**5. Frequency and amplitude modification ratio**

To overcome the difficulty to obtain better velocity models through physical parametrization of the underground structure at the sites without observed records, we would like to propose a different but simple approach here.

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

$$H\text{SAF}\_{\text{mod}}\left(f\right) = \text{AMR} \* H\text{SAF}\_{\text{the}}\left(f \mid F\text{MR}\right),\tag{2}$$

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

$$\text{RES}\left(F\text{MR, AMR}\right) = \sum\_{f=f\text{min}}^{f\text{max}} \frac{\left|\log\mathbf{10}\left(H\text{SAAF}\_{\text{mod}}\right) - \log\mathbf{10}\left(H\text{SAAF}\_{\text{obs}}\right)\right|}{f} \tag{3}$$

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.

#### **5.1 Grid search scheme**

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

$$\text{GOF} = \text{RES}\_{\text{min}} / \text{RES} \{ \text{FMR}, \text{AMR} \} \* \text{COR} \{ \text{FMR} \} / \text{COR}\_{\text{max}} \tag{4}$$

*Earthquakes - From Tectonics to Buildings*

hill, a valley, or a cliff (see Kawase [26]).

amplification due to basin-induced surface waves.

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

*are well reproduced, however, the theoretical hSAF tends to underestimate the observed hSAF.*

*Observed hSAF extracted by GIT and 1D theoretical hSAF from UVM after [18]. Fundamental characteristics* 

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

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

**126**

**Figure 4.**

where *RESmin* is the minimum residual in the searching range, *RES*(*FMR*, *AMR*) is the residual shown in Eq. (2) as a function of *FMR* and *AMR*, *COR*(*FMR*) is the correlation coefficient between *HSAFmod* and *HSAFobs*, which is a function of only *FMR*, not a function of *AMR*, and *CORmax* is the maximum correlation coefficient in the searching range. Thus 1.0 is the maximum of GOF.

We set the searching range for *FMR* depending on the original correlation without frequency modulation, which is *COR*(1.0), as follows:

$$\begin{aligned} \text{If } 0.6 \le \text{COR}(1.0), 0.80 \le \text{FMR} \le 1.25\\ \text{If } 0.4 \le \text{COR}(1.0) < 0.6, 0.67 \le \text{FMR} \le 1.50\\ \text{If } \text{COR}(1.0) < 0.4, 0.50 \le \text{FMR} \le 2.00 \end{aligned} \tag{5}$$

We use these searching ranges because if the correlation of the original model is sufficiently high, we should not modify its frequency characteristics so much. For *AMR* we set the searching range to be 0.333 ≤ AMR ≤ 3.00 irrespective of *COR*(1.0) because there are a few tens of sites with those amplitude differences as high as 3 times or as low as 1/3 and *AMR* does not alter *COR*(1.0). To efficiently search the best *FMR* and *AMR* with the precision of two digits, we employ the two-step grid search; first with every 0.1 increments, then with every 0.01 increments around the best *FMR* and *AMR* in the first step.
