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Application of a simplified calculation for fullwave microtremor H/V spectral ratio based on the diffuse field approximation to identify underground velocity structures
Earth, Planets and Space volume 69, Article number: 162 (2017)
Abstract
Under the diffuse field approximation, the fullwave (FW) microtremor H/V spectral ratio (H/V) is modeled as the square root of the ratio of the sum of imaginary parts of the Green’s function of the horizontal components to that of the vertical one. For a given layered medium, the FW H/V can be well approximated with only surface waves (SW) H/V of the “caplayered” medium which consists of the given layered medium and a new larger velocity halfspace (cap layer) at large depth. Because the contribution of surface waves can be simply obtained by the residue theorem, the computation of SW H/V of caplayered medium is faster than that of FW H/V evaluated by discrete wavenumber method and contour integration method. The simplified computation of SW H/V was then applied to identify the underground velocity structures at six KiKnet strongmotion stations. The inverted underground velocity structures were used to evaluate FW H/Vs which were consistent with the SW H/Vs of corresponding caplayered media. The previous study on surface waves H/Vs proposed with the distributed surface sources assumption and a fixed RayleightoLove waves amplitude ratio for horizontal motions showed a good agreement with the SW H/Vs of our study. The consistency between observed and theoretical spectral ratios, such as the earthquake motions of H/V spectral ratio and spectral ratio of horizontal motions between surface and bottom of borehole, indicated that the underground velocity structures identified from SW H/V of caplayered medium were well resolved by the new method.
Introduction
The microtremor H/V spectral ratio (H/V) is the most popular approach to estimate the predominant period of a site (Nakamura 1989, 2000; Lermo and ChávezGarcía 1993). In what has been controversial, Nakamura (1989, 2000) proposed to consider this ratio to be the empirical transfer function. Using a numerical simulation, Lachet and Bard (1994) verified that the natural period of the sedimentary site coincides well with the predominant period of H/V. These issues have raised an assortment of opinions on the characteristics and origin of features of H/V, i.e., predominant period and amplitude, being a SH waves transfer function (Nakamura 1989, 2000; Lermo and ChávezGarcía 1993), the role of ellipticity of fundamental mode or high modes of Rayleigh waves and the emergency of Airy phase of Love waves (Lachet and Bard 1994; Konno and Ohmachi 1998; Fäh et al. 2001; Arai and Tokimatsu 2004; BonnefoyClaudet et al. 2006a; Baan 2009). Other researchers, such as Tokimatsu and Tamura (1995), Tamura (1996), and BonnefoyClaudet et al. (2008), concluded that the origin is not unique, and it depends on the impedance ratio, distribution of sources (or cultural condition), and the relative proportion of body (SH, PSV waves) and surface waves (Rayleigh and Love waves). Accordingly, various methods to identify underground velocity structures have been proposed for threecomponent station measurements, depending on different assumptions on the origin, e.g., ellipticity of fundamental mode of Rayleigh waves (Fäh et al. 2001), higher modes of surface waves (Arai and Tokimatsu 2004), or SHwave transfer function (Lin et al. 2014).
Since microtremor consists of both body and surface waves, a fullwave description is desirable to interpret the H/V. Diffuse field approximation is one of the promising theories to accomplish it. The diffuse field concept originated in acoustics and was firmly established for elasticity by Weaver (1982). But the roots of the idea can be traced back to the pioneering work of Aki (1957) who used ambient seismic noise (ASN) in the SPAC method (for spatial autocorrelation). The shortrange crosscorrelations of noise in a small array and its azimuthal average allow, through the argument of the Bessel function of zero order, to retrieve the phase velocity of Rayleigh waves. Aki’s pioneering work addressed first ideas and issues that were later formalized by Weaver (1982) as diffuse fields. The connection with 2D elasticity was pointed out by SánchezSesma and Campillo (2006). It has been proposed that the coda of earthquakes can be seen as a realization of a diffuse field (Shapiro et al. 2000; Hennino et al. 2001; Margerin 2009). In several instances, the ASN allows Green’s function retrieval from the average of crosscorrelation between two receivers within a diffuse seismic field. Shapiro and Campillo (2004), Sabra et al. (2005), and Shapiro et al. (2005) confirmed that ASN is generally diffuse. We assert that the key of this fact lies in multiple scattering. The properties and the conditions of emergence of a diffuse field have been explored by many researchers worldwide (e.g., Weaver and Lobkis 2001; SánchezSesma and Campillo 2006; SánchezSesma et al. 2008). For the applications, the diffuse field assumption allows to get useful approximations of the Green’s Function from crosscorrelation of recorded motions. Even Mulargia (2012) who claims that ASN is not diffuse accepts that it holds significant imaging properties. Based on the diffuse field approximation, the directional energy density is proportional to the imaginary part of the Green’s function when source and receiver coincide with each other both in position and direction (SánchezSesma et al. 2008). The FW H/V is then expressed as the square root of the sum of imaginary parts of the horizontal components of Green’s function to that of the vertical counterpart (SánchezSesma et al. 2011). In the diffuse field description, the imaginary part of the Green’s function incorporates naturally the contributions of both body (BW) and surface waves (SW). Therefore, an explanation on the origin of H/V is achieved by fullwave.
Lontsi et al. (2015) used the FW description to invert H/V recorded and processed from different depths to identify underground velocity structures at a couple of sites in Germany. They used a straightforward calculation inspired on the conventional discrete wavenumber method to evaluate the Green’s functions. This fullwave approach is time consuming and limits the application of inversion of FW H/V to some extent. GarcíaJerez et al. (2013) evaluated the imaginary parts of the Green’s functions, accounting for the separate contributions of body and surface waves using the Cauchy’s contour integration method, for which the computation of surface waves corresponding to residues is fast. They further developed two faster codes for inversion of both the FW H/V and joint inversion, adding the velocity dispersion curve of fundamental mode of Rayleigh waves, and then implemented these codes to identify underground velocity structures at some sites (GarcíaJerez et al. 2016; PiñaFlores et al. 2017). That has been a major accomplishment, but the computation of body waves which contribute little to the fullwave consumes most of the computing time. Also, the efficiency of their code (opened to public) for forward calculation of FW H/V reduces significantly, when the number of frequencies of H/V is larger than 1000 and the number of layers is more than 5. To circumvent those drawbacks, we explore other ways in which we see room for improvement. Harvey (1981) applied the lockedmode approximation method to approximate the Green’s functions in time domain for fullwave with superposition of normal modes (surface waves) by adding a large velocity halfspace to the bottom of given velocity structures. He referred to this large velocity halfspace as “cap layer.” Because the computation of surface waves is quite fast and the computation of body waves is not needed, the computational efficiency is better than the one achieved with other methods, such as reflectivity, Thomson–Haskell propagation matrix, and global matrix, all within the discrete wavenumber summation scheme.
In this study, we adopt the diffuse field approximation (DFA) and investigate the applicability of the caplayered medium to compute the H/V from the surface waves of that waveguide. Our approach allows identifying the underground velocity structures between the surface and bottom of boreholes at six KiKnet stations where the PS logging data are not reliable enough to explain the observed H/Vs. In the framework of SW H/V, we investigate the differences between SW H/V of our approach and SW H/V of the previous study on microtremor H/V spectral ratio developed by Arai and Tokimatsu (2004). We finally discuss the validity of identified underground velocity structures by comparing observed results with theoretical results, such as earthquake motions H/V spectral ratio developed by Kawase et al. (2011) and spectral ratio of horizontal motions between surface and bottom of borehole.
Formula of microtremor H/V spectral ratio based on diffuse field approximation
Based on the diffuse field approximation, the FW H/V is expressed as follows:
where ImG is the imaginary part of the Green’s function when both source and receiver coincide with each other both in position and direction (SánchezSesma et al. 2011), and the subscripts 11 and 22 mean two horizontal components, and 33 the vertical one. The imaginary part of the Green’s functions “detects” energies that are both radiated from and coming back to the source. Im G _{11}, Im G _{22}, and Im G _{33} are proportional to directional energy densities associated with the three degrees of freedom. Other components, such as G _{12} (or G _{21}), G _{13} (or G _{31}), and G _{23} (or G _{32}), are not needed, and the imaginary parts of them are zero for coincident source and receiver. The right side of Eq. (1) only depends on the underground velocity structure, such as density, P and Swave velocity, and thickness of each layer, regardless of sources distribution and proportion of body and surface waves, which makes it the signature of the site. The corresponding code (SánchezSesma et al. 2011) was inspired on the discrete wavenumber method to carry out the integrations in the wavenumber domain.
GarcíaJerez et al. (2013) proposed to evaluate the Green’s functions by the Cauchy’s contour integration method, because the Green’s functions for surface waves can be quickly evaluated as the sum of residues. For horizontally layered media, the imaginary part of the Green’s functions and the SW H/V is expressed as follows:
where A _{R} and A _{L} are the medium responses of Rayleigh and Love waves, respectively (Harkrider 1964), u/w is the ellipticity of Rayleigh waves, m and n are the order of modes for Rayleigh and Love waves, M and N are the largest order of higher modes for Rayleigh and Love waves. Because the medium responses of higher modes, e.g., larger than the fifth mode, are much smaller than those of fundamental mode and only contribute in a short period range, without loss of generality, M and N are set to be 5 in the following calculations. Regarding the calculation of surface waves, the ellipticity, the medium response terms A _{R} of Rayleigh waves, and A _{L} of Love waves should be evaluated from the fundamental to the fifth higher mode, and for a given layered medium, Saito and Kabasawa (1993) proposed the “compound matrix” to overcome the numerical instability at high frequencies of the Thomson–Haskell propagation matrix. We improved their code and enabled it to evaluate Eq. (4). For a layered medium consisting of less than 10 layers, the computation of SW H/V by our code on a desktop computer (64bit, Windows 8.1, Intel Core i7, 3.20 GHz, RAM 8.0 GB) takes no more than 1 s even if the number of frequencies reaches 2000. It is significantly faster than the computation of FW H/V by discrete wavenumber method (SánchezSesma et al. 2011) which usually takes several or tens of minutes. It is also faster than the MATLAB runtime environment developed by GarcíaJerez et al. (2016) based on contour integration method which for the same problem takes several or tens of seconds. Furthermore, the numerical instability that can be observed at high frequencies of certain higher modes of Rayleigh waves for the GarcíaJerez et al. (2016)’s code is avoided.
Approximation of FW H/V for twolayered medium
Harvey (1981) proposed that by adding a large velocity cap layer to a given layered medium and fixing the bottom, the Green’s function of fullwave could be approximated with superposition of normal modes of surface waves. However, the effects of (1) depth of the upper interface, (2) the largeness of velocity of the cap layer, and (3) the impedance ratios of target layered medium, on the applicability of this approximation, were not discussed. In this section, we add a large velocity cap layer at large depth to examine the feasibility of approximation of the Green’s function of fullwave with the Green’s function of surface waves. This consideration is implicitly involved in the study of Arai and Tokimatsu (2004) who added two thick layers with large velocity to reach the engineering bedrock. They only formulated the H/V of surface waves based on distributed surface sources assumption.
Table 1 lists the parameters of a twolayered medium consisting of a single layer and a halfspace. We assume the Swave velocity (V _{s}) in the halfspace is three times larger than that of the first layer, the density is fixed (2.0 g/cm^{3}) in each layer so that the impedance ratio is the same as velocity contrast, and Pwave velocity (V _{p}) is twice that of V _{s} in each layer, which implies a Poisson ratio of 1/3. In any case, density and Pwave velocity show much less sensitivity than Swave velocity to H/V (e.g., Arai and Tokimatsu 2004; Lontsi et al. 2015). We regard this layered medium as “target” twolayered medium, and the layered medium composed of target layered medium and a cap layer as “caplayered” medium.
First, we investigate the effect of depth of upper interface of cap layer on the applicability of approximating the FW H/V of target twolayered medium with SW H/V of caplayered medium. Table 2 lists the parameters of caplayered media with three different depths (i.e., 2.5, 5, and 10 times the wavelength λ _{0} at fundamental frequency or period) of upper interface of cap layers. For target twolayered medium, λ _{0} is just four times the thickness of first layer, whatever the velocity and frequency is. The velocity of the cap layer is fixed, twice as large of the layer above the cap layer. As shown in the left panel of Fig. 1, the SW H/V of the target twolayered medium is almost the same as FW H/V in the period smaller than the predominant period of fundamental Swave resonance modeTp_{0}, namely 0.4 s, according to one quarter wavelength law. This indicates that the contribution of fullwave in the short period range is nearly the same as that of surface waves in the first layer. As shown in the middle panel of Fig. 1, the SW H/V of caplayered medium approximates well to the FW H/V of target twolayered medium in the period ranging from 0.1Tp_{0} to 5Tp_{0} of the target medium if the depth of the upper cap layer is larger than 5λ _{0}. The reason is that the large velocity cap layer in the deep converts the body waves that are leaked from the first layer of target twolayered medium into higher modes of surface waves. It is obvious from Fig. 2 that the largest peak of SW H/V of target twolayered medium is caused by a local trough in the vertical component which corresponds to the cutoff of vertical medium response of the first higher mode of Rayleigh waves. The cutoff of horizontal medium response has little effect on the largest peak, because its amplitude is not only smaller than that of fundamental mode of Rayleigh waves, but much smaller than that of Love waves. Due to the addition of cap layer, the vertical medium responses of the second, third and fourth higher modes of Rayleigh waves just fill in the local trough caused by the first higher mode, and this leads to a good approximation of FW H/V of target twolayered medium with SW H/V of caplayered medium. Therefore, although the main advantage of SW H/V is its computation efficiency over FW H/V, the calculation of medium responses of surface waves gives insight into the different contribution from fundamental and higher modes of both Rayleigh and Love waves.
Second, we investigate the effect of the cap layer velocity on the feasibility of approximating the FW H/V of target twolayered medium with SW H/V of caplayered medium. Table 3 lists the parameters of caplayered media with different velocities of the caplayer. The depth of upper interface of cap layer is set as 10λ _{0} (i.e., 400 m here). As shown in the right panel of Fig. 1, the SW H/V of caplayered medium agrees well with the FW H/V of target twolayered medium if the velocity of cap layer is larger than 1.5 times the velocity of the second layer (V _{s2}). It is considered that the small velocity of cap layer (e.g., 1.2V _{s2}) is not enough for totally converting the body waves into surface waves.
Third, we investigate the effect of the impedance ratio of twolayered medium on the feasibility of approximating the FW H/V of target twolayered medium with SW H/V of caplayered medium. Table 4 lists the parameters of target twolayered media with different impedance ratios. The depth of upper interface of cap layer is set to be 10λ _{0}, and the velocity of cap layer of these target twolayered media is twice as large of V _{s2}. Figure 3 shows that the amplitudes of FW and SW H/Vs increase with the impedance ratio. This fact prompted Nakamura to consider H/V as a transfer function. Obviously, the impedance ratio is relevant. Ellipticity of the fundamental mode of Rayleigh waves is different from FW H/V, not only the amplitude but also the predominant period. For the Rayleigh’s fundamental mode, the conspicuous period of ellipticity coincides with the wellknown local minimum in the vertical component. It is gradually close to that of FW H/V as the impedance ratio increases. This fact was carefully studied in numerical simulations (BonnefoyClaudet et al. 2008). Similarly, for the SW H/V, the predominant periods of SW H/Vs of twolayered media for which the peaks coincide with the troughs of higher mode or fundamental mode (if the impedance ratio is large) of Rayleigh waves in the vertical component vary with impedance ratio. Figure 3 also shows that when the impedance ratio is not larger than 6, the SW H/V of caplayered medium agrees well with the FW H/V of twolayered medium in the period range, e.g., 0.1Tp_{0}–5Tp_{0}, even if only the contributions from fundamental to the fifth higher modes of surface waves are included in the SW H/V. Nolet et al. (1989) concluded that the evaluation of the Green’s function requires 50–100 higher modes for lockedmode approximation. The site they selected is rather hard because the V _{s} in the first layer is as large as 1500 m/s. In contrast, the sites in our study are much softer, e.g., the V _{s} is about 300 m/s in the first layer and is smaller than 1000 m/s from the surface downward to 100 m in depth. This suggests that the Nolet et al. (1989)’s study was barely related with the results presented herein. The amplitudes of tiny fluctuations at very short periods, e.g., below 0.1Tp_{0}, for SW H/V are slightly smaller than for FW H/V due to the loss of modes higher than 5 for the target twolayered media. This small discrepancy between SW and FW H/Vs is difficult to detect for realistic multilayered media, such as those sites studied by Arai and Tokimatsu (2004) and Lunedei and Albarello (2009). Considering the temporal variation of microtremors, this small discrepancy seems negligible, e.g., smaller than one standard deviation of observed H/V, which we discuss later for one site. Besides, it only affects the accuracy of uppermost velocity structures (e.g., thinner than 1 m) which is far from engineering interest. The amplitude of SW H/V is twice as large as that of FW H/V around the predominant period so that the difference cannot be neglected if the impedance ratio is larger than 6.
Figure 4 shows the comparison between FW H/V of target twolayered medium (V _{s2} = 8V _{s1}) and SW H/V of caplayered medium for which the depth of upper interface of cap layer is 10λ _{0} (i.e., 400 m), as well as the imaginary parts of the Green’s functions and vertical medium responses from fundamental to the fifth higher modes. Due to cutoff effect of higher modes, the vertical medium responses from modes higher than third have no contributions at periods longer than the predominant period. Therefore, summation of higher modes larger than the fifth cannot improve the approximation of FW H/V of target twolayered medium with SW H/V of caplayered medium at periods longer than the predominant period. We further place the cap layer at a great depth, such as 100λ _{0} (i.e., 4000 m here) to investigate whether the SW H/V of caplayered medium can approximates well with FW H/V. Figure 5 shows the comparison between FW H/V of target twolayered medium (V _{s2} = 8V _{s1}) and SW H/V of caplayered medium for which the depth of upper interface of cap layer is as large as 100λ _{0}, as well as the imaginary parts of the Green’s functions and vertical medium responses from fundamental to the 25th higher modes of Rayleigh waves. The discrepancy between FW H/V of target medium and SW H/V of caplayered medium around the predominant period still cannot be neglected, as the vertical medium responses of higher modes of Rayleigh waves do not totally fill in the trough caused by the vertical medium response of fundamental mode. Therefore, even if the cap layer is located at a great depth and more modes are considered, the FW H/V of target medium with large impedance ratio cannot be well approximated with SW H/V of caplayered medium. Tokimatsu and Tamura (1995) pointed out that if the impedance ratio is too large, the contribution of body waves is much larger than that of surface waves near the predominant period. It is thus considered that the cap layer fails to totally convert body waves into surface waves in this case. Applying rigid boundary condition (Margerin 2009) at the upper interface of cap layer is considered an effective solution to totally convert body waves into surface waves.
Another aspect that is of interest in practice is the effect of damping ratio or quality factorQ which is used in the calculation of FW H/V but not included in SW H/V. In the FW H/V, some damping was introduced by adding a small imaginary part to frequency and also using some material damping. Although the purpose was mainly to stabilize the results and to smooth the integrands in the wavenumber domain (SánchezSesma et al. 2011), the practical interest of damping in realistic soil configurations led us to explore the effects of small material damping using a quality factorQ. This is not accounted for in the theory, but in some cases adding small amount of damping allows for results closer to observations (e.g., Lawrence and Prieto 2011). We explore the effects of quality factorQ on FW H/Vs of twolayered media with three different impedance ratios shown in Fig. 6. It suggests that the difference of FW H/Vs caused by three different Q values (50, 100 and 500) is negligible. Thus, the discrepancy between FW H/V of target medium with large impedance ratio and SW H/V cannot be attributed to the damping ratio or Q values.
Finally, it should be noted that the large velocity cap layer at large depth is just an artificial layer used to convert the body waves leaked from the bottom of given layered medium into surface waves of caplayered medium. Meanwhile, the SW H/V of caplayered medium is regarded a simplified calculation of FW H/V of layered medium, and we agree that microtremor consists of fullwave, but not surface waves only. Based on the aforementioned discussions, we conclude that the FW H/V of target layered medium should be well approximated with SW H/V of caplayered medium in the period range, e.g., 0.1Tp_{0}–5Tp_{0}, under the following conditions: The impedance ratio of target layered medium is not larger than 6, and the Swave velocity of cap layer is about twice as large as that in the halfspace of target layered medium, the depth of upper interface of cap layer is about (5–10)λ _{0}. However, it should be noted that Q = 100 might be too large for FW H/V at soft sediment sites, and thus further investigation is needed.
Approximation of FW H/V for multilayered medium
In Japan, PS logging data between surface and bottom of boreholes at most KiKnet strongmotion stations are available, so the underground velocity structures of PS logging data are adopted to generate multilayered media. With respect to the corresponding caplayered media, we set the velocity of the cap layer as about two times larger than that in the above layer. Because one cannot precisely estimate the Tp_{0} or λ _{0} for a given layered medium, we define the “apparent” predominant period of Swave fundamental mode (Tp ^{A}_{0} ) as the ratio of the “apparent” wavelength of fundamental mode (λ ^{A}_{0} ) for the given layered medium to the thicknessweighted average Swave velocity (Fäh et al. 2001). The “apparent” wavelength of fundamental mode (λ ^{A}_{0} ) is four times the depth from surface to the uppermost interface of halfspace, but not four times the thickness of first layer anymore. Both Tp ^{A}_{0} and λ ^{A}_{0} become the “real” Tp_{0} and λ _{0} for the twolayered medium (single layer over a halfspace) and tend to be the “real” ones if the velocity contrast at the interface of halfspace is large enough for multilayered medium. We find that if the depth of cap layer is set to be about (5–10) λ ^{A}_{0} , the SW H/V of caplayered medium approximates well to the FW H/V of target multilayered medium. Six KiKnet stations were selected after a careful scrutiny that accounts for local geology, seismicity, and site effects. They do not locate near the edge between basin and mountain nor close to active faults so that the 3D effects are considered minor. Figure 7 shows the Swave velocity profiles in the top panels at six KiKnet stations. Table 5 lists the basic parameters, such as Tp ^{A}_{0} , thicknessaveraged V _{s}, and λ ^{A}_{0} for velocity profiles at these stations. The numerical values show the Tp ^{A}_{0} at the interface of halfspace of target layered medium in black and that at the uppermost interface of cap layer in red. It is obvious that the cap layer at large depth produces a Tp ^{A}_{0} about 5–10 times as larger as that for target layered medium. These 1D vertical profiles are obtained from PS logging data. In most circumstances, that 1D profile is verified by studying the measured amplification from an earthquake, such as comparing the observed surface/bottom of borehole spectral ratio with theoretical 1D transfer function (e.g., Thompson et al. 2012). Another approach is to use microtremor to accomplish such verification by comparing the observed microtremor H/V with the theoretical H/V, considering that the soil models are 1D or only depth dependent at these stations. The underground velocity structures consisting of PS logging data are used to evaluate four theoretical H/Vs in the bottom panels, i.e., ellipticity of fundamental mode of Rayleigh waves, FW H/V and SW H/V of the multilayered medium, and SW H/V of the corresponding caplayered medium. The coincidence between FW and SW H/Vs of multilayered media in the short period range at all stations again indicates that the contribution of surface waves which are trapped in the shallow layers is overwhelmingly dominant. By just adding a cap layer to the bottom of borehole, the SW H/Vs of caplayered media approximate the FW H/Vs well at almost all the stations except KMMH14 at which a relatively large impedance ratio near the bottom of borehole could be responsible for the discrepancy between FW H/V of multilayered medium and SW H/V of caplayered medium. However, this discrepancy is negligible given that the microtremor is subjected to the temporal variance (Okada 2003).
The ellipticities of fundamental mode of Rayleigh waves at all stations have two peaks which are caused by the underground velocity structures of shallow layers, and deep layers, respectively. The sharp peak at periods shorter than 0.5 s at TCGH16 station corresponds to the large impedance ratio in the shallow layers as shown in the Swave velocity profiles. The sharp peaks at periods longer than 0.5 s at other stations correspond to the large impedance ratio in the deep layers near the bottom of boreholes. Both peaks of ellipticity of fundamental mode of Rayleigh waves at GIFH28 are moderate. This is attributed to the moderate impedance ratio from surface to bottom of borehole.
The observed microtremor H/Vs at six KiKnet stations are also shown in bottom panels of Fig. 7. We conduct the microtremor measurement with a threecomponent velocity seismometer, a GPS receiver, and a recorder with 24 bits. We conduct the measurement for 30–60 min with 100 Hz sampling at all stations except TCGH16 where we observed as long as 12 h. We remove those time windows (Tw = 40.96 s) including extremely large amplitude which is caused by nearby human activities, such as jogging, vehicles, before obtaining the power spectra in three components. These power spectra in three components are then averaged over the number of time windows. The observed H/V is obtained from the square ratio of average power spectra in the horizontal components (arithmetic summation of north–south (NS) and east–west (EW) components) to that in the vertical component, just as proposed by Arai and Tokimatsu (2004) and numerically verified by Albarello and Lunedei (2013). Smoothing with b = 50 by Konno and Ohmachi (1998) is finally implemented on the observed H/V. Thus only one H/V without temporal variability can be obtained at one site. As the record is long at TCGH16, we divide it into 24 parts each of which is used to evaluate H/V to investigate the temporal variation of observed H/V. We take the average H/V as the observed one at TCGH16 and find the standard deviation at each period. The average H/V is shown in thick black curve, and the ones obtained from the average plus and minus one standard deviation are shown in dashed curves in Fig. 7. Temporal variation can be observed, yet it is very small. The mismatch between the FW H/V and observed H/V suggests that the underground velocity structures of PS logging data are not fully reliable and thus should be improved through inversion of H/V.
Identification of underground velocity structures from SW H/V
Since the computation of medium responses of surface waves is significantly faster than the computation of fullwave, and FW H/V can be approximated with SW H/V by just adding a cap layer to the bottom of given layered medium, we prefer to identify the underground velocity structures from SW H/V. We perform the identification by simulated annealing algorithm which is elaborated in Satoh (2006). Generally, this algorithm finds many local minima at the beginning and gradually converges to a global minimum. The temperature scheme is expressed as T _{ k } = T _{0}∙exp(− ck ^{α}), where k is the step of cooling, T _{ k } is the temperature at the kth step, T _{0} is the initial temperature, and c and α are key parameters controlling the cooling speed. We set T _{0}, c, and α to be 1.0, 1.0, and 0.6, respectively, iterations at one step to be 5, and total steps 1000. The objective function E _{ m } is defined as follows: (Saguchi et al. 2009)
where (H/V) ^{SW}_{ f } denotes the SW H/V of caplayered medium, (H/V) ^{obs}_{ f } denotes the observed H/V, f denotes frequency (equally spaced precisely as it is in FFT), f _{min} and f _{max} denote the minimum and maximum frequencies of the target fitting frequency range, respectively. The f _{min} is determined in consideration of the depth of underground velocity structures from surface to bottom, e.g, it may be not deep enough to explain peak at low frequency or long period. The f _{max} is the Nyquist frequency—50 Hz, if sampling frequency is 100 Hz, whereas f _{max} = 20 Hz is better for some sites. In this study, we set f _{min} to be 0.25–0.5 Hz, and f _{max} to be 20–50 Hz. The weight function is usually employed in the objective function to guarantee a good fitting over a broad frequency range. Typically, the expressions for the objective or cost functions are weighted sums of residuals. The choice of the weights is a matter of taste. Certainly, a common selection aimed to smooth out the results is to use frequency so that larger frequencies with their multiple oscillations have less constraining power. Our previous studies (Wu et al. 2012, 2016; Wang et al. 2017) suggested that the linear “frequency” worked well as the weight function, while Satoh (2006) suggested a logarithmic frequency to be another weight function. The temporal variation of observed H/V is considered small, and it is not included in the objective function. The discussion about the effect of different weight functions on the objective function and how to incorporate the temporal variation in the objective function is important, yet beyond the scope of this study.
If there is not any a priori knowledge about the underground velocity structures, the identified velocity structures are not unique by inversion of H/V only. It is a good practice to implement the joint inversion of both H/V and dispersive curve, such as Parolai et al. (2005), Picozzi et al. (2005), Arai and Tokimatsu (2005), and PiñaFlores et al. (2017). An alternative method to uniquely identify the underground velocity structures by the inversion of H/V only is to make use of a priori knowledge. In this study, we refer to the PS logging data to be the initial underground velocity structures rather than those generated by genetic algorithm (PiñaFlores et al. 2017). It is well known that the accuracy of velocity structures of PS logging data in shallow layers is generally not reliable enough. The velocities of soil layers near the surface in most cases are unstable and fluctuate, and the interface between two adjacent layers is not easy to detect and so is the layer thickness. As shown in Fig. 7, the H/Vs for initial caplayered media match well with the observed ones at relative long periods, such as around 0.6 s at TCGH16, 0.6 s at KMMH03, 2.0 s at SZOH26, and 0.8 s at KMMH14, respectively. This is attributed to the fact that thickness and average velocity from surface downward to the layer boundary which has a remarkable velocity contrast at each site is clearly detectable for PS logging data. It implies that the velocity structures in moderate depth accounting for the H/Vs at relative long periods are reliable. Thus we fix the average velocity down to the moderate depth (including the bottom of the boreholes) during the identification. Moreover, considering the density and V _{p} are insensitive to H/V, we fix the densities and the Poisson ratios so that V _{p} can be evaluated from identified V _{s}. Under these constraints, we are able to uniquely identify the velocity structures in the shallow layers from our SW H/V for caplayered media. Although there is a tradeoff relationship between V _{s} and thickness, identification of both V _{s} and thickness in each layer converges faster than that of only V _{s} or only thickness in each layer at some stations.
Figure 8 shows all the profiles of identified V _{s} structures between the surface and bottom of boreholes as well as the initial ones, and the observed and theoretical H/Vs. The initial underground velocity structures are composed of those PS logging data and a cap layer. Additional file 1: Tables A1–A6 list the initial and identified underground velocity structures at six KiKnet stations, while the PS logging data used to construct the initial underground velocity structures are available online (see “Availability of data and materials”). The large variance of velocity profiles and corresponding SW H/Vs (shown in light pink) is explained that the simulated annealing algorithm is likely to find many solutions at local minima before the solution converges toward a global minimum. The black numerical values in the left bottom of H/V panels are the objective function Em between observed H/Vs and SW H/Vs for initial underground velocity structures. The red numerical values in the left bottom of H/V panels are the objective function Em between observed H/Vs and SW H/Vs for identified underground velocity structures. The smaller Em values suggest that the SW H/Vs for identified underground velocity structures agree better with the observed H/Vs at these stations than SW H/Vs for initial underground velocity structures. Meanwhile, the FW H/Vs, evaluated with identified underground velocity structures without cap layer, are in good agreement with SW H/Vs of corresponding caplayered media. It in turn suggests that the SW H/V of caplayered medium can be considered as a simplified calculation of FW H/V of layered medium without cap layer, to identify the underground velocity structures.
Comparison with previous study on H/V and validation of identified underground velocity structures
Regarding SW H/V, we studied the differences between the diffuse field approximation (DFASW H/V) and the Arai and Tokimatsu’s (2004) proposal that considers surface waves from a random distribution of surface sources (DSSSW H/V). Figure 9 shows the comparison between (DFA) SW and DSSSW H/Vs. Just as we did on the observed H/Vs, smoothing with b = 50 by Konno and Ohmachi (1998) is implemented on the DSSSW H/Vs, but not on (DFA) SW H/Vs as they are already smoothed. Both of these two SW H/Vs at six stations show a qualitative agreement. It is attributed to the fact that both of these two studies incorporate the contributions of both fundamental and higher modes of surface waves (e.g., the medium response terms of A _{R} and A _{L}), despite of different premises. On the other hand, the amplitudes of DSSSW H/Vs at peaks are a bit smaller than those of SW H/Vs. It agrees with the study of Kawase et al. (2015) who compared the DSSSW and FW H/Vs by use of underground velocity structures identified from DSSSW H/Vs. The constant RayleightoLove waves amplitude ratio for horizontal motions (R/L) in a period range of 0.1–5 s which is assumed to by Arai and Tokimatsu (2004) suggests that the shape of DSSSW H/V is the same as H/V evaluated from the medium responses of fundamental and higher modes of Rayleigh waves, and the different amplitude between them only relies on the value of R/L. In fact, the R/L may vary with site condition, as reviewed by BonnefoyClaudet et al. (2006b). The apparent peaks of DSSSW H/V around 0.07 s at TCGH16 and 0.4 s at KMMH14, and sharp troughs imply the limitation of R/L which should be variable with period rather than constant. Arai and Tokimatsu (2004) assumed that the sourcetoreceiver distance is larger than one wavelength. Thus their DSSSW H/V involves the power spectra of displacements in the horizontal components (G _{13} and G _{23}) excited by vertical loading force, and the power spectra of displacement in the vertical component (G _{31} and G _{32}) excited by horizontal loading force. In contrast, the imaginary parts of G _{13}, G _{23}, G _{31}, and G _{32} which are not required for evaluating FW H/V in Eq. (1) are zero when source and receiver coincide with each other both in position and direction. It is considered another reason for the difference between (DFA) SW and DSSSW H/Vs. Lunedei and Albarello (2009) followed the assumption of Arai and Tokimatsu (2004) that surface waves were generated by a random distribution of surface sources, and improved the formulation for surface waves approximation by considering the effects of damping and discarding the R/L assumption. They further released the limitation on the sourcetoreceiver distance (e.g., r = 0) to develop the formulation for fullwave (Lunedei and Albarello 2010). For the same layered medium, DSSFW H/V is found to be similar to FW H/V (GarcíaJerez et al. 2012; Lunedei and Albarello 2015), but the computation is not faster than that of FW H/V evaluated by discrete wavenumber method.
The available strongmotion records at KiKnet stations can be used to evaluate the Earthquake motions H/V Spectral Ratio (EHVSR) and Spectral Ratio of Horizontal motions between Surface and Bottom of borehole (SRHSB). Therefore, we validate the underground velocity structures identified from SW H/V, if the observed and theoretical EHVSRs and SRHSBs have a good consistency. Data analysis about earthquake motions is similar to that of Ducellier et al. (2013). The Swave parts of strongmotion records for small events (e.g., peak ground acceleration < 100 gal; magnitude < 6; epicenter distance < 200 km) are used to evaluate the power spectra in three components, i.e., north–south, east–west, and up–down. Smoothing with b = 50 by Konno and Ohmachi (1998) is implemented on the power spectra in three components before finding the square root of power spectral ratios, EHVSR and SRHSB, for each small event. Finally, the observed EHVSRs and SRHSBs are arithmetically averaged over the number of small events. The average EHVSRs and SRHSBs, as well as one standard deviation plus and minus the average one, are shown in Fig. 10. Additional file 2: Tables B1–B6 list the information about small events at six KiKnet stations. Although the number of strongmotion records at a couple of stations is not adequate due to low seismicity, we provide the standard deviations of spectral ratios as rough indicators of variability. The underground velocity structures from surface to bottom of boreholes are used to evaluate the theoretical SRHSBs by the propagation matrix method (Haskell 1960). The underground velocity structures from surface to halfspace without cap layer are used to evaluate the theoretical EHVSR which is proportional to the square ratio of transfer functions between S and Pwaves based on diffuse field approximation (Kawase et al. 2011). Smoothing with b = 50 by Konno and Ohmachi (1998) is also implemented on the theoretical SRHSBs and transfer functions.
The effects of attenuation are present in real data, but using damping within the diffuse field approximation is not fully supported by theory. Snieder et al. (2007) showed that the Green’s function can be retrieved from the response to random forcing for a variety of conditions, including the extreme case of the diffusion equation. In many practical circumstances, small amounts of damping allow stabilizing results so that the retrieved Green’s functions are consistent with observations (e.g., Lawrence and Prieto 2011). Here we assume the quality factor for Swave, Qs = 100, the same as that for FW H/V. Figure 10 depicts the comparison between observed and theoretical EHVSRs and SRHSBs. The theoretical counterparts were computed for both initial and identified underground velocity structures. The black numerical values in the left bottom of EHVSR and SRHSB panels are the objective functions between observed and theoretical spectral ratios for initial underground velocity structures, and the red ones are the objective function between observed and theoretical spectral ratios for identified underground velocity structures. The smaller objective functions at stations other than GIFH29 and KMMH14 suggest that EHVSRs and SRHSBs for identified underground velocity structures agree better with the observed counterparts. This indicates that the underground velocity structures identified from SW H/V are reasonable. The disagreement between observed and theoretical EHVSRs at GIFH29 and KMMH14, and between observed and theoretical SRHSB at KMMH14 in the short period range, might imply that the assumed damping (Qs = 100) is not appropriate. This issue requires further scrutiny.
Conclusions
Based on the diffuse field approximation, FW H/V is expressed with the imaginary parts of the Green’s functions which incorporate the contributions of both body and surface (Rayleigh and Love) waves. SW H/V is expressed with medium responses of Rayleigh and Love waves according to the Cauchy’s contour integration method. For a given layered medium, the SW H/V was almost the same as the FW H/V in the short period range. It suggested that the contribution of surface waves trapped in shallow layers was overwhelmingly dominant compared with that of body waves. The largest peak of SW H/V was caused by a local trough of fundamental or higher modes of vertical medium responses of Rayleigh waves. The predominant period of SW H/V is close to but not identical to that of FW H/V unless the impedance ratio is large enough. Therefore, the calculation of medium responses of Rayleigh and Love waves gave insight into the contribution of surface waves to fullwave, and the relative contribution between fundamental and higher modes of surface waves.
By adding a cap layer to the bottom of given layered medium, the SW H/V of caplayered medium agreed well with the FW H/V of layered medium in the period range of engineering interest (e.g., 0.02–5.0 s). The depth of upper interface of the cap layer is about (5–10)λ ^{A}_{0} (product of thicknessweighted average Swave velocity from surface to halfspace of target layered medium and “apparent” predominant period of fundamental mode of target layered medium), and the velocity of the cap layer is about twice as large as the one in the halfspace of given layered medium. Because the computation of surface waves was significantly fast, SW H/V of caplayered medium, as a simplified calculation of FW H/V of layered medium without cap layer, was preferable to be applied to identify underground velocity structures.
We applied the SW H/V of caplayered medium to identify the underground velocity structures between surface and bottom of boreholes (about several hundred meters in depth) at some KiKnet stations where the FW H/Vs did not match with the observed H/Vs. We employed the identified underground velocity without the cap layer to evaluate the FW H/V. Good consistence between FW H/V with SW H/V of the corresponding caplayered medium was confirmed. It in turn suggested that SW H/V could be regarded as a simplified calculation of FW H/V to identify underground velocity structures. We further investigated the difference between SW H/V of our approach and DSSSW H/V proposed on a different premise. Although the amplitude of DSSSW H/V was a bit smaller than our SW H/V, the similarity of these two SW H/Vs can be confirmed to some extent. Finally, we employed the identified underground velocity structures to evaluate earthquake motions of H/V spectral ratios based on diffuse field approximation (EHVSR) and spectral ratios of horizontal motions between surface and bottom of boreholes (SRHSB). The good agreement between theoretical and observed EHVSR, and between theoretical and observed SRHSB, indicated that the underground velocity structures identified from SW H/V of caplayered medium were well resolved by the new method.
This research implied verifying many details in order not to leave dead ends. The results are quite satisfactory because efficient, faster procedures for modeling and inversion of DFA H/V were developed. Some fundamental issues and significant questions are still open, such as the effect of high damping in soft sediments, the sensitivity to the objective function, and the temporal variance of observations. We need to clarify these issues and open questions before our approach becomes a practical tool for field practitioners. This will require further scrutiny and we are committed in this noble endeavor.
Abbreviations
 DFA:

diffuse field approximation
 H/V :

microtremor H/V spectral ratio
 FW:

full–wave
 SW:

surface waves
 DSSSW:

surface waves based on distributed surface sources assumption
 EHVSR:

earthquake motions of H/V spectral ratio
 SRHSB:

spectral ratio of horizontal motions between surface and bottom of borehole
 Tp_{0} :

predominant period of fundamental mode
 λ _{0} :

wavelength of fundamental mode
 Tp ^{A}_{0} :

“apparent” predominant period of fundamental mode
 λ ^{A}_{0} :

“apparent” wavelength of fundamental mode
 V _{s} :

Swave velocity
 V _{p} :

Pwave velocity
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Authors’ contributions
HW designed this study and drafted the manuscript. KM and HW conducted the microtremor measurement at strongmotion stations. KI participated in all discussions throughout the study. FS participated in the discussions and interpretation. All authors read and approved the final manuscript.
Acknowledgements
The authors wholeheartedly thank two anonymous reviewers and the Guest Editor PY Bard, for their detailed review comments and most valuable editing suggestions of the admittedly rough, initial manuscript. Special thanks to the National Research Institute for Earth Science and Disaster Resilience (NIED) for providing PS logging data and strongmotion records at KiKnet stations. We thank E. Plata, G. Sánchez, and their team of the Unidad de Servicios de Información (USI) of the Institute of Engineering—UNAM, for locating useful references. This work was supported in part by the GrantsinAids for Scientific Research (B) (P.I.: Hidenori Kawabe, Grant Number: 16H03144) from the Ministry of Education, Culture, Sports, Science and Technology of Japan, by AXA Research Fund and by DGAPAUNAM under Project IN100917. Figures are plotted with Generic Mapping Tools (GMT) (Wessel et al. 2013).
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
Strongmotion records at studied KiKnet stations can be downloaded from the website “http://www.kyoshin.bosai.go.jp/kyoshin/” after a free registration. PS logging data (thickness, Pwave velocity, Swave velocity in each layer) are available from the Web site “http://www.kyoshin.bosai.go.jp/kyoshin/db/index.html?all”.
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Additional files
Additional file 1: Table S1.
Underground velocity structure model at TCGH16. Table S2. Underground velocity structure model at KMMH03. Table S3. Underground velocity structure model at GIFH28. Table S4. Underground velocity structure model at GIFH29. Table S5. Underground velocity structure model at SZOH26. Table S6. Underground velocity structure model at KMMH14.
Additional file 2: Table S1.
Small events at TCGH16. Table S2. Small events at KMMH03. Table S3. Small events at GIFH28. Table S4. Small events at GIFH29. Table S5. Small events at SZOH26. Table S6. Small events at KMMH14.
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Wu, H., Masaki, K., Irikura, K. et al. Application of a simplified calculation for fullwave microtremor H/V spectral ratio based on the diffuse field approximation to identify underground velocity structures. Earth Planets Space 69, 162 (2017). https://doi.org/10.1186/s4062301707468
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DOI: https://doi.org/10.1186/s4062301707468
Keywords
 Diffuse field approximation
 Microtremor H/V spectral ratio
 Fullwave
 Surface waves
 Underground velocity structures