**4. Observed and theoretical SAF**

As mentioned in the previous section, the UVM of NIED is considered to be the most reliable velocity model for the strong motion simulation because the UVM combines all the available geophysical information to date related to the velocity structures from the ground surface to the seismological bedrock as densely sampled as possible. However, the actual S-wave SAF at a specific site, as shown in **Figure 1**, can be significantly different from the theoretical one calculated by a simple 1D S-wave multi-reflection theory for a stack of layers [20–22]. To see the difference, we plot in **Figure 4** comparisons of the 1D theoretical hSAF with the observed hSAF at the same four sites in **Figure 1**. We calculate the theoretical hSAF as the 1D soil response on the surface of a combined velocity structure of the shallower- and deeper-parts with respect to the outcrop of the seismological bedrock motion (i.e., twice of the input at the bottom of the deeper part). Except for the site SZOH31, the theory tends to underestimate the observation.

The major reasons for discrepancy are twofold; one is due to an inevitable inaccuracy of the derived velocity structure, and the other is due to a too simplistic assumption of the 1D horizontally-flat layered model. In the former there are two possible causes; one is the inaccuracy of the referenced values to delineate the

**Figure 4.**

*Observed hSAF extracted by GIT and 1D theoretical hSAF from UVM after [18]. Fundamental characteristics are well reproduced, however, the theoretical hSAF tends to underestimate the observed hSAF.*

structure including the assumed Q-values, and the other is the rapid spatial variations within the 250 m grid. In the latter there are two possible causes; one is the additional amplification due to the basin-induced surface waves generated at the edge of two or three-dimensional (2D or 3D) basins (see for example [23–25]) and the other is the topographic effects near the surface of irregular shapes such as a hill, a valley, or a cliff (see Kawase [26]).

To account for the effects of the basin-induced surface waves inside sedimentary basins, Nakano [8] and Nakano et al. [27] proposed to use an empirical ratio called the whole-wave-to-S-wave ratio (WSR), where the spectral ratios of the whole duration with respect to the S-wave portion with relatively short duration (5 to 15 s depending on the source magnitude as used in GIT) are averaged over all the observed events at a site. They found that the WSR tends to be close to unity irrespective of frequency for a site on hard rock, whereas it can easily exceed 10 in the lower frequency range for a site inside a soft sedimentary basin. Even for such a site, WSR will converge to unity in a higher frequency range above a few Hz. Because the spatial variation of WSR at one specific frequency highly correlates with that of the basin depths, as seen in Nakano et al. [27], Nakano [8] proposed a scheme to interpolate WSRs to make it possible to calculate a scenario-type hazard map with much higher spatial density than those of the strong motion observation sites. This WSR correction is a simple, empirical way to account for the additional amplification due to basin-induced surface waves.

However, other than the contributions of basin-induced surface waves through the modeling of WSR on top of hSAF as proposed by Nakano [8], it is difficult to account for the physical cause of the discrepancy between observation and theory at every sites as seen in **Figure 4**. We have been attempting to fill the gap

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*S-Wave Site Amplification Factors from Observed Ground Motions in Japan: Validation…*

motion simulation should be close enough to the observed hSAF.

we would like to propose a different but simple approach here.

Here is a simple way of correction for the theoretical hSAF, *HSAFthe*:

between *HSAFmod* and the observed hSAF (*HSAFobs*), minimum:

*RES FMR AMR*

**5.1 Grid search scheme**

*f fmin*

constraint. Then the target function to be maximized, *GOF*, becomes.

**5. Frequency and amplitude modification ratio**

by inverting the velocity structures from observed horizontal-to-vertical spectral ratios (HVSRs) of earthquakes under the diffuse field assumption [28–30], which is quite successful to reproduce observed HVSRs (and consequently also hSAF). This approach can provide us an equivalent 1D structure that will reproduce the observed hSAF at the target site quite precisely, however, the method is valid only for a site with either earthquake or microtremor records. We need a different strategy to evaluate hSAF as precisely as possible at an arbitrary site without any records. Because the velocity structure in the UVM of NIED is obtained with the spatial density of the 250 m grid, we want to reflect the fundamental characteristics of the theoretical transfer function of that structure, yet the resultant hSAF for strong

To overcome the difficulty to obtain better velocity models through physical parametrization of the underground structure at the sites without observed records,

Suppose that a theoretical 1D hSAF from the UVM at a certain site deviates from the true hSAF by one of the aforementioned causes or their combination, the difference of the theoretical hSAF would be manifested in both the frequency axis and the amplitude axis. If we have stiffer or thinner layers in reality than the assumed profile in UVM, then the frequency characteristics would be all shifted towards the higher side. Or if we have a stronger velocity gradient within layers in reality, then the peak amplitudes would be all shifted towards the higher side. Thus, we have two different ways to adjust the theoretical hSAF to make it closer to the observed hSAF, one in the frequency axis and the other in the amplitude axis.

*HSAF f AMR HSAF f FMR mod* ( ) = ∗ *the* ( / ,) (2)

*fmax mod obs*

*HSAF HSAF*

<sup>−</sup> <sup>=</sup> ∑ (3)

where *FMR* is the frequency modification ratio and *AMR* is the amplitude modification ratio. *HSAFmod* is the resultant hSAF after both modifications as a function of frequency *f*. We need to determine *FMR* and *AMR* to make *RES*, the residual

( ) log10( ) log10( ) ,

*f* <sup>=</sup>

where *fmin* and *fmax* are the minimum and maximum frequencies of interest and we set them 0.12 Hz and 20 Hz, respectively. Here we use frequency *f* as a weight because the higher the frequency the denser the data in the linear space.

Because the calculation of Eq. (2) is quite easy, we use the grid search to obtain the best *FMR* and *AMR* combination. However, after several experiments, we found that the evaluation function in Eq. (2) seems too weak to determine *FMR* and *AMR* in a reasonable range because there is a trade-off between them. Therefore, we introduce the correlation function between *HSAFmod* and *HSAFobs* as an additional

*GOF RES RES FMR AMR COR FMR COR* = *min* ( ) ∗ ( ) *max* /, / (4)

*DOI: http://dx.doi.org/10.5772/intechopen.95478*

*S-Wave Site Amplification Factors from Observed Ground Motions in Japan: Validation… DOI: http://dx.doi.org/10.5772/intechopen.95478*

by inverting the velocity structures from observed horizontal-to-vertical spectral ratios (HVSRs) of earthquakes under the diffuse field assumption [28–30], which is quite successful to reproduce observed HVSRs (and consequently also hSAF). This approach can provide us an equivalent 1D structure that will reproduce the observed hSAF at the target site quite precisely, however, the method is valid only for a site with either earthquake or microtremor records. We need a different strategy to evaluate hSAF as precisely as possible at an arbitrary site without any records. Because the velocity structure in the UVM of NIED is obtained with the spatial density of the 250 m grid, we want to reflect the fundamental characteristics of the theoretical transfer function of that structure, yet the resultant hSAF for strong motion simulation should be close enough to the observed hSAF.
