The propagation paths of the HFLPs were investigated using a source-scanning algorithm (SSA) (Kao and Shan 2004). The conventional SSA evaluates the location of an energy radiation source based on spatiotemporal changes in brightness values computed from the amplitude records of seismic waves. Using this SSA feature, we searched for passing points representing the distribution of scatterers explaining the observed peak delays of the HFLPs.

Following Hasemi and Horiuchi (2010), we assumed that each HFLP is a single scattered wave. The brightness, \({b}_{ijk}\), of an earthquake–station pair is defined as

$${b}_{ijk}={E}_{ij}\left({\tau }_{ijk}\right),$$

(1)

where \({E}_{ij}\) is the mean-square (MS) envelope amplitude of the \(i\)th earthquake observed at the \(j\)th station, and \({\tau }_{ijk}\) is the delay time of the scattered wave passed at the \(k\)th grid point from direct S-wave arrival. The delay time is given by \({\tau }_{ijk}={\tau }_{\mathrm{L}}-{\tau }_{\mathrm{S}}\), where \({\tau }_{\mathrm{S}}\) and \({\tau }_{\mathrm{L}}\) are the travel times of the direct S wave and the scattered wave, respectively. The brightness, \({B}_{k}\), at the \(k\)th grid point is calculated from the values of \(b\) all of the available earthquake–station pairs as follows:

$${B}_{k}=\frac{1}{N}\sum_{i=1}\sum_{j=1}\overline{{b}_{ijk}},$$

(2)

where \(\overline{{b}_{ijk}}\) is the brightness value normalized by the maximum \(b\) for each earthquake–station pair, and \(N\) is the number of available pairs. Normalizing the \(b\) value reduces the bias due to source and site-dependent effects in the SSA analysis (e.g., Kao and Shang 2004). The grid on which \({B}_{k}\) is maximized represents the common passing point that can explain the peak-amplitude delays of the HFLPs.

Applying the SSA technique, Kosuga (2014) found the source location of the reflected waves observed for local inland earthquakes in NE Japan. In that study, the background coda-wave amplitude was subtracted from the observed records by the synthesized theoretical amplitude based on the single scattering model (Sato 1977; Sato et al. 2012). This procedure efficiently highlights the amplitude of the targeted later phases, but requires spatial homogeneity of the seismic wave attenuation. Therefore, it is inapplicable to the NE Japan subduction zone, in which attenuation properties are spatially variable (e.g., Tsumura et al. 2000; Takahashi 2012; Nakajima et al. 2013). Rather than Kosuga’s approach, we considered the spatial changes of the velocity and attenuation structures in the brightness evaluations, as detailed below.

The present study considered the effects of both geometrical spreading and path attenuation of the scattered waves. In the single scattering model (Sato 1977; Sato et al. 2012), the background coda-wave energies are obtained by summing the wave energies of the singly scattered waves from all scatterers having a common delay time. The scatterers distributed on one isochrone of delay time form one scattering shell. As the scattering shell expands in time, the number of scatterers on the corresponding shell increases and the amplitude of the scattered wave from each scatterer is reduced by geometrical spreading. Thus, the contributions of the individual scatterers on the shell to the background coda-wave amplitude should decrease over time. To express the expansion effect of the scattering shell relative to that of a direct S wave, we define the following weight, \({w}_{\mathrm{T}}\):

$${w}_{\mathrm{T}}=\frac{1}{{\tau }_{\mathrm{L}}^{2}}/\frac{1}{{\tau }_{\mathrm{S}}^{2}}=\frac{{\tau }_{\mathrm{S}}^{2}}{{\tau }_{\mathrm{L}}^{2}}.$$

(3)

We also define the path attenuation effect, \({w}_{\mathrm{Q}}\), on each scattered wave relative to that of the direct S wave:

$${w}_{\mathrm{Q}}=\mathrm{exp}\left[-2\pi f\left({t}_{\mathrm{L}}^{*}-{t}_{\mathrm{S}}^{*}\right)\right],$$

(4)

where \({t}_{\mathrm{S}}^{*}\) and \({t}_{\mathrm{L}}^{*}\) denotes the path attenuations of the direct S wave and scattered wave, respectively. The path attenuation, \({t}^{*}\), is obtained by integrating the travel time, \(\tau\), and the quality factor, \(Q\), of the attenuation along a given ray path:

$${t}^{*}={\int }_{\mathrm{path}}\frac{1}{V}\frac{1}{Q}\mathrm{d}s,$$

(5)

where \(V\) is the seismic velocity. Multiplying \({w}_{\mathrm{T}}\) and \({w}_{\mathrm{Q}}\) by the MS amplitude, the brightness of an earthquake–station pair given by Eq. (1) is modified to

$${b}_{ijk}^{w}={w}_{\mathrm{T}}{w}_{\mathrm{Q}}{E}_{ij}\left({\tau }_{ijk}\right).$$

(6)

Using Eq. (2) together with Eq. (6) in the SSA, we estimated the locations of passing points of HFLPs in the attenuating medium. The inclusion of \({w}_{\mathrm{Q}}\) often yielded unrealistic estimates of \({B}_{k}\), because the attenuation structure was oversimplified as described below. This ill-posed behavior was improved by limiting \({w}_{\mathrm{Q}}\) to the 0.1–10.0 range.

Characteristic structures, such as the subducting Pacific slab and upwelling mantle flow, are commonly known to be distributed along NE Japan (e.g., Nakajima et al. 2001, 2013; Takahashi 2012). Therefore, the present SSA analysis was conducted on two-dimensional vertical cross-sections across the NE Japan arc, labeled as cross-sections A–E in Fig. 1. The model space was divided into four layers by three structural boundaries: the Conrad and Moho discontinuities (Katsumata 2010), and the upper boundary of the Pacific slab (Nakajima et al. 2009). The S-wave velocity in each layer was taken from recent tomographic models (e.g., Nakajima et al. 2001; Matsubara et al. 2017). We also incorporated the low-velocity subducting oceanic crust in the uppermost part of the Pacific slab (e.g., Tsuji et al. 2008; Shiina et al. 2013). In NE Japan, a high S-wave attenuation (\({Q}_{\mathrm{S}}\)) zone in the mantle wedge has been constrained to the depth of about 60 km (e.g., Takahashi et al. 2009; Takahashi 2012). This high-attenuation zone has also been detected as a P-wave attenuation (\({Q}_{\mathrm{P}}\)), and is continuously distributed into a deeper position of the backarc mantle wedge (e.g., Tsumura et al. 2000; Nakajima et al. 2013). Based on the results of P waves and assuming that \({Q}_{\mathrm{P}}/{Q}_{\mathrm{S}}=2\) (e.g., Shito and Shibutani 2003; Nakajima et al. 2013), we adopted the high-attenuation mantle wedge extending below the backarc region. The assumed attenuation structure reproduces the path-averaged properties of direct S waves, as confirmed in the Discussion Section. For example, the velocity and attenuation structures over cross-section D are shown in Fig. 4.

The travel times and path attenuations of the direct S and scattered waves through the assumed velocity and attenuation structures were computed using the ray-tracing method of Zhao et al. (1992). Grid points were set at 2.5-km intervals in the horizontal and vertical directions.

The brightness values in the frequency range of 8–16 Hz were determined following Hasemi and Horiuchi (2010), who reported delayed high-frequency waves of forearc intraslab earthquakes in NE Japan. From the two horizontal components of the seismograms, the MS amplitudes were synthesized with a 1-s moving average. The noise amplitude was calculated as the 5-s average of the MS amplitudes before the P-wave arrival. The MS envelopes were visually checked, and then accepted if their amplitude at 2.5 times the lapse time from the direct S-wave travel time was twice the noise amplitude.