Numerical Modeling of Seismic Noise in Canonical Structures

Transcription

1 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

2 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

3 Contents 1. Introduction 1 2. Site effects Seismic ground motion at a site Site effects during earthquakes Examples of site effects Local geological structure Basic types of local geological structures and associated site effects Flat free surface Topographical structures Sedimentary structures Methods of investigation of site effects Ambient seismic noise Introduction Nature of the noise wave-field Methods of investigation H/V spectral ratios Partial conclusive remarks Numerical modeling of seismic ground motion The role of numerical modeling / simulation Review of methods Advantages and disadvantages of numerical modeling 30 III

4 5. The finite-difference method The principle Finite-difference approximations Formulations of the equation of motion Finite-difference spacetime grids Homogeneous and heterogeneous finite-difference schemes The goal of the diploma thesis Models and methods of computation and analysis Computations Design of the models and space-grids for the finite-difference computations Generation of point sources of seismic noise and finite-difference modeling Filtering in the finite-difference computations Models Analysis of synthetic seismograms Results Homogeneous halfspace One layer over halfspace Discussion Conclusion 78 References 79 Appendices Appendix I. The seismic response of one layer over rigid halfspace to a vertically incident SH wave. 85 Appendix II. 3D 4 th -order displacement-stress staggered-grid finite-difference scheme 87 IV

5 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. Earth s surface Site Site Local geological structure S O U R C E Medium between source and site 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

6 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 5 th 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

7 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, u i, u i t, ²u i t², F[u i ], ui F[ ], t 2 u i F[ ] t² - anomalies in the maximum ground motion and/or its Fourier transform, u, i,max F[u i,max ], (2) - anomalies in the differential motion in time and frequency domain, ui F[u i] x i, x i (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,...). (1) 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, , 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, , Mw = 8.1 This is perhaps the best known example of a site effect. More than people were killed and more than 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

8 Some more examples of recent earthquakes during which site effects were observed: 1976 Friuli, Italia 1979 Imperial Valley, California 1980 Guerrero, Mexico 1981 Irpinia, Italy 1985 Chile 1988 Spitak, Armenia 1989 Loma Prieta, California 1990 Iran 1990 Philippines 1994 Northridge, California 1995 Kobé, Japan 1995 Southern Mexico 1999 Izmit, Turkey 1999 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 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 T < 0.05, 10 > sec. In Austria, the range is f < 0.5, 5 > Hz or f < 2, 10 > Hz The range of the S-wave velocity in near surface geological structures is about 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

9 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 10 1 metres 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 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

10 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) 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 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

11 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

12 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 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

13 The sequence of resonant frequencies for a vertical incident S-wave for a soft layer over rigid half-space is given by f n = ( 2n+1 )β 1 / 4 H ; n=0,1, 2,..., (6) Where β 1 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: f n = ( 2n+1 )β 1 / (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 )+( ρ 1 β 1 /ρ 2 β 2 )² sin²( ωh/β1 ) } -.5 (8) Here, ω is the angular frequency, β 1, β 2 and ρ 1, ρ 2 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 Uw ( ) 4 2 Fig w 90 Amplification factor for the vertical incident of a plane SH-wave in a flat layer with H=60m, β 1 =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 ( f n,d =2nβ 1 /4H ), and the value of 2 ρ 2 β 2 /ρ 1 β 1, which is two times the impedance contrast between soft and stiff material ( f n,c =(2n+1)β 1 /4H ). 9

14 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: t ts s(t) = exp[ [ ωp ]²]cos[ ωp(t t s) +Ψ] γ, (9) as the source-time function of the incoming wave-field. By using different parameters ω p, t s 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

15 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

16 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

17 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

18 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= 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

19 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 f 0 =1.31 f h (where f h is the 1D fundamental frequency) and f 1 =2.3 f h. 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 (f 0 =1.3 f h ) 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

20 The first higher mode shows displacement nodes at x/l= 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) Anti-plane shear SH resonance mode. ii) SV resonance, which is an in-plane shearing pattern and may be thought of as rocking of the whole valley around its central axis. iii) 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 t 0 and t /f 0 (solid lines). In-plane bulk mode (top), in-plane shear mode (middle) and anti-plane shear mode (bottom).(reproduced from Bard & Bouchon, 1985) 16

21 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

22 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, f p =1.23Hz, Ψ=π/2 and γ=6.5, f p =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 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

23 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

24 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

25 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 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

26 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 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

27 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 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 (S NV ) and horizontal (S NH ) part of the noise amplitude spectrum can be represented as S NH (f) = S b H (f) + S s H (f) = H t (f). R b H (f) + S s H (f), (10) S NV (f) = S b V (f) + S s V (f) = V t (f). R b V (f) + S s V (f). (11) 23

28 Here H t (f) and V t (f) represent the true transfer functions for horizontal and vertical components, respectively, and R b H (f) and R b V (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 S NH and S NV (A NHV ) can be written after some algebra as A NHV = [H t. A r NHV + β. A s ] / [V t + β] (12) where A r NHV 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 = S s V (f) / R b V (f) and A s is the horizontal to vertical ratio due only to surface waves, i.e., A s (f) = S s H (f) / S s V (f). Nakamura (1996) then imposes four assumptions: a) the vertical component is not amplified at f H0 b) the H/V ratio on rock is equal to 1 at f H0 c) β is much smaller than 1 at f H0 d) β. A s (f H0 ) is much smaller than H t (f H0 ) Accordingly he then comes to the final result A NHV = H t (f H0 ) (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 S s V (f) vanishes around f H0. Assumption d), i.e., β.a s (f H0 ) = β. S s H (f H0) / S s V (f H0) = S s H (f H0). R b V (f H0) << H t (f H0 ) 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 S- wave 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

29 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 S E (ω) = H s (ω) / H b (ω) (14) which is the Fourier spectrum of horizontal component of motion of body waves on the surface, i.e., H s (ω), divided by motion at the base of the layer, H b (ω). (see chapter 2.6. Reference site techniques). To compensate Eq.14 by the source spectrum A S (ω), a modified site effect spectral ratio is defined as S M (ω) = S E (ω) / A S (ω) = [ H s (ω) / V s (ω) ] / [ H b (ω) / V b (ω) ] (15) In a final assumption H b (ω) / V b (ω) is set to 1 for all frequencies of interest, which was experimentally verified with bore-hole measurements by Nakamura (1989). Then S E (ω) ~ S M (ω) ~ H s (ω) / V s (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

30 u j (ω,r) = [L(ω)/2] [u /w ] j A j H 1 2 (k j r) ω, (17a) w j (ω,r) = [L(ω)/2] A j H 0 2 (k j r) 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), A j is medium response and k j 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, V SH /V SL 2.5) particle motion is retrograde for all frequencies. In type 2 models with higher velocity contrast (V SH /V SL ~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 (V SH /V SL 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: V L is 250m/s for type 1, 200m/s for type 2 and 50m/s for type 3. V H 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

31 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 S- waves 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) Partial conclusive remarks 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 S- waves (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,