Universität Wien
Formal- und Naturwissenschaftliche Fakultät
Institut für Meteorologie und Geophysik
Christian Stotter
Numerical Modeling of Seismic Noise
in Canonical Structures
Diplomarbeit
zur Erlangung des Akademischen Grades
“Magister der Naturwissenschaften”
Wien, Jänner 2003
Acknowledgements
I would like to thank all who helped me during the preparation of the Diploma
Thesis:
•
•
•
•
Prof. RNDr. Peter Moczo, DrSc., supervisor,
Mgr. Jozef Kristek, PhD., supervisor-specialist,
Mgr. Miriam Kristeková and
Dr. Bruno Meurers.
My thanks also go to my parents and my grandparents who have been
helping, supporting and encouraging me during the whole university studies
and to Manuela and Andrea for enduring my bad moods and tenseness.
II
Contents
1. Introduction
1
2. Site effects
2
2.1. Seismic ground motion at a site
2
2.2. Site effects during earthquakes
3
2.3. Examples of site effects
3
2.4. Local geological structure
4
2.5. Basic types of local geological structures and associated site effects
5
2.5.0. Flat free surface
5
2.5.1. Topographical structures
6
2.5.2. Sedimentary structures
8
2.6. Methods of investigation of site effects
3. Ambient seismic noise
18
21
3.1. Introduction
21
3.2. Nature of the noise wave-field
21
3.3. Methods of investigation
22
3.4. H/V spectral ratios
23
3.5. Partial conclusive remarks
28
4. Numerical modeling of seismic ground motion
28
4.1. The role of numerical modeling / simulation
28
4.2. Review of methods
29
4.3. Advantages and disadvantages of numerical modeling
30
III
5. The finite-difference method
31
5.1. The principle
31
5.2. Finite-difference approximations
35
5.3. Formulations of the equation of motion
39
5.4. Finite-difference spacetime grids
41
5.5. Homogeneous and heterogeneous finite-difference schemes
44
6. The goal of the diploma thesis
46
7. Models and methods of computation and analysis
46
7.0. Computations
48
7.1. Design of the models and space-grids
for the finite-difference computations
49
7.2. Generation of point sources of seismic noise and
finite-difference modeling
50
7.3. Filtering in the finite-difference computations
52
7.4. Models
52
7.5. Analysis of synthetic seismograms
54
8. Results
55
8.1.
Homogeneous halfspace
55
8.2.
One layer over halfspace
59
9. Discussion
73
10. Conclusion
78
References
79
Appendices
Appendix I. The seismic response of one layer over rigid halfspace
to a vertically incident SH wave.
Appendix II. 3D 4th-order displacement-stress staggered-grid
finite-difference scheme
IV
85
87
1. Introduction
Since human beings have been living on our planet, they were facing natural
disasters. Volcanoes, landslides, thunderstorms, tornadoes and earthquakes can
destroy thousands of lives and livelihoods. Until now mankind has not developed
proper technologies to prevent or even forecast these events properly.
Earthquakes can cause particularly severe damages.
We are still not able to forecast earthquakes, but even if we could do so, it
would still be of great importance to predict earthquake ground motion at a site or in a
region during future earthquakes and estimate the impacts of the future earthquakes.
There are a lot of examples that show, that site effects are the main reason for the
most severe damages even in moderate seismicity countries like Austria.
Site effects are local anomalies in the seismic ground motion i.e., the motion is
in a contradiction with what could be expected from the source and seismic wave
propagation in a homogeneous medium. The seismic ground motion at a site is
influenced mainly by three factors:
1. The wave-field radiated by the seismic source at the epicentre.
2. The medium between the seismic source and the site.
3. The local geological conditions at the site.
Sit e
Eart h’s surface
Sit e
Local geological st ruct ure
SO URC E
Medium bet ween source
and sit e
Fig.1
The seismic ground motion at a site is determined by the wave-field excited by the source, the
medium between source and site and the local geological structure (very schematic).
Site effects occur if the effect of local geological conditions on seismic motion is
significant. Site effects will be treated in detail in chapter 2.
If we want to know how strong the site effects in different regions will be during a
future earthquake, we should not wait for the next earthquake to measure them, but
we want to predict them.
In principle we can
1. Measure small earthquakes,
2. Measure seismic ground motion generated by artificial sources (e.g. explosions),
3. Analyse (historical) macro-seismic effects,
1
4. Physically model earthquake ground motion,
5. Numerically model earthquake ground motion,
6. Measure and analyse ambient seismic noise.
The latter technique seems very attractive since it requires relatively simple, low-cost
measurements and analysis. However, although used extensively in Japan, it is very
much debated in the “western world”. There are some authors that report good
results (e.g., Konno & Ohmachi 1998, Milana et al. 1996) while others are very critical
(e.g., Dravinski et al. 1996, Coutel & Mora 1998)
What most of the authors believe in, is, that it should be possible to estimate at least
the fundamental frequency of the local geological structure. There are not many
evidences that there is any reliable information on higher frequency modes or
amplitude factors of the amplification of seismic ground motion during an earthquake
in the ambient seismic noise.
The biggest problem in using seismic noise is that we know too little about the
nature of the noise field. Does noise consist mainly of surface or of body waves?
To light up this open question the 5th FP project SESAME (Site Effect Studies Using
Ambient Excitations) was launched. This project will try to fill the lack of information
concerning the understanding of the real nature of noise. By developing and using
numerical tools to generate noise synthetics for different geological structures it will
try to assess the ability of different noise techniques to provide information on
transfer properties of local geological structures.
In this diploma thesis the author wants to contribute to this project by modeling noise
synthetics by the finite-difference method and analysing them in different geological
structures:
1. homogenous half-space,
2. one layer over half-space.
The synthetics will be analysed by the H\V technique. The H\V ratios will be
compared with the theoretical transfer functions calculated for the above models of
geological structures.
2. Site effects
2.1.
Seismic ground motion at a site
Seismic ground motion at a site is determined mainly by three factors: (Fig.1)
a) the wave-field radiated by the source,
b) the medium between the source and site,
c) the local geological structure at a site.
2
2.2.
Site effects during earthquakes
If the observed seismic ground motion is somehow in contradiction to the motion that
would be determined only by the source itself and the homogenous medium, we
speak of a site effect.
• Any anomaly in the seismic wave field can be regarded as site effect:
e.g. anomalies in the displacement field, the velocity- or the acceleration field, or
their Fourier transforms ,
∂ui
∂ ²ui
∂u
∂ 2ui
ui ,
,
,
(1)
F[
]
F[ i ] ,
F[ui ] ,
∂t
∂t
∂t²
∂t²
-
anomalies in the maximum ground motion and/or its Fourier transform ,
ui,max ,
-
F[ui,max ] ,
(2)
anomalies in the differential motion in time and frequency domain,
∂ui
∂F[ui ]
∂xi ,
∂xi
(3)
-
anomalies in the duration of the seismic ground motion,
•
anomalies in the macro-seismic field,
•
secondary motion induced by vibrational seismic motion (e.g. landslides,...).
2.3.
Examples of site effects
To illustrate the importance of estimating site effects let us mention only some
examples of very disastrous events during which site effects were observed:
GEDIZ, Turkey, 22.3.1970, Ms = 7
Some buildings of a car factory, which was more than 135km away from the
hypocentre of the earthquake, were damaged due to mutual resonance of the
buildings and the local geological structure.
MICHOACÁN, MEXICO, 19.5.1985, Mw = 8.1
This is perhaps the best known example of a site effect. More than 10000 people
were killed and more than 50000 left homeless in Mexico City. The estimated
damages amounted up to 4 billions of US Dollars.
Mexico City was about 360 km away from the seismic focus. The main reason for the
severe damage was resonance in unconsolidated lake deposits and man-made land
layers.
3
Some more examples of recent earthquakes during which site effects were observed:
1976
1979
1980
1981
1985
1988
1989
1990
1990
1994
1995
1995
1999
1999
Friuli, Italia
Imperial Valley, California
Guerrero, Mexico
Irpinia, Italy
Chile
Spitak, Armenia
Loma Prieta, California
Iran
Philippines
Northridge, California
Kobé, Japan
Southern Mexico
Izmit, Turkey
Chi-Chi, Taiwan
So we can see that site effects occurred during nearly all recent earthquakes which
hit populated area and they are responsible for the largest damages. This is because
sites that are most prone to site effects on one hand (e.g. sedimentary basins,
sedimentary layers over hard rock, sedimentary valleys...) are very attractive for
people to live there on the other hand. Free surface of sedimentary basins is suited
well to agricultural activities, building big cities, etc., because it is flat and relatively
vast.
2.4.
Local geological structure
Next we have to define what is meant when we talk about a local geological
structure. We can define a local geological structure as that part of the Earth which
influences seismic ground motion at a site in the frequency range we are interested
in.
Obviously, the frequency range has to cover
•
the fundamental and higher-order resonant frequencies of man-made structures.
The frequencies correlate with the size and type of the buildings. Typically,
f ∈ < 0.1 , 20 > Hz
which corresponds to
In Austria, the range is f ∈ < 0.5 , 5 > Hz or
•
T ∈ < 0.05, 10 > sec.
f ∈ < 2 , 10 > Hz
The range of the S-wave velocity in near surface geological structures is about
100–3000 m/s.
The lower border is due to by unconsolidated sediments whereas the upper one is
due to granite.
•
Because the frequency, the velocity and the wavelength of seismic waves are
directly coupled through the relation
c=λ.ν,
(4)
4
we can get typical wavelengths and, at the same time rough estimates of the
dimensions of local geological structure:
l = 10 – 10³ m.
Thinking about the upper border, we can imagine structures like dams and bridges
that can almost reach dimensions in the order of kilometres. On the other hand small
and stiff buildings correspond to wavelengths of the order of 101 metres.
2.5.
Basic types of local geological structures and associated site
effects
Most of the theoretically and observed site effects can be attributed to anomalous
wave field in “typical” geological settings that are characterised by their geometrical
and mechanical parameters. The two main configurations that give rise to site effects
are
topographical structures,
surface sedimentary structures.
In both groups we can observe characteristic features and corresponding site effects.
Few studies indicate combined topographic-sedimentary effects due to interaction of
the wave fields in the topographic structure and adjacent sedimentary structures.
2.5.0. Flat free surface
Flat free surface neither can be addressed as topography nor is there any
sedimentation. Since it does not fit in these two categories, it should have an extra
chapter.
The reaction of flat free surface to vertically incident S-waves is comparable to
the reflection of a wave in a rope with a free end. From basic physics we know that
this will generate amplification of factor 2. For the SH-waves the amplification of 2
applies to all incident directions preventing any local variation caused by flat free
surface.
However, for the SV-waves this is not true. The SV-wave incident can develop
an extremely complex pattern of local seismic ground motion and therefore in
damage distribution (Sammis et al. 1987) especially for near critical incident where
the incident angle γ is given by sin γ = β/α . Here β is the S-wave velocity and α is the
P-wave velocity. In this case the horizontal slowness of the S- and P-waves is the
same and strong coupling occurs, leading to generation of SP-waves that propagate
along the surface. In Fig.2 the dependence of the amplification factor on the angle of
incidence of the plane SV-wave is illustrated. There is a very strong amplification
(factor 5) of the horizontal component for a relatively narrow (~1°) range of incident
angles near the critical incidence angle (Aki 1988). The peak amplification depends
on the Poisson’s ratio.
5
Fig.2
Amplitude of horizontal (solid line) and vertical (broken line) component displacement at the free
surface due to plane SV-wave incidence as function of the incident angle. Poisson’s ratio is 0.25
(Reproduced from Aki 1988)
2.5.1. Topographical structures
Different authors reported an increasing amount of damages on hill tops compared to
the damaged buildings at the base. Some examples can be found in Levret et al.
1986, Siro 1982 and Celebi 1987. Theoretical and numerical studies also predict a
systematic amplification of seismic ground motion on “convex” topographies (see
Fig.3c) like hills or ridge crests and deamplification over “concave” parts (see Fig.3b)
such as valleys and foothills ( Boore 1972 , Bouchon 1973 , Bard 1982 , Bard and
Tucker 1985 , Sánchez-Sesma 1985 ).
For a wedge-shaped ridge or valley there even exists a surprisingly simple,
exact solution for the motion at the vertex of the structure due to incident SH-waves
polarized in the direction of the vertex. As shown by Sánchez-Sesma (1985) the
displacement amplification at the vertex is 2/n when the angle of wedge is
n*π (0<n<2).
(5)
Fig.3
Typical topographical structures that give rise to anomalies in the seismic ground motion. a)
Step like faults are strong lateral discontinuities and origin of diffracted waves.
b)
Concave
structures (e.g. valleys) generally deamplify seismic ground motion. c) Convex structures (e.g.
ridge crests, hills) usually focus and therefore amplify seismic motion.
6
Despite the theoretical and intuitive vividness of this focusing/defocusing effect, there
has not been any instrumental proof to it, due to the lack of dense 3D seismological
arrays on topographical features (Bard 1995).
The second basic phenomenon related to topographic structures is the generation of
diffracted body and surface waves that propagate downwards and outwards from a
topographic feature. During their propagation they interfere with the incident wavefield. The amplitudes of waves propagating along the surface are generally smaller
than those of the direct waves and have been reported for the first time by LeBrun
(1993).
In theory they had been predicted long before. As an example the study of
Kawase (1987a) can be mentioned. Kawase calculated the response of a cylindrical
canyon to normally incident SH-waves. An example of the calculated synthetics is
shown in Fig.4.
Fig 4.
Time domain solution of the seismic ground motion in a cylindrical canyon due to normally
incident SH-waves. The incident waveform is a Ricker wavelet with predominant frequency of
f=2 Hz. (Reproduced from Kawase 1987a).
The scattered wave-field at the surface is made up of three different types of waves:
the direct incident wave-field, reflected and (compared to the flat surface) deamplified
waves, and, finally diffracted waves that are generated at both edges of the canyon.
Diffracted waves that travel at the surface of the canyon are the main motions
observed inside the canyon after the arrival of the direct wave. As can be seen in
Fig.4, large differential motion has to be expected near the edges of the canyon.
Furthermore one has to say that these phenomena described above are very
sensitive to the nature of the incident wave-field, which means that they will be
significantly different for different wave types, different incidence and azimuth angles.
From various numerical studies it can be inferred that the amplification is larger on
the horizontal components (S-waves) than for the vertical component (P-wave).
(Boore 1972, Bouchon 1973, Bard 1982, Wong 1982, Bard and Tucker 1985,
Sánchez-Sesma 1985, Kawase 1987a) The amplification also seems roughly linked
to the “sharpness” of the topography (Bard 1995). The steeper the average slope, the
higher the top amplification.
Speaking about spectral behaviour of these effects it is clear that they are
frequency dependent or, strictly taken, band-limited. As instrumental and numerical
results show, the biggest effects appear at wavelengths comparable to the horizontal
extent of the topographic feature.
7
Despite the fact that there is a qualitative agreement between theory and
observations, there are significant differences from a quantitative viewpoint. Some
studies (Pedersen et al. 1994b, Rogers et al. 1974) report a reasonable fit between
numerical and instrumental results, while others (e.g., Nechtschein et al. 1994, Géli
et al. 1988) do not.
As said before, more detailed and dense instrumental studies should be
performed to get a deeper insight in the physics of topographic site effects especially
in Europe because of populated mountainous regions.
2.5.2. Sedimentary structures
For a very long time the effects of a relatively soft soil deposits on the local seismic
ground motion have been recognized. If we have geological structures that consist of
a relatively soft sedimentary cover over bedrock, we have to expect different site
effects due to geometry of the sediment-bedrock interface. Generally, we can
distinguish between 4 basic 2D models of local geological structures and, associated
to each, a typical anomaly in seismic ground motion. The 4 settings are shown in
Fig.5 and will be discussed in detail in this chapter.
Fig.5
Basic types of sedimentary 2D structures a) layer over half-space b) semi-infinite layer over
half-space (strong lateral discontinuity), c) shallow sedimentary basin d) deep sedimentary
basin. Grey shaded areas correspond to sediments, having velocities α1 and β1, density ρ1,
and quality factors Qα1 and Qβ1, whereas white areas correspond to rock that has velocities α2
and β2, density ρ2 and quality factors Qα2 and Qβ2, respectively.
a) One layer over half-space (Fig.5a) causing 1D vertical resonance
If we have a sediment layer over half-space this means that we have a material with
generally lower density and S-wave velocity (ρ1 and β1) over a denser and faster
material (ρ2 and β2). We can expect that incident S-waves will be trapped in the low
velocity layer which will lead to 1D vertical resonance. This has been proved
theoretically and observationally in Japan in the early 1930 by Ishimoto and Sezawa
(Aki 1988).
8
The sequence of resonant frequencies for a vertical incident S-wave for a soft layer
over rigid half-space is given by
fn = ( 2n+1 )β1 / 4 H
;
n=0,1, 2, ...,
(6)
Where β1 is the S-wave velocity of the layer and H is the depth of the layer. The
derivation of this formula can be found in the Appendix. If the incident angle (the
angle between the direction of propagation and the vertical) is 0≤θ<π/2 , than the
sequence is given by:
fn = ( 2n+1 )β1 / (4 H cos θ ) .
(7)
There also exists an exact formula (Aki and Richards 1980) for the amplification
factor of surface displacement due to SH-waves normally incident on a soft surface
layer from the bottom:
|U(ω)| = 2{ cos²( ωH/β1 )+( ρ1β1/ρ2β2 )² sin²( ωH/β1 ) }-.5
(8)
Here, ω is the angular frequency, β1, β2 and ρ1, ρ2 are the S-wave velocity and
density for the soft and hard material, respectively and H is the thickness of the layer.
|U(ω)| is displayed in Fig.6.
7
6
4
U( w)
2
0
0
0
20
40
0
Fig.6
60
w
80
90
Amplification factor for the vertical incident of a plane SH-wave in a flat layer with H=60m,
β1=400m/s. Peak amplification is twice the impedance contrast, which is 3 in this case.
Looking at Fig.6 and Eq.8 it is clear, that the amplification approaches 2 for
wavelengths much longer than the layer thickness ( ωH/β1~0 ). Then it varies
periodically between 2 for destructive interference ( fn,d =2nβ1/4H ), and the value of
2 ρ2β2/ρ1β1 , which is two times the impedance contrast between soft and stiff material
( fn,c=(2n+1)β1/4H ).
9
b) Lateral discontinuities causing strong differential motion
There have been a large number of observations consistently reporting an increase
in damage intensity distribution on narrow stripes in the vicinity of strong lateral
discontinuities, i.e., contact of two materials with different impedance, like ancient
faults, debris zones or steep edges of alluvial valleys. These phenomena have been
observed during the 1868 Hayward, California, earthquake (Prescott, 1982), the 1909
Provence, France, earthquake (Levret et al. 1986), and many more earthquakes.
(Weischet 1963, Siro 1983, Ivanovic 1986). All of them reported a significant increase
of intensity on the softer side of such a lateral discontinuity.
Since these phenomena can not be explained by a simple 1D model, Moczo &
Bard (1993) addressed this problem in a 2D study. They modelled a semi-infinite
layer of soft sediments over hard rock (see Fig.5b) using a generalisation of the finitedifference technique of Moczo (1989) in which they incorporated absorption based on
a method by Emmerich & Korn (1987). For detailed explanation of the finitedifference technique see chapter 5.
After Moczo & Bard had tested their method on a “canonical” sedimentary
basin (Boore et al.1971) and found good agreement with the Aki-Larner method (Aki
and Larner 1970), they performed a parameter study in which they considered a
series of 12 semi-infinite layers differing in the impedance contrast, damping factors
and the thickness of the layer. They analysed models with constant velocity as well
as models with a velocity gradient. They studied the case of a vertically incident SHwave assuming a Gabor impulse, given by:
s(t) = exp[ −[ωp
t − ts
]²]cos[ωp (t − t s ) + Ψ ]
γ
,
(9)
as the source-time function of the incoming wave-field. By using different parameters
ωp, ts and Ψ they could account for a “Dirac” like (I1) as well as for a quasimonochromatic signal (I2).
As a result of their computations they got seismograms at receivers located on
the free surface of the model from –19.2m to 230.4m with the discontinuity at 0m. In
addition to that they calculated Fourier transfer functions (FTF) by dividing the Fourier
transform of the local response of each receiver through the Fourier spectrum of the
input signal I1. Next they calculated the response due to input signal I2 and at last
they computed the differential motion both in time and frequency domains. Results
for the C-5-50 model (constant velocity in the layer, impedance contrast of 5 and
quality factor of 50) are shown in Fig.7.
It can be seen that at receivers at some distance away from the contact of the
two media there exist two separated wave phenomena. The first one corresponds to
the well known 1D vertical resonance of the primary arrival and its subsequent
reverberations due to up- and downwards reflected waves in the soft layer. The
second is linked to waves diffracted away from the discontinuity.
In the immediate vicinity of the contact the wave-field is very complex. It
consists of a combination of oblique waves radiated directly by the discontinuity, and
waves reflected at the horizontal surface and at the interface between layer and rock.
At some distance away from the discontinuity, because of damping and geometrical
spreading, the direct diffracted waves die out and the multiple reflected waves
propagate along the surface as Love waves.
So there are two important effects. First, the classical 1D resonance effect,
and second, the lateral diffraction at the discontinuity. At sites far away from the
10
contact, 1D resonance is the major effect and differential motion results only from
passage of the Love waves, but places near the discontinuity undergo significant
differential motion that is largest over a very narrow zone located on the soft side of
the discontinuity. In the frequency domain these effects can be seen as a
superposition of the 1D resonance pattern with oscillatory patterns corresponding to
interference between direct (body) and diffracted (surface) waves.
Fig.7.
Results for the C-5-50 model. a) Response to I1 Gabor pulse at 14 equi-distant surface sites. b)
Fourier transfer functions at the same sites. c) Response to I2 Gabor pulse d) Surface
differential motion in time domain for I2 pulse e) Surface differential motion in frequency domain.
Numbers to the right represent peak values of the corresponding quantity for each site. The five
curves on top of each column display the spatial variations of these peak values along a cross
section of the model (Reproduced from Moczo & Bard, 1993).
After performing the simulation for SH-wave incidence on models with different
damping, impedance contrast and velocity gradient configurations, Moczo & Bard
(1993) concluded that the effect of impedance contrast on the amplitude of
amplification is dominant. It affects both translational and differential motion in
approximately the same way. The existence of a gradient amplifies the direct waves
and makes the Love waves slower, but nevertheless all those effects are relatively
small. The effect of damping is proportional to the ratio between travel distance and
wavelength and affects the Love waves much more than the direct wave. Because of
this it is of particular importance for the differential motion. The main conclusion,
however, is, that a very sharp peak can be found in differential motion close to the
lateral discontinuity, regardless of different values of the mechanical parameters. The
amplitude of this peak is controlled mainly by the impedance contrast and only
slightly by the presence of a velocity gradient in the layer. Different values of
attenuation only affect the decay of this peak but not the amplitude.
11
Even if this model is very simple (“step” like shape of contact, plane SH-wave vertical
incidence…) it gives an insight into the basic phenomena that occur near a strong
lateral discontinuity.
c) Shallow sedimentary valleys generating surface waves
Seismologists had recognized the effect of increased damage along alluvial valleys
or sedimentary basins many years ago. Many studies (a review of which is given in
Gutenberg 1957) were performed, compiling data on damages from great
earthquakes and instrumental records, resulting in the qualitative statement that
sedimentary basins are generally exposed to greater amplification of displacement,
longer duration of the ground motion and usually very complex wave-fields.
Generally one may expect different effects in sedimentary valleys: First,
displacement amplification due to transition of the wave-field from a more rigid
medium to a softer medium. Second, the classical 1D vertical resonance described
above. Third, influence of the lateral geological heterogeneities and topography.
Because of the scarcity of appropriate data Bard & Bouchon (1980) decided to
theoretically compute these effects using the Aki-Larner method. First they tested the
method on a cosine-shaped valley by comparing it with the finite-difference (Boore et
al. 1971), the finite-element (Hong & Kosloff) and the Glorified Optics (Hong &
Helmberger, 1978) methods, and it showed reasonable good agreement.
After the verification of the applicability of the method, Bard & Bouchon (1980)
investigated different configurations of shallow sedimentary valleys. They considered
two shapes of a valley: A one cycle cosine-shaped valley and a plane layer bounded
on each side by a half-cycle cosine-shaped interface. (Fig.8)
For those models they computed the response to vertical and oblique
incidence of a plane SH-wave. The incoming wave was represented by a Ricker
wavelet. For detailed parameters of the velocities, densities and wavelet see Bard &
Bouchon (1980).
Fig.8
Models used by Bard & Bouchon for their study of site effects in shallow sedimentary basins.
Type 2 is a plane layer with half-cycle cosines-shaped borders and Type 1 is a one cycle
cosine-shaped valley. (Reproduced from Bard & Bouchon 1980)
They performed a parameter study accounting for high and low impedance
contrasts, for different incident angles and for different geometry parameters, i.e.
different h/D values. Here h is the maximum valley depth and D is the half width.
At first Bard & Bouchon concentrated on vertical incidence on high-impedance
valleys. For Type 1 valleys they observed two phenomena. In the beginning of the
12
record they found 1D vertical resonance, but with increasing time they saw the
generation of a wave disturbance at the edge of the valley and its lateral
propagation towards the other edge. The phase velocities observed were in good
agreement with the ones of the fundamental Love wave in a flat layer with
thickness equal to the maximum depth of the valley.
Also the evolution of displacement with depth at the centre of the basin shows
the characteristics of fundamental Love wave modes. These Love waves are
excited as soon as the frequency of the incoming wave reaches the first resonant
frequency of the equivalent flat layer (Eq.6 with n=0).
Bard & Bouchon (1980) originally wanted to show the focusing effect of a
cosine-shaped valley, but this focusing does not affect direct arrival of the incident
signal but acts basically on the generation of Love waves on the edges.
As this surface wave reaches the other edge of the high-impedance contrast
valley, it is reflected and travels backwards. The amount of energy reflected
seems to increase with increasing mean slope of the interface.
This interesting result can even better be seen in the Type 2 valleys (Fig.9) as
may be expected because four fifths of the valley are now of uniform thickness,
which favours pure Love waves to be generated and to propagate over the whole
width of the basin. In the simulation, the first three modes of Love waves were
excited and simultaneously the first three resonant frequencies of the equivalent
flat layer case could be observed. Love waves are generated at each slope of the
basin by deflection of the incident, vertical propagating wave towards the
horizontal direction by the oblique interface. Subsequently they propagate back
and forth within the valley, being reflected at the valley edges.
The amplitude of the frequency response remains relatively constant over the
whole valley because of the constant sediment thickness. The peak amplitudes of
ground motion are biggest in the centre of the valley where symmetrical Love
wave trains meet each other.
Comparing Type 1 and Type 2 valleys Bard & Bouchon (1980) found out that
the Love wave amplitude is bigger in cosine-shaped valleys whereas the longer
travel path and the better trapping of Love waves in the “flat layer” valleys
prolongs the duration of ground motion in the latter models.
By simulating models with lower impedance contrast Bard & Bouchon (1980)
found principally the same phenomena, although slightly lower in amplitude and
duration. At the edges they found parasitic signals outside the valley due to the
inefficient trapping of Love waves in the case of low impedance contrast.
At last they simulated the effect of oblique incidence. Here they recognized
only one significant difference. The nonzero horizontal wave number of the
incoming signal favours the Love wave generated at the edge first reached by the
incident wave resulting in peak amplitude about 30 per cent greater than that for
the vertical incidence. Obviously, also the meeting zone of the two Love wave
trains is shifted towards the edge of the basin first hidden by the incoming signal.
13
Fig.9
Example of the response of a “flat layer” valley (Type 2) to an incident vertical SH- Ricker
wavelet with characteristic period of t=0.732. The valley’s shape ratio h/D is 0.1 in this case.
a) Displacement at surface receivers spaced from 0 to 6.2 km from the valley centre. Bottom
trace is the incident wavelet. b) Space (x) – time (t) evolution of surface displacement for the
receivers (indicated by dots) in a. The 1D vertical resonance and the multiple reflected Love
waves can easily be seen c) Depth distribution of displacement at a receiver in the very centre
of the valley. Because of the symmetry of the problem only half of the valley is shown
(Reproduced from Bard & Bouchon, 1980)
d) Deep sedimentary valleys showing global 2D resonance
One may intuitively think that the generation of surface waves and their propagation
back and forth can also be observed in deeper valleys. In 1984 King and Tucker
performed an instrumental study in the Garm region (former USSR) where they
measured the response of the Chusal Valley, a small, sediment-filled valley, to
regional and teleseismic events, having acceleration values from 10-5 to 10-3 g. They
observed significant features that did not depend on the earthquake hypocenter or
source characteristics.
Unexpectedly large, frequency dependent amplification (about a factor of 5)
across the whole width of the valley was measured. A spectral peak at 2-3 Hz that
grew smoothly going from the edge to the valley centre was observed. The amplitude
but not the frequency of the peak changed depending on the position within the
valley. This feature could be observed in all three component recordings, although at
different frequencies. The frequency of the gravest response peak was different from
that for a 1D model of vertically reflected P- or S-waves with layer thickness of the
deepest part of the valley.
Concluding all those observations it can be said that deep sedimentary basins
exhibit resonance patterns with specific characteristics:
i)
the frequency of the gravest peak is the same in the whole valley,
regardless of the sediment thickness.
14
ii)
the amplification is biggest in the centre of the valley, decaying smoothly
from the centre to the edges.
iii)
the ground motion is in phase across the whole valley at this frequency.
(Bard and Tucker, 1985)
These experimental results motivated Bard & Bouchon (1985) to investigate these
effects numerically, using the Aki-Larner technique. They computed the spectral
transfer functions for sine-shaped valleys of different shape ratio. (Maximum depth
divided by valley half-width, i.e., h/l). The amplitude of the gravest peak depends
strongly on the shape ratio and is biggest for 0.4, where it is 3.5 times larger than that
predicted by a 1D model. The frequency of this gravest peak, which is relatively close
to the value for the fundamental 1D resonant frequency, decreases as the valley
depth increases.
As next step they computed the time-domain response of a valley with shape
ratio of 0.4 and observed some striking features for a vertically incident plane SH,
transient, quasi- monochromatic signal at the frequencies f0=1.31 fh (where fh is the
1D fundamental frequency) and f1=2.3 fh. Results are shown in Fig.10.
Fig.10
Time domain response of a sine-shaped valley, excited at the frequencies of the first two SH
resonance modes. The parameters of the valley are displayed at the bottom. Each column
represents the surface ground motion at locations regularly spaced from the centre (x/l=0) to the
edge of the valley. Fundamental mode is shown left and first higher mode right. The vertical bar
on the left of each seismogram corresponds to an amplification range of [-1,+1]. (Reproduced
from Bard & Bouchon, 1985)
What can be seen in Fig.10 is that the fundamental model (f0=1.3 fh) concerns mostly
the central part of the valley and is in phase across the whole valley. The amplitude
of amplification is about 8 times larger at the centre and decreases regularly to the
edges. Time duration is large (partly owing to the fact that damping is zero in this
case).
15
The first higher mode shows displacement nodes at x/l=+-0.16 and three
amplitude maxima with large amplification (up to 5.0). This mode affects a wider area
of the valley and its frequency is not corresponding at all to the first higher mode of
1D resonance.
To explain this 2D resonance phenomenon the similarity to the shallow
sedimentary valley has to be considered. In the vertical direction there are up and
down going waves that are trapped inside the valley, comparable to the 1D case. At
the edges of the valley surface waves are generated and propagate laterally through
the valley. In the case of shallow sedimentary valleys these effects are separated in
time. If, however, the valley width and thickness are comparable to the wavelength of
lateral waves, generated at the valley edge, the result is lateral interference. These
lateral interferences, together with 1D vertical interferences, produce specific 2D
resonance patterns in deep sedimentary valleys.
As next step Bard & Bouchon (1985) extended their models to the P and SVwave incidence, and observed qualitatively very similar results, despite the fact that
resonant frequencies were different and peak values could be observed at shape
ratios different from that for the SH-wave incidence.
Compiling all results they showed the existence of specific 2D resonance
patterns in relatively deep sedimentary valleys. These patterns can be classified in
three categories (Fig.11):
i)
ii)
iii)
Anti-plane shear SH resonance mode.
SV resonance, which is an in-plane shearing pattern and may be
thought of as rocking of the whole valley around its central axis.
P resonance that is a succession of expansions and contractions, which
may be thought of as “respiration” of the valley.
Fig.11. The three fundamental modes for the corresponding critical shape ratio of a sine-shaped valley,
displayed as motion along the surface and along a vertical line in the valley centre, at two times
of maximum motion t0 and t0+0.5/f0(solid lines). In-plane bulk mode (top), in-plane shear mode
(middle) and anti-plane shear mode (bottom).(Reproduced from Bard & Bouchon, 1985)
16
Subsequently, Bard & Bouchon (1985) performed a parameter study to analyse the
sensitivity of this global 2D resonance phenomena to parameters like impedance
contrast, damping, excitation and geometrical parameters.
They found out that the amplification increases with growing impedance
contrast, but the peak value always occurs at the same frequency. The effect of
damping is similar. Damping reduces the amplitude (especially of higher harmonics)
and the signal duration, but does not affect the resonant frequency. Oblique
incidence of the incoming signal can favour the excitation of particular modes of 2D
resonance. However, the frequencies of each mode are not affected at all by the
value of the incidence angle.
Moczo et al. (1996), inspired by the realistic geological conditions beneath the
Collosseum in Rome (Moczo et al., 1995), extended the approach of Bard &
Bouchon by considering not only a deep sediment valley, embedded in a
homogenous, relatively hard half-space, but also models of deep valleys with
different geological parameters embedded in a medium with a horizontal surface
layer (with different geological parameters) and a model of a surface layer with a
through at the bottom of the layer.
They used a finite-difference algorithm on a combined rectangular grid for the
SH-waves to compute ground motion for a parabolic shape valley-base interface,
lying in a horizontal surface layer (Moczo et al., 1996). As input signal a Gabor
wavelet was used.
In their study they found 2D resonance, as had been expected, for the model
of a valley in stiff medium. What, however, seemed very surprising, was that this
global resonance was excited even below an existence value that had been
proposed by Bard & Bouchon (1985). The presence of a surface layer, the thickness
of which is not bigger than half the maximum valley depth, does not change
significantly the global resonance effect, caused by the deep sedimentary valley.
When they considered models that consisted of a surface layer with a through
at its bottom (see Fig.12), Moczo et al. (1996) found that this setting gives rise to the
fundamental mode of the 2D resonance. The resonant frequency, spectral
amplification and maximum time domain differential motion turned out to be close to
that of a closed valley imbedded in a homogenous medium. Thus, a through at the
bottom of a horizontal layer exhibits more the response characteristics of a deep
sedimentary valley (large amplification at the centre of the valley, in-phase motion
over the whole valley, long duration, large differential motion) than 1D vertical
resonance, which is, or at least should be, an important result for engineering
practise, since 1D models are the most used even today.
17
Fig.12
Representation of the 2D fundamental resonance mode in a layer with a through at the bottom
(sketch at the bottom) in a higher (100m/s) and a lower (200m/s) velocity contrast model. Input
signal was a Gabor wavelet (Equation 9) with γ=4, fp=1.23Hz, Ψ=π/2 and γ=6.5, fp=2.44Hz,
Ψ=π/2 respectively. (Reproduced from Moczo et al. 1996)
Finally it remains to mention, what marks the border line between shallow and deep
sedimentary valleys. Although being fuzzy, this border line can be drawn by a rule of
thumb:
valleys with a shape ratio of h/l < 0.2 can be regarded as shallow valleys (with
the corresponding wave phenomena), whereas valleys with h/l > 0.3 can be classified
as deep valleys.
2.6.
Methods of investigation of site effects
Since we know the importance of site effects as main reason for anomalous seismic
ground motion and corresponding damage distribution, it is of big significance for
land use planning or design of critical facilities to account for site effects during future
earthquakes in certain regions.
There are several techniques to achieve this goal:
a) Measuring small earthquakes
b) Measuring seismic ground motion generated by artificial sources (e.g.
explosions)
c) Analysis of (historical) macro-seismic effects
d) Physical modeling
e) Numerical modeling of earthquake ground motion
f) Measuring and analysing ambient seismic noise
18
Looking at this compilation of methods we can basically distinguish between two
large groups: direct in situ observations and/or measurements and modeling
(numerical or physical) based on available geotechnical data.
The various ground shaking effects have all been proven to be frequency
dependent. Because of this the analysis of macro-seismic intensity distribution,
although containing some information about site effects, seems to be too inaccurate.
The amount of research, done on physically modeling site effects, remains poor.
There are only a few attempts with foam rubber (Anooshepoor & Brune, 1989), or
gelatine water gel (Stephenson & Barker, 1991) models. The main reason not to use
physical models is, apart from the difficulties, arising in the construction of
heterogeneous models with foam rubber, probably an economic one. The costs of
physical models are relatively high compared to a computer run. So it is much easier
to perform parameter studies on computers than with physical models.
As numerical modeling and the use of ambient seismic noise will be
addressed in a separate chapter, the only technique that will be described here in
some detail is instrumental measurement and observation.
The most challenging problem in estimating site response from measurements
is to remove the source and path effects from the instrumental records (Field &
Jacob, 1994). For this reason several methods have been developed, which can be
divided into two large groups, depending on whether or not they need a reference
site, in respect to which the particular effects are estimated (Bard, 1995).
Reference site techniques
The most common procedure consists in comparing records of seismic ground
motion from the site of interest with that of a nearby site, that is believed to be free of
any site effects (i.e., a site on a hard rock) through spectral ratios. The source and
path effects of these two sites should be approximately the same because of the
geometrical proximity of the two sites, which is true at least for sources far enough
away. This technique was proposed by Borcherdt (1970) and is still widely used,
despite the obvious disadvantage that a relatively dense, local array has to be
installed.
Andrews (1986) proposed a generalisation of this method to use it in local or
regional arrays. By solving a large inverse problem both source and path effects are
eliminated simultaneously. Disadvantages of this generalised inversion scheme are
the relatively big number of events needed (Field and Jacob, 1994), the dependce on
the weighting scheme for the least square inversion and the a priori law that has to
be set for the path term (for example 1/r).
19
Non reference site techniques give estimates of site response by using
measurements at a single site, which is advantageous because adequate reference
sites are not always available.
One technique assumes the source and path effects through formulae
providing the spectral shape as a function of a few parameters like corner frequency,
seismic moment, Q factor, which are estimated together with the site response
factors again in a generalised inversion scheme (Boatwright et al., 1991b). This
inversion scheme is generally even more complex than the generalised inversion
approach of reference sites, since the dependence on some parameters is nonlinear.
Another extremely simple technique has been proposed: it just consists of
taking the spectral ratio between horizontal and vertical components of the shear
wave part of seismic records, following a proposal of Nakamura (1989) to use this
ratio on seismic noise recordings. Lermo & Chávez-García (1993) first used this
method and they found significant similarities between classical spectral ratios and
these H/V ratios. Both the resonant frequencies and the corresponding amplitudes
agreed well between both techniques. The H/V spectral ratios also showed
experimental stability and seemed to be little sensitive to source or path effects.
Fig.13 shows a comparison between different techniques of site effect
estimation at a particular site in Oakland, California.
Fig.13. Comparison between different site response transfer function estimation techniques for two
sites in Oakland, California. a) Traditional spectral ratio b) Generalised inversion (GI) scheme
where all data have been given unit weight c) GI spectral ratios when only data with a signalnoise ratio of bigger than 3 are kept d) Parameterised inversion estimates e) Horizontal-tovertical spectral ratios of the S-wave part of earthquakes f) Nakamura’s horizontal-to-vertical
spectral ratio of seismic noise. a) to c) are reference site techniques , d) to f) are non reference
site techniques (Adapted from Field & Jacob, 1994)
20
3.
Ambient seismic noise
3.1.
Introduction
Common sense tells us that the surface of the Earth, the ground we are standing on,
usually is at rest. However, very sensitive instruments show that the contrary is true.
Ground is never standing still. There is always some ground shaking, which is called
seismic noise. While this ambient seismic noise is more or less a hindrance for
accurate applied seismic or geo-technical measurements (signal-to-noise-ratio), it
has been supposed since several decades, especially by Japanese authors (Kanai et
al. 1954, Akamatsu 1961), that seismic noise contains some useful information on
the soil characteristics and on the earthquake response of a site.
This approach has long been mistrusted by “western” scientists, because of
reported discrepancies between earthquake and noise recordings (Bard, 1999), but it
has received renewed attention after the Guerrero-Michoacán event of 1985, where
the information provided by simple, low-cost noise measurements was consistent
with the strong-motion observations. Considering the increased emphasis on
microzonation on one hand, and the small budget available for such studies,
especially in developing and low-to-moderate seismicity countries (like Austria), this
low-cost technique seems attractive, even though the theoretical background is not
unambiguous and no experimental consensus could have been reached yet (Bard,
1999). This is, apart from financial compulsion, because of several reasons: seismic
noise measurements can be performed anywhere and at any time, the instruments
and analysis are simple and seismic noise measurements do not generate any
environmental trouble.
3.2.
Nature of the noise wave-field
A question that, even though it is essential for the right interpretation and, because of
that, for the usefulness of the noise method, is still doubtful is the nature of the noise
wave-field. Kanai (1983) assumed that the noise wave-field mainly consists of
vertically incident S-waves, and is therefore very similar to earthquake signals, which
was turned down by several studies (Aki 1957, Milana et al. 1996, Chouet et al.
1998) that found a large proportion of the noise to consist of surface waves. The
good success of these studies seems an indirect proof that this assumption may be
true.
Speaking about the origin of the noise wave-field, we can distinguish between
long-period (T>1s) and short-period (T<1s) noise. The first one of which is usually
called “microseisms”, whereas the latter one is called “microtremors”. Because
microseisms are caused by ocean waves (action of ocean waves on the coast, nonlinear interaction between ocean waves) at long distances for periods below 0.3 to
0.5 Hz, they are stable over a few hours and well correlated with the large scale
meteorological conditions on the ocean. Microseisms at intermediate periods, from
0.3 to 1 Hz, are generated by coastal waves and by the wind. Because of this their
temporal stability is much smaller (Seo 1997, Kamura 1997, Seo et al. 1996).
Microtremors are mainly generated by artificial sources, e.g. wind-structure
interaction, traffic and vibrations from machines and pumps. They are linked to
human activities and therefore they reflect human cycles. Because of the fuzzy
borderline between microseisms and microtremors probably the best way to
21
distinguish between them, are continuous broad-band measurements. The part of the
record that shows no significant daily amplitude variations is generated by
microseisms (Seo 1996).
Thus on one hand we know relatively much about the origin of the noise, but
on the other hand the question of composition of noise wave-field is still controversial.
For the further practical use of noise recordings it is of essential need to gain further
insight into the noise nature. A decomposition of the noise wave field into body and
surface waves, and among the latter in Love and Rayleigh waves at different sites
with different geological conditions seems the only way to reach this goal.
3.3.
Methods of investigation
Microtremors are used in principally 4 different ways:
a) absolute spectra,
b) spectral ratios between noise records at the site of interest and a reference site,
c) H/V ratios,
d) inversion of the velocity structure through array recordings.
The latter one is rather a geophysical exploration technique than a direct method to
investigate earthquake site effects and therefore will not be addressed here. The H/V
ratios will be discussed in 3.4. The remaining methods will be discussed in the
following.
a) Absolute spectra
The use of absolute noise spectra was first proposed by Kanai (1954). The two
assumptions he had to impose were, that the noise wave-field corresponds to vertical
incident S-waves and that the incident spectrum was white. In such a case noise
spectra would directly reflect the transfer function for S-waves of the surface layers.
Even if only the first assumption was true, the noise wave-field would at least be
similar to that of real earthquakes.
It has been shown by several authors (Aki 1957, Chouet et al. 1998,
Yamanaka et al. 1996, Milana et al. 1996) that neither the first nor the second
assumption is true. There is a significant amount of surface waves in the
microtremors and anthropic noise contains various band-limited components
(machines, buildings, etc.). Nevertheless absolute noise spectra are still used in
different ways. The crudest use of noise spectra consists in attributing a qualitative
“soil index” to sites, depending on the peak frequency of the spectrum. According to
several Japanese scientists (Bard 1999) short predominant periods (T<0.2s) indicate
rather stiff rock, while softer and thicker deposits correspond to larger periods.
Peak frequencies of noise spectra have often been taken as fundamental
resonant frequency of a site. This can only be true, if the site effects at a given site
are big enough to surpass every other effect. In the long period range (T>1s) many
authors (Lermo et al. 1988, Yamanaka et al. 1993, Field et al. 1990) supported this
assumption, especially when the impedance contrast in the investigated area was
large. In this case surface and body waves are trapped, and there is a conspicuous
spectral peak at the resonant frequency, not depending on the origin of the noise.
(Zhao et al. 1996, 1998).
Even if the original assumptions of Kanai (1954) have proven to be wrong,
absolute noise spectra are still in use, providing, under some conditions, reliable
22
information on site effects. Since this technique lacks systematic rules and relies
much on “expert” judgement, it is not accepted in the “western” scientific community.
A sound guideline for potential users and some more research on this topic however
could probably make it more attractive even amongst non-Japanese users.
b) Site to reference site spectral ratios
Similar to the reference site method in site effect estimation (see 2.6.) this technique
is also used with seismic noise, simply by replacing earthquake measurement by
noise measurements. This technique relaxes the assumption of white spectrum, but it
implicitly assumes, that the incoming noise wave-field is the same for site and
reference, and that the wave-field is spatially uniform at least within the area of
interest. Since high frequency microtremors are generated by many different artificial
sources, the assumption of a uniform incoming wave-field becomes more reasonable
for larger periods and smaller distances between site and reference. Owing to this
fact Kudo (1995) reports reliability of this method for periods larger than 1s where
noise origin is the same for both sites. A good correlation with site geology and
theoretical transfer function is reported by Milana et al. (1996) for station pairs
spaced not more than 500m.
3.4.
H/V spectral ratios
In 1971 Nogoshi and Igarashi introduced the H/V ratio in seismology. They measured
three components of seismic ground motion at one site, and then divided the
spectrum of the horizontal part by that of the vertical part. They showed the
relationship of their H/V ratio to the ellipticity curve of Rayleigh wave and thus they
were able to estimate the fundamental resonant frequency, because the lowest
frequency maximum of the H/V ratio curve and the fundamental resonant frequency
of S-waves are very close to each other.
This method was revisited by Nakamura (1989, 1996) who claimed, by making
some semi-theoretical assumptions, that the H/V ratio was a direct estimation of the
site transfer function. This method looked so much attractive because of its simplicity
and cheapness that its application spread all over the world in very short time.
Nevertheless, or because of this very fact, there is an urgent need to critically look at
the assumptions and theoretical background that led Nakamura to his hypothesis.
a) First interpretation of the H/V ratio
Nakamura’s interpretation of the H/V ratio is based on the assumption, that the effect
of surface waves on the H/V ratio can either be neglected or “eliminated”, so that the
result directly corresponds to the transfer function for S-waves (Bard 1999).
Nakamura (1996) separates the noise into body (b) and surface waves (s).
Then the vertical (SNV) and horizontal (SNH) part of the noise amplitude spectrum can
be represented as
SNH(f) = SbH(f) + SsH(f) = Ht(f) . RbH(f) + SsH(f) ,
(10)
SNV(f) = SbV(f) + SsV(f) = Vt(f) . RbV(f) + SsV(f) .
(11)
23
Here Ht(f) and Vt(f) represent the “true” transfer functions for horizontal and vertical
components, respectively, and RbH(f) and RbV(f) is the spectrum of the body wave
part of the noise at a reference site free of any site effects.
The H/V ratio between SNH and SNV (ANHV) can be written after some algebra as
ANHV = [Ht . ArNHV + β . As] / [Vt + β]
(12)
where ArNHV is the H/V ratio of noise at the reference site, β is the relative proportion
of surface waves in the noise measurements of the vertical component, i.e.,
β = SsV(f) / RbV(f) and As is the horizontal to vertical ratio due only to surface waves,
i.e., As(f) = SsH(f) / SsV(f).
Nakamura (1996) then imposes four assumptions:
a) the vertical component is not amplified at fH0
b) the H/V ratio on rock is equal to 1 at fH0
c) β is much smaller than 1 at fH0
d) β . As (fH0) is much smaller than Ht(fH0)
Accordingly he then comes to the final result
ANHV = Ht(fH0)
(13)
If all these assumptions were true whatever the frequency then the H/V ratio of
ambient seismic noise would directly reflect the true transfer function of a site.
Assumption c) and d) seem very controversial even for the fundamental frequency.
Assumption c) may be valid for high impedance contrast sites where SsV(f) vanishes
around fH0.
Assumption d), i.e., β.As (fH0) = β. SsH(f H0) / SsV(f H0) = SsH(f H0) . RbV(f H0) <<
Ht(fH0) implies that the ratio of the horizontal amplitude of surface waves to the
vertical amplitude of body waves at the rock site is small, compared to the true Swave amplification. There is no straightforward reason to admit this (Bard 1999).
b) Second interpretation of the H/V ratio
As a result Nakamura’s explanation, that noise consists mainly of body waves,
seems questionable indeed. While Nakamura assumes most of the noise wave-field
to consist of body waves to justify his H/V ratio, we know from chapter 3.2 that a
relatively large proportion of noise is made up by surface waves. This leads to
another interpretation of the H/V ratio, assuming that noise consists predominantly of
surface waves. Many authors (Nogoshi & Igarashi 1971, Field & Jacob 1993, Konno
& Ohmachi 1998) then agree on the following arguments:
a) Because of the predominance of Rayleigh waves in the vertical
component, the H/V ratio is basically related to the ellipticity of Rayleigh
waves.
b) The ellipticity is frequency dependent, and, because of the vanishing of the
vertical component, corresponding to the reversal of the rotation sense of
the fundamental mode of the Rayleigh wave from clockwise to counter
clockwise, exhibits a sharp peak at the fundamental resonant frequency, at
least for sites with a high enough impedance contrast.
An example of Rayleigh wave ellipticity curve is given in Fig.14
24
Fig.14. Example of ellipticity curve for Rayleigh
waves in a stratified half-space. The H/V
ratio for the first five modes is plotted as a
function of frequency. Peaks correspond to
vanishing of vertical, and troughs to
vanishing of the horizontal component
respectively.
The arguments above can be justified if we think about the noise wave-field as
consisting of Rayleigh waves travelling in a single layer over half-space.
Argument a)
If motion is entirely due to near surface sources, deep sources are neglectable and
microtremor motion at the base of the soil layer is not affected by local sources.
An estimate for site effects in engineering is given by the ratio
SE(ω) = Hs(ω) / Hb(ω)
(14)
which is the Fourier spectrum of horizontal component of motion of body waves on
the surface, i.e., Hs(ω), divided by motion at the base of the layer, Hb(ω). (see chapter
2.6. Reference site techniques). To compensate Eq.14 by the source spectrum
AS(ω), a modified site effect spectral ratio is defined as
SM(ω) = SE(ω) / AS(ω) = [ Hs(ω) / Vs(ω) ] / [ Hb(ω) / Vb(ω) ]
(15)
In a final assumption Hb(ω) / Vb(ω) is set to 1 for all frequencies of interest, which was
experimentally verified with bore-hole measurements by Nakamura (1989). Then
SE(ω) ~ SM(ω) ~ Hs(ω) / Vs(ω)
(16)
which is Nakamura’s ratio (Lermo & Chávez-Garcia 1994).
Argument b)
If we believe noise to consist mainly of Rayleigh waves, we can take a look at the
Rayleigh waves propagating in a single layer over half-space (Konno & Ohmachi
1998). At the ground surface the horizontal and vertical component of the j-th
Rayleigh wave mode excited by a vertical point source L(ω) can be expressed as
(Harkrider, 1964)
25
u’j(ω,r) = [L(ω)/2] ּ [u’/w’]j ּ Aj ּ H12(kj r) ּ ω ,
(17a)
w’j(ω,r) = [L(ω)/2] ּ Aj ּ H02(kjr) ּ ω
(17b)
where u’j(ω,r) and w’j(ω,r) are radial and vertical velocity amplitudes at the surface,
ω is angular frequency, r is distance between point source and observation site,
[u’/w’]j is the H/V ratio at a large distance r as defined by Haskell (1953), Aj is medium
response and kj is wave-number. Because of the similarity to observed spectra of
microtremors the relation
L(ω)
≈ ω-2i
(18)
has been proposed by Konno & Ohmachi (1998). The authors showed that three
different types of particle orbits exist during the propagation of the fundamental mode
of the Rayleigh wave in a layer over half-space, depending on the velocity contrast
between layer and half-space. All types are illustrated in Fig.15. For type 1 (low
velocity contrast, VSH/VSL≤2.5) particle motion is retrograde for all frequencies. In type
2 models with higher velocity contrast (VSH/VSL~2.5) motion changes with increasing
frequency from retrograde to vertical only, then to prograde, vertical only and back to
retrograde. For a very high velocity contrast (VSH/VSL≥2.5) type 3 motion appears.
With an increasing period the particle motion is in order, retrograde, vertical only,
prograde, horizontal only and retrograde. The horizontal only part corresponds to the
singularity in the H/V ratio.
Fig.15
Three types of H/V ratio of
fundamental mode of the Rayleigh
waves: VL is 250m/s for type 1, 200m/s
for type 2 and 50m/s for type 3. VH is
500m/s in all models. (Reproduced
from Konno&Ohmachi, 1998)
If the wave-field consists not only of Rayleigh but also of Love waves, this does not
change the frequency of the first peak because Love waves have no vertical
component to influence the H/V ratio at the fundamental frequency. Because the Airy
phase of Love waves appears at a frequency very close to the fundamental S-wave
resonant frequency, Love waves, in fact, even strengthen the amplitude of the peak.
(Konno & Ohmachi, 1998). On the other hand P- and SV-waves do consist of vertical
components and their appearance in the noise wave-field may prevent such an
interpretation as stated above.
26
If, however, the assumption that noise consists of surface waves was true, one
could ask what the H/V ratio may tell us about the amplitude and frequency
behaviour of the transfer function of a certain site. Speaking first about the spectral
information we can gather from the H/V ratios, it is clear that we cannot get more
than the fundamental frequency peak, because as can be seen in Fig.14 the
fundamental mode of Rayleigh waves is the only one for which the vertical
component is really zero leading to a sharp peak. For all other modes there is a lower
order mode with not vanishing vertical component at their peak frequency.
Considering the impedance contrast threshold above which the peaks are
conspicuous, the significant value varies from paper to paper but a threshold of 3
should be reasonable (Bard 1999). Konno & Ohmachi (1998) in their study also
analysed the trough which in many noise records appears at almost twice the
fundamental frequency. This trough corresponds to the vanishing of the horizontal
component and exists even for a lower impedance contrast.
The other important question concerns the amplitude of the H/V peak. Since
the ellipticity peak for a vanishing vertical component should theoretically be infinite,
there cannot exist a relation between the fundamental S-wave resonant frequency
and the peak of the ellipticity curve. Nevertheless some authors (Konno & Ohmachi
1998, Miyadera & Tokimatsu 1992, Chouet et al. 1998) tried to correlate the
amplitude of the ellipticity peak to the peak of the transfer function at the fundamental
frequency. Their results are not consistent and depend on their assumption of
proportion of Rayleigh and Love waves in the noise wave-field.
As an example of the possible use of the H/V ratios, Fig.16 shows a
comparison between the H/V ratios of microtremors, the transfer functions of Swaves and the H/V ratios of the fundamental and first higher modes of Rayleigh
waves. It clearly can be seen that many (depending on the site) curves correlate
satisfactorily at least for the position of the first peak (however not as good for the
amplitude).
3.5.
Partial conclusive remarks
27
Fig.16
Examples of ellipticity curves
for a collection of realistic
profiles. Observed H/V ratios
of microtremors (thick solid
lines +-standard deviation),
the transfer functions of Swaves (thin solid lines) and
the H/V ratios of the
fundamental and first higher
modes of Rayleigh waves
(thick respectively thin dotted
lines) are compared.
Reproduced from Konno &
Ohmachi, 1998
To conclude this small overview on ambient seismic noise some points stated by
Bard (1999) will be quoted and summarised in which he infers some rather well
established conclusions.
-
Experimental results show that the H/V ratio does point out the fundamental
frequency of soft soils, whatever the theoretical background.
-
It is easier to measure the H/V ratio than absolute or site to reference site
spectra, especially in geological environments with low impedance contrast.
-
Experimental results yield differences between the H/V curve and the true site
amplification function for incident S-waves, in particular the H/V ratio does not
give estimates about the bandwidth over which seismic ground motion is
amplified.
-
Up to now there is no generally accepted theoretical or experimental link
between the peak H/V amplitude and the maximum spectral amplification.
-
Based on empirical and instrumental studies the inequality
A0NHV ≤ AHp
(19)
has been established, where A0NHV represents the peak amplitude of the
spectral ratio between the horizontal and the vertical noise spectrum and AHp
is for the maximum site amplification. This however might not be true without
exceptions and has to be used with great caution.
4. Numerical modeling of seismic ground motion
4.1.
The role of numerical modeling / simulation
The word simulation in science has the meaning of replacing an, in most cases very
complex, process, which one wants to investigate, by another, more simplified one,
where all parameters and conditions can be set by the simulator. The environment in
which a simulation takes place is a model. No need to say, the simulation should
have some relation to the real process it replaces, and so the model should on one
hand be rather simple and easy to understand and on the other hand it should not be
too far away from complex reality.
Some decades ago computing power was very expensive, and programs were
not as sophisticated as today. Therefore many simulations were based on physical
modeling. Some examples for that are wind channel tests, or foam rubber models.
Besides the fact that physical models are still useful and valuable, nowadays, as
computing capabilities are easier to be accessed, numerical modeling plays a bigger
and bigger role in all science. This is likely to correspond to the fact, that it is probably
less complicated to develop one single computer code for a model with which then
many different settings of parameters can be computed, than to build a new physical
model, e.g. made of foam rubber, for every single parameter, that is changed.
28
Especially in seismology the importance of modeling and simulation is very
high and still increasing, since the real process in this field, the earthquake, is
disastrous and its appearance is stochastic. Thus seismologists do not want to wait
for a real earthquake to take place, but they want to simulate earthquakes and their
effects on the computer.
Early studies in the seventies of the 20th century considered only a onedimensional model, where material parameters only changed with depth and that
were laterally homogeneous. These simple models are still used routinely in
engineering practice (Bard 1995), but can not account for the complex
heterogeneous three dimensional geological and geometrical situations we observe
in nature. So during the last decades many different methods and techniques have
been developed to account for very complex structures and rheologies.
In the following, a short introduction to the field of numerical modeling of seismic
ground motion together with a review over the different methods for solving
numerically the equation of motion and a concluding statement on the advantages
and restrictions of numerical modeling will be given.
4.2.
Review of methods
The main and fundamental problem in seismology is to solve the equation of motion,
which in its most general form reads
ρui,tt = τij,j + fi
(20)
∂ ² τij
∂ ²ui
τ
=
ij,j
ui,tt =
∂x j ² .
∂t²
where
and
ui is displacement, ρ is density, τij is the stress tensor and fi is a force term and
Einstein’s summation convention for repeated subscripts i and j is assumed.
In an isotropic medium the stress tensor is given by Hooke’s law:
τij = λ uk,k δij + µ ( ui,j + uj,i )
where
λ = λ(xi) and µ = µ(xi)
ui,j =
∂ ui
∂x j
;
i,j ∈ {1,2,3}
(21)
are Lamé’s elastic coefficients and
.
From Eq.21 it is clear that stress tensor τij is symmetric: τij = τji
Dozens of different methods have been developed for solving the equation of motion.
Generally they can be divided into two groups:
a) Exact (analytic)
b) Approximate
29
Exact methods are applicable only in the case of a homogeneous medium or in
simple heterogeneous models - e.g., 1D vertically heterogeneous models or
spherically symmetrical models. Separations of variables or matrix methods are
typically applied.
Approximate solutions, on the other hand, can be classified as high-frequency
or low- frequency methods. High-frequency methods are crucially important in
structural seismology and in seismic oil exploration. The most important method is
the ray method (or asymptotic ray theory – ART). For the simulation of earthquake
ground motion low-frequency methods are by far more important. Low frequency
methods can be grouped into three classes:
•
•
•
domain methods (e.g., Finite difference method – FDM, Finite element
method – FEM, Spectral element method – SPEM , ...)
boundary methods (e.g., Boundary integral equation method – BIEM,
Boundary element method – BEM ...)
hybrid methods (e.g., Finite difference-Finite element method FD – FEM,
Discrete wavenumber-Finite difference method DWN –
FDM, Alexeev-Mikhailenko method A-M M ...)
Despite the fact that boundary methods are generally more accurate than domain
methods, they are practically applicable only to models with two or three
homogeneous layers or blocks, because computer memory and time requirements in
the case of more complex models are too large.
Domain methods are generally less accurate than boundary methods but allow
computing seismic motion in relatively complex models. Because of this, finitedifference methods play a dominant role in the recent modeling of earthquake ground
motion.
Hybrid methods are designed to overcome the limitations or drawbacks of
individual methods. They combine two or three methods and utilize one particular
method to solve dependence on some of the independent variables and another
method to solve dependence on the remaining independent variables. They can also
be used to compute the wave-field in one part of the model by one method and the
remaining part of the computational region is treated by another method. Therefore
they are usually more computationally efficient but imply a more difficult algorithm.
4.3.
Advantages and disadvantages of numerical modeling
The basic advantage of numerical modeling of seismic ground motion is obvious: if
the knowledge of geological and geometrical parameters is sufficient to build a
reasonable model that is not too far away from reality, and if the method that is used
to compute seismic ground motion within this model is stable and accurate for a
given wave-field, then numerical modeling provides a valuable tool to estimate
seismic ground motion in a certain area.
With an accurate model and method it is possible to perform
phenomenological and parameter studies, that can show the sensitivity of ground
shaking to a change in certain parameters like incident wave-field (S-, P-SV-,
Rayleigh, Love wave incidence with different incident angles), or distribution of
parameters like impedance contrast, damping, etc. In addition to that, numerical
modeling allows an estimation of the uncertainty in the response of a given site, given
30
the uncertainty of input geometrical and mechanical parameters. Indeed, these
techniques have revealed many physical processes and have led to a significant
breakthrough in the understanding of site effects during the last two or three decades
(Bard, 1995). Processes like the site effects described above (e.g. 2D resonance in
deep sedimentary valleys, generation of surface waves at material discontinuities
etc.) would never have been discovered without the use of numerical modeling. In
addition to that, numerical models also provide useful information for the planning of
field experiments, which can be designed in order to measure the specific effects
predicted by numerical modeling.
Nevertheless, the routine use of numerical modeling for the a priori estimation of site
effects raises several questions:
-
The most important feature of numerical modeling is that it can predict seismic
ground motion at a site, without waiting for a seismic event (like all
instrumental methods of estimating site effects do). However there is, until
recently, relatively weak evidence that this statement is true. This is because
most of the numerical site effect studies have been performed a posteriori,
which means that the “predictors” already knew, from instrumental
measurements, what they had to find.
-
All of the different methods can only be used in their validity domain. A model
has to be capable of predicting and computing all possible effects. For
example a 1D model is totally unable to show 2D resonance patterns of deep
sedimentary basins or the generation of surface waves at lateral
discontinuities.
-
Another important restriction concerns the funding of such numerical
simulations. Even if the computations on modern standard computers may be
quite cheap, intense geotechnical measurements have to be performed, in
order to get detailed geophysical information about the site underground and
“neighbourhood” to construct a proper model.
5. The finite-difference method
(Following lecture notes of Moczo, 1998)
5.1. The principle
The finite-difference method is a tool for approximately solving partial differential
equations (like the equation of motion). Approximate solutions are required because
analytical methods do not provide solution of partial differential equations for complex
or sufficiently realistic, heterogeneous models.
The application of the FD-method consists of
a) Construction of a discrete finite-difference model of the problem:
the computational region has to be covered by a grid
31
-
the derivatives of the differential equation have to be approximated by
the finite-difference formulae at the grid points
the functions involved in the differential equation have to be
approximated at the grid points
initial and/or boundary conditions have to be approximated
construction of a system of the finite-difference (i.e., algebraic)
equations
b) Analysis of the finite-difference model:
consistency and order of approximation
stability
convergence
c) Numerical computations
Grid
Computations are performed in a domain D = DI ∪ DB , where DI denotes the interior
and DB the boundary of the domain respectively. This domain is lying in the four
dimensional space of variables (x,y,z,t). Cover this space by a grid of discrete points
(x i ,y k ,zl ,t m )
given by
xi = x 0 + i∆x ,
yk = y 0 + k∆y ,
zl = z0 + l∆z ,
t m = t 0 + m∆t
i,k,l = 0, ±1, ±2,...,.
m = 0,1,2,...
∆x, ∆y and ∆z are called grid spacings and ∆t is called time step, since t usually
represents time. If we consider a Cartesian coordinate system, the corresponding
spatial grid is a rectangular grid.
A function u(x,y,z,t) is to be approximated at the grid points by a grid function
m
m
m
U(xi,yl,zk,tm). Denote a value of u(xi,yl,zk,tm) by uikl and approximation of uikl as Uikl .
Depending on the problem under consideration the most appropriate spatial grid
should be chosen. In many applications the rectangular grid with the grid spacings
∆x = ∆y = ∆z = h will be a reasonable choice, while other types of grids will be
chosen, if they better accommodate the geometry of the problem or if they simplify
the finite-difference approximations of derivatives.
Approximation of Derivatives
Let us consider a function Φ(x). We can use Taylor’s expansion of the function to
derive different approximations of the first and higher derivatives of the function.
Denoting h as grid spacing ∆x, we can write Taylor’s expansions of the function Φ at
x+h and x-h as
1
1
Φ(x + h) = Φ(x) + Φ '(x)h + Φ ''(x)h² + Φ '''(x)h³ + ...,
2
6
(22)
32
Φ(x − h) = Φ(x) − Φ '(x)h +
1
1
Φ ''(x)h² − Φ '''(x)h³ + ...
2
6
(23)
From Eq.22 we get
Φ(x + h) − Φ(x) = Φ '(x)h +
1
1
Φ ''(x)h² + Φ '''(x)h³ + ...,
2
6
1
Φ '(x) = [Φ(x + h) − Φ(x)] − O(h)
h
and subsequently the forward-difference formula
1
Φ '(x) [Φ(x + h) − Φ(x)]
h
.
(24)
From Eq.23 we get
1
1
Φ(x) − Φ(x − h) = Φ '(x)h − Φ ''(x)h² + Φ '''(x)h³ + ...,
2
6
1
Φ '(x) = [Φ(x) − Φ(x − h)] + O(h)
h
and approximation of derivative as backward-difference formula
1
Φ '(x) [Φ(x) − Φ(x − h)]
h
.
(25)
By subtracting Eq.22 and Eq.23
Φ(x + h) − Φ(x − h) = 2Φ '(x)h² +
Φ '(x) =
2
Φ '''(x)h³ + ...,
6
1
[Φ(x + h) − Φ(x − h)] − O(h²)
2h
we obtain the central-difference formula
1
[Φ(x + h) − Φ(x − h)]
Φ '(x) 2h
(26)
Eq.24 and Eq.25 are approximations of first order since the leading term of the
approximation is proportional to h, whereas Eq.26 is a second-order accurate
approximation.
A second-order accurate approximation of the second derivative can be obtained by
summing up Eq.22 and Eq.23.
1
Φ ''(x) [Φ(x + h) − 2Φ(x) + Φ(x − h)]
h²
(27)
Finite-Difference Scheme
Consider f(P) as a function defined on DI. If L(u) represents a linear operator, then
(28a)
L(u(P)) = f(P)
;
P∈DI
33
denotes a linear partial differential equation for unknown u(P).
Initial and boundary conditions can be represented by the equation
P∈DB .
B(u(P) = g(P) ;
(28b)
Consider problems for which a unique, smooth and bounded solution u exists for any
data in some class of smooth functions {f,g}.
I
B
We can denote grid points interior to D by D∆ , and boundary grid points by D∆ . Let U
be a solution of the system of finite-difference equations (or finite-difference
scheme)
L ∆ (U) = f(P) ;
P ∈ DI∆
(29a)
P ∈ DB∆
B∆ (U) = g(P) ;
(29b)
Eq.29a,b can be considered as finite-difference approximation to Eq.28a,b and U as
finite-difference approximation to u respectively.
If we want the finite-difference scheme to be accurate, U has to be close to the
solution u at the corresponding grid points for all data that are sufficiently smooth and
U has to be uniquely defined by the scheme (29). In order to give a reasonable
approximation to the solution of the partial differential equation and adjoined
initial/boundary condition, some properties have to be set on the finite-difference
scheme.
Properties of a Finite-Difference Scheme
It is reasonable and fundamental for the finite-difference scheme to be consistent,
convergent and stable in order to give good approximation to the solution of the
partial differential equation.
Let Φ(P) be any smooth function in D. For each such a function a local truncation
error can be defined:
τ{Φ(P)} ≡ L(Φ(P)) − L ∆ (Φ(P)) ; P ∈ DI∆
(30a)
B
β{Φ(P)} ≡ B(Φ(P)) − B∆ (Φ(P)) ; P ∈ D∆
(30b)
The difference problem (29) is consistent with the problem (28) if
τ(Φ ) → 0
(31a)
and
β(Φ ) → 0
(31a)
for
∆x → 0,..., ∆t → 0 ,
where
stands for the appropriate norms.
The difference problem is conditionally consistent with the problem (28), if relations
(29) are satisfied only when a certain relationship among ∆x,..,∆t is satisfied.
34
The solution of problem (28) is convergent to the exact solution u if
u(P) − U(P) → 0 ;
P∈D
(32)
for
∆x → 0,..., ∆t → 0 .
If the difference solution is convergent for all data in some class of smooth functions
{f,g}, the corresponding finite-difference scheme is convergent.
A scheme that is determined by linear difference operators L∆ and B∆ is stable if
there exists a finite positive quantity K, independent of the grid spacings, such that
U ≤ K( L ∆ (U) + B ∆ (U) )
(33)
for all grid functions U on D.
If Eq.33 is valid only for some restricted family of grid spacings in which ∆x,..,∆t may
be arbitrarily small, the scheme is conditionally stable.
Finally an important theorem, connecting consistency, stability and convergence of
the finite-difference schemes, reads:
Let L∆ and B∆ be linear difference operators which are stable and consistent with L
and B on some family of grids in which ∆x,..,∆t may be arbitrarily small. Then the
difference solution U of (29) is convergent to the solution u of (28).
5.2.
Finite-difference approximations
Fourth-order Finite-difference Approximations
Up to this point we have used only a second-order approximation of the spatial
derivatives (Eq.22-27). Now we will consider the more accurate fourth-order
approximation. Its approximation error is considerably smaller than that of the second
order since it is proportional to h4. As a consequence, the fourth-order finitedifference schemes allow reducing the spatial sampling of the wavelength which is to
be propagated without grid dispersion. While second-order schemes require at least
10 samples per wavelength, the fourth-order schemes only require 5 samples.
We want to find a fourth-order approximation of the derivative Ψx. Taylor’s expansion
for Ψ(x+h/2) and Ψ(x-h/2) are
1
1
1
Ψ(x + 21 h) = Ψ(x) + hΨ '(x) + h² Ψ ''(x) +
h³ Ψ '''(x) + ...
2
8
48
(34a)
1
1
1
Ψ(x − 21 h) = Ψ(x) − hΨ '(x) + h² Ψ ''(x) −
h³ Ψ '''(x) + ...
2
8
48
(34b)
and
Subtracting Eq.34b and Eq.34a leads to
35
Ψ(x + 21 h) − Ψ(x − 21 h) = hΨ '(x) +
1
h³ Ψ '''(x) + O(h5 )
24
(34c)
In order to remove the Ψ’’’-term we need an independent approximation of Ψ’ which
includes the Ψ’’’-term. This can be obtained from Taylor’s expansion for, e.g.
Ψ(x+3h/2) and Ψ(x-3h/2),
3
9
9
Ψ(x + 32 h) = Ψ(x) + hΨ '(x) + h² Ψ ''(x) + h³ Ψ '''(x) + ...
2
8
16
(35a)
3
9
9
Ψ(x − 32 h) = Ψ(x) − hΨ '(x) + h² Ψ ''(x) − h³ Ψ '''(x) + ...
2
8
16
(35b)
and
We subtract Eq.35b from Eq.35a and arrive at
9
Ψ(x + 32 h) − Ψ(x − 32 h) = 3hΨ '(x) + h³ Ψ '''(x) + O(h5 )
8
(35c)
Next we multiply Eq.34c by 9/8 and subtract Eq.35c multiplied by 1/24 and obtain
Ψ '(x) =
1
1
9
{− [ Ψ(x + 32 h) − Ψ(x − 32 h)] + [Ψ(x + 21 h) − Ψ (x − 21 h)]} + O(h5 )
h 24
8
Let x=xi.
Then x+3/2h = xi+3/2 ,
x-3/2h = xi-3/2,
x+1/2h = xi+1/2,
(36)
x-1/2h = xi-1/2 .
Approximation Eq.36 can now be written as
Ψ x |i 1
1
9
{− [Ψ i+ 3 − Ψ i− 3 ] + [Ψ i+ 1 − Ψ i− 1 ]}
2
2
2
2
h 24
8
Let x=xi+1/2.
Then x+3/2h = xi+2 ,
x-3/2h = xi-1,
x+1/2h = xi+1,
(37)
x-1/2h = xi .
Now approximation Eq.36 can now be written as
Ψ x |i +1/ 2 1
1
9
{− [Ψ i+ 2 − Ψ i−1 ] + [ Ψ i+1 − Ψ i ]}
h 24
8
(38)
Approximation of traction-free surface
For solving differential equations we have to provide initial and boundary conditions.
One boundary for which we have to set a boundary condition is earth’s free surface.
K
Consider a horizontal surface at z=0 and let n = (0, 0, -1) be a unit normal vector to
this surface. Furthermore we consider the surface to be a traction-free surface, i.e.
K
T(n) = 0 .
36
Since
Ti = τijn j
we have
τ31 = 0 ,
or
τzx = 0 ,
τ32 = 0 ,
τ33 = 0 ,
τzy = 0 ,
τzz = 0 .
Consider for simplicity only the P-SV case and the second-order velocity-stress finitedifference scheme. Then
τzx = µ(uz + w x ) = 0
(39a)
τzz = λ(ux + w z ) + 2µw z = 0 ,
(39b)
and
which we can write as
and
τzx = 0
:
uz = -wx
or
uz = -wx
τzz = 0
:
wz = -(λ/(λ+2µ) ux
or
wz = -(λ/(λ+2µ) ux
(40a)
.
(40b)
xx
zz
Localize the free surface so that Ti0 ,Ti0 and Ui+1/2 0 are located on the free surface.
We have to update U, Txx, W and Txz at the same time to assure that Tzz = 0 and
Txz = 0 on the free surface.
Recalling Eq.38ab, in the P-SV version, omitting the body force term, i.e.
ρut = τxx,x + τxy,y + τxz,z
ρwt = τxz,x + τzz,z
+
λ wz
τxx,t = [λ +2µ] ux
+
[λ +2µ] wz
τzz,t = λ ux
τxz,t = µ ( uz +wx )
(41a)
(41b)
(41c)
(41d)
(41e)
we find no problem with the x-derivatives, but we have to look at the approximation of
the z-derivatives.
In Eq.41c we can replace wz by -(λ/(λ+2µ) ux according to Eq.40b and solve for
xx
Ti0 .
Since τzz=0 on the free surface, we do not need Eq.41d. We simply prescribe
zz
i0
T
for all time levels.
In Eq.41a we need to find an approximation to τxz,z. We will extend the grid
above z = 0 (i.e., l = 0) and image τxz as an odd function with respect to z = 0
(Levander, 1988). Then obviously τxz=0 at z=0. Then
Ti+xz1,− 1 = −Ti+xz1,+ 1
2
2
2
2
and
37
τ xz,z |z =0 2 xz
T 11
h i+ 2 , 2 .
There is no problem to approximate Eq.41b and Eq.41e since both W and Txz are
localized half grid spacing below the free surface.
Summarizing, the scheme to update Txx, Tzz and U on the free surface is
m+ 1
m− 1
2
2
Ui+ 1,02 = Ui+ 1,02 +
∆t 1
xx,m
(Ti+xx,m
+ 2Ti+xx1, 1 )
1,0 − Ti,0
2 2
h ρi+ 1,0
2
Ti,0xx,m+1 = Ti,0xx,m +
∆t 4µ(λ + µ )
m+ 1
m+ 1
|i,0 (Ui+ 1,02 − Ui− 1,02 )
2
2
h λ + 2µ
Ti,0zz,m+1 = 0.
(42)
Approximation of non-reflecting boundaries
The spatial finite-difference grid is bounded by artificial boundaries. In an ideal case
these boundaries should be perfectly transparent for any wave impinging on the
boundary. Generally we can only approximate transparency. Many different
techniques were developed to simulate the so-called absorbing or non-reflecting
boundaries.
One simple technique (Cerjan et al., 1985) simulates transparent boundaries
by introducing a boundary zone that is placed around the useful part of the spatial
grid. In this artificial damping zone displacement and/or particle velocity values are
multiplied by an attenuating function A(i).
Another approach (Sochaki et al. 1987) suggests a different type of the
boundary zone. It is based on inclusion of a damping term in the equation of motion.
It is also possible to apply a paraxial (one-way) wave equation at the artificial
grid boundary since such equation permits energy propagation only in one direction –
out of the computational region. Paraxial equations can be replaced by the finitedifference schemes which are then applied at the boundary (e.g., Clayton & Engquist
1977).
There are several other techniques to simulate non-reflecting boundaries.
Numerical experience with different types of the absorbing boundary conditions
indicates that there is no best absorbing boundary condition which would be
universally (i.e.., in all wave-field configurations) both sufficiently accurate and stable.
Stability condition and spatial sampling criterion
Moczo et al. (2000) performed a detailed analysis on stability and grid dispersion.
They investigated these phenomena in the 3D fourth-order in space, second-order in
time, staggered-grid finite-difference scheme and arrived at
h≤
cβ
λmin
=
6
6 ⋅ fmax
∆t ≤
and
6
h
7 3 cβ
(43a,b)
where h is the grid spacing, ∆t is the time step, cβ is the S-wave velocity, fmax is the
maximum frequency that should be propagated through the grid with sufficient
accuracy and λmin is the smallest wavelength, respectively.
38
5.3. Formulations of the equation of motion
The fundamental relation in seismology, i.e. the equation of motion, can be noted in
three different ways.
K
K K
ρ
(x)
u(x,t)
Let us consider a Cartesian coordinate system (x1,x2,x3). Denote
density,
K
KK
(x,t);i,
j = 1,2,3 stress
τ
f(x,t)
ij
displacement vector,
body force per unit volume,
K
tensor, and λ(xi) and µ(xi) Lamé’s elastic coefficients, x meaning position and t
time.
The displacement-stress formulation is represented in Eq.20 and 21
ρui,tt = τij,j + fi ;
τij = λ uk,k δij + µ ( ui,j + uj,i ) ;
i, j ∈ {1,2,3}
The velocity-stress formulation reads
ρui,t = τij,j + fi
K
τij (x,t) = λuk,k δij + µ(ui,j + u j,i )
(44)
∂ui
where u,i,t = ∂t .
The Displacement formulation is
ρui,tt = (λuk,k ),i +(µui,j ),j + fi
(45)
It may be useful to choose to use an alternative notion instead of the concise
subscript notation above:
x = x1
,
y = x2
,
z = x3
,
u = u1
,
v = u2
,
w = u3
,
τxx = τ11
,
τyy = τ22
,
τzz = τ33
,
τxy = τ12
,
τxz = τ13
,
τyz = τ23
,
fx = f1
,
fy = f2
,
f z = f3
,
u = u1
,
v = u2
,
w = u3
,
39
and
ux = u1,1
,
uy = u1,2
,
uz = u1,3
,
vx = u2,1
,
vy = u2,2
,
vz = u2,3
,
wx = u3,1
,
wy = u3,2
,
wz = u3,3
,
and analogously for derivatives of the stress-tensor components (e.g., τxy,x = τ12,1).
Using this we can write the three formulations of the equation of motion in the 3D
case as follows.
K
u(u( x, y, z, t ), v ( x, y, z, t ), w ( x, y, z, t ))
τ ξη ( x , y , z ); ξ, η ∈ { x , y , z }
K
f ( f x ( x , y , z , t ), f y ( x , y , z , t ) , f z ( x , y , z , t ) )
ρ ( x, y, z ), λ ( x, y, z ), µ ( x, y, z )
K
u (u ( x, y, z, t ),v ( x, y, z, t ),w ( x, y, z, t ))
Displacement-stress formulation
ρutt = τxx,x + τxy,y + τxz,z + fx
ρvtt = τxy,x + τyy,y + τyz,z + fx
ρwtt = τxz,x + τyz,y + τzz,z + fz
(46a)
τxx = [λ +2µ] ux
+
λ vy
+
λ wz
τyy = λ ux
+
[λ +2µ] vy
+
λ wz
τzz = λ ux
+
λ vy
+
[λ +2µ] wz
τxy = µ ( uy +vx )
τxz = µ ( uz +wx )
τyz = µ ( vz +wy )
(46b)
40
Displacement formulation
ρ utt = ([ λ + 2µ) ux )x
( λ wz )x
+
+
ρ vtt =
(µ vx )x
( λ ux )y
+ ([ λ + 2µ) vy )y
+
(λ wz )y
ρ wtt =
(µ wx )x
( µ vz )y
+
+
( µ uy )y
( µ vx )y
( µ wy )y
( λ ux )z
+
+
( µ uz )z
( µ wx )z
+
+
( λ vy )x
fx
+
+
+
( µ vz )z
( µ wy )z
+
+
( λ uy )x
fy
+
+ ([ λ + 2µ) w)z
+
( λ vy )z
+
+
( µ uz )x
fz
+
(47)
Velocity-Stress Formulation
ρut = τxx,x + τxy,y + τxz,z + fx
ρvt = τxy,x + τyy,y + τyz,z + fx
ρwt = τxz,x + τyz,y + τzz,z + fz
(48a)
τxx,t = [λ +2µ] ux
+
λ vy
+
λ wz
τyy,t = λ ux
+
[λ +2µ] vy
+
λ wz
τzz,t = λ ux
+
λ vy
+
[λ +2µ] wz
τxy,t = µ ( uy +vx )
τxz,t = µ ( uz +wx )
τyz,t = µ ( vz +wy )
(48b)
5.4. Finite-difference spacetime grids
Let us consider the x-z plane and cover it by a rectangular grid with the spacing ∆x =
= ∆y = ∆z = h. We can either use a conventional grid or a staggered grid. For the
construction of a finite-difference scheme a conventional grid or a staggered grid
can be utilized. While a conventional grid is usual in the displacement formulation, in
41
the displacement-stress formulation a staggered grid is used commonly. In a
conventional grid all displacement components and material parameters are defined
at each grid point. In a staggered grid, different displacement components, stresstensor components, and material parameters are defined in different grid positions.
The two types of a spatial grid are displayed in Fig.18. Displacement finite-difference
schemes on a conventional grid have been used since the late sixties for modeling
seismic wave propagation, eg. Alterman & Karal (1968).
Since the P-SV case is simpler than the 3D case, it will be used in order to
explain the construction of a finite-difference scheme. We will restrict ourselves to the
displacement stress formulation for the staggered grid as an example.
Fig.18
Two types of a spatial grid are shown. (Reproduced from Moczo, 1998)
Displacement-stress scheme on a staggered grid
Recall the equation of motion (see Eq.20) and approximate the first spatial
derivatives using the central-difference formula applied over one grid spacing and
start with the equations for the diagonal stress-tensor components. Both equations
contain terms proportional to ux and wz and therefore it is reasonable to localize Txx
and Tzz, the discrete approximations to τxx and τzz, in the same grid position, say
I+1/2 l+1/2. For the time level m we obtain
1 m
1
Ti+xx,m
(Ui+1/ 2,l+1/ 2 − Ui,lm+1/ 2 ) + λi+1/ 2,l+1/ 2 (Wim+1/ 2,l+1/ 2 − Wim+1/ 2,l )
1/ 2,l+1/ 2 = [λ + 2µ ]i+1/ 2,l+1/ 2
h
h
and
1 m
1
Ti+zz,m
(Ui+1,l+1/ 2 − Ui,lm+1/ 2 ) + λ i+1/ 2,l+1/ 2 (Wim+1/ 2,l+1 − Wim+1/ 2,l )
1/ 2,l+1/ 2 = λ i+1/ 2,l+1/ 2
h
h
(49)
where U and W stand for discrete approximations to u and w, respectively.
Next we localize Txz, a discrete approximation to τxz, at the grid position il, and obtain
the finite-difference approximation of the equation for τxz at the time level m:
1
Ti,lxz,m = µil (Ui,lm+1/ 2 − Ui,lm−1/ 2 + Wim+1/ 2,l − Wim−1/ 2,l )
h
.
42
(50)
The second time derivative, e.g., of the u-component, at time level m and position
I,l+1/2 can be approximated by
1 m+1
(ui,l+1/ 2 − 2ui,lm+1/ 2 + ui,lm+−1/1 2 )
∆ ²t
.
(51)
Then the finite-difference approximation of the equation for the u-component at the
time level m and position i,l+1/2 is
1
1
xx,m
xx,m
xx,m
(Umi,l++1/1 2 − 2Ui,lm+1/ 2 + Ui,lm+−1/1 2 ) = (Ti+xx,m
) + Fi,lx+1/ 2
1/ 2,l+1/ 2 − Ti−1/ 2,l+1/ 2 + Ti,l+1 − Ti,l
h
∆ ²t
,
(52)
x
where Fi,l+1/ 2 is a discrete approximation to fx(xi,yl+1/2,tm) .
Now we could write the complete displacement-stress finite-difference scheme
for the P-SV waves, but because in computer codes it is reasonable to consider
integer values of grid indices, let us consider a so called finite-difference cell il
(shown in Fig.19):
zz,m
m
m
{Ti,lxz,m ,Ti+xx,m
1/ 2,l+1/ 2 ,Ti+1/ 2,l+1/ 2 ,Ui,l+1/ 2 ,Wi+1/ 2,l }
At last we can re-index all components of the finite-difference scheme, in order
to get indexes showing the actual finite-difference cell indices. The rule for reindexing is simple: 1. an index having an integer value does not change, 2. 1/2 has to
be subtracted from an index which does not have an integer value. If we assume a
homogeneous medium inside the finite-difference cell we can now write a finitedifference scheme ready for programming:
Umi,l +1 = 2Umi,l − Umi,l −1 +
∆ ²t x,m ∆ ²t 1 xx,m
xz,m
xz,m
Fi,l +
(Ti,l − Ti−xx,m
)
1,l + Ti,l+1 − Ti,l
ρi,l
ρi,l h
Wi,lm+1 = 2Wi,lm − Wi,lm−1 +
∆ ²t z,m ∆ ²t 1 zz,m
xz,m
xz,m
Fi,l +
(Ti,l − Ti,lzz,m
)
−1 + Ti+1,l − Ti,l
ρi,l
ρi,l h
1
{[λ + 2µ]i,l (Uim+1,l − Ui,lm ) + λi,l (Wi,lm+1 − Wi,lm )
h
1
= {λi,l (Uim+1,l − Ui,lm ) + [λ + 2µ]i,l (Wi,lm+1 − Wi,lm )
h
1
= µil (Ui,lm − Ui,lm−1 + Wi,lm − Wim−1,l )
h
Ti,lxx,m =
Ti,lzz,m
Ti,lxz,m
43
(53)
Fig.19
Field variables and material parameters entering the displacement-stress finite-difference
scheme. Dashed lines indicate a finite-difference cell il. (Reproduced from Moczo, 1998)
5.5.
Homogeneous and heterogeneous finite-difference schemes
(Following Moczo et al., in press)
Motion in a smoothly heterogeneous elastic continuum is governed by the equation
of motion. The equation can be solved by a proper finite-difference scheme and very
good accuracy can be achieved at a reasonable price. If the medium contains a
material discontinuity, i.e., an interface between two homogeneous or smoothly
heterogeneous media, at which density and elastic moduli change discontinuously,
the equation of motion still governs motion outside the discontinuity but boundary
conditions apply at the discontinuity. Then a natural approach to use the finitedifference method is to apply either a finite-difference scheme for the smoothly
heterogeneous medium at grid points outside the discontinuity or a finite-difference
scheme obtained by a proper discretization of the boundary conditions at grid points
at or near the discontinuity. This approach is sometimes called homogeneous. The
approach is suitable for simple geometry of discontinuities. If the material
discontinuity has complex shape, the approach is obviously impractical.
A complex shape of material discontinuities is inevitable if a computational
model is to be close to reality. Therefore, an alternative approach, sometimes called
heterogeneous, has been much more popular since the beginning of the seventies.
In the heterogeneous approach only one finite-difference scheme is used for all
interior grid points (points not lying on boundaries of a grid) no matter what their
positions are with respect to the material discontinuity. The presence of the material
discontinuity is accounted for only by values of elastic moduli and density. The
obvious questions are:
1.
Is the heterogeneous approach justified? In other words, is it possible to find a
heterogeneous formulation of the equation of motion?
2.
How are the values of the material parameters at grid points at and near the
discontinuity determined?
44
Zahradník & Priolo (1995) formulated the first question. Assuming a
discontinuity in material parameters they obtained an expression whose dominant
term is equivalent to the traction continuity condition. This result was interpreted as
justification of the finite-difference schemes constructed purely from equations of
motion (without explicit treatment of the traction continuity).
In order to solve the second question, several attempts were made, but none
of them was built on the boundary conditions at the material discontinuity. This is why
they do not result in proper averaging of material parameters and field quantities at
the discontinuity. Moczo et al. (in press) developed the first heterogeneous finitedifference scheme that was based on the boundary conditions at the material
discontinuities. In their approach they start with the boundary conditions at the
material discontinuity – continuity of displacement and continuity of traction. First they
consider a simple 1D physical model of the contact of two elastic media. As a result
they get harmonic averaging of elastic moduli, arithmetic averaging of densities and
arithmetic averaging of the corresponding field quantities at the discontinuity. Next
they formulate and prove a simple mathematical theorem on boundary conditions at
the discontinuity. They conclude the 1D case with a heterogeneous formulation of the
equation of motion and a possible finite-difference scheme. They then apply the
approach to the 3D case and obtain a possible finite-difference scheme (that is given
in the Appendix).
The accuracy of this scheme was tested by numerical comparisons with the
discrete-wavenumber method. Numerical comparison revealed a very good accuracy
of the heterogeneous finite-difference scheme.
45
6. The goal of the diploma thesis
In the previous chapters the importance of estimating site effects during future
earthquakes has been outlined. Even moderate earthquakes can cause anomalous
seismic ground motion that might result in severe damage to structures, property and
lives. Since detailed studies on site effect estimation are expensive and the majority
of the techniques in use depend on measuring small earthquakes, such studies are
commonly not performed in low to moderate seismicity countries like Austria. The
analysis of historical earthquake may indicate site effects but cannot replace analysis
of possible effects of site conditions on the earthquake ground motion. In this respect
there is a strong need for a cheap technique that does not depend on real
earthquakes. Such a technique should provide quantitative results, it should be easy
and fast to perform, even or especially in an urban environment, and it should be
trustworthy. The analysis of ambient seismic noise seems to fit perfectly the first
demands: it is easy and quick to perform, does not depend on any source, apart from
noise that is always present, is extraordinarily cheap and provides quantitative
characteristics, i.e., estimates of the fundamental frequency and amplification. On the
other hand this method is very much doubted within the scientific community and its
reliability is very much debatable.
In order to lighten up this question the 5th FP project SESAME (Site Effect
Studies Using Ambient Excitations) was launched. In this project a set of canonical
models was defined, that represent basic geological conditions (e.g., one layer over
halfspace, two layers over halfspace, shallow and deep sedimentary valley, etc.) that
are known to effectuate site effects. In this thesis only noise in a halfspace and in one
layer over halfspace is considered. The propagation of seismic noise in these models
is simulated by the finite-difference method. The noise synthetics are then analysed
and the spectral ratio between the horizontal and vertical part (H/V ratio) is
computed. The results are compared to the theoretical transfer function of S-waves in
each model. The goal of the thesis is to contribute to the effort to find out whether or
not there is a direct relationship between the H/V ratios of simulated noise and the Swave transfer function both for the spectral value of the peak frequency and the
amplitude of this peak respectively. The existence of a close relationship between
H/V ratios and transfer functions would be an argument for the use of ambient
seismic noise recordings in site effect studies.
7. Models and methods of computation and analysis
The simulation of ambient seismic noise is not a simple straightforward process, but it
requires several steps that are interdependent. If the results are not reasonable or
numerically stable, one might have to go back to previous processes, or even right to
the beginning, and make changes in some input parameters. In the following a
flowchart will be given to see, what processes are needed to create realistic noise
synthetics. Subsequently a short description of each step and a description of the
computer-cluster at which the computations were performed, will be given.
46
Model parameters
Grid
parameters
Parameters for
noise sourcetime-functions
Design of a model for finitedifference computations
Construction of a spatial
grid for finite-difference
computations
Computation of noise
sources
Input parameters
for finitedifference
computation
Finite-difference
simulations
Noise Synthetics
Definition of
a suitable
filter
1
47
Changes in model
parameters
1
No
Changes in source
parameters
Yes
Are the results
numerically stable ?
No
Changes in finitedifference input
parameters (filter)
Yes
Do the synthetics
comply with the
characteristics of
noise?
Computation of H/V ratios
Fig.20
Flowchart of the different steps which are essential for the computation and analysis of seismic
noise, and subsequently for the computation of spectral H/V ratios of noise synthetics.
7.0.
Computations
All computations have been performed on the Schrödinger I. cluster. This
professional high performance cluster for numerically intensive calculations (e.g., in
physics and chemistry, simulations...) has been installed in order to create sufficient
computing performance for the research at the Faculty of Natural Sciences and
Mathematics at the University of Vienna in 2001. High floating point performance and
the use of modern and proven concepts (e.g., Beowulf clusters) are likewise
objectives as simple administration, maintenance, reliability, scalability and
expansion. In the first stage of development 160 nodes are controlled by one
administration server and two central servers. Each node or each group of nodes can
be monitored and maintained by the central administration server. Users have to
login at the login-server by secure shell and jobs are submitted to the nodes by the
OpenPBS queuing system. All nodes are equipped with an AMD Athlon XP-1700+
processor, 1 GB RAM and 40 GB local disk storage. The operating system is SuSE
7.2 Linux and compilers for the majority of programming languages are available. To
illustrate the computation power of the cluster, the overall performance data is given
by
•
•
•
204,5 GFlops,
RAM > 162 GB,
disk space > 6 TB.
48
7.1.
Design of the models and space-grids for finite-difference
computations
The first step for creating noise synthetics is to create a model in which the finitedifference modeling is then performed. Martin Gális (Slovak Academy of Sciences)
has written a program in Fortran95 to construct a computational region that is a
volume of a parallelepiped with the top side representing a planar free surface,
and bottom, rear, front, left and right sides representing model boundaries. The
parallelepiped is divided into cubes. The size of the cubes is the spatial grid step for
the finite-difference computations. In order to reduce memory requirements and
calculation time, the model consists of an upper part with a grid spacing h and a
bottom part with grid spacing 3*h.
Inside this computational region it is possible to prescribe two homogeneous
parts, each with different material parameters (representing a layer over halfspace).
These two homogeneous parts have to have a planar horizontal boundary. Within the
first homogeneous part it is possible to prescribe a heterogeneous part (representing
a sediment structure), that may have an arbitrary shaped bottom boundary and
whose material parameters may change with depth.
In order to construct a model it is necessary to define the material and
computational parameters of the structure under consideration.
The values for density, the P-wave and the S-wave velocity, and attenuation
have to be prescribed in input files. In another file all model parameters have to be
set. A list of the most important input parameters with description of each parameter
is given in Tab.1.
VARIABLE
EXPLANATION
VARIABLE TYPE
H
Grid spacing in the coarser spatial grid
REAL(8)
key_q
= true: damping coefficients are calculated
LOGICAL
Fvp
Name of the input file with the P-wave velocities
Fvs
Name of the input file with the S-wave velocities
Frho
Name of the input file with densities
Fqsp
Name of the input file with quality factors Qp and Qs
CHARACTER(20)
Xbmin
Xbmax
Ybmin
Boundaries of the computational model
Ybmax
REAL(8)
Zbmin
Boundary of the model in Z-direction
Zhom
Depth of the top of the second homogeneous part of the
model
Zhf
Depth of the top of the coarser spatial grid
Tab.1
Parameters for the design of models for finite-difference simulations
49
In order to avoid unacceptable effects of the grid dispersion, the grid-spacing h has to
be chosen according to
h≤
cβ
λmin
=
6
6 ⋅ fmax ,
(43a)
where λmin is the minimum wavelength that should be propagated in the model with a
sufficient accuracy, cβ is the S-wave velocity, and fmax is the maximum frequency up
to which the simulation should be sufficiently accurate.
It is crucial to design a model that is as big as possible, because the
approximately non-reflecting boundaries are never strictly non-reflecting and a certain
proportion of the wave-field that reaches the boundaries is always reflected and can
perturb the noise synthetics. As a rule of thumb the model extension should be at
least the length of the longest wavelength that has to be propagated through the
model.
These two conditions, as small grid step as possible on the one hand, and
grids as big as possible, result in a considerable big model and possibly in large time
computer memory and time. One has to find a compromise to get realistic noise
synthetics in a reasonable amount of time.
The depth of the top of the coarser spatial grid has to lie at least four grid
spacings beneath the last inhomogeneity in the model.
Output files of the model program serve as an input for the program
GRIDCELLS (Gális, 2001). In this program a spatial grid is constructed from the
model that is suitable for the finite-difference computations.
7.2.
Generation of point sources of seismic noise and finite-difference
modeling
(Adapted from the user’s guide to the program package NOISE)
For the simulation of ambient seismic noise Kristek and Moczo (2000) have
developed the Fortran95 program package NOISE. The program package NOISE is
designed for numerical generation and simulation of seismic noise in 3D
heterogeneous surface geological structures with planar free surface due to surface
and near-surface random sources.
The program package consists of two programs written in Fortran95: program
RANSOURCE and program FDSIM.
Program RANSOURCE
The program RANSOURCE is designed for random space-time generation of point
sources of seismic noise (Kristek et al., 2002). The output files serve as input files for
the program FDSIM. The algorithm of random noise generation assumes regular
spatial distribution of potential point sources inside of a specified source volume. The
spatial distribution is controlled by
• the prescribed minimum distance between two neighbour point sources,
• minimum distance between a point source and a receiver and
• maximum distance between a point source and a receiver.
50
The temporal distribution of point sources is controlled by the prescribed
minimum and maximum numbers of point sources acting at the same time. For each
generated position of a point source,
• a direction of acting single body force at the position,
• time function and
• maximum amplitude
are randomly generated. The time function is either delta-like signal or pseudomonochromatic signal (a harmonic carrier with the Gaussian envelope), whose
source-time function is given by
s(t) = exp{−[ω(t − t s ) / γ ]2 } cos[ω(t − t s ) + θ],
ω = 2πfp ,
t ∈ 0,2t s ,
t s = 0.45 γ / fp ,
(55)
where fp is the predominant frequency, γ controls the width of the signal and its
spectrum, and θ is a phase shift. Spectrum of the delta-like signal is low-pass filtered
in order to fit the prescribed frequency range. In the case of the pseudomonochromatic signal, first its duration, then its predominant frequency are randomly
generated. The maximum amplitude of the signal is randomly generated from the
interval (0,1> according to a chosen distribution. The program has to be run before
the finite-difference simulation of the noise itself.
The program generates two files for all delta-like sources and as many
files as the number of generated pseudo-monochromatic sources. All the files serve
as the input data files for the program FDSIM.
Program FDSIM
The Program FDSIM is designed for the finite-difference simulation of seismic wave
propagation and seismic ground motion in a 3D surface heterogeneous viscoelastic
structure with a planar free surface. The computational algorithm is based on the
explicit heterogeneous finite-difference scheme solving equations of motion in the
heterogeneous viscoelastic medium with material discontinuities (Moczo et al., in
press, Kristek et al., 2002). The scheme is 4th-order accurate in space and 2nd-order
accurate in time. The displacement-velocity-stress scheme is constructed on a
staggered finite-difference grid. The computational region is represented by a volume
of a parallelepiped with the top side representing a planar free surface, and bottom,
rear, front, left and right sides representing either non-reflecting boundaries or planes
of symmetry. Different types of non-reflecting boundaries can be chosen on different
sides of the computational region. The discontinuous spatial grid is used to cover the
computational region. The upper part of the grid has three times smaller grid spacing
than the lower part. Each part itself is a uniform rectangular grid. The rheology of the
medium corresponds to the generalized Maxwell body. This makes it possible to
account both for spatially varying quality factors of the P- and S-waves and for
arbitrary Q-omega law. The wave-field is excited by a set of randomly generated
point sources, each representing one single force acting in an arbitrary direction. The
core memory optimisation is applied in order to significantly reduce requirements of
the computer’s core memory.
51
7.3. Filtering in the finite-difference computations
The first finite-difference computation for a given model is performed without filtering.
In the program RANSOURCE a distinct frequency band is defined, i.e., noise sources
are prohibited to contain frequencies outside this range. On the other hand noise
simulations are specific for the presence of many sources at the same time which
interfere with each other. Therefore also higher frequencies than that defined in the
frequency band for noise sources are to be expected and numerical instabilities will
occur.
To overcome these high frequency instabilities, filtering has to be included in
the finite-difference computations. Since the results should be accurate up to a
specific frequency, all higher frequencies have to be suppressed. A suitable method
to design a low-pass filter is Parks-McCellan FIR Filter Design. The filter coefficients
have to be given in the file FC.DAT which is an input file for the program FDSIM. The
interval at which data is filtered is given by the variable IFILT and should be small
enough to be able to suppress instabilities before they are visible.
While the results of the finite-difference computations can be improved by lowpass filtering, still, low-frequency instabilities can appear and a band-pass filtering
has to be applied to the finite-difference computations. The use of band-pass filtering
significantly improves the results of the finite-difference computations.
7.4.
Models
During the SESAME task C meeting in August 2001 a set of canonical models has
been defined, for which seismic noise simulations are to be performed. These
canonical models include
• homogeneous halfspace,
• single layer over halfspace,
• dipping layer,
• semi-infinite layer,
• single layer with a rough layer-halfspace interface,
• deep sedimentary valley,
• single layer with a trough at the bottom
• buried fault,
• two layers over halfspace and
• gradient layer.
This diploma thesis treats the models of a homogeneous halfspace and single layer
over halfspace. The two configurations are illustrated in Fig.21 and the material and
computational parameters are listed in Tab.2.
52
M1 Homogeneous halfspace
M2 Single layer over halfspace
1
h
2
Fig.21
Model
label
M1
M2.1
M2.2
M2.3
Model
label
M1
M2.1
M2.2
M2.3
A Schematic picture of the two canonical models for which noise simulations have been
performed in this diploma thesis. M1 is a homogeneous halfspace and M2 is a single layer over
halfspace.
α1 [m/s]
β1 [m/s]
α2 [m/s]
β2 [m/s]
α1/β1
β2/β1
ρ1 [kg/m³]
2000
500
1350
1350
1000
250
250
667
2000
2000
2000
2000
1000
1000
1000
1000
2
2
5,4
2,0240
1
4
4
1,4993
2500
1900
1900
1900
ρ2 [kg/m³]
Qp1
Qs1
Qp1
Qs2
H [m]
h [m]
2500
2500
2500
2500
100
50
50
50
50
25
25
25
100
100
100
100
50
50
50
50
0
31,25
31,25
83
16
4
4
8
Tab.2
Parameters of the models used for the simulation of seismic noise. M1represents the model of a
homogeneous halfspace, whereas M2.1, M2.2 and M2.3 stand for one single homogeneous
layer over homogeneous halfspace. α1 and α2 are the P-wave velocities for the layer and
halfspace respectively. β1 and β2 are the S-wave velocities and r1 and ρ2 is density of the layer
and the halfspace respectively. The level of attenuation is represented through the quality
factors Qp1 and Qp2 for P-wave attenuation and Qs1 and Qs2 for S-wave attenuation,
respectively. H is the thickness of the layer and h is the grid step for the finite-difference
modeling.
M1 is the model of a homogeneous halfspace. M2.1, M2.2 and M2.3 are a series of
1D models representing a single, infinite, horizontal, softer layer (referred to as
“sediments”) over a homogeneous, harder halfspace (referred to as “bedrock”). The
contact between bedrock and sediments is horizontal, giving rise to the classical 1D
vertical resonance. The fundamental frequency of 1D resonance is f0=βσ/4h (see
Eq.6). In all M2 models the fundamental frequency is 2Hz in the perfectly elastic
medium. The attenuation quality factors are the same for all models. In models M2.1
and M2.3 the velocity contrast between the P- and S-waves in the sediment layer,
i.e., α1/β1 is 2, whereas the velocity contrast is higher (α1/β1=5.4) in model M2.2.
The S-wave velocity contrast between sediments and bedrock is higher (β2/β1=4) in
models M2.1 and M2.2 than in model M2.3. Here β2/β1 ~ 1.5. In order to preserve
the same fundamental 1D resonant frequency for all models, layer thickness in
models M2.1 and M2.2 is bigger (H=31.25m) than in model M2.3, were H=83m. For
the homogeneous halfspace a grid step of h=16m and a time step of ∆t=0.0035s are
53
chosen. The two high S-wave velocity models, i.e. M2.1 and M2.2, are computed with
h=4 and ∆t=0.0009s, whereas for the low S-wave velocity contrast model M2.3, h=8
and ∆t=0.0019s are taken.
16 receivers are placed, in each model, in the central surface area of the
computational grid. The array of 4x4 receivers is placed equally spaced 20m apart on
a square in the centre of the computational region. Receiver 1 is located on the upper
left corner and receiver 16 on the lower right corner, respectively. Seismic noise
sources are generated in a volume defined by the minimum and maximum distance
between point source and receiver and the maximum source depth. At one time step
there are between 10 and 120 active point sources. The frequency range for the
generation of seismic noise sources is 0.5-10 Hz in all models. Point sources with
delta-like signal and pseudo-monochromatic signal are equally generated for all
models.
The output of the finite-difference computations are three component ground
velocity and displacement seismograms for each receiver. The velocity seismograms
are chosen for analysis, following the proposal of Bard (1999). The first 20 seconds
of each seismogram are then selected for analysis.
7.5.
Analysis of synthetic seismograms
The ‘raw’ results of the finite-difference computations are time-domain seismograms
for all specified receivers. The synthetic seismograms computed by the program
package NOISE have to be processed in order to extract information for the
estimation of site effects. The analysis of the data consists of several steps, which
are
• autocorrelation analysis,
• crosscorrelation analysis,
• use of synthetic seismograms from different receivers in order to get a
sufficient number of time windows,
• computation of the H/V ratios for seismic noise and
• comparison of the H/V ratios with theoretical transfer functions for vertically
incident SH-waves.
Autocorrelation
In order to confirm that the finite-difference synthetics actually represent random
seismic noise in a reasonable way, the first step in analysis is to determine, whether
or not the noise synthetics are really random signals. Autocorrelation is a tool to
verity this requirement. For a perfectly random signal autocorrelation is zero, expect
for t=0 where it is 1.
Crosscorrelation
Signals from different receivers, placed apart from each other, should bear no
similarity, if the wave-field represents random noise. No common signal should be
recognized in any receiver pair. To quantify this demand, a mathematical tool for
measuring the amount of common signals in multi-channel data is needed. There are
several coherence measures used in seismology, a review of which is given in
Neidell & Turhan Taner (1971). The most familiar of the likeness criteria is
crosscorrelation. Miriam Kristeková (Slovak Academy of Sciences) has developed a
Fortran95 code to perform correlation analysis for noise synthetics. This program is
capable of computing
54
•
•
•
•
•
•
autocorrelation for one receiver,
crosscorrelation of receiver pairs with geometric normalisation,
crosscorrelation of receiver pairs with arithmetic normalisation,
multi-channel crosscorrelation with geometric normalisation,
multi-channel crosscorrelation with arithmetic normalisation and
semblance.
If autocorrelation confirms the assumptions that the noise synthetics represent
random seismic noise, at least in the given frequency band, the individual files are
truncated to segments of a distinct length for each receiver and component. The
duration of 20 seconds is chosen in order to make sure that the lowest frequency that
is propagated in the model, i.e., 0.5Hz, is undoubtedly represented. Then these
segments are added together to one single file with three columns representing the
three components of the displacement-vector. This file then serves as input file for
the H/V computation.
Computation of the H/V ratios
If all data is checked for correlation and prepared in the right format, the H/V ratios
can be computed. Miriam Kristeková has written a Fortran95 code to carry out this
task. First the X and the Y component of the seismograms are combined to one
horizontal component. The H/V ratios are computed for different time windows,
representing different receivers. The individual H/V ratios are then averaged and a
single H/V curve is computed. The H/V curves are smoothed according to Konno &
Ohmachi (1998).
8. Results
8.1.
Homogeneous halfspace
Model layout
The layout for the simulation of seismic noise in a homogeneous halfspace is
displayed in Fig.22. The receivers are placed in the middle of the computational
region. These receivers are surrounded by a source volume in which both pseudomonochromatic and delta-like sources are distributed randomly. The source volume
is defined by a minimum distance between the receiver at position (150,150) of 40
grid spacings, the maximum distance of 100 grid spacings between a receiver and a
point source and the maximum source depth, which is 5m. Grid spacing is 16m in this
grid.
Autocorrelation and crosscorrelation
16 receivers record the ground velocity of the finite-difference noise simulation.
Fig.23 shows an example of a ‘raw’ seismogram at receiver 11 for the model M1 that
represents homogeneous halfspace. It can be seen that the seismograms show no
obviously visible trend. To verify this supposition the seismograms have to be
analysed. The first step in the processing is the correlation analysis. The
seismograms from all receivers have to be checked for autocorrelation and
crosscorrelation. A summarising plot of the correlation analysis is given in Fig.24.
55
300
280
260
240
220
200
180
160
140
120
100
80
60
40
20
0
0
Fig.22
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
Layout of model M1. Crosses represent point sources and triangles receivers, respectively. The
labels of the axes are in grid spacings. Grid spacing is 16m in model M1.
X -c o m p o n e n t
0 .0 6
0 .0 4
V[m/s]
0 .0 2
0
-0 .0 2
-0 .0 4
-0 .0 6
0
2
4
6
8
10
12
14
16
18
20
Y -c o m p o n e n t
0 .0 8
V[m/s]
0 .0 4
0
-0 .0 4
-0 .0 8
0
2
4
6
8
10
12
14
16
18
Z -c o m p o n e n t
0 .1 5
0 .1
V[m/s]
0 .0 5
0
-0 .0 5
-0 .1
-0 .1 5
0
2
4
6
8
10
12
14
16
18
Time [sec]
Fig.23
Example of synthetic seismograms obtained from finite-difference noise simulations for model
M1 (i.e., homogeneous halfspace). Top, middle and bottom figures show X, Y and Z component
of ground velocity at receiver 1 for the first 20 seconds of the computation.
56
20
Autocorrelation with arithmetic and geometric normalisation is performed in order to
verify that the signals recorded at the individual receivers are random signals. It is
remarkable that the autocorrelation of all receivers show very close characteristics
(compare to Fig.24b). The value of all components is between -0.2 and 0.2 for all t
except t=0 where it is 1 (Fig.24a). This result indicates a random signal.
As next step crosscorrelation between receiver pairs is performed. In addition
to this a multi-channel correlation with arithmetic and geometric normalisation, as well
as semblance is computed. Fig.24c and Fig.24d show crosscorrelation for the Xcomponent of two receivers placed 56m and 28m apart, respectively. The maximum
is about 0.2 in both figures. For better comparison the crosscorrelation for the Xcomponent between 5 receiver pairs is illustrated in Fig.25.
The crosscorrelation analysis for the Y and Z-component does not differ
substantially from that for the X-component. The range of the crosscorrelation
maximum between receiver pairs is [-0.3, 0.38] for the Z-component, [-0.26, 0.4] for
the Y-component and [-0.3, 0.31] for the X-component. The peak value of the
crosscorrelation is not depending on the distance between the receivers used for
crosscorrelation. This can be seen in Fig.25. where no substantial difference can be
observed between the different receiver pairs, despite the fact that the minimum and
maximum distance between the two receivers is 20m and 113m, respectively.
As last step of the correlation analysis multi-channel correlation analysis is
performed. Multi-channel correlation with arithmetic and geometric normalisation as
well as semblance is computed and given in Fig.24f and 24g, respectively. No
significant difference between the three methods of multi-channel correlation can be
observed for model M1. All three measures vary between -0.02 and 0.02 which is
final evidence that the noise synthetics do not include any common signal.
Looking at these results the claim of the computations to generate random
noise seems justified and the individual recordings of the receivers may be combined
for the computation of H/V spectral ratios.
H/V spectral ratio
After having demonstrated the fact, that the results of the finite-difference simulations
in fact represent the characteristics of ambient seismic noise, the next step in
analysis is the computation of H/V spectral ratios. The individual H/V ratios for single
receivers, i.e., time windows of 20 seconds, are computed, as well as a final H/V
spectral ratio that is an average over all individual spectral ratios. Results of these
computations are displayed in Fig.26.
All individual time windows show a nearly flat characteristic. In almost all H/V
ratios a slight maximum can be observed, but none is very prominent. These maxima
are on different positions for all time windows, lying between 0.5 and 4Hz. In the
averaged H/V spectral ratio a small maximum can be seen at 0.5Hz and the H/V ratio
is decreasing with increasing frequency. Nevertheless both, individual and averaged
H/V ratios, remain very close to the value of 1.
These results may have been expected owing to the fact that model M1 is
homogeneous halfspace. In such an environment no frequency dependence of
ground amplification should arise and both horizontal and vertical motion should be
distributed equally for all frequencies, leading to a constant H/V spectral ratio of 1.
57
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-20
-15
-10
-5
0
5
10
15
20
-20
a
-10
0
10
20
b
0.3
0.3
0.2
0.2
0.1
0.1
0
0
-0.1
-0.1
-0.2
-0.2
-20
c
-15
-10
-5
0
5
10
15
-20
20
-15
-10
-5
0.03
0.03
0.03
0.02
0.02
0.02
0.01
0.01
0.01
0
0
0
-0.01
-0.01
-0.01
-0.02
-0.02
-0.02
-0.03
-0.03
-20
-15
e
-10
-5
0
5
10
15
20
0
5
10
15
20
d
-0.03
-20 -15
-10
-5
f
0
5
10
15
20
-20 -15
-10
-5
0
5
10
15
20
g
Fig.24
A summary of different correlation measures for the X-component for model M1 (homogeneous
halfspace). X-axis is time [sec] in all figures. Y-axis is the value of the correlation measure. a)
Autocorrelation function of receiver 1. Autocorrelation is in the interval [-0.2, 0.2], except for
values near t=0. b) A comparison of the autocorrelation functions of the first 6 receivers. The
peak value is 1 for all receivers corresponding to the length of the line on the left of each series.
A striking similarity can be observed. c) and d) Crosscorrelation functions with geometric
normalisation for two receivers, placed 28m and 56m apart, respectively. e) Multi-channel
crosscorrelation analysis with geometric normalisation for all 16 receivers. f) Multi-channel
crosscorrelation analysis with arithmetic normalisation for all 16 receivers. g) Semblance
computed for all 16 receivers.
Fig.25
Illustration of the crosscorrelation analysis for the X-component for different distances between
the receivers. 4 results of crosscorrelation are displayed. Numbers under each trace indicate the
two receivers used for crosscorrelation. The array of 4x4 receivers is placed equally spaced
20m apart on a square in the centre of the computational region. Receiver 1 is located on the
upper left corner and receiver 16 on the lower right corner, respectively. The vertical line on the
left of each trace represents a value of 1.
58
Averaged H/V ratio
H/V ratios for time windows of M1
10
H/V
H/V
10
1
1
0.1
0.1
4
Fig.26
8.2.
8
frequency [Hz]
12
16
4
8
frequency [Hz]
12
16
H/V spectral ratios for model M1, plotted on a lin-log scale. The figure on the left shows H/V
ratios for time windows of 20sec, representing individual receivers. On the right is the H/V
spectral ratio, averaged over all individual H/V ratios.
One layer over halfspace
Model M2.1
Model layout
The first model in the series of models of a single layer over halfspace, i.e. model
M2.1, is characterised by a velocity contrast of 2 between the P- and the S-wave in
the sediment layer and by a velocity contrast of 4 for S-waves in the layer and the
bedrock, respectively. The theoretical fundamental 1D vertical resonant frequency for
vertical SH-wave incidence is 2Hz in the perfectly elastic medium, having a peak
value of 8, which is two times the impedance contrast between sediments and
bedrock (see Eq.6 and Eq.8).
The layout of the finite-difference simulation and an example of a synthetic
seismogram generated for model M2.1 are displayed in Fig.27 and Fig.28,
respectively. The grid spacing is 4m in models M2.1 and M2.2. Therefore the
minimum distance of 40 grid spacings between the receiver at (150,150) and the first
point source means 160m and the maximum distance of 80 grid spacings between a
point source and the receiver (150,150) means 320m. The grid consists of 300x300
grid points in the finer grid. This means an area of 1200mx1200m.
Autocorrelation and crosscorrelation
The autocorrelation analysis gives results similar to that of model M1. The
autocorrelation function is in the interval [-0.2, 0.2] for all components and receivers,
except for t=0 where it is 1. Because all receivers show the same characteristics,
autocorrelation is shown only for receiver 1 in Fig.29a,b,c.
59
Crosscorrelation of all possible receiver pairs is low in the whole time interval
of [-20s, 20s] for all times and components. The maximum crosscorrelation between
two receivers is randomly distributed in the interval [-0.4, 0.4] for the X component, [0.43, 0.47] for the Y-component and [-0.4, 0.5] for the Z-component. This peak value
in the crosscorrelation function occurs at different times for every receiver pair and is
not depending on the distance between receivers. An example is given in Fig.29d.
300
280
260
240
220
200
180
160
140
120
100
80
60
40
20
0
Fig.27
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
Layout of model M2.1. Crosses represent point sources and triangles receivers, respectively.
The labels of the axes are in grid spacings. Grid spacing is 4m in model M2.1.
X -c o m p o n e n t
1 .5
1
V[m/s]
0 .5
0
-0 .5
-1
0
2
4
6
8
10
12
14
16
18
Y -c o m p o n e n t
1
0 .5
V[m/s]
0
-0 .5
-1
-1 .5
0
2
4
6
8
10
12
10
12
14
16
18
Z -c o m p o n e n t
0 .8
0 .4
0
V[m/s]
-0 .4
-0 .8
0
2
4
6
8
14
16
18
Time [sec]
Fig.28
Example of synthetic seismograms obtained from finite-difference noise simulations for model
M2.1 (i.e., 1st model of a single layer over halfspace). Top, middle and bottom figures show X,
Y and Z component of ground velocity at receiver 1 for the first 20 seconds of the computation.
60
b) Autocorr Vy - rec1
a) Autocorr Vx - rec1
1.2
1.2
0.8
0.8
0.4
0.4
0
0
-0.4
-0.4
-0.8
-0.8
-20
-10
0
time [sec]
10
20
-20
-10
0
10
time [sec]
20
d) Crosscorr Vx1-Vx16
c) Autocorr. Vz - rec.1
0.2
1.2
0.8
0.1
0.4
0
0
-0.1
-0.4
-0.2
-0.8
-20
-10
Fig.29
0
time [sec]
10
-20
20
-10
0
time [sec]
10
20
Example of crosscorrelation and autocorrelation analysis for model M2.1. a) Autocorrelation of
the X-component of velocity at receiver 1. b) Autocorrelation of the Y-component of velocity at
receiver 1. c) Autocorrelation of the Z-component of velocity at receiver 1. d) Crosscorrelation of
the XY-component of velocity between receiver 1 and receiver 16, spaced 84m.
In the multi-channel analysis semblance has turned out to be a reliable tool for
coherency analysis (Neidell & Turhan Taner, 1971). Because semblance has shown
some advantages and no substantial drawback compared to multi-channel
crosscorrelation, it has been chosen for multi-channel correlation analysis.
Semblance for the X, Y and Z component of model M2.1 is shown in Fig.30. All three
measures remain low for the whole time interval. The peak value of the X-component
is 0.06, that of the Y-component 0.04 and that of the Z-component 0.07. Semblance
differs somewhat between the three components. Nevertheless it still remains at a
very low level. The maximum value is attained near t=0.
61
0.06
Semb
Vx
0.02
-0.02
-0.06
-20
-15
-10
-5
0
5
10
15
20
-15
-10
-5
0
5
10
15
20
-15
-10
-5
0
5
10
15
20
time [sec]
0.04
Semb
Vy
0.03
0.02
0.01
0
-0.01
-0.02
-20
time [sec]
0.08
Semb
Vz
0.04
0
-0.04
-20
time [sec]
Fig.30
Semblance analysis for model M2.1.Top is semblance for the X-component of the ground
velocity, middle and bottom figures are the Y- and Z-components of the ground velocity for
model M2.1, respectively.
H/V spectral ratio
The computation of H/V spectral ratios is performed according to Konno & Ohmachi
(1998), who assumed noise to consist mainly of Rayleigh waves. As mentioned
earlier, the H/V ratios of fundamental-mode Rayleigh waves become infinite at some
peak periods. This makes it difficult to find a correlation between amplification factors
for S-waves and the H/V ratios. This difficulty can be avoided when a smoothing
process is introduced before calculating the H/V ratio. The authors introduce a
logarithmic smoothing function, given by
WB (f,fc ) = [sin(log10 (f / fc )b / log10 (f / fc )b ]4 ,
(56)
where b, f, and fc are a coefficient for band width, frequency, and a centre frequency,
respectively. They show that such a smoothing function does not introduce any
amplitude distortion in the H/V ratios, like it would be the case when smoothing would
be performed with a Parzen window. In the program for the computation of spectral
H/V ratios, both the length of the time window and the exponent b of the logarithmic
smoothing function have to be set.
In order to analyse the stability of the results of H/V computations for a single
layer over halfspace, H/V ratios are computed with different exponents of the
smoothing function. The results are displayed in Fig.31 and in Fig.32, respectively.
Fig.31 shows H/V spectral ratios computed with different length of the time window.
The H/V ratios for individual time windows show no substantial difference. In all plots
a maximum of the H/V ratio can be observed near 2Hz. From 0.5 to 4Hz all H/V ratios
look similar. For higher frequencies in the H/V ratios, computed with 10s and 20s,
respectively, a slight second high is visible at 6-7Hz. For time windows of 30s and
62
40s, this second maximum is not observable any more. After this small analysis it
seems justifiable to compute the H/V spectral ratios with a time window of 20s as well
as an H/V ratio averaged over all individual time windows.
time window 10s
time window 20s
10
H/V
H/V
10
1
1
0.1
0.1
4
8
12
frequency [Hz]
4
16
16
10
H/V
H/V
12
time window 40s
time window 30s
10
1
0.1
1
0.1
4
Fig.31
8
frequency [Hz]
8
12
frequency [Hz]
16
4
8
frequency [Hz]
12
16
H/V spectral ratios for model M2.1, computed with different length of the time window. The
lengths of the individual time windows are 10s, 20s, 30s and 40s, indicated on top of each plot.
The axes of the graphs are lin-log. A value of b=15 has been chosen for all H/V ratios.
Fig.32 shows the dependency of the H/V spectral ratios on the choice of the
exponent of the smoothing function, proposed by Konno & Ohmachi (1998). The
computation of H/V ratios without any smoothing shows many peaks, reaching peak
values as large as 100. All information is totally distorted. Therefore the use of a
smoothing function is essential. The dependency of the H/V spectral ratio on the
choice of the exponent for the smoothing is clearly visible in Fig.32. Using a bigger
exponent, i.e. b=30, leads to an H/V ratio with two prominent maxima, one slightly
smaller than 2Hz and one slightly higher. Since we know that the theoretical SHwave transfer function has its fundamental peak at 2Hz, this result may not be
correct. The choice of a smaller smoothing parameter, i.e. b=20 or b=15, leads to a
single, prominent peak at about 2.3Hz. Regarding this results, an exponent with the
value of 15 has been chosen for further computations.
63
b = 30
No smoothing
10
100
H/V
H/V
10
1
1
0.1
0.1
4
8
12
frequency [Hz]
16
4
b = 20
12
16
b = 15
10
10
H/V
H/V
8
frequency [Hz]
1
0.1
1
0.1
4
Fig.32
8
12
frequency [Hz]
16
4
8
frequency [Hz]
12
16
H/V spectral ratios for model M2.1, computed without smoothing and with different exponents of
the parameter b of the smoothing function. The type of smoothing is indicated on top of each
plot. The axes of the graphs are lin-log.
H/V spectral ratios are computed for both, individual receivers and all
receivers averaged. The results are displayed in Fig.33. The H/V ratios remain low,
i.e., between 0.6 and 0.9, under 1Hz for all time windows. Between 2 and 2.5Hz a
first prominent maximum can be observed for all individual time windows with a peak
value between 3 and 5. With increasing frequency all H/V ratios decrease and reach
a minimum between 4 and 10Hz. Some H/V ratios then increase again and reach a
second maximum at about 6Hz. However, there is no generally visible trend in the
individual H/V ratios at frequencies above 4Hz. Some H/V ratios have their second
maximum at the same frequencies, where others reach a minimum value. H/V ratios
are between 1.1 and 2.3 for frequencies above 4Hz.
In the averaged H/V ratio all important features of the individual H/V ratios can
be observed. The averaged H/V ratio is smaller than 0.8 for frequencies under 1Hz.
Then it is permanently increasing, attaining a prominent maximum with a value of
3.73 at 2.2Hz. A minimum value of 1.5 is attained at 4.3Hz. With increasing
frequency the averaged H/V ratio is slightly increasing, however remaining smaller
than 2.
64
Averaged H/V ratio
H/V ratios for time windows of M2.1
10
H/V
H/V
10
1
1
4
Fig.33
8
12
frequency [Hz]
16
4
8
12
frequency [Hz]
16
Spectral H/V ratios for model M2.1. Left are the H/V ratios for the individual time windows, left
is the H/V ratio averaged over all individual time windows. The curves are displayed in Lin-Log
style.
Model M2.2
Model layout
Model M2.2 is the second model in the series of models of a single layer over
halfspace. It is characterised by a velocity contrast of 2 between the P- and the Swave in the sediment layer, which is the same as for model M2.1. The velocity
contrast for S-waves in the layer and the bedrock, respectively, however is bigger
than that of model M2.1. In model M2.2 this velocity contrast is 5.4. The theoretical
fundamental 1D vertical resonant frequency for vertical SH-wave incidence is 2Hz in
the perfectly elastic medium, having a peak value of 8, like in model M2.1.
The layout of the finite-difference simulation and an example of a synthetic
seismogram generated for model M2.2 are displayed in Fig.34 and Fig.35,
respectively. The grid spacing is 4m. The minimum distance between the receiver at
(150,150) and the first point source is 160m and the maximum distance between a
point source and the receiver (150,150) is 320m. The grid consists of 300x300 grid
points in the finer grid. This means an area of 1200mx1200m.
Autocorrelation and crosscorrelation
Similar to the models above the autocorrelation function is in the interval [-0.2, 0.2]
for all components and receivers, except for t=0 where it is 1. Because all receivers
show the same characteristics, autocorrelation with arithmetic normalisation is shown
only for receiver 1 in Fig.36a,b,c. These results indicate that the source generation
for model M2.2 fits the requirements of random noise.
In order to justify the combination the synthetic seismograms from all receivers
to a single record for the computation of H/V spectral ratios, crosscorrelation and
multi-channel crosscorrelation functions have to be computed. Crosscorrelation of all
possible receiver pairs is low in the whole time interval of [-20s, 20s] for all times and
components. The maximum crosscorrelation between two receivers is generally in
65
the interval [-0.35, 0.4] for all three components of ground motion velocity. Only for
the X-component and the Z-component of 3 receiver pairs the maximum value of the
crosscorrelation function is 0.6. The peak value in the crosscorrelation function
occurs at different times for every receiver pair and is not depending on the distance
between receivers. An example is given in Fig.36d.
300
280
260
240
220
200
180
160
140
120
100
80
60
40
20
0
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
Fig.34 Layout of model M2.1. Crosses represent point sources and triangles receivers, respectively.
The labels of the axes are in grid spacings. Grid spacing is 4m in model M2.2.
X-component
0.6
0.4
V[m/s]
0.2
0
-0.2
-0.4
-0.6
0
2
4
6
8
10
12
14
16
18
10
12
14
16
18
Y-component
0.8
0.6
0.4
V[m/s]
0.2
0
-0.2
-0.4
-0.6
-0.8
0
2
4
6
8
Z-component
0.4
0.3
0.2
V[m/s]
0.1
0
-0.1
-0.2
-0.3
-0.4
0
2
4
6
8
10
12
14
16
Time [sec]
Fig.35
Example of synthetic seismograms obtained from finite-difference noise simulations for model
M2.2 (i.e., 2nd model of a single layer over halfspace). Top, middle and bottom figures show X,
Y and Z component of ground velocity at receiver 1 for the first 20 seconds of the computation.
66
18
b) Autocorr Vy - rec.1
a) Autocorr Vx - rec.1
1.2
1.2
0.8
0.8
0.4
0.4
0
0
-0.4
-0.4
-0.8
-0.8
-20
-10
0
time [sec]
10
20
-20
c) Autocorr Vz - rec.1
-10
0
time [sec]
10
20
d) Crosscorr Vx1-Vx16
1.2
0.2
0.8
0.1
0.4
0
0
-0.1
-0.4
-0.8
-0.2
-20
Fig.36
-10
0
time [sec]
10
20
-20
-10
0
time [sec]
10
20
Example of crosscorrelation and autocorrelation analysis for model M2.2. a) Autocorrelation of
the X-component of velocity at receiver 1. b) Autocorrelation of the Y-component of velocity at
receiver 1. c) Autocorrelation of the Z-component of velocity at receiver 1. d) Crosscorrelation of
the X-component of velocity between receiver 1 and receiver 16, spaced 84m.
Multi-channel coherency analysis is performed by the use of semblance. The
semblance function for the X, Y and Z component of model M2.2 is shown in Fig.37.
All three measures remain low for the whole time interval. The peak value of the Xcomponent is 0.1, that of the Y-component 0.09 and that of the Z-component 0.3.
These values are bigger than the semblance maxima of model M2.1 and might be
caused by the 3 receiver pairs with significantly higher crosscorrelation maxima.
Nevertheless it is small enough to justify the evidence that the finite-difference noise
synthetics do not contain any common signal. The maximum value is attained near
t=0.
0.12
Semb
Vx
0.08
0.04
0
-0.04
-20
-15
-10
-5
0
5
10
15
20
-10
-5
0
5
10
15
20
-10
-5
0
5
10
15
20
time [sec]
0.08
Semb
Vy
0.04
0
-0.04
-20
-15
time [sec]
0.4
0.3
Semb
Vz
0.2
0.1
0
-0.1
-20
-15
time [sec]
Fig.37
Semblance analysis for model M2.2. Top is semblance for the X-component of the ground
velocity, middle and bottom figures are the Y- and Z-components of the ground velocity for
model M2.2, respectively.
67
H/V spectral ratio
H/V spectral ratios are computed for both, individual receivers and all receivers
averaged. The results are displayed in Fig.38. Some striking differences to the H/V
ratios of model M2.1 can be observed, despite the fact, that only the impedance
contrast between the P- and the S-wave velocity is different in this model. The H/V
ratios increase right from the start, i.e., 0.5Hz, starting from a relatively high value of
1-2 for all time windows and attain a prominent maximum between 2 and 3Hz with a
peak value between 3 and 8.5. With increasing frequency all H/V ratios decrease and
reach a minimum between 4 and 10Hz. In contrast to model M2.1 the H/V ratios tend
more to oscillate in state of approaching a constant value. In general it can be said
that the scatter in the individual H/V spectral ratios is bigger than for model M2.1.
Both the peak values and the values for higher frequencies are bigger in model M2.2.
The averaged H/V ratio starts with a value of 1.56 at 0.5Hz. It increases
constantly and reaches a maximum of 5.68 at 2.1Hz. Then it is permanently
decreasing with increasing frequency, up to 4Hz, where a very small local minimum
is visible. Subsequently the H/V spectral ratio remains at a nearly constant value. Up
to 8Hz the averaged H/V ratio remains under 3. Then it increases slightly with
increasing frequency, however remaining under 3.2.
H/V ratios for time windows of M2.2
Averaged H/V ratio
10
H /V
H /V
10
1
1
4
Fig.38
8
12
frequency [Hz]
16
4
8
12
frequency [Hz]
16
Spectral H/V ratios for model M2.2. Left are the H/V ratios for the individual time windows, right
is the H/V ratio averaged over all individual time windows. The curves are displayed in lin-log
style.
68
Model M2.3
Model layout
Model M2.3 is the third model in the series of models of a single layer over halfspace.
It is characterised by a velocity contrast of 2.024 between the P- and the S-wave in
the sediment layer, which is approximately the same as for model M2.1. The velocity
contrast for S-waves in the layer and the bedrock, respectively, is 1.499. This velocity
contrast is smaller than that of models M2.1 and M2.2 where it is 4. The theoretical
fundamental 1D vertical resonant frequency for vertical SH-wave incidence is 2Hz,
like in the models M2.1 and M2.2. The value of this peak theoretically should be two
times the impedance contrast between the S-wave velocities in the layer and in
bedrock, respectively, i.e., 2 * 1.5 = 3 for model M2.3.
The layout of the finite-difference simulation and an example of a synthetic
seismogram generated for model M2.3 are displayed in Fig.39 and Fig.40,
respectively. The grid spacing is 8m model M2.3. Therefore the minimum distance of
40 grid spacings between the receiver at (150,150) and the first point source means
480m and the maximum distance of 80 grid spacings between a point source and the
receiver (150,150) means 640m. The grid consists of 300x300 grid points in the finer
grid. This means an area of 2400mx2400m.
300
280
260
240
220
200
180
160
140
120
100
80
60
40
20
0
Fig.39
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
Layout of model M2.3. Crosses represent point sources and triangles receivers, respectively.
Autocorrelation and crosscorrelation
The autocorrelation function is in the interval [-0.18, 0.18] for all components and
receivers, except for t=0 where it is 1. Because all receivers show the same
characteristics, autocorrelation with arithmetic normalisation is shown only for
receiver 1 in Fig.41a,b,c. These results, like those for the previous models, indicate
69
that the generation of point sources in model M2.3 really produces random noise
signals.
In order to combine the synthetic seismograms from all receivers to a single
record for the computation of H/V spectral ratios, crosscorrelation and multi-channel
crosscorrelation functions have to be computed to look for any common signal that is
present in the synthetic seismograms for different receivers. Crosscorrelation of all
possible receiver pairs is low in the whole time interval of [-20s, 20s] for all times and
components. The maximum crosscorrelation between two receivers is between -0.37
and 0.34 for the X-component, between –0.47 and 0.51 for the Y-component and –
4.1 and 4.0 for the Z-component of seismic ground velocity. These values, however,
are maximum values. The mean value of the peak crosscorrelation is about 0.2 for all
components. This peak value in the crosscorrelation function occurs at different times
for every receiver pair and is not depending on the distance between receivers. An
example is given in Fig.41d.
X-component
0.2
V[m/s]
0.1
0
-0.1
-0.2
0
2
4
6
8
10
12
14
16
18
20
12
14
16
18
20
12
14
16
18
20
Y-component
0.3
0.2
V[m/s]
0.1
0
-0.1
-0.2
-0.3
0
2
4
6
8
10
Z-component
0.2
V[m/s]
0.1
0
-0.1
-0.2
0
2
4
6
8
10
Time [sec]
Fig.40
Example of synthetic seismograms obtained from finite-difference noise simulations for model
rd
M2.3 (i.e., 3 model of a single layer over halfspace). Top, middle and bottom figures show X,
Y and Z component of ground velocity at receiver 1 for the first 20 seconds of the computation.
Multi-channel coherency analysis is performed by the use of semblance.
Semblance for the X, Y and Z component of model M2.3 is shown in Fig.42. All three
measures remain very low for the whole time interval. The peak value of the Xcomponent is 0.027, that of the Y-component 0.032 and that of the Z-component
0.045. The maximum value is attained near t=0. These values are smaller than all
previous semblance maxima for the models of a single layer over halfspace. This is
strong evidence that the finite-difference noise synthetics do not contain any common
signal.
70
b) Autocorr Vy - rec.1
a) Autocorr Vx - rec.1
1.2
1.2
0.8
0.8
0.4
0.4
0
0
-0.4
-0.4
-0.8
-0.8
-20
-10
0
time [sec]
10
20
-20
c) Autocorr Vz - rec.1
-10
0
time [sec]
10
20
d) Crosscorr Vx1-Vx16
1.2
0.2
0.8
0.1
0.4
0
0
-0.1
-0.4
-0.8
-0.2
-20
-10
0
time [sec]
Fig.41
10
20
-20
-10
0
time [sec]
10
20
Example of crosscorrelation and autocorrelation analysis for model M2.3. a) Autocorrelation of
the X-component of velocity at receiver 1. b) Autocorrelation of the Y-component of the ground
velocity at receiver 1. c) Autocorrelation of the Z-component of the ground velocity at receiver 1.
d) Crosscorrelation of the XY-component of the ground velocity between receiver 1 and receiver
16, spaced 90.5m.
0.03
0.02
Semb
Vx
0.01
0
-0.01
-0.02
-20
-15
-10
-5
0
5
10
15
20
-15
-10
-5
0
5
10
15
20
-15
-10
-5
0
5
10
15
20
time [sec]
0.04
Semb
Vy
0.02
0
-0.02
-20
time [sec]
0.05
Semb
Vz
0.03
0.01
-0.01
-0.03
-20
time [sec]
Fig.42
Semblance analysis for model M2.3.Top is semblance for the X-component of the ground
velocity, middle and bottom figures are the Y- and Z-components of the ground velocity for
model M2.3, respectively.
71
H/V spectral ratio
H/V spectral ratios are computed for both, individual receivers and all receivers
averaged. The results are displayed in Fig.43. Differences to the H/V ratios of the
previous models can be observed. All individual H/V ratios start with a value between
0.7 and 1.6 and most of them have a small minimum between 0.5 and 0.9Hz. Then
all H/V ratios increase permanently and attain a prominent maximum between 2 and
2.6Hz with a peak value between 1.8 and 3.2. With increasing frequency all H/V
ratios decrease and most of them reach a small minimum around 4 followed by a
minor maximum at 10Hz. Only one H/V ratio from an individual time window shows
another characteristic. It attains its global maximum at 1.8Hz and a relatively sharp
minimum at 3.5Hz. In general it can be said that the scatter in the individual H/V
spectral ratios is smaller than for models M2.1 and M2.2.
In the averaged H/V ratio starts with a value of 1.01 at 0.5Hz. It increases
constantly and reaches a maximum of 2.445 at 2.15Hz. Then it is permanently
decreasing reaching a small local minimum of 1.5 at 6Hz. With further increasing
frequency the averaged H/V spectral ratio slightly increases, reaching a value of 1.64
at 9.79Hz.
H/V ratios time windows of M2.3
Averaged H/V ratio
5
4
3
H/V
H/V
1
0.1
2
1
4
Fig.43
8
12
frequency [Hz]
16
4
8
12
frequency [Hz]
16
Spectral H/V ratios for model M2.3. Left are the H/V ratios for the individual time
windows, right is the H/V ratio averaged over all individual time windows. The curves are ploted
on a lin-log scale.
72
9. Discussion
In chapter 8 the results of the finite-difference noise simulations for the four models
under consideration have been presented. The layout of the models has been
described and the synthetic seismograms obtained by noise simulations have been
presented and analysed. The seismograms have been checked for their randomness
by using the autocorrelation function. Synthetic seismograms have been shown to
represent random signals. The seismograms from different receivers have been
investigated for common information content by two-receiver crosscorrelation with
arithmetic and geometric normalisation and by multi-channel crosscorrelation and
semblance. Individual and averaged H/V ratios with different time window length and
with a different exponent of the smoothing function have been computed. The
averaged H/V spectral ratios of the different models are the basis for further
discussion. Therefore, a summarising plot of all averaged H/V ratios is given in
Fig.44.
b) Averaged H/V ratio of M2.1
10
H/V
H/V
a) Averaged H/V ratio of M1
10
1
0.1
0.1
4
8
12
frequency [Hz]
4
8
12
frequency [Hz]
16
16
d) Averaged H/V ratio of M2.3
10
H/V
c) Averaged H/V ratio of M2.2
10
H/V
1
1
1
0.1
0.1
4
8
12
frequency [Hz]
Fig.44
4
8
12
frequency [Hz]
16
16
Averaged H/V spectral ratios for all models in which seismic noise has been numerically
simulated by the finite-difference method. a) Averaged H/V ratio for model M1, i.e., a
homogeneous halfspace. b) Averaged H/V ratio for model M2.1, i.e., a single layer over
halfspace with α1/β1 = 2 and β2/β1 = 4. c) Averaged H/V ratio for model M2.2, i.e., a single layer
over halfspace with α1/β1 = 5.4 and β2/β1 = 4. d) Averaged H/V ratio for model M2.3, i.e., a
single layer over halfspace with α1/β1 = 2.024 and β2/β1 = 1.4993. α1 and β1 are the P- and
the S- wave velocities in the sediment layer, respectively, and β2 is the S-wave velocity in
bedrock.
73
Model M1 – Homogeneous halfspace
The averaged H/V ratio for the homogeneous halfspace, i.e., model M1, is almost
flat. There is a small maximum at 1.1Hz with a peak value of 1.27. The H/V ratio then
is slightly decreasing approaching 0.75. Generally one can say that the H/V ratio is
around 1 for all frequencies in model M1. This means that the horizontal part of the
spectrum of the ground motion velocity is almost equal to the vertical part of the
spectrum of the ground velocity. This result seems reasonable since there is no
reason that would favour ground motion in the horizontal direction compared to
motion in the vertical direction, and vice versa, in a homogeneous halfspace. The
theoretical amplification for the horizontal SH-wave incidence in a homogeneous
halfspace is 2 for all frequencies, i.e., reflection on the free surface. If we compare
the averaged H/V spectral ratio with the theoretical transfer function we find out that
the latter is twice as big as the first. The H/V spectral ratio of ambient seismic noise
underestimates the amplification of seismic ground motion by a factor of 2.
In chapter 3.4 two different interpretations of the H/V ratio have been
proposed. Nakamura’s interpretation (Nakamura 1996) of the H/V ratio is based on
the assumption that seismic noise consists mainly of body waves and that the effect
of surface waves on the H/V ratio can either be neglected or “eliminated”. In this
interpretation the H/V spectral ratio of ambient seismic noise would correspond
directly to the transfer function of S-waves (see Eq.13). Finite-difference simulations
of seismic noise, however, show that the H/V spectral ratio is nearly constant near to
a value of 1. So, for a homogeneous halfspace H/V ratios are not reflecting the Swave transfer function directly but underestimate it by a factor of 2.
The second interpretation of the noise H/V ratio (Konno & Ohmachi 1998)
assumes most of the noise wave-field to consist mainly of surface waves and the H/V
ratio is basically related to the ellipticity of the Rayleigh waves. In a homogeneous
halfspace, however, the ellipticity of the Rayleigh waves is constant for all
frequencies. In chapter 3.4 the ratio of the spectrum of the horizontal component of
ground motion in bedrock divided by the spectrum of the vertical component in
bedrock was set to 1 (see Eq.16). This assumption was experimentally verified with
the bore-hole measurements by Nakamura (1989). This corresponds directly to the
resulting H/V ratio of the finite-difference noise simulations for the homogeneous
halfspace, where the spectrum of the horizontal part of seismic ground velocity and
the spectrum of the vertical component are equal, leading to an H/V ratio of 1 for all
frequencies (compare with Fig.44a). This result may be a confirmation for the second
interpretation of seismic noise H/V spectral ratios, which assumes noise to consist
mainly of surface waves.
Models M2.1, M2.2 and M2.3 – Single layer over halfspace
In the discussion of the series of models of a single layer over halfspace first all
results will be analysed and discussed separately by comparing them to the transfer
functions for a vertically incident plane SH-wave. Then a comparison between the
different models will be made.
Fig.45 shows a comparison between the averaged H/V spectral ratio of model
M2.1 and the transfer function for a vertically incident plane SH-wave. The amplitude
of the H/V ratio has a small minimum at 0.8Hz with a value of 0.7. Then the H/V ratio
is permanently rising, reaching a distinct maximum. The peak value of this maximum
in the H/V ratio is at the frequency of 2.22Hz and its amplitude is 3.7.
74
M 2.1
Module
10
1
0.1
2
Fig.45
4
6
frequency [Hz]
8
Comparison between the averaged H/V spectral ratio for model M2.1 computed from noise
synthetics and the module of the transfer function for the vertically incident plane SH-wave
computed by the matrix method. The solid represents the H/V ratio and the dotted line SH-wave
transfer function, respectively. The axes are plotted in lin-log style.
Up to 4Hz the H/V ratio is then decreasing, attaining a nearly constant value of about
1.7 for frequencies between 4 and 8Hz. The SH-wave transfer function, computed
with the matrix method has its fundamental frequency at 1.95Hz. The module is 8.8
at this frequency. At 4Hz the SH-wave transfer function has its first minimum with a
module of 2 and attains its second maximum with a value of 6.44 at 5.79Hz. The H/V
ratio resulting from the finite-difference noise simulation for model M2.1 therefore,
compared to the SH-wave transfer function slightly overestimates the fundamental
frequency of the site (by about 0.2Hz). The peak value of the fundamental resonant
frequency on the other hand is underestimated by a factor of 2.38. Although in some
H/V spectral ratios of individual time windows a minimum at 4Hz and a second
maximum at 6Hz has been visible, none of those features occurs in the averaged H/V
ratio. Looking at the averaged H/V ratio we can not tell anything about higher order
resonant frequencies.
Fig.46 shows the comparison between the averaged H/V spectral ratio for
model M2.2 and the transfer function for a vertically incident plane SH-wave, which is
identical to the transfer function in M2.1 since models M2.1 and M2.2 differ only in
the P-wave velocity. From 0.5Hz the H/V ratio is constantly rising, reaching a distinct
maximum of 5.7 at a frequency of 2.11Hz. Up to 4Hz the H/V ratio is decaying, and
then it has a nearly constant value of about 2.7. In this model the H/V spectral ratio
better approximates the SH-wave transfer function. The frequency of the first
maximum is very closely resolved (with a difference of 0.16Hz). The amplitude of this
maximum on the other hand is still underestimated by a factor of 1.54 and the second
maximum, as well as the first minimum, is not present in the H/V ratio at all.
75
M 2.2
Module
10
1
2
Fig.46
4
6
frequency [Hz]
8
Comparison between the averaged H/V spectral ratio for model M2.2 computed from noise
synthetics and the module of the transfer function for the vertically incident plane SH-wave
computed by the matrix method. The solid represents the H/V ratio and the dotted line SH-wave
transfer function, respectively.
M 2.3
Module
10
1
2
Fig.47
4
6
frequency [Hz]
8
Comparison between the averaged H/V spectral ratio for model M2.3 computed from noise
synthetics and the module of the transfer function for the vertically incident plane SH-waves
computed by the matrix method. The solid represents the H/V ratio and the dotted line SH-wave
transfer function, respectively.
76
In Fig.47 the comparison between the averaged H/V spectral ratio for model
M2.3 and the transfer function for a vertically incident plane SH-wave is shown. At
low frequencies the H/V ratio is constantly rising, arriving at a distinct maximum of
2.44 at a frequency of 2.1Hz. Then the H/V ratio is decreasing. At frequencies higher
than 4Hz it is nearly constant with a value of about 1.5. The vertically incident plane
SH-wave transfer function reaches its first maximum at 1.94Hz. The value of this
maximum is 3.76. The first minimum with a module value of 1.994 is reached at
3.96Hz. At 5.98Hz a second maximum with a peak value of 3.19 is visible. In the
case of model M2.3 the H/V spectral ratio overestimates the fundamental frequency
by 0.16Hz and underestimates the amplitude by a factor of 1.54. The second
maximum as well as the first minimum of the SH-wave transfer function is not present
in the averaged H/V spectral ratio.
The reported results are consistent with the paper of Bard (1999), who gives a
list of studies during which seismic noise was measured. In those studies good
correlation between the value of the first maximum of the H/V spectral ratio and the
fundamental frequency of the SH-wave transfer function were found. The amplitudes,
however, were always underestimated. All models of the series of a single layer over
halfspace have been designed in order to have their fundamental S-wave resonant
frequency at 2Hz in the perfectly elastic medium. This fundamental frequency is
resolved with good accuracy in all models, although it is always slightly
overestimated. In model M2.1 the difference between the fundamental peak
frequency in the H/V ratio and the fundamental resonant frequency of for a plane
vertically incident SH-wave, computed by the matrix method, is 0.2Hz. In model M2.2
it is 0.16Hz and in model M2.3 it is also 0.16Hz.
On the other hand the maximum amplitude of the SH-wave transfer function is
not well established by the H/V maximum amplitude. The amplitude of the
fundamental resonant frequency, computed by the matrix method is underestimated
in all three models. In model M2.1 it is underrated by a factor of 2.38, in model M2.2
by a factor of 1.54 and in model M2.3 by a factor of 1.54 as well. This
underestimation may be caused by the choice of the parameter b in the smoothing
function (see Eq.56) proposed by Konno & Ohmachi (1998). This smoothing function
has been introduced in order to avoid singularities in the H/V ratios. As pointed out
before, the H/V ratios of the fundamental-mode Rayleigh waves become infinite at
some peak periods. These difficulties can be avoided when a smoothing process is
introduced before calculating the H/V ratio. The amplitude of the H/V peak then
implies no direct relationship to the amplitude of the S-wave resonant frequency, but
rather is a function of the parameter b in the smoothing function. This can be seen in
Fig.32 where the peak value of the H/V spectral ratio decreases for smaller
parameters b of the smoothing function. In this study the value of the parameter b
has been chosen equal to 15 for all models. The H/V spectral ratios computed with
this value of the parameter b generally underestimate the amplitude of the
fundamental S-wave resonant frequency. A very interesting fact is that the amplitude
of the fundamental S-wave resonant frequency is underestimated by different factors
in model M2.1 and in model M2.2, though theoretically the transfer functions of model
M2.1 and model M2.2 are exactly the same, since these models differ from each
other only by the P-wave velocity and the S-wave transfer function is only influenced
by the ratio between the S-wave velocity in the layer and in halfspace, i.e., β2/β1,
respectively. If we compare models M2.2 and M2.3, which have different ratios
between the P- and the S-wave velocity in the layer, i.e., α1/β1, but the same ratio
β2/β1, we find out that the fundamental S-wave resonant frequency is underrated by
77
the same factor, i.e., 1.54. These results indicate that not only the ratio between the
S-wave velocities in the layer and in the halfspace, but also the ratio between the Pand the S-wave velocity in the layer, affects the amplitude of the H/V spectral ratio.
This makes a direct correlation between the peak value of the H/V spectral ratio and
the amplitude of the transfer function of vertically incident SH-waves even more
controversial. What can be said clearly is that the peak value of the H/V spectral ratio
is lower than the value of the fundamental peak of the SH-wave transfer function in
all models and independently of the parameter b of the smoothing function. This
result corresponds with the paper of Bard (1999) who refers to 16 different studies
which all report an H/V spectral ratio smaller than the SH-wave transfer function.
10. Conclusion
In this diploma thesis site effects have been described as a major reason for damage
during earthquakes. The importance of estimating site effects during future
earthquakes was pronounced. A review of possible site effects during an earthquake
has been given. The site effects have been related to different geological structures.
Topographical effects and effects of the alluvial cover have been described.
Different techniques for the estimation of site effects during future earthquakes
have been presented. The H/V spectral ratios of seismic noise have been introduced
as a possible, low-cost technique for site effect estimation, although bearing some
restrictions and obscurities.
The role of numerical simulation in seismology has been stressed and the
finite-difference method has been identified as the most used simulation method in
the earthquake seismology and it has been described in some detail.
Seismic noise has been simulated in 3 different models of a sediment layer
over bedrock by the finite-difference method. The synthetic seismograms from these
simulations have been analysed for autocorrelation and crosscorrelation. The H/V
spectral ratios of seismic noise have been computed and discussed. Averaged H/V
ratios have been compared to SH-wave transfer functions computed by the matrix
method.
The results indicate that the fundamental S-wave resonant frequency can be
detected by the H/V spectral ratios of seismic noise with satisfactory accuracy. A
direct relationship between the most prominent peak of the H/V spectral ratio and the
amplitude of the first maximum in the S-wave transfer function, on the other hand,
could not be found. The amplitudes of the H/V ratios have shown to depend on the
choice of the parameter b in the smoothing function for the H/V ratios, proposed by
Konno & Ohmachi (1998). The results indicate that not only the ratio between the Swave velocities in the layer and in bedrock, respectively, but also the ratio between
the P- and the S- wave velocities in the layer affects the amplitude of the H/V spectral
ratio. This, however, is not the case for the SH-wave transfer functions.
78
References
Aki. K., 1957. Space and time spectra of stationary stochastic waves with special
reference
to microtremors, Bull. Earthq. Res, Inst., 35, 415-456.
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Appendix I.
The seismic response of one layer over rigid halfspace
to a vertically incident SH wave.
Remark: This derivation was made with the program Mathcad2000, where ‘ := ’ means the equality sign ‘ = ’.
In this model we have a layer with finite depth H over an infinitely deep rigit halfspace HS.
Depth is z.Both, layer and halfspace show no lateral heterogenity. v is y component of
displacement and
µ is the second Lamè parameter for the layer.
Displacement v due to vertical incident SH
wave in the layer is a superposition of upand downgoing harmonic waves with
amplitudes v´ and v´´ respectively.
The stress tensor is defined by the only
non-zero component
τ zy .
v :=
 v´ ⋅ exp  i⋅ ω ⋅  t − ( z − H )   + v´´ ⋅ exp  i⋅ ω ⋅  t + ( z − H )   







β1   
β1






τ zy := µ ⋅
∂
∂z
( 1)
v
( 2)
Inserting 1 in 2 yields:
−1
 ⋅ i⋅ ω ⋅ exp  i⋅ ω ⋅  t − ( z − H )   + v´´ ⋅ i⋅ ω exp  i⋅ ω ⋅  t + ( z − h )   






β1
β1   
 β1 





τ zy := µ ⋅  v´ ⋅ 

τ zy := µ ⋅ i⋅
ω
β1
denote
⋅  v´ ⋅ exp  − i⋅ ω ⋅

(z − H)

β1
( 3)
 − v´´ ⋅ exp  i⋅ ω ⋅ ( z − H )   ⋅ exp ( i⋅ ω ⋅ t )



β1  


τ zy := τ
traction free surface boundary condition
τ ( 0 ) := 0
τ ( H ) := µ ⋅ i⋅
ω
β1
⋅ ( v´ − v´´ ) ⋅ exp ( i⋅ ω ⋅ t )
( v´ − v´´ ) :=
( 4)
τ ( h ) ⋅ β1
µ ⋅ i⋅ ω ⋅ exp ( i⋅ ω ⋅ t )
( 5)
What we want to find is v(0)
v ( H ) := ( v´ + v´´ ) ⋅ exp ( i⋅ ω ⋅ t )
( 6)
impose 2 conditions: 1) time independence
2) v´´=1 unit displacement in HS
then
v ( H ) := ( v´ + 1 )
v ( H ) − 1 := v´
 ( i⋅ ω ⋅ H ) 
v ( 0 ) := v´ ⋅ exp 

v ( 0 ) :=
cos ( ω ⋅ H )
β1
β1
 ( − i⋅ ω ⋅ H ) 
 + v´´ ⋅ exp 


⋅ ( v´ + v´´ ) + i⋅


β1
sin ( ω ⋅ H )
β1
( 7)
⋅ ( v´ − v´´ )
insert 5 and 6 ( both at t=0) into 8 and arrive at
85
( 8)
v ( 0 ) := cos
 ( ω ⋅ H )  ⋅ v ( H ) + sin  ( ω ⋅ H )  ⋅  τ ( h ) ⋅ β 1 




 β1 
 β1   µ ⋅ω ⋅i 
( 9)
if we impose the rigit halfspace condition then:
=>
v ( H ) := 0
v´ := − v´´
( 10 )
inserting in 9 leads to:
v ( 0 ) := sin
 (ω ⋅ H )  ⋅  τ ( H ) ⋅ β 1 


 b1   µ ⋅ ω ⋅ i 
( 11 )
insert 5 into 11
v ( 0 ) := sin
 ( ω ⋅ H )  ⋅ 2 ⋅ v´´ ⋅ exp ( i⋅ ω ⋅ t )


 β1 
( 12 )
it follows from condition 2 that v´´ = 1 and so:
 ( ω ⋅ H )  ⋅ 2 ⋅ exp ( i⋅ ω ⋅ t )


 β1 
 (2⋅ π f ⋅ H ) 
H ( f ) := 2 sin 

 β1 
v ( 0 ) := sin
let
( 13 )
be amplitude variation of harmonic SH wave
ω := 2 ⋅ π ⋅ f
then :
v ( 0 ) := H ( f ) ⋅ exp ( 2 ⋅ π ⋅ i⋅ f )
( 14 )
∂
Maxima of H(f) are at
∂f
∂
∂f
H ( f ) := 4 ⋅ π ⋅
so
H
β1
(2⋅ π ⋅ f ⋅ H )
β1
=>
⋅ cos
f n :=
:=
( 15 )
 (2⋅ π ⋅ f ⋅ H ) 


β1


( 2⋅ n − 1)
2
( 2⋅ n + 1) β 1
4
H ( f ) := 0
H
( 16 )
( 17 )
π
f 1 :=
β1
4⋅ H
f 2 :=
3β 1
4⋅ H
f 1 :=
fn are frequency maxima of the response of a single layer over rigit halfspace to vertical
incident SH wave with unit amplitude.
86
5⋅ β 1
4⋅ H
etc.
Appendix II.
3D 4th-order displacement-stress staggered-grid
finite-difference scheme
(Adapted from Moczo et al., in press)
87
88
89
90
91