.
The dispersion curve represents phase velocity corresponding to frequencies
in geotechnical characteristics for surface wave analysis. Multi-station recording
permits a single survey of a broad depth range, high levels of redundancy with
a single field configuration and the a bility to adjust the offset (Lin
and Chang, 2004). Park et al. (1999a, b,
2000) extracted the dispersion curve with modal discrimination
using frequency domain analysis for MASW method. This development is also important
for representing the mode separation in dispersive outcome. But, the noise reduction
performance is not improved through this conventional MASW method. Miller
et al. (2000) has compared shear-wave velocity profiles from usual
MASW method with Borehole Measurements and resulted in differences between inverted
S-wave velocities between the MASW method and borehole measurements to be as
low as 18%, with potential improvement as low as 9%. The overall difference
between Shear (S) wave velocities derived from the MASW (multi-channel analysis
of surface wave) technique and borehole measurements shown by
Xia et al. (2002a, b) is about 15%.
Due to the analysis corresponding to the new method, the extracted phase velocity
data is achievable to compare and adjust if any irregularities occur. After
the comparing and irregularities adjustment, the final dispersion curve shown
in Fig. 11 is obtained through proposed method regarding
complex Morlet wavelet transform. Usually, the noise reduction technique or
irregularities adjustment is not maintained for the final phase velocity representation
(Park et al., 2000).
However, in the conventional MASW method, the phase velocity is extracted using
Fourier transformation and cross spectral density corresponding to all received
signal which is tedious for analysis. Some noise and interferences added during
Fourier transformation shown in Fig. 13 as well as wrapped
and unwrapped phase angle extraction shown in Fig. 14. Furthermore,
the Fourier transformation of each receiver corresponding signal, cross spectral
density between each pair of receivers and additional filtering technique of
noises make the conventional method slower.
On the contrary, the proposed method is robust with faster, easier and noise
reduction performance during analysis for using WT avoiding Fourier transformation
and conventional phase velocity extraction.
||Representation of noises during Fourier transformation of
||Presence of noises in (a) wrapped phase angle, (b) unwrapped
For the utilization of the wavelet transform in multichannel analysis, the
performance is superior because wavelet theory is one of the most successful
tools to analyze, visualize and manipulate complex non-stationary data, for
which the traditional Fourier methods cannot be applied to directly (Yves
et al., 1992). In each of these fields, the wavelets are applied
for data compression, noise removal, feature extraction, classification and
regressions face (Saito, 2004).
For multichannel analysis in the surface wave method with Morlet wavelet transform, the phase velocity outcomes for each pair of receivers are compared. In the WT, for each value of the scale used, the correlation between the scaled wavelet and successive segments of the data stream is computed. Unless reconstruction is needed, there are no restrictions in the WT as to how many scales are used, nor of the spacing between the scales. If needed, a high resolution spectrum can be generated for a narrow range of frequencies. The unique success of the proposed method is estimated by the rapidness, easiness and noise reduction performance during analysis rather than conventional surface wave method.
In this study, a wavelet decomposition technique is implemented on MASW method to obtain an evaluation of dispersion curve. A more stable and consistent outcome is obtained with reference to the de-noising performance of Morlet wavelet analysis. In the evaluation method, the phase and group velocity are determined in correspondence to multiple receivers at each frequency component using the information obtained around the time at which the signal energy is concentrated. Therefore, the new method is less affected by noise, effect of body wave or higher mode surface wave. Furthermore, the imaging of soil layer profile in geotechnical subsurface exploration can be developed inaugurating wavelet analysis in MASW method.
This research is sponsored by Research Project of Science Fund No. 01-01-02-SF0338 from Ministry of Science, Technology and Innovation of Malaysia.