--Manuscript Draft-- Diffuse Field Concept to Previously Observed Data. Kyoto University Uji, Kyoto Pref. JAPAN. Shinichi Matsushima, Ph.D.

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1 Bulletin of the Seismological Society of America Applicability of Theoretical Horizontal-to-Vertical Ratio of Microtremors Based on the Diffuse Field Concept to Previously Observed Data --Manuscript Draft-- Manuscript Number: Article Type: Section/Category: Full Title: Corresponding Author: Corresponding Author's Institution: Corresponding Author Order of Authors: BSSA-D R2 Article Regular Issue Applicability of Theoretical Horizontal-to-Vertical Ratio of Microtremors Based on the Diffuse Field Concept to Previously Observed Data Hiroshi Kawase, Ph.D. Kyoto University Uji, Kyoto Pref. JAPAN Kyoto University Hiroshi Kawase, Ph.D. Shinichi Matsushima, Ph.D. Toshimi Satoh, Ph.D. Francisco J. Sánchez-Sesma, Ph.D. Abstract: Author Comments: Horizontal-to-vertical spectral ratios of microtremors (MHVRs) have been interpreted as representing either the Rayleigh-wave ellipticity or the amplitude ratio of the sum of Rayleigh and Love waves in a horizontally layered structure. However, based on the recently established diffuse field concept, the theoretical form of MHVR has been proposed to be the square root of the ratio between the imaginary part of the horizontal Green's function on the surface and that of the vertical one. The theory assumes that the energy of a wave field inside the earth will be equipartitioned among the various states in three-dimensional space. In the case of microtremors, this may occur for randomly applied point-force loadings on the surface after sufficient lapse time to allow multiple scattering. Recent works on diffuse fields suggest that equipartition may arise in several ways, but understanding the emergence of equipartition in realistic settings requires further scrutiny. In the meantime, the resulting formula is quite simple and its meaning has theoretical support from deterministic exact solutions. Here we use as references observed microtremor data from several sites that were reported previously and validate the diffuse field method (DFM) as an alternative method to explain observed MHVR. We use only sites with reliable velocity structures in order to compare different methods quantitatively. As a result, we found that the DFM solutions with the corresponding one-dimensional layered structures well explain the observed MHVRs for most of the sites. Thus, we believe that MHVR can be used to invert a onedimensional velocity structure by using DFM as a theoretical tool. Dear Editor of BSSA: Thank you for your time and effort. Suggestions from AE are valuable for our future research. Best regards, Hiroshi Kawase, DPRI, Kyoto University Suggested Reviewers: Opposed Reviewers: Response to Reviewers: Powered by Editorial Manager and ProduXion Manager from Aries Systems Corporation

2 Manuscript Click here to download Manuscript: BSSA-D-00134r1-Text&Table&Caption(final).docx 1 2 Applicability of Theoretical Horizontal-to-Vertical Ratio of Microtremors Based on the Diffuse Field Concept to Previously Observed Data Hiroshi Kawase, Shinichi Matsushima, Toshimi Satoh, and Francisco J. Sánchez-Sesma Corresponding author Hiroshi Kawase kawase@zeisei.dpri.kyoto-u.ac.jp Disaster Prevention Research Institute, Kyoto University Gokasho, Uji, Kyoto , Japan TEL: FAX:

3 17 18 Abstract Horizontal-to-vertical spectral ratios of microtremors (MHVRs) have been interpreted as representing either the Rayleigh-wave ellipticity or the amplitude ratio of the sum of Rayleigh and Love waves in a horizontally layered structure. However, based on the recently established diffuse field concept, the theoretical form of MHVR has been proposed to be the square root of the ratio between the imaginary part of the horizontal Green s function on the surface and that of the vertical one. The theory assumes that the energy of a wave field inside the earth will be equipartitioned among the various states in three-dimensional space. In the case of microtremors, this may occur for randomly applied point-force loadings on the surface after sufficient lapse time to allow multiple scattering. Recent works on diffuse fields suggest that equipartition may arise in several ways, but understanding the emergence of equipartition in realistic settings requires further scrutiny. In the meantime, the resulting formula is quite simple and its meaning has theoretical support from deterministic exact solutions. Here we use as references observed microtremor data from several sites that were reported previously and validate the diffuse field method (DFM) as an alternative method to explain observed MHVR. We use only sites with reliable velocity structures in order to compare different methods quantitatively. As a result, we found that the DFM solutions with the corresponding one-dimensional layered structures well explain the observed MHVRs for most of the sites. Thus, we believe that MHVR can be used to invert a one-dimensional velocity structure by using DFM as a theoretical tool

4 41 Introduction Horizontal-to-vertical spectral ratios of microtremors (MHVRs) are useful for finding the dominant frequency of a site, and are therefore useful for site characterization and microzonation because single-station measurement of microtremors is much easier than measurement of earthquake ground motions and easier than any other measurements of exploration methods with active sources, such as reflection and refraction surveys. The most important issue associated with the use of MHVR as a simple and easy exploration method lies in its theoretical interpretation. They have been traditionally interpreted as representing either the Rayleigh-wave ellipticity (Horike, 1980; Lermo and Chávez-García, 1994; Satoh et al., 2001b; Satoh, Kawase, Iwata et al., 2001; Malischewsky and Scherbaum, 2004) or the S-wave amplification directly (Nakamura, 1989; Nakamura, 2000; Bonnefoy-Claudet et al., 2008; Herak, 2008) for a horizontal stack of layers. Konno and Ohmachi (1998) used a strong smoothing function to obtain a finite peak amplitude from Rayleigh-wave ellipticity and found that the theoretical peak amplitude expressed as a mixture of fundamental modes of Love and Rayleigh waves yielded a similar value to S-wave amplification. Then, the interpretation of MHVR as ratios of horizontal and vertical component of surface waves with a mixture of multiple modes of Love and Rayleigh waves was used to determine the subsurface velocity structure by Arai and Tokimatsu (2004). However, such approaches require estimating either the relative power of horizontal and vertical microtremor sources or the relative amplitude ratio between Rayleigh and Love waves; these depend on the site environment and the frequency of interest in general, although Picozzi et al. (2005) showed that theoretical MHVRs are quite stable on the choice of the ratio for a wide range of variation 3

5 in the case of their target sites. We also need to assume that relative excitation of higher modes should follow the medium response for a source on the surface. It is difficult to prove the validity of the necessary assumptions, and it is more computationally laborious to calculate higher modes of surface waves and sum their contributions, compared with simply using the fundamental mode of Rayleigh-wave ellipticity. In follow-up studies (e.g., Parolai et al., 2005; Arai and Tokimatsu, 2005; Picozzi et al., 2005; Picozzi and Albarello, 2007; Arai and Tokimatsu, 2008; Castellaro and Mulargia, 2009) the fixed ratios of Love and Rayleigh waves have been used irrespective of the site environment. As evidenced by significant numbers of studies using array measurements of microtremors (e.g., Aki 1957; Horike 1980, 1985; Matsushima and Okada, 1990; Kawase et al., 1998; Satoh et al., 2001a), it would be unreasonable to deny that major contributions to microtremors are actually being made by surface waves. Theoretical simulations for random surface sources have also provided supporting evidence (e.g., Cornou et al., 2004; Albarello and Lunedei, 2010; Lunedei and Albarello, 2010). Reviews of array methods for exploring underground structures can be found in Bard (1998), Okada (2003), and Bonnefoy-Claudet et al. (2004). Thus, the approach by Arai and Tokimatsu (2004) seems reasonable as a theoretical interpretation of MHVR. However, MHVR gives an indication of direct S-wave amplification, as many previous studies have shown after the initial proposal by Nakamura (1989). In fact, such an interpretation has been used for various studies and practical designs because of its simplicity. Through extensive comparisons of the spectral ratios of earthquakes and microtremors in Kushiro, Japan, Horike et al. (2001) found that MHVR yields similar characteristics to horizontal-to-vertical spectral ratios of earthquake ground motions (EHVR), although at several sites MHVRs tend to be a little smaller in amplitude. They also found that horizontal 4

6 spectral ratios of microtremors with respect to a reference rock site (horizontal-to-horizontal ratio (HHR) or standard spectral ratio (SSR)) in a small (several hundred meters) array or ratios of MHVRs share more or less similar characteristics to HHRs of earthquake ground motions. However, the correspondence of EHVR or SSR to MHVR or its ratios does not necessarily mean that MHVR can be considered as the site amplification factor of S-waves. This is because if there is a strong impedance contrast among soil layers, then resonance phenomena will emerge at similar frequencies for either surface wave incidence or S-wave incidence (e.g., Lachet and Bard, 1994; Malischewsky and Scherbaum, 2004). Thus, we cannot rely on phenomenological evidence to support the interpretation of MHVR as being directly related to the site amplification factor of S-waves. After a long history of such microtremor studies, however, it has been recently suggested that the MHVR corresponds to the square root of the ratio of the sum of the imaginary parts of horizontal displacements for horizontal unit harmonic loads, Im[G 11 ] + Im[G 22 ], and the imaginary part of vertical displacement for a vertically applied unit harmonic load, Im[G 33 ] (Sánchez-Sesma, Rodríguez, et al., 2011), based on the diffuse field concept (Perton et al., 2009). This formula is derived from the fact that the autocorrelation of the observed wave-field component corresponds to the energy radiated into the medium by a unit load in the same direction, which can be exactly represented by the imaginary part of the Green s function at the observed location, x, for a given direction i, as Im[G ii (x, x; ω)]. This implies energy equipartition of the three-dimensional wave field in space for a distribution of random sources. The meaning of the formula has theoretical support from deterministic exact solutions (Sánchez-Sesma, Weaver, et al., 2011). The proposed formula is also valid for observation at depth, as shown in a recent work by Lontsi 5

7 et al. (2015) where the MHVRs observed by borehole receivers at different depths were used to invert S- and P-wave velocity structures. Another recent work on diffuse fields in layered media (García-Jerez et al., 2013) showed that their theoretical predictions for low- and highfrequency asymptotes from the separate calculation of various states (including P- and S- body waves and Love- and Rayleigh-waves as well) suggest that equipartition may arise in microtremors. For seismic codas, equipartition has been recognized in carefully planned experiments (e.g., Hennino et al., 2001; Margerin et al., 2009). In the latter, the ratio of V/H was shown to converge to the theoretical prediction including surface waves. For EHVR of the whole S- wave part, Kawase et al. (2011) successfully derived the corresponding formula for body waves incident to a layered half-space, by applying a concept and derivation process similar to MHVR. In that case we have a very simple formula as a ratio between a transfer function for horizontal motion on the surface induced by vertically incident S-waves at the bedrock level and a transfer function for vertical motion induced by vertically incident P-waves. The square root of the bedrock s P-wave and S-wave velocity ratio is used as a correction coefficient for the incident wave amplitude. The validity of the method used to derive underground S-wave structures from EHVRs has been reported by Ducellier et al. (2013) and Nagashima et al. (2014). On the other hand, Salinas et al. (2014) compared observed EHVR and MHVR at one site inside the Mexico City Basin and found that the observed EHVR is similar to the theoretical MHVR, especially in the high frequency range. This can be interpreted as a consequence of strong damping of waves traveling inside the deep sediments there. It has been suggested that seismic noise does not seem to be a diffuse field (Mulargia and Castellaro, 2008; Mulargia, 2012). These papers argue that observed microtremors do 6

8 not exhibit total isotropy on the horizontal plane. However, Matsushima et al. (2014) recently showed the applicability of the two- and three-dimensional (2D/3D) irregular ground extensions of the MHVR method for a site with strong lateral heterogeneity. For a laterally heterogeneous underground structure, the medium responses due to point forces are different for different directions (Guéguen et al., 2007; Uebayashi et al., 2012), and so horizontal motions will be different even for isotropic input. Thus, anisotropy of observed microtremors does not necessarily indicate violation of the diffuse field assumption in general. In any event, understanding the emergence of equipartition in realistic settings requires further scrutiny. In one such effort, Pilz and Parolai (2014) recently analyzed statistical properties of microtremors as time-evolutionary characteristics of the squared average displacement and found that over very short durations (<0.1 s) microtremors show a highly correlated (ballistic) nature, but that over longer durations (> 1.0 s) they show a weakly correlated nature with diffusive characteristics. They also reported that at all of their target sites in Tajikistan they cannot reject the hypothesis of directional isotropy based on the same procedure as in Mulargia and Castellaro (2008) and Mulargia (2012). In this paper, we would like to revisit previous studies and check the applicability of the newly proposed theoretical formula for MHVR, since we need to provide as many examples as possible in order to establish the proposed MHVR technique as an easy-to-use yet reliable method for geophysical exploration. It is not easy to prove (or disprove) the applicability of the theoretical prediction through simple comparison with observed data, because we need a reliable velocity model with which we can calculate the theoretical prediction, preferably down to the bedrock. We typically have a velocity structure based on the logging data from boring in a shallow part of a target site; however, such data may not be appropriate for reproducing observed MHVR, depending on the depth, geology, and 7

9 frequency, due to various practical reasons (differences in location, method used, insufficient depth, 2D/3D effects, and so on). Besides, it is quite rare to have boring data down to bedrock, and so we need to use velocity data derived by some indirect methods, although this again may not be appropriate for reproducing observed MHVR. Still, we depend solely on the information provided by the literature throughout this paper and compare the theoretical MHVRs with the observed values without any modification, because our aim here is to show applicability through fair comparison. Once we establish the validity of the proposed method, we can invert S-wave velocity structures from observed MHVRs, which can be used to quantify S-wave amplification factors Theoretical MHVR Formula Since theoretical formulation of MHVRs has already been reported by Sánchez-Sesma, Rodríguez et al. (2011), we show only its essence in this section, doing so for the sake of completeness. Within a 3D diffuse equipartitioned field, the average cross correlations of displacement between two points x A and x B can be written as (Sanchez-Sesma et al., 2008): u i (x A, ω)u j(x B, ω) = 2πE S k 3 Im[G ij (x A, x B, ω)] (1) where x A and x B are position vectors, ω is circular frequency, u i is displacement in 180 direction i, * indicates the complex conjugate, angular brackets denote the azimuthal 181 average, E S = ρω 2 S 2 is the energy density of S-waves, k = ω β is the wavenumber of 8

10 S-waves, β is the S-wave velocity, and S 2 is the average spectral density of S-waves. Green s function G ij (x A, x B, ω) gives the displacement at x A in direction i produced by a unit load applied at x B in direction j (Sánchez-Sesma, Rodríguez et al., 2011). To calculate the theoretical energy density at a given point x A, we rewrite Equation 1 assuming x A = x B as E(x A ) = ρω 2 u m (x A )u m(x A ) = 2πμE S k 1 Im[G mm (x A, x A )] (2) where μ is shear modulus. We can express MHVRs in terms of energy density. For instance, E 1 (x, ω) is proportional to u 2 1 = H 2 1. It is common practice to eliminate the angular brackets, writing the expression for the average MHVR as 192 H(ω) V(ω) = E 1(x, ω) + E 2 (x, ω) E 3 (x, ω) (3) where E 1, E 2 and E 3 are the energy densities and the subscripts refer to degrees of freedom (1 and 2 are horizontal; 3 is vertical). Equation 3 is the form adopted by Arai and Tokimatsu (2004). Using Equation 2 we can write: 198 H(ω) V(ω) = Im[G 11(x, x; ω)] + Im[G 22 (x, x; ω)] Im[G 33 (x, x; ω)] (4) 9

11 This equation (Sánchez-Sesma, Rodríguez et al., 2011) shows that MHVR corresponds to the square root of the ratio of the sum of the imaginary parts of two horizontal displacements for horizontally applied unit harmonic loads, Im[G 11 ] and Im[G 22 ], and the imaginary part of vertical displacement for a vertically applied unit load, Im[G 33 ] In an equipartitioned field, the formulation will be valid even if we take only one horizontal component of Equation 3, so the directional MHVR can be derived as 206 H m (ω) V(ω) = E m(x, ω) E 3 (x, ω) (m = 1, 2) (5) Then again, from the relation between the energy densities of a diffuse field with the imaginary part of Green s function, we can obtain the directional MHVR as follows: 210 H m (ω) V(ω) = Im[G mm(x, x; ω)] Im[G 33 (x, x; ω)] (m = 1, 2) (6) Actual calculations of the imaginary parts of Green s functions in these equations are performed by the conventional discrete wavenumber summation method of Bouchon (1981), with an imaginary part of 0.7 /Tw in the circular frequency, where the duration Tw is set to be 50 s. Because of the artificial damping due to this imaginary frequency component, we assume a small amount of material damping (h = 1.1% or Q = 45)

12 218 Comparison with MHVRs in Satoh et al. (2001b) In a previous study, Satoh et al. (2001b) examined differences of empirical site characteristics, including HVRs and HHRs, among the time segments of P waves, S-waves, P- and S-wave codas, and microtremors. They used records at 20 sites in and around the Sendai Basin in Tohoku, Japan. They reported that HVRs for S-wave codas become similar to MHVRs with increasing elapsed time at a frequency range lower than 3 Hz at soft soil sites. By contrast, at a rock site and at two hard soil sites, HVRs for S-wave codas agree well with EHVRs. By using whole sedimentary structures above the bedrock predating the Tertiary age, as determined from boring data for the shallower (<80 m) part and the array method of microtremors for the deeper part (Satoh et al., 2001a), they found that theoretical HVRs for the fundamental mode of Rayleigh-waves were consistent with the observed MHVRs. Although it is interesting to compare EHVRs and MHVRs in both theory and observation, we focus our attention here mainly on MHVRs. Figure 1 shows a map of observation stations, at which both microtremors and earthquakes were recorded. There were three different station networks operated by three different institutions, but we show only two networks used in this paper. From among the thirteen sites, we use only seven sites from the Building Research Institute (BRI) network (Kitagawa et al., 1994) and one site from the K-NET (Kinoshita, 1998) because only these sites have velocity structures down to the bedrock (Satoh et al., 2001a; Satoh, Kawase et al., 2001). Tables 1 to 8 show velocity and density profiles at eight target sites, and Figure 2 shows their S-wave velocity profiles. The shallower parts are based on logging data from boring (Kitagawa et al., 1994; Kinoshita, 1998), while the deeper parts are determined by the 11

13 Rayleigh-wave dispersion from array measurements of microtremors (TAMA, TRMA, ARAH, OKIN, and MYG015) or by matching the peak frequency of the Rayleigh-wave ellipticity (modifying thickness with a prefixed velocity profile) to the MHVR peak frequency less than 1 s (NAKA, NAGA, and SHIR). TAMA has the hardest (shallowest) soil structure, while OKIN has the deepest basin structure. Figures 3 and 4 show comparisons between the observed MHVRs and the theoretical MHVRs determined by the diffuse field method (DFM) of Sánchez-Sesma, Rodríguez et al. (2011) at eight target sites in Sendai. For comparison we also plot HVR curves calculated by the simple Rayleigh-wave ellipticity, which are labeled as the surface wave method (SWM). First, at the TAMA site, both DFM and SWM failed to reproduce the observed HVR, which does not show any significant peaks or troughs. Since this site is in a hilly zone with strong topography and less impedance contrast, microtremors there may not be characterized strongly by the one-dimensional underground structure. At ARAH, NAKA, OKIN, and SHIR, the agreement between DFM values and observed values is quite good. When we compare DFM and SWM, SWM values show peaks of infinite amplitude and troughs of zero amplitude and so we cannot use amplitude as a basis for matching observed data. In contrast, DFM can reproduce the observed amplitudes quite well. Especially at ARAH, it is worth mentioning that SWM gives the first peak frequency as lower than the observed peak, but that DFM gives a fair match for the observed values, including both peak and trough frequencies and their amplitudes. At TRMA, overall matching is obtained but the first peak frequency from DFM is a bit too high and the amplitude fluctuation is too small. At MYG015 and NAGA, overall correspondence can be seen, but matching in peak frequency and amplitude is not quite satisfactory. Note that a sharp peak at 6 Hz at NAGA may be due to a local, specific source 12

14 of noise. In summary, we can see fairly good matching between theoretical MHVRs by DFM and observed MHVRs, both for amplitudes and for peak and trough frequencies when using an independently estimated velocity profile. Prediction by SWM is also good for most of the sites, though only for reproducing peak and trough frequencies. As expected, when we have smaller peak amplitudes, it becomes more difficult to reproduce observations because sites with smaller peak amplitudes do not have strong velocity contrasts within layers Comparison with MHVRs in Arai and Tokimatsu (2004) Arai and Tokimatsu (2004) proposed a multi-mode Love- and Rayleigh-wave summation method (hereinafter, the MMLR method). They need to assume contribution (i.e., strength) ratios between Love- and Rayleigh-wave amplitudes, and the excitation coefficients for different modes are assumed to be their medium responses. Inversion analysis using the observed MHVR and the MMLR method was then performed to estimate S-wave velocity profiles of subsurface soils. In the study they assumed that the S-wave velocity values of the shallow soil layers were reliably known a priori to avoid trade-offs between velocities and thicknesses. The inversion analyses were performed using the MHVRs observed at six sites as targets, and their shallow S-wave profiles were estimated. See Arai and Tokimatsu (2004) for details of the data and analyses used. Although only S-wave velocities were shown in their original paper, we used the same P-wave velocities and densities as the MMLR method given by the authors. The assumed soil profiles at six target sites are summarized in Tables 9 to 14 and S-wave velocity profiles are plotted in Figure 5. In Figure 6 we compare observed MHVRs (open circles) and theoretical MHVRs 13

15 (solid black lines) calculated by the MMLR method from Figure 6 of Arai and Tokimatsu (2004) with theoretical MHVRs (gray lines) calculated by DFM at the six target sites in the Kanto area of Japan. For comparison, we also plot HVR curves (broken lines) based on the simple Rayleigh-wave ellipticity as SWM, as was used for the Sendai case. We can see fairly good overall matching between the DFM and MMLR methods. Matching of the DFM with the observed data is not as good as the matching of the MMLR method, which is a natural consequence of optimization using the MMLR method. The same level of matching could be achieved if we were to optimize velocity structures by using the theoretical predictions from DFM. The peak MHVR amplitude of DFM is mainly controlled by the velocity contrast within layers. As a result, the peak amplitude will be decreased if there are layers with gradual velocity fluctuations and increased if there is a clear discontinuity. The controlling factor of the MMLR method is essentially the same as that of DFM, but it should work in a different manner so that peak DFM and MMLR amplitudes are different at some sites. Note that even though many modes are summed in the MMLR calculation, we can see significant troughs at frequencies where the fundamental mode of the Rayleigh wave (SWM) shows close to zero amplitude. In contrast MHVRs from DFM always show gentle troughs at these frequencies, as the observed MHVRs do Discussions and Conclusions In order to check the validity of the recently proposed interpretation of MHVR, which is based on the diffuse field concept, we compared theoretical prediction of MHVRs with the observed MHVRs presented in previous studies (Satoh et al., 2001b; Arai and Tokimatsu, 2004). In the comparison with data from the Sendai Plain (Satoh et al., 2001b) we observed 14

16 fairly good amplitude and frequency matching between DFM-predicted MHVRs and observed MHVRs. Prediction by SWM was also good for most of the sites, although it reproduced only peak and trough frequencies, not amplitudes. In the comparison with data from the Kanto Plain (Arai and Tokimatsu, 2004) we also observed fairly good amplitude and frequency matching between DFM-predicted MHVRs and observed MHVRs. Although the MMLR method proposed by Arai and Tokimatsu (2004) showed better matching with the observations in general, this is natural because the S-wave velocity structures used for theoretical calculations are those inverted by the MMLR method. As compared to the observed MHVRs, a notable discrepancy found in the MMLR method but not in DFM was an excessively sharp trough (dip) immediately behind the fundamental peak. This occurred because such a trough in the MMLR method is due to a high-amplitude vertical component associated with the fundamental mode of the Rayleigh-wave, as can be seen in SWM, and contribution from different modes of Rayleigh- and Love-waves cannot diminish the sharpness. In DFM we have contributions from all the possible wave states, including horizontally polarized S-waves generated by a point source, so the trough sharpness is effectively suppressed as in the observed MHVRs. Among the six sites used in Arai and Tokimatsu (2004), peak amplitudes are all well reproduced by DFM, except for Site E, where the peak amplitude from DFM is 1.5-fold the observed peak amplitude. If we look at the S-wave velocity profile of the inverted structure at Site E and compare it with that of the original P S logging (Figure 7 in Arai and Tokimatsu, 2004), it turns out that the third layer of the inverted structure used a common value from averaging over three layers with different velocities. This kind of simplification may not be reflected in the peak frequency, but it may be reflected in the peak amplitude because the peak amplitude of MHVR is mainly controlled by the velocity contrast within layers. Such a 15

17 tendency is the same for both the MMLR method and DFM, but its quantitative effect on the peak amplitude could not be the same. To check this we compared theoretical MHVRs from DFM for the inverted and P S logging structures. The structure used from P S logging is listed in Table 15. Figure 7 shows a comparison of the theoretical MHVRs, together with the observed data and the solution given by the MMLR method. As expected, the P S logging structure yields a much smaller peak amplitude, while its peak frequency is almost the same. This suggests that a similar level of matching to the observed MHVRs would be obtained if we inverted the S-wave velocity structures by using DFM, which will be a primary target of our future work. In summary we have compared theoretical and observed MHVRs at fourteen sites, for which we have both reliable observed data of microtremors and S-wave velocity structures from the literature. This allowed us to validate theoretical MHVRs calculated by diffuse field theory. We found that results for almost all the sites showed quite good correspondence to predictions from diffuse field theory, except for one site with high S-wave velocity where no notable peak in the MHVR was observed Data and Resources Microtremor data used in this study were all previously reported in the literature (Satoh et al., 2001b; Arai and Tokimatsu, 2004). Velocity structures at eight sites in Sendai were obtained from Dr. Satoh, and those at six sites in Kanto were obtained from Dr. Arai. Figure 1 was made with Generic Mapping Tools version ( Wessel and Smith, 1998)

18 362 Acknowledgements The authors thank Dr. Arai of the Building Research Institute for permission to use his microtremor and velocity data, and to Lic. G. Sánchez N. and her team at the Unidad de Servicios de Informacion (USI) of the Institute of Engineering-UNAM for locating useful references. We also thank the anonymous reviewers for their invaluable comments. Partial support for this work from a MEXT Grant-in-Aid for Scientific Research (A) No , , from the AXA Research Fund, and from DGAPA-UNAM under Project IN is gratefully acknowledged References Aki, K. (1957). Space and time spectra of stationary stochastic waves, with special reference to microtremors, Bull. Earthquake Res. Inst., Tokyo Univ. 35, Albarello D., and E. Lunedei (2010). Alternative interpretations of horizontal to vertical spectral ratios of ambient vibrations: new insights from theoretical modeling, Bull. Earthq. Eng., 8, , doi: /s Arai, H., and K. Tokimatsu (2004). S-Wave velocity profiling by inversion of microtremor H/V Spectrum, Bull. Seism. Soc. Am., 94:1, Arai, H. and K. Tokimatsu (2005). S-wave velocity profiling by joint inversion of microtremor dispersion curve and horizontal-to-vertical (H/V) spectrum, Bull. Seism. Soc. Am., 95:5, Arai, H. and K. Tokimatsu (2008). Three-dimensional Vs profiling using microtremors in Kushiro, Japan, Earthq. Eng. Structural Dyn., 37:6,

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24 Seism. Soc. Am., 91:2, Satoh, T., H. Kawase, and S. Matsushima (2001b). Differences between site characteristics obtained from microtremors, S-waves, P-waves, and codas, Bull. Seism. Soc. Am., 91:2, Satoh, T., H. Kawase, T. Sato, and A. Pitarka (2001). Three-dimensional finite-difference waveform modeling of strong motions observed in the Sendai Basin, Japan, Bull. Seism. Soc. Am., 91: Satoh, T., H. Kawase, T. Iwata, S. Higashi, T. Sato, K. Irikura, and H.-C. Huang (2001). S- wave velocity structure of the Taichung Basin, Taiwan, estimated from array and single-station records of microtremors, Bull. Seism. Soc. Am., 91:5, doi: / Uebayashi, H., H. Kawabe, and K. Kamae (2012). Reproduction of microseism H/V spectral features using a three-dimensional complex topographical model of the sedimentbedrock interface in the Osaka Sedimentary Basin, Geophys. J. Int., doi: /j X x. Wessel, P. and W.H.F. Smith (1998). New, improved version of the generic mapping tools released, EOS Trans. AGU, 79, Authors affiliations and addresses H. K. and S. M.; Disaster Prevention Research Institute, Kyoto University; Gokasho, Uji, Kyoto , Japan. T. S.; Institute of Technology, Shimizu Corp.; Etchujima, Tokyo , Japan. 23

25 F. J. S.-S.; Instituto de Ingeniería, Universidad Nacional Autónoma de México; C.U., Coyoacán D.F., Mexico. 24

26 531 Tables Table 1. Velocity structure at TAMA No. Thickness Depth Vp Vs Table 2. Velocity structure at TRMA. Density (g/cm 3 ) No. Thickness Depth Vp Vs Density (g/cm 3 )

27 No. Table 3. Velocity structure at ARAH. Thickness Depth Vp Vs Density (g/cm 3 ) No. Table 4. Velocity structure at OKIN. Thickness Depth Vp Vs Density (g/cm 3 )

28 579 Table 5. Velocity structure at MYG No. Thickness Depth Vp Vs Table 6. Velocity structure at NAKA. Density (g/cm 3 ) No. No. Thickness Table 7. Velocity structure at NAGA. Thickness Depth Depth Vp Vp Vs Vs Density (g/cm 3 ) Density (g/cm 3 )

29 No. Table 8. Velocity structure at SHIR. Thickness Depth Vp Vs Table 9. Velocity structure at Site A. Density (g/cm 3 ) No. No. Thickness Table 10. Velocity structure at Site B. Thickness Depth Depth Vp Vp Vs Vs Density (g/cm 3 ) Density (g/cm 3 )

30 Table 11. Velocity structure at Site C No. Thickness Depth Vp Vs Density (g/cm 3 ) Table 12. Velocity structure at Site D No. Thickness Depth Vp Vs Density (g/cm 3 ) Table 13. Velocity structure at Site E No. Thickness Depth Vp Vs Density (g/cm 3 )

31 Table 14. Velocity structure at Site F No. Thickness Depth Vp Vs Density (g/cm 3 ) Table 15. Velocity structure at Site E originally from borehole P S logging No. Thickness Depth Vp Vs Density (g/cm 3 )

32 636 Figure captions Figure 1. Map of strong-motion sites in and around the Sendai Basin. The eleven sites shown as triangles were deployed by the BRI (Kitagawa et al., 1994), while the other two sites shown as squares were deployed by the Natural Research Institute for Earth Science and Disaster Prevention (Kinoshita, 1998). Among these, only the eight stations with station codes are used in this study. The geological classification shown comes from Satoh et al. (2001a), namely, H, Holocene in Quaternary; Q3 and Q2, Pleistocene in Quaternary; N3, Pliocene in Neogene (Tertiary); N2 and N1, Miocene in Neogene (Tertiary); TR1, early Triassic in Mesozoic (pre-tertiary); and CM, Cambrian in Paleozoic (pre-tertiary). Figure 2. S-wave velocity profiles at eight sites in Sendai (Satoh et al., 2001a). Figure 3. Comparison of theoretical MHVRs from DFM (gray lines) with observed MHVRs (solid black lines) at four sites in Sendai, TAMA, TRMA, ARAH, and OKIN. We also plot Rayleigh-wave ellipticity, labeled as SWM for comparison (dashed lines). The deeper parts of the velocity models at these sites are based on the array measurement of microtremors (Satoh et al., 2001a). Figure 4. Comparison of theoretical MHVRs from DFM (gray lines) with observed MHVRs (solid black lines) at four sites in Sendai and Iwanuma: MYG015, NAKA, NAGA, and SHIR. We also plot Rayleigh-wave ellipticity, labeled as SWM for comparison (dashed lines). The deeper part of the velocity model at MYG015 is based on array measurements of microtremors (Satoh et al., 2001a), while those of the other sites are based on peak frequency matching using SWM (Satoh, Kawase, Sato et al., 2001). Figure 5. S-wave velocity profiles (shallower part) at six Kanto region sites (Arai and 31

33 Tokimatsu, 2004). The deeper part is common for all the sites, as shown in Tables 9 to 14. Figure 6. Comparison of theoretical MHVRs calculated by DFM (gray lines) with those calculated by the MMLR method (black lines). The structures used were optimized by Arai and Tokimatsu (2004) to the observed MHVRs (open circles). We also plot Rayleigh-wave ellipticity for the fundamental mode labeled as SWM for comparison (dashed lines). Figure 7. Comparison of the theoretical MHVR calculated by DFM (gray line) for the structure optimized by the MMLR method with that calculated by DFM (dotted line) for the original P S logging structure from boring data shown in Table 15. The theoretical MHVR calculated by the MMLR method (black line) and the observed MHVR (open circles) are also plotted as in Figure 6(e). 32

34 Figure 1 Black & White Click here to download Figure: BSSA-D-00134r1-Fig.1-BW.pdf