• dispersion curves
• filter bank
• polarization maps
• SVD
• rayleigh waves
• quaternions
• singular value decomposition

Separation of seismic wavefields is an important topic in geophysical research. One of the most challenging targets is the correct and complete
separation of the body wavefield from the surface wavefield. When surface waves and body waves are superposed in the offset-time domain, the part

of the signal constituting surface waves and the part of signal constituting body waves must be inferred from prior information or directly from the data. In general, real seismic waveforms are determined by a large number
of factors and the correct wavefield separation is an ambitious target.


Applications of such separation are manifold. The extracted surface
waves can be used as input of the inverse problem of determining the S-waves
velocities of the near surface ground layers. Vice versa, surface waves can
be extracted in order to be removed from the seismic section. In reflection

seismic, for example, only body waves constitute the signal and surface waves
are considered noise.


In particular, this thesis focuses on the extraction of Rayleigh waves,
which are a particular type of surface waves with elliptical polarization, i.e.,
they induce ground particles to move on elliptic trajectories. Such waves
require multicomponent sensors to be correctly recorded, as elliptic motion

needs intrinsically at least bidirectional recording sensors. In this thesis we
will use mainly only two components: the x and the z directions. Rayleigh
waves, in spite of their multicomponent nature, are usually processed component wise, repeating the analysis for the x, y and z components. In order
to introduce a more compact and effective multicomponent approach, I propose in this thesis a quaternion approach for the extraction of Rayleigh
waves.

In mathematics, quaternions are a number system that extends the complex numbers. They were first introduced by Sir W. R. Hamilton in 1843
[5], and they are commonly used for calculations involving three-dimensional
rotations such as in three-dimensional computer graphics. Each quaternion
is constituted by a scalar component and three different imaginary components. The imaginary components can be imagined as a vector part.
I will use quaternions in order to represent the multicomponent signals
recorded on triaxial geophones. In the geophysical field, quaternion algebra
has been used to implement multicomponent Wiener deconvolution, to
perform multicomponent velocity analysis via the implementation of quaternion matched filters, and to detect the wavefronts arrivals directions via
the implementation of a biquaternion version of the MUSIC algorithm.

The structure of the thesis is outlined in the following paragraphs.

The first chapter is constituted by this introduction.
In chapter two the theoretical propagation of the free modes of Rayleigh
waves in elastic media is introduced. Two simple examples of ground are
considered: the former is a homogeneous half space, which shows no dispersion, the latter is a
fluid layer over a half space, which is one of the

simplest ground model which shows dispersion characteristics; for this dispersive ground model, the dispersion curves equations are derived.

In chapter three, first SVD is introduced, then a polarization method is
described. This method uses a parameter inferred by the SVD analysis (the
inverse ellipticity parameter) in order to determine the degree of ellipticity
of the signal for local time windows. In order to improve the discrimina-

tion between circularly/elliptically polarized signals and incoherent noise a
filtering process that changes the phase-difference between two spatial components of the data is applied to the polarization maps. Furthermore, in
this chapter the two datasets that are used in this thesis are introduced.
They are a synthetic dataset generated for a 1D elastic ground model, and

a real dataset acquired in China.

In chapter four, the dispersion characteristics are derived for the synthetic and the real dataset. The dispersion characteristics are described by
two dispersion curves, respectively the phase velocity dispersion curve and
the group velocity dispersion curve. Phase velocity is the velocity of the

wave ripples, and group velocity is the velocity of the wave packet. The
phase velocity dispersion curves are determined in the frequency-(ray parameter) domain, also called f-p domain. No equivalent domain exists in
the standard processing for computing the group velocity dispersion curves.
I implemented an ad hoc transform in order to determine the group velocity
and I called it frequency-(group slowness) transform. In this chapter an
effective method for separating a mode of the Rayleigh wave will also be
shown. It consists in the muting in the f-p domain of the area outside of a
selected zone surrounding the mode. The computed dispersion curves are
compared with the theoretical dispersion curves for the synthetic data. For
the real data two modes of Rayleigh waves are identified.

In chapter five quaternions and matrices of quaternions are introduced.
The main definitions, properties, and different representations of quaternions
are introduced. In addition, the unit quaternion representation of rotation
and its relation with the 3 x 3 standard rotation matrix are described. Regarding quaternion matrices, the complex adjoint, and the definition of rank and SVD are introduced.

In chapter six the representation of Rayleigh waves with quaternion SVD
is studied. Firstly an example of signal of quaternion rank 1 is introduced,
its form and its polarization characteristics are described. Next three test
signals with different polarizations are introduced. They represent the three
cases of circular, elliptic and linear polarizations. The different representations in quaternion autoimages for the three polarizations are examined.
In particular, the first quaternion autoimage truncation of the elliptic test
signal is shown to have circular polarization. I called this phenomenon circularization of the elliptic wave. Next, I will briefly point out dispersion
effects and I will explain the motive for the introduction of the filter bank
in the first QSVD extraction method.

In chapter seven, the extraction method using a filter bank and QSVD
is described. The filter bank is used to separate different frequency bands of
the signal. In this chapter, firstly FIR filters and filter banks are introduced;
then, the specific filter bank used for the extraction method is described and
the procedure of the extraction method is outlined; finally the results applied
on the synthetic and the real datasets are exhibited.

In chapter eight, I will present the extraction method that performs the
QSVD analysis on local portions of the seismogram in the offset-time domain. The results of this method applied on the synthetic and the real
datasets are exhibited.

The final chapter will reveal conclusions. In this chapter the results of
the different extraction methods will be evaluated, and a global view of the
thesis is given.