We employed the two-station method of Isse et al. (2006) and Suetsugu et al. (2009) to measure dispersion curves of fundamental-mode Rayleigh waves. When two stations are located on approximately the same great circle from an earthquake, the phase velocity dispersion between stations can be determined by computing the phase differences of surface waves (Fig. 2). This method allows us to ignore the effects of phase shifts due to source excitation and lateral heterogeneities far outside the inter-station path. If the source location is far enough from the stations, the wave front of a surface wave can be treated as a plane wave. Under these conditions, the phase differences of surface waves between two stations are caused by the differential distance (AB′) between a farther station (A) and a point (B′) projected from the nearer station (B) onto the great circle path (AE) from the source to the far station (Fig. 2), so that the measured phase velocities are averages over the differential distance (AB′). We selected station pairs whose azimuthal differences from the source (α) were less than 5°, and this met the condition that the difference between the great circle distance from the event to the far station and the distance to the far station via the near station was less than 25 km. To remove phase velocity measurements near the nodal directions of the surface wave radiation, we calculated radiation patterns at the source locations of fundamental mode of Rayleigh waves using the Global CMT. We used only data with a normalized radiated amplitude of >0.4. We then measured the phase velocity dispersion curves of fundamental Rayleigh waves for periods between 30 and 140 s, whose RMS errors (Aki and Richards 2002) were less than 0.02 km/s.
A total of 1127–1934 surface wave paths were collected in this period range (Fig. 3a). The ray distributions of the obtained phase velocities are shown in Fig. 3b. We inverted the measured phase velocities between station pairs for two-dimensional phase velocity maps using a method developed by Montagner (1986), in which a smoothness constraint can be applied by introducing a covariance function.
In the present study, the covariance function (C
p
) is defined as
$$C_{p} \left( {M_{1} ,M_{2} } \right) = \sigma \left( {M_{1} } \right)\sigma \left( {M_{2} } \right)\exp \left[ {\frac{\cos \varDelta - 1}{{L_{{M_{1} }} L_{{M_{2} }} }}} \right]$$
where Δ is the distance between points M
1 and M
2 on the Earth’s surface. The a priori parameter error σ gives a constraint on the strength of the velocity perturbation. In constructing phase velocity maps with varying σ values, we chose 0.10 km/s, which provides a best fit to the data. As patterns of obtained velocity maps with varying σ are similar, the choice of σ has little effect on our results except for the strength.
The correlation lengths \(L_{{M_{1} }} ,L_{{M_{2} }}\) control the smoothness of the model. Two correlation lengths (100 and 200 km) were examined with synthetic data to determine an optimal value of correlation length. We then inverted the dispersion curves for the shear wave velocity model at each grid, using the linearized relationship between the period dependence of surface wave phase velocity and the depth variation of shear wave velocity (e.g., Takeuchi and Saito 1972), as follows:
$$\begin{aligned} \frac{{\delta c(\omega )}}{c} & = \int_{0}^{R} {\left\{ {K_{\rho } (\omega ,z)\frac{{\delta \rho (z)}}{\rho } + K_{\alpha } (\omega ,z)\frac{{\delta \alpha (z)}}{\alpha }} \right.} \\ & \quad \left. { + K_{\beta } (\omega ,z)\frac{{\delta \beta (z)}}{\beta }} \right\}{\text{d}}z, \\ \end{aligned}$$
where δc is the perturbation of the phase velocity; δρ, δα, and δβ are the density, P-wave velocity, and shear wave velocity, respectively; R is the radius of the Earth; and K
ρ
, K
α
, and K
β
are sensitivity kernels, which represent the partial derivatives of phase velocity with respect to each model parameter. We fixed the density and P-wave velocity structure at the reference model’s values and solved only for shear wave velocity, as the effects of density and P-wave velocity on Rayleigh wave phase velocity perturbations are not significant (Nataf et al. 1986). The iterative least squares inversion technique proposed by Tarantola and Valette (1982) was used for the inversion; this nonlinear inversion procedure has been used in many previous surface wave studies (e.g., Nishimura and Forsyth 1989). Our reference one-dimensional model was modified from PREM (Dziewonski and Anderson 1981) by smoothing the 220-km discontinuity. We adopted the CRUST2.0 model (Bassin et al. 2000) for the crust. We chose an a priori parameter error of 0.10 km/s and an a priori data error of 0.05 km/s. Changing the a priori data error does not influence shear wave velocity models significantly. The vertical correlation length was 5 km at depths shallower than 30 km, and 20 km at greater depths. In these calculations, we corrected an anelastic effect caused by the attenuation of seismic waves by using PREM, so that the reference frequency of the obtained model was 1 Hz.
Resolution test
To assess the lateral resolution of tomographic models and select appropriate horizontal correlation lengths, we performed ray-theoretical checkerboard resolution tests. We calculated the synthetic data from input checkerboard models with 8 % anomalies at a period of 80 s, with a cell size that varied from 3° to 8°. We added random errors with amplitudes up to 0.02 km/s, a value comparable to measured RMS errors, to the synthetic data. We then inverted the synthetic data for a two-dimensional phase velocity map using correlation lengths of 100 and 200 km. Figure 4 shows a recovery of the input checkerboard pattern of 3° and 5°. The 5° checkerboard pattern is well recovered in the whole studied region with the correlation length of 200 km (Fig. 4c), except for the southwest region, where ray paths are sparse. Using a correlation length of 100 km, the input pattern was well recovered in the vicinity of the Society hotspot (latitudes 12°–25°S and longitudes 141°–153°W; herein called the “Society region”) and in the vicinity of the Samoa hotspot, where seismic stations were densely distributed (Fig. 4b). On the other hand, the retrieved patterns were distorted outside the Society region (herein called “the outer region”). This suggests that a correlation length of 200 km is appropriate in the outer region, and the best possible resolution in the outer region is ~5°.
Next, we investigated recovery of the checkerboard patterns with a cell size of 3° (Fig. 4e, f). Using a correlation length of 200 km, the recovered pattern was smeared throughout the whole studied region (Fig. 4f). A correlation length of 100 km yielded good recovery in the Society region, but the pattern was only poorly recovered in the outer region (Fig. 4e).
Because our tests suggest that lateral resolution in the studied region is not uniform, a single choice of a correlation length may be inappropriate. Therefore, to optimize resolution in both the Society and outer regions, we use a correlation length of 100 km in the Society region and 200 km in the outer region. A third checkerboard test, using cell sizes of 3° in the Society region and 5° in the outer region (Fig. 5a), achieved satisfactory recovery in both regions (Fig. 5b). In the present study, we inverted for phase velocity maps using a correlation length of 100 km in the Society region and 200 km in the outer region.
The results of these checkerboard tests suggest that lateral resolution in the Society region is about 300 km and that in the outer region is 500 km, which is a higher resolution than the previous studies achieved. The amplitude of the recovered patterns is also better in the present study. The dense coverage of the TIARES network in the Society region is likely to contribute to the improvement of the horizontal resolution.
To assess the vertical resolution of the model, we performed spike tests (Fig. 6a–d). We created four synthetic models (dashed lines in Fig. 6) with 5 % fast anomalies in a narrow depth range of 60–180 km. We calculated synthetic phase velocities of Rayleigh waves from these models and inverted for shear wave profiles (red solid lines in Fig. 6). Although the shapes of the recovered spikes are vertically smeared, due to the long wavelength of the surface waves, the input anomaly is well recovered for the target depths of 60 and 100 km. At depths of 140 and 180 km, the shape of the recovered spike is largely smeared out, so the vertical resolution at these depths is worse than that at shallower depths.
To assess the sensitivity to the initial model, we created a synthetic model using a modified PREM with a seafloor depth of 4.2 km, crustal thickness of 6.6 km, and ±3 % uniform velocity perturbations. The results suggest that shear wave velocities at depths shallower than 50 km are not well recovered if the initial model is substantially different from the synthetic model (Fig. 6e, f). Rayleigh waves in the range of periods analyzed are less sensitive to such depths, so final models resemble the initial model at shallow depths. Small misfits to the synthetic models, less than 0.03 km/s, are also observed at depths greater than 50 km, which compensate the misfits shallower than 50 km. In the present study, we focus on shear wave models at depths greater than 50 km.
