1. 1
Estimating the shear wave velocity profile in marine sediments.
Presentation and comparison of wavelet techniques and evaluation
of an inversion code.
Nikos Papantonopoulos
Master thesis for the M.Sc in Sound and Vibration, Royal Institute of Technology (KTH), Stockholm,
Sweden

2. 2
Acknowledgments
This work has been my first insightful interaction within the general field of Underwater
Acoustics. It comprises the master thesis part of an M.Sc degree in ¨Sound and Vibration¨ offered by
the Royal Institute of Technology (KTH) in Stockholm. An initial frustration has transformed into a
rewarding satisfaction, which made me become certain that this is a field that I would love to
continue learning about and researching into. Because of this understanding that occurred to me, I
feel the need to thank my supervisor Prof. Hefeng Dong of the Acoustics Group of NTNU for her
guidance and patience and Prof. Jens M. Hovem for making me feel welcome to the Group. I would
also like to thank Prof. Anders Nilsson from KTH for his support and concern during the whole
period of my M.Sc studies. The study, programming and computations of this master thesis were
performed at the Acoustics Group of NTNU in Trondheim.

3. 3
Abstract
Knowledge of shear wave velocity as function of depth in the top-layer of the sediment is very
important for the applications in the seafloor. This information can be obtained by geoacoustic
inversion. A possible approach for estimation of shear wave velocity as function of depth is based on
the dispersion characteristics of seismic interface waves propagating along the water-seafloor
boundary. A dispersion estimation of seismic interface waves can be obtained by time-frequency
analysis of wavelet transform (WT). In order to achieve a more accurate estimation of the dispersion
of the interface wave, an adaptive wavelet which is driven by the data is presented. The results of the
dispersion analysis through the use of the adaptive wavelet are compared with the results from a
common wavelet method utilizing the Morlet wavelet and by using both experimental and synthetic
data.
An inversion algorithm is evaluated with the use of synthetic data. A general presentation of
the mathematical background is attempted, as being aware of the theory behind the current
methodology of the problem proves to be vital for the reader.

4. 4
Contents
1. INTRODUCTION 5
2. THEORETICAL BACKGROUND: SOME PROPERTIES OF SURFACE WAVES AND METHODS FOR
THEIR ANALYSIS. 6
A. Surface waves 6
B. Dispersion 7
C. The Continuous Wavelet Transform 9
D. The Adaptive Wavelet 11
E. Inversion 12
F. Singular Value Decomposition of a matrix 13
G. Synopsis 14
3. ESTIMATION OF THE SHEAR WAVE VELOCITY PROFILE: PRESENTATION OF THE KEY FACTORS
OF THE PROBLEM. COMPARISON OF THE WAVELET TECHNIQUES 14
A. Data analysis 15
A1. Preprocessing 15
A2. Dispersion analysis of experimental data 16
B. Comparison with the aid of OASES 19
C. Synopsis 27
4. INVERSION OF DISPERSION DATA. SHEAR WAVE VELOCITY PROFILE ESTIMATION. 28
A. Tests 29
B. Synopsis 38
REFERENCES 40

5. 5
1. Introduction
Knowledge of the geoacoustic properties of the sea-bottom is of great importance for the
correct prediction of underwater acoustic propagation. The geoacoustic properties are considered the
compressional (P) and shear (S) wave velocities and corresponding attenuation expressed as a
function of depth within the sediments. As the medium in underwater acoustic propagation could be
thought off as a waveguide with an elastic bottom where acoustic energy generated in the water
column interacts with the seafloor and the sub-bottom layers, the properties of the sea bottom have to
be accurately known for the correct determination of the transmission loss in an acoustic field. From
the total of the geoacoustic properties, it is the shear wave velocity that is usually the most important
one while difficult to measure. Furthermore as the shear wave velocity is related with the shear
strength of a sediment, its knowledge is deemed important for uses in the geotechnical and offshore
industry. The installations made upon the seabed by this type of applications, including
communication cables, gas and oil cables and other types of underwater constructions, require the
ability to predict the amount of load that the seabed is able to support. Thus there is an increased
importance in an accurate estimation of the shear wave velocity profile with depth within the
sediments. An estimation of the variation with depth of the P and S-wave velocities is also of great
importance in rock physics e.g. for establishing velocity-depth trends i.e. the way that sound speed
tends to vary with depth in a specific type of sediment.
Determining the compressional parameters and density is a task that utilizes conventional
reflection and refraction seismic methods. A common method to estimate the shear wave velocity
variation with depth is by sampling or coring with subsequent laboratory measurements. Another
method for deriving the S-wave velocity information is through the use of S-wave sources placed at
the seafloor and the accompanying traveltime and refraction measurements. The operational effort
and technical difficulties that are embedded in the above treatments together with the limited spatial
coverage that these methods can provide, calls for an alternative approach in measuring the shear
structure of the seabed. This approach utilizes the established relation between the dispersion of
interface waves that propagate along the seafloor and the geoacoustic parameters that characterize the
marine sediments [1,2,3,4,5]. An analytical treatment that gives an insight in the aforementioned
relation is presented in [6]. Surface waves exhibit the phenomenon of dispersion when propagating
along the interface with a solid inhomogeneous medium. In the marine seismics case these waves are
known as Scholte waves and the requirement for extracting their dispersive characteristics involves
the selection of an appropriate time frequency representation (TFR) method to be applied on the
recorded signals. The present treatment makes use of the continuous wavelet transform (CWT) for the
purposes of estimating the dispersion curves of the interface waves in terms of the group velocity of
the interface waves. An alternative approach for extracting dispersion data is a multi-sensor method
which is able to come up with an estimate of the phase velocity variation with frequency. After the
dispersion data have been obtained, an inversion algorithm is required providing as output an estimate
of the shear wave velocity as a function of depth.
In this thesis two different wavelet techniques are compared using as input both experimental
and synthetic data. The first of the wavelet techniques is using the Morlet wavelet in the WT, while
the other makes use of the Karhunen Loeve Transform for the purposes of creating an adaptive set of

6. 6
wavelet functions. The behavior of the wavelet methods in relation with the signal under analysis is
explained through the comparison results. The extracted dispersion data that originate from synthetic
surface wave data are used as input to an inverse algorithm and the algorithm is evaluated for a
number of cases.
2. Theoretical background: Some properties of surface waves and
methods for their analysis.
There exist three basic types of seismo-acoustic waves each of them with their own uses in
seismic exploration.
The most widely used, in partial accordance with their earliest discovery, are the pressure or
primary waves indicated more conveniently with the symbolism P-waves. These waves are
dilatational waves where the elemental volumes of a solid or a liquid that undergo motion due to their
existence are exhibiting only a change in their volume while keeping their shape unaltered. Pressure
waves have a polarization parallel to the propagation direction and from this property arises another
designation that symbolizes them, the one of longitudinal waves. By polarization we mean the
direction of displacement of the elementary volumes of the medium in which the wave propagates.
We may note the important fact that primary waves possess a higher speed of propagation when
compared to the other types of waves.
The waves that are related to the shearing motion of solids are known as shear or secondary
waves (S-waves). The term secondary was used as these waves were appearing –initially on
seismograms- after the passage of the faster P-waves. Such waves are polarized in a direction
orthogonal to the direction of propagation. Two distinct modes of polarization can be observed, the
shear vertical (SV) mode and the shear horizontal (SH) mode. As liquids can not support a static shear
stress, these waves exist mainly in solids, although for high frequencies and short distances shear
waves can propagate in a liquid medium.
The third type of waves is the one of surface or interface waves and some additional focus on
their qualitative properties will be given on the next section.
A. Surface waves
Surface acoustic or, as sometimes mentioned, interface acoustic waves are waves that are
propagating along an interface between two media of different acoustic impedances. The names used
to describe them are providing us with the information of the material state of the associated media.
The subsequent presentation takes into account the seismo-acoustic nature of the problem treated in
this thesis. Interface waves are categorized as,
• Rayleigh waves, waves that propagate at an interface between air and a solid. The
motion associated with this type of wave is of P-SV type. There is a coupling
between the longitudinal and shear waves that in this case are traveling with a
common velocity, the Rayleigh velocity. Lord Rayleigh’s (John William Strutt)

7. 7
intuition led to the prediction of these waves in 1885. Although the classical
treatment of the description of this wave type is assuming a solid in vacuum, in
practical cases it is the air which replaces the vacuum.
• Love waves propagate along a boundary between a fluid (e.g. water) and a solid (e.g.
marine sediments). The main difference between this type of waves and the Rayleigh
waves is that Love waves are horizontally polarized. Love waves do not exist in a
homogeneous half-space and the presence of a layer over a half-space is essential for
the existence of these waves. It was A.E.H. Love who developed a mathematical
model for the waves that carry his name since 1911.
• Scholte waves propagate along the interface between water and sediments. These
waves are characterized by a motion of SV type. The recorded or synthetic waves that
will be analyzed in the present work are of this type. A wave of this type is an
inhomogeneous wave with amplitude decaying away from the surface, on both sides
of it, though more slowly in the fluid half-space than in the solid one. Seismic
interface waves at the water-sediment interface are of the Scholte type.
• Stoneley waves, in a general sense, are waves that travel along the boundary between
two elastic media. These waves consist of P-SV inhomogeneous waves and in the
case of two solids they exist only for certain ranges of density and sound velocity
ratios.
A common property of the total of the surface waves, is that their amplitude decays
exponentially with distance from the surface and experimental experience shows that they can be
detected at a distance of magnitude of approximately one wavelength from the surface. Surface waves
exhibit the property of dispersion in the case where at least one of the two half-spaces that are
separated by the interface is layered. In the marine seismics case, the sediment environment consists
of a number of sediment layers of different geoacoustical properties each. This property of the sea-
bottom causes the Scholte waves to be dispersive. The concepts of phase and group velocity are then
entering to help with the characterization of dispersion. The phenomenon of dispersion of seismic
interface waves is strongly linked with the shear wave velocity that characterizes each one of the
layers. In addition, the existence of interface waves could be attributed to the effects of shear in the
elastic medium along the boundary of which these waves propagate.
B. Dispersion
Surface waves are non-dispersive only in the case where they propagate along an interface
with a homogeneous solid halfspace. If the solid is inhomogeneous, as is the case when one or more
of the acoustic parameters (i.e. parameters that enter the acoustic wave equation) are not constant
inside the volume of the solid, the surface waves become dispersive. In general terms, this means that
different frequency components will travel with different speeds as the wave progresses. This causes
the shape of the wave to spread in time and become more extended as the wave propagates.
Dispersion is a property of the wave and it is expressed through the terms of phase and group velocity
which are both dependent on frequency. In figure 2.1 a dispersive interface wave is shown through the
vertical particle displacement components in 4 different offsets (distances from the source). The
spreading of the time-length of the signals as one moves away from the source is an effect of the
dispersion.

8. 8
Figure 2.1. The manifestation of dispersion in the time records of particle velocity component
As the approach of analyzing dispersion involves the establishment of a relation between the
group or the phase velocity and frequency, an explanation of these notions should be attempted. A
rather simple but illuminating example that describes a part of the concept of the phase velocity is
given bellow.
Let us take the case of two plane waves that possess equal amplitudes and different frequencies and
wavenumbers,
u1
(x,t) = Aei(!1t"k1x)
u2
(x,t) = Aei(!2
t "k2
x )
(2.1)
where,
!1 = !0 +"!, !2 = !0 #"!, k1 = k0 +"k, k2 = k0 #"k
The superposition of the two waves provides us with a new wave described by,
u(x,t) = 2Acos(!" t #!k x)ei("0t#k0x)
(2.2)
When the waves are close together in frequency and wavenumber, the above equation represents two
waves. One is ¨visible¨ through the exponent and is the one which travels with a velocity c=ω0/k0 , the
phase velocity. The second wave is the one represented by the cosine factor, with a velocity equal to,
V =
!"
!k
=
d"
dk
= c + k
dc
dk (2.3)
The above V is the group velocity. What equation (2.2) tells us is that the wave with velocity V is
modulating the wave with velocity c. The latter wave is known as the carrier wave.
The previous approach will help us understand the more general case of a dispersive wave. In this
case the wavenumber is a function of frequency and thus symbolized by k=k(ω). The Fourier
transform theory supplies us with the possibility of expanding an arbitrary wave into a superposition

9. 9
of an infinite number of plane waves each one having an infinite number of single frequency
components. This is described by,
u(x,t) = A(!,k)ei(!t"k(! )x)
## d!dk (2.4)
keeping with the one dimensional space for reasons of simplicity. The condition,
!t " k(!)x = # (2.5)
where λ a constant phase, is known as the dispersion relation. From the above the phase velocity is
derived, representing the propagation velocity of a surface with constant phase. In this case,
c =
!
k(!)
(2.6)
The group velocity expresses the velocity of propagation of the wave packet or equivalently
the speed of propagation of each energy component (of frequency ω) of the wave and is given by,
Vg
=
d!
dk(!)
(2.7)
For the purposes of the estimation of shear properties it will be necessary to extract the
dispersion curves i.e. the curves that give us the variation of phase or group velocity (consider also the
fact that the group velocity is the derivative of the phase velocity) with frequency in a wave. In order
to realize this, we resort to certain methods that can provide us with the time-frequency representation
of a certain signal. The continuous wavelet transform is the starting element and a short presentation
of the transform will be given next. However there are several ways for estimating group and phase
velocities. Some of them include Prony’s method, multiple filter analysis (Gabor matrix) and the 2D
DFT among others.
C. The Continuous Wavelet Transform
We are facing the problem of determining in a robust and efficient way the dispersion relation
of a wave, in our case a surface acoustic wave. The mathematical definition of the continuous wavelet
transform (CWT) is,
W (b,a) =
1
a!"
+"
# x(t) w$ t ! b
a
%
&'
(
)* dt (2.8)
where W(b,a) symbolizes the wavelet transform of the signal x(t), w is a function of time known as the
mother wavelet and the symbol * denotes the complex conjugate. The wavelet is assumed to be a
function well localized in the time and frequency domain and obeying the oscillation condition,
w(t)dt = 0
!"
+"
# (2.9)

10. 10
It is more convenient in practice to compute the wavelet transform in the frequency domain
through the Fourier transform (FT) of the signal under analysis, and by using the following relation,
W (b,a) =
!"
+"
# ˆx( f ) ˆw$
af( ) ei2% fb
df (2.10)
where the symbol ^ accounts for the FT of the signal and wavelet.
We will try to give a description on how the wavelet methods work. The first step in
obtaining the CWT of a signal x(t) is to select a mother wavelet. This mother wavelet is an oscillating
function in time and for a common type of wavelets named as the Morlet wavelet, this function is
defined as,
w(t) = !"1/4
ei2! fct
e"t
2
/2
(2.11)
In the previous equation fc is the central frequency of the mother wavelet. What equation (2.11)
reveals is the fact that the division with the scale α of the wavelet’s independent variable t,
corresponds with a multiplication with the same scale when we turn into the frequency domain. This
stems from the basic properties of the FT. For a wavelet’s center frequency fc and a physical
frequency f, the relation between the scale and the frequency f is given by,
a =
fc
f
(2.12)
In this way we can select the value of the scale in order to search for a desired frequency in a signal.
A helpful presentation on how a wavelet operates on a signal, should be based on a visual
representation of these operations. In figure 2.2 a wavelet is shown shifted in time in three different
positions over a seismic signal.
Figure 2.2. A seismic signal (particle velocity component) scanned by a wavelet of a specific scale
What a wavelet of a given frequency actually does, is to scan the signal in time. When applying the
wavelet transform to a signal, we are requested to select a number of frequencies and time shifts for
the different wavelets. A discrete number of scales will correspond to the same number of different

11. 11
daughter wavelets. The time shifts correspond to the quantity b in the wavelet transform.
Simplistically, we have a wavelet of a certain frequency that moves in discrete steps in time, from the
start until the end of the time signal that we are analyzing. This operation will resolve the frequency
content of the signal with the basic limitation addressed by the famous Heisenberg uncertainty
principle. As frequency is a measure of a repetition occurring in time, there is the need of performing
measurement in a finite time length in order to measure frequency. It is therefore not possible to speak
about frequency content of a time signal at a single moment in time.
We have to remind that the implementation involves multiplication between the FT of each of
the daughter wavelet vectors and the signal vector and then the use of the inverse FT in order to
acquire the CWT matrix. This matrix will be a ready interpretation of the frequency content in time of
the signal.
D. The Adaptive Wavelet
The choice of a wavelet function is crucial for the derived TFR of the seismic signal of
interest. The adaptive wavelet is based on the Karhunen Loeve Transform (KLT, named after Kari
Karhunen and Michel Loève) or Singular Value Decomposition (SVD) as it is most commonly
known. Other names include those of the Principal Component Transformation and Hotelling
Transformation. In this case, the functions that are going to serve as wavelets are derived directly
from the signal features.
The KLT can be applied to a seismic section that consists of a number of M traces each one
having N points, thus comprising a matrix of an M x N dimension. In the present work every seismic
trace is analyzed individually in order to estimate the group velocity variation with frequency of the
surface waves. A vector-based presentation of the KLT is thus more appropriate. For a N x 1 random
vector x that in our case represents a sampled trace, we are able to express x as a linear combination
of orthonormal basis vectors,
xi
= aj
wji
j=1
N
! i = 1,2,..., N (2.13)
where w = (w1, w2, … , wN ) is the set of basis vectors and aj are random coefficients. In matrix
formulation equation (2.13) is equal to,
X = W A (2.14)
The KLT will provides us with a set of basis vectors A that follow from the above equation if
rewritten as,
A = WT
X (2.15)
The matrix W is derived from the covariance matrix of the process x (i.e. the signal under analysis) as
described next. Starting with the definition of the cross covariance as the cross correlation of mean
removed sequences designated by,
xcov(µ) = E xn+m
! µx( ) yn
! µy( )"
{ } (2.16)
where x and y are two stationary random processes with mean values µx and µy respectively and E is
the expected value operator.

12. 12
In the case where only one process is involved, the function which interests us is the autocovariance
which is expressed as,
acov(m) = xn+m
! µx( )
n=0
N ! m !1
" xn
*
! µx
*( ) (2.17)
where * is for complex conjugation. The autocovariance (mean removed autocorrelation) of a random
process x is actually a matrix where each entry represents the covariance value of x with its time
shifted self. For a time series represented by the process x, the autocovariance is a measure of how
well the signal correlates with different versions of the same signal that are just shifted in time with
the zero lag signal. After calculating the autocovariance matrix, we must solve the eigen equation for
the autocovariance matrix which is the following,
C = W Λ WT
(2.18)
where C is the autocovariance matrix, W is a matrix whose columns are the eigenvectors of C and Λ
the matrix whose elements are the eigenvalues of C. Returning our attention to equation (2.15) we
observe now how the matrix W is derived.
For the purposes of creating a set of basis vectors that will be used for the expansion of our
signal, we assign the eigenvector of W corresponding to the largest eigenvalue of Λ, as the first basis
vector in the expansion. The second basis vector is the eigenvector that corresponds to the second
largest eigenvalue and by following the same concept we formulate the complete set of the basis
vectors. In this way, noise elements of a signal which by nature have a low degree of correlation with
the signal, correspond to numerically small eigenvalues. By arranging Λ in the way of placing the
eigenvalues in descending order in the diagonal of Λ, we are also placing the eigenvectors of less
importance in the last columns of W. We can then choose only a certain number of eigenvectors as
basis vectors, minimizing in this way the effect of noise.
E. Inversion
Inverse problems have a wide range of classification. We will refer in a wider sense to the
concept of inversion here. What we are dealing with, in science and engineering is the establishment
of a specific relation between a cause and the effects of that cause on a system. This relation can be
described as,
T (!) = " (2.19)
In the above relation, by α, we characterize the model which is actually a set of physical parameters
(density, wave speed, etc) that describe that model efficiently. The term on the right hand side is
usually a set of data recovered from a number of observations. The above relation is very general and
can be used to refer to a number of mathematical relations depending on the nature of α and β. For
example if these terms are symbolizing functions then T is an operator like the case of an ordinary
differential equation. The operator T could also be a linear or nonlinear system of algebraic equations.
Usually, T is mentioned as the forward operator and equation (2.19) as the mathematical model [4].
The forward problem is to find β when α is given. The inverse problem is finding α given β.
We are requested to find a model α which will fit the data β either in an exact or in a close to exact
way. The approximate solution is termed as a least-squares solution in a great number of problems. In
the case where the forward operator is a matrix, we are requested to factorize that matrix and try to

13. 13
come out with a generalized inverse. A common method to achieve this is by the use of the Singular
Value Decomposition (SVD) of the matrix T. There are many issues that enter in the process of the
solution, like the existence (the case where no model exactly fits the data), uniqueness (when more
than one exact solutions exist) and instability (how a small change in the measurements can cause the
solution to vary). Attention has to be paid to the fact that, when depending on measurements in order
to obtain a data set β, noise will inevitably enter into the data set, making β a superposition of the
¨correct¨ set i.e. the one that satisfies exactly the mathematical model, and the noise data. This will
make the inverse problem a lot harder to tackle. As the SVD is the key factor in the inversion
algorithm that is used in this thesis, a short introduction to this highly important mathematical tool is
required
F. Singular Value Decomposition of a matrix
Following the discussion of the previous sector, we consider the matrix T with the dimension
m x n. The SVD of T is given by,
T = U!VT
(2.20)
where U,V are orthogonal matrices of m x m and n x n dimensions respectively and Σ is an m x n
diagonal matrix, while T
indicates the transpose. The diagonal entries of Σ are called the singular
values of T and the matrix analysis requires them to be arranged in a descending order, such that the
first singular value σ1 in Σ is the largest one. The singular values can be positive or zero. A question
arises on how do we compute the above matrices. It can be shown that for the singular values and the
singular vectors i.e. the columns of U and V the following relations are true,
Tvi
= !i
ui
TT
ui
= !i
vi
(2.21)
where ui , vi are the ith
columns of U and V respectively and σι is the ith
singular value of T.
Furthermore, the singular values σι can be proved to be the positive square roots of the eigenvalues of
the matrices TTT
and TT
T. Using these facts we are able to compute the SVD of the matrix T. There
exists a number of efficient algorithms that may assist us in this task.
With the help of the SVD we can create a generalized inverse matrix that can be used for the
solution of an inverse problem. This matrix is called the Moore-Penrose pseudoinverse and is equal
to,
T
†
= V!
"1
U
T
(2.22)
leading us to the solution,
!†
= T†
" = V#$1
UT
" (2.23)
The above solution can be proven to be a least-squares solution. This solution exists always as the
pseudoinverse matrix always exists as well. The SVD is used for solving ill-posed inverse problems,
that is problems where instability becomes an issue. The numerically larger singular values and their
inter-connected singular vectors are those with the highest significance as it comes to the factorization
elements of the matrix. The lowest in order singular values and usually those arithmetically close to
zero are representing the effects of discretization in the formulation of the forward problem. An

14. 14
integral equation for example should first be discretised (discretization of the domain of integration)
in order to come up with the inverse problem which contains discrete measurement data. The decay
rate of the singular values and the possibility of separating the discretization-related singular values
from the ones that are a part of the real solution is what defines the ill posedness of a problem. The
presence of noise will severely affect ill-posed problems. Regularization is applied in these cases for
stabilizing the solution of inverse problems.
G. Synopsis
In chapter 2, a presentation took place of the basic theory that underlies the treatment of the
problem of estimation of shear wave velocity in seafloor sediments. It is the dispersive properties of
interface waves and their connection with the shear properties of the sediments that are providing us
with a possibility of a remote-sensing of the shear parameters. We have seen that dispersion is
actually quantified with the variation of phase or group velocity with frequency for a dispersive wave.
The way that this variation is presently analyzed is by the use of the CWT. The theory behind two
different wavelet techniques, the comparison of which is included in the next chapter, was presented.
The relation between the geoacoustic parameters and the phase velocity dispersion is well-
known and so is the forward problem. Inversion is required for estimating the shear properties from
the dispersion of interface waves. As the theory behind inverse problems proves to be complex, we
tried to give a short presentation of the general concepts.
3. Estimation of the shear wave velocity profile: Presentation of the
key factors of the problem. Comparison of the wavelet techniques
Initially, there is a clearly defined goal of generating and recording the surface waves on the
sea-floor that lies above the area under investigation. There exists a general methodology used in this
kind of experiments. A seismic source is required for the generation of a pressure wave in the water
column and operated in close proximity with the seafloor. In order to generate a multimodal surface
wave the source should be placed as close to the seafloor as possible. Explosive charges are used as
sources and the requirement is a high coupling of the energy of the source into the sea-bottom. For
acquiring the data either a hydrophone array, a geophone array or Ocean Bottom Seismometers (OBS)
can be used. The OBS are units that integrate a hydrophone together with a three-axial geophone and
is the type of sensors that were used in the Tommeliten experiment. The experimental data used in
this thesis are acquired at this particular experiment. The Tommeliten experiment is a Subsea seismic
(SUMIC) survey acquired in 1993 by Statoil at the Tommeliten field located in the Norwegian sector
of the North Sea. The data that have been used in the current work are the vertical and horizontal
particle velocity components that were recorded by the OBS. In that experiment, two vessels were
deployed, one for positioning and controlling the data acquisition equipment and one for operating the
seismic source. The data acquisition procedure applied at Tommeliten was the use of a small number
of geophones along a short section of the survey line and the firing of the seismic source a large

15. 15
number of times along a long segment of the survey line [1,2]. At present there is no need to get more
involved with the technicalities of the experimental procedure. It is sufficient to be aware of the
physics involved in the generation of the surface waves on the sea-floor. In this context a
simplification of the configuration used in this type of experiments is shown in figure 3.1.
Figure 3.1. Basic configuration of data collection for a seismic survey intended for the recording of interface
waves
A. Data analysis
Data analysis involves the pre-processing of the raw data obtained from the transducers on the
seafloor and the application of signal processing techniques in order to extract the dispersion relation
in terms of the group velocity or phase velocity variations with frequency. At the end of the analysis a
set of data ready for inversion should be extracted.
A1. Preprocessing
After the data have been obtained, we are facing the problem of separating in an efficient way
the surface waves from the rest of the waves that exist in the data set. Some pre-processing is required
before the stage of the dispersion analysis. The total of the processing of the raw data is done within
MATLAB. The code that provides the wavelet analysis has been developed by Alexander Kritski and
Trygve Carlsen for the case of the adaptive wavelet analysis [12]. A number of modifications were
added for plotting and selecting the dispersion data that will subsequently be used for inversion. A
presentation of the processing steps of experimental data will be given at first, as it suits best for
explaining the practical issues involved.
Every geophone’s record constitutes a trace and by plotting the total of the traces on the same
graph we obtain a first indication of the existence of P and S waves together with the interface waves
that might exist. The faster propagating pressure waves are usually easy to discriminate from the rest,
while the shear and surface waves tend to overlap in time. To create a set of traces which will contain
as possible the surface waves, a low pass filter is initially applied to the total of the traces. A time-
equal spacing
geophone array
explosive source
water column
solid seabed
interface wave

16. 16
variable gain and offset scaling is applied next and the signals are Fourier transformed into the
frequency domain where multiplied with a window function. The inverse Fourier transform provides
us with the optimized trace signals containing the surface waves. It is worth noting that in the
experiment at Tommeliten the source which was towed from a vessel, was fired at a depth of 6m
while the total depth of the water column was 70m, resulting in low amplitude-surface waves in
comparison with the primary waves. An appropriate pre-processing phase was deemed of great
importance in that case. In figure 3.2 the raw and pre-processed data of the horizontal (X) and vertical
(Z) particle velocity components are presented. It is evident that the surface waves are present in the
data set and are dispersive. The slope of the red lines indicates an approximate group velocity range
for the velocity components, a range that extends from 300 to 596 m/sec for the Z component and
from 327 - 655 m/sec for the X component. The higher group velocity values for the horizontal
component are justified since it is usually the horizontal component that reveals the presence of
higher order modes in the surface waves. Another expected outcome of the dispersion analysis is the
stronger presence of the low order mode in the vertical particle velocity component.
Figure 3.2. Recorded data from the Tommeliten experiment. The unprocessed (raw) and filtered data of both
the particle velocity components are shown. The lines (red) are indicating the dispersion with slopes that give
us an indication of the limits of the group velocity range of the surface waves.
Consequently, a set of traces optimized for the purposes of dispersion analysis is obtained. As we may
observe on the previous plots, the higher frequency primary wave has been filtered out together with
noise components of high frequency content. Dispersion is also visible as the spreading in time of the
signals with distance from the source.
A2. Dispersion analysis of experimental data
The need for processing the seismic data in such a way that high resolution dispersion curves
can be extracted, leads us to the use of the CWT. In the present context, dispersion is presented as the
variation of group velocity versus frequency. This treatment offers the possibility for determining also

17. 17
horizontal variations of the shear parameters. A thesis work [11] that has treated the comparison
between two different wavelet functions in the CWT, the Morlet and the Mexican hat wavelet
respectively, has indicated that, from these standard wavelets it is the Morlet wavelet which provides
the best results. More recently, in [13], it was shown that the adaptive wavelet can provide an equal or
a better time-frequency resolution when compared to the Morlet wavelet, especially in the high
frequency region. Additional papers on the topic are [13,14,15].
The attention is focused again on a comparison between the adaptive and Morlet wavelet
methods. When first dealt with the topic, the records acquired at the experiment at the Tommeliten
field in Norway were used. The comparison between the Morlet and adaptive wavelet was in favor of
the adaptive wavelet as indicated previously. Some additional comments could be made on the issue.
Keeping in mind that the CWT is given by equation 2.10 and repeated here for the sake of facilitation,
W (b,a) =
1
a!"
+"
# x(t) w$ t ! b
a
%
&'
(
)* dt (3.1)
With the use of the Morlet wavelet as a mother wavelet function, we are requested to select a number
of scales in order to construct the different daughter wavelets, each one scanning the signal x(t) and
searching for a specific frequency component. In the analysis performed, the scale values were
equally spaced and the number of scales was chosen sufficiently large in order to obtain a detailed
time-frequency analysis. Increasing the number of scales after a certain number of approximately 200
scales (depending on the frequency content of the signal) was not providing us with better results. A
higher number of scales were required in the case of a low frequency content of the time signal.
Whereas the Morlet wavelet analysis was using a predetermined function from which the total of the
wavelets were formed, the analysis with the adaptive wavelet utilized the properties of the signal for
creating the required wavelets. This adaptive property is the main reason for the improved
performance of the adaptive wavelet analysis in a number of cases. As the wavelet functions in this
case are derived from the eigenvectors of the autocovariance matrix of the signal, the presence of
noise is possible to be less of a trouble. The solution to the eigen-problem of the autocovariance
matrix, will inherently connect the noise-related eigenfunctions with eigenvalues of low magnitude,
thus enabling us not to select these column vectors as wavelet functions. However, in a highly noisy
environment a speculation can be formed, on whether the signal can provide the sufficient number of
wavelet functions for a quality time-frequency analysis.
In figures 3.3 (Morlet wavelet) and 3.4 (adaptive wavelet) a selected trace from the Tommeliten data
is wavelet transformed and the matrix of the transform is plotted to provide us with the dispersion of
the signal in terms of the group velocity. Each figure contains a colormap plot and a contour plot of
the same WT matrix for both of the particle velocity components. In the time-frequency analysis, time
is converted to group velocity as the distance between source and receiver is known.
There is a clearer representation of the relation of group velocity versus frequency when the
adaptive wavelet is used. The Morlet wavelet seems to spread the energy of the signal across a wider
frequency range, and provides a possibly distorted image of dispersion compared with the one derived
from the adaptive wavelet. Both wavelet methods resolve the same number of modes in each velocity
component, however it is the adaptive wavelet which is able to provide a clearer separation between
the energy in each mode. Noise is efficiently removed with the use of the adaptive wavelet, a feature
which is the main advantage of this technique. Another important observation includes the fact that
with the analysis of the horizontal component of the surface wave, higher order modes were favored
in contrast with the lower order modes. On the other hand, from the vertical component we obtained
usually the two lower order modes, (presumably the zero order and first order mode in respect). From
the wavelet analysis of the horizontal component a second order mode was clearly visible, while the
zero order mode was rather faint. The above is an expected outcome as also described in [1].

18. 18
Fig 3.3. Dispersion curves generated with the Morlet wavelet analysis for the vertical and the horizontal
particle velocity components of the trace number 2 the right side are contour diagrams with the selected
dispersion data for each of the interface wave modes.

19. 19
Fig. 3. Dispersion curves generated with adaptive wavelet analysis for the vertical and the horizontal particle
velocity components of the trace number 25 of the Tommeliten data. The plots on the right side are contour
diagrams with the selected dispersion data for each of the interface wave modes.
B. Comparison with the aid of OASES
The comparison procedure between the two wavelet techniques that can be termed efficient,
requires the use of synthetic data. These data were generated with the OASES code, a computer code
for modeling seismo-acoustic propagation in horizontally stratified media and which constitutes a
continuation of the SAFARI code (developed at SACLANTCEN). The method used by OASES to
solve the problem of wave propagation in horizontally stratified media is by the use of the depth-
separated wave equation and the solution of this equation with wavenumber integration and the Direct
Global Matrix technique. The basic mathematical relations involved are presented next as they
represent a classical approach for the solution of wave propagation problems in range independent
environments. Assuming a horizontally stratified environment as shown in figure 3.5.
Figure 3.5. A horizontally stratified environment, the principle idea for the solution of the depth-separated
wave equation
Layer 1
Layer 2
interface N
interface 1
Layer N
Layer N-1
Upper halfspace
Lower halfspace
r
z

20. 20
By introducing cylindrical co-ordinates as well as the prerequisite that the source distribution follows
the direction of the z-axis, the field in each layer must satisfy the Helmholtz equation,
!2
+ kl
2
(z)"# $%ul
(r,z) = F(z,&)
'(r)
2(r
(3.2)
where u is the displacement potential, kl is the medium wavenumber for each of the layers, and F=0
for source-less layers. By introducing the forward Hankel transform Eq. 3.4, in the above equation,
!(r,z) = !(kr
,z)J0
(kr
r)kr
dkr
0
"
# (3.3)
!(kr
,z) = !(r,z)J0
(kr
r)rdr
0
"
# (3.4)
we obtain the depth-separated wave equation which is an ODE in z,
d2
dz2
+ kr
2
! kl
2
(z)"# $%
"
#
&
$
%
'ul
(kr
,z) =
F(z)
2(r
(3.5)
The solution to this equation is given as,
ul
(kr
,z) =
)
ul
(kr
,z) + Al
+
ul
+
(kr
,z) + Al
!
ul
!
(kr
,z) (3.6)
also known as the depth-dependent Green’s function, where ul is a particular solution and ul
+/-
the
homogeneous solutions, while A+/-
are coefficients that will be determined by the boundary conditions
applied at the interfaces. These coefficients are calculated first and the total field can be found at a
specific frequency and at any range by evaluating the inverse Hankel transform (Eq. 3.3). The inverse
Fourier transform can then supply us with the field distribution in the time domain. In the numerical
solution, the wavenumber domains are discretised. The depth-dependent Green’s function is found at
a selected number of horizontal wavenumber values and the wavenumber integral is evaluated. After
a repetition of these processes for a number of frequencies, the frequency integral (inverse FT) will
give us the response in the time domain.
In practice, and for the problem treated at present, OASES requires an input file with the
following information included,
• The stratification of the medium i.e. number of layers and depth of each layer,
• the geoacoustic parameters (i.e. density, P and S wave velocity, attenuation),
• the source center frequency and depth,
• the wavenumber integration limits and discretization of the domain,
• the frequency range of interest
• the receiver’s co-ordinates.
The code is then able to generate the time responses at the points of every different receiver and for
the vertical and horizontal component of the wave. These time responses were imported into
MATLAB in order to extract the dispersion curves through the adaptive and Morlet wavelet. Some
modifications had to be made in the original .m files to account for the properties of the sets of data
generated with OASES. It is evident that the data do not contain noise other than the numerical one
generated through the processes of computation. In the present work, the OASES-OASP module was

21. 21
used, which calculates the depth-dependent Green’s function for a selected number of frequencies and
determines the transfer function at the receiver positions by evaluation of the wavenumber integral.
The OASES output used in this case was the particle velocities of the water-solid bottom interface
points corresponding to any of the receiver’s position. For the purposes of wavelet comparison and
evaluation of the inverse algorithm, a number of environmental models were created in OASES.
Provided that the input parameters in OASES were set properly, the program was able to generate the
time records of the particle velocity in both the horizontal and vertical direction. The time records
consisted of the primary wave plus the surface and shear waves with their later arrivals. The data sets
were then used as input to the .m files which generated the dispersion curves through the use of the
wavelet transform.
An environmental model in which the parameters approach the geoacoustical information
consistent with the Tommeliten field was used initially. The parameters used are listed in Table 3.1.
In the input files for OASES the first layer is set as the vacuum. This layer is omitted from the
following tables. Referring to the rest of the input parameters, the recorders were placed in the same
co ordinates as in the real experiment. The source was placed at a depth of 65m in order to allow for
strong excitation of the surface waves. The source center frequency was set at 10 Hz. The frequency
band of interest was chosen from 0 to 25 Hz. With OASES computing the horizontal and vertical
velocity components of each receiver point, a data set suitable for dispersion analysis is derived. The
particle velocity data generated from the input model of Table 3.1 have been contaminated with white
noise. The noise PSD was higher in the low frequency region below the cut off of the low-pass filter
of the pre-processing stage and of an amplitude that could be considered reasonable of an underwater
seafloor environment. In figure 3.6 the wiggle plots of the 52 traces are shown for the raw and filtered
data.
Layer
No
Depth of
interface
(m)
P-wave
velocity
(m/sec)
S-wave
velocity
(m/sec)
Density
(g/cm3
)
Compr.
Attenuation
(dB/Λ)
Shear
Attenuation
(dB/Λ)
1 0 1500 0 1.0 0 0
2 70 1700 215 1.8 0 0
3 75 1700 230 1,8 0 0
4 80 1700 270 1.8 0 0
5 85 1800 320 1.8 0 0
6 90 1850 360 1.9 0 0
7 100 1900 365 1.9 0 0
8 110 2000 380 1.9 0 0
9 125 2000 415 2.0 0 0
10 150 2000 440 2.0 0 0
11 170 2000 455 2.1 0 0
12 190 2100 480 2.1 0 0
13 210 2100 520 2.15 0 0
14 230 2100 570 2.2 0 0
15 250 2100 615 2.2 0 0
16 270 2100 655 2.3 0 0
Table 3.1 . The environmental model parameters used in the present analysis.

22. 22
Figure 3.6. Trace plots of the modeled environment for the vertical particle velocity and the horizontal particle
velocity components. The dispersion is manifested through the spreading in time of the waves with distance from
the source. The data have been contaminated with white noise.
The dispersion is observable in both plots. Two distinct wave trains are seen, each of them
representing a mode of the surface wave with the earliest arriving wave train possibly associated with
the higher order mode. The fact that two modes are present is more evident through the horizontal
component. Rune Allnor in his thesis [1] has referred to the horizontal and vertical particle velocity
components with some remarks, showing that the fundamental mode is appearing at least through the
vertical particle velocity data. It could be stated that the fundamental mode would probably possess
less energy in the horizontal component, while the higher order modes are expected to be visible in
the horizontal component. This seems to be the case in every dispersion analysis attempted in the
present thesis. Keeping in mind that the useful frequency range during the computations with OASES
was between 0 and 25 Hz, the FFT of the particle velocity components of the trace analyzed are
shown in figure 3.7. The selected part of the entire signal is the one that contains the surface wave.
The knowledge of the frequency content serves as an indication of the cut-off frequency needed to be
applied on the pre-processing phase.
The created model was not anticipated to produce quite similar results with the experimental data. The
two wavelet techniques produced the following results as shown in figure 3.8. The contour plots are
omitted since focus is given at the resolution properties of each of the wavelet techniques. The results
indicate that the adaptive wavelet proves to be more efficient in terms of noise removal. We observe
though that the energy of the surface waves resides in really low frequencies below 8 Hz for both
cases. For the synthetic data both wavelet techniques yield good results while there is a concentration
of the modal energies within the same frequency ranges. The difference lies within the shape of the
modal energy in the time-frequency representation, as the adaptive wavelet gives more regular-shaped
representations. Similar results are obtained on every trace analyzed, the difference being a more
distorted plot with distance from the source.

23. 23
Figure 3.7. The time signal of both components (with the horizontal in the left side plot) and their frequency
content
Figure 3.7. The Morlet wavelet analysis (upper plots) and the adaptive wavelet analysis (lower plots) for the
noisy synthetic data. On the left the results of the vertical particle velocity component while are shown, while
the horizontal component is presented on the right side plots

24. 24
The same environmental model but with a source center frequency of 6 Hz and 2 Hz respectively was
used next as input for the wavelet analysis. In this case no noise has been added in the data sets
computed with OASES. The resulted plots are shown in figures 3.8 and 3.9.
Figure 3.8. Morlet (upper plots)and adaptive wavelet (lower plots) analysis of the synthetic data (Table 3.1)
for both velocity components of the interface waves, where the source center frequency was set equal to 6 Hz.

25. 25
Figure 3.9. Morlet wavelet analysis (upper plots) and adaptive wavelet analysis (lower plots) of synthetic data
with a source center frequency of 2 Hz.
By using different center frequencies of the source pulse signal, we observe a shifting of the energy in
modes towards lower frequencies. The fact that low energy source signals are needed for determining
the sediment properties in greater depths is obvious. The Morlet wavelet analysis can provide with
better results when lower frequencies are involved. For both methods there is the indication of better
results when the source frequency band is centered around lower frequencies.
In the wavelet techniques comparison, another environmental model is created within
OASES. The parameters of the model are listed on Table 3.2. The source frequency is centered
around 15 Hz and the depth of the source is at 95m. The frequency band of interest is between 0 and
20 Hz. The trace records are shown on figure 3.10. The number of receivers was chosen to be small
though their separation distance is larger than the previous model in order for the dispersion to be
exhibited clearly. The particle velocity data generated from the input model of Table 3.2 have been
contaminated with white noise. Both methods supply good results with the Morlet wavelet giving a
somewhat distorted image of the low order mode and especially at the low frequency range of that
mode. The extracted dispersion curves are presented in figure 3.11 for the Morlet wavelet and in
figure 3.12 for the adaptive wavelet analysis.
Layer
No
Depth of
interface
(m)
P-wave
velocity
(m/sec)
S-wave
velocity
(m/sec)
Density
(g/cm3
)
Compr.
Attenuation
(dB/Λ)
Shear
Attenuation
(dB/Λ)
1 0 1450 0 1.0 0 0
2 100 1700 200 1.8 0 0
3 110 1700 270 1,8 0 0
4 120 1800 320 1.8 0 0
Table 3.2. The second environmental model used for creation of the synthetic data within OASES.

26. 26
Figure 3.10. The dispersive wave trains of the synthetic data (Table 3.2) with added noise for both particle
velocity components. The higher amplitude Vz-component (left side plots) is less influenced by the presence of
noise.
Figure 3.11. Morlet wavelet analysis for vertical (upper plots) and horizontal (lower) particle velocity
components of the synthetic data (Table 3.2) with added noise. Manual selection of the dispersion data is
sometimes necessary in cases when the automatic selection (highest WT matrix values) seems unreasonable.

27. 27
Figure 3.12. The adaptive wavelet analysis results of the synthetic data used in the current comparison. The
vertical particle velocity component analysis is presented at the upper plots while the horizontal component is
shown at the lower plots. A degree of ambiguity in the selection of the final dispersion data still exists. Noisy
elements of the velocity data have been efficiently removed in this adaptive approach of the WT.
The Morlet wavelet analysis generates what seems to be a more distorted dispersion plots when
compared with the results obtained from the adaptive wavelet analysis, while it is the Morlet wavelet
again which provides us with an energy content spread in a wider frequency band. The same
observations were made in the case of the experimental data. The ability of the adaptive wavelet
method to remove noise is again observed.
C. Synopsis
The two wavelet methods have been used in order to extract the dispersion curves of
experimental and synthetic particle velocity data of interface waves. The adaptive wavelet has an
inherent property that is able to effectively remove noisy elements present in a data set. This property
is observed through the results of the wavelet analysis. The performance of the wavelet method based
on the Morlet wavelet is affected when noise enters the data set in a relevantly high degree. As an
underwater environment contains a number of natural and artificial noise sources, removal of the
unwanted parts of a signal is highly important. The pre-processing stage is able to remove high
frequency noise, however there exist a number of low-frequency sources that are in the useful

28. 28
frequency range of a seismic-interface wave signal. That is the reason for the adaptive wavelet to gain
preference as a tool for the dispersion analysis of seismic interface waves. An examination of the
frequency content of the wavelet functions generated with the adaptive method, reveals the fact that
there exists a sufficient number of wavelet functions that should analyze the frequency-time content
of the signal with the desired quality. In the Morlet wavelet method, an increase in the number the
wavelet functions after a specific point does not provide us with better results. The increased
computational effort that the adaptive wavelet method requires is not a disadvantage within the limits
of a modern processor.
We can conclude that the adaptive wavelet seems to behaves better when higher frequencies
are included in the signal under analysis. The somewhat more distorted dispersion plots of the Morlet
wavelet is an indication of a poorer time-frequency resolution, though the dispersion data are
expected to be accurate enough for a number of applications. For the synthetic data used in the
comparison both methods supply us with similar results, the advantage being in the case of the
adaptive wavelet the noise-removal as stated earlier.
4. Inversion of dispersion data. Shear wave velocity profile
estimation.
The time-frequency analysis was used in order to come up with a set of data suitable for
inversion. We have to remind that the wavelet transform is a matrix with each column corresponding
to a specific frequency component of the signal and its progression in time. This matrix is plotted in
order to give us a ready interpretation of the time-frequency distribution of the signal. To come up
with a set of dispersion data, we select the maximum values of a number of columns in the CWT
matrix where each column corresponds to a specific frequency in Hz. We are able to identify the
different order modes and select the dispersion data of each mode separately. Finally, we end up with
a vector of group velocity data for each mode, together with the corresponding frequency vector.
These data will be imported into the inverse algorithm. The general frame of the algorithm is
presented in [6]. The general structure is the following.
The starting point is the general relation between the group velocity vector g and the shear
wave velocity vector s. The forward operator is symbolized with T , in this case a rectangular matrix,
which connects the two vectors through the equation,
g = T (s) (4.1)
We want to determine the vector s under the condition,
se
= min g ! T (s)
2
(4.2)
where se is the estimated shear wave velocity vector and g the group velocity vector. The norm is the
Euclidean norm and the solution is a least-squares solution to the inverse problem.
The work done by Haskell and Thomson is the fundament for the creation of the forward
model. An instructive paper that describes the way that phase velocity of the interface waves depends
on the parameters of the stratified environment is [9]. There exists a number of similar simplifications

29. 29
on that treatment, as exist for the assumptions of the inverse algorithm. The OASES code solves the
propagation problem by utilizing those simplifying assumptions, the most important being the
horizontally homogeneous seafloor. Density, P-wave velocity and depth of interfaces are not changed
throughout the inversion. It is only the shear wave velocity of each layer that is updated on every
iteration. An initial model is required, on which consecutive updates are generated up until
convergence is reached. The problem is nonlinear and linearization is required. After linearization has
been made on T in the neighborhood of the initial guess, the solution of the inverse problem takes
place with the use of the SVD. The theory behind the SVD has been presented in chapter 2. The
linear problem will then be,
g = Ts (4.3)
with the solution being,
s = (T †
T )!1
T †
g (4.4)
where T †
the pseudoinverse of the matrix T . Dealing with the ill-conditioning of the problem
requires the use of some additional constraints that the solution has to obey. The regularized solution
is restated as,
se
= min{ g ! T (s)
2
+ " Hs
2
} (4.5)
with H being the regularizing operator and λ the regularization parameter. Both can be chosen as
desired. The operator H is a differential one and thus not favoring highly oscillatory solutions. The
shear wave velocity tends to vary linearly or exponentially with depth in a sediment of the same type.
Usually there is an increase of the shear wave velocity with depth as the pressure acting upon deeper
buried sediments tends to be higher with increasing depth. That is not always the case, and a decrease
in shear wave velocity is possible with increasing depth, though the variation will be generally
smooth.
A. Tests
The use of synthetic data enables us to have prior knowledge of the shear wave velocity of
each of the layers and thus by inverting to evaluate which wavelet method can provide with better
dispersion data. We have also to consider the behavior of the inverse algorithm to different data sets
in this case.
A model with three sediment layers was used first. This is actually the same environmental
model that is displayed on table 3.2 of the previous chapter. The parameters of the model are listed
again for convenience on table 4.1.
Layer
No
Depth of
interface
(m)
P-wave
velocity
(m/sec)
S-wave
velocity
(m/sec)
Density
(g/cm3
)
Compr.
Attenuation
(dB/Λ)
Shear
Attenuation
(dB/Λ)
1 0 1450 0 1.0 0 0

30. 30
2 100 1700 200 1.8 0 0
3 110 1700 270 1,8 0 0
4 120 1800 320 1.8 0 0
Table 4.1. The environmental model used for extraction of dispersion data and inversion.
The inversion produced the following results as shown in figures 4.1 and 4.2 after the
dispersion has been estimated with both the wavelet methods. The initial model chosen for the
inversion followed the same depth layering, while the passive parameters of the inversion i.e. P-wave
velocity, density and attenuation, were set equal to the OASES input parameters. The parameter
lambda (λ) of the inversion algorithm has been selected in order for the numerically zero singular
values (and their corresponding basis vectors or principal components) of the matrix T not to be
included in the final estimation of the shear-wave velocity.
In the end plots of the inversion code, the upper left plot is the one showing the measured-
extracted dispersion data (red dots) together with the modeled curve of the group velocity (blue line).
On the upper right plot the singular values from the SVD of the T matrix are shown and the blue
dotted line separates the singular values included in the inversion computations from the ones
omitted. The singular values on the left of the blue dotted line are larger than the regularization
parameter. The plot on the lower left shows the shear wave velocity variation with depth which is the
final result of the inversion with an error estimate calculated by considering an uncertainty range of
+/- 15 m/s in the group velocity. The included principal components of the T matrix are shown with
blue on the down right plot while the omitted ones are presented with red.
(a)

31. 31
(b)
Figure 4.1 Inversion results of dispersion data from Morlet wavelet analysis of the model presented in table
4.1. (a) vertical particle velocity component and (b) horizontal particle velocity component of the same trace.
(a)

32. 32
(b)
Figure 4.2 The environmental model of table 4.1 is used for inversion of the group velocity data. Results for
(a) data from the vertical particle velocity component and for (b) data from the horizontal particle velocity
component analysis with the adaptive wavelet.
The shear wave velocity values resulted from the inversion are also shown on table 4.2. The source
center frequency in this set up was equal to 10 Hz.
Wavelet technique –
particle velocity
component
Shear wave velocity
(m/s) below interface at
100 m
Shear wave velocity
(m/s) below interface at
110 m
Shear wave velocity
(m/s) below interface at
120 m
Adaptive-Vx 184.3 209.17 214.2
Adaptive-Vz 187.4 212.26 218.8
Morlet–Vx 196.5 261.1 307
Morlet-Vz 196.7 259.2 305.3
Table 4.2. Reporting the inversion data shown at figures 4.1 and 4.2
What the inversion results reveal is that the dispersion data obtained with the Morlet wavelet
analysis provided us with a more accurate estimation of the shear wave velocity of each layer after the
inversion. This could be contradictory to the existing idea that the adaptive wavelet provides a better
dispersion analysis. When the source center frequency was shifted to 15 Hz similar results were
obtained from inversion of the group velocity data, it was the group velocity data extracted with
Morlet wavelet analysis that gave better inversion results. Shifting down the frequency of the source
into 2 Hz the following dispersion curves (figure 4.3) and inversion results (figure 4.4) were
extracted.

33. 33
Figure 4.3. Dispersion curve extracted with Morlet wavelet analysis (upper plots) and adaptive wavelet
analysis (lower plots). The horizontal particle velocity component is shown on the left side plots while the
vertical one on the right side.
(a)

34. 34
(b)
Figure 4.4. Inversion results from dispersion data extracted with (a) Morlet wavelet analysis and (b) adaptive
wavelet analysis. X-component is shown on the left side plots while Z-component on the right side plots.
When the data from the adaptive wavelet analysis were used for inversion the shear wave
velocity of the second layer was estimated accurately. The 3rd
layer of 320 m/s s-wave velocity was
underestimated at a value of 40 m/s for both components of particle velocity. The first layer was
overestimated, though the correct value of s-wave velocity was in the error estimation range.
The group velocity vector which originated from the wavelet analysis with the Morlet wavelet
has been inverted. The estimates were of the same accuracy, with the first layer being determined
more accurately, while the second layer was estimated less accurately compared with the adaptive
wavelet data set. The data sets from both wavelet methods seem to give similar accuracy estimates of
the shear wave velocity of the layers.
By comparing the above results with the ones from the same synthesized environment but
with a higher source center frequency, we observe that the inversion code provides better results for
when the source center frequency is lower for the case of the data generated from the adaptive
wavelet. For the Morlet wavelet case, the inversion algorithm can determine the shear wave velocities
of the solid halfspace layers with similar accuracy when the source center frequency is varied.
A two-layers environmental model was used next, with a high jump in the shear wave
velocity value between the two layers. The parameters are listed on table 4.3.
Layer
No
Depth of
interface
(m)
P-wave
velocity
(m/sec)
S-wave
velocity
(m/sec)
Density
(g/cm3
)
Compr.
Attenuation
(dB/Λ)
Shear
Attenuation
(dB/Λ)
1 0 1450 0 1.0 0 0
2 100 1700 200 1.8 0 0
3 110 2100 600 2.1 0 0
Table 4.3. Environmental model parameters for the present inversion
The wavelet analysis plots of the model are shown on figure 4.5 and the inversion results on figure
4.6.

35. 35
(a)
(b)
Figure 4.5. (a) Morlet and (b) adaptive wavelet analysis of the trace generated from the modeling of the
environmental model listed on table 4.3.
The Morlet wavelet analysis produced some distortion in the group velocity versus frequency plots.

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(a)
(b)
Figure 4.6. (a) Inversion results of group velocity data from Morlet wavelet analysis (the left plot shows the
vertical velocity component while the right plot shows the horizontal one). (b) Inversion results from the
adaptive wavelet analysis.
The inversion results are accurate for the shear wave velocity of the first layer. There is an
incapability of the algorithm to estimate the shear wave velocity of the second layer. This behavior
was observed in cases where the shear wave velocity undergoes a high jump in its value between two
consecutive layers. The regularization applied in this algorithm could be deemed responsible for this
fact.
The synthetic particle velocity data of the environmental model of table 3.1 supplied us with
the following dispersion curves.

37. 37
(a)
(b)
Figure 4.7. (a) Dispersion data extracted with the Morlet wavelet from a trace of the OASES input model of
table 3.1. (b) The adaptive wavelet dispersion data of the same environmental model. The horizontal particle
velocity component is shown at the left for both methods, while on the right side of the figure the vertical
component is presented.
The inversion of the data depicted in figure 4.7 produced the results shown at figure 4.8. Inversion
results are shown only for the vertical particle velocity component, as the dispersion data from the
vertical component could not be handled properly from the inversion code. We have to note at this
point that the inversion code was not able to produce results with a number of sets of group velocity
data, while tracing the problem was not always feasible. The forward model used in the general
algorithm could be the reason for some of those cases.

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Figure 4.8. Inversion results of the group velocity data from Morlet wavelet (left side plot) and adaptive
wavelet (right side plot) analysis. Only the vertical particle velocity component is used for analysis.
With inversion of the group velocity data generated from Morlet wavelet analysis, the shear
wave velocities of the solid layers were sometimes more accurately determined, in comparison with
the results from inversion of the adaptive wavelet data. This occurred mainly in situations of low
frequency content of the surface waves, with the main part of their energy being present below 5 Hz.
When the frequency content (usually with the use of a higher frequency source) was higher the two
wavelet methods provided us with data that seemed to give inversion results of similar accuracy. It is
not certain that the more distorted image of the group velocity-frequency plots obtained with the
Morlet wavelet analysis, is an indication of the data set will be of less quality especially within the
lower frequency ranges. We have to note that in this procedure the dispersion data were usually
selected automatically. The elements of the WT transform matrix that had the largest values in terms
of their group velocity and in every frequency. There is the ability to manually select the dispersion
data for inversion, but this brings in a certain degree of ambiguity. As the forward problem (the
prediction of the group velocity from a given shear wave velocity profile) is done with a code written
in Fortran, a possible further involvement with the present topic is the implementation of the
calculation of the dispersion within Matlab. Many problems will be solved in this way, as the
computational processes will be controlled and viewed in a more understandable way.
B. Synopsis
The use of synthetic data from a computational model (OASES) provided the possibility of
acquiring dispersion data of interface waves, in an environment of known geoacoustic parameters.
Inversion of these data has been attempted. Inversion of the data generated from the Morlet wavelet
method resulted in a number of times in an estimation of the shear wave velocity that was more
accurate than the estimation with inversion of adaptive wavelet data. As the adaptive wavelet
theoretically is able to resolve the time-frequency content of a signal with a better quality, these
results could be attributed to the problems that enter in the selection of the dispersion data from the
dispersion plots. A sophisticated approach on this issue could resolve the problems.
The inversion algorithm showed an incapability in estimating large steps in the S-wave
velocity parameters. The structure could be improved and different regularization could be added. As

39. 39
the whole methodology of the problem of estimating the S-wave velocity parameter is structured,
selecting a more evolved inversion code may provide better results.

40. 40
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