In the GIT-based spectral decomposition algorithm, \(Path\left(f,{R}_{n}\right)\) obtained by solving the matrix in Fig. 5 was first calibrated to obtain rational values. This was because the path effect was not affected by the source and site effects according to \({R}_{H}\) (Oth et al., 2010; Bindi and Kotha, 2020). The calibration factor was calculated to scale \(Path\left(f,{R}_{n}\right)\) at the nearest station from a given earthquake source to be one. Since the \({R}_{n}\) values ranged from 5 km to 200 km, 5 km was used as the reference distance, and a calibration factor was determined, which made \(Path\left(f, 5\text{k}\text{m}\right)\) equal to one. The calibration factor was applied to all \(Path\left(f,{R}_{n}\right)\) values for the same earthquake. The calibrated \(Path\left(f,{R}_{n}\right)\) values were smoothed according to Castro et al. (1990). Figure 6 shows \(\text{ln}Path\left(f,{R}_{n}\right)\) before and after the smoothing process.
Once \(Path\left(f,{R}_{n}\right)\) was determined, Eq. 3 could be expressed as Eq. 6 after removing \(Path\left(f,{R}_{n}\right)\) from Eq. 3. The GIT-based spectral decomposition algorithm could also be used to solve Eq. 6.
$$\text{l}\text{n}{FAS\left({R}_{H},f\right)}_{ij,w/o path}=\text{l}\text{n}{FAS\left({\left({R}_{H}\right)}_{ij},f\right)}_{ij}-{a}_{n}\text{l}\text{n}Path\left(f,{R}_{n}\right)-{a}_{n+1}\text{l}\text{n}Path\left(f,{R}_{n+1}\right)=\sum _{k=1}^{{N}_{\text{e}\text{v}\text{e}\text{n}\text{t}}}{\delta }_{ik}\text{l}\text{n}{Source}_{k}\left(f\right)+\sum _{l=1}^{{N}_{\text{s}\text{t}\text{a}\text{t}\text{i}\text{o}\text{n}}}{\delta }_{jl}\text{l}\text{n}{Site}_{l}\left(f\right)$$
6
where \({FAS\left({R}_{H},f\right)}_{ij,w/o path}\) is the \(FAS\) for earthquake \(i\) and seismic station \(j\) after removing \(Path\left(f,{R}_{n}\right)\). Figure 7 shows the matrix formulation for Eq. 6.
The \(Source\left(f\right)\) and \(Site\left(f\right)\) values can be estimated at the same time by solving the matrix in Fig. 7. In this case, accurate values of these two parameters may not be obtained because they are mutually affected by the matrix formulation.
To solve this problem, \(Site\left(f\right)\) was calibrated to have a rational value. First, reference stations with insignificant site effects were collected. The mean value of \(Site\left(f\right)\) for the group of reference stations was set to 0, and the \(Site\left(f\right)\) values of other individual stations were calibrated using factors obtained for the reference stations (Ameri et al., 2011; Bindi and Kotha, 2020).
However, it was generally not an easy task to determine a reference station solely based on site information (Ren et al., 2013). Amplification was often observed even at very hard rock sites, particularly at their resonant frequency (Steidl et al., 1996; Ren et al., 2013). In this study, the procedure proposed by Bindi and Kotha (2020) for selecting reference stations was adopted: (1) The average of the \(Site\left(f\right)\) values of all stations at each frequency (0.2–30 Hz) is calibrated to be 1 and the cumulative distribution function was calculated for the normalized \(Site\left(f\right)\) values, (2) The reference stations were determined when p50 < 0.9, p95 < 1, p05 > 0.1, and |p95-p05|<0.4 where p95, p50, and p05 are the 95th, 50th and 5th percentile values of the normalized \(Site\left(f\right)\). Table 1 lists the 61 reference stations. Figure 8 shows the normalized \(Site\left(f\right)\) values of three example stations, of which station BGDB was classified as a reference station because it met the criteria. The average \(Site\left(f\right)\) value for the 61 reference stations was calibrated to be 1 for each frequency, by which the calibration factor value was estimated at each frequency. The factor was applied to the \(Site\left(f\right)\) values of all other stations.
Table 1
Reference stations in the Korean peninsula selected to scale\(Site\left(f\right)\)
No. | Code | Lon [\(^\circ \text{E}\)] | Lat. [\(^\circ \text{N}\)] | No. | Code | Lon [\(^\circ \text{E}\)] | Lat. [\(^\circ \text{N}\)] |
1 | BGDB | 125.9469 | 34.7726 | 32 | JMJ | 128.7561 | 37.8816 |
2 | BON | 127.7981 | 36.5482 | 33 | JMJ2 | 128.7561 | 37.8816 |
3 | BUS | 129.1125 | 35.2487 | 34 | JNUA | 127.2005 | 36.6908 |
4 | BUYB | 126.9206 | 36.2726 | 35 | KH2B | 127.2758 | 34.6186 |
5 | CEA | 127.2575 | 36.8231 | 36 | KKDA | 127.1223 | 34.4557 |
6 | CEA2 | 127.2574 | 36.823 | 37 | KOHB | 127.2758 | 34.6185 |
7 | CHDA | 127.2496 | 34.2368 | 38 | KOJ2 | 127.1447 | 36.4708 |
8 | CHJ | 127.9748 | 36.873 | 39 | KWJ2 | 126.9911 | 35.1599 |
9 | CHJ2 | 127.9748 | 36.873 | 40 | MGY | 128.0608 | 36.6552 |
10 | CHJ3 | 127.9748 | 36.873 | 41 | MGY2 | 128.0608 | 36.6538 |
11 | CHRB | 128.4779 | 35.5342 | 42 | MND | 126.4242 | 35.8043 |
12 | CHYB | 128.9145 | 36.944 | 43 | MNDB | 127.1609 | 34.9665 |
13 | CWO | 127.5205 | 38.0835 | 44 | MOGA | 127.5825 | 37.6962 |
14 | CWO2 | 127.5205 | 38.0834 | 45 | NAJA | 126.8265 | 35.026 |
15 | ECDB | 125.9797 | 36.1184 | 46 | OYDB | 126.0757 | 36.2294 |
16 | GGTA | 126.8991 | 37.7686 | 47 | PORA | 126.5575 | 36.3278 |
17 | GICA | 128.1016 | 36.0813 | 48 | SCH | 127.2406 | 35.065 |
18 | GLSA | 126.545 | 36.5256 | 49 | SECA | 126.7518 | 36.1421 |
19 | GOCB | 126.5982 | 35.3485 | 50 | SEHB | 128.2525 | 38.2686 |
20 | GODA | 126.2916 | 37.7895 | 51 | SEO3 | 126.9171 | 37.4939 |
21 | GSGA | 126.8444 | 37.5516 | 52 | SGNA | 127.1831 | 37.4462 |
22 | HAWA | 126.3283 | 34.671 | 53 | SH2B | 128.2525 | 38.2686 |
23 | HESA | 127.9564 | 37.5407 | 54 | SHHB | 126.7039 | 37.3488 |
24 | HWCB | 127.6707 | 38.2215 | 55 | SMKB | 126.5561 | 35.6891 |
25 | HWDB | 127.3404 | 37.6322 | 56 | SNNA | 128.7111 | 36.0502 |
26 | ICN | 127.4167 | 37.2907 | 57 | UJBA | 127.1064 | 37.7548 |
27 | ICN2 | 127.4167 | 37.2908 | 58 | YALB | 128.1934 | 35.7279 |
28 | IJDB | 126.0651 | 35.1028 | 59 | YC2B | 126.9258 | 38.0399 |
29 | INC | 126.6239 | 37.4776 | 60 | YEGA | 126.4777 | 35.2838 |
30 | JEO2 | 127.2928 | 35.9379 | 61 | YNCB | 126.9258 | 38.0398 |
31 | JLSA | 127.648 | 35.3575 | - | - | - | - |
Figure 9 shows \(Site\left(f\right)\) values estimated for 8 of the 367 seismic stations used in this study. In this figure, the mean value of the 61 reference stations was also plotted. As shown in Fig. 9a, the \(Site\left(f\right)\) values of the reference stations were close to one, whereas for those of stations other than the reference station, \(Site\left(f\right)\) varied greatly according to frequency and also differed greatly from 1.
Once \(Site\left(f\right)\) was obtained, the source effect could be determined using Eq. 7, which is the equation after removing \(Site\left(f\right)\) from Eq. 6.
$$\text{l}\text{n}{FAS\left({R}_{H},f\right)}_{ij,w/opath and site}=\text{l}\text{n}{FAS\left({R}_{H},f\right)}_{ij,w/opath}-\sum _{l=1}^{{N}_{\text{s}\text{t}\text{a}\text{t}\text{i}\text{o}\text{n}}}{\delta }_{jl}\text{l}\text{n}{Site\left(f\right)}_{l} =\sum _{k=1}^{{N}_{\text{e}\text{v}\text{e}\text{n}\text{t}}}{\delta }_{ik}\text{l}\text{n}{Source}_{k}\left(f\right)$$
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where \({Source}_{k}\left(f\right)\) is the source effect at frequency \(f\) for earthquake event \(k\). In this study, \({Source}_{k}\left(f\right)\) values were determined for 771 earthquake events that occurred in the Korean peninsula between 2016 and 2017, including the 2016 Gyeongju earthquake foreshock (\({\text{M}}_{\text{L}}=5.1\)), the 2016 Gyeongju earthquake main shock (\({\text{M}}_{\text{L}}=5.8)\), and the 2017 Pohang earthquake main shock (\({\text{M}}_{\text{L}}=5.4\)). Figure 10 shows \({Source}_{k}\left(f\right)\) estimated from this study. To minimize the effects of noise contained at low frequencies, \({Source}_{k}\left(f\right)\) started from a specific frequency was determined for each earthquake event (Yenier and Atkinson, 2015). In this study, an allowable minimum frequency (\({f}_{lb}\)) that was not affected by noise was calculated according to Yenier and Atkinson (2015) [Eq. 8].
$$\text{l}\text{o}\text{g}{f}_{lb}=\text{m}\text{a}\text{x}\left(-1,\frac{3-M}{3}\right)$$
8