Tesi Stabile Tony Alfredo

Tesi Stabile Tony Alfredo
Alma Mater Studiorum
Università di Bologna
Dottorato di ricerca in Geofisica - XX Ciclo
Tesi di Dottorato
Settore scientifico-disciplinare FIS/06
High frequency seismic and underwater
acoustic wave propagation and imaging
techniques
Dottorando:
Dr. Tony Alfredo Stabile
Tutor:
Prof. Aldo Zollo
Coordinatore:
Prof. Michele Dragoni
Esame finale anno 2008
2
i
To my wife and my son,
my everlasting love.
ii
Contents
Introduction
ix
1 Wave propagation theory
1.1 Elastodynamic equations . . . . . . . . . . . .
1.2 Synthetic Seismogram Methods . . . . . . . .
1.3 Ray-Theory . . . . . . . . . . . . . . . . . . .
1.3.1 Mathematical derivation . . . . . . . .
1.3.2 Ray theory general validity conditions
1.4 Rayleigh Scattering . . . . . . . . . . . . . . .
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2 Method for rapid high-frequency seismogram calculation:
code
2.1 Aim of a multiphase dynamic ray-tracing code . . . . . . .
2.2 Method description . . . . . . . . . . . . . . . . . . . . . .
2.2.1 General concepts . . . . . . . . . . . . . . . . . . .
2.2.2 Selection Criteria . . . . . . . . . . . . . . . . . . .
2.2.3 Structure of the COMRAD code . . . . . . . . . . .
2.3 Validation of the method . . . . . . . . . . . . . . . . . . .
2.4 Results and discussions . . . . . . . . . . . . . . . . . . . .
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the COMRAD
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3 Applications of the COMRAD code in seismology
3.1 Forward modelling of active seismic data: the case study of Campi Flegrei
caldera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 A brief description of Serapis active seismic survey . . . . . . . . .
3.1.2 Comparisons between synthetic sections and real sections . . . . . .
3.1.3 Conclusions and discussions . . . . . . . . . . . . . . . . . . . . . .
3.2 Forward modelling of seismic sources . . . . . . . . . . . . . . . . . . . . .
3.2.1 Faulting sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Synthetic seismograms for a vertical strike-slip point source . . . . .
3.2.3 Conclusions and discussions . . . . . . . . . . . . . . . . . . . . . .
19
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4 High frequency underwater acoustic propagation in the Gulf of Naples 51
4.1 Physical properties of the Gulf of Naples . . . . . . . . . . . . . . . . . . . 52
iv
CONTENTS
4.2
4.3
4.4
4.1.1 Analysis of the weather conditions . . . . . . . . . . . . . . . . . . .
4.1.2 One-dimensional velocity and geoacoustic model of the Gulf of Naples
4.1.3 Noise sources up to a frequency of 100 kHz . . . . . . . . . . . . . .
Simulations of acoustic signal propagation at 100 kHz . . . . . . . . . . . .
4.2.1 Methodology used for the simulation of the signal propagation . . .
4.2.2 How to calculate the signal-to-noise ratio (SNR) . . . . . . . . . . .
Calculation of the SNR in the Gulf of Naples at 100 kHz . . . . . . . . . .
Conclusions and discussions . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Very high frequency Submarine Acoustic Imaging
5.1 The STSS-500 Project . . . . . . . . . . . . . . . . . . . .
5.2 Development of an Acoustic Imaging Numerical Simulator
5.2.1 Forward modelling: Rayleigh scattering . . . . . . .
5.2.2 Imaging: Beamforming techniques . . . . . . . . . .
5.3 Three-dimensional imaging of submerged objects . . . . . .
5.4 Conclusions and discussions . . . . . . . . . . . . . . . . .
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53
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Conclusions
91
Acknowledgements
93
A The COMRAD Code
A.1 User Guide . . . . . . . . . . .
A.1.1 What the program does
A.1.2 Program Manual . . . .
A.2 The source code . . . . . . . . .
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95
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B Fundamentals of underwater acoustics
B.1 The sonar equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Sound attenuation in seawater up to a frequency of 1 MHz . . . . . . . . .
B.3 Calculation of the transmission reliability . . . . . . . . . . . . . . . . . . .
113
113
115
117
List of Figures
1.1
1.2
An example of seismogram recorder by a three component station. . . . . .
Subcritical, critical, and postcritical angles of incidence. . . . . . . . . . . .
5
14
2.1
2.2
Tree structure of ray strings stopped to the fourth generation. . . . . . . .
Tree structure of ray strings stopped to the fourth generation as in Figure
2.1, but using all of the (a) to (g) constraints . . . . . . . . . . . . . . . . .
Integral of scattering coefficients for an incident P-wave. . . . . . . . . . .
Integral of scattering coefficients for an incident S-wave. . . . . . . . . . . .
Block scheme of the Comrad.f computer code. . . . . . . . . . . . . . . . .
T F EM (t, f ) and T F P M (t, f ) plots for a receiver at 1 km distance from
the source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
As in Figure 2.6 but referring to the receiver at 30 km distance from the
source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.3
2.4
2.5
2.6
2.7
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.1
4.2
4.3
4.4
4.5
4.6
An image of the Campi Flegrei caldera. . . . . . . . . . . . . . . . . . . . .
Comparison between a real section and a synthetic section for a 1-D Campi
Flegrei model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Three different real sections in which the multiple reflection of Figure 3.2 is
clear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison between the observed and synthetic sections for the average
1-D model of Table 3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
AVO analysis (PS-to-PP ratio) to the second interface of the velocity model.
Extended source and point source conditions in far-field approximation. . .
Standard definition of fault-plane and slip vector orientation parameters. .
Synthetic seismograms for R2 and R4 receivers. . . . . . . . . . . . . . . .
Focal mechanism obtained from FPFIT program. . . . . . . . . . . . . . .
Bathymetric and topographic image of the Gulf of Naples. . . . . . . . . .
Mean air temperature and wind speed in the Gulf of Naples during one year.
Monthly frequency of the Wind direction in the Gulf of Naples. . . . . . .
Salinity profiles in the Gulf of Naples from the sea surface to the bottom. .
Temperature profiles in the Gulf of Naples from the sea surface to the bottom.
Velocity profiles in the Gulf of Naples calculated from the salinity and temperature data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
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61
vi
LIST OF FIGURES
4.7
4.8
Mean velocity profiles for the summer and winter seasons. . .
The signal-to-noise ratio for the summer and winter months in
Naples with the transmitter positioned at a depth of 1 m. . . .
4.9 The signal-to-noise ratio for the summer and winter months in
Naples with the transmitter positioned at a depth of 190 m. .
4.10 The signal-to-noise ratio for the summer and winter months in
Naples with the transmitter positioned at a depth of 299 m. .
4.11 Monitoring system prototype developed during SisMa Project.
5.1
5.2
5.3
5.4
5.5
5.6
5.7
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the Gulf of
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the Gulf of
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the Gulf of
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Sound attenuation in seawater against range and frequency. . . . . . . . .
Scheme of the beamforming technique used for the imaging process. . . .
Source-receiver geometry used for the first example. . . . . . . . . . . . .
Synthetic data and images obtained for the first example. . . . . . . . . .
Two different configuration used for the acquisition system. . . . . . . . .
Images of the target obtained using the configurations of Figure 5.5. . . .
Real shape of the object and its image obtained by the imaging process. .
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88
List of Tables
2.1
2.2
2.3
2.4
2.5
3.1
3.2
4.1
4.2
4.3
4.4
5.1
Different velocity models to study the effect of the boundary between a
half-space that contains the incident wave and a half-space that contains
the transmitted wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Weights assigned for each region, defined by z parameter, and for each type
of phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The crustal velocity model used for the simulations. . . . . . . . . . . . . .
Computing time of COMRAD and core codes . . . . . . . . . . . . . . . .
Computing time of COMRAD code with constraints . . . . . . . . . . . . .
Average 1-D model for Campi Flegrei as derived by Maercklin and Zollo
(2008). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Model used to compute synthetic seismograms for a vertical strike-slip point
source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
25
30
30
31
42
48
Mean annual temperature in the Gulf of Naples from 1997 to 2006. . . . .
Monthly total rain fallen in the Gulf of Naples (mean value from 1997 to
2006). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Values of the physical parameters of the propagation medium for the Gulf
of Naples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sound velocities of the summer and winter models for increasing depths. .
54
63
63
Physical properties of the cubic block and the background medium used for
the second example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
55
B.1 The sonar parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
viii
LIST OF TABLES
Introduction
The direct measurement of elastic properties of the Earth interior encounters many problems due both to the vast dimension of our planet and to the technological difficulty to
perform measurements in depth of its physical properties. In fact, for the solid Earth the
only means of in situ measurements is by drilling, which however can perforate few kilometres of the crust and provides only a few point measurements. For this reason drillings
are insufficient to describe the global structure of the Earth and elastic waves constitute
the solution of the problem.
The elastic waves are generated by natural or human made sources. Earthquake generated waves may travel through the Earth depending to the source energy and carry
on the surface the information (seismograms) about its physical properties which can be
interpreted by seismologists. This approach is based on the use of indirect methods to
reconstruct the internal structure and physical properties of the Earth (e.g. tomography,
seismic reflection, and seismic refraction) or to study the processes that have generated
the observed wavefield.
The most part of the Earth surface is covered by water, so there is the necessity to
install seismic instruments also on the seafloor. Geophysical observation at sea present
different difficulties with respect to on-land where seismic observation are routinely made.
The main problems are the transmission of data on the surface, the installation and the
maintenance of underwater stations, the usage of instruments in the seawater and the
remote control of the submerged instrumentation, so the observations at sea become very
expensive.
The problem related to the transmission of data in seawater can be solved by the use
of acoustic wavefield, since both light and radio waves propagates hardly in the water
respect to acoustic waves as a consequence of a higher attenuation. Recently there has
been a rapid extension of the use of digital underwater acoustic communication due to the
development of underwater acoustic wireless communication systems.
All what said above outline the importance of studying both the elastic wavefield
propagation in complex 3-D media and the acoustic wavefield propagation in seawater. In
particular the forward modelling, which is the calculation of theoretical wavefield starting
from a given model, is an important tool for both seismological and underwater acoustic
problems. For seismological problems the forward modelling can be of great support for
the interpretation of data and to construct or to refine models of both the propagation
medium and of the source. The calculation of the theoretical wavefield propagation in the
x
INTRODUCTION
seawater is, instead, widely used for studying the effects on the data transmission produced
by the variability in space and time of both the underwater environment and its boundaries
(sea-surface and seafloor).
Many methods have been developed during the years in order to calculate the complete
wavefield (i.e. solve the elastodynamic equations) both in simple structures, such as a
layered medium, and in complex 3-D structures. In Chapter 1, section 1.2, a summary of
these methods will be reported.
In this work we are interested to the high frequency elastic and acoustic wavefield
propagation, which can be computed by approximate high-frequency asymptotic methods
(such as ray methods). The high-frequency methods are preferable to direct numerical
methods for many applications, both for the shorter computing times (direct numerical
methods require large quantities of memory, computing time, and processor capacity) and
for the full interpretability of the seismograms, since all the phases on a seismogram are
relative to a priori known body waves. High frequency methods are preferable whenever
a fast synthetic seismogram computation is needed, as in inversion procedures where an
iterative forward modelling is required. In seawater ultrasonic frequencies are typically
used to transmit data, so ray-theory method are widely used to study the propagation of
these high-frequency signals.
Although methods based on ray theory give only an approximate solution of the elastodynamic equation, and they can be applied only if some general conditions are satisfied
(see section 1.3.2), the computing time is order of magnitude less than that required by
complete wavefield methods. Moreover the differences between the ray theory solution and
the direct numerical solution of the elastodynamic equation become less and less as the
frequency f tends to infinity (f → ∞). The ray theory will be described in Chapter 1; a
complete treatment of the theory can be found in Červený (2001).
Since the ray theory solution of the elastodynamic equation is composed of elementary
body waves that travel through the medium from the source to the receiver, the wavefield
will be computed only for the chosen body waves. The problem starts when we want
to produce synthetic data as complete as possible by considering a large number of body
waves. In this case the computation of the wavefield propagation using ray theory becomes
tedious if an automatic body waves (raypaths and phases) generation will not be used.
In order to resolve the problem of generating a comprehensive set of raypaths and phases
to be computed, we have developed and quantitatively tested a method that uses a hierarchical order of rays and seismic phases generation, taking into account some existence
constraints for a ray-path and some physical constraints. The method has been implemented in the COMRAD code (from the Italian: “COdice Multifase per il RAy-tracing
Dinamico”), written in Fortran77 language. The program’s core is the dynamic ray-tracing
method developed by Farra and Madariaga (1987). In Chapter 2 the description of the
method with a discussion of its limits and its advantages will be presented. A quantitative
test will be also reported. Further information of the COMRAD code (the user guide, the
source code, and where they should be downloaded) can be found in Appendix A.
In Chapter 3 some applications of the COMRAD code in seismology will be presented.
In particular applications to seismic reflection data, obtained from the SERAPIS (Zollo
xi
et al., 2003) active survey, and the computation of synthetic seismograms generated by
faulting sources will be discussed.
In Chapter 4 we will present a study on the physical properties of the seawater in
the Gulf of Naples, considering both their spatial and temporal variability. We will show
the effects they produce on the underwater acoustic digital communication at 100 kHz
frequency and the limit of the transmission reliability in space and time. The underwater
acoustic wave propagation in the geoacoustic model developed for the Gulf of Naples was
computed using the COMRAD code. The study presented is a work carried out during
this thesis by Stabile et al. (2007) for the Pon-MIUR SisMa project.
As we said before, the interpretation of the information that the wavefield propagation
brings at the receiver is used to study the interior of the medium. So petroleum and mineral
resource prospects, subsurface water, environmental pollution, archaeology objects, etc.
can be found in the Earth and/or in the seawater, and imaging techniques can be used to
reconstruct their shape.
In the seawater, the main advantage of acoustic imaging systems over the optical ones
is that they properly work beyond the optical visibility range. Moreover they provides
images of submerged object also in turbid waters. Thus, the development of an acoustic
camera system, which gives a 3-D image of the insonified object, has becomes recently an
active research field.
In Chapter 5 we will study the detection and imaging of objects submerged in the
seawater, using acoustic signals around a frequency of 500 kHz, for the development of
an acoustic imaging sysmet. Since we want to reconstruct a high resolution 3-D image of
submerged not smoothed objects (such as blocks with corners or rough surfaces), the raytheory is not valid yet. In this case scattering occurs and we use the Rayleigh approximation
for the amplitude computation of the scattered wave. Using scattering theory we compute
synthetic data as echoes to a receiver produced by a target modelled as a sum of little (i.e.
much smaller than the characteristic wavelength of the wavefield) scattering point having
the same physical properties of the target. The synthetic data are hence considered as
real data to reconstruct the image of the target using beamforming techniques. This is an
on-going work funded on Pon-MIUR STSS-500 project, which will be described in section
5.1.
xii
INTRODUCTION
Chapter 1
Wave propagation theory
Elastic waves are generated whenever a transient stress imbalance is produced within or
on the surface of an elastic medium. Almost any sudden deformation or movement of a
portion of the medium results in such a source, that can be external to the solid Earth or
internal and can be natural of artificial. Except in the immediate vicinity of the source,
the ground returns to its initial position after the transient motion has subsided. Vibration
of this type involves small elastic deformations, or strains, in response to internal forces in
the rock, or stresses. The properties of elastic waves are governed by the physics of elastic
solids, which is fully described by the theory of elastodynamics.
In this chapter we briefly derive the elastodynamic equations and the wave equations. It
is well known that analytical solutions to the elastodynamic equation can be found only for
simple cases, but there are several techniques that can solve numerically the elastodynamic
equations for more complex media. We will show an overview of these methods in the
section 1.2. In particular we present in the section 1.3 the ray theory, which provides the
solution to the elastodynamic equation in the high-frequency approximation.
Finally, when we want to take into account the presence in the medium of small1 discrete
inclusions, the scattering theory must be considered. In the last section of this chapter
we present the scattering theory and, in particular, the Rayleigh scattering approximation
that occurs when the size of heterogeneities is much smaller than the wavelength of the
incident wave field.
1.1
Elastodynamic equations
The basic concepts and equations of linear elastodynamics have been explained in many
textbooks, such as Atkin and Fox (1980), Ben-Menahem and Singh (1981), Lay and Wallace
(1995), and Aki and Richards (2002).
Here we shall use the Lagrangian description of motion in an elastic continuum, in
which the motion of a particular particle is followed as a function of time and space. This
1
With small we intend that the size of the inclusion is of the order (or less) of the wavelength of the
incident wave field.
2
Ch.1 WAVE PROPAGATION THEORY
is a natural system for seismology, because seismograms are essentially records of particle
motions at near-surface sensors as seismic waves pass by. A continuum is a continuous
distribution of particles; thus a vector field, u(x, t), is required to describe the motions
of every point in the medium, where we are free to choose a convenient reference system.
Because we wish to make the treatment as simple as possible, we shall use Cartesian
coordinates xi , i = 1, 2, 3, and Cartesian tensors only.
Assume that the particle is located at the position described by Cartesian coordinates
xi at the reference time. The vector distance of a particle at time t from position x at the
reference time is called the displacement vector u = u(x, t).
We denote the Cartesian components of the stress tensor by τij (x, t) and the Cartesian
components of the strain tensor by eij (x, t). Both tensors are considered to be symmetric
τij = τji ,
eij = eji .
(1.1)
The strain tensor can be expressed in terms of the displacement vector as follows:
eij =
1
(ui,j + uj,i ) ,
2
(1.2)
where the comma between subscripts is used for spatial derivatives (e.g. ui,j = ∂ui /∂xj ).
The stress tensor τij (x, t) fully describes the stress conditions at any point x. It can be
used to compute traction T acting across a surface element of arbitrary orientation at x:
Ti = τij nj ,
(1.3)
where n is the unit normal to the surface element under consideration.
The elastodynamic equation relates the spatial variations of the stress tensor with the
time variations of the displacement vector and its expression is:
τij,j + fi = ρüi ,
i = 1, 2, 3.
(1.4)
Here fi denote the Cartesian components of body forces (force per volume), and ρ is the
density. The term with fi in elastodynamic equation (1.4) will also be referred to as the
source term. Quantities üi = ∂ 2 ui /∂t2 , i = 1, 2, 3, represent the second partial derivatives
of ui with respect to time (that is, the Cartesian component of particle acceleration ü). In
a similar way, we shall also denote the Cartesian components of particle velocity ∂ui /∂t
by vi or u̇i . The introduced quantities are measured in the following units:
• stress τij and traction Ti in pascals (Pa),
• components of body forces fi in newtons per cubic metre (N/m3 ),
• density ρ in kilograms per cubic metre (kg/m3 ),
• displacement components ui in metres (m),
• strain components eij dimensionless.
1.1 Elastodynamic equations
3
In a linear, anisotropic, perfectly elastic solid, the constitutive stress-strain relation is
given by the generalized Hooke’s law (Aki and Richards, 2002; Červený, 2001):
τij = cijkl ekl ,
(1.5)
where cijkl are components of the elastic tensor, which has in general 81 components. These
components2 , however, satisfy the following symmetry relations:
cijkl = cjikl = cijlk = cklij ,
(1.6)
which reduce the number of independent components of the elastic tensor from 81 to 21.
The components of the elastic tensor are measured in pascals (Pa), the same units as the
stress components.
If we express ekl in terms of the displacement vector components, see equation (1.2),
and take into account symmetry relations (1.6), we can also express equation (1.5) in the
following form:
τij = cijkl uk,l .
(1.7)
Now, inserting relation (1.7) into elastodynamic equation (1.4), we obtain the elastodynamic equation for an unbounded anisotropic, inhomogeneous, perfectly elastic medium:
(cijkl uk,l ),j + fi = ρüi ,
i = 1, 2, 3.
(1.8)
The elastodynamic equation (1.8) represents a system of three coupled partial differential equations of the second order for three Cartesian components ui (xj , t) of the
displacement vector u.
Alternatively, the elastodynamic equation may be expressed in terms of 12 partial
differential equations of the first order in three Cartesian components vi (xj , t) of the particle
velocity vector v = u̇, and 9 components τij of the stress tensor (see Červený, 2001).
The elastodynamic equation (1.8) can be expressed also in the frequency domain by
applying the Fourier transform operator F to u(x, t) and f (x, t), and taking into account
the generalized derivative property of the Fourier transform3 . Hence, in the frequency
domain the equation (1.8) becomes:
(cijkl Uk,l ),j + Fi = −ρω 2 Ui ,
i = 1, 2, 3.
(1.9)
where U(x, ω) and F(x, ω) are the Fourier transform of u(x, t) and f (x, t), respectively.
2
The components cijkl of the elastic tensor are also called elastic constants, elastic moduli, elastic
parameters, or stiffnesses.
3
Considering g(t) an integrable function of time t, G(ω) its Fourier transform, and F the Fourier
transform operator, the Fourier transform of the nth derivative of g(t) satisfies the following property:
n
d (g(t))
F
= (iω)n G(ω),
dtn
with ω the angular frequency.
4
Ch.1 WAVE PROPAGATION THEORY
In the case of an isotropic medium the number of the independent components cijkl of
the elastic tensor is reduced from 21 to only 2, infact:
cijkl = λδij δkl + µ(δik δjl + δil δjk )
(1.10)
where λ and µ are the Lame’s elastic parameters used to describe an isotropic medium,
and δij is the Kronecker’s symbol. In particular µ is the rigidity or shear modulus and it
is equal to zero in fluids.
Taking into account the equation (1.10), the elastodynamic equation (1.8) in an isotropic
medium becomes:
(λuj,j ),i + [µ(ui,j + uj,i )],j + fi = ρüi ,
i = 1, 2, 3.
(1.11)
The solution of the elastodynamic equations (1.8) and (1.11) for anisotropic and isotropic
unbounded media, respectively, results in a term that behaves like r−2 and in terms that
behave like r−1 . The terms that behave like r−1 becoming dominant (over r−2 ) as r → ∞,
and therefore they are called far-field terms. On the other hand, since the r−2 term dominates over r−1 terms as r → 0, it is called the near-field term. All these terms are body
(or volume) waves that travel through the interior of the unbounded medium. If there are
discontinuities in the medium and/or boundaries, additional waves will be generated, such
as reflected/transmitted/converted body waves, surface waves (which travel on the surface
of the medium), headwaves, inhomogeneous waves, etc. Details can be found in Lay and
Wallace (1995) and Aki and Richards (2002) books.
1.2
Synthetic Seismogram Methods
The seismogram (Figure 1.1) is the recording, during the time t, of the ground motion by
instruments called seismometers installed in a certain point x of the propagation medium.
In particular, in Figure 1.1 the principal seismic phases are indicated. On the vertical
component (top) we indicate the P-wave (also called longitudinal or compressional wave)
which in general corresponds to the first arrival4 on the seismograms. The P-wave is linearly
polarized longitudinally to the direction of the wavefield (or equivalently, perpendicularly to
its wavefronts). The P-wave can be used for several problems, such as tomography, source
localization, focal mechanisms of the seismic source, earthquake magnitude estimation.
For the same problems it is possible to use S-waves (the first S-wave arrival is indicated
on the E-W component of the seismogram plotted in Figure 1.1) instead or in addition
to P-waves. The S-wave (also called shear wave) is polarized in the plane tangent to the
wavefronts, its velocity is always smaller than the P-wave velocity, and its polarization is in
general elliptical. Brune (1970) demonstrated that the spectra of seismic shear waves (not
4
For large offsets (horizontal distance between the source and the receiver) the first arrival can be a
headwave and not the direct P-wave. In particular cases (layers with slow velocities above layer with very
high velocities) and for large offsets also some reflected waves to the deepest interfaces should arrive before
the direct P-wave.
1.2 Synthetic Seismogram Methods
5
only the first S-wave arrival) can be used to calculate the seismic moment M0 . Generally,
P and S wavefields are not separated in inhomogeneous structures.
Between P and S waves, and after S waves, there are several reflected/refracted and(or)
converted waves which carry important information on the medium crossed by the wavefield. This phases are used is seismic reflection to find in the medium gas-bearing formations, reservoirs, discontinuities and interfaces.
Finally, also the coda of the seismic signal (evidenced on the E-W component of the
seismogram in Figure 1.1), and other type of phases (i.e. the surface waves or the headwaves) are used to study the medium properties and/or the source.
In summary the entire seismogram waveform can be studied to reconstruct both the
characteristics of the source that has generated the wavefield and the properties of the
medium the wavefield has propagated throughout. For this reason it is important to treat
mathematically the process of excitation, propagation, and recording of seismic waves in
order to correlate the shape and the amplitude of the recorded seismic waves to their
physical meaning.
Because of the nature of the Earth, which is considered as a linear stationary filter for
elastic waves, it is possible to treat mathematically the process of excitation, propagation,
and recording of seismic waves as a sequence of linear filters that combine to produce
Figure 1.1: An example of seismogram for an earthquake of magnitude M=4.0 recorded by a
three component station of the ISNet network (Weber et al., 2007) installed in the Irpina region
(Southern Italy). The principal seismic phases are indicated. From top to bottom the vertical
component, the N-S component, and the E-W component are shown, respectively.
6
Ch.1 WAVE PROPAGATION THEORY
observed seismograms. In other words, an observed ground displacement history, u(t), can
be expressed as the result of a source function, s(t), operating on a propagation function,
g(t), combined with an instrument recording function, i(t), by the convolution operation:
u(t) = s(t) ∗ g(t) ∗ i(t)
(1.12)
It is also possible to model each of these effects mathematically, and develop a procedure to predict the character of a seismogram in a realistic model of the Earth. Such a
mathematical construction is known as a synthetic seismogram. The formalism of comparing synthetic and observed seismograms is known as waveform modelling. Waveform
modelling has become one of the most powerful tools available to seismologists for refining Earth structure models and understanding fault rupture processes. For this reason a
method for rapid high frequency seismogram calculation is developed during this thesis
work and it will be described in Chapter 2.
Here we summarize the most important methods used to compute synthetic seismograms with their advantages and disadvantages:
• The Normal Mode summation method utilize the dispersion relations and amplitude functions (eigenvalues) for Rayleigh or Love motion to determine the amplitude and phase spectrum for a layered (1-D) half space at a given distance. Fourier
transformation then yields the synthetic surface wave seismograms. A large number
of modes can be included. Body wave arrivals can be calculated if sufficiently high
mode number is used. This implies that the method must sum a large number of
modes to approximate broadband data and body waves. Numerical effects due to
high mode number and long period require large number of layers in model which
can increase computational effort significantly.
• The Discrete wavenumber method is based on a discrete horizontal wave-number
representation of seismic source wave fields with the seismic source embedded in a
layered (1-D) medium. The principle of the method is founded of the fact that the
steady-state radiation from a line source in an infinite homogeneous medium can be
represented as a cylindrical wave or, equivalently, as a continuous superposition of
homogeneous and inhomogeneous plane waves (Bouchon and Aki, 1977). Therefore,
denoting by x and z the horizontal and vertical axes in the plane normal to the
source axis, any observable such as displacement or stress can be written in the
form of an integration with respect to the horizontal wave number. The integral
is then transformed into a summation of functions of discrete wavenumbers. The
computation time increases with increasing frequency since more wavenumbers have
to be considered in the summation. A program based on this method will be used
in the Chapter 2 as reference for testing the COMRAD code developed during this
thesis work.
• The finite-difference method belongs to the so-called grid-point methods. In the
grid-point methods a computational domain is covered by a space-time grid and each
1.2 Synthetic Seismogram Methods
7
function is represented by its values at grid points. The space-time distribution of
the grid points may be, in principle, arbitrary, but it significantly affects the accuracy
of the approximation. Usually, no assumption is made about the values in-between
the grid points. A derivative of a function is approximated by the finite-difference
formula which uses values of the function at a specified set of the grid points. Grid
spacing (10 points per wavelength) and time step (small step to insure stability)
requirements result in very large computation time and storage use. This method
is applicable in 3-D models with arbitrary complex local structures and the solution
obtained should contain all the possible waves propagating in the given medium
within the given time window.
• The finite-element method, as the finite-difference method, provides the direct
numerical solution of partial differential equations (i.e. the wave equation) by finiteelement matrix methods. The solution approach is based either on eliminating the
differential equation completely (steady state problems), or rendering the partial differential equations into an approximating system of ordinary differential equations.
The finite-element method is a good choice for solving partial differential equations
over complex domains, when the domain changes, or when the desired precision varies
over the entire domain. The method needs the discretization of the space in elements
(in general triangles in 2-D or tetrahedrons in 3-D) and piecewise polynomial (in general linear) basis functions in order to replace the infinite dimensional problem with
a finite dimensional version. Finite element matrices are typically banded (elements
are locally connected) and the storage requirements for the system matrices typically grow quite linearly with the problem size, so for models with high complexity
large banded matrices must be inverted. Methods that use higher degree piecewise
polynomial basis functions are often called spectral element methods, especially
if the degree of the polynomials increases as the triangulation size goes to zero.
• The boundary element method is another numerical computational method of
solving partial differential equations which have been formulated as integral equations (i.e. in boundary integral form) that are mathematically equivalent to the
original partial differential equations. The essential re-formulation of the partial differential equation consists of an integral equation that is defined on the boundary
of the domain and an integral that relates the boundary solution to the solution at
points in the domain. The boundary element method is often more efficient than
other methods for problems where there is a small surface/volume ratio but, on the
other hand, the method formulations typically give rise to fully populated matrices.
This means that the storage requirements and computational time will tend to grow
according to the square of the problem size.
• The asymptotic ray method is an approximate high frequency method based
on an asymptotic solution of the elastodynamic equation. The solution is sought
in the form of the ray-series and considering zero-order terms which dominates over
the high order terms as the frequency tends to infinity. For this reason the solution
8
Ch.1 WAVE PROPAGATION THEORY
of the elastodynamic equation is only approximate, but the errors become small if
the frequency is high (asymptotic solution). The method is very general since in
can be applicable both to anisotropic and isotropic media containing curved interfaces, separating layers or block in the space, to arbitrary 3-D variations of elastic
parameters and density, to very general types of body waves, and to an arbitrary
source-receiver configuration. On the other hand some kind of waves (such as surface
waves, inhomogeneous waves, near-field terms, head-waves) cannot be considered by
the method. Although the method is in principle incomplete, it can be applicable to
a large variety of problems and it is the most effective method in term of computing
time. In fact the computing time does not depend from frequency and the results
accuracy increases with increasing frequency.
1.3
Ray-Theory
This section presents a simple treatment of the ray method, applicable to high-frequency
body waves propagating in complex 3-D laterally varying media. Here we follow and use
the notations of Červený (2001), and we discuss the mathematical derivation of the zeroorder ray solution (we remind to Brokešová (2006) for a complete treatment of zero-order
ray solution, and Červený (2001) for a consistent treatment of the ray methods).
1.3.1
Mathematical derivation
The high frequency asymptotic methods are based on an asymptotic solution of the elastodynamic equation. They can be applied to compute not only rays and travel times
(kinematic part) but also the vectorial complex-valued amplitudes of the displacement
vector (dynamic part).
The final ray solution of the elastodynamic equation is composed of elementary waves
(direct, reflected, refracted, multiply reflected, converted, and the like) corresponding to
various rays connecting the source to the receiver. In most cases, the numerical modelling
of seismic wavefields has been based on the zeroth-order leading term of the ray series.
In practice ray theory is based on the assumption that the displacement vector u
is assumed in the form of the so-called ray series. Hence, the ansatz solution for the
displacement vector u of the elastodynamic equations (1.8) or (1.11) at point x and time
t will read
∞
X
u(x, t) =
Uk (x)Fk (t − τ (x))
(1.13)
k=0
th
where Uk is the k vectorial amplitude coefficient (a complex-valued function), τ is the
eikonal (a real-valued function), and Fk (k = 1, . . . , ∞) represent high-frequency analytical5 signals. The analytical signals in wave propagation problems are just the way to
allow for suitable compact notation and to make mathematics simpler; moreover at the
5
The analytical signal in any complex-valued function whose real and imaginary parts form a Hilbert
1.3 Ray-Theory
9
end of an analytical signal computation one must take only the real part of the analytical
signal to obtain the physically meaningful quantity for which the problem is solved (i.e.
the displacement). Finally, the advantage of the analytical signal is that it has a one-sided
spectrum since its Fourier transform vanishes for negative frequencies.
Using the ray series the Uk (k = 1, . . . , ∞) and the eikonal τ become the functions to
be determined under the condition (1.8) in anisotropic media, or the condition (1.11) in
isotropic media. These quantities are assumed to change only slowly in space. Since we
are interested to the wavefield solution in a certain point x far from the source, we shall
not consider the source term (fi = 0, i = 1, . . . , 3) in the elastodynamic equations (1.8)
and (1.11).
The zero-order ray solution is obtained considering only the first term (k = 0) of the
ray series (1.13), which is equivalent to the asymptotic high frequency solution of the
elastodynamic equation:
−iωτ (x)
F [uk=0 (x, t) = U0 F0 (t − τ (x)] = U0 (x)e
≡ lim e
ω→∞
−iωτ (x)
∞
X
Uk (x)
(−iω)k
k=0
(1.14)
where the last quantity is the Fourier transform of the ray series (1.13).
Omitting the index “0” in F0 and U0 , the zero-order trial ray solution can be written
af follows:
u(x, t) = UF (t − τ (x))
(1.15)
where the eikonal function τ is interpreted as travel-time of the elementary wave for which
the solution is constructed. Moreover, surfaces τ (x) = const represent wavefronts while
the gradient of the eikonal p = ∇τ is the slowness vector and it is perpendicular to the
wavefront:
nτ (x)
, i = 1, . . . , 3
(1.16)
pi (x) = τ,i (x) = i
c(x)
with nτ the unit vector perpendicular to the wavefront and c the phase velocity.
Substituting now the zero-order ray ansatz (1.15) in the equation (1.8) or (1.11) without
the source term and gathering the terms with the same derivative of F, we have the
following equation for both the anisotropic and isotropic cases:
F̈ Ni (U, ∇τ ) − Ḟ Mi (U, ∇τ ) + F Li (U) = 0,
i = 1, . . . , 3
(1.17)
where the vectorial coefficients N , M , and L have different expressions for anisotropic and
isotropic cases. In particular for anisotropic structures the three coefficients are given by
pair:
F (t) = f (t) + iH[f (t)]
where f (t) is a real-valued function and H is the Hilbert’s transform operator. As an example the
exponential function exp(ωt) = cos(ωt) + i sin(ωt) is the special case of the analytical function (with
sin(ωt) = H[cos(ωt)]).
10
the relations
Ch.1 WAVE PROPAGATION THEORY


Ni (U, ∇τ ) = cijkl τ,l τ,j Uk − ρUi
Mi (U, ∇τ ) = cijkl τ,j Uk,l + (cijkl τ,l Uk ),j


Li (U, ∇τ ) = (cijkl Uk,l ),j
while for isotropic media they are given by the relations

Ni (U, ∇τ ) = (λ + µ)Uj τ,i τ,j + µ)Ui τ,j τ,j − ρUi



M (U, ∇τ ) = (λ + µ)(U τ + U τ + U τ ) + µ(2U τ + U τ )+
i
j,i ,j
j,j ,i
j ,ij
i,j ,j
i ,jj

+λi Uj τ,j + µ,j Ui τ,j + µ,j Uj τ,i



Li (U, ∇τ ) = (λ + µ)Uj,ij + µUi,jj + λ,i Uj,j + µ,j (Ui,j + Uj,i )
(1.18)
(1.19)
The equation (1.17) must be satisfied at any time t and any point x and, because of
different orders of derivatives of F, it will be done only setting to zero both N, M, and
L coefficients. In general the second (M) and third (L) term of equation (1.17) do not
vanish, but in high frequency approximation the error is presumably small although the
equation of motion can not be satisfied completely.
In the frequency domain the equation (1.17) can be written, away from the source term
and considering the derivative property of the Fourier transform, as
− ω 2 Ni (U, ∇τ ) − iωMi (U, ∇τ ) + Li (U) = 0
(1.20)
hence the first two terms dominate over the third if the frequency is high, so only the first
two terms are used to determine the eikonal and the ray amplitude.
Taking into account this consideration, we have to solve the following system:
(
N(U, ∇τ ) = 0
(1.21)
M(U, ∇τ ) = 0
The first condition, N = 0, results in the eikonal equation, a nonlinear partial differential equation for τ (x) that constrains also the polarization of U. The magnitude of U
(one scalar quantity) can be determined from the transport equation, which is derived by
projecting the vector M into the direction (polarization) of U.
Eikonal equation
b with
We now solve the eikonal equation N = 0 introducing the Christoffel matrix Γ
elements Γik as
cijkl
τ,j τ,l
(1.22)
Γik =
ρ
for anisotropic media (note that τ,i , i = 1, 2, 3, is the ith component of the slowness vector
p) while for isotropic media they can be written as
Γij =
λ+µ
µ
τ,i τ,j + δij τ,k τ,k
ρ
ρ
(1.23)
1.3 Ray-Theory
11
b has many useful properties (see Červený (2001)) and simplifies
The Christoffel matrix Γ
both the eikonal equations for isotropic and anisotropic media into the simple compact
form
(Γij − δi,j )Uj = 0,
i = 1, 2, 3
(1.24)
Representing the ray amplitude U by the use of the scalar amplitude factor A and the
corresponding unit polarization vector g, so Ui = Agi , the equation (1.24) yields
(Γij − δi,j )gj = 0
(1.25)
that can be viewed as a typical eigenvalue problem for the Christoffel matrix
(Γij − Gδi,j )gj = 0
(1.26)
with G being a general eigenvalue. The eikonal equation (1.25) is equivalent to the equab
tion (1.26) under the condition G = 1, thus determining the eigenvalues of the matrix Γ
and equating them to unity, we obtain the eikonal equation. The Christoffel matrix has,
in general, three eigenvalues G1 (xi , pi ), G2 (xi , pi ), and G3 (xi , pi ), and three eigenvectors
g(1) (xi , pi ), g(2) (xi , pi ), and g(3) (xi , pi ).
Once the eigenvalues and eigenvectors are evaluated, we can see that for a given
anisotropic medium three independent linearly polarized body waves may propagate in
it. These are the quasi-P (qP ) and quasi-S (qS1 and qS2 ) waves. Their amplitudes are
given as
U(m) (x) = A(m) (x)g(m) (x)
(1.27)
and their travel times τ (m) must satisfy the eikonal equation
Gm (xi , pi ) = 1
(1.28)
In the case of an isotropic medium, the Christoffel matrix has again three eigenvalues
but two of them coincide. The expressions of these eigenvalues are
G1 (xi , pi ) = G1 (xi , pi ) = β 2 (xi )pi pi ,
where α(xi ) and β(xi ) are given by the relations
s
λ(xi ) + 2µ(xi )
,
α(xi ) =
ρ(xi )
G3 (xi , pi ) = α2 (xi )pi pi
s
β(xi ) =
µ(xi )
ρ(xi )
(1.29)
(1.30)
The eigenvector g(3) corresponding to the eigenvalue G3 has the same direction as the
slowness vector p, while the eigenvectors g(1) and g(2) corresponding to the two coinciding
eigenvalues G1 = G2 are mutually orthogonal unit vectors, also perpendicular to p, but
they cannot be determined uniquely from the equation (1.26). Once the eigenvalues and
the eigenvector are evaluated, for the isotropic case we have a P wave with the amplitude
U(P ) (x) = A(3) (x)g(3) (x)
(1.31)
which is linearly polarized perpendicularly to its wavefronts, and the S wave which is
elliptically polarized in the plane tangent to the wavefronts with the amplitude
U(S) (x) = A(1) (x)g(1) (x) + A(2) (x)g(2) (x)
(1.32)
12
Ch.1 WAVE PROPAGATION THEORY
Transport equation
Now we solve the transport equation which can be derived from the second equation of the
system (1.21) inserting into equation M = 0 the expression (1.27) for anisotropic media
or, in the case of isotropic media, the expression (1.31) and the expression (1.32) for the P
wave and the S wave, respectively. This means that the only quantities to be determined
are the scalar, complex-valued, amplitude factors A(m) .
(m)
For anisotropic media, multiplying the obtained equation by gi , after some algebra
(see Červený (2001)) we have
(m)
Mi (A(m) g(m) )gi
(m) (m) (m)
gk A,l
= 2cijkl pj gi
(m) (m)
+ A(m) (cijkl pl gk gi
),j = 0
(1.33)
where the index m = 1, 2, 3. Moreover,introducing the vector v(m) with components
(m)
vi
1
(m) (m)
= cijkl pl gk gj
ρ
(1.34)
the equation (1.33) becomes
(m)
(ρA(m) A∗(m) vi
),i = 0
(1.35)
where A∗(m) is the complex conjugate of A(m) and the vector v(m) has the meaning of a
group velocity. The vector in parentheses in the equation (1.35) is the energy flux, so that
the equation (1.35) represents conservation of energy along rays.
Because the anisotropic medium includes as special case the isotropic medium, the
procedure for deriving the transport equation in isotropic structures is analogous and
finally we have transport equations for the P wave and the S wave in the form similar to
equation (1.35). In particular the transport equation for the P wave will be
(ρA(3) A∗(3) α2 pi ),i = 0
(1.36)
with α the P-wave phase velocity, and its expression is given by the first equation in (1.30),
while the system of transport equations for the S wave will be
(
(ρA(1) A∗(1) β 2 pi ),i = 0
(1.37)
(ρA(2) A∗(2) β 2 pi ),i = 0
with β the S-wave phase velocity, and its expression is given by the second equation in
(1.30).
It is possible to demonstrate6 that the solution of the transport equation (1.35) in
anisotropic media can be written in a form that relates the scalar amplitude factor A(m) (R)
at a given point R on the ray (R stands for the receiver point) to the scalar amplitude
factor A(m) (S) at a reference point S on the same ray (S stands for the source point):
s
ρ(S)v (m) (S)J(S)
,
m = 1, 2, 3
(1.38)
A(m) (R) = A(m) (S)
ρ(S)v (m) (S)J(S)
6
We remind the reader to Červený (2001) for the demonstration. Moreover, a step by step demonstration will be found in Brokešová (2006).
1.3 Ray-Theory
13
where v (m) is the group velocity along the ray of the considered wave, ρ is the density,
and J is the ray Jacobian7 of transformation from coordinates xi to ray-coordinates γi .
The above equation is the most common form of the continuation formula in anisotropic
models. The equation (1.38) is valid also for isotropic media, substituting in the equation
the group velocity v (m) with α for P waves or β for S waves.
Finally we want to drive the attention on the regularity of the ray-coordinates system,
which is guaranteed whenever the Jacobian J does not vanish. In complex media the
Jacobian vanishes on caustics so, from equation (1.38), the ray amplitude in a caustic
point grows to infinity. This means that ray methods fail in caustics.
1.3.2
Ray theory general validity conditions
The asymptotic ray theory is based on the asymptotic ray series for high frequencies
(ω → ∞), or alternatively, for small wavelengths (λ → 0). In practical seismic applications
the maximum frequency is of a finite value, so it would be very useful to know the validity
conditions under which the ray method can be applied to compute seismic wavefields.
The applicability of the asymptotic ray theory is limited by several qualitative conditions,
usually formulated as certain inequalities for the frequency of the wavefield or alternatively,
for the wavelength.
The following three general validity conditions, formulated using the wavelength λ in
a qualitative way, are presented by Červený (2001) in the following way:
1. The wavelength λ of the wave under consideration must be considerably smaller than
any characteristic quantity of length dimension li (i = 1, . . . , n, with n the number
of these quantities) in the problem under study:
λ l1 , . . . , ln
(1.39)
Among these quantities there are, for example, dimensions of the model, sourcereceiver distance, layer thicknesses hk (k = 1, . . . , K with K being the number of
layers in the model), radii of curvature of structural interfaces Kj (j = 1, . . . , K − 1),
but also the scale length of inhomogeneities of the medium of type v/|∇v| (with
v being α or β, the P or S wave speed, respectively), ρ/|∇ρ|, where ρ denotes
density, and so on. The condition (1.39) expresses the high-frequency character of
the wavefield.
2. The ray method fails in the vicinity of surfaces S along which the ray field of the wave
under study is not regular, like a caustic surface or a shadow zone boundary. Hence,
the wavefield has to be computed not closer to these surfaces, but at a distance r
from the surface S which satisfies the following condition:
λr
7
The square root
√
J of the Jacobian is called Geometrical Spreading.
(1.40)
14
Ch.1 WAVE PROPAGATION THEORY
3. The ray method is not applicable when the length L of the ray trajectory of the wave
under consideration between the source and the receiver is too large. The estimates
based on the theorem of mean value lead to the condition
λ l02 /L
(1.41)
where l0 has the same meaning as lj in the first condition.
Because of the high-frequency character of the wavefield, it is impossible to remove these
limitations. It is only possible to specify in a more quantitative way these three qualitative
conditions. In particular for the condition (1.39), some authors like Beydoun and Keho
(1987) propose the value of order of units (up to ten times) that guarantee the inequalities
to be satisfied. This evaluation is still commonly used in modelling the medium for high
frequency studies.
Figure 1.2: Subcritical (1), critical (2), and postcritical (3) angles of incidence i0 . For the
subcritical angle of incidence i0 , the R/T angle i and the corresponding ray are real-valued, see
(1). For the critical angle of incidence i0 = i∗ , a headwave will be generated, see (2). For the
postcritical angle of incidence i0 , the R/T angle i and the corresponding ray are complex-valued,
see (3).
A brief summary of the most common situations in which the ray theory may fail is
now reported:
• Postcritical incidence and inhomogeneous waves. Indicating with i0 the angle
of incidence of the wavefield (which propagates with a velocity v0 ) at an interface Σ
and with i the angle of reflection or transmission of the wavefield (which propagates
with a velocity v) at that interface, by the standard Snell’s law we have sin i =
(v/v0 ) sin i0 . For sin i0 > v0 /v, both the angle i, the eikonal and the components
of the slowness vector pi become complex-valued. The waves relative to this case,
called inhomogeneous or evanescent, exist but cannot be computed by the standard
ray method and the angle of incidence i0 is postcritical (see Figure 1.2).
• Ray tangent to an interface locally or globally. Using the same notation of
the previous point, for sin i0 = v0 /v, the angle of incidence i0 is critical8 , and the ray
8
The angle of incidence i0 such that the reflected angle i = π/2 (and so sin i0 = v0 /v) is called critical
angle, and is generally written as i∗ .
1.3 Ray-Theory
15
is globally tangent to the interface (see Figure 1.2). The generated waves propagate
with velocity v along the interface Σ, and they are the so-called headwaves. For
this kind of waves the rays could be formally traced but their amplitudes cannot be
computed by the standard zero-order ray method as the waves belong to the class
of high-order waves. In the case of a ray locally tangent to an interface Σ (when
the angle of incidence i0 = π/2) the ray can be traced, but the wavefield is singular
beyond the point at which the ray touches the interface. A shadow zone is usually
formed beyond the tangent point, in the region between the ray and the interface,
and the ray itself becomes a boundary ray separating the shadow and illuminated
region. Various waves with diffractive nature penetrate into the shadow zone, but
the standard ray method is not capable to handle them.
• Caustics. Caustic surfaces are envelopes of rays. We have seen at the end of section
1.3.1 that the Jacobian J of the transformation into ray coordinates becomes equal
to zero at a caustic point, so the ray theory predicts infinite amplitudes at this point.
At any point situated close to the caustic surface on its illuminated side, there are
always at least two intersecting rays, one approaching the caustics ant the other
leaving it. The wavefield is anomalous in the vicinity of the caustic point but not
along the whole ray, in fact at larger distances from this point the wavefield is again
regular.
• Fine layers and regions of high velocity gradient. If in a model there are layers
too thin with respect to the wavelength, the ray theory general validity conditions
are not fulfilled. These conditions are not satisfied also in regions of high velocity
gradient. Although in the latter case the ray can be formally traced, the wavefield
computations will be low.
• Large curvature of interfaces. The presence of interfaces with large curvature
with respect to the wavelength does not meet the ray theory general validity conditions. The rays of reflected/transmitted waves can be traced but the wavefield
computations will be inaccurate.
• Objects with dimensions smaller or comparable to the wavelength. In the case of
the presence in the model of small objects with dimensions equal or small than
the wavelength under consideration, these objects act as scatterers. The scattered
wavefield cannot be calculated by the ray methods.
Much of these problems can be treated using extensions of the ray method. As an example,
the headwaves can be treated using the high order terms of the ray series (1.13).
In the next section, instead, we will discuss the case of objects in the medium which act
as scatterers. In particular, the calculation of the amplitudes for the Rayleigh scattering
case will be treated.
16
1.4
Ch.1 WAVE PROPAGATION THEORY
Rayleigh Scattering
Seismic waves are scattered at small-scale inhomogeneities (with respect to the wavelength)
when propagating through complex media. These heterogeneities include velocity and
density fluctuations of smaller scales down to microscopic pores, fractures and cracks.
Such small-scale structures cannot be resolved deterministically by seismic surveys but,
nevertheless, they have significant effects on seismic wave propagation because they scatter
the incident seismic wave into all directions.
Indicating with a the characteristic length of a subsurface structure and with k = 2π/λ
the wavenumber relative to the dominant wavelength λ of the wavefield, the ray theory
validity condition (1.39) gives ka 10. The quantity ka is called the normalized frequency.
For regions such that their characteristic length is of the order of the wavelength of the
incident wavefield (ka < 10), the assumptions of the ray approximation break down and
scattering occurs. Hence, as the value of ka becomes smaller and smaller, subsurface
structures will tend to scatter rather then reflect the seismic wave field.
In the case the wavelength λ is of the order to the size a of the object (ka ≈ 1), we
enter the regime of resonance or Mie scattering.
If the wavelength of the incident wave field is much greater than the inclusion itself
(ka 1), the scattering can be described by the Rayleigh scattering theory (Müller,
2000). In the particular case of a weak (low parameter contrast between the object and the
medium) single scattering, it is possible to treat this case also using the Born approximation
(Gubernatis et al.,1977; Bohlen,2004).
Let us now consider small spheres in the medium with a diameter 2a such that 2ka 1
and with arbitrary contrasts in the material parameters. The scattered wavefields up and
us for an incident P-wave, using Rayleigh approximation (Müller, 2000), are respectively:
(
up = Rp (A cos2 θ + B cos θ + C)r̂
us = Rs (D sin 2θ − B sin θ)θ̂
(1.42)
where r is the distance and θ is the scattering angle. Rp and Rs indicate the quantities
Rp = kp2
V e−ikp r
,
4π r
Rs = ks2
V e−iks r
4π r
(1.43)
with V = 34 (πa3 ) the volume occupied by the spherical inclusion. The dynamic character-
1.4 Rayleigh Scattering
17
istics are governed by the parameters A, B, C, and D, which are explicited as follows:
0
2
µ
γ
A = −2
−1
µ
Γ
0
ρ
B =
−1
ρ
2
1 23 (λ0 − λ) + (µ0 − µ) 2 µ0
γ
C = −
−
1
+
1 3 0
2
3 µ
Γ
λ + µ0 + µ
2
0 2 µ
γ
D =
−1
µ
Γ
(1.44)
where the prime stands for inclusion characteristics. Finally, γ and Γ are
vs
2 µ0
γ= ,
Γ=1+
− 1 (3 + 2γ 2 )
vp
15 µ
(1.45)
The upper limit for the Rayleigh approximation is not exactly known, but Gritto et al.
(1995) demonstrate that inclusions with relative size of a/λ ≤ 17% generally carry an error
of less than 5%.
18
Ch.1 WAVE PROPAGATION THEORY
Chapter 2
Method for rapid high-frequency
seismogram calculation: the
COMRAD code
The calculation of synthetic seismograms has recently become a useful tool in seismological
research, and a wide variety of techniques have been developed. Forward modelling, which
means in this case the generation of synthetic seismograms, represent an important part
of many seismological studies, such as seismic tomography or the kinematic inversion of
source parameters (Pengcheng and Archuleta, 2004). The solution of an inverse problem
requires the repeated solving of the forward problem, so that speed is the most stringent
condition that the method must supply.
For horizontally stratified Earth models the complete wave field computation can be
performed by using the theory of normal mode (Aki and Richards, 2002; Rosenbaum, 1960)
or the discrete wave number method (Bouchon and Aki, 1977). Methods based on direct
numerical solution of the elastodynamic equation by finite difference (Madariaga, 1976;
Virieux, 1986), finite elements (Marfurt, 1984), spectral elements (Festa and Vilotte, 2005;
Komatitsch and Vilotte, 1998), or discontinuous Galerkin method (Kaser and Dumbser,
2006) can be applied to study the seismic wave field in complex laterally varying structure.
As an alternative it is possible to obtain the solution to the wave equation by using
approximate high-frequency methods (Červený, 2001). The solution of the elastodynamic
equation is composed of elementary body waves that correspond to the various rays connecting the source to the receiver.
2.1
Aim of a multiphase dynamic ray-tracing code
Although the computation of ray synthetic seismograms is only approximate and the ray
method can fail in certain situations, the high-frequency methods are preferable to direct
numerical methods for many applications, both for the shorter computing times and for
the full interpretability of the seismograms. The problems arise when we want to use
Ch.2 METHOD FOR RAPID HIGH-FREQUENCY SEISMOGRAM
CALCULATION: THE COMRAD CODE
20
a synthetic seismogram as similar to the real one as possible in the high-frequency approximation, such that the seismogram should be relatively complete, although it is not
necessary that it contains every feature of the full elastic wave field. It is becoming necessary to select from all of the rays connecting the source to the receiver only those that
produce appreciable amplitude on the seismogram. The problem of generating a comprehensive set of rays was approached for the first time by Hron (1971, 1972; Hron et al.,
1986) for layered media, by grouping individual rays into families of kinematic equivalents.
Afterwards, Clarke (1993aa, 1993bb) developed a technique for computing synthetic seismograms based on a ray-generation algorithm that involved the symbolic manipulation of
complete wavefield expressions from reflectivity theory, which were truncated to produce
a finite ray series.
Here we propose a new technique for the rapid definition of an exhaustive set of rays
that is based on the hierarchic generation of strings that describe the ray paths and the
phase types. The string generation is subjected to physical constraints that are related
to the propagation medium and the source-receiver geometry. The ray sets will represent
the input of a kinematic or dynamic ray-tracing algorithm (i.e. Červený and Hron, 1980;
Farra and Madariaga, 1987; Snieder and Spencer, 1993; Virieux, 1991). In particular,
the technique developed has been implemented in the multiphase dynamic ray-tracing
COMRAD code (from the Italian: “COdice Multifase per il RAy-tracing Dinamico”) that
uses as its core the dynamic ray-tracing code provided by Farra and Madariaga (1987).
2.2
Method description
Our goal is to carry out an algorithm that rapidly generates an exhaustive number of
seismic-phases to calculate a high frequency seismogram as complete as possible. In particular, the computing time is a critical parameter when the computation of synthetic
seismograms has to be repeated many times in the inversion processes. The amplitudes,
the raypaths and the travel times of the seismic-phases are computed by the dynamic
ray-tracing code provided by Farra and Madariaga (1987). In this section we describe our
method and the discretization of the propagation model.
2.2.1
General concepts
It is possible to use an arbitrary medium as long as it is discretized by M ordered elements
between a free surface and a halfspace. The sequential elements are separated by a surface.
Each element can have an arbitrary shape and is characterized by the following properties:
• V pi : compressive wave velocity (P wave);
• V si : shear wave velocity (S wave);
• ρi : density;
• Qpi : P-wave quality factor (optional);
2.2 Method description
21
• Qsi : S-wave quality factor (optional);
• Ti : top surface of the element;
• Bi : bottom surface of the element (= Ti+1 : top surface of the next element);
• i = 1, . . . , M : index of the element.
where the elastic wave velocities, densities and quality factors are functions of spatial
coordinates.
Both the source and the receivers take the index i of the element in which they are
included. Each ray goes from the source to the receiver and is divided into L curves if
it passes through L elements. Each ray is described by a numerical string composed of
L numbers, and each one is the index of the element crossed by the ray. For a ray there
are also 2L phase numerical strings, because in each element crossed by the ray the wave
can have two different polarizations (P wave or S wave). A numerical string for a phase is
composed of L numbers and each one can be 1 (to indicate a P wave) or 2 (to indicate an
S wave).
As example, for a medium where the elements are three horizontal, parallel layers, the
source is inside the third layer, and the receiver is inside the first layer, the ray string
relative to the direct wave is 3-2-1 at which correspond 23 phase strings.
Although the algorithm can generate only body waves, it is possible to calculate headwaves travel times using an artifact. We can consider a tiny element between two elements,
with the same properties of the second one, and generate a ray string relative to a wave
that propagates inside the tiny element. The travel time obtained for that wave is about
equal to that for a headwave, but the amplitude is not correct.
2.2.2
Selection Criteria
Once the length L of the numerical ray string is fixed, there are M L ray strings of the same
length if the medium is discretized by M elements. However, not all of the ray strings
describe a real ray, because there are some existence constraints for a ray path:
(a) The ray starts at the source;
(b) The ray arrives at the receiver;
(c) The ray path is bordered by the free surface and the half space;
(d) If a ray points towards the top surface of the ith element, it can remain in the same
element (with opposite direction) or go to the (i − 1)th element (with the same direction);
(e) If a ray points towards the bottom surface of the ith element, it can remain in the same
element (with opposite direction) or go to the (i + 1)th element (with the same direction);
(f) If the receiver is on the free surface, the ray must come from below;
(g) If the source is on the free surface, the starting direction of each ray cannot be up.
22
Ch.2 METHOD FOR RAPID HIGH-FREQUENCY SEISMOGRAM
CALCULATION: THE COMRAD CODE
To generate only the ray strings that describe real ray paths, we use a hierarchic order of ray generation, where the root (first generation) is the index of the element that
contains the source, and the number of the generations is the length of the numerical
strings of the rays. The hierarchic order of ray generation satisfies itself for the (a), (b),
and (c) constraints. Indicating with S the source index and with R the receiver index, the
generation that produce the ray string of minimum length, that is the direct ray, is given
by the number Lmin = |S − R| + 1. An example of hierarchic ray generation taken to
the fourth generation for a source in the third element (S = 3) and a receiver in the first
element (R = 1), where the medium is discretized by M = 5 elements, is shown in Figure
2.1. In this Figure, the last component of the strings that satisfy for each generation the
(b) constraint is evidenced with a grey circle. The selected string will be composed by all
the numbers in the tree structure from the root to the evidenced number. In this case we
have, on the whole, four ray strings (3-2-1, 3-2-1-1, 3-3-2-1, and 3-2-2-1) up to the fourth
generation that satisfy the (a), (b), and (c) constraints.
Figure 2.1: Tree structure of ray strings stopped to the fourth generation. The root, that is
the source position, is the number three. The grey circles identify, for each generation, the last
component of the strings that satisfy the (b) selection criterion. We have four ray strings up to
the fourth generation.
Using the additional (d), (e), (f), and (g) constraints, we discard some branches inside
the numerical string tree. In particular, putting the receiver on the free surface the ray
strings selected are shown in Figure 2.2. Now we have only two ray strings (3-2-1 and
3-3-2-1) up to the fourth generation. If we put the receiver inside the first element instead
of on the free surface, we will have the additional ray string 3-2-1-1 because there is no (f)
constraint yet, and consequently the ray can arrive at the receiver also from above.
Since the number of phase strings becomes greater and greater if the number of generations increases, and consequently the computing time rises, it is possible to introduce
some constraints on the propagation in order to reduce the number of phases. For this
2.2 Method description
23
reason, we created the following additional constraints on ray-paths and on phases:
(I ) Maximum number of generations that means the truncation of ray-series;
(II) Maximum number of reflections in each element;
(III) Phase selection based on their expected amplitude values.
In agreement with ray-theory there are infinite rays that propagate form the source
Figure 2.2: Tree structure of ray strings stopped to the fourth generation as in Figure 2.1, but
using all of the (a) to (g) constraints. There are only two ray strings up to the fourth generation
because the receiver is on the free surface. If the receiver is inside the first element, there will also
be the string 3-2-1-1. The ray-paths relative to the two selected ray strings in the layered velocity
model are shown on the right. The white star is for the source and the white triangle is for the
receiver.
to the receiver, but there is also a loss of the initial ray energy whenever it has a reflection/transmission on a discontinuity, besides the energy lost for geometric and anelastic
attenuation. After a certain number of generations, the ray strings represent those rays
that have encountered so many discontinuities as to have a negligible energy. For this
reason it is possible to truncate the ray series at a certain generation without significant
loss in energy; this is the principle on which is founded the (I) constraint. In particular
the default maximum number of generation is fixed by the number
Lmax = Lmin + 2 · M
(2.1)
which guarantees a complete investigation of the propagation medium.
The (II) constraint assumes that a ray trapped inside an element has a loss in energy
bigger than the energy lost by untrapped rays, especially for deeper elements. Using the
24
Ch.2 METHOD FOR RAPID HIGH-FREQUENCY SEISMOGRAM
CALCULATION: THE COMRAD CODE
(II) constraint, a ray string will be accepted only if is satisfies the condition:
(n − 1) · i ≤ RIF M
(2.2)
with n the number of consecutive multiple reflection of the ray inside the ith element, i
the element index, and RIF M a number that controls the multiple reflections (the default
value is RIF M = 8). Using the RIF M default value we can have maximum 9 consecutive
reflections in the first element (the shallowest), 5 consecutive reflections in the second, 3
in the third and the fourth, 2 from the 5th to the 8th , and only one for the others. The
(III) constraint is developed following this scheme:
• For each k th component of the phase string a weight wk , from 0 to 1, is assigned;
• Once the weight is assigned, the phase string will be accepted only if it satisfies the
condition:
L
Y
wk ≥ P HS
(2.3)
k=1
where L is the length of the phase string and P HS is a number that represents a percentage of the weights product assigned to the direct P-wave (the default percentage
is 1 · 10−4 %).
Here we describe how the weights wk have been calculated. For each incident phase
(P or S) on the boundary between two elements, we have to calculate each time the
scattering coefficient relative to one of the 8 phases that can be generated for reflection/transmission/conversion (PrP, PrS, PtP, PtS, SrP, SrS, StP, StS; where “r” indicates
the reflection and “t” the transmission). These coefficients are functions of the incidence
angle and the physical parameters (P-wave velocity, S-wave velocity, and density) of the
elements crossed by the wave. Their calculation requires both computing time and the values of the incidence angles, but the latter are known only after ray-tracing. To overcome
the problem, the weights wk are defined for each type of interface and for each type of
phase independently on the incidence angle. Indeed, to define the weights, we have calculated, for each one of the 8 scattering coefficients, the integral on the whole interval of the
incidence angle, dividing the result for the same interval. The values obtained for the four
phases that an incident wave can generate (PrP, PrS, PtP, PtS for a P-wave; SrP, SrS, StP,
StS for a S-wave) are normalized such that their sum is equal to one. This computation
is carried out varying the interface parameters, that is the values of the P-wave velocities,
S-wave velocities, and densities of the elements separated by the discontinuity (Table 2.1).
Precisely, S-wave velocities have been determined from P-wave velocities using different
values of the V p/V s ratio, while densities have been calculated using the following empirical
formula of Ludwig et al. (1970):
ρ = 0.7 + 2.23v − 0.598v 2 + 0.0704v 3 − 0.00283v 4
(2.4)
with ρ the density in g/cm3 and v the P-wave velocity in km/s. The results show that
for an incident P-wave the value of the integral, defined previously, depends essentially on
2.2 Method description
25
the V p1 /V p2 ratio1 . Likewise for an incident S-wave the integral depends on the V s1 /V s2
ratio. In fact, as showed in Figure 2.3 and 2.4, the curves relative to the different cases
reported in Table 2.1 have a similar shape for each type of phase (i.e. PtP, PrP, etc. . .).
Velocity model Vp1 (km/s) Vp1 /Vs1
A
2.50
1.73
B
7.00
1.73
C
3.40
1.73
D
4.00
3.50
E
4.00
2.00
Vp2 (km/s) Vp2 /Vs2
1.8-7.0
1.73
1.8-7.0
1.73
1.8-7.0
3.50
1.8-7.0
1.73
1-8-7.0
2.00
Table 2.1: Different velocity models to study the effect of the boundary between a half-space
that contains the incident wave (labelled by the number 1) and a half-space that contains the
transmitted wave (labelled by the number 2).
To avoid the construction of an indexed table for each phase and for each value of
V p1 /V p2 and V s1 /V s2 we have selected only 5 intervals of these ratios. The weight
assigned to each phase will correspond to the maximum of the corresponding integral
curve inside each region. This allows the realization of a smaller indexed table (Table 2.2)
which contains the weights to assign to the components of the phase strings. Moreover,
the weight assigned to a component of the phase string is equal to zero if in the ith element
the component indicates an S wave and V si < 0.1 m/s. From equation (2.3) this implies
that the whole phase string is rejected.
Type
of phase
ptp
prp
pts
prs
sts
srs
stp
srp
Region I
Region II
Region III
Region IV
z < 0.5 0.5 < z < 0.8 0.8 < z < 0.95 0.95 < z < 1.2
0.17
0.45
0.73
1.00
0.70
0.69
0.37
0.24
0.21
0.24
0.21
0.25
0.51
0.35
0.19
0.12
0.50
0.92
0.98
1.00
0.90
0.66
0.19
0.10
0.01
0.02
0.02
0.02
0.17
0.07
0.03
0.03
Region V
1.2 < z
0.71
0.35
0.27
0.21
0.96
0.16
0.17
0.05
Table 2.2: Weights assigned for each region, defined by z parameter, and for each type of phase.
The z parameter is equal to the V p1 /V p2 ratio for an incident P-wave or equal to the V s1 /V s2
ratio for an incident S-wave.
1
The indexes 1 and 2 in the velocities ratio indicate the medium that contains the incident wave and
the medium that contains the transmitted wave, respectively.
26
Ch.2 METHOD FOR RAPID HIGH-FREQUENCY SEISMOGRAM
CALCULATION: THE COMRAD CODE
Figure 2.3: Integral of scattering coefficients for an incident P-wave, as a function of the
V p1 /V p2 ratio between the P-wave velocity in the medium 1 (that contains the incident wave)
and P-wave velocity in the medium 2 (that contains the transmitted wave). The colour of curves
refers to different type of phases, while the style is relative to different velocity models reported in
Table 2.1. The five regions I, II, III, IV, V, separated by vertical lines, are defined in Table 2.2.
Figure 2.4: As for Figure 2.3 but referring to an incident S-wave as a function of the V s1 /V s2
ratio between the S-wave velocity in the medium 1 and S-wave velocity in the medium 2.
2.2 Method description
2.2.3
27
Structure of the COMRAD code
The Comrad.f code is the Fortran77 version of multiphase code that, based on selection
criteria, generates an exhaustive set of ray and phase strings allowing the computation
of a seismogram as complete as possible. In order to compute synthetic seismograms,
that is travel times, amplitudes, and ray paths relative to the selected phases, it uses as
its core the dynamic ray-tracing code developed by Farra (Farra and Madariaga, 1987)
which needs a medium discretized by flat tilted layers. However, the COMRAD code
can be easily implemented with other dynamic ray-tracing programmes which can work
on 3-D models with irregular discontinuities. The code is accompanied by the bash script
multiphase.sh, which optimizes the whole calculation process and runs both the COMRAD
and core programmes.
Figure 2.5: Block scheme of the Comrad.f computer code.
28
Ch.2 METHOD FOR RAPID HIGH-FREQUENCY SEISMOGRAM
CALCULATION: THE COMRAD CODE
The structure of the code is shown in Figure 2.5, where we outline its most important
features. The code requires the following input data:
• the creation.inp file which contains the medium information;
• the xxxx.dis file (where xxxx represents a four characters word) which contains the
source-receiver geometry;
• the maximum number of generations (Lmax), which truncates the ray series. The
user can choose to use the default value (Lmax = Lmin + 2 · m);
• the multiple reflections parameter (RIF M ), which allows the ray to have a finite
number of reflections in each element. The user can choose to use the default value
(RIF M = 8);
• the phase selection parameter (P HS), which discards phases with negligible amplitudes. The user can choose to use the default value (calculated on the approximated
amplitude value of the direct P wave and fixed at 1 · 10−4 % of this value);
• the initial polarization of the phases (P or S).
Using the hierarchical generation method with the constraints described previously, the
code generates the ray and phase strings and creates the following output data:
• the kernel.inp file which contains the input parameters for the core;
• the two xxx1.dis and xxx2.dis files (where xxx represents a three character word)
inside which the Comrad.f code will write all of the ray and phase strings to be used
by the core programme for the computation of the synthetics. The first file describes
rays where their starting direction is up, and the second, where it is down. These
two files are very important also to search the type of phase identified on the output
seismograms, since they contains the list of the ray and phase strings.
The existence of the two xxx1.dis and xxx2.dis files assures the calculations of both
the P/P\ (where S/ or P/ indicates an upcoming S or P wave and S\ or P\ a downgoing
S or P wave, respectively) and P\P/ phases (and similar), although they are described by
the same ray and phase strings if the source and the receiver are both in the same element
(i.e. the ray string is 1-1 both for P/P\ and P\P/ phases if the source and the receiver
are in the first element of the medium). Moreover, if the source is on the free surface, the
xxx1.dis does not exist because of the (g) constraint.
In order to select the appropriate weight for each type of phase we have transformed
the Table 2.3 in a 3D-matrix of indexes Z, RT , and P OL. The RT index indicates
if there is a reflection or a transmission; it is equal to 1 for a transmission and 2 for
a reflection. The P OL index indicates the polarization of the phase before and after
a reflection/transmission; it is equal to 1 for PP, 2 for PS, 3 for SP and 4 for SS. Z
indicates the number of the region (see Figures 2.3 and 2.4) containing the V p1 /V p2 ratio
2.3 Validation of the method
29
(for an incident P-wave) or V s1 /V s2 ratio (for an incident S-wave) values. To avoid the
introduction in the comrad.f program a great number of control instructions to verify
the index values, we have searched two relations (equation (2.5) and (2.6)) that directly
determine the values of the RT and P OL indexes. The value of the RT index is obtained
from the equation (2.5):
RT =| ud1 − ud2 | +1
(2.5)
where ud1 and ud2 are the directions (1 for up and 2 for down) of the ith component and
the (i + 1)th component of the ray string, respectively. The value of the P OL index is
obtained instead from the equation (2.6):
P OL = 2a + b − 2
(2.6)
where a and b are the polarizations (1 for a P-wave and 2 for a S-wave) of the ith component
and the (i + 1)th component of the phase string, respectively.
After the COMRAD and core programmes have finished, the output consists of files
containing the Green function2 and a list of raypaths, travel times and amplitudes for all
the phases. If more than one receiver is considered it is possible to produce a synthetic
section for each component (X, Y, and Z).
2.3
Validation of the method
The higher the number of generations we consider, the higher the number of phases the
core must calculate. To make the best choice on the number of generations, we have to
compare the calculation time with the complexity we need for the synthetics.
Several tests have been carried out both to set the default parameters and to understand the limits beyond which the method should fail. We have computed synthetics for
different velocity models and compared qualitatively and quantitatively the results with
those obtained using complete wavefield programs. In particular, here we will describe in
details one of these validation tests.
We compute synthetic seismograms for two receivers (R1 and R2 ) respectively at 1 km
and 30 km distance from the epicentre of an explosive source. The source function is a
triangle with duration of 0.1 s, but it is possible to choose other source functions (i.e.
Sinusoidal, Gaussian, Ricker, . . .) and change the duration and the frequency. We use a
crustal velocity model Bernard and Zollo (1989), where the elements are considered in this
case as horizontal parallel layers. The model is described in Table 2.3, while the results
of the simulations are given in Table 2.4. Only the geometrical spreading is considered in
the computation of synthetics and no headwaves are considered since we can compute only
their travel times and not the amplitudes.
The source depth is 4 km, and both of the receivers are at 1 m in depth. For this sourcereceiver geometry, the source is in the second layer, the receivers are in the first layer and
2
The Green function is the displacement due to a unidirectional unit impulse, which is localized precisely
in both space and time.
30
Ch.2 METHOD FOR RAPID HIGH-FREQUENCY SEISMOGRAM
CALCULATION: THE COMRAD CODE
Interfaces Depth (km) Vp (km/s) Vs (km/s) ρ (g/cm3 )
1
0
2.30
1.33
2.2
2
3
5.30
3.06
2.3
3
7
6.00
3.46
2.4
4
10
6.28
3.63
2.6
5
20
6.54
3.78
2.8
Table 2.3: The crustal velocity model used for the simulations.
the first real phases start at the second generation. In Table 2.4, for each generation,
the time needed by both the COMRAD and core programmes for the calculations can
be seen, along with the total number of phases and the RM S calculated for both of the
receivers and for both of the X and Z components of the seismogram. The RM S is a
measure of the similarity between two signals and in the Table 2.4 it is calculated between
the synthetic seismogram obtained at each generation (S(t)) and that obtained for the
previous generation (SREF (t)). The RM S is defined as follows:
sP
| S(t) − SREF (t) |2
tP
(2.7)
RM S =
2
t | SREF (t) |
where t is the time. The amplitudes of the seismograms are normalized with respect to their
maximum amplitudes. The calculations are carried out using a PC with an AMD-3GHz
processor and 2Mb SD-RAM memory.
N. of
GEN.
2
3
4
5
6
7
8
9
10
11
12
COMRAD
TIME
0m 00.05s
0m 00.05s
0m 00.05s
0m 00.05s
0m 00.05s
0m 00.05s
0m 00.05s
0m 00.11s
0m 00.27s
0m 01.18s
0m 03.39s
CORE
TIME
0m 00.06s
0m 00.15s
0m 00.29s
0m 01.13s
0m 04.42s
0m 16.84s
1m 07.45s
4m 18.82s
16m 18.15s
1h 00m 23.47s
3h 40m 56.38s
N. of
PHASES
2
10
34
114
370
1266
4210
14706
49522
174450
590194
RM S
R1 -CompX
1.037
0.108
0.107
0.027
0.026
0.008
0.008
0.002
0.002
0.001
RM S
R1 -CompZ
0.992
0.104
0.103
0.144
0.028
0.010
0.010
0.004
0.004
0.002
RM S
R2 -CompX
4.628
0.980
0.262
0.440
0.398
0.186
0.131
0.087
0.077
0.037
RM S
R2 -CompZ
4.789
0.932
0.340
0.494
0.399
0.252
0.220
0.109
0.092
0.058
Table 2.4: Computing time of COMRAD and core codes, number of phases, and RMS computed
at each generation for both of the receivers R1 and R2 , at 1 km and 30 km distance from the source,
respectively. The horizontal (CompX) and vertical (CompZ) components of the seismograms are
considered.
We show the results up to the 12th generation since the RMS becomes negligible at the
expense of an excessive computing time. From Table 2.4, it is clear that the RM S is lower
2.3 Validation of the method
31
and lower as a function of the number of generations. Moreover, starting from the RM S
calculated between the 9th and 10th generations, the RM S decrease rate at 30 km distance
becomes stable for both the X and Z components. This means that if we truncate the ray
series at the 10th generation we obtain little values both for the RM S and the computing
time.
If we take into account also the propagation constraints it is possible to reduce further
the number of phases created at each generation. The selection on the phase strings is
controlled by the RIF M and the P HS parameters discussed in the previous paragraph. In
order to evaluate the changes on the seismogram due to the application of these constraints,
we calculate the RM S, the computing time, and the number of phases for different values
of RIF M and P HS parameters. In particular we used RIF M = ∞, when the constraint
is not used, and the default value RIF M = 8. The test is performed on the seismogram
obtained at the 10th generation for both receivers R1 and R2 of the previous model. The
RM S is computed using as reference signal both the same seismogram but complete with
all phases and the seismogram computed at the 12th generation complete with all phases.
As we can see in the Table 2.5 the use of the RIF M parameter can reduce the number of
phases (and then the computing time) from 2% to 20%, but this reduction depends on the
choice of the P HS value (it is bigger for lower P HS values). On the other hand P HS is
a crucial parameter able to reduce considerably the number of phases without losing the
phases with bigger amplitude. If we choose for P HS a value equal to 1.4% of the relative
direct P-wave value, the computing time is drastically reduced (from 16m 18.15s to 1.77s)
whereas the RM S becomes significant for the receiver more distant (R2 ). The best values
for RIF M and P HS parameter can be chosen on the basis of the required precision.
P HS
(%)
0
1.4·10−4
1.4·10−4
1.4·10−2
1.4·10−2
1.4
1.4
RIFM
∞
∞
8
∞
8
∞
8
COMRAD
TIME
16m18.42s
8m 52.96s
8m 18.91s
1m 25.20s
1m 18.31s
0m 01.99s
0m 01.77s
N. of
PHASES
49522
29300
24218
4648
4195
137
134
RM S-10th
R1 -CompZ
0.000
0.000
0.000
0.000
0.000
0.015
0.015
RM S-10th
R2 -CompZ
0.000
0.035
0.035
0.298
0.298
0.694
0.694
RM S-12th
R1 -CompZ
0.005
0.005
0.005
0.005
0.005
0.017
0.017
RM S-12th
R2 -CompZ
0.109
0.116
0.116
0.319
0.319
0.703
0.703
Table 2.5: Computing time of COMRAD(+core) code, number of phases, and RM S computed at
each generation for both of the receivers R1 and R2 , at 1 km and 30 km distance from the source,
respectively. The vertical component (CompZ) of the seismogram is considered. RM S-10th is the
RM S relative to the seismogram computed at the 10th generation without constraints. RM S-12th
is the RM S relative to the seismogram computed at the 12th generation without constraints.
To better understand if our choice is appropriate, we can calculate the complete
wavefield for the case under study by the AXITRA programme, which uses the discrete
wavenumber method developed by Bouchon (Bouchon, 1981; Coutant, 1989). Afterwards,
we can compare the AXITRA results with those obtained using the COMRAD programme
32
Ch.2 METHOD FOR RAPID HIGH-FREQUENCY SEISMOGRAM
CALCULATION: THE COMRAD CODE
stopped at the 10th generation, where the amplitudes of the synthetics are expressed in
velocity and are normalized with respect to their maximum amplitude. The AXITRA
computation time (up to the frequency of 25 Hz) is 53 minutes and 41.6 seconds, 3-4-fold
longer than the COMRAD+core computation time obtained at the 10th generation (see
Table 2.5). For a quantitative comparison between the synthetics, we use the misfit criteria
in time and frequency that was developed by Kristeková et al. (2006). In fact the standard RM S matches the single-valued envelope misfit only in the case of a pure amplitude
modification of the signal while in all other cases RM S considerably overestimates the
misfits and does not recognize what causes the difference (Kristeková et al., 2006). The
time frequency envelope misfit (T F EM ) is calculated according to the following equation:
T F EM (t, f ) =
∆E(t, f )
maxt,f (| WREF (t, f ) |)
(2.8)
while the time frequency phase misfit is calculated according to equation (2.9):
T F P M (t, f ) =
∆P (t, f )
maxt,f (| WREF (t, f ) |)
(2.9)
where t is the time, f is the frequency, ∆E(t, f ) is the local time-frequency envelope
difference, ∆P (t, f ) is the local time-frequency phase difference, and W REF (t, f ) is the
time-frequency representation of the reference signal SREF (t) based on the continuous
wavelet transform. In this case the difference is computed between the COMRAD and
AXITRA seismograms.
In Figures 2.6 and 2.7, the T F EM and the T F P M misfits for the vertical component
of the seismograms computed at the receivers at 1 km and 30 km distance from the source
are shown. Red colour in the T F EM and the T F P M images indicates greater amplitude
and a positive phase shift in the COMRAD seismogram respect to AXITRA seismogram.
Blue colour represents the opposite case. For the receiver at 1 km distance the T F EM
and the T F P M values suggest a good agreement between the synthetics computed using
the two methods. In fact the maximum T F EM value, observed at 1.5 s, is about 4%
while the maximum T F P M value is about 0.6%.This agreement slowly decreases with the
source-receiver distance as shown in Figure 2.7 for the receiver at 30 km distance from
the source. In Figure 2.7 we can see that there is a maximum envelope misfit, of about
20%, in correspondence of 7.5 s and 11 s. These misfits are due to differences in amplitude
between the two seismograms, as evidenced in the plots of the synthetics in Figure 2.7.
Over 16 s COMRAD seismogram amplitudes are generally underestimated probably for
the absence of the surface waves calculated by AXITRA and not by COMRAD code.
Moreover, from the T F P M plot in Figure 2.7, the phase misfit is less than 7%. In general,
the differences in amplitude and phase between the synthetic seismograms computed by
the two codes decrease at higher frequencies as expected by theory. The misfits are shown
only for frequencies greater than 3 Hz because the COMRAD code is valid only in the high
frequency approximation.
2.3 Validation of the method
33
Figure 2.6: (Top) Time frequency envelope misfit T F EM (t, f ) and (Middle) time frequency
phase misfit T F P M (t, f ) plots for the vertical components of the seismogram computed for a
receiver at 1 km distance from the source. (Bottom) The two synthetic seismograms computed
by AXITRA and COMRAD codes. The amplitudes are normalized with respect to the maximum
amplitude of each seismogram.
34
Ch.2 METHOD FOR RAPID HIGH-FREQUENCY SEISMOGRAM
CALCULATION: THE COMRAD CODE
Figure 2.7: As in Figure 2.6 but referring to the receiver at 30 km distance from the source.
2.4 Results and discussions
2.4
35
Results and discussions
We have developed a method for rapid high frequency seismogram computation based on
a hierarchical order of ray and phase strings generation subjected to physical constraints.
The method is used in a code (Comrad.f) to compute an exhaustive set of phases able to
produce a complete body waves seismogram. It uses as core the dynamic ray-tracing code
developed by Farra and Madariaga (1987).
We have numerically tested the method in two steps using a PC with an AMD-3GHz
processor and 2Mb SD-RAM memory. First we evaluated, generation by generation, both
the computing time and the RMS between the COMRAD synthetics obtained in each
generation and those obtained in the previous generation. This allowed us to choose the
number of generation at which we can truncate the ray series that is the 10th generation.
After fixing the number of generations, we compared the COMRAD-derived synthetic
seismograms with AXITRA-derived synthetic seismograms, the latter being a programme
that calculates the complete wavefield. The numerical tests were carried out by the quantitative misfit criteria developed by Kristeková et al. (2006).
The seismograms computed by the COMRAD and AXITRA programmes are in good
agreement at least up to 30 km source-receiver distance. For a receiver 30 km distant from
the source, where we have greater misfits, the maximum time frequency envelope misfit is
about 20% only in correspondence of 7.5 s and 11 s on seismograms, due to the differences
in amplitudes.
Finally, we have compared the computation times for both of the programmes, and
we can be sure that COMRAD code (with 10 generations without constraints) is 3-4fold faster than AXITRA code (up to a frequency of 25 Hz). If we want to compute the
synthetics at higher frequencies, the AXITRA code needs more time, while the computation
time of the COMRAD code remains the same or decreases if we take into account the
additional constraints on the phases. These constraints are based on the maximum number
of reflections in each element of the model and on the expected amplitude value for each
phase. We have evaluated the effects of these constraints on the seismogram computed
at the 10th generation in terms of number of phases, computing time, and accuracy of
the signal. The results confirm that the constraints are effective, reducing the number
of phases to be computed without losing the phases with appreciable amplitude on the
seismogram.
The source code and a user guide of the COMRAD program are reported in Appendix
A.
36
Ch.2 METHOD FOR RAPID HIGH-FREQUENCY SEISMOGRAM
CALCULATION: THE COMRAD CODE
Chapter 3
Applications of the COMRAD code
in seismology
In this chapter we will show some applications of the COMRAD code in seismology. In
particular in the first section some applications to seismic reflection data recorded during SERAPIS active seismic survey will be described. In the second section synthetic
seismograms for an earthquake faulting source will be computed, considering the focal
mechanism of the source. In particular, an example of a vertical strike-slip mechanism will
be considered. Finally, the extended source case will be planned.
Each section has a conclusive paragraph, where the limits and the potentialities of this
code for such problems will be discussed.
3.1
Forward modelling of active seismic data: the
case study of Campi Flegrei caldera
Seismic data are used for the seismic exploration of the medium in order to deduce information about the rocks from the observed arrival times and from variations in amplitudes,
frequency, phase and wave shape in the recorded seismogram. In particular structural
information is derived from headwaves paths (refraction seismic) and reflected paths (reflection seismic). In seismic exploration both passive and active seismic data (Telford et al.,
1990) can be used.
When an earthquake occurs, the rocks rupture generates seismic waves that travel
outward from the fracture surface. These waves, as they spread throughout the propagation
medium, are recorded by seismic instruments, called seismometers, and seismologists use
the data to deduce information about the nature of the rocks or the properties of the
seismic source. In this case we use the term of passive seismic, where both the position
and characteristics of the source are unknown.
Active seismic involves basically the same type of measurements as passive seismic, but
the sources are human-made, controlled and movable. Moreover, the distances between
the source and the receivers are relatively small.
38
Ch.3 APPLICATIONS OF THE COMRAD CODE IN SEISMOLOGY
A useful tool for the seismic exploration is the calculation of synthetic seismograms for
a given structure to be compared with the active seismic data recorded for that structure.
The comparison can help the operator to understand the goodness of the model produced
for the structure under investigation and to the identification of second arrivals of phases
in the observed seismograms. The model can be further optimized using step by step
procedures based on the comparison between synthetic and real seismograms and the
minimization of their differences.
Here we show an application of the COMRAD code to reflection seismic studies on the
Campi Flegrei caldera (Figure 3.1). Since the code is based on the ray theory (see Chapter
2), it is very useful whenever headwaves and surface waves are not required to study the
medium, as in the case of seismic reflection when the reflected body waves are the most
important waves to be considered.
Figure 3.1: An image of the Campi Flegrei caldera (Campania region, southern Italy). This
image was produced using Google Maps tools.
Campi Flegrei (Figure 3.1) is an active volcano located at the eastern margin of the
Tyrrhenian Sea, wherein a large part of the city of Naples is located. Campi Flegrei is a
partly submerged caldera and its shape is believed to be a consequence of two huge areal
collapses generated by large explosive eruptions occurred in the past. The structure of
3.1 Forward modelling of active seismic data: the case study of Campi Flegrei
caldera
39
the caldera has been investigated previously via several methods and in 2001 an extended
active survey was performed in order to study its internal structure. On the following
section a brief description of this experiment is shown.
3.1.1
A brief description of Serapis active seismic survey
SERAPIS (SEismic Reflection/Refraction Acquisition Project for Imaging complex volcanic Structures) is the acronym of an extended active survey performed during September 2001, in the gulfs of Naples and Pozzuoli (Zollo et al., 2003) with the aim to provide
new insights on the Campi Flegrei caldera, Southern Italy, and to investigate its feeding
system. During the SERAPIS experiment, seismic signals produced by a battery of 12,
16-liters air-guns mounted on the oceanographic vessel NADIR (IFREMER) were recorded
at a dense array of three-component, sea bottom and on land seismographs installed in
the bays of Naples and Pozzuoli.
Seventy-two, three-component stations were installed on-land in the areas of Campi
Flegrei, Mt.Vesuvius and on the islands of Ischia and Procida. Sixty, sea bottom seismographs (OBS) were installed in the gulfs of Naples and Pozzuoli. A total number of 5000
shots were fired during the experiment, with an average spacing of 125 m.
Based on the acquisition layout of the experiment, a specific trace gathering scheme,
that took advantage of the potential of the 3D source-receiver geometry, was developed
and applied to improve the detection and resolution of reflected/converted phases.
The recorded seismic waveforms were arranged in 3D Common Midpoint (CMP) gathers
following the procedure described below. The Pozzuoli Bay area was subdivided into
square cells and the midpoint position between the corresponding source and receiver was
calculated for each trace considered, taking as maximum offset a distance of 9000 m. All of
the records with the midpoint falling in a given cell have therefore been grouped into the
same CMP, independent of the source-receiver azimuth. This is equivalent to assuming a
near vertical variation in the velocity distribution in a cone, pointing down in the Earth
at the cell centre, and having a diameter of 9000 m at the surface (Zollo et al., 2003).
After several trials, a cell size of 500 x 500 m was chosen based on the criteria of having
a sufficiently high number of traces in each CMP gather (> 30) and a dense coverage of
the explored area, while maximum offset was chosen based on the duplex criteria that
the lateral variations of the velocity had to be negligible with respect to those vertical,
and along the sections the morphology of the interface had to be more or less planar and
horizontal.
3.1.2
Comparisons between synthetic sections and real sections
Starting from a 3-D tomographic model of Campi Flegrei (Zollo et al., 2003) obtained from
data recorded during the SERAPIS experiment, a 1-D average model is derived using also
information on physical properties of rocks measured on samples from Agip wells (Rosi
and Sbrana, 1987; Giberti et al., 2006).
40
Ch.3 APPLICATIONS OF THE COMRAD CODE IN SEISMOLOGY
The 1-D average model has been used to calculate synthetic sections in order to find
multiple reflections in the real sections. In Figure 3.2 the 1-D average model and the
comparison between a real section with a synthetic section is reported.
Figure 3.2: (Right) 1-D average model for Campi Flegrei. (Left) Comparison between a real
section of SERAPIS experiment and the relative synthetic section (only vertical component) computed with COMRAD code for the 1-D model reported in the figure. Reflections to the interfaces
of the model are indicated, and the multiple reflection to the first interface is evidenced in blue.
A blue colour evidences in Figure 3.2 the multiple reflection to the first interface at a
depth of 600 metres. Moreover, also the P-to-P reflections to the three interfaces of the
model with the relative rms velocity1 are indicated in Figure 3.2.
1
Considering a medium composed by a number of horizontal layers of different velocities (the velocity
is constant within each layer), the travel time t of the reflected wave to the nth interface can be calculated
using the root-mean-square (rms) value Vrms of its velocity, so that:
2
t2 = t20 + x2 /Vrms
3.1 Forward modelling of active seismic data: the case study of Campi Flegrei
caldera
41
The COMRAD code up to the 6th generation has been used to compute the vertical
component of the synthetic seismogram of Figures 3.2 and 3.3, with the source-receiver
geometry equal to the source-receiver geometry of the SERAPIS experiment. In practice,
we have considered an explosive source (with a peak frequency of about 10 Hz) at a depth
of 10 m and the receivers depth equal to 90 m, while the maximum offset has been fixed
at a distance of 2000 meters. The processing has been performed using an automatic
gain control with a 0.5 s window, and amplitude normalization for both the real and the
synthetic sections. Moreover, an 8-15 Hz band-pass filtering was applied on real sections in
order to remove low frequency waves, which are not taken into account by the COMRAD
code.
Figure 3.3: Three different real sections (CMP 785, CMP 907, CMP 949) in which the multiple
reflection of Figure 3.2 is clear. The multiple reflection to the interface at 600 m is evidenced with
blue colour both on real and synthetic sections (vertical components), while the single reflection
to the same interface is evidenced with red colour only on the synthetic section.
where x is the distance between the source and the receiver (offset), t0 is the travel time for x = 0, and
Vrms is given by
sP
n
2
i=1 Vi ti
P
Vrms =
n
i=1 ti
Vi and ti are, respectively, the average velocity in and the traveltime through, the ith layer.
42
Ch.3 APPLICATIONS OF THE COMRAD CODE IN SEISMOLOGY
The multiple reflection shown in Figure 3.2 is clear also on other real sections, as
reported in Figure 3.3. This confirms that the phase is really a multiple reflection to the
first interface of the model and not a reflection to a deeper interface (that can be found at
about 1500 meters in depth, according to the travel-time of the phase under investigation).
The COMRAD code has been used also to compute a synthetic seismic section to
simulate one of the CMP gathers of SERAPIS survey in order to make quantitative analyses
on reflected waves to the interfaces of a 1-D model for Campi Flegrei. For this case we used
the average 1-D model for the Campi Flegrei caldera, based on ray kinematic modelling
of PP and PS travel times and on PS-to-PP amplitude ratios vs. offset, as obtained by
Maercklin and Zollo (2008) from the analysis of real data. The model is shown in Table
3.1.
Interfaces Depth (km)
V p (km/s)
V p/V s
1
0.0
1.6
3.40
2
0.6
3.5
1.70
3
2.7
from 4.7 to 6.0
1.58
4
4.0
6.0
1.65
5
7.5
2.8
2.35
√
Halfspace
8.7
6.5
3
ρ (g/cm3 )
2.2
2.6
2.6
2.6
2.6
2.6
Table 3.1: Average 1-D model for Campi Flegrei (Southern Italy) as derived by Maercklin and
Zollo (2008). In the third layer there is a positive gradient velocity with depth, while the 5th layer
is an area (probably the magma chamber) with low P-velocity and high V p/V s ratio.
For these simulations we have reproduced, like the previous case, the acquisition geometry used during the SERAPIS experiment, where the source depth is 10 m and the depth
of the receivers is 90 m. We have considered a Gaussian source function with a peak frequency of 11 Hz. Radial and vertical synthetic seismograms have been computed stopping
now the hierarchic ray generation to the 11th generation and considering a maximum offset
of 4000 m.
The synthetic sections have been compared with the real ones relative to the same CMP.
In Figure 3.4 the vertical and radial components for both the synthetic and real seismic
sections are shown. Red and green lines indicate the travel time of PP and PS reflection
to the 2nd interface of the model, respectively. The gather is formed by 190 recordings.
The processing has been performed using 8-15 Hz band-pass filtering, an automatic gain
control with a 1 s window (only on synthetic sections), and an amplitude normalization.
Although a processing has been performed, it is difficult to see the PP reflection to the
2nd interface on the radial components of the sections. This is normal, since the P-wave is
polarized along the direction of the raypath which is near vertical at each receiver due to
a high contrast of elastic parameters at the 1st interface of the model (see Table 3.1).
In summary, the qualitative comparison shows that the synthetic sections well reproduce the main seismic phases. This confirms the goodness of the 1-D velocity model used
to calculate synthetics.
3.1 Forward modelling of active seismic data: the case study of Campi Flegrei
caldera
43
Once the synthetic sections are calculated for the model of Table 3.1, we can use these
sections as real sections to make some analyses in order to estimate the elastic parameter
of the model. As an example amplitude variation with offset (AVO) analyses are often
used to estimate elastic parameter contrasts at a given reflector.
Maercklin and Zollo (2008) propose a variant of these techniques that evaluates amplitude ratios between PS and PP reflected at the same interface. Here we use the PS-to-PP
amplitude ratio variation with offset to check the amplitudes of the COMRAD synthetics.
In Figure 3.5 we show a comparison between the amplitude ratios measured from the
synthetic seismograms (green dots) and the theoretical curve (grey line) for the corresponding interface at 2.7 km depth (see Table 3.1). Blue and red lines are theoretical curves
for alternative models with higher and lower P-velocity in the layer below the reflector,
respectively (2% interval).
Figure 3.4: Comparison between the observed (top) and synthetic (bottom) sections for the average 1-D model of Table 3.1. Both radial component (left) and vertical component (right) are
reported. Red and green lines indicate the travel time of PP and PS reflection to the 2nd interface
of the model, respectively.
44
Ch.3 APPLICATIONS OF THE COMRAD CODE IN SEISMOLOGY
Generally, the measured amplitude ratios are consistent with the theoretical curves up
to an offset of about 3 km. The scatter around the expected curve can be explained by
the contamination with other phases, mainly multiples from the strong reflector above the
target reflector. Around the second minimum of the theoretical curves (offsets greater than
3 km), the measured ratios are better fit using a model with a higher P-velocity contrast
at the reflector (blue lines), and the rapid amplitude increase toward 4 km offset has not
been reproduced. Since the second minimum of the PS-to-PP curve indicates the distance
of critical reflection, the observed amplitude discrepancy may be related to inaccurate
post-critical amplitudes in the synthetic seismograms.
Figure 3.5: AVO analysis (PS-to-PP ratio) to the second interface of the velocity model. Blue
and red lines are theoretical curves for alternative models with higher and lower P-velocity in the
layer below the reflector, respectively.
3.1.3
Conclusions and discussions
In this section, applications of the COMRAD code to seismic reflection data illustrate how
such a method can be a useful tool for the waveform analysis. We compute a synthetic
section reproducing the acquisition geometry of SERAPIS survey and considering first the
1-D model of Campi Flegrei plotted in Figure 3.2 and after the 1-D obtained by Maercklin
and Zollo (2008), reported in Table 3.1.
In the first case qualitative comparisons between synthetic and real seismic section are
used to search multiple reflections. In the second case the computation of the complete
high frequency synthetic section and the comparison with the real data allowed us to verify
3.2 Forward modelling of seismic sources
45
how the 1-D velocity model of Table 3.1 well reproduces the complexity of the observed
waveforms both in qualitative (Figure 3.4) and quantitative (Figure 3.5) way. Both these
approaches could be used to refine a preliminary velocity model for complex structures.
Moreover, we have found that the evaluation of PS-to-PP amplitude ratios as a function
of offset generally verified pre-critical amplitudes of the COMRAD synthetic seismograms,
but also pointed to some problems in the post-critical offset range.
3.2
Forward modelling of seismic sources
In this section the COMRAD code is used to compute synthetic seismograms for an earthquake faulting source (a single-particle fault), considering the radiation pattern of the
source. For these simulations the core (Farra and Madariaga, 1987) of the COMRAD code
has been modified in order to take into account of the radiation pattern of the seismic
source. In particular we have to consider in the model the strike, dip, and slip (or rake)
angles, which give the fault orientation and the direction of the sliding motions on the
fault.
However, we have to bear in mind that only far-field body waveform modelling can be
carried out by ray-theory methods, such as the COMRAD code. Indicating with R and λ
the source-receiver distance and the wavelength, respectively, this means that R λ.
3.2.1
Faulting sources
Unlike an underground explosion, most seismic sources do not have spherical symmetry
since the outgoing waves are influenced by the strain distribution near the source. The
geometric description of the initial motion and amplitude near the source of P and S
wavefields is called radiation pattern. The low-order symmetry for such a source provides
relations between the fault-plane orientation and the radiation pattern.
Figure 3.6: Extended source (left) and point source (right) conditions in far-field approximation
(R λ). In the figure the source dimension is indicated by D, the source-receiver distance by R,
and the wavelength by λ.
46
Ch.3 APPLICATIONS OF THE COMRAD CODE IN SEISMOLOGY
When an earthquake occurs, the sliding motion start at a certain point called the earthquake hypocenter (or focus). A slip front expands outward over the fault with a rupture
velocity vr . Considering L the length of the fault, and λ the characteristic wavelength of
the wavefield, the rupture area and the source volume can be approximated as a point
source if L λ, else the source must be considered as extended. Figure 3.6 shows both
the point and the extended source cases, considering also the far-field condition.
The mathematical representation of the geometric nature of faulting is now introduced.
We consider faulting as a slippage between two blocks of material2 , where the slip is
constrained to lie in the plane connecting the two blocks (see Figure 3.7a). We call the
strike θ of the fault the azimuth measured from north of the projection of the fault onto
the surface; the dip δ of the fault is the angle measured downward from the surface to
the fault plane in the vertical plane perpendicular to the strike; the slip λ is the angle
measured in the plane of the fault from the strike direction to the slip vector (see Figure
3.7b). These three parameters describe the orientation of the fault plane in geographic
coordinates. Finally, the magnitude of the slip vector is given by the total displacement D
of the two blocks. The total displacement D is important to determine the seismic moment
M0 since:
M0 = µAD
(3.1)
with µ the rigidity modulus, and A the fault area.
In general θ, δ, λ, and D can vary over the finite fault surface, but their average values
can be used for simple model, as we have done in our simulations.
Figure 3.7: Standard definition of fault-plane and slip vector orientation parameters. (a) The
slip vector lies in the plane connecting the two blocks (the hanging wall and the footwall). (b)
The strike θ, dip δ, and slip λ angles are defined. The figure was modified from Lay and Wallace
(1995).
2
Conventionally, the block above the fault is called the hanging wall and the block below is called the
footwall.
3.2 Forward modelling of seismic sources
47
The motions on the fault can be divided in three basic categories:
1. Strike slip fault: When the two sides of a fault slip horizontally relative to each
other, the motion is called pure strike slip, with λ equal to 0◦ or 180◦ , and if the dip
δ = 90◦ the geometry is called vertical strike slip. In particular, for λ = 0◦ the block
of the other side of the fault moves to left, so the fault movement is called left-lateral
slip; for the opposite case (λ = 180◦ ) the fault movement is called right-lateral slip.
2. Thrust fault: When the slip vector is parallel to the dip direction, the motion
is called dip-slip; if the dip δ = 90◦ the geometry is called vertical dip-slip. For
λ = 90◦ , the hanging wall moves upward, causing thrust (or reverse) faulting. The
thrust faults are generated by a compressive force system acting against the two
blocks.
3. Normal fault: When a dip-slip motion occurs with λ = 270◦ , the hanging wall
moves downward, causing normal faulting. The normal faults are generated by dilatational force system acting against the two blocks.
In general, λ will have a value different than these special cases, and the motion is then
called oblique slip.
It was demonstrated (Lay and Wallace (1995); Aki and Richards (2002)) that an appropriate equivalent body-force system for an earthquake dislocation is the double-couple
model. In fact the displacement field can be substituted by an equivalent field due to a
distribution of equivalent double couples that are placed in a medium without any dislocation. This is one of the most important concepts in seismology, and it is used both for
static and dynamic displacement modelling. Hence, in the following sections we intend the
seismic source as a double-couple force system.
3.2.2
Synthetic seismograms for a vertical strike-slip point source
Here we will show and example of computing synthetic seismograms for faulting sources
using the COMRAD code. This example considers a sigle-particle fault (a point-source)
with a vertical strike-slip geometry (θ = 0◦ , δ = 90◦ , λ = 0◦ ) and a left-slip movement,
according to the convention introduced in section 3.2.1.
We have considered a simple propagation medium with two layers above a halfspace,
as reported in Table 3.2. For each layer and the halfspace the elastic parameters have been
set.
The source-receivers geometry is the following. We consider a source of 7 km in depth
with the epicentre (its projection onto the surface) falling in the centre of an 8x8 km square.
Eight receivers are positioned as follows: 4 receivers in the vertexes of the square 4 receivers
inside the square; all the receivers are on the surface. If we consider the epicentre of the
source as the origin of the coordinate system, the relative positions of the receivers are:
R1 =(3,-3,0), R2 =(4,4,0), R3 =(-1,-2,0), R4 =(-4,4,0), R5 =(2,3,0), R6 =(-4,-4,0), R7 =(-2,3,0),
R8 =(4,-4,0), where all quantities are expressed in km.
48
Ch.3 APPLICATIONS OF THE COMRAD CODE IN SEISMOLOGY
Interfaces Depth (km) V p
1
0
2
3
Halfspace
10
(km/s) V p/V
√ s
4.5
√3
6.0
√3
7.0
3
ρ (g/cm3 )
2.6
2.7
2.7
Table 3.2: Model used to compute synthetic seismograms for a vertical strike-slip point source.
The source lies in the second layer, since it is at 7 km in depth.
We have used the following convention for the X,Y, and Z coordinates: X is equal to
the North, Y corresponds to the East, and Z is the vertical component with the positive
axis downward (the same as in Figure 3.7b).
Since we were interested in the polarities of the first P-wave arrival (which are used
to compute the focal mechanism of the source) and not to reproduce a seismogram as
complete as possible, we have stopped the COMRAD’s hierarchic ray generation to the
5th , leaving the other parameters to the default value.
Figure 3.8 shows the vertical component of the synthetic seismograms (the displacement velocity) for R2 and R4 receivers. The polarity of the P-wave (the first arrival on
the synthetic seismogram) at the R2 receiver is down. This means that the P arrival is
compressional (since Z in positive downward), then there is a motion away from the source.
The opposite case is found at the R2 receiver, where the P arrival is dilatational as a consequence of a motion toward the source. In the simulations we have considered as source-time
function a Gaussian with duration of 0.1 seconds in order to include temporal and spatial
fault finiteness. In general the source-time function is more clearly approximated by a
trapezoid (Lay and Wallace (1995); Aki and Richards (2002)).
Figure 3.8: Synthetic seismograms (Z comp.) for R2 and R4 receivers in the time window between
1.5 and 4.5 seconds after the origin time of the earthquake.
3.2 Forward modelling of seismic sources
49
As a trivial exercise, we have considered the synthetics as real data in order to perform
a localization of the seismic source and the study of its focal mechanism. The localization
was obtained using the HYPO71 program (Lee and Lahr, 1975), while the focal mechanism
is obtained using the FPFIT program (Reasenberg and Oppenheimer, 1985).
The HYPO71 program localizes the source at the point of coordinates S=(0.03,0.00,7.11)
km. The focal mechanism obtained from FPFIT program gives θ = 5◦ , δ = 80◦ , λ = 0◦
with a left slip movement, as shown in Figure 3.9. This geometry is similar to the vertical strike-slip geometry used for the simulation. The result can become better if a large
number of stations will be used.
It is possible to model also an extended source by considering N elementary point
sources with the same focal mechanism of the extended source. In other words, the seismic
source process can be thought of a series of N little earthquakes, each one starts according
Figure 3.9: Focal mechanism obtained from FPFIT program. The result obtained is similar to
the geometry of the model used for the simulations. The + sign represents a compressional wave
(motion away from the source), while the sign represents a dilatational wave (motion toward
the source).
50
Ch.3 APPLICATIONS OF THE COMRAD CODE IN SEISMOLOGY
to the time history of the extended source process. The first point-source activates at the
origin time, the rupture propagates with a velocity vr and the ith source activates at the
time ti such that ti = δi /vr , with δi the distance between the ith and the (iP− 1)th source.
Moreover, the total seismic moment M0 must be given by the sum M0 = N
i=1 Mi0 , with
Mi0 the seismic moment of the ith source.
3.2.3
Conclusions and discussions
In this section we have shown an application of the COMRAD code earthquakes by considering the radiation pattern of the faulting source that generates the earthquake. A vertical
strike-slip geometry for a single-particle fault has been considered in the example.
As a trivial exercise, we have considered the synthetics as real data in order to find
the focal mechanism of the source and to localize it. The results show that the forward
modelling of the seismic source using the COMRAD code is correct. Finally, the extended
source case has been planned.
Although the code produces good results for the far field modelling of seismic sources,
some work has to be done to assess the amplitudes for a given earthquake. In fact the
synthetics do not have the exact amplitude values for a given seismic moment of the source,
since this parameter is not included yet. We are plannig to provide in the future synthetic
seismograms with amplitudes related to a given seismic moment.
Chapter 4
High frequency underwater acoustic
propagation in the Gulf of Naples
Over the last ten years, there has been a rapid extension of the use of digital underwater
acoustic communication from mainly military purposes to those commercial, to satisfy the
growing needs for underwater acoustic wireless communication systems in real time and
with the necessity to oversee the submerged instrumentation without the direct intervention
of man.
Underwater acoustic wireless sensor networks are an emerging topic of research and
are envisioned to enable applications on the surveillance and monitoring of sea areas
(Adams et al., 2000; Sozer et al., 2000; Akyildiz et al., 2005; Benson et al., 2006; Jaffe and
Schurgers, 2006). There is, in fact, significant interest in monitoring aquatic environments
for scientific, environmental, commercial, safety, and military reasons such as detect tectonic movements, incoming tsunamis, water pollution, global warming, distributed tactical
surveillance, mine reconnaissance, and many other facts that are bound to affect our lives
(Badia et al., 2006; Pompili et al., 2006).
The main problem of underwater acoustic communication is the complexity of the marine acoustic channel: background noise from the environment and from human activities;
unevenness in the water caused by the presence of solid and gaseous particles; the presence
of spatial and temporal sound velocity gradients that can arise from salinity, pressure and
temperature variations; multipath effects caused by multiple reflections on the sea surface
and bottom; reverberations caused by the agitation of the sea and the topography of the
sea bottom; and marine currents. In particular, with a digital communication system,
the multipath that occurs with the propagation of signals results in intersymbol interference (ISI), which limits the capacity of the underwater acoustic communication channel.
All of these phenomena are magnified in the situation of acoustic transmission in shallow
water with respect to transmission in deep water, due to the depth of the sea and the
closeness to the coast (Cazzolato et al.; Smith et al., 2003), so the shallow water acoustic
communication channel is characterized by strong signal degradation.
The use of high frequencies in acoustic communication improves the velocity of digital
transmission of data, but also causes a greater attenuation of the signal (Thorp, 1967;
Ch.4 HIGH FREQUENCY UNDERWATER ACOUSTIC PROPAGATION
52
IN THE GULF OF NAPLES
Fisher and Simmons, 1977), with the consequent decrease in the range of distance, i.e. in
the ability to transmit data over distance.
To be able to complete any underwater acoustic communication system, it is necessary
to know the physical properties of the transmission channel to be able to understand what
the limits on signal transmission are and how its complexity can influence the quality of
the information transmission.
In this Chapter we will show a work we have carried out to studying the physical
properties of the water layer of the Gulf of Naples, which can be used as an acoustic
communication channel for high-frequency digital transmissions (around 100 kHz). We
have analyzed also the influence of the seasonal variations in its physical parameters on
underwater transmission and the limits that the channel forces upon this communication.
For this purpose we were interested to understand the effects produced by the underwater
acoustic channel on a high frequency digital transmission to choose the best configuration of an underwater communication system that can be installed in the Gulf of Naples
(Southern Italy). This kind of system shall extent offshore the geophysical monitoring of
the Neapolitan volcanoes, Mt.Vesuvius and Campi Flegrei caldera.
In this study we did not take into account the effects of bubble scattering and absorption
because the layer that contains bubbles is near the surface (Preisig, 2006) and, as our
models use wind speeds less than 6 m/s, bubbles cannot significantly attenuate propagating
signals in this layer (Dahl, 2004). We also have considered the sea as a steady propagation
environment with smooth sea surface and seafloor during the transmission of data.
The simulations of the high frequency wavefield propagation in the Gulf of Naples have
been carried out using the COMRAD code described in Chapter 2.
4.1
Physical properties of the Gulf of Naples
The Gulf of Naples is sited along the south-western coast of Italy, between 40◦ 35’ and
40◦ 50’ latitude North, and between 14◦ 00’ and 14◦ 30’ longitude East. The depth of the sea
in the area of interest is not deeper than 300 m, except for the northern part of the Dohrn
canyon (Figure 4.1), in which the sea depth reaches 400 m. Since the Gulf of Naples is
not deeper than 300 m, it can be considered as a shallow water acoustic communication
channel. This implies a bigger complexity on the sound propagation because in shallow
water the surface, the volume, and the bottom properties are all important. Moreover
the seasonal variation in sound-speed structure is significant with winter condition being
nearly isospeed (Jensen et al., 2000).
In the following pages we will describe the physical properties of the Gulf of Naples
considered as a one-dimensional (1D) communication channel in shallow water. We will
construct a velocity model for the seawater, considering that the average depth of this
water layer is less than 300 m, and a geoacoustic model for the sea bottom. Moreover the
potential noise sources are described, and the noise levels (NL) expected in the Gulf of
Naples at the 100 kHz frequency are calculated.
4.1 Physical properties of the Gulf of Naples
4.1.1
53
Analysis of the weather conditions
In order to study the physical properties of the Gulf of Naples it is important to know the
weather conditions because, as it will be shown in sections 4.1.2 and 4.1.3, the weather has
an important influence on both the sea-water temperature and the underwater noise level.
Here we show the analysis of the air temperature, precipitations, wind speed, and wind
direction in the Gulf of Naples. The weather data have been downloaded from the NOAA1
website at the link http://www.arl.noaa.gov/ready/amet.html, taking the temperature
(in ◦ C) at 2 meters above the sea-level, the precipitations (in mm), the wind direction (in
degrees respect to the North), and the wind-speed (in knots) at 10 meters above the
sea-level. The database is sampled every 6 hours from the 1st January 1997 to the 31st
December 2006, collecting a total number of 14549 data for each meteorological parameter.
Figure 4.1: Bathymetric and topographic image of the Gulf of Naples. A bright ellipse indicates
the area of interest for our studies, where the researchers of the Anton Dohrn Zoological Station
carried out measurements of the physical properties of the sea water in the period from February
2002 to February 2003.
1
National Oceanic and Atmospheric Administration, USA.
Ch.4 HIGH FREQUENCY UNDERWATER ACOUSTIC PROPAGATION
54
IN THE GULF OF NAPLES
We have calculated from the data the mean values and their standard deviations for the
air temperature and the wind speed in the Gulf of Naples during the year. In Figure 4.2
the mean air temperature at 2 meters above the sea level (on the left) and the mean wind
speed at 10 meters above the sea level (on the right) in the Gulf of Naples are reported.
The mean values are plotted with a red line, while the ± one deviation standard curves are
plotted with blue and green lines, respectively. From the left side of Figure 4.2 it is possible
to note that in general there is a thermal excursion between coldest months (January and
February) and the hottest months (July and August) of about 15-20 degrees. The lowest
temperature in the data is 2.4 ◦ C and it was reached at the midnight of the 31st January
1999, while the highest is 31.1 ◦ C and it was reached at the noon of the 9th August of the
same year. These variations influence the water temperature in the surface layer of the sea
since there is a thermal exchange between the sea and the air above. In table 4.1 the mean
annual air temperature in the Gulf of Naples from 1997 to 2006 is reported. The coldest
year was 2005, with a mean temperature of 16.3 ◦ C, and the hottest year was 2003, with
a mean air temperature of 17.6 ◦ C. No linear trends of the air temperature are founded
during this period in the Gulf of Naples.
Year
1997 1998 1999 2000 2001 2002 2003 2004 2005 2006
◦
T( C) 16.8 16.7 16.4 17.1 17.4 17.3 17.6 17.0 16.3 16.7
Table 4.1: Mean annual temperature in the Gulf of Naples from 1997 to 2006.
Looking to the right side of Figure 4.2 it is possible to see that the mean value of the
wind speed does not change so much from winter to summer in the Gulf of Naples. In
Figure 4.2: Mean air temperature and wind speed in the Gulf of Naples during one year, calculated
from 1996 to 2006. The red curves indicate the mean values of air temperature and wind speed,
while the blue curves and the green curves are the +1σ and the −1σ curves, respectively.
4.1 Physical properties of the Gulf of Naples
55
fact the mean wind speed varies between about 7 knots in June and nearly 12 knots in
December. According to the Beaufort Scale2 the sea conditions change on average from
gentle Breeze (force 3, equivalent to 7-10 knots) in summer to moderate Breeze (force 4,
equivalent to 11-16 knots) in winter. From 1997 to 2006 the lowest wind velocity was equal
to 0.1 knots (force 0 → Calm) while the highest wind velocity was equal to 40.9 knots (force
8-9 → Gale). As shown in Figure 4.2 the standard deviation of the wind speed is about
5 knots for each month, a very high value that implies a big variation of the sound speed
independently of the season taken into account. The Figure 4.3 shows the frequency of the
wind direction for each month. The wind direction is expressed in degrees respect to the
North. From April to September the most common wind direction is 270◦ (West wind),
with a probability of 30%. In the same period there is also a probability of 15 − 20% that
the wind direction is 315◦ ; this is the Mistral. In particular in July and August both the
Mistral and the West wind can occur with the same probability. Finally from January to
March in the 50 − 60% of cases the wind can blow with an angle of 45◦ (East Wind), 270◦
(West Wind), or 315◦ (Mistral). During the other periods of the year no favourite wind
direction is observed.
The wind speed is partially correlated to the precipitation since the latter indicates bad
weather conditions, when the wind speed reaches high values. In fact the most dry months
are May, June, and July (when the wind speed reaches the lowest values) with a total
precipitation less than 30 mm, while from September to December the total precipitation
is greater than 70 mm, as shown in Table 4.2. In particular the rainiest month is December
with a total precipitation of 100 mm. Between 1997 and 2006 there were also big differences
in total precipitation from one year to each other, with a minimum of 383 mm in 2001 and
a maximum of 991 mm in 2005.
Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Rain 67.2 46.8 45.5 53.2 27.4 23.2 21.2 34.8 81.6 77.3 83.4 100.0
(mm)
Table 4.2: Monthly total rain fallen in the Gulf of Naples (mean value from 1997 to 2006). The
most rainiest period is from September to January, while the most driest period is from May to
August.
In summary, both the wind speed and the air temperature are important weather
parameter to be considered because they influence the water temperature and, as it will
be discussed in section 4.1.2, the sound velocity in the seawater depends strongly on the
water temperature. Moreover, the wind speed produces also noise in the seawater due to
the breaking of the waves.
2
The Beaufort scale is an empirical measure for describing wind velocity based mainly on observed sea
conditions.
Ch.4 HIGH FREQUENCY UNDERWATER ACOUSTIC PROPAGATION
56
IN THE GULF OF NAPLES
Figure 4.3: Monthly frequency of the Wind direction respect to the North in the Gulf of Naples.
The most frequent winds from January to March are the North-east Wind, the West Wind, and
the Mistral, while from April to September are the West Wind and the Mistral.
4.1 Physical properties of the Gulf of Naples
4.1.2
57
One-dimensional velocity and geoacoustic model of the Gulf
of Naples
The essential characteristic of sea water is, in fact, the velocity at which the sound can be
propagated through it. The square of the velocity of the sound depends then on the ratio
between the density and the bulk modulus of the propagation medium. In particular, in
the sea the density is a function of the pressure, temperature and salinity of the sea, while
the bulk modulus is a function of the pressure and temperature of the sea; for this reason,
the propagation velocity of acoustic energy in the sea can be described as a function of
three physical parameters: pressure, temperature and salinity (Jensen et al., 2000). It is
not only necessary to know the mean values of these three physical parameters for the sea,
but it is also important to know their spatial and temporal variabilities, as these variations
significantly modify the propagation velocity of the acoustic waves in the sea. Over the
years, various formulae have been suggested that link the sound velocity in the sea to the
three parameters of pressure, temperature and salinity (Coppens, 1981; Mackenzie, 1981;
Kinsler et al., 2000), and in particular, the UNESCO algorithm (Chen and Millero, 1977;
Wong and Zhu, 1995) and the equation of Del Grosso (Del Grosso, 1974) are commonly
used.
If the sea is considered to be a resting fluid with a constant density ρ, the pressure is
solely a function of the depth. There are also some very accurate, empirical relationships
between sea depth and pressure. The sea temperature and salinity, however, depend upon
the ambient conditions of the location, such as the solar irradiation, evaporation, the
atmospheric conditions, the water circulation, the wind, the presence of rivers, and the
geographical position.
The characterization of the Gulf of Naples starts, therefore, from the knowledge of the
values and variabilities of these parameters in space and time. For this, the temperature
and salinity data were analyzed from measurements carried out by the researchers of the
Anton Dohrn Zoological Station (Ribera, personal communication, 2005) in the Gulf of
Naples from the sea surface to a depth of 300 m, in the period from February 2002 to
February 2003. The sound velocity data, calculated by the same with the equation of
Del Grosso (Del Grosso, 1974) from these salinity and temperature measurements, were
analyzed.
Figure 4.4 shows the salinity profiles of the sea between the surface and the bottom
at 300 m in depth, with a sampling rate of 0.1 m, across 15 different periods of the year.
The salinity went through small variations during the whole year, with a value of about
38 ppm. It can be seen from Figure 4.4 that there was an increase in the salinity with
depth of about 0.5 ppm, from the surface to the bottom, and also rapid oscillations in the
salinity values in the first 100 m in the non-winter periods due to surface marine currents,
where at some depths the salinity can change of about 2 ppm from the mean value. In the
winter periods, the strong agitation of the sea causes a mixing of the water down to great
depth, resulting in stability of the salinity values.
Figure 4.5 shows the temperature profiles measured at the same time as the salinity at
the same depths. The temperature varies with time, and on large spatial scales, in con-
Ch.4 HIGH FREQUENCY UNDERWATER ACOUSTIC PROPAGATION
58
IN THE GULF OF NAPLES
Figure 4.4: Salinity profiles in the Gulf of Naples from the sea surface to the bottom, at 300 m
in depth, measured across 15 different periods between February 2002 and February 2003.
4.1 Physical properties of the Gulf of Naples
59
Figure 4.5: Temperature profiles in the Gulf of Naples from the sea surface to the bottom, at 300
m in depth, measured across 15 different periods between February 2002 and February 2003. The
main temperature variations are above 50 m depth. For depths greater than 160 m the seasonal
temperature variation rarely reaches 0.1 ◦ C.
Ch.4 HIGH FREQUENCY UNDERWATER ACOUSTIC PROPAGATION
60
IN THE GULF OF NAPLES
comitance with the daily and seasonal variations of solar irradiation; fine spatio-temporal
variations are determined locally by turbulent movements of the seawater. The variability
in the temperature of the sea depends on the variability of the thermal content of the water
as a consequence of the heat transfer between the surface and the adjacent atmosphere
(De Dominicis Rotondi, 1995). From Figure 4.5, it can be seen that at the surface the
higher temperatures are reached in the summer months, with maximum values of 27 ◦ C in
August, while the lowest temperatures are reached in the winter months, with minimum
values of 13 ◦ C in February; in the deepest water, the mean temperature is lower and the
temperature variations between the summer and winter seasons are greatly reduced. In
summer, water mixings hardly and results in a large negative temperature gradient in the
first 75 m of depth, where the temperature can drop from 27 ◦ C at the surface to 14 ◦ C
at 75 m in depth. In autumn and winter, the mixing of the water is very intense, which
generates isothermic surface layers that reach down to a depth of 50 m in October to 160 m
in February. For depths greater than 160 m the mean temperature is 14 ◦ C and its seasonal
variation rarely reaches 0.1 ◦ C; at this depth the temperature is no longer influenced by
the outside meteorological conditions.
Figure 4.6 shows the sound velocity profiles calculated with the equation of Del Grosso
(Del Grosso, 1974), starting from the salinity and temperature data. In this figure, it is
possible to infer that the highest variations in the sound velocity are at the surface, and
are between 1508 m/s and 1541 m/s, equivalent to a percent variation of ±1% of the mean
velocity value. At increasing depths, the seasonal variations decrease until they become
less than 5 m/s below 100 m and negligible below 160 m. A comparison between the sound
velocity profiles with those of temperature reveal an overall correlation at all depths for all
the periods of the year, except for a positive linear trend followed by the sound velocity
profiles for increasing depths, caused by a raise in pressure with growing depth. From
160 m to 300 m in depth, where the temperature variations are negligible, the velocity
only depends upon the pressure variations, increasing by around 2 m/s. The comparison
between the salinity and sound velocity profiles shows that the rapid salinity oscillations
in the hottest months, and for depths less than 100 m, do not correspond to appreciable
variations in the sound velocity profiles, since the latter are less then 0.5 m/s (that is
0.03% of the mean sound velocity in the seawater). Consequently, the rapid variations in
salinity have not been considered in the construction of a sound velocity model of the Gulf
of Naples.
The knowledge of the effects of the three physical parameters of temperature, pressure
and salinity on the sound velocity profiles has allowed the construction of a multiparameter
function of depth z that simulates the sound velocity profiles throughout the year and for
depths from 0-300 m in the Gulf of Naples, assuming that the variations seen over the
sampling period are stationary. The parameters of the function were obtained through the
inversion of the 15 sound velocity profiles using the non-linear Simplex global optimization
algorithm (Nelder and Mead, 1965; Press et al., 1992) and creating a function of time
for each parameter that interpolates the values obtained from the inversion. The chosen
4.1 Physical properties of the Gulf of Naples
61
Figure 4.6: Velocity profiles in the Gulf of Naples calculated from the salinity and temperature
data obtained from the measurements carried out between February 2002 and February 2003.
Ch.4 HIGH FREQUENCY UNDERWATER ACOUSTIC PROPAGATION
62
IN THE GULF OF NAPLES
multiparameter model is described by the following function:
V (z) =
V0
+ α + βz
1 + (z/zc )γ
(4.1)
The linear dependence with depth (βz) expresses the increase in the mechanical pressure. The parameter α represents the value of the sound velocity in sea water at a temperature of 15 ◦ C and a pressure of 1 atm (depth of z = 0 m); the parameter V0 represents
the variation with respect to α in the sound velocity at the sea surface for a different temperature. The parameter zc gives information on the thickness of the surface isothermic
layer. The dimensionless parameter γ amplifies the z/zc ratio and determines the negative
slope of the function V(z) for z ≈ zc .
Using equation (4.1), a mean velocity profile was constructed for the summer and winter
months, with the aim of studying the behaviour of the acoustic propagation according to
the variations in the sound velocity in the sea; the two velocity profiles are shown in
Figure 4.7. We have also calculated (Figure 4.7) the residuals for the summer and winter
months as the difference between the values of the mean velocity profiles constructed using
equation (4.1) and those calculated using the profiles of Figure 4.6. The residuals are less
Figure 4.7: Mean velocity profiles (left) for the summer and winter seasons calculated from
equation (4.1) and residuals (right) evaluated respect to the mean velocity profiles computed using
the 15 velocity profiles obtained from data. Below 50 m depth the residuals are less than 0.5 m/s
for both summer and winter mean velocity profiles.
than 0.5 m/s for both summer and winter mean velocity profiles, except for the first 50 m
4.1 Physical properties of the Gulf of Naples
63
depth in summer profile where the residuals can reach a maximum value of 1.8 m/s (only
0.1% of the sound velocity in the water).
Once the velocity model of the seawater is constructed, it is important to characterize
also the sea bottom that means to define a geoacoustic model of the seafloor. The sediments
samples taken from the first few meters of the sea bottom in the Gulf of Naples show that
it is composed by clayey silt sediments in the first five meters and sand-silt-clay sediments
below, because of lower percentage of silt. From Hamilton studies on physical properties
of marine sediments (Hamilton, 1979a, 1979b, 1980) we can therefore define a geoacoustic
model of the seafloor in the Gulf of Naples assigning, for each solid layer, the thickness,
the sound velocity (P waves), the V p/V s ratio, and the density.
Stratum typology
Sea water
Clayey silt
Sand-silt-clay
Thickness
(m)
300
5
∞
Sound velocity Vp
(m/s)
1510-1534
1550
1650
Vp/Vs
Density
(g/cm3 )
∞
1.03
13.0 ± 2
1.40
9.4 ± 2
1.80
Table 4.3: Values of the physical parameters of the propagation medium for the Gulf of Naples.
The propagation medium for the Gulf of Naples is described assuming a sea water layer
and two solid layers (sea bottom sediments). The values of the physical parameters of the
propagation medium are given in Table 4.3. We also consider a variation of the V p/V s
ratio for both the solid layers since water saturation and porosity3 may be different for the
same sediment composition.
Depth (m)
Summer model
Vp (m/s)
Winter model
Vp (m/s)
0
10
20
30
40
50
60
80
100
160
300
1534
1533
1527
1523
1517
1514
1512
1511
1510
1511
1513
1514
1514
1514
1514
1514
1514
1513
1512
1511
1511
1513
Table 4.4: Sound velocities of the summer and winter models for increasing depths. The main
differences between the two models are above 50 m depth..
The discontinuity surfaces between the layers are flat and horizontal and no rough sea
surface and seafloor are considered. Within the water layer, the sound velocity is assumed
to vary continuously with depth, following the trend given in Figure 4.7. The model that
describes the characteristics of the propagation medium in the summer months essentially
differs from the one relative to the winter months in terms of the sound velocity values in
3
The change of sediment porosity and water saturation influence the rigidity of the sediment structure,
then the V p and V s velocities. The reduction in the value of V p/V s is due to open, dry or undersaturated
cracks, but the V p/V s ratio increases as these cracks fill with water.
Ch.4 HIGH FREQUENCY UNDERWATER ACOUSTIC PROPAGATION
64
IN THE GULF OF NAPLES
the water layer above the cut-off depth of about 50 m. Table 4.4 gives the sound velocity
values for the two models at increasing depths.
4.1.3
Noise sources up to a frequency of 100 kHz
The noise signal of an underwater acoustic sensor is composed by electronic and acoustic
noise, generated by the pressure fluctuations in the sea water due to the action of ambient
sources (De Dominicis Rotondi, 1996).
Here, we consider only acoustic noise, and in particular we calculate the noise level
around a frequency of 100 kHz in the Gulf of Naples. Analytically, the noise level (N L)
will be expressed in decibels (dB) as the ratio between the noise pressure (or intensity)
revealed with omnidirectional hydrophones and a reference pressure (or intensity) of a
plane acoustic wave in the same frequency bandwidth. Typically, the reference pressure
level used in underwater acoustics is 1 µPa.
In the sea, there are various noise sources that are both artificial and natural, which
have intensities that vary in time, space and frequency. The natural sources have physical
origins (molecular thermal agitation, sea tides and turbulence), meteorological (wind, rain,
hail, storms, wave backwash, glaciers), terrestrial (earthquakes, volcanoes, distant storms)
and biological (cetaceans, crustaceans, fish); the artificial sources are due to human activities, like seaside traffic, coastal industrial works, ports, boreholes for research into energy
sources, and underwater acoustic communications besides those of interest.
This noise covers a vast frequency range that goes from frequencies lower than 1 Hz
to frequencies higher than hundreds of kHz. For the various frequency ranges, the trend
of the spectrum shows particular characteristics as a consequence of the peculiarities of
the various sources: each source has its own dominant frequency band in which it has a
predominant aspect with respect to the other noise sources; sometimes, different sources
can predominate in the same frequency band.
The range going from a few kHz to 100 kHz is of interest in our study and is dominated
by the noise generated by the impact on the sea surface of drops and water spray that
arise from the breaking of the waves. A classical representation of the spectral distribution
in the range from 100 Hz to 25 kHz comes from the historic Knudsen curves (Knudsen
et al., 1948), which came from a vast measurement campaign carried out during the Second
World War by a group of researchers guided by V.O. Knudsen, along the coastal waters
of various ocean regions. The curve, parameterized according to the condition of the sea
or equivalently as a function of the wind velocity (in knots), shows a linear trend on a
logarithmic frequency scale with a slope of around -20 dB/dec. A series of subsequent
investigations in the Knudsen range, carried out by Wenz (Wenz, 1962) and Urick (Urick,
1986), established a correlation between wind speed and the ambient noise spectrum level.
Using the empirical formulae given by Coates (Coates, 1990; Stojanovic, 2006), it is possible
to calculate the noise level (N L), from 1 Hz to 1 MHz, as a function of frequency (in kHz),
wind speed w (in m/s) and shipping activity factor s, whose value ranges between 0 and 1
for low and high activity, respectively. From these formulae it is possible to infer that the
shipping activity factor s is negligible at frequencies of about 100 kHz.
4.2 Simulations of acoustic signal propagation at 100 kHz
65
The calculation of the noise level of the Gulf of Naples, for a frequency of 100 kHz,
was carried out using Coates formulae and the mean wind velocity in the Gulf of Naples
for both the summer and winter months. The wind velocities at 10 m above see surface
were considered with the READY program starting from the FNL model (the model and
program are available on-line on the website of the National Oceanic and Atmospheric
Administration) they were sampled every six hours from February 2002 to February 2003,
the period in which the measurements of the physical parameters of the sea were taken.
The mean wind velocity calculated for the summer months was 3.9 m/s, while for the
winter months it was 5.6 m/s. The noise levels in the Gulf of Naples at the frequency of
f = 100 kHz are then 28 dB/µPa in summer and 30 dB/µPa in winter.
4.2
Simulations of acoustic signal propagation at 100
kHz
The knowledge of the physical parameters and the use of the function designed to simulate
the sound velocity variations with depth and time in the Gulf of Naples has allowed us
to carry out various simulations of high frequency acoustic propagation in the Gulf of
Naples. Below, we describe the calculation method used for the simulation of the signal
propagation and for the calculation of the signal-to-noise ratio.
4.2.1
Methodology used for the simulation of the signal propagation
Using the model described in the Table 4.3 of the section 4.1.2, we have calculated synthetic
seismograms for different source-receiver configurations. Moreover, we have changed the
V p/V s ratio for both the solid layers following the values reported in the model of Table
4.3 to understand if it influences the signal propagation.
The simulations were carried out using the COMRAD code with modifications in its
core program4 to allow for the anelastic attenuation in the sea, which is considered by
multiplying for each phase (direct P, multiple P reflections in the water) the amplitude of
the non-attenuated wave A0 by a negative exponential:
A = A0 exp(αx)
(4.2)
where α is the attenuation coefficient and has the dimensions of inverse length and x is
the length of the ray-path.
From the empirical Thorp’s formula (Thorp, 1967; Fisher and Simmons, 1977; Jensen
et al., 2000):
0
−3
α (dB/km) = 3.3 · 10
4
44f 2
0.11f 2
+
+ 3.0 · 10−4 f 2
+
2
2
1+f
4100 + f
The core program is the 3D dynamic ray-tracing code developed by V. Farra (see chapter 2).
(4.3)
Ch.4 HIGH FREQUENCY UNDERWATER ACOUSTIC PROPAGATION
66
IN THE GULF OF NAPLES
the attenuation is expressed as a loss in decibels per unit of distance as a function of the
frequency f (in kHz) of the wave that propagates in the sea and it is indicated by α0 (Jensen
et al., 2000), instead of α (see Appendix B.2 for more details on the sound attenuation in
seawater). From the equation (4.3) and considering a frequency of 100 kHz we find that
α0 is equal to 34.3 dB/km. Taking into consideration equation (4.2), we can obtain α from
α0 as follow:
α0
A
0
' 8.686αx ⇒ α '
= 3.95km−1
(4.4)
α x ≡ −20 log
A0
8.686
To study the reliability of the acoustic communication, we simulated the propagation
from the transmitter to the receivers of a signal consisting of a sinusoidal transient of 1 s
duration and with a frequency of 100 kHz.
The transmitter, i.e. the source of the signal, is omnidirectional and was positioned
at the three different depths of 1 m, 190 m and 299 m from the sea surface; for each
transmitter position, the reception of the signal was simulated on a grid of 21x12 receivers,
disposed in a vertical plane inside the area of interest shown in Figure 4.1 by a bright
ellipse. The horizontal distance between the receivers is ∆x = 100 m from the vertical
axis through the source for up to 2 km in distance, while the vertical distance between the
receivers is ∆z = 25 m, from 5 m in depth to 280 m in depth.
For each receiver, the particle displacement velocity v = (vx , vy , vz ) was simulated, from
which it is possible to deduce the intensity of the signal, information that is useful for the
determination of transmission losses. Indeed, the instantaneous acoustic intensity I of field
produced by force F that acts normally on a unitary surface A is I = F v/A = pv, where p
is the pressure and v is the particle displacement velocity along the normal to the surface
A. Considering that the time-scale of oceanographic changes is order of magnitude greater
than the time-scale of acoustic propagation, it can be assumed that the density ρ and the
sound velocity c are independent of time. With this assumption, both the pressure p and
the velocity potential Φ (v = ∇Φ) satisfy the wave equation, and using the linearized
Euler equation (Jensen et al., 2000), this gives:
p = ρcv ⇒ I = ρcv 2
(4.5)
where v is the particle displacement velocity along the direction of acoustic wave propagation.
Calculating therefore the particle displacement velocity v as a function of the time t
in a certain point of the space, it is possible to obtain the intensity I of the field using the
equation (4.5).
4.2.2
How to calculate the signal-to-noise ratio (SNR)
An object that radiates with a source level (SL) will be registered by a receiver with a
lower signal level (S) due to the transmission loss (T L) that it undergoes in its passage
4.3 Calculation of the SNR in the Gulf of Naples at 100 kHz
67
through the propagation medium. The signal-to-noise ratio (SN R) for this receiver is
obtained by subtracting the noise level (N L) from the signal level (S), such that:
SN R(dB) = S − N L = SL − T L − N L
(4.6)
where all of the quantities under consideration are expressed in dB (see also Appendix B).
To determine the limits beyond which the transmission is no longer reliable, using
equation (4.6) we can construct the SN R maps for a determined source level (SL) starting
from the knowledge of the transmission loss (T L) at each receiver and the noise level (N L)
for the Gulf of Naples. The noise level (N L) was discussed above in section 4.1.3, so we
now consider the determination of the source level (SL) and calculate the transmission
loss (T L) at the frequency of 100 kHz on the whole grid of 21x12 receivers. The source
level is determined according to the relationship (De Dominicis Rotondi, 1990):
SL(dB) = 170.5 + 10 log(W e) + 10 log(η) + DI
(4.7)
where W e represents the electrical power supplied to the antenna, η the electroacoustic
efficiency, and DI the directivity index. Considering a realistic example where the transmitter has an efficiency of 50% and the current supplied to the antenna is 185 mA with
a tension of 3.24 V, thus giving it an electrical power of 0.6 W, from equation (4.7) the
source level will be SL = 165 dB. The transmission loss undergone by the signal at any
specific receiver are obtained instead from the ratio between the mean intensity I of the
signal simulated at each receiver and the mean intensity I0 of the signal simulated at one
meter distance from the transmitter, giving the value in dB according to the relationship:
hIi
(4.8)
T L(dB) = −10 log
hI0 i
which, taking into account equation (4.5), becomes:
2 hvx2 + vy2 + vz2 i
hv i
T L(dB) = −10 log
= −10 log
2
2
2
hv02 i
hv0x
+ v0y
+ v0z
i
(4.9)
Using equation (4.9), it is therefore possible to calculate for each receiver the transmission loss knowing the mean value of the particle displacement velocity v = (vx , vy , vz ).
The mean value is calculated across a period of time equal to the duration of the signal
sent by the transmitter.
4.3
Calculation of the SNR in the Gulf of Naples at
100 kHz
Using equation (4.6) we have constructed the maps that give the signal-to-noise ratio
(SN R) for the summer and winter months, for three different transmitter positions, considering both the multiple reflections and the direct phase only. The calculation is carried
Ch.4 HIGH FREQUENCY UNDERWATER ACOUSTIC PROPAGATION
68
IN THE GULF OF NAPLES
out from the knowledge of the transmission loss in the acoustic communication channel
at the frequency of 100 kHz, and from the knowledge of the noise level in the Gulf of
Naples, and considering a standard digital data transmission apparatus with a source level
of 165 dB. The data transmission apparatus considers an omnidirectional transmitter that
delivers sinusoidal acoustic signals for the transmission of a digital binary sequence, with
a velocity of 2.4 kbit/s at a dominant frequency of 100 kHz, in an operative band of 10
kHz. The power delivered to the antenna was assumed to be 0.4 W with an electroacoustic
efficiency of 50%.
From the Shannon’s Capacity Theorem for a communication channel (Shannon, 1948)
it is possible to calculate (see Appendix B.3) that the signal-to-noise ratio must be greater
than 22 dB for reliable information transmission.
Figures 4.8, 4.9, and 4.10 indicate in black the regions in the section of water in which
the communication is not reliable, as in those zones of the channel in which the signal-tonoise ratio is less than 22 dB. Where the transmission should be reliable (SN R > 22 dB) a
different colour has been assigned to each SN R value calculated, in decreasing order from
red to violet.
Figure 4.8 shows the signal-to-noise ratio for a transmitter positioned at 1 m in depth.
A comparison between the summer and winter maps relative to the direct phase only shows
that if the source is positioned at a low depth, the effect of the negative velocity gradient
that is seen in summer does not favour surface communication for increasing distances.
If, however, a comparison is made between the summer and winter maps that take into
account multiple reflections between the surface and the bottom, we see that the multipath
effect predominates over the velocity gradient effect. Figures 4.9 and 4.10 show the signalto-noise ratio for transmitters positioned at 190 m and 299 m in depth, respectively. From
a comparison of the summer and winter maps, relative to the direct phase only, it can be
seen that for these source depths the negative velocity gradient of the summer months no
longer influences communication.
Comparing Figures 4.8, 4.9, and 4.10 for the cases in which multiple reflections are
taken into account, it can be seen that if the source is a long way from the bottom and
from the surface of the sea, the communication is less disturbed for up to 1.0 to 1.2
km, because of the less influence of the surfaces that delimit the communication channel.
Above this distance, the effects produced by the bottom and the free surfaces are similar,
independently of the position of the source. The V p/V s ratio does not influence the
communication. In fact changing the V p/V s ratio in the range illustrated in Table 4.3 we
obtain the same results as reported in Figures 4.8, 4.9, and 4.10. Moreover, from these
three Figures (4.8, 4.9, and 4.10), it can be seen that the transmission is slightly favoured
in the summer period due to a lower noise level from agitation of the sea. It is nevertheless
important to bear in mind that on average the difference in the noise levels between the
summer and winter months due to the conditions of the sea is only 4 dB in the Gulf of
Naples.
Considering also the information of Figure 4.7, we can see that below 50 m depth the
mean sound velocity profiles for summer and winter season are similar, so also the SN R
is stable during the year for transmitter-receiver connections below this depth.
4.3 Calculation of the SNR in the Gulf of Naples at 100 kHz
69
Figure 4.8: The signal-to-noise ratio for the summer and winter months in the Gulf of Naples
with the transmitter positioned at a depth of 1 m. The grid of receivers is plotted with white points.
A comparison between the summer and winter maps relative to the direct phase only shows that
in summer the effect of the negative velocity gradient does not favour surface communication for
increasing distances. Considering also multiple reflections, the multipath effect predominates over
the velocity gradient effect.
Ch.4 HIGH FREQUENCY UNDERWATER ACOUSTIC PROPAGATION
70
IN THE GULF OF NAPLES
Figure 4.9: The signal-to-noise ratio for the summer and winter months in the Gulf of Naples
with the transmitter positioned at a depth of 190 m. The grid of receivers is plotted with white
points. For this source depth the negative velocity gradient of the summer months no longer
influences communication; moreover the communication is less disturbed for up to 1.0 to 1.2 km
because of the less influence of both the sea and bottom surfaces, since they are far from the
source.
4.3 Calculation of the SNR in the Gulf of Naples at 100 kHz
71
Figure 4.10: The signal-to-noise ratio for the summer and winter months in the Gulf of Naples
with the transmitter positioned at a depth of 299 m. The grid of receivers is plotted with white
points. For this source depth the negative velocity gradient of the summer months no longer
influences communication, but the influence of the seafloor is magnified.
Ch.4 HIGH FREQUENCY UNDERWATER ACOUSTIC PROPAGATION
72
IN THE GULF OF NAPLES
Over the reliability of the communication against distance and depth, it is important
to bear in mind that the simulated signal at each receiver is perturbed by the multipath
effects, arising from the multiple reflections from the surface and the bottom of the sea.
Because of this effect, with the transmission of digital data, the coding signals applied
to the preceding bit are superimposed on the coding signals of the succeeding bit upon
reception, causing an increase in the intersymbol interference (ISI).
Fortunately, acoustic communication has been progressively improving over the course
of the years, both in terms of the distances and the data transmission capacity, due to
the development of new techniques of signal modulation. In more recent years, the feasibility of the modulation techniques for underwater communication and telemetry has
been demonstrated (Catipovic, 1990; Coates, 1993; Stojanovic et al., 1993, 1994, 1995;
Stojanovic, 1996, 2005; Hursky et al., 2006), with the minimizing of the ISI with the aim
of improving the capacity of the transmission channel.
4.4
Conclusions and discussions
In this Chapter a work carried out by Stabile et al. (2007) during this thesis work for the
MIUR (Ministry of Education, University and Research) PON-SisMa project is reported.
We have simulated the propagation of sinusoidal acoustic signals sent by an omnidirectional transmitter at the frequency of 100 kHz for the binary sequential digital transmission
of information, in a propagation medium with the physical characteristics of the Gulf of
Naples. We have used the COMRAD code, described in Chapter 2, to calculate the propagation of the wavefield in the Gulf of Naples. Three different transmitter positions have
been considered, at 1 m, 190 m and 299 m (1 m above the bottom) in depth. The reception
of the transmitted signal was simulated, for all three cases, by a grid of 21x12 receivers in
the sea of 300 m in depth, for a range of 2 km; from the simulation, the SNR maps have
been constructed for all of the cases under consideration.
Considering a transmitter with a source level of 165 dB and a velocity of 2.4 kbit/s,
for a band pass of 10 kHz, it is seen that the transmission of information is reliable up to
distances of 1400-1600 m between source and receiver below 50 m depth and in all seasons
of the year, because the mean sound velocity profiles for summer and winter months are
similar below this depth. The summer seasonal effects on the sound velocity cannot be
overlooked when the transmitter is positioned at the surface or at a low depth. The presence
of the sea surface and bottom produce, moreover, important multipath effects that result
in errors in the decoding process of the bit sequence of the transmitted information, and
particularly if the transmitter is positioned near the surfaces limiting the acoustic channel.
Although the sea surface and bottom produce multipath effects, the V p/V s ratio of the
sedimentary layers does not influence the communication.
If the transmitter is supplied with directionality, the source level can then be increased
up to 40 dB, and in this case it is possible to reach a distance of 2000 m while minimizing
the multipath effects due to the multiple reflections. Figures 4.8, 4.9, and 4.10 that are
relative to the direct phase only give indications as to the SN R that would be present in
4.4 Conclusions and discussions
73
the case in which the transmitter is directional; these values must then be increased by
the factor of directionality (DI) of the source, which in the case examined here is equal to
zero, because it involves a non-directional source.
This study represents a feasibility test of using the acoustic communication to transmit
data acquired on the seafloor of the Gulf of Naples. In this region a complex geophysical
monitoring system operates for the surveillance of the Neapolitan volcanoes (Mt.Vesuvius
and Campi Flegrei caldera). All the instruments are installed in land, while on the seafloor
only temporary experiments have been carried out. This study offers a new perspective
to the future developments of the monitoring system (such as in Figure 4.11) by the
installation of geophysical stations on the seafloor with data transmission by acoustic link
to a moored buoy that provides connectivity back to the land by a radio link.
Figure 4.11: Monitoring system prototype developed during SisMa Project. Geophysical stations
installed on the seafloor collect data, which are transmitted to a moored buoy (secondary master) by acoustic link; then a radio communication connects the buoy to the on-land data centre
(primary master).
Ch.4 HIGH FREQUENCY UNDERWATER ACOUSTIC PROPAGATION
74
IN THE GULF OF NAPLES
Chapter 5
Very high frequency Submarine
Acoustic Imaging
In the turbid, saline water of the sea, the attenuation of both light and radio waves is orders
of magnitude greater than the attenuation of the sound, which is the best form of radiation
that travels through the sea. For this reason the acoustic (sound) wave propagation is
applied to transmit data through the sea channel or to explore the underwater ambient.
The use of the sound for data transmission in the seawater was discussed in Chapter 4.
Here we will apply the underwater acoustic wave propagation to the imaging of submerged
objects. This is an on-going work started in the third year of this thesis work and funded
on the STSS-500 project that will be described in section 5.1.
Acoustic imaging is nowadays an active and attractive research field, which draws the
attention of many companies, organizations, institutes, research groups, and universities
in different countries with the aim to study and develop techniques for the processing of
signals acquired by acoustic systems in order to reconstruct the image of the insonified
medium. The STSS-500 project belongs to this field.
Before describing this work, we want to introduce the concept of passive and active
sonar:
• Passive (or listening) sonar systems use sound radiated by the target. Only one-way
transmission through the sea is involved, so the system is centred on the hydrophone,
which is used to listen to the target sound. These kinds of systems are usually used to
the detection and localization of ships, submarines, torpedoes, or everything produces
sound in the water.
• Active sonar systems generate the sound by a projector; the sound propagates
through the seawater to the target, and comes back as echoes to the hydrophone
of the system. These kinds of systems are usually used to reconstruct the image of
the target (e.g. submerged objects, sea-bottom topography, ships).
• Hybrid forms of sonar systems use a projector and hydrophone at both ends of the
76 Ch.5 VERY HIGH FREQUENCY SUBMARINE ACOUSTIC IMAGING
acoustic communication path. These kinds of systems are employed for communication, telemetry, and control applications.
For each of this type of sonar systems there is an appropriate equation. In Appendix B.1
the different sonar equations will be discussed. A complete treatment on the principles of
underwater sound and its applications can be found in Urick (1983).
5.1
The STSS-500 Project
The MIUR (Ministry of Education, University and Research) Pon STSS-500 (from the
Italian: “Sonar Tridimensionale Stereoscopico Simultaneo a 500 kHz”) project is an ongoing project whose aim is to research and to study the technologies and the processes
that can be applied to the sonar imaging field, in order to build a prototype of underwater
acoustic camera.
Although underwater optical vision provides images with finer resolution respect to
the acoustical one, as we said before its range is limited to a few meters, in particular in
turbid water. Using very high frequencies acoustic signals from about one kilohertz to few
megahertz the resolution of acoustic imaging systems can become fine too.
Using antennas composed by high-technology electroacoustic transducers, the prototype of acoustic camera can be able to reconstruct the image of submerged objects in the
seawater, in particular for turbid water conditions. The implementation of this system can
be used to the remote management of several activities in the seawater (e.g., monitoring
and surveillance of maritime activities, monitoring industrial pollution along coast and
offshore, detection of archaeological objects in the sub-bottom, exploration of the seabed).
The main characteristics of the image produced by the acoustic camera will be:
• complete image of the target
• three-dimensional image
• stereoscopic effect of the image
This is an innovative imaging instrument, since commonly the 3D image is actually produced by moving the system along a specific trajectory, and stereoscopy is not taken into
account yet.
The prototype of acoustic camera belongs to the class of active sonar, since the system
will be composed by a projector and a receiver apparatus. The projector generates sound
in the frequency band between 400 kHz and 600 kHz with a central frequency of 500 kHz,
a frequency that can guarantee both a high resolution of the image and a good visibility
range. The receiver apparatus is composed by an array of 128 receivers electrically and
mechanically insulated (diced1 array). The maximum planned distance for the detection
of the target is about 50 m.
The most important expected results for the research in this field are:
1
The diced technology consists in a method able to produce transducers by dividing a ceramic block
into small separated little blocks insulated each to other.
5.2 Development of an Acoustic Imaging Numerical Simulator
77
• to study the materials properties in order to find the most appropriate materials for
this kind of array and these high frequencies
• to choose the best technological configuration of the system that gives a 3-D stereoscopic image of the target
• to simulate the behaviour of the chosen configuration both respect to the material
response and respect to the physical parameters of the surrounding underwater environment.
• to optimize the transducer parameters (e.g. resolution, performance).
The partners of this project are the WASS2 industry of Monterusciello (Pozzuoli,
Naples), the University of Naples3 Federico II, and the ENEA4 research institute.
5.2
Development of an Acoustic Imaging Numerical
Simulator
Our role in this project is to study the high frequency propagation of the sound in seawater, from the generation of the sound by the projector to the recording of the echoes
from submerged objects in the seawater by the array of receivers, taking into account the
influence of the environment parameters on the sound propagation.
Another important aspect we are interested is the processing of the backscattered echoes
in order to create an image of the insonified scene.
For these reasons, our work has been implemented in a program which simulates the
propagation of the sound in the seawater with modelled objects inside (forward modelling)
and reconstructs the image of the target (imaging) from its echoes using beamforming
techniques. This simulator is of crucial importance since it can be used for the choice of
the best configuration of the acoustic camera system.
Additional tools are included in the numerical simulator in order to improve interactivity with the user. It is possible to create plane and solid objects (as rectangles, circles,
spheres, parallelepipeds) that can be used to model simple targets, and import ASCII files
describing the shape of both the target and the acquisition system (i.e. the source-receiver
geometry) in 3-D. The object can be translated along each direction and can be rotated
in the three-dimensional space around each axis using the matrix


1
0
0
 0 cos θ − sin θ 
(5.1)
0 sin θ cos θ
2
Whitehead Alenia Sistemi Subacquei.
Dipartimento di Scienze Fisiche dell’Università degli Studi di Napoli Federico II.
4
Ente per le Nuove tecnologie, l’Energia e l’Ambiente; offices in Brindisi, Casaccia (Rome), and Frascati
(Rome).
3
78 Ch.5 VERY HIGH FREQUENCY SUBMARINE ACOUSTIC IMAGING
with θ the rotation angle (anti-clockwise). The synthetic data provided by the forward
modelling part of the simulator can be given in the standard SeismicUnix format (Cohen
and Stockwell, 2007). The output of the imaging part of the simulator can be an ASCII
file or plots in postscript format along horizontal planes of fixed values of z. Postscript
plots of the acquisition geometry should be also provided.
Following we explain in details both the forward modelling and the imaging part of the
numerical simulator.
5.2.1
Forward modelling: Rayleigh scattering
The sea contains inhomogeneities of many different kinds both within itself and on its
boundaries. These inhomogeneities form discontinuities in the physical properties of the
medium reradiate the incident wave into all directions whenever their dimension results
comparable or much smaller than the characteristic wavelength of the incident wavefield.
This reradiation of energy is called scattering, and the superposition of the scattering
contributions from all scatterers is called reverberation.
In the sea there are three different classes of reverberation:
1. Volume reverberation, which occurs when the waves are scattered by matter distributed inside the seawater (e.g. marine life, suspended inanimate materials);
2. Sea-surface reverberation, which is produced by scatterers located near or on the sea
surface;
3. Bottom reverberation, which is produced by scatterers located near or on the sea
bottom.
Both the sea-surface and the bottom reverberation may be analytically considered as surface reverberation.
The forward modelling in acoustics has as objective the computation of the scattered
pressure field for given incident acoustic waves and the physical properties of the object.
Studies of forward scattering problems have been reported in a large number of publications, and both exact analytical solutions5 and approximate methods have been developed
(Liu et al., 2003).
The forward modelling part of the numerical simulator developed in this study is more
general, since it considers both the acoustic and the elastic cases. The forward modelling
is actually based on single scattering from a spatial distribution of scatterers inside a
homogeneous medium. The scatterers are considered as spheres with a diameter 2a such
that 2ka 1, with k = 2π/λ the wavenumber relative to the dominant wavelength λ of the
incident wave. The contrast between the material parameters of the scatterers and those
of the surrounding medium can be arbitrary. In this situation the Rayleigh approximation,
5
The exact analytic solutions can be derived only when the technique of separation of variables can
be used, such as objects like spheres, spheroids, and cylinders. For objects having more general shape,
approximate methods must be used.
5.2 Development of an Acoustic Imaging Numerical Simulator
79
explained in section 1.4, can be used in order to compute the amplitude of the scattered
waves.
In the case of objects having large dimensions respect to the dominant wavelength λ of
the incident wave, they are modelled with a large number of this kind of little spheres. The
scattered wavefield is hence a superposition of single scattering produced by each sphere.
This allows the modelling of objects with arbitrary shape (thus objects having corners,
rough surfaces, thin components).
In summary the following quantities have to be given in input:
1. The physical properties of the background homogeneous medium that contains the
scatterers, such as the P-wave velocity V p0 , the S-wave velocity V s0 , and the density
ρ0 .
2. The position of the ith source Si = (xi , yi , zi ), with i = 1, · · · , M , and M the total
number of the sources.
3. The position of the j th receiver Rj = (xj , yj , zj ), with j = 1, · · · , N , and N the total
number of the receivers.
4. The position of the k th scatterer Pk = (xk , yk , zk ) and its physical properties (P-wave
velocity V pk , S-wave velocity V sk , density ρk ), with k = 1, · · · , L, and L the total
number of the scatterers.
5. The waveform and the duration of the signal emitted by the source.
Moreover, it is possible to compute P-P and/or P-S scattering (equations 1.42) and to
project (or not) the scattering response onto one of the two horizontal (X or Y) components
or onto the vertical (Z) component.
In the case of underwater acoustic propagation only P-P scattering can be computed,
since no S-waves propagates inside the water, and the following background parameters
can be used
V p0 = 1500 m/s;
V s0 ≈ 0 m/s;
V p0 = 1500 kg/m3
where V s0 ≈ 0 m/s means a very little6 value for S waves.
The numerical simulator computes also the geometrical spreading of the wavefield and,
in the case of underwater acoustic propagation, the attenuation for absorption α is taken
into account by using the empirical equation (B.6) described in Appendix B. The two
effects in underwater acoustic can be tied together giving the following expression for the
total attenuation A in decibels per kilometre (Stojanovic, 2006):
A(dB/km) = k · 10 log r + r · α
6
(5.2)
It is well known that the S-wave velocity must be equal to zero in fluids, but in the program the
S-wave velocity must be set to a little value in order to avoid computational problems due to the division
by zero (see section 1.4).
80 Ch.5 VERY HIGH FREQUENCY SUBMARINE ACOUSTIC IMAGING
with r the range and k the spreading factor, which describes the geometry of propagation.
The commonly used values for the spreading factor are k = 2 for spherical spreading, k = 1
for cylindrical spreading, and k = 1.5 for the so-called practical spreading.
The spherical spreading occurs when the power generated by the source is radiated
equally in all directions, so it is equally distributed over the surface of a sphere surrounding
the source having radius equal to the range r. The cylindrical spreading occurs when the
medium is bounded by two plane-parallel surfaces, so beyond certain distances the power
radiated by the source is equally distributed over the surface of a cylinder having radius
equal to the range r and height equal to the distance h between the two surfaces. Finally,
the practical spreading is used in the case of hybrid (one half spherical and one half
cylindrical) spreading.
Figure 5.1 shows curves of the total sound attenuation A in seawater against range and
frequency, considering a spherical spreading and the physical parameters of the Gulf of
Naples7 . The range is from 0 km to 1 km, and the plots are for 10 kHz, 100 kHz, 500 kHz,
and 1 MHz of frequency, respectively.
400
Sound attenuation in seawater (dB)
350
300
z
H
1M
250
200
Hz
500 k
150
100 kHz
100
50
10 kHz
0
−50
0
0.2
0.4
Range (km)
0.6
0.8
1
Figure 5.1: Sound attenuation in seawater against range and frequency, and considering a spherical spreading. Plots relative to frequencies of 10 kHz, 100 kHz, 500 kHz, and 1 MHz are reported,
respectively.
7
Typical values of physical parameter of the Gulf of Naples are: salinity of 38.5 ppm, pH of 8.1, water
temperature of 14.2 ◦ C, depth of 200 metres, and sound velocity of 1512 m/s.
5.2 Development of an Acoustic Imaging Numerical Simulator
5.2.2
81
Imaging: Beamforming techniques
When a signal is recorded by a receiver it is possible to see in its waveform many phases
that differ each to other both in amplitude and in shape. The main phases of a seismogram
or, in general, of a recorded signal are not always of interest. The identification of weak
phases such as reflections, convertions, or scattered waves from the discontinuities inside
the medium plays an important role. The identification of these weak phases usually
requires good-quality registrations and a good coverage of the area under investigation by
stations.
Thus having a large number of traces, the weak phases can be detected by their coherent
appearance on different traces. A classical method is the stack, where the traces are
summed along assumed traveltime curves. If the traveltime curve is correct, the signal is
summed coherently while the noise amplitude decreases as a consequence of his incoherency.
Various techniques (Yilmaz, 2001) have been developed for extracting information from
the observed wave field either in time or in frequency domain in order to detect coherent
signals. Moreover, different processing and migration techniques applicable to marine and
vertical-seismic-profiling acquisition geometries have been discussed by Müller (2000).
A class of imaging methods is the beamforming, which is a signal processing technique
used with arrays of receivers. It involves combining or sum delayed signals from each
receiver at slightly different times, so that the signals will be aligned at exactly the same
time and produce a strong signal as it would come from a very sensitive receiver. When
beamforming is used on both the source and the receiver array side simultaneously, the
technique is called the “double-beam method” (Krüger et al., 1996; Scherbaum et al.,
1997).
The imaging part of the numerical simulator developed in this study considers the same
imaging technique applied by Maercklin et al. (2004) to image the Dead Sea Transform
with scattered seismic waves. This is a beamforming technique that can be used for both
the elastic and the acoustic case.
Figure 5.2 on the next page will help us to explain the method. We consider a medium
with an inhomogeneity (a scatterer) in its interior and one source with and array of receivers
on the free-surface (Figure 5.2, on the left). The energy excited by the source travels
through the medium and is thus recorded by the receivers. The first arrival is the direct
P wave, and a second arrival (labelled PxP) is due to scattering from the inhomogeneity.
The first panel of traces in Figure 5.2 shows the result of the forward modelling for this
case, with a P arrival around 3.6 s and PxP around 4.3 s.
Direct P traveltimes can be applied to these traces as static shifts to align the first
onset. This yields the second panel in Figure 5.2, labelled P. For the second arrival, the
PxP wave, we consider in the medium a grid of possible scattering point. For each possible
scatterer of the grid and for each receiver we compute the traveltime from the source to the
scatterer and from the scatterer to the corresponding receiver. Applying these moveout
times as shifts to the initial recordings aligns the the PxP phase as shown in the right
panel in figure 5.2.
As we said before, generally real data are contaminated with noise. Thus weak phases
82 Ch.5 VERY HIGH FREQUENCY SUBMARINE ACOUSTIC IMAGING
Figure 5.2: Scheme of the beamforming technique used for the imaging process. The semblance S
is maximum in correspondence of coherent phases (P and PxP panels). The figure was modified
from Maercklin (2004).
like PxP have often small amplitudes compared to the P-wave. In these cases the use of
coherency measures are preferable to a simple stack of the traces. We have used in this
work the semblance S as a coherence measure, but other measures such as the phaseweighted stack (Schimmel and Paulssen, 1997) could be used. The coherency is measured
in a time window centred at the predicted arrival time (0 s in Figure 5.2).
The semblance S is defined to be the normalized output/input energy ratio, where the
output trace is a simple compositing or sum of the input traces. In mathematical terms it
is given by:
2
Pk+(N/2) PM
f
i=1 i,j
j=k−(N/2)
(5.3)
S=
Pk+(N/2) PM 2
M j=k−(N/2) i=1 fi,j
in which M aligned traces are analysed in a time window of N samples centred at sample
k.
From equation (5.3) we can see that 0 ≤ S ≤ 1, where 0 means complete incoherency
and 1 perfect agreement between traces, respectively. A high semblance value observed for
PxP aligned traces indicates the presence of a scatterer at the corresponding grid point,
as shown on the left of Figure 5.2. Thus, we can assign for each value of the semblance
a probability meaning: the higher the value is the higher the probability to find in that
point the scatterer will be.
Scanning the entire volume, it is possible to reconstruct an image of the medium that
localizes scatterers by the use of high values of semblance.
The following quantities have to be given in input to the imaging part of the numerical
simulator:
1. The physical properties of the background medium that contains the scatterers, such
as the P-wave velocity V p0 , the S-wave velocity V s0 , and the density ρ0 . These are
5.3 Three-dimensional imaging of submerged objects
83
taken automatically from the forward modelling part of the numerical simulator
2. The length of the time window, in which the coherency measure of traces using
semblance will be computed.
3. The definition of the image volume. Thus, we have to define the coordinates of
the bottom-left point (xmin , ymin , zmin ) of the grid, the number of points along the X
direction (nx), Y direction (ny), and Z direction (nz), and the sampling rates (dx, dy,
and dz) along these three directions, respectively. One notes that the construction of
the grid is more general, since it is possible to sample the medium differently along
the three directions.
The output of the imaging process is a 4-column file containing the grid points coordinates
(first three columns) and the semblance value (4th column), respectively.
5.3
Three-dimensional imaging of submerged objects
In this section some examples on 3-D imaging using the numerical simulator described in
the previous section will be reported.
The first example considers the case of 4 balls in a deep ocean, having a diameter
of 1 m and coordinates B1 = (3, 4, −1) km, B2 = (3, 3, −1) km, B3 = (4, 4, −1) km,
B4 = (4, 3, −1) km. We suppose we want to perform a localization of these balls using a
system composed by one source and 50 receivers. The system is placed on the sea-surface.
Figure 5.3 shows the position of both the system (source+receivers) and the scatterers
from two different views (lateral and top). To avoid a strong attenuation we consider a
Figure 5.3: Source-receiver geometry used to perform a localization of the four scatterers. (Left)
Positions of the source, the receivers, and the scatterers from a lateral view. (Right) Projection of
scatterers onto the source-receivers plane. The source, the receivers and the scatterers are plotted
using ∗, +, and × symbols, respectively.
84 Ch.5 VERY HIGH FREQUENCY SUBMARINE ACOUSTIC IMAGING
Gaussian source having a dominant frequency f of 20 Hz. These balls satisfies the Rayleigh
condition ka 1, so Rayleigh scattering occurs. In fact we have:
ka =
2πaf
2πa
=
= 0.08 1
λ
c
(5.4)
where c = 1500 m/s is the sound velocity in seawater.
The echoes recorded by the receivers are reported in figure 5.4 (left side). The traces are
ordered by offset. In the same figure (right side) the image of the scatterers on the plane
z = 1 km in depth (where the four scatterers are) is reported. Two different sampling rate
of 20 m and 200 m are used to define the same image volume. For both the choices the
imaging process found the scatterers in their exact position, but the resolution becomes
Figure 5.4: (Left) Synthetic data, ordered by offset, obtained for the model of Figure 5.3. (Topright) Image of the four scatterers, defining the image volume by a sampling rate of 20 m.
(Bottom-right) The same image but using a sampling rate of 200 m.
5.3 Three-dimensional imaging of submerged objects
85
better for smaller sampling rates (top right of Figure 5.4). In particular, the scatterers are
localized by the use of semblance values, reported in Figure 5.4 in different colours from
blue to red. Blue colour indicates a low semblance value, while red colour indicates a high
semblance value that is a high probability to find a scatterer in that point.
The second example considers the case of a granite block submerged in the seawater.
The block is a cube having dimension of 50x50x50 cm. Table 5.1 shows the physical
properties of the background medium and the cubic block used to compute synthetic data
for the case under consideration. One notes that there is a strong contrast between the
Type
Material
Background
Water
Cubic block Granite
Vp (m/s) Vs (m/s) Density (kg/m3 )
1500
0.0001
1000
4500
2600
3000
Table 5.1: Physical properties of the cubic block and the background medium used for the forward
modelling in the second example. The cubic block has dimension of 50x50x50 cm and is made of
granite. The background medium is the water.
cubic block and the surrounding water. The block is considered with water inside and
its surfaces (made of granite) are modelled by 15000 spherical scattering points (with a
sampling rate of 1 cm) having the physical properties of Table 5.1. We insonify the block
by a system having one projector (a non directional point source) above a grid of receivers
(acquisition system). The projector emits a Gaussian signal with a dominant frequency
of 100 kHz; the signal scattered by the cubic block is recorded by each receiver, thus the
image of the target (i.e. the cubic block) is reconstructed using the beamforming technique
described in section 5.2.2.
Two different configurations for the acquisition system have been used. The first is a
rectangle of 2x1 m in size, composed by 200 receivers having 0.1 m distance each to other.
The second configuration is a more complex system of 245 receivers. In Figure 5.5 both
the configurations are shown.
First
configuration
1
0.8
Z (m)
0.6
0.4
0.2
0
Z (m)
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0.8
0.6
0.6
0.4
0.2
−1
0
−0.5
−0.2
0
X (m)
0.5
−0.4
1−0.6
0.4
Second
configuration
−0.05
Y (m)
0.2
0
0
0.05
X (m)
0.1
0.15
0.2
−0.2
−0.4
−0.6
−0.8
0.25
Y (m)
Figure 5.5: Two different configuration used for the acquisition system. (Left) 200 receivers are
arranged on a 1x2 m rectangle. (Right) 245 receivers are arranged in a more complex configuration
composed by five not parallel panels.
86 Ch.5 VERY HIGH FREQUENCY SUBMARINE ACOUSTIC IMAGING
Figure 5.6: Images of the target in the planes z = 0 m, 0.25 m, 0.5 m and 1 m, respectively,
obtained using the second configuration of Figure 5.5 (left slices) or the first configuration of
Figure 5.5 (right slices).
5.3 Three-dimensional imaging of submerged objects
87
We suppose the projector-receivers system to be rigid. This system can rotate around
the cubic block at a fixed distance (5 metres) from the centre of the block, which is
considered the origin of the axes. Images of the block at different planes z = const as
functions of semblance values are reported in Figure 5.6. Here again blue colour indicates
a small value of semblance, while red colour indicates a high semblance value. The image
volume, that is the grid of points where the semblance is computed, has been built using the
following sampling rates along X, Y, and Z direction respectively: dx = 0.01 m, dy = 0.01
m, and dz = 0.25 m.
As shown in Figure 5.6, the second configuration of Figure 5.5 gives high resolution
images (slices A1, B1, C1, D1) respect to those obtained by the first simple configuration
(slices A2, B2, C2, D2). Moreover, the corners of the block are well evidenced and the
correct dimension of the target is obtained from the imaging process. Unfortunately as
shown in the slices C1-C2 and D1-D2 on the bottom of Figure 5.6, fuzziness in the image
of the object still remains.
Using the second configuration of Figure 5.5, we have also reconstructed a 3-D image
of the cubic block by eliminating in the plot all the points P of the image volume that
have the corresponding semblance value S less than 0.9, hence the condition imposed is:
The point P is plotted if ⇒ S(P ) ≥ 0.9
(5.5)
Figure 5.7 shows a comparison between the real image of the cubic block (left side) and
the image obtained by the system under consideration (right side).
Figure 5.7: (Left) Real shape of the block and (right) its image obtained using the second configuration of Figure 5.5 and taking into account the condition (5.5).
88 Ch.5 VERY HIGH FREQUENCY SUBMARINE ACOUSTIC IMAGING
5.4
Conclusions and discussions
In this Chapter we have discussed the detection and imaging of objects submerged in the
seawater with very high frequency acoustic signals. We have used the scattered theory
with the Rayleigh approximation that is valid when the scatterer is much smaller than
the dominant frequency of the incident wave. The amplitudes are computed for such a
scatterer with spherical shape. Large objects have been modelled as a sum of these little
scatterers, each of them contributes to the total scattered field.
We have developed and tested (by the use of synthetic examples) a numerical simulator
in order to reproduce the entire process from the generation of the signal to the processing
of echoes by the target. Images of the insonified environment have been reproduced.
The numerical simulator will be a useful tool in the development of a 3-D stereoscopic
underwater acoustic camera, which is the final product of the STSS-500 project described
above. In fact, the simulator can be used both to reproduce synthetic data for the acquisition system and to help the engineers to the development of the system apparatus
configuration (shape of projectors and antennas, distances, directivities).
In section 5.3 we have shown some imaging examples until 100 kHz. We have demonstrated that a simple improvement of the receivers’ geometry (i.e. the acquisition system)
provides a better resolution of the object insonified. Further improvements of the acquisition system will give better images, reducing fuzziness and improving resolution. Moreover,
stereoscopy will be taking into account by the use of two of such a system (as we do with
two eyes).
Due to problems on the maximum length of traces (65536 samples) in Seismic Unix
format (Cohen and Stockwell, 2007), the imaging with 500 kHz signals actually can be
carried out only for very short distances because, to avoid aliasing, the sampling rate must
be almost 1 mus. We are working both in the direction to extend the trace length and in
cutting the portion of the trace that does not contains information.
In the future we will consider also the effect of the medium (especially sedimentary
layers) on the image reconstruction as it may improve or disorganize the image coherence. There will be an evaluation of the accuracy on velocity variations in the sedimentary
medium for such reconstruction as well as the defocusing effect of a smooth velocity variation in the water column coming from currents modifying the temperature gradient.
Conclusions
In this thesis work we have studied the high-frequency seismic and underwater acoustic
wave propagation in complex media. Moreover, imaging techniques are applied in different
research field, from the exploration geophysics to the acoustic imaging.
A method for rapid high-frequency seismogram computation has been presented in this
thesis. The most important features of this method are summarized as follows:
1. Short computing time that does not depend on the frequency, since the method is
based on ray theory.
2. Each phase on the synthetic seismogram is associated at a known body wave, that is
both the travel-time, the raypath, and the amplitude of each wave is computed and
it is possible to isolate all the phase each to other.
3. Automatic definition of an exhaustive set of reflected/transmitted/converted body
waves in a short time (about one second). This is based on the hierarchic generation
of strings that describe the ray paths and the phase types to be computed.
4. Constraints on the wavefield propagation can be set in order to reduce (or not) the
number of phases. More complex synthetic seismograms requires longer computing
time since thousands amplitudes, travel-time and ray-paths have to be computed;
simpler synthetic seismograms, including only the direct waves and few reflections,
require only few seconds to be computed.
5. Easy implementation with other ray-theory programs, since the criterion used to
define the body waves to be computed is almost the same for all of them.
Some work is carried out also to correlate the computing time to the complexity of the
synthetics, as function of the values chosen for the constraint parameters. An important
series of quantitative tests has been used to validate the method. In particular in this text
we have described a quantitative comparison, according to a time-frequency misfit criteria
based on the continuous wavelet transform of the signals, between synthetic seismograms
computed using our method against those obtained by a discrete wavenumber method,
which computes the complete wavefield.
The method proposed cannot be used whenever surface waves, near field terms, or in
general low-frequency seismogram computations are required.
90
Conclusions
On the other hand, lots of applications of the method have been shown in this text,
both in seismology and in underwater acoustics. Seismological problems discussed in this
thesis are: identification of seismic phases on real data by the comparison with synthetics;
the study of multiple reflections on seismogram in order to develop and refine model obtained with other methods; quantitative analyses on seismic phases to estimate the elastic
parameter of given models; forward modelling of faulting sources and analyses of their focal mechanisms. Underwater acoustic problems discussed in this text are: analysis of the
physical properties of the seawater, in particular for the Gulf of Naples, and their effects
on the underwater acoustic propagation; definition of geoacoustic models for underwater
environments; high-frequency underwater digital communications, studying their limits
and the problems the acoustic channel produces; high frequency imaging of submerged
objects; forward modelling of the acoustic wave propagation in the seawater in presence of
scattering inhomogeneities.
These works have given the opportunity to study a large variety of problems in geophysics. Moreover, a contribution to different research project, as the SisMa project, the
STSS-500 project, and the INGV-DPC V4 project, has been produced during the three
years of this thesis work.
For the INGV-DPC V4 project, the method developed for rapid high-frequency seismogram computation, and its implementation in the COMRAD code, results one of the
final products of the project.
For the SisMa project this study offers a new perspective to the future developments
of the monitoring system considering wireless acoustic two-way communication between
geophysical stations on the seafloor and moored buoys on the sea surface that provides
connectivity back to the land by a radio link.
Finally, for the on-going STSS-500 project, a numerical simulator has been implemented
in order to help the engineers in the development of a 3-D stereoscopic underwater acoustic
camera. In particular, the simulator ties together the forward modelling and the imaging
process of submerged objects.
Acknowledgements
I would like to thank all members of the Rissc-Lab Group, who have given me their
availability during this thesis work and helpful suggestions. Many of them are not only
colleagues but also good friends.
Moreover, I thank my tutor, Prof. Aldo Zollo, and Prof. Giovanni Iannaccone, for
giving me the opportunity to work in this interesting research topic with an advanced
research group.
Thank you, particularly, to Prof. Raffaella de Matteis for her continuous advice in the
development of the high frequency method used during this thesis work. The work with
her was constructive and successful, and it is opening new research opportunities.
Finally, I thank the engineers of the Whitehead Alenia Sistemi Subacquei of Monterusciello (Naples) for providing the technical specifications of underwater instrumentation
and for the useful discussions on the topics of the SisMa and the STSS-500 projects.
92
Acknowledgements
Appendix A
The COMRAD Code
In this Appendix the User Guide and the source code of the COMRAD code are reported.
The program can be downloaded from the Computer & Geosciences (Elsevier) website. A
paper (Stabile et al., 2008) on the method developed during this thesis work is in press
under Computer & Geosciences journal.
A.1
User Guide
The propagation of seismic body waves in complex, laterally varying 3-D layered structures
is a complicated process. Analytical solutions of the elastodynamic equations for such types
of media are not known. The most common approaches to the investigation of seismic
wavefields in such complex structures are:
1. methods based on direct numerical solutions of the elastodynamic equation, such as
the finite-difference and finite-element methods
2. approximate high-frequency asymptotic methods (such as ray methods)
Both the methods are very useful for solving certain types of seismic problems, have
their own advantages and disadvantages, and supplement each other suitability (Červený,
2001).
This program is based on ray-theory, where the solution of the elastodynamic equation
is composed of elementary body waves that correspond to the various rays connecting
the source to the receiver. Although the computation of ray synthetic seismograms is
only approximate and the ray method can fail in certain situations, the high-frequency
methods are preferable to direct numerical methods for many applications, both for the
shorter computing times and for the full interpretability of the seismograms. The problems
arise when we want to use a synthetic seismogram as similar to the real one as possible in
the high-frequency approximation, such that the seismogram should be relatively complete,
although it is not necessary that it contains every feature of the full elastic wave field. It
is becoming necessary to select from all of the rays connecting the source to the receiver
only those that produce appreciable amplitude on the seismogram.
94
Appendix A : THE COMRAD CODE
A.1.1
What the program does
This program makes use of an algorithm to automatically generate an exhaustive set of
seismic phases with appreciable amplitude on the seismogram. The method uses a hierarchical order of rays and seismic phases generation, taking into account some existence constraints for a ray-path and some physical constraints. To compute synthetic seismograms,
the COMRAD code (from the Italian: “COdice Multifase per il RAy-tracing Dinamico”)
uses as its core the dynamic ray-tracing code provided by Farra and Madariaga (1987).
Both the programs are written in FORTRAN77 language. For detailed information about
the method used to build this program and its validation see Stabile et al, 2007.
A.1.2
Program Manual
This section covers installation and usage of comrad, including some shell script examples.
The knowledge of the dynamic ray-tracing code provided by Farra and Madariaga (1987)
and UNIX shell scripts is assumed here.
Installation
To compile the program you have to go into the source code directory SRC and type:
$ make
in fact the makefile in this directory compiles the source file comrad.f and moves the
executable comrad in the bin directory created outside the SRC directory. Following is
shown the structure of the makefile script:
#
#
#
#
COMRAD makefile
Author: Tony Alfredo Stabile
Date:
07 September 2007
BINDIR = bin
PROGS = comrad
install:
$(PROGS)
if [ -n ../$(bindir) ]; then \
mkdir -p ../$(BINDIR); \
fi
mv $(PROGS) ../$(BINDIR)
.SUFFIXES: .f
.f:
f77
$*.f -o $*
A.1 User Guide
95
Input files
The program needs the input parameters and the two core input files, one (creation.inp)
indicating the physical properties of the propagation medium and one (xxxx.dis, where xxxx
indicates a four characters name) setting the source(s)-receiver(s) geometry. An example
of creation.inp 1 file is the following:
test
# name of the xxxx.dis file
0
# overwrite the existing file: 1=no, 0=yes (CORE)
3
# number of layers
0.0 5.0 10.0
# top layer depths
0.0 0.0 0.0 0.0 0.0 0.0 # tilt of the layers (CORE)
15.0
# depth of the last layer
0.0 30.0 0.0 30.0 # x0 x1 y0 y1 dimensions of the medium (CORE)
2
# number of nodes (of splines) 1st layer (CORE)
0
# velocity gradient: yes=1 no=0 (CORE)
2
# numbers of sampling points in this layer (CORE)
0.0 5.0
# depths of the sampling points (CORE)
4.5 4.5
# Vp values (at each sampling point)
3
# number of nodes (of splines) 2nd layer (CORE)
0
# velocity gradient: yes=1 no=0 (CORE)
3
# numbers of sampling points in this layer (CORE)
5.0 7.0 10.0
# depths of the sampling points (CORE)
6.0 6.5 6.8
# Vp values (at each sampling point)
2
# number of nodes (of splines) 3rd (CORE)
1
# velocity gradient: yes=1 no=0 (CORE)
7.0
# Vp value (top)
7.2
# Vp value (bottom)
1.732 1.732 1.732 # Vp/Vs ratios in each layer
2.2 2.2 2.5 2.6 2.7 2.75 # top and bottom densities in each layer
the xxxx.dis must take the name indicated in the first command line of the creation.inp
file, so in this example it becomes test.dis. An example of the xxxx.dis file, i.e. test.dis 2
file, is the following :
0.0 0.0 6.0
# x,y,z of the source
0
# iplan (CORE)
0.0 10
# depth z of the stations, number of stations
2 4 6 8 10 20 22 24 26 28 # x stations
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. # y stations
1
If in the comments of the command line there is the word CORE this means that the line is important
only for the core and not for comrad
2
If in the comments of the command line there is the word CORE this means that the line is important
only for the core and not for comrad
96
Appendix A : THE COMRAD CODE
0.25
0.0 180.0 1800
0.0 0.0 1
0.3 0.15
0.1 0.2 100
---------------3
1 1
1
1
4 0
1 2 2 1
1 1 2 2
4 0
1 2 2 1
1 1 1 1
# dtau (CORE)
# TETAmin, TETAmax, nTETA (CORE)
# PHImin, PHImax, nPHI (CORE)
# distm, distm2 (CORE)
# DIST accepted, DIST tolerated,ITMAX (CORE)
BELOW WILL BE REPLACED BY THE COMRAD OUTPUT
# number of phases
# ray-string length, idir (1=direct wave, 0=other waves)
# id layer that contains the component of the ray-string
# direct P-wave: wave polarization (1=P, 2=S)
# ray-string length, idir (1=direct wave, 0=other waves)
# id layer that contains the 1st,2nd,3rd,4rt component
# P\S/ bottom of the second layer: wave pol. (1=P, 2=S)
# ray-string length, idir (1=direct wave, 0=other waves)
# id layer that contains the 1st,2nd,3rd,4rt component
# P\P/ bottom of the second layer: wave pol. (1=P, 2=S)
it is clear from the second part of this file that the user must indicate the number of phases
the core must calculate, writing also for each phase the ray-path and the polarization.
This becomes a tedious work if we want to calculate synthetic seismograms as complete as
possible, i.e. thousands of phases in the input file. The comrad program is developed to
do this automatically and quickly (about a second).
How to run the program
You can run the program simply typing:
$ comrad < comrad.inp
where the input file comrad.inp has the following command lines:
2
1
12
1
1e-4
1
8
2
0
# 1 P and S waves from the source; 2 only P waves from the source
# 1 select manually the max length of the ray-strings; 2 automatic
# maximum length of the ray-string
(Cancel this line if you choose automatic selection)
# 1 type manually the value of the phase parameter; 2 automatic
# value of the phase parameter - % of direct P-wave amplitude
(Cancel this line if you set the automatic value)
# 1 set manually the multiple reflection parameter; 2 automatic
# value of the multiple reflection parameter
(Cancel this line if you set the automatic value)
# 0 travel-times; 1 ray-paths; 2 seismograms
# 0 no free-surfaces effects; 1 free-surfaces effects
You can also run the program typing:
A.1 User Guide
97
$ comrad
but in this case you have to insert manually the parameters. However, the comrad program
will read automatically the information contained in the creation.inp and the xxxx.dis input
files.
Output files
As explained in the section A.1.2, the second part of the xxxx.dis file will be replaced by
the comrad output. Here the output files will not be shown because of their length (you
can imagine a file like the second part of the xxxx.dis but with thousands of command
lines). After the comrad program is run, it creates six output files:
1. 1nfasi.tmp (total number of phases for rays which start direction is up, i.e. first
command line of the second part of the xxxx.dis file))
2. 1raggi.tmp (codification of ray-strings and phase strings, i.e. second part of the
xxxx.dis file)
3. 1core.inp (input file for the core program)
4. 2nfasi.tmp (total number of phases for rays which start direction is down, i.e. first
command line of the second part of the xxxx.dis file))
5. 2raggi.tmp (codification of ray-strings and phase strings, i.e. second part of the
xxxx.dis file)
6. 2core.inp (input file for the core program)
the calculation of travel-times, ray-paths or synthetic seismograms is then divided in two
steps. A first run of the core program is done using (1), (2) and (3) files, and then a second
run of the core program is done using (4), (5) and (6) files. Below is shown the procedure
step by step for the first run:
• cut the second part of the xxxx.dis file (here test.dis) and indicate in the 7th command
line the initial and final starting angle for a ray3
$ sed -n 1,6p test.dis > cuttest.dis
$ echo 90.0 180.0 1800 >> cuttest.dis
$ sed -n 8,10p test.dis >> cuttest.dis
• concatenate cuttest.dis, 1nfasi.tmp and 1raggi.tmp files:
$ cat cuttest.dis 1nfasi.tmp 1raggi.tmp > test.dis
3
Put 90◦ -180◦ if starting direction is up and 0◦ -90◦ if starting direction is down.
98
Appendix A : THE COMRAD CODE
• run the core programs4 (creation and rayintpl2pt) with the core.inp and creation.inp
input file to overcome the program questions (test.dis input file will be read automatically)
$ creation < creation.inp
$ rayintpl2pt < core.inp
the core output should be a xxxx.dat ASCII file, if we choose to calculate the travel-times,
a xxxx.rai binary file, if we choose to calculate the ray-paths, and a xxxx.sis ASCII file, if
we choose to calculate the synthetic seismograms.
In particular the xxxx.sis ASCII file gives the Green functions and so it is necessary to
use a convolution program to have the synthetic seismograms. There are available several
freeware convolution programs (i.e. sismos developed by Véronique Farra).
Script example
This script does automatically all the instructions to create synthetic seismograms. It runs
the comrad program, its core and uses sismos program for the convolution.
#! /bin/bash
#
#
#
Bash script "multiphase.sh"
Author: Tony Alfredo Stabile
comrad < comrad.inp
creation < creation.inp
nome=‘sed -n 1p creation.inp|awk ’{print $1}’‘
a1=90
for i in 1 2
do
a2=$(echo "scale=3; $a1 + 90" | bc)
nome1=‘sed -n 1p ${i}raypt.inp|awk ’{print $1}’‘
nome2=${nome1}.tmp
nome3=${i}nfasi.tmp
nome4=${i}raggi.tmp
nome5=${nome1}.dis
if [ $i -eq 1 ]
4
The creation program generates another input file (xxxx.mil ) for the dynamic ray-tracing code rayintpl2pt. Contact Véronique Farra to have the core programs.
A.1 User Guide
then
green1=$nome1
else
green2=$nome1
fi
cp ${nome}.mil ${nome1}.mil
sed -n 1,6p ${nome}.dis > $nome2
iteraz=‘sed -n 7,7p ${nome}.dis | awk ’{print $3/2}’‘
echo $a1 $a2 $iteraz >> $nome2
sed -n 8,10p ${nome}.dis >> $nome2
cat $nome2 $nome3 $nome4 > $nome5
rayintpl2pt < ${i}raypt.inp
a1=0
done
for k in 1 3
do
(
echo "sismos.inp"
echo "0"
echo "1"
echo "${green1}.sis"
echo "1"
echo "${green2}.sis"
echo "0"
echo "${k}"
echo "2"
echo "0.03"
echo "0"
) > macro.tmp
sismos < macro.tmp
rm *.tmp
done
rm 1tes.mil 2tes.mil
exit
99
100
A.2
Appendix A : THE COMRAD CODE
The source code
Following is shown the file dimensioni.inc, in which are defined the values of the parameters and the dimensions of variables, arrays, and matrices used in the comrad code.
c
c
c
c
c
c
c
c
c
c
c
c
c
c
file "dimensioni.inc"
Multiphase code COMRAD
AUTHOR: Tony Alfredo Stabile
**************************************************************
*
*
*
Dimensions
*
*
*
*
dst: maximum number of elements
*
*
dnr: maximum number of rays
*
*
dl: maximum length of ray and phase strings
*
*
dl2: maximum length of ray string + counter + multiple *
*
reflections + direction (up/down)
*
*
df: maximum number of phases of each ray
*
*
*
**************************************************************
integer dst,dnr,dl,dl2,df
parameter (dst=100,dnr=500000,dl=25,dl2=dl+2,df=4100)
Here is shown the source code of the COMRAD program. It is written in Fortran77.
c**************************************************************************
c*****
c***** COMRAD PROGRAM (COdice Multifase per il RAy-tracing Dinamico)
c***** AUTHOR: Tony Alfredo Stabile
c***** EMAIL:
[email protected]
c***** VERSION: 1.3
c*****
c***** THIS PROGRAM NEEDS THE INPUT FILES creation.inp AND ----.dis
c*****
c***** THE OUTPUT IS SAVED ON THE FOLLOWING FILES:
c***** 1nfasi.tmp, 1raggi.tmp, 2nfasi.tmp, 2raggi.tmp
c*****
c***** THE PROGRAM MAKE USE OF THE FOLLOWING SUBROUTINES:
c***** legginpdis, genray
c*****
c***** THE PROGRAM WRITE ALSO THE FILES 1raypt.inp AND 2raypt.inp
c***** WHICH GIVE THE INPUT PARAMETERS FOR THE rayintpl2pt PROGRAM
c***** (rayintpl2pt IS THE CORE PROGRAM DEVELOPED BY V. FARRA)
c*****
A.2 The source code
101
c***** READ THE USER GUIDE FOR ADDITIONAL INFORMATION
c**************************************************************************
program comrad
implicit none
integer lmin,lmax,p,nf,ud,att,maxrefl
integer j,l,m,r,s,opz,suplib
character*4 nomefile
character*10 rag,fas,inp
real rif,frac
common /perc/ frac
common /rmul/ maxrefl
att=0
rif=0.
call legginpdis(nomefile,m,r,s)
write(*,*)’Only P waves from the source (YES=2)?’
read(*,*)p
lmin=abs(s-r)+1
write(*,*)’Choose one of the two following options:’
write(*,*)’1. Insert the maximum length of the ray strings’
write(*,*)’2. Choose the default length (=Lmin+2m)’
read(*,*)opz
if (opz.eq.1) then
write(*,*)’Insert the length (almost’,lmin,’)’
read(*,*)lmax
else
lmax=lmin+2*m
endif
write (*,*)’Plan the phase selection:’
write (*,*)’1. Manual plan’
write (*,*)’2. Default plan (0.0001% of the direct P wave)’
read (*,*)opz
if (opz.eq.1) then
write(*,*)’Insert the value’
read(*,*)frac
else
frac=1e-4
endif
write (*,*)’Set the multiple reflection parameter:’
write (*,*)’1. Manually’
write (*,*)’2. Default value (RIFM = 8)’
read (*,*)opz
if (opz.eq.1) then
write(*,*)’Insert the value’
102
100
200
300
400
Appendix A : THE COMRAD CODE
read(*,*)maxrefl
else
maxrefl=8
endif
if (lmax.gt.25) lmax=25
write(*,*)’Choose one of the three following options:’
write(*,*)’0. Calculate travel times’
write(*,*)’1. Calculate ray-paths’
write(*,*)’2. Calculate seismograms’
read(*,*)opz
write(*,*)’Do you want to take into account’
write(*,*)’of free surface effects (YES=1, NO=0)?’
read(*,*)suplib
do ud=1,2
write(inp,100),ud
format(i1,’raypt.inp’)
write(rag,200),ud
format(i1,’raggi.tmp’)
write(fas,300),ud
format(i1,’nfasi.tmp’)
OPEN(70,FILE=rag,FORM="FORMATTED")
OPEN(71,FILE=inp,FORM="FORMATTED")
write(71,400)ud,nomefile
format(i1,a3)
write(71,*)opz
write(71,*)suplib
call genray(lmax,lmin,p,nf,ud,m,r,s,rif,att)
CLOSE(70)
CLOSE(71)
OPEN(70,FILE=fas,FORM="FORMATTED")
write(70,*)nf
CLOSE(70)
enddo
end
c
c===========================================================
c**********************************************************
c***** THE legginpdis SUBROUTINE READS THE INFORMATION
c***** CONTAINED IN THE "creation.inp" AND "----.dis" FILES.
c*****
c***** THE TWO FILES MUST BE IN THE FOLDER WHERE THE PROGRAM
c***** WILL RUN
c*****
A.2 The source code
c***** THIS SUBROUTINE HAS TO BE SUBSTITUTED IF ANOTHER CORE
c***** IS USED INSTEAD OF rayintpl2pt
c**********************************************************
SUBROUTINE legginpdis(nomefile,m,r,s)
include ’dimensioni.inc’
integer m,r,s
integer i,j,h,tre(3),nstaz,diff(2)
character*4 nomefile
character*8 nomeintero
real zs(3),zr(3)
real spess(dst),dens(dst,2),vp(dst,2),vs(dst,2)
real vpvs(dst),appo,appovet(205),prf(2)
common /info1/ spess,dens,vp,vs
common /info2/ zr,zs
OPEN(8,FILE=’creation.inp’,STATUS=’OLD’,FORM="FORMATTED")
read(8,*)nomefile,appo,m
read(8,*) (appovet(i), i=1,m)
do i=2,m
spess(i-1)=appovet(i)-appovet(i-1)
enddo
appo=2*m+5
read(8,*) (appovet(i), i=1,appo)
do j=1,m
read(8,*)tre(1),tre(2),tre(3)
do h=1,2
read(8,*) (appovet(i), i=1,tre(3))
enddo
vp(j,1)=appovet(1)
vp(j,2)=appovet(tre(3))
enddo
read(8,*) (vpvs(i), i=1,m)
do i=1,m
vs(i,1)=vp(i,1)/vpvs(i)
vs(i,2)=vp(i,2)/vpvs(i)
enddo
read(8,*) (dens(i,1),dens(i,2), i=1,m)
CLOSE(8)
write(nomeintero,500),nomefile
500
format(a4,’.dis’)
OPEN(9,FILE=nomeintero,FORM="FORMATTED")
read(9,*) (zs(i), i=1,3)
read(9,*) (appovet(i), i=1,3)
zr(3)=appovet(2)
103
104
Appendix A : THE COMRAD CODE
c...TO BE MODIFIED IF THE Y COORDINATE OF THE RECEIVERS ARE NOT EQUAL TO 0
zr(2)=0.
c...END MODIFICATION
nstaz=int(appovet(3))
read(9,*) (appovet(i), i=1,nstaz)
diff(1)=abs(appovet(1)-zs(1))
diff(2)=abs(appovet(nstaz)-zs(1))
if (diff(1).gt.diff(2)) then
zr(1)=diff(1)
else
zr(1)=diff(2)
endif
CLOSE(9)
prf(1)=0.
do i=1,m
prf(2)=prf(1)+spess(i)
if ((zs(3).ge.prf(1)).and.(zs(3).lt.prf(2))) s=i
if ((zr(3).ge.prf(1)).and.(zr(3).lt.prf(2))) r=i
prf(1)=prf(2)
enddo
m=m-1
return
end
c
c============================================================
c************************************************************
c***** THE genray SUBROUTINE GENERATES RAY-STRINGS AND WRITES
c***** THE INFORMATION TO BE ADDED IN THE "----.dis" FILE
c*****
c***** THIS SUBROUTINE MAKE USE OF THE FOLLOWING SUBROUTINES:
c***** controlla, fase
c**********************************************************
SUBROUTINE genray(lmax,lmin,p,nf,ud,m,r,s,rif,att)
include ’dimensioni.inc’
integer lmax,lmin,p,nf,ud,m,r,s,a(dl2,dnr,2),av(dl2),f(dl2,df)
integer sw1,sw2,n,appo,cont,c,dec(4),att
integer l,j,i,k,h,tot
real zs(3),zr(3),rif
common /info2/ zr,zs
nf=0
sw1=1
sw2=2
n=1
A.2 The source code
a(1,1,sw1)=s
a(3,1,sw1)=0
a(4,1,sw1)=ud
if ((zs(3).le.0.000001).and.(ud.eq.1)) return
if ((s.eq.r).and.((zs(3).ne.0).or.(zr(3).ne.0))) then
if ((zs(3).ge.zr(3)).and.(ud.eq.2)) goto 600
if ((zs(3).le.zr(3)).and.(ud.eq.1)) goto 600
write(70,*) ’1 1’
write(70,*) s
write(70,*) ’1’
nf=nf+1
if (p.ne.2) then
write(70,*) ’1 1’
write(70,*) s
write(70,*) ’2’
write(71,*) ’1’
nf=nf+1
endif
600
endif
do l=2,lmax
cont=0
do j=1,n
do i=-1,1
a(l,j,sw1)=a(l-1,j,sw1)+i
do k=1,4
dec(k)=a(l-2+k,j,sw1)
enddo
call controlla(m,dec,c)
if (c.eq.1) then
cont=cont+1
do k=1,l
a(k,cont,sw2)=a(k,j,sw1)
enddo
do k=3,4
a(l-1+k,cont,sw2)=dec(k)
enddo
c**********************************************************
c****** START OUTPUT FOR THE RAYCODE **********************
c**********************************************************
if ((a(l,cont,sw2).eq.r).and.((zr(3).ne.0).or.((zr(3)
&.eq.0).and.(a(l+3,cont,sw2).eq.1)))) then
write(*,*) (a(k,j,sw1), k=1,l)
tot=0
105
106
Appendix A : THE COMRAD CODE
h=l+2
do k=1,h
av(k)=a(k,j,sw1)
enddo
call fase(m,tot,l,av,p,f,nf,ud,rif,att)
if (tot.ne.0) then
do h=1,tot
if (l.eq.lmin) then
write(70,*) l,’ 1’
else
write(70,*) l,’ 0’
endif
write(70,*) (a(k,j,sw1), k=1,l)
write(70,*) (f(k,h), k=1,l)
c ... START COMMAND TO INDICATE A SV WAVE
if (f(1,h).eq.2) write(71,*) ’1’
c ... END COMMAND TO INDICATE A SV WAVE
nf=nf+1
enddo
endif
endif
c**********************************************************
c****** END OUTPUT FOR THE RAYCODE ***********************
c**********************************************************
endif
enddo
enddo
n=cont
appo=sw1
sw1=sw2
sw2=appo
enddo
return
end
c
c===========================================================
c**********************************************************
c***** THE SUBROUTINE controlla VERIFIES IF THE RAYS
c***** SATISFY THE SELECTED CONSTRAINTS ON RAYS. IF A RAY
c***** DOES NOT SATISFY THE CONSTRAINTS, THE c VARIABLE
c***** WILL BE 0, ELSE IT WILL BE 1
c**********************************************************
SUBROUTINE controlla(m,dec,c)
A.2 The source code
include ’dimensioni.inc’
integer m,dec(4),c,updw,rifmult,maxrefl
real spess(dst),dens(dst,2),vp(dst,2),vs(dst,2)
common /info1/ spess,dens,vp,vs
common /rmul/ maxrefl
c=1
if ((dec(2).gt.m).or.(dec(2).lt.1)) then
c=0
return
endif
if ((dec(4).eq.2).and.(dec(2).lt.dec(1))) then
c=0
return
endif
if ((dec(4).eq.1).and.(dec(2).gt.dec(1))) then
c=0
return
endif
if (dec(2).eq.dec(1)) then
dec(3)=dec(3)+1
if (dec(4).eq.1) then
if (vp(dec(2),1).eq.vp(dec(2)-1,2)) then
c=0
return
endif
updw=1
else
if (vp(dec(2),2).eq.vp(dec(2)+1,1)) then
c=0
return
endif
updw=-1
endif
else
dec(3)=0
updw=0
endif
rifmult=dec(3)*dec(2)
if (rifmult.gt.maxrefl) c=0
dec(4)=dec(4)+updw
return
end
c
107
108
Appendix A : THE COMRAD CODE
c===========================================================
c***********************************************************
c***** THE fase SUBROUTINE GENERATES 2^L PHASES FOR EACH RAY
c***** CREATED BY THE decray SUBROUTINE. THIS SUBROUTINE
c***** VERIFIES IF EACH PHASE SATISFIES THE SELECTED
c***** CONSTRAINTS ON PHASES. IF A PHASE DOES NOT SATISFY
c***** THE CONSTRAINTS, THE okko VARIABLE WILL BE 0,
c***** ELSE IT WILL BE 1
c**********************************************************
SUBROUTINE fase(m,tot,l,av,p,f,nf,ud,rif,att)
include ’dimensioni.inc’
integer m,tot,l,av(dl2),p,fa(dl2,df),f(dl2,df),nf,ud,att
integer i,k,sw1,elevmax,okko,tb(dl2),tb2,inc,pol,w,rt
real novs,rif,cf(4,5,2),imp1,imp2,cnt,frac
real spess(dst),dens(dst,2),vp(dst,2),vs(dst,2)
parameter (novs=0.001)
data (cf(1,j,1),j=1,5)/0.17,0.45,0.73,1.00,0.71/,
&(cf(2,j,1),j=1,5)/0.21,0.24,0.21,0.25,0.27/,
&(cf(3,j,1),j=1,5)/0.01,0.02,0.02,0.02,0.17/,
&(cf(4,j,1),j=1,5)/0.50,0.92,0.98,1.00,0.96/,
&(cf(1,j,2),j=1,5)/0.70,0.69,0.37,0.24,0.35/,
&(cf(2,j,2),j=1,5)/0.51,0.35,0.19,0.12,0.21/,
&(cf(3,j,2),j=1,5)/0.17,0.07,0.03,0.03,0.05/,
&(cf(4,j,2),j=1,5)/0.90,0.66,0.19,0.10,0.16/
common /info1/ spess,dens,vp,vs
common /perc/ frac
tb(1)=ud
do i=2,l
tb(i)=abs(abs(av(i)-av(i-1))-1)+tb(i-1)
if (tb(i).eq.3) tb(i)=1
enddo
write(*,*)’...the phase strings will be generated’
elevmax=2**l
do k=1,elevmax
c********* START CREATION OF ALL THE PHASES **********
if (k.eq.1) then
do i=1,l
fa(i,k)=1
enddo
else
sw1=0
do i=1,l
if (sw1.eq.1) fa(i,k)=fa(i,k-1)
A.2 The source code
if (sw1.eq.2) then
if (fa(i,k-1).eq.1) then
sw1=1
fa(i,k)=2
else
fa(i,k)=1
endif
endif
if ((sw1.eq.0).and.(fa(i,k-1).eq.1)) then
fa(i,k)=2
sw1=1
endif
if ((sw1.eq.0).and.(fa(i,k-1).eq.2)) then
fa(i,k)=1
sw1=2
endif
enddo
endif
c******** END CREATION OF ALL THE PHASES ********
c******** START SELECTION OF PHASES ********
if (((p.eq.2).and.(fa(1,k).eq.1)).or.(p.eq.1)) then
okko=0
prod=1
do i=1,l-1
if ((vs(av(i),1).lt.novs).and.(fa(i,k).eq.2)) then
okko=1
goto 700
endif
rt=abs(tb(i)-tb(i+1))+1
pol=2*fa(i,k)+fa(i+1,k)-2
if (tb(i).eq.1) then
tb2=2
inc=-1
else
tb2=1
inc=1
endif
if (pol.le.2) then
imp1=vp(av(i),tb(i))*dens(av(i),tb(i))
imp2=vp(av(i)+inc,tb2)*dens(av(i)+inc,tb2)
else
imp1=vs(av(i),tb(i))*dens(av(i),tb(i))
imp2=vs(av(i)+inc,tb2)*dens(av(i)+inc,tb2)
109
110
Appendix A : THE COMRAD CODE
endif
cnt=imp1/imp2
if (cnt.le.0.5) w=1
if ((cnt.gt.0.5).and.(cnt.le.0.8)) w=2
if ((cnt.gt.0.8).and.(cnt.le.0.95)) w=3
if ((cnt.gt.0.95).and.(cnt.le.1.2)) w=4
if (cnt.gt.1.2) w=5
prod=prod*cf(pol,w,rt)
if (prod.lt.rif) okko=1
enddo
if (att.eq.0) then
att=1
rif=prod*frac/100
endif
if ((vs(av(l),1).lt.novs).and.(fa(l,k).eq.2)) okko=1
700
if (okko.eq.0) then
tot=tot+1
do i=1,l
f(i,tot)=fa(i,k)
enddo
endif
endif
c********
END SELECTION OF PHASES
********
enddo
return
end
Appendix B
Fundamentals of underwater
acoustics
In this Appendix some concepts and equations used commonly in underwater acoustics
will be introduced. Here we illustrate briefly the concept of micropascal reference unit and
the Decibel.
The level of an acoustic wave is defined as the number of decibels by which its intensity
I differs from the intensity of a reference sound wave I0 :
I
(B.1)
I(dB) = 10 log
I0
Since I = p2 /ρc, where p is the pressure, ρ is the density, and c is the sound velocity, if ρc
can be considered as constant (as usual), the above equation can be written as follows:
p
I(dB) = 20 log
(B.2)
p0
The unit of intensity in underwater acoustics is the intensity of a plane wave having a
pressure of 1 µPa, or equivalently 105 dyne per square centimetre (dyn/cm2 ). Thus, a
wave having intensity 1000 times that of a plane wave of pressure 1 µPa, using equation
(B.1), it would be said to have a level of 30 dB re 1 µPa.
Following the sonar equations, the sound attenuation in seawater, and the calculation
of the transmission reliability will be discussed in separated sections.
B.1
The sonar equations
The many phenomena and effects that are related to the sound propagation in the seawater
can be grouped together quantitatively in a small number of sonar parameters, which are
related to the sonar equations. These equations tie together the medium, the target and
the equipment effects and are used to set one of the sonar parameters if the others are
known.
112
Appendix B : FUNDAMENTALS OF UNDERWATER ACOUSTICS
The sonar equations are founded on a basic equality between the desired and the
undesired portions of the received signal. In fact, the portion of the total acoustic field
at the receiver which is desired is called the signal while the portion undesired is called
the background. In sonar equations the background is either noise (undesired waves in the
seawater) or reverberation (return of own acoustic wave due to scatterers in the seawater).
The objective of the sonar equations is to find means for increasing the response of the
sonar system and decreasing the response of the system to the background.
The sonar parameters are levels in unit decibels relative to the standard reference
pressure of a 1-µPa plane wave. Table B.1 shows the sonar parameters determined by the
equipment, the medium, and the target, respectively. The sonar parameters that represent
Equipment
Projector Source Level: SL
Self Noise Level: N L
Directivity Index: DI
Detection Threshold: DT
Medium
Transmission Loss: T L
Ambient-Noise Level: N L
Reverberation Level: RL
Target
Target Source Level: SL
Target Strength: T S
Table B.1: Sonar parameters determined by the equipment, the medium, and the target, respectively.
the desired portion of the acoustic wavefield at the receiver are SL, T S, and DI, while
the undesired portion of the acoustic wavefield at the receiver is represented by T L, N L,
and RL. The difference between the first group of parameters and the second is the
signal-to-background ratio, commonly called simply signal-to-noise ratio SN R. Finally,
the Detection Threshold DT parameter gives an indication of the presence or the absence
of the target. In other words, when the input signal-to-noise ratio SN R is above a certain
DT fulfilling certain probability criteria, the target is considered present, else the target is
absent. Now we will obtain the sonar equations for active and passive sonar.
In the case of active sonars a sound source acting also as receiver produces by some
means a source level of SL decibels at unit distance (1 m). When the sound reaches
the target it is reduced by the transmission loss T L. On reflection or scattering by the
target, the reflected or backscattered level has a gain at a unit distance from the target
equal to the target strength T S. When the sound propagates back toward the source it
is again reduced by T L. Moreover, in the seawater there is a noise level (neglecting the
reverberation) which is reduced by the directivity index DI of the transducer acting as a
receiver. Taking into account these quantities, the equation for active sonar becomes:
SN R = SL − 2T L + T S − (N L − DI) = SL − 2T L + T S − N L + DI
(B.3)
A modification is required when the background is reverberation instead of noise. In
this case the terms N L − DI of equation (B.3) is replaced by an equivalent plane-wave
reverberation level RL:
SN R = SL − 2T L + T S − RL
(B.4)
B.2 Sound attenuation in seawater up to a frequency of 1 MHz
113
where DT has a different value than DT for noise.
In the case of passive sonar, the target itself produces the signal with a source level
of SL decibels at 1 meter distance. Moreover, the target strength T S is not considered
yet and one-way transmission loss is involved. So for passive sonar the equation (B.3)
becomes:
SN R = SL − T L − N L + DI
(B.5)
which is the equation (4.6) used in section 4.2.2 without the directivity index since in that
simulation omnidirectional sources and receivers have been considered.
The SN R value can be finally compared with the detection threshold DT in order to
make a decision on the presence (SN R ≥ DT ) or absence (SN R < DT ) of the target.
B.2
Sound attenuation in seawater up to a frequency
of 1 MHz
The geometrical spreading is not the unique cause of the sound attenuation in seawater:
the underwater acoustic wave propagation is subjected also to scattering and absorption
effects, which produce additional attenuation of the sound energy. The absorption of sound
in seawater is considered to be that of pure water plus an excess absorption due to small
amounts of various salts in the sea. In particular, the sound absorption is due by the
combination of the two following different processes:
1. viscous dissipation of energy (pure water absorption)
2. chemical relaxation of magnesium sulphate (M gSO4 ) and boric acid (H3 BO3 )
Each process dominates compressional wave attenuation in seawater in a particular range
of frequencies. From 50 Hz to 10 kHz the boric acid contribution becomes the most
important, although this is only 0.055% of the total molality of seawater. Also magnesium
sulphate is present in little quantities equal to 0.84% of the total molality of seawater, but
its relaxation process dominates the wave attenuation in the band of frequencies between
10 kHz and 200 kHz. At frequencies above 200 kHz, pure water absorption (due to shear
and bulk viscosity of freshwater) is the largest contributor (Jackson and Richardson, 2007;
Jensen et al., 2000).
Several studies carried out between the 1950’s and the 1980’s demonstrated that the
sound attenuation in the water increases according to the square of the frequency and it
depends from the processes described above. The equation for the absorption coefficient
α at frequency f in seawater is usually written, following the notation of (Fisher and
Simmons, 1977), as
α(dB/km) =
A1 P1 f1 f 2 A2 P2 f2 f 2
+ 2
+ A3 P3 f 2
f12 + f2
f2 + f2
(B.6)
where indexes 1, 2, and 3 indicate the dependence from the chemical relaxation of boric
acid, the chemical relaxation of magnesium sulphate, and the water viscosity, respectively.
114
Appendix B : FUNDAMENTALS OF UNDERWATER ACOUSTICS
The dependencies from the pressure in the three processes are indicated with P1 , P2 , and
P3 , respectively. Finally, the two relaxation frequencies (expressed in KHz) are indicated
with f1 and f2 . A1 , A2 , and A3 terms were originally considered as constants, but it was
experimentally demonstrated (Francois and Garrison, 1982ba,b) that they vary with the
water properties. A1 and A2 terms are both expressed in dB/(km · kHz), while A3 term is
expressed in [dB/(km · kHz)]2 .
A simplified expression of equation (B.6) was formulated by Thorp (1967) and optimized
later by Fisher and Simmons (1977). The Thorp’s formula is still commonly used for the
calculation of the sound attenuation α in seawater up to a frequency of 100 kHz. Its
expression is the equation (4.3) used in the section 4.2.1, which is reported here again for
completeness:
α(dB/km) = 3.3 · 10−3 +
44f 2
0.11f 2
+
+ 3.0 · 10−4 f 2
1 + f 2 4100 + f 2
(B.7)
with the frequency f in kHz. Francois and Garrison (1982aa, 1982bb) obtained in the
1982 a more complete expression for the seawater sound attenuation α (in dB/km) with
depends not only from the frequency f (as for the Thorp’s formula), but considering the
terms A1 , A2 , and A3 as functions of the seawater parameters like the temperature, the
depth (or pressure), the salinity, and the pH. Following, the values obtained by Francois
and Garrison for the terms of the equation (B.6) are reported:

8.86

 A1 =
· 10(0.78pH−5)
c
Boric acid contribution
P1 = 1


f1 = 2.8(S/35)0.5 · 10(4−1245/θ)

S

A2 = 21.44 (1 + 0.025T )



c
P2 = 1 − 1.37 · 10−4 D + 6.2 · 10−9 D2
Magnesium sulphate contribution


8.17·(8−1990/θ)

 f2 =
1 + 0.0018(S − 35)

P3 = 1 − 3.83 · 10−5 D + 4.9 · 10−10 D2




For T ≤ 20◦ C




A3 = 4.937 · 10−4 − 2.59 · 10−5 T

+9.11 · 10−7 T 2 − 1.50 · 10−8 T 3
Water viscosity contribution


For T > 20◦ C




A3 = 4.937 · 10−4 − 2.59 · 10−5 T



+9.11 · 10−7 T 2 − 1.50 · 10−8 T 3
where c is the sound velocity (m/s) which can be calculated by the use of one of the various
expressions in literature (e.g., the equation of Del Grosso (1974)), T is the temperature
(◦ C), θ = 273 + T , S is the salinity (ppm), D is the depth (m), and f is the frequency
(kHz).
B.3 Calculation of the transmission reliability
115
The empirical equation of Francois and Garrison is valid for frequencies between 200
Hz and 1 MHz, for seawater temperature from -2 ◦ C to 30 ◦ C, for typical values of the
seawater1 pH (from 7.6 to 8.3), for salinity between 30 ppm and 40 ppm, and for depth
also bigger than 3000 m.
B.3
Calculation of the transmission reliability
The velocity of a binary sequential source which emits K bits in a time T , is defined by
the ratio
K
(B.8)
Rs ≡
T
and it is expressed in bit/s, or in kbit/s. The capacity of the communication channel is
related to the bandwidth w of the transmission medium and to the signal-to-noise ratio
by the following expression:
S
C ≡ w log2 1 +
(B.9)
N
The Shannon’s capacity theorem (Shannon, 1948) states that error-free transmission in a
digital communication system is possible as long as the transmitter velocity Rs is less then
the channel’s capacity C, hence
Rs ≤ C
(B.10)
If we want to know the value of the signal-to-noise ratio in decibels (dB), below which the
digital communication is no longer reliable, taking into account that
S
S
(dB) = 10 log10
(B.11)
N
N
and using the equations (B.9) and (B.10), we have the following result:
Rs
S
(dB) = 10 2 w + 1
N
(B.12)
As an example, if the bandwidth of the transmission channel is equal to 10 kHz and the
transmitter velocity is equal to 2.4 kbit/s, the signal-to-noise ratio must be
S
≥ 21.8 dB
N
for a reliable (error-free) transmission in a digital communication system with these characteristics.
1
In particular in the Gulf of Naples the seawater pH is about 8.1-8.2.
116
Appendix B : FUNDAMENTALS OF UNDERWATER ACOUSTICS
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