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Influence of lowvelocity superficial layer on longperiod basininduced surface waves in eastern Osaka basin
Earth, Planets and Space volume 75, Article number: 55 (2023)
Abstract
The longperiod strong ground motions with periods above 1 s have, in the case of farther or deeper earthquakes, potential to cause serious damage to structures with low eigen frequency, such as long bridges, oil tanks, or artificially damped structures, such as highrise buildings. This work focuses on wave propagation due to the deep large earthquake representing rare deep damaging events of the region, with relatively sparse data coverage, studied for simple geological models computed by 2D finite differences. We model the wave propagation by finite differences using uptodate 3D structural model of the Osaka basin. The strong surface waves in the region are not directly generated by these deep sources, but they originate by refraction mostly at the edges of the bedrock–sediments interface. The objective of this research is to model observed surface Love wave generated in the eastern part of the basin that propagates approximately westwards and is recorded by several surface stations. At these stations, the 3D finitedifference modeling provides a good fit with the observed surface wave in terms of waveform, amplitude, and arrival time for the most detailed 3D velocity model that contains topmost 50–250 m structure with the lowest Swave velocities of 250 m/s. The semblance analysis of the synthetic wave field reveals that the respective synthetic surface wave is a result of interfering waves arriving in OSA and WOS stations from NE and SE directions. Performed tests reveal that such a synthetic wave field is extremely sensitive to the presence of the superficial 50—250 m thick lowvelocity structure which is only a small fraction of the propagating surface wave length and occupies only part of the surface area. The ability to model the surface wave in terms of amplitude and time arrival validates the 3D structural model for longperiod Osaka Bay earthquake scenario computations.
Graphical Abstract
Introduction
The importance of the longperiod ground motions with long durations in earthquake hazard and risk mitigation is manifested by serious damages to large structures in period bands longer than 1 s. Such longduration ground motions have relatively strong impact within the first few hundred kilometers of epicentral distance. They potentially affect all longperiod structures (Furumura et al. 2008 and references therein) such as highrise buildings, long bridges (Fujino and Siringoringo 2013), oil tanks (Hatayama 2008), and artificially damped structures (Ismail 2018 and references therein). For the Osaka region, the longperiod motions are relevant between 1 and 10 s, namely, for shallow and large offshore events.
An important high casualty example due to seismic hazard can be well known case of Mexico City damaged at 400 km epicentral distance from Michoacán event, 1995 (Beck and Hall 1986). Among many damaging Japanese events with longperiod effects, such as 2003 Tokachioki, Japan, earthquake (Koketsu et al. 2005; Hatayama 2008) with oil tanks damaged in the Yufutsu basin situated as far as 250 km from the epicenter—the damages reported in Tomakomai were observed on structures with eigen periods between 1.5 s and 15 s. Out of those severe damages (all in the Tomakomai west port area) were on structures with eigen periods between 7 s and 8.5 s and one case even at 12 s. Another example is the 2004 Kii peninsula earthquake (Miyake and Koketsu 2005; Iwata and Asano 2005—modeling) stimulating longperiod ground motions in Omaezaki region (Shizuoka prefecture), Kanto (with Tokyo city), Osaka, and Nobi (with Nagoya city) basins representing a wide area of Honshu, Japan, with epicentral distances from 200 to 250 km.
The Osaka metropolitan area (Fig. 1) is densely populated and threatened by earthquake hazards. The longperiod ground motions modeling according to contemporary 2D or 3D Osaka basin models was done by numerous authors (Hatayama et al. 1995; Kagawa et al. 2004; Sekiguchi et al. 2008; Iwaki and Iwata 2008, 2010; Asano et al. 2016) and also the contributions to the 1998 meeting Effects of Surface Geology (Irikura et al. 1998). Yet, the studies have not concentrated on research combining the rare deep event with the temporary OSA array recoding and uptodate 3D Osaka basin model. The observations or modeling of basininduces surface waves amplified inside the sedimentary area of basin at various locations over the world were studied by, e.g., Frankel (1993), Koketsu and Kikuchi (2000), Kawase and Aki (1989).
The focus of the present study is the interpretation of locally generated surface waves in terms of the ground motions due to deep earthquakes using uptodate 3D velocity models of the Osaka basin. The deep 1993/12/10 M_{JMA} = 7.1 earthquake (388 km hypocentral depth) resulted in longperiod ground motions of the Osaka basin and was recorded by a relatively sparse temporal seismic network, see Table 1 and Fig. 1. Nevertheless, these recordings present a unique dataset, as there have been no strong regional deep earthquakes array recordings in the area since then. Because the hypocentral depth of the event is much larger than the surface wave eigen function reach for investigated frequencies, strong surface waves are not directly generated by this deep source. They originate as refractions mostly at the edges of the bedrock–sediments interface. Previous analysis of these recordings identified a strong longperiod wave phase (called SL1), which was interpreted as Love wave locally generated at the Eastern edge of the Osaka basin (Hatayma et al. 1995). In particular, the SL1 was generated by the Swave impeding the eastern Osaka basin edge close to ISK station, and continuing along YAE, MKT OSA, and WOS stations, sequentially (Fig. 1). The interpretation was based on a simplified 2D modeling of the seismic wave propagation, which could not explain the SL1 arrival time and amplitude at OSA and WOS, missing the 3D surface wave field complexity (Hatayma et al. 1995). Therefore, in the present study, we reanalyze the data with the semblance technique, as described later, and we do the same with the synthetic wave field. For this we use the uptodate 3D velocity model of the Osaka basin and its larger locality (Sekiguchi et al. 2013, 2016). We use hybrid formulation (Oprsal et al. 2009) applied in the SW4 software 3D FD package (Petersson and Sjögreen 2017b) modified as rayFD approach (Oprsal et al 2002; also see Table 2) for impeding Swave. The study can be considered as one of the validation tests of the current 3D velocity model.
The method
Finitedifference computations, hybrid formulation
The numerical simulation of the source generating seismic waves, regional wave propagation, and the local site effects is carried through by finite differences (FD). Out of presently available established seismic wave propagation codes (Chaljub et al. 2010; and references therein), we use the freely available SW4 (Seismic Waves, 4th order) code of Petersson and Sjögreen (2012; 2015; 2017a,b). After careful consideration, the SW4 code appears to be optimal in time efficient for in the computerdemanding Osaka basin wave propagation simulations. The code can simulate seismic wave generation and propagation in a topographic 3D inhomogeneous viscoelastic anisotropic medium. Its design enables a distributedmemory parallel run. The governing viscoelastic equations are approximated by fourth order of accuracy in space using a gridnodebased finitedifference approach and by explicit time formulation. The SW4 may apply a curvilinear grid to incorporate curved topography, local vertical mesh refinement to cover well computation of high elastic moduli areas inside the hard rock and very low moduli in the soft sediments at the same model. The method is stable for high Poisson’s ratios and high Poisson’s ratio contrasts and allows for a realistic model of seismic attenuation. The autogenerated meshing is based on specific numerical requirements.
The parallel computations presented in this study are required to be accurate in the period range of 1 – 100 s with the lowest Swave velocities 250 m/s located in the superficial parts of the model. The size of the complete allinone computational FD model, including the 388 km deep source with epicentral distance around 380 km, would be rather large. Because of the computertime and memory requirements of such settings for complete P and Swave filed included in one computation, we consider only the Swave impeding the Osaka basin in a twostep hybrid formulation (Oprsal et al. 2009) in hybrid rayFD variant (Oprsal et al. 2002; and see Appendix 1). In the first step, the ray method for source and path effects up to regional structure is considered. Following second step is using local FD with Osaka basin structural model included. The secondstep results then contain source, path, and site effects for the Swave part of the wave field impeding to the basin.
The code has been thoroughly tested and validated against analytical and numerical methods (Petersson and Sjögreen 2017a; and references therein), while the hybrid formulation adopted inside SW4 was tested against fullscale computation by the SW4 by replication tests generalized by Oprsal et al. (2009).
Geological structure and FD computational model approximation
There have been many geological and geophysical techniques extensively investigating the 3D Osaka basin structure on various size scales and locations over the past decades (e.g., Horikawa et al. 2003; Kagawa et al. 2004; Iwata et al. 2008; Iwaki and Iwata 2011). We use the most uptodate 3D Osaka velocity structure model (Sekiguchi et al. 2016) with recent updates (Iwata and Sekiguchi, pers. comm. 2021). The size of the presently used Osaka basin area geological model is 85 km × 90 km, respectively (left panel of Fig. 2). As to the depth, the lowest layer crustal interface between 30.86 km and 34.58 km is modified to be horizontal at a 32 km depth. Deeper structure is adopted from the PREM isotropic model (Dziewonski and Anderson 1981), as nondecreasing function (Bormann 2012, Datasheet DS 2.1, Table 1).
In order to represent the 3D continuous velocity model in SW4 properly, it is transformed into FD computational grid of effective parameters with a grid step of 50 m following the procedure shown in Oprsal and Zahradnik (2002, par. 2.5.3). The detailed velocity model (Fig. 2) is completely included in terms of horizontal dimensions and goes approximately to a depth of 2.7 km. The size of the computational model is further enlarged to NS x EW x DEPTH = 118 km x 123 km × 3.1 km by adding the nonreflecting boundaries. The lowest Swave velocity in the superficial parts of the model (Sekiguchi et al. 2016) is 248 m/s. It serves as input for computational model with finest gridding of 50 m. The lowest Swave velocity of the computational model is 250 m/s. For the numerical experiments, we also consider another model with minimum Swave velocity Vs = 500 m/s for each computational cell with original Vs < 500 m/s. Hatayama et al. (1995) used the 500 m/s superficial layer to improve the arrival time and amplitude of the SL1 fit. Moreover, it allows to show dependence of the results on the superficial layers’ velocity. These models are hereinafter called Vsmin250 and Vsmin500, respectively. The topography is flat and no water medium is considered. The topography and seabottom upper structure is shifted into flat topography while keeping the upperbasin structure vertical profile under each such FD horizontal surface point unchanged according to original topographical model vertical profile. The Q_{p} and Q_{s} are realized by viscoelastic modeling using three standard linear solid mechanisms (Petersson and Sjögreen 2012; 2017a) providing correct damping for periods of 0.4 – 40 s.
The seismic source
The M_{JMA}7.1 event of October 12, 1993 was the only strong deep regional event recorded during temporary operation of the OSA array, while also recorded at ISK, YAE, MKT, and WOS stations (Hatayama et al. 1995). The point source located at Lat = 32.017N, Lon = 138.233E, depth = 388 km (after Hatayama et al. 1995) is represented by a double couple of [ strike, dip, rake] = [ 171, 82, 81] degrees (JMA nodal plane solution, referred to as 1993/10/11 15:54:20.90 event, ID447083). The generated waves are propagated by ray method in 1D layered medium in the first hybrid step. We compute only Swave ray from source up to the excitation box boundary underlying the whole Osaka basin computational area. The time history of the raypropagated Swave pulse is integrated Ricker wavelet with ω = 0.50 Hz (Petersson and Sjögreen 2017a). To represent the complexity of the source function, the complete 3D FD hybrid response is then convolved with EW horizontal component of nearbedrock ISK station recording of the 1993/10/12 M_{JMA} = 7.1 event. The simulated results (and data) are band passed between 0.05 and 0.4 Hz. The applied band encompasses the frequency range where the observations match the dispersion curve of the fundamental mode of the Love wave, which is the main feature of SL1 (Hatayama et al. 1995).
Semblance and polarization analysis
We adopt the semblance technique (Neidell and Taner 1971) to determine the apparent wave propagation velocity and direction at a given point and time, see Fig. 3. It is applied to bandpassed time histories of a respective array. The frequency range that we utilize for the semblance calculation is from 0.20 to 0.38 Hz. This range is specifically chosen because it encompasses the frequency band where the energy is the strongest (Hatayama et al. 1995). Additionally, narrowing the frequency band for this particular analysis also helps stabilize the results. The necessary time window for each semblance central time, determined by the frequency content of the narrowband signal, is 1.9 s; hence, the semblance sensitivity (or uncertainty) in time is approximately 1 s. The semblance value threshold is specified in the results. The FD synthetics are analyzed for respective arrays placed at location of each of the five stations (ISK, YAE, MKT, OSA, and WOS); the aperture of each such virtual array is 400 m.
The time–frequency polarization analysis is then used to characterize particle motion at a given point. This motion is, in general, elliptical for a short time window and is described by the orientation of the semimajor axis of the ellipse in space (Burjánek et al. 2010).
Results
The presumed surface wave generated in the bedrockbasin border in the vicinity of the ISK station due to incident Swave and structural discontinuity close to Ikoma fault propagates along the free surface approximately in the WNW direction. The semblance analysis for station locations ISK, YAE, MKT, OSA, and WOS are shown in Fig. 4, upper right panel. The value of the semblance threshold is 0.80. The minimum observed semblance values (SB_{station_name}) for the Lovewave arrivals are as follows: SB_{YAE} > 0.85 (typically 0.90), SB_{MKY} > 0.88 (typically 0.90), SB_{OSA} > 0.9 (typically 0.95), and SB_{WOS} > 0.93. Figure 4 shows apparent slowness and polarization vectors of the horizontal ground motion at distinct times to indicate the presence, direction, and apparent velocity of the surface Love wave (mostly visible on NS component, i.e., transverse direction to slowness vector). The velocity of the surface Love wave, present between times of 39 and 41 s, is mostly between 650 and 800 m/s being slightly higher than for the data, heading SW and turning to WNW after 5 s with velocity between 1000 and 1500 m/s, which is mainly due to the reason that two wave groups are arriving at the OSA station at a similar time. This is also manifested by inconsistent polarization azimuth with respect to obtained slowness vector. The animation and respective snapshots suggest that the surface waves were created at the eastern edge of the Osaka basin, contoured by the Ikoma fault system (Figs. 1, 2), and were directed differently due to the curved shape of the basin edge and the 3D structure in the area. Figure 4 shows the semblance analysis also for YAE, MKT, and WOS with the Lovewave velocities are between 650 and 1000 m/s. Prevailing propagation at YAE and MKT is in a westward direction, while at WOS the wave propagates toward a WNW direction. The synthetics are shown for stations and arrays corresponding to their real positions. The synthetic solution shows relatively strong and longlasting oscillations at OSA station; however, the longer wave train does not really correspond to sole SL1’ surface Love wave as depicted in Fig. 4. The amplitude of the SL1’ wave is stronger also at the WOS station, with short duration and little earlier arrival than the observed SL1. For semblance analysis of data recorded at OSA array see Fig. 3. The comparison of the synthetic to data is shown in Fig. 5. The timing and fit of the impeding Swave synthetics and data are very good in terms of timing and amplitudes at ISK station, and the OSA data are more oscillatory than synthetics. Arrival of SL1’ at all stations is satisfactory enough to visually correlate the same SL1 on data for YAE, MKT, and WOS. The amplitudes at OSA station are a bit lower in the synthetics. Later arrivals (after 59 s) at WOS belong to another wave group arriving from SES (see also Fig. 4) that is generated at the basin edge located approximately 25 km SES of the WOS station (see attached fullscale animation, Figs. 1, 2, and 12).
We have done an additional numerical experiment for the same setting with modified 3D FD model called VSmin500 (Fig. 6) to approximate modeling cases of previous research of Hatayama et al. (1995) who added a superficial layer of 500 m/s to the model to emphasize the surface wave velocities and to add some oscillatory part to the wave field. On the other hand, this model is simpler and “faster” in the vicinity of the free surface sedimentary parts of the Osaka basin, than the Vsmin250 model. As a result of the higher propagation velocities and lower attenuation, the singletoplayered Vsmin500 model results are showing a different SL1’ hodochrone (read from waveforms) and are less complex. The slowness vector has changed direction from WNW to NNW at WOS at about 54 s compared to VSmin250 case. OSA station shows predominantly NWpropagating surface wave of 700 m/s at 44 s contrary to the same case in Vsmin250, where it propagates in SW direction at 46–50 s at 1100 m/s with much more complex arriving wave field. The difference of the SL1’ phase apparent velocity and propagation direction is substantial. Because the semblance technique is able to pick only one apparent slowness or velocity vector at a time, its sensitivity in this case indicates relatively high sensitivity of the wave field amplitudes and directions on the superficial part of the model in terms of generation of the surface waves, their amplitudes, and propagation.
Comparison of the best solution of the 2D FD modeling (Hatayama et al. 1995), the Vs250 and the Vs500 3D FD modeling are then shown in Fig. 7. Basininduced Love waves SL1 are compared at the five stations. The model with low velocities (Vsmin250) preserves the SL1’ phase in OSA and WOS stations. The Vsmin250 and Vsmin500 model results do not differ much in the earliest times after the Swave arrival thanks to prevailing lowfrequency content of the signal. The differences of the hodochrones in Vsmin250 and Vsmin500 solutions is due to slightly different average group velocity of the SL1’ phase for different models.
Conclusions
The wave propagation in the Osaka basin, including the surface waves, is essentially a 3D wave propagation problem. The lowperiod Swave of the respective event with simple pulselike character generates surface waves at the eastern and southeastern edges of the model. The waves propagate throughout laterally inhomogeneous Osaka basin structure, where they mutually interfere. The 2D numerical modeling of the case does not provide the complexity of the surface wave field, where the structural complexity trades off the amplitudes with arrival times.
In 3D modeling, we have used the uptodate 3D structural model of the Osaka basin called Vsmin250. Added to that we also tested its modification called Vsmin500 to reflect the fact that the 2D modeling of SL1’ Love surface waves had improved only partially in time arrivals without a good amplitude fit by adding the superficial Vs = 500 m/s layer of 400 m thickness atop of the basin (Hatayama et al. 1995). Both of the two 3D models have local superficial strata of various thickness and minimum Vs being 250 and 500 m/s, respectively, the latter simplifying heterogeneous nearsurface velocity structure.
The thin lowvelocity parts of the model are important even in lowfrequency bands. Counterintuitively, the superficial 50 m thick low Swave velocity areas occupying only 1/20 to 1/10 of Swave length are reasonably influencing the wave field in terms of surface wave propagation. It is due to the fact that the respective wave spends several periods when traveling though such a thin lowvelocity stratum patch of the surface. Thus, the propagation of strong Love waves inside the basin is conditioned by presence of nearsurface lowvelocity structure, or as an approximation, by thin lowvelocity structure of 50–200 m thickness which is a small fraction of the Swave length.
The 3D FD modeling validates present 3D Osaka basin structural model (Vsmin250) by the synthetic SL1’ phase generated at the Eastern part of the Osaka basin. This wave exists for both structural models with Vsmin = 250 m/s and Vsmin = 500 m/s. A nearsurface layer being fraction of wavelength thick reasonably influences the wave field in terms of prevailing surface waves. The amplitude of the SL1’ phase depends strongly on the azimuth of the impeding Swave. Therefore, the present computational 3D velocity model Vsmin250 prepared within this study (based on Asano et al. 2016) is suitable for longperiod Osaka Bay earthquake scenario computations.
Wave propagation in both 3D models, Vsmin250 and Vsmin500, contains the SL1’ phase in OSA and WOS stations in lowfrequency band. Vsmin500 model SL1’ has earlier arrivals and lower amplitudes in further stations (OSA and WOS).
The differences in semblance result for Vsmin250 and Vsmin500 model synthetics (in band 0.2–0.38 Hz) are caused by scattered wave field arrivals at higher frequencies contained in the Vsmin500 model arriving from different directions. The semblance analysis for the OSA array shows Love wave propagating mostly in NWN to SWS directions for data and for both FD synthetic models. The reason for earlier synthetic Lovewave arrival at WOS station may be that the computational model’s lowest Swave velocity was kept above 500 m/s limit. This clearly manifests strong sensitivity of the longperiod wave field to the presence of small volumes of low propagation velocities in the 3D model.
Modeled 3D wave field at OSA station at the time of SL1’ phase arrival is a mixture of several surface waves traveling in SWW and WNW directions, while the semblance technique picks only one of the directions from the whole arriving surface wave field. This suggests that the real observed SL1 surface wave phase is not generated solely at the eastern edge of the Osaka basin and does not arrive at the stations purely from an EW direction. The simulations support the possibility that these surface waves traveling in different directions come across at OSA at the time of SL1. There is high possibility that both are generated along Ikoma fault system (eastern edge of the Osaka basin) propagating in slightly different directions because of the bent shape of the basin edge line and 3D structure between the basin edge and OSA.
Availability of data and materials
The data shall not be shared, and they are not publicly available.
Abbreviations
 3D:

Three dimensional
 FD:

Finite difference/ finite differences
 SL1:

Surface Love wave (data)
 SL1’:

Surface Love wave (synthetics)
 M_{JMA} :

Magnitude by Japan Meteorological Agency
 NS:

North–south (a direction parallel with lines of longitude)
 EW:

East–west (a direction parallel with lines of latitude)
 NW:

Northwest
 SW:

Southwest
 SSE:

Southsoutheast
 WNW:

Westnorthwest
 NNW:

Northnorthwest
 Lat:

Latitude
 Lon:

Longitude
 s:

Second
 Hz:

Hertz
 m, km:

Meter, kilometer
 VSmin250:

Model with minimum Vs = 250 m/s
 VSmin500:

Model with minimum Vs = 500 m/s
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Acknowledgements
In terms of the computer resources, this work was supported by the Ministry of Education, Youth and Sports (MEYS) of the Czech Republic through the eINFRA CZ (ID:90140) of IT4Innovations Centre, Technical University, Ostrava. The seismic ground motion data in this study are observed by the temporal strong motion observations maintained by the authors of Hatayama et al. (1995).
Funding
This work was financially supported by Czech Science Foundation (GACR), project no. GA2015818S; and by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, under its The Second Earthquake and Volcano Hazards Observation and Research Program (Earthquake and Volcano Hazard Reduction Research).
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IO contributed to conceptualization, analysis of the data and synthetics, coding the data processing software, interpretation of the results, visualization, and drafting the manuscript including figures. HS was involved in creation of the velocity model, supporting the research by knowledge of the local conditions, data curation, and supervision; TI contributed to conceptualization, creation of the velocity model, supporting the research by knowledge of the local conditions, data curation, approving the station positions and data, formal analysis, and supervision; JB was involved in methodology, draft corrections, enhancing paper by the interpretation of results, and supervision. All the authors read and approved the final manuscript.
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Supplementary Information
Additional file 1. The animation of total horizontal ground motion of the scattered wave field of the Osaka basin's response to the Swave of the 1993/10/12 M_{JMA}7.1 event for Vsmin250 model and frequency range of 0.050.4 Hz. The lines originating at each of the stations show slowness vector of prevailing apparent wave propagation direction in the horizontal plane. The time is same as in Figures 47, the amplitude is normalized.
Additional file 2. The animation of total horizontal ground motion illustrating the complexity of the scattered wave field of the Osaka basin's response to the Swave of the 1993/10/12 M_{JMA}7.1 event for Vsmin250 model and frequency range of 0–1 Hz. The shown surface size is SN x WE = 85 km x 90 km (without damping area). The time is same as in Figures 47, the amplitude is normalized.
Appendices
Appendix 1
The hybrid box technique can inject wave field containing source and path effects inside local or regional FD hybrid model while being permeable for any wave field – scattered inside the box or backscattered into the box from outside. The following shows the hybrid method (Oprsal et al. 2009) applied for case of planar Swave traveling arbitrarily from below with purely horizontal displacements, which is strongly simplifying the Eq. 22 (Ibid) to be used in the following example. The hybrid box is realized by horizontal double FD plane throughout the whole model. Horizontal forces added to the elastodynamic equation at neighboring FD points (as in Fig. 8) of the box realize the traction discontinuity. The application of sole forces assures permeability of the box.
Let \({u}_{FD1}\left({t}_{ray}\right)\) and \({u}_{FD2}\left({t}_{ray}\right)\) be timeintegrated histories of displacements computed by ray method in a certain point, where \({t}_{ray}\) is time of the histories (Fig. 8). Let further \({t}_{FD}\) be the time history in the FD method and time shift,
be a constant shift, which is a usual case when FD computation starts at \({t}_{FD}=0\) while \({t}_{ray}\) reflects the travel time from source. For brevity of the demonstration, let us consider only horizontalmotion planar Swave. Then the time history of, e.g., horizontal forces (for SW4 method) prescribed at FD grid point 1 (scattered wave field area, Fig. 8) is
and the time history of, e.g., horizontal forces prescribed at FD grid point 2 (scattered wave field area, Fig. 8) is
where \(\upmu\) is Lame’s parameter, v_{s} is the Swave velocity, and other variables are depicted in Fig. 8, \(\overrightarrow{{s}_{s1}}=\overrightarrow{{s}_{s2}}\) for planar wave, thus having the FD point 1 ray method wave field values signreversed and being delayed by \(\overrightarrow{{s}_{s2}}\cdot \overrightarrow{{n}_{2}} DZ\) to be applied in FD point 2. In this case, the firststep ray (paraxial) computation \({f}_{FD2}\) can be expressed as
The same result could be obtained using the wave equation. For incident planar wave impeding vertically (4) simplifies further to
Hence, paradoxically, the prescribed force realized at the scatteredwavefield nodal point 2 is delayed by \(\frac{DZ}{{v}_{s}}\) and has the opposite sign to the force realized at the completewavefield nodal point 1 (Fig. 8). The opposite signs of the \({f}_{FD}\) force are finitedifference realizations of (point) traction discontinuity \({f}_{FD2}  {f}_{FD1}\) on the permeable hybrid boundary. Thus, general traction discontinuity realized by voluminal forces \({f}_{FD2}  {f}_{FD1}= f\left({u}_{FD}\right)\), where \({u}_{FD}\) is displacement at the hybrid boundary computed in the first step (here by the ray method). Both, \({u}_{FD1}\) and \({u}_{FD2}\), have to be computed to approximate \({\nabla u}_{FD}\) and \({\nabla \bullet u}_{FD}\) in the Eq. 22 and paragraph 2.5.1 of Oprsal et al. (2009). The mentioned Eq. 22 also shows that applying \({u}_{FD1}\) and \({u}_{FD2}\) as described in the above example produces time wave field being derivative of the impeding \({u}_{FD1}\) at the inner boundary. In the SW4 code, the voluminal force discontinuity is realized, as FD approximation, in every two grid points (upper and lower) of two neighboring horizontal grid planes placed under the whole regional computational model including the damping area. The amplitudes of the forces are tapered to zero within the damping area at the edges, where the upper and lower hybrid box planes go through. Hence, there are no sides to the box in this case. There are around 180 000 automatically generated singleforce point forces (or 90 000 traction discontinuity dipoles) with negligible influence on computational speed. The point forces are being active from 7 to 25 s of FD computation time history corresponding to complete coverage of the area by the obliquely impeding Swave. Complete ASCII SW4 input for parametric forces is less than 18 MB of data in our case.
Appendix 2
Computational Vsmin250 model Vs velocity of the 50–100 m and 100–150 m depths is shown in Fig. 10.
Excerpt of the computational Vsmin250 model Vs velocity of the 50–100 m and 100–150 m depths is presented in Fig. 11.
Horizontal ground motion at band 0–1 Hz. The red arrows show propagating surface waves in the eastern Osaka basin is shown in Fig. 12.
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Oprsal, I., Sekiguchi, H., Iwata, T. et al. Influence of lowvelocity superficial layer on longperiod basininduced surface waves in eastern Osaka basin. Earth Planets Space 75, 55 (2023). https://doi.org/10.1186/s40623023018049
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DOI: https://doi.org/10.1186/s40623023018049