We have first tried SHS processing of the three sets of GPS network data. The results (not shown) are somewhat more noisy and less clean than those obtained via the OSE-processing scheme (which we report below). That is not surprising because the performance of SHS relies more on the global distribution of the stations than does OSE (Ding and Chao 2015a), while GEONET and PBO are non-global regional networks.
The OSE scheme yields one single composite time series from multiple-station time series upon optimal weighting with respect to designated spatial function, in our case the scalar or vector surface spherical harmonics as the case may be, as described above. We apply OSE to the near-field network GEONET, and obtain the power spectra of the composite time series targeted for designated Ylm, for the horizontal and vertical components. Figure 4a gives the power spectra for l = 1 and m = − 1, 0, 1, for spheroidal modes in the U-component. Ideally, if the OSE operation is perfectly effective, we expect to see the (m = − 1, 0, 1) singlets of the nth radial overtones nS1 that are excited, with all others suppressed to the extent dependence upon the global-ness of the station distribution.
Figures 4, 5, 6 and 7 present selected examples of Fourier power spectra of the OSE stacks, for spheroidal vs. toroidal modes, for the low-degree l = 1, 2, 3 with respective order m’s, and for the vertical (U) vs. horizontal (N/E) components, from different GPS networks of GEONET, PBO or IGS. The most striking difference of these time-domain stacking results from the power spectral stacking (Figs. 2, 3) is the enhanced SNR for the normal modes and hence the heightened spectral peaks, which match remarkably well in general against the vertical solid/dashed lines that indicate the PREM eigenfrequencies of the spheroidal/toroidal fundamental modes.
Figure 4 targets the spheroidal modes from the GEONET network with 968/910 stations—3(a) for U-component for l = 1 (m = − 1, 0, 1); 3(b) U-component for l = 2 (m = − 2, − 1, 0, 1, 2); 3(c) N/E-component for l = 2 (m = − 2, − 1, 0, 1, 2); 3(d) N/E-component for l = 3 (m = − 3, − 2, − 1, 0, 1, 2, 3). One sees that essentially all the spheroidal fundamental modes between 1.5 and 5 MHz, i.e., 0S9–0S43, stand out prominently. Lower-frequency modes are less excited by this earthquake (unlike the 2004 Sumatra earthquakes, see Park et al. 2005), where only 0S4 and 0S5 can be seen relatively well. The overtone modes are barely excited to a detectable level at the GPS sensitivity. The only overtone that is identifiable consistently is the cluster 1S3/3S1 at ~ 0.94 MHz.
The results in Fig. 4 further demonstrate an inadequacy in the sensitivity for discriminating the modes by OSE (or any spatial stacking pending global distribution)—that is, the GEONET network being only regional and far from global in either latitude (pertaining to degree l) or longitude (pertaining to order m), all low-frequency modes stay near constant in phase in the limited region because of their long wavelengths. In such a case the OSE procedure is effective in enhancing the SNR of the spatially coherent signals but cannot effectively discriminate and suppress the non-target modes. That is evident in Fig. 4, where all the well-excited modes, namely the fundamental modes (especially spheroidal) whether or not targeted, would show up in the OSE stacks, although weighted differently dependent on the l and m.
The same things happen with respect to the OSE stacks for the toroidal modes. Figure 5 shows the results in the horizontal N/E-components. Here one sees clear toroidal fundamental modes 0T2–0T12 on the left portion of the spectra, as well as the series of spheroidal fundamental modes from 0S10 up to 0S42 for l = 2 in 4(a), and up to 0S31 for l = 3 in 4(b).
Figure 6 gives the OSE stacks for the western USA PBO network using 685/585 GPS stations, targeted for l = 3 (m = − 3, − 2, − 1, 0, 1, 2, 3) and (a) U-component of spheroidal modes; (b) N/E-component of spheroidal modes; (c) N/E-component of toroidal modes. Again, the results show clear detection of the fundamental modes, albeit not as strong as in the GEONET case. The latter is not surprising as stated above, because the PBO network is no longer near the anti-nodes of each of the excited normal modes. In spectrum 5(a) for the U-component, one sees spheroidal 0S8–0S15 (in frequency band 1.4–2.4 MHz), plus a few toroidal modes of 0T3, 0T5, 0T8, and 0T9. The appearance of toroidal modes in the U-component results from their coupling with nearby spheroidal modes (e.g., Masters et al. 1983). The large peak at ~ 0.4 MHz coincides with that of the overtone 2S1, but cannot be further identified in our analysis. In Fig. 6b, c for the N/E-components, most of the toroidal 0T4–0T22 and spheroidal 0S6–0S22 (0.7–3.3 MHz) show up in the spectrum.
Figure 7 gives some examples of the corresponding results for the global IGS network (see Fig. 1) using 146/143 stations for l = 1 (m = − 1, 0, 1): (a) U-component of spheroidal modes; (b) N/E-component of spheroidal modes; (c) N/E-component of toroidal modes. The results fall short of what are expected despite the relatively worldwide distribution of the IGS network stations. Significant spectral peaks are much fewer and weaker compared to those from the GEONET and PBO networks. Whereas a few peaks in the lowest-frequency end do not correspond to any PREM-modeled eigenfrequencies, only a handful peaks are identifiable against the PREM eigenfrequencies of the fundamental modes (e.g., 0T2–0T4 and 0S3). This presumably can be attributed to the sparsity and small number of available IGS stations and the fact that the suppression of non-target modes is relatively more effective given the global-ness of the network.