**5.2 Evaluated** *FMR* **and** *AMR*

**Figure 5** shows examples of the resultant *HSAFmod* in comparison with *HSAFobs* and *HSAFthe* at the same K-NET, KiK-net, or JMA Shindokei network sites shown in **Figures 1** and **4**. As we can see, *HSAFmod* determined by the grid search are quite close to *HSAFobs* in both frequency fluctuations and amplitudes.

**Figure 6** shows the resultant optimal values of *FMR* and *AMR* at all the 546 sites used. We can see a very weak correlation between them. As for the searching range of *AMR*, namely 1/3 to 3, looks sufficient. On the other hand, we see a significant concentration of sites near the searching range boundary, 1/2 or 2 for *FMR*. This means that we could obtain better residuals and correlations if we search the optimal *FMR* in the frequency range wider than those limits. However, when we increase the searching range for *FMR* too much, we will see some cases where the reverberated fluctuations by the 1D resonance within the sediments seem to be shifted to the next overtone.

#### **5.3 Correlation and residual improvement**

If no improvement to the matching with the observed hSAF is achieved, our correction method does not have any merit. Therefore, we need to check if we can see significant improvement or not.

**Figure 7** shows the distributions of the obtained correlation and the averaged residuals between *HSAFmod* and *HSAFobs*. We can see most of the site shows residuals less than 1.5 and correlations higher than 0.5. **Table 2** shows the percentage of the sites in different categories in terms of the goodness of fit to *HSAFobs*. When we compare the matching seen in **Figure 5** and the distribution of these data in **Figure 7**, we can see that the average residual is more important than the correlation in terms of the quality of the modified hSAF because the correlation can be deteriorated easily by small fluctuations at different frequencies.

**Figure 8** shows significant improvements in the correlations from the original ones to the modified ones. There is no data with decreased correlations, however,

**129**

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

there still remain 7 sites with the correlations less than zero, which was decreased from 84. **Figure 9** also shows correlation improvements but as a function of *FMR*. We can see a clear concentration of *FMR* near the boundaries at 0.8 and 1.25, the boundaries of the searching range if the original correlation is larger than 0.6 as shown in Eq. (4). We can see a smaller improvement in this *FMR* range for higher

*Modified hSAF (*HSAFmod*) by using* FMR *and* AMR *in comparison with observed hSAF (*HSAFobs*) extracted by GIT and 1D theoretical hSAF (*HSAFthe*) from the velocity structure taken from UVM after [18] at the same sites in Figures 1 and 4. The optimal values of* FMR *and* AMR *for each site are shown in the upper-right* 

Now we have more than 500 sites in the Kanto and Tokai regions where we have determined modification ratios for frequency and amplitude, *FMR* and *AMR*. There are various ways to utilize these ratios for the prediction of site amplifications with much denser spatial resolutions based on the UVM in the 250 m grid. One way is to establish relationships of these modification ratios with respect to a site proxy or proxies such as Vs30 or Z1.0 as mentioned in the introduction. As seen in a lot of previous studies for site effects based on such relationships, however, we need to accept large deviations from the average relationships from site to site because it is the nature of the site amplification. We also face the possibility of the inaccurate choice of a site proxy in UVM for an arbitrary site used for modification ratios. Therefore, we decide to use a direct spatial interpolation scheme as Nakano [8] proposed for WSR. In this scheme, we employ first GMT's "surface" function [31, 32] in which the curvature minimization scheme is used together with the

correlation sites, in comparison to those with lower correlations.

**5.4 Spatial interpolation**

**Figure 5.**

*corner.*

*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*

#### **Figure 5.**

*Earthquakes - From Tectonics to Buildings*

best *FMR* and *AMR* in the first step.

**5.2 Evaluated** *FMR* **and** *AMR*

shifted to the next overtone.

**5.3 Correlation and residual improvement**

by small fluctuations at different frequencies.

see significant improvement or not.

in the searching range. Thus 1.0 is the maximum of GOF.

without frequency modulation, which is *COR*(1.0), as follows:

close to *HSAFobs* in both frequency fluctuations and amplitudes.

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

We set the searching range for *FMR* depending on the original correlation

If 0.6 1.0 , 0.80 1.25 If 0.4 1.0 0.6, 0.67 1.50 If 1.0 0.4, 0.50 2.00

*COR FMR COR FMR COR FMR*

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

**Figure 5** shows examples of the resultant *HSAFmod* in comparison with *HSAFobs* and *HSAFthe* at the same K-NET, KiK-net, or JMA Shindokei network sites shown in **Figures 1** and **4**. As we can see, *HSAFmod* determined by the grid search are quite

**Figure 6** shows the resultant optimal values of *FMR* and *AMR* at all the 546 sites used. We can see a very weak correlation between them. As for the searching range of *AMR*, namely 1/3 to 3, looks sufficient. On the other hand, we see a significant concentration of sites near the searching range boundary, 1/2 or 2 for *FMR*. This means that we could obtain better residuals and correlations if we search the optimal *FMR* in the frequency range wider than those limits. However, when we increase the searching range for *FMR* too much, we will see some cases where the reverberated fluctuations by the 1D resonance within the sediments seem to be

If no improvement to the matching with the observed hSAF is achieved, our correction method does not have any merit. Therefore, we need to check if we can

**Figure 7** shows the distributions of the obtained correlation and the averaged residuals between *HSAFmod* and *HSAFobs*. We can see most of the site shows residuals less than 1.5 and correlations higher than 0.5. **Table 2** shows the percentage of the sites in different categories in terms of the goodness of fit to *HSAFobs*. When we compare the matching seen in **Figure 5** and the distribution of these data in **Figure 7**, we can see that the average residual is more important than the correlation in terms of the quality of the modified hSAF because the correlation can be deteriorated easily

**Figure 8** shows significant improvements in the correlations from the original ones to the modified ones. There is no data with decreased correlations, however,

≤ ≤ < ≤≤ < ≤≤

(5)

( ) ( ) ( )

**128**

*Modified hSAF (*HSAFmod*) by using* FMR *and* AMR *in comparison with observed hSAF (*HSAFobs*) extracted by GIT and 1D theoretical hSAF (*HSAFthe*) from the velocity structure taken from UVM after [18] at the same sites in Figures 1 and 4. The optimal values of* FMR *and* AMR *for each site are shown in the upper-right corner.*

there still remain 7 sites with the correlations less than zero, which was decreased from 84. **Figure 9** also shows correlation improvements but as a function of *FMR*. We can see a clear concentration of *FMR* near the boundaries at 0.8 and 1.25, the boundaries of the searching range if the original correlation is larger than 0.6 as shown in Eq. (4). We can see a smaller improvement in this *FMR* range for higher correlation sites, in comparison to those with lower correlations.

#### **5.4 Spatial interpolation**

Now we have more than 500 sites in the Kanto and Tokai regions where we have determined modification ratios for frequency and amplitude, *FMR* and *AMR*. There are various ways to utilize these ratios for the prediction of site amplifications with much denser spatial resolutions based on the UVM in the 250 m grid. One way is to establish relationships of these modification ratios with respect to a site proxy or proxies such as Vs30 or Z1.0 as mentioned in the introduction. As seen in a lot of previous studies for site effects based on such relationships, however, we need to accept large deviations from the average relationships from site to site because it is the nature of the site amplification. We also face the possibility of the inaccurate choice of a site proxy in UVM for an arbitrary site used for modification ratios.

Therefore, we decide to use a direct spatial interpolation scheme as Nakano [8] proposed for WSR. In this scheme, we employ first GMT's "surface" function [31, 32] in which the curvature minimization scheme is used together with the

#### **Figure 6.**

*Resultant values of* FMR *and* AMR *after the grid search for 546 sites in the Kanto and Tokai regions. Red triangles are those for the sites in Figure 5.*

#### **Figure 7.**

*Distributions of the correlation of* HSAFmod *versus the averaged residual (the averaged spectral ratio between*  HSAFmod *and* HSAFob*s) for the same 546 sites shown in Figure 6. Triangles are values at the sites shown in Figure 5.*

smoothing constraint of an elastic shell with the tension factor of 0.25. In this Step-1 interpolation, we use the 3 km equal-spaced grid. Then in Step-2, we use the 250 m grid to interpolate further by using the modified Shepard's method [33].

**131**

**Figure 8.**

*at the sites shown in Figure 5.*

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

*Residual and correlation improvement in terms of the number of sites in each range before and after the* 

**Residual Before After Correlation Before After** <1.25 5 43 <0.0 84 7 1.25–1.50 175 389 0.0–0.4 227 86 1.50–2.00 286 108 0.4–0.6 116 186 2.00–3.00 77 6 0.6–0.8 85 193 3.00< 3 0 0.8–1.0 34 74 Total 546 546 Total 546 546

**Figure 10** shows the comparison of interpolated values in Step-1 with those used as targets for (a) *FMR* and (b) *AMR*. Interpolation in *AMR* is better because its spatial variation is smoother than *FMR* as shown later. Although we see tens of sites in **Figure 10a** away from the 1:1 line, the average deviation from unity for *FMR i*s about 10% (1.1 or 0.9 times) and 91% of the interpolated values are within the range between 1/1.25 and 1.25 times of the original *FMR*. The number of outliers is misleading because we use the "blockmedian" command of GMT to refer to only the

*Improvement from the original correlations of* HSAFthe *to the modified ones of* HSAFmod*. Triangles are values* 

Similarly, **Figure 11** shows the comparison of interpolated values in Step-2 with those used as targets from Step-1 for (a) *FMR* and (b) *AMR*. In Step-2 the

median value if we have plural sites in the same 3 km grid.

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

**Table 2.**

*correction.*

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


**Table 2.**

*Earthquakes - From Tectonics to Buildings*

**130**

**Figure 7.**

**Figure 6.**

*triangles are those for the sites in Figure 5.*

*Figure 5.*

smoothing constraint of an elastic shell with the tension factor of 0.25. In this Step-1 interpolation, we use the 3 km equal-spaced grid. Then in Step-2, we use the 250 m grid to interpolate further by using the modified Shepard's method [33].

*Distributions of the correlation of* HSAFmod *versus the averaged residual (the averaged spectral ratio between*  HSAFmod *and* HSAFob*s) for the same 546 sites shown in Figure 6. Triangles are values at the sites shown in* 

*Resultant values of* FMR *and* AMR *after the grid search for 546 sites in the Kanto and Tokai regions. Red* 

*Residual and correlation improvement in terms of the number of sites in each range before and after the correction.*

#### **Figure 8.**

*Improvement from the original correlations of* HSAFthe *to the modified ones of* HSAFmod*. Triangles are values at the sites shown in Figure 5.*

**Figure 10** shows the comparison of interpolated values in Step-1 with those used as targets for (a) *FMR* and (b) *AMR*. Interpolation in *AMR* is better because its spatial variation is smoother than *FMR* as shown later. Although we see tens of sites in **Figure 10a** away from the 1:1 line, the average deviation from unity for *FMR i*s about 10% (1.1 or 0.9 times) and 91% of the interpolated values are within the range between 1/1.25 and 1.25 times of the original *FMR*. The number of outliers is misleading because we use the "blockmedian" command of GMT to refer to only the median value if we have plural sites in the same 3 km grid.

Similarly, **Figure 11** shows the comparison of interpolated values in Step-2 with those used as targets from Step-1 for (a) *FMR* and (b) *AMR*. In Step-2 the

#### **Figure 9.**

*Improvement from the original correlations of* HSAFthe *(before) to those of* HSAFmod *(after) as a function of* FMR*. Data after the correction are concentrated near the boundaries of the searching range (0.8 and 1.25) when the original correlation is more than 0.6 because of the setting used. Note that* FMR *in the vertical axis of this figure is common for both before and after the correction.*

#### **Figure 10.**

*Comparison of the original* FMR *and* AMR *used as targets of interpolations and those of interpolated values in Step-1 at strong motion observation sites. Red broken lines are linear regression lines whose inclinations and coefficients of determination R2 are listed inside. (a) FMR and (b) AMR.*

interpolation is performed from the 3 km grid in Step-1 to the 250 m grid. The linearity of interpolation in Step-2 is much higher than that in Step-1 in the case of *FMR*, whereas that of Step-2 is as high as that of Step-1 in the case of *AMR*.

We can see the spatial stability of the interpolation scheme as a gross picture shown in **Figure 12** for *FMR* and *AMR*. These are the results of Step-2 with a spatial resolution of 250 m. Apparently, *AMR* is much smoother than *FMR* in terms of spatial variability so that the interpolation for *AMR* should be much easier and precise than *FMR*. On average, the Kanto region needs smaller correction values in *FMR* than those in the Tokai region, although it needs relatively larger correction values in AMR inside the whole soft-sedimentary areas in the north of Tokyo Bay.

**133**

**5.5 Validation**

*resolution of 250 m.*

**Figure 12.**

**Figure 11.**

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

*Comparison of Step-1* FMR *and* AMR *used as targets of interpolations in Step-2 interpolation and those of interpolated values in Step-2 at strong motion observation sites. Red broken lines are linear regression lines* 

 *are listed inside. (a) FMR and (b) AMR.*

So far we show that a simple two-step scheme of interpolation works to calculate

**Figure 13** shows contour maps of *FMR* and *AMR* without four points used as examples in **Figure 5**. We can see smooth interpolation is achieved at these four points. **Figure 14** shows the correspondence of original and interpolated values of *FMR* and *AMR* for the cases with and without four points. In case of *FMR*, the

modification ratios for both frequency and amplitude, namely, *FMR* and *AMR* from 546 strong motion stations in the Kanto and Tokai regions in a grid as small as 250 m. Because the UVM in these regions has a spatial resolution of 250 m, we can directly use these interpolated modification ratios once we calculate 1D theoretical hSAF at any of these grid points. To validate the method, we take four sites shown in **Figure 5** out from the control points used for interpolation and let the program

*Interpolated contour maps of* FMR *(left) and* AMR *(right) after the Step-2 interpolation with the spatial* 

interpolate the modification ratios there and see how it works.

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

*whose inclinations and coefficients of determination R2*

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

#### **Figure 11.**

*Earthquakes - From Tectonics to Buildings*

*this figure is common for both before and after the correction.*

**132**

Tokyo Bay.

**Figure 10.**

**Figure 9.**

*coefficients of determination R2*

interpolation is performed from the 3 km grid in Step-1 to the 250 m grid. The linearity of interpolation in Step-2 is much higher than that in Step-1 in the case of

 *are listed inside. (a) FMR and (b) AMR.*

*Comparison of the original* FMR *and* AMR *used as targets of interpolations and those of interpolated values in Step-1 at strong motion observation sites. Red broken lines are linear regression lines whose inclinations and* 

*Improvement from the original correlations of* HSAFthe *(before) to those of* HSAFmod *(after) as a function of* FMR*. Data after the correction are concentrated near the boundaries of the searching range (0.8 and 1.25) when the original correlation is more than 0.6 because of the setting used. Note that* FMR *in the vertical axis of* 

We can see the spatial stability of the interpolation scheme as a gross picture shown in **Figure 12** for *FMR* and *AMR*. These are the results of Step-2 with a spatial resolution of 250 m. Apparently, *AMR* is much smoother than *FMR* in terms of spatial variability so that the interpolation for *AMR* should be much easier and precise than *FMR*. On average, the Kanto region needs smaller correction values in *FMR* than those in the Tokai region, although it needs relatively larger correction values in AMR inside the whole soft-sedimentary areas in the north of

*FMR*, whereas that of Step-2 is as high as that of Step-1 in the case of *AMR*.

*Comparison of Step-1* FMR *and* AMR *used as targets of interpolations in Step-2 interpolation and those of interpolated values in Step-2 at strong motion observation sites. Red broken lines are linear regression lines whose inclinations and coefficients of determination R2 are listed inside. (a) FMR and (b) AMR.*

#### **Figure 12.**

*Interpolated contour maps of* FMR *(left) and* AMR *(right) after the Step-2 interpolation with the spatial resolution of 250 m.*

#### **5.5 Validation**

So far we show that a simple two-step scheme of interpolation works to calculate modification ratios for both frequency and amplitude, namely, *FMR* and *AMR* from 546 strong motion stations in the Kanto and Tokai regions in a grid as small as 250 m. Because the UVM in these regions has a spatial resolution of 250 m, we can directly use these interpolated modification ratios once we calculate 1D theoretical hSAF at any of these grid points. To validate the method, we take four sites shown in **Figure 5** out from the control points used for interpolation and let the program interpolate the modification ratios there and see how it works.

**Figure 13** shows contour maps of *FMR* and *AMR* without four points used as examples in **Figure 5**. We can see smooth interpolation is achieved at these four points. **Figure 14** shows the correspondence of original and interpolated values of *FMR* and *AMR* for the cases with and without four points. In case of *FMR*, the

#### **Figure 13.**

*Interpolated contour maps of* FMR *(top) and* AMR *(bottom) after the Step-2 interpolation without four points shown by triangles. Inside the triangles interpolated values at these sites are shown by color-coded circles.*

original and interpolated values are close to the 1:1 line and the pure interpolation values at three sites out of four are close to the original ones. We should note that the worst site of the interpolated *FMR*, AIC001, and the JMA site \_\_\_E34 were very close to each other as shown in **Figure 13**. In case of *AMR*, at three sites the

**135**

**Figure 14.**

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

interpolation values without four sites were not as good as those with four sites. Still the deviation from the original values are within the range of 1.5 or 1/1.5 times. Finally, we compare the corrected hSAF at those four sites not used in the spatial interpolation as references but used as the targets of the interpolation with the observed hSAF in **Figure 15**. Although the corrections by the original *FMR* and *AMR* seen in **Figure 5** are much better than the corrections by these

*Comparison of the original* FMR *and* AMR *used as targets of interpolations and those of interpolated values in Step-2 with and without four example sites in Figure 5. Black crosses are original values at four sites and red circles are interpolated values without referring to those original values, that is to say, purely interpolated* 

*values. Except for one site for* FMR *(AIC001), our interpolation scheme works.*

*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*

#### **Figure 14.**

*Earthquakes - From Tectonics to Buildings*

**134**

**Figure 13.**

original and interpolated values are close to the 1:1 line and the pure interpolation values at three sites out of four are close to the original ones. We should note that the worst site of the interpolated *FMR*, AIC001, and the JMA site \_\_\_E34 were very close to each other as shown in **Figure 13**. In case of *AMR*, at three sites the

*Interpolated contour maps of* FMR *(top) and* AMR *(bottom) after the Step-2 interpolation without four points shown by triangles. Inside the triangles interpolated values at these sites are shown by color-coded circles.*

*Comparison of the original* FMR *and* AMR *used as targets of interpolations and those of interpolated values in Step-2 with and without four example sites in Figure 5. Black crosses are original values at four sites and red circles are interpolated values without referring to those original values, that is to say, purely interpolated values. Except for one site for* FMR *(AIC001), our interpolation scheme works.*

interpolation values without four sites were not as good as those with four sites. Still the deviation from the original values are within the range of 1.5 or 1/1.5 times.

Finally, we compare the corrected hSAF at those four sites not used in the spatial interpolation as references but used as the targets of the interpolation with the observed hSAF in **Figure 15**. Although the corrections by the original *FMR* and *AMR* seen in **Figure 5** are much better than the corrections by these

#### **Figure 15.**

*Modified hSAF (*HSAFmod*) by using* FMR *and* AMR *after interpolation (without using these four site) in comparison with observed hSAF (*HSAFobs*) extracted by GIT and 1D theoretical hSAF (*HSAFthe*) from the velocity structure taken from UVM at the same sites in Figures 1, 4 and 5. The values of* FMR *and* AMR *by interpolation for each site are shown in the upper-right corner.*

interpolated *FMR* and *AMR* in this figure, the interpolated corrections still make theoretical hSAF closer to the observed hSAF.
