Evaluation of accuracy of synthetic waveforms for subduction-zone earthquakes by using a land–ocean unified 3D structure model

Abstract

Seismic wave propagation from shallow subduction-zone earthquakes can be strongly affected by 3D heterogeneous structures, such as oceanic water and sedimentary layers with irregular thicknesses. Synthetic waveforms must incorporate these effects so that they reproduce the characteristics of the observed waveforms properly. In this paper, we evaluate the accuracy of synthetic waveforms for small earthquakes in the source area of the 2011 Tohoku–Oki earthquake (M_JMA 9.0) at the Japan Trench. We compute the synthetic waveforms on the basis of a land–ocean unified 3D structure model using our heterogeneity, oceanic layer, and topography finite-difference method. In estimating the source parameters, we apply the first-motion augmented moment tensor (FAMT) method that we have recently proposed to minimize biases due to inappropriate source parameters. We find that, among several estimates, only the FAMT solutions are located very near the plate interface, which demonstrates the importance of using a 3D model for ensuring the self-consistency of the structure model, source position, and source mechanisms. Using several different filter passbands, we find that the full waveforms with periods longer than about 10 s can be reproduced well, while the degree of waveform fitting becomes worse for periods shorter than about 10 s. At periods around 4 s, the initial body waveforms can be modeled, but the later large-amplitude surface waves are difficult to reproduce correctly. The degree of waveform fitting depends on the source location, with better fittings for deep sources near land. We further examine the 3D sensitivity kernels: for the period of 12.8 s, the kernel shows a symmetric pattern with respect to the straight path between the source and the station, while for the period of 6.1 s, a curved pattern is obtained. Also, the range of the sensitive area becomes shallower for the latter case. Such a 3D spatial pattern cannot be predicted by 1D Earth models and indicates the strong effects of 3D heterogeneity on short-period (${ \lesssim }\;10 {\text{s}}$) waveforms. Thus, it would be necessary to consider such 3D effects when improving the structure and source models.

Introduction

Seismic wave propagation from shallow subduction-zone earthquakes can be strongly affected by three-dimensionally heterogeneous structures, such as oceanic water and sedimentary layers with irregular thicknesses (e.g., Okamoto 2002; Nakamura et al. 2012, 2015). In waveform analyses of earthquake sources and structural heterogeneities, these effects due to three-dimensional (3D) heterogeneities must be properly incorporated in the synthetics, or otherwise the analyses could result in erroneous solutions (e.g., Okamoto and Takenaka 2009). Thus, we need to evaluate the accuracy of synthetic waveforms using a realistic 3D structure. Such studies require accurate source parameters and source locations. However, estimating these parameters for oceanic areas is often a challenge because of the presence of the above-mentioned 3D heterogeneities and the one-sided distribution of seismic stations with respect to a trench (e.g., Okamoto 2002; Takemura et al. 2016). It is not satisfactory to address the problem by using only long-period waveforms, because surface waves are dominant in the long-period records, and therefore, the body waves, which provide information such as P-wave travel times to deal with the biases, become insignificant. To deal with this problem, we recently developed a method called first-motion augmented moment tensor (FAMT) analysis (Okamoto et al. 2017). FAMT analysis uses the initial part of the short-period waveforms in addition to the long-period full waveforms in order to augment the information based on the body wave first motions. The detailed source time function can also be inferred by using short-period waveforms (4–40 s).

In this paper, we investigate the accuracy of synthetic waveforms for small suboceanic earthquakes in the Japan Trench area, in connection with the 2011 Tohoku–Oki earthquake (M_JMA 9.0: Fig. 1a), as the effect of 3D heterogeneities on the waveforms would be different depending on the source location on the vast fault plane of the 2011 Tohoku–Oki earthquake.

Fig. 1

a The study area: northeastern Japan and Japan Trench. The GCMT solutions in the oceanic area from 01/01/1996 to 30/09/2017 $(M_{\text{W}} \ge 5.8 \;{\text{and}}\; {\text{depth}}\; \le 50 \;{\text{km}})$ are drawn. The GCMT solutions of the 01/11/2003 event (EV2003) and of the 25/12/2007 event (EV2007) are indicated by yellow arrows. The station locations are denoted by red triangles. The directions of the horizontal axes (X and Y) of the finite-difference grid are indicated by white arrows: the X axis is ${\text{N}}20^\circ {\text{E}}$, and the Y axis is ${\text{N}}110^\circ {\text{E}}.$ b Example of a vertical cross section of the land–ocean unified 3D structure model: the cross section along the line A–B shown in a is displayed. Blue color denotes the ocean water layer. The epicenters of EV2003, EV2007, and the 2011 Tohoku–Oki earthquake (M_JMA 9.0) determined by JMA are projected (green triangles)

Data

We selected two events whose hypocentral parameters determined by the Japan Meteorological Agency (JMA) are: 01/11/2003, 22:10:08.33 (JST), 37.7742°N, 143.2713°N, depth of 46 km, $M_{\text{JMA}}$ 6.2 ($M_{\text{W}} 5.8$), and 25/12/2007 23:04:32.85 (JST), 38.4833°E, 142.1537°N, depth of 40.29 km, $M_{JMA}$ 5.6 ($M_{W} 6.1$), respectively (Fig. 1a; the $M_{W}$ values were taken from the Global Centroid Moment Tensor (GCMT) catalog). Both events had inter-plate thrust mechanisms and occurred in the source region of the 2011 Tohoku–Oki earthquake. The waveforms from the 01/11/2003 event (referred to as EV2003 hereafter) and the 25/12/2007 event (referred to as EV2007), respectively, represent examples of “basic” waveforms from shallow and deep parts of the fault responsible for the 2011 Tohoku–Oki earthquake. We also note that, with the exception of EV2003, inter-plate thrust earthquakes with magnitudes of about 6 or greater are sparse around the GCMT location, which represents the major source region of the 2011 Mw9.1 Tohoku–Oki earthquake (Fig. 1a). We used local full waveform data recorded at the K-NET, KiK-net, and F-net stations (Fig. 1a) operated by the National Research Institute for Earth Science and Disaster Resilience (NIED).

First-Motion Augmented Moment Tensor (FAMT) Analysis

In FAMT analysis, we combine a nonlinear waveform inversion method to estimate the moment tensor and the source time function of a point source simultaneously with a space–time grid search method to infer the best point source position and origin time correction that minimize the squared residual:

$${F = \frac{1}{T}\sum\limits_{i = 1}^{{N_{W} }} {\int\limits_{0}^{{T_{i} }} {\frac{{\left[ {D_{i} \left( {x_{i} ,t} \right) - S_{i} \left( {x_{i} ,t} \right)} \right]^{2} }}{{\left( {\frac{1}{{w_{i} }}\sqrt {\frac{1}{{T_{i} }}\int_{0}^{T_{i}} {\left[ {D_{i} \left( {x_{i} ,t} \right)} \right]^{2} {\text{d}}t} } } \right)^{2} }}{\text{d}}t,} } }$$

(1)

where $D_{i} \left( {\varvec{x}_{i} ,t} \right)$ and $S_{i} \left( {\varvec{x}_{i} ,t} \right)$ are the ith observed and synthetic waveforms, respectively, $T_{i}$ is the length of the dataset, $N_{\text{W}}$ is the number of waveforms, $T = \mathop \sum \nolimits_{i = 1}^{{N_{\text{W}} }} T_{i}$, and $w_{i}$ is weight on the ith datum (for details, see Okamoto et al. (2017) and its Additional file 1).

In the present analysis, as in Okamoto et al. (2017), passbands of 10–40 and 4–40 s are used for the “long-period” and the “short-period” bandpass-filtered waveforms, respectively. The end time of the short-period waveforms is 10 s after onset of the P-wave, which is a sufficient duration to estimate the source time functions of the analyzed events. The weights ($w_{i}$ in Eq. 1) are set to 3.0 for short-period waveforms and 1.0 for long-period waveforms. The space–time grid searches are performed on 3528 spatial grid points for EV2003 ($21 \times 21$ in horizontal and 8 in vertical), and 7938 points for EV2007 ($21 \times 21$ in horizontal and 18 in vertical), and on seven time points from − 3 s to + 3 s with an increment of 1 s for the origin time corrections. The spatial increments are 2.4 km in horizontal and 1.2 km in vertical, respectively, for both cases.

The synthetic waveforms, $S_{i} \left( {\varvec{x}_{i} ,t} \right)$ in Eq. (1), are composed of Green tensor generated with a triangular moment rate function with a width of 2 s. In order to incorporate the structural effects in the Green tensor, we use a land–ocean unified 3D structure model for the Japan Trench (Fig. 1b). This model was built from models of land–ocean topography (Kisimoto 2000), subsurface structures in the Japan Seismic Hazard Information Station V1 model (Fujiwara et al. 2009), the crust (Baba et al. 2006), the subducting Pacific plate (Baba et al. 2006), and the subducting Philippine Sea plate (Nakamura et al. 2010).

The Green tensor was computed using the same reciprocal method as in Okamoto et al. (2017), in which we applied our heterogeneity, oceanic layer, and topography (HOT)-FDM scheme (Nakamura et al. 2012) with the correct fluid–solid boundaries. The spatial grid interval and maximum frequency of the FDM simulations were 150 m and 0.611 Hz, respectively. The simulations were performed on the TSUBAME-2.5 and -3.0 supercomputers at the Tokyo Institute of Technology, Japan, using 150–640 graphic processing units (GPUs) as accelerators (Okamoto et al. 2010, 2013). For the details of the FDM parameters, see Additional file 1: Table S1.

Results of FAMT Analysis

We find that the best origin time correction to the JMA origin times is 0 s for both events (Additional file 1: Fig. S1). The inferred FAMT parameters are listed in Table S2 and S3 in Additional file 1. For both EV2003 and EV2007, the distributions of the residuals (F in Eq. 1) display well-defined spatial minimums (Fig. 2a, b). In the case of EV2003, the reddish area with small residuals is confined to layer 2 of the oceanic crust, and in the case of EV2007, the upper limit of the reddish area is nearly along the inclined plate interface: these features show the sensitivity of the waveforms to the source positions in the 3D model. In both cases, the FAMT locations are almost at the plate interface of the 3D model and are thus consistent with the inter-plate mechanisms of these events. Figure 2c, d compares various estimated source depths: the GCMT and the JMA-CMT are based on waveforms with periods of 40 s or longer, and the F-net MT is based on waveforms with periods of 20 s or longer. Along with these routinely determined solutions are plotted the positions of LPMT (long-period MT) that denote the best locations of the point moment tensor sources determined in this study by using the same procedure as in the FAMT analysis but with only the long-period (10–40 s) data. The estimates based on the long-period waveforms tend to be deeper than the plate interface (except the JMA-CMT of EV2003). It is unlikely that these differences in the estimated depths are the result of the source finiteness, as the fault planes are considered to be located along the plate interface. Thus, we consider that these differences are caused by the differences between the assumed structures of the respective analyses and by the degraded resolution of the long-period (long-wavelength) signals. The FAMT solutions demonstrate the importance of the 3D structure for ensuring the “self-consistency” of the structure model, source locations, and source mechanisms. The horizontal scatter of the source locations is within a few kilometers for EV2003 and within 20 km for EV2007 (Additional file 1: Fig. S2).

Fig. 2

a Vertical cross sections of the spatial distribution of the residual (Eq. 1) for EV2003 obtained by the spatial grid search performed with the best (0 s) origin time correction. The cross section through the best (FAMT) source position (middle), the cross section shifted from the middle one in the positive X direction by 9.6 km (top), and the cross section shifted from the middle one in the negative X direction by 9.6 km (bottom) are shown. Red corresponds to the small (good) residuals, and the best point is indicated by the white cross on the middle figure. Gray curves denote the interfaces of the 3D structure model. The plate interface (Plate Int.), layer 2 (OC-L2), and layer 3 (OC-L3) of the oceanic crust are indicated. b Same as a but for EV2007. c Various results of source position estimations for EV2003. The JMA hypocenter (JMA) and the positions of the F-net MT (best depth of a point moment tensor), the GCMT and the JMA-CMT (best locations of the centroids), and the FAMT (this study) are projected on the best cross section displayed in a. The “LPMT” (long-period MT) determined in this study is also projected (see text). d Same as c but for EV2007

Figure 3 compares the observed velocity waveforms with the synthetics determined in the FAMT analyses. The synthetic waveforms fit the observed waveforms generally well in both cases. The simple short-period waveforms (the initial P-waves) of EV2003 are well reproduced with an approximately triangular source time function. On the other hand, a somewhat complex source time function is estimated for EV2007 on the basis of the multiple (split) peaks in the observed short-period waveforms.

Fig. 3

a Observed and synthetic waveforms for EV2003. The top trace in each pair, denoted DATA, is the observed ground velocity waveform; the bottom trace, denoted as SYN, is the corresponding synthetic computed for the FAMT parameters. The horizontal components with respect to the FDM grid are used in the FAMT analysis so that the last characters X and Y in the component name denote the directions of the horizontal ground motions (Fig. 1a). The character U denotes upward ground motion. The first two columns from TYS-X to HRO-U display the short-period waveforms with a passband of 4–40 s; the remaining columns display the long-period waveforms with a passband of 10–40 s. The numbers next to each pair denote the maximum amplitude ($10^{ - 3}$ cm/s) of the observed waveform. The estimated moment magnitude and the normalized residual (F in Eq. 1) are shown at the top. The obtained source time function (STF) and the moment tensor are also plotted in the bottom-right corner. b Same as a but for EV2007. c Detailed comparisons of the observed velocity waveforms (DATA: red) and the synthetic waveforms computed using 3D (blue) and a flat-layered structure model (1D: gray) at station TYS and HRO. d Same as c but for EV2007

We compare the observed waveforms with the synthetics computed using the 3D model and a flat-layered (1D) structure model for a period band of 10–40 s in Fig. 3c, d. The 1D synthetics were computed using the 1D structure model listed in Table 3 of Wu et al. (2008) and the discrete wavenumber method program (Takeo 1985) with the FAMT parameters. The 1D model was constructed for the analysis of the 1978 and 2005 Miyagi–Oki earthquakes that occurred near the coast; thus, the fit of the 1D synthetics to the observations is better for EV2007 (Fig. 3d) than for EV2003 (Fig. 3c). The land-based 1D structure model is not appropriate for EV2003, which occurred far off the coast; this can be seen from the large misfit of the 1D synthetic surface wave packets to the observations. Even for EV2007, the major phases (surface waves) show phase shifts and amplitude mismatches relative to the observations, even in the applied long-period band (10–40 s).

Waveform fitting

To evaluate the degree of waveform fitting, we compare the observed and synthetic waveforms with several passbands. Figure 4a, b plots waveforms from two selected stations (TYS and HRO; all the waveforms are plotted in Additional file 1: Figs. S3 and S4). In these analyses, the FAMT parameters are used to generate synthetic waveforms; in so doing, the biases due to inappropriate source mechanisms and locations are minimized. For the shortest passband (2–4 s), the observed waveforms show prolonged ground motions in some components (e.g., HRO in Fig. 4b and MYGH06 in Additional file 1: Fig. S3a), but the corresponding synthetics do not reproduce such characteristics. For the passbands of 4–7 and 7–10 s, which are included in the passband of the “short-period” waveforms in the FAMT analysis, the later large-amplitude surface waves are not well reproduced by the synthetics, while the body waveforms of the initial P-waves are reproduced (Fig. 3). The observed waveforms with periods of around 10 s or longer, especially the major part of the surface wave packets, are generally well reproduced, as shown in the fourth pairs of Fig. 4a, b.

Fig. 4

a Comparisons of the observed and synthetic velocity waveforms with several different passbands for EV2003. Here, we show the three components from two of the F-net broadband stations (TYS and HRO); for all thirty-three components, see Additional file 1: Fig. S3. The FAMT parameters are used to generate the synthetic waveforms. The passbands for each pair from top to bottom are 2–4, 4–7, 7–10, 10–15 and 4–40 s, respectively. The symbols “P” and “S” at the top traces denote the onsets of the P- and S-wave packets, respectively. b Same as a but for EV2007. For all forty-two components, see Additional file 1: Fig. S4. c The normalized residuals (F in Eq. 1) for EV2003. The residuals are computed using all the full waveforms of the 33 components with all the weights set to 1. Narrow-band cases are denoted by red squares, and broadband cases are denoted by blue squares. d Same as c but for EV2007. The residuals are computed using all the full waveforms of the 42 components

Figure 4c, d plots the normalized residuals (F in Eq. 1) computed using the full waveforms of the 33 components in the case of EV2003 and the full waveforms of the 42 components in the case of EV2007, respectively. Here, we set all the weights to 1.0 in order that a residual value of 1.0 is assigned to a synthetic trace with “null” (zero) amplitude: that is, a residual larger than 1 (depicted with green lines in Fig. 4c, d) indicates a synthetic worse than the “null trace.” For both events, the residuals for passbands shorter than 10 s are larger than 1, while those for passbands longer than 10 s are smaller than 1: these values indicate that the synthetics with periods shorter than 10 s do not fit the corresponding observations, as described above. The residual of the 4–40 s passband also exceeds 1 for EV2003, while it is (slightly) smaller than 1 for EV2007, which shows that the synthetics with the 4–40 s passband fit the observations to some extent for the latter event. This fit can be visually verified in Fig. 4b (bottom pairs of traces) and in Additional file 1: Fig. S4e. We consider that this difference between EV2003 and EV2007 is due to the different source locations: the EV2007 event occurred close to land at a relatively deep plate interface, where the effect of heterogeneous shallow structures would be weaker than in the case of EV2003, which occurred far off the coast at a shallow plate interface.

We note that the normalized residual depends on the selected time window. Limiting the time window to the body wave part dramatically reduces the residuals (Additional file 1: Figs. S5 and S6), indicating that the P-waves are reproduced well at short periods down to around 4 s. The residuals for the passband of 2–4 s are not sufficiently reduced in our analysis, because the passband is outside that of the FAMT analysis and the waveforms are not well constrained.

Discussion: sensitivity kernels

Through the FAMT analyses of the subduction-zone earthquakes, we have concluded that the seismic waveforms cannot be reproduced in the periods shorter than about 10 s, and the degree of fitting depends on the source locations, with better fittings for deep sources near land. These results indicate the necessity of improving the 3D structure model, especially for short-period waveforms from shallow sources far off the coast like EV2003. For this purpose, it would be helpful to study the regions where the waves propagate in the heterogeneous structure. In order to gain insights into the characteristics of such wave propagations, we employ an analysis based on the sensitivity kernel.

The sensitivity kernel has been used to study heterogeneities in the Earth’s interior (Tarantola 1984; Tanimoto 1990; Tromp et al. 2005; Zhao et al. 2005; Yoshizawa and Kennett 2005). We use the method of Tanimoto and Okamoto (2014) to compute the waveform sensitivity kernels. The perturbation in a single component of the displacement in the frequency domain ($\delta u\left( \omega \right)$) can be expressed using the sensitivity kernels $K_{\rho }$, $K_{\lambda }$, and $K_{\mu }$ as follows:

$$\begin{array}{*{20}c} {\delta u\left( \omega \right) = \int\limits_{V} {\left[ {K_{\rho } \left( {x,\omega } \right)\left\{ {\frac{\delta \rho \left( x \right)}{\rho \left( x \right)}} \right\} + K_{\lambda } \left( {x,\omega } \right)\left\{ {\frac{\delta \lambda \left( x \right)}{{\lambda_{0} \left( x \right)}}} \right\} + K_{\mu } \left( {x,\omega } \right)\left\{ {\frac{\delta \mu \left( x \right)}{{\mu_{0} \left( x \right)}}} \right\}} \right]dx,} } \\ \end{array}$$

(2)

with

$$\begin{array}{*{20}c} {\left. {\begin{array}{*{20}l} {K_{\rho } \left( {x,\omega } \right) = \rho \left( x \right)\omega^{2} \sum\limits_{i} {u_{i}^{A} \left( {x,\omega } \right)u_{i} \left( {x,\omega } \right),} } \hfill \\ {K_{\lambda } \left( {x,\omega } \right) = - \lambda_{0} \left( x \right)\Delta^{A} \left( {x,\omega } \right)\Delta \left( {x,\omega } \right),} \hfill \\ {K_{\mu } \left( {x,\omega } \right) = - 2\mu_{0} \left( x \right)\sum\limits_{ij} {e_{ij}^{A} \left( {x,\omega } \right)e_{ij} \left( {x,\omega } \right)} ,} \hfill \\ \end{array} } \right\}} \\ \end{array}$$

(3)

where $\rho$ denotes the density, $\lambda_{0}$ and $\mu_{0}$ the reference elastic constants (relaxed moduli), $\mu_{0}$ the rigidity, $e_{ij}$ the strain tensor, and $\Delta = \mathop \sum \nolimits e_{ii}$ (the above definitions are slightly different from that of Tanimoto and Okamoto (2014)). These kernels are computed from two wavefields: the forward wavefield computed with the earthquake source and the adjoint wavefield, which is denoted with the superscript $A$ and computed with a single (impulsive) force placed at the station location (the direction of the force is the same as that of $\delta u$).

Here, we discuss the sensitivity kernels for the vertical component at station TYS for EV2003. We apply the FAMT parameters for the forward modeling of the seismic wavefield with a triangular moment rate time function with a base width of 2 s and a vertical single force with a triangular time function with a base width of 2 s at the station location for the adjoint wavefield. In computing the wavefield, we use the same method (GPU-accelerated HOT-FDM) as in the FAMT analyses. To avoid the singularity at the source location, we use a cosine-tapered finite source centered at the FAMT best location and distributed along the plate interface with a radius of 5 km for the forward wavefield. Further, we apply spatial smoothing to the raw kernels with a running average using a cell of $11.7 \times 11.7$ km horizontally and 6.3 km vertically. We apply a Fourier transform to the full synthetic waveforms with a duration of 250 s from the onset of the source application to obtain the components in the frequency domain. The real parts of the smoothed, normalized kernels thus computed are shown in Fig. 5. The kernel for a period of 12.8 s (Fig. 5a) displays a horizontal distribution nearly symmetric with respect to the straight path between the source and the station. This feature suggests a moderate effect of the heterogeneous structure, which results in the synthetics having a good fit to the observations for the long-period band (${ \gtrsim }\,10 {\text{s}}$) waveforms. Also, the sensitive area extends to a relatively deep part (about 21 km deep), while the sensitivity in the shallow part (about 2 km deep) is relatively weak. On the other hand, the short-period kernel (6.1 s) shows a bend and has a spatially asymmetric pattern, which indicates a strong effect due to the 3D heterogeneous structure. A strong, shallow sensitive area in the depth range from 2 to 11 km for the period of 6.1 s indicates that shallow structures, such as sedimentary layers, are important in modeling short-period waveforms. Such a spatial pattern of sensitivity cannot be predicted with 1D Earth models and indicates the strong effects of 3D heterogeneity on short-period (${ \lesssim }10 {\text{s}}$) waveforms. In addition, the cross-branch coupling between different mode branches would have a significant effect on the propagation of short-period surface waves in the period range treated in this study (e.g., Kennett and Nolet 1990; Marquering and Snieder 1995). Thus, it would be necessary to consider 3D effects such as curved paths and cross-branch coupling when improving the structure models. The finite-difference-based adjoint kernels displayed in this study incorporate all these effects and would be suitable for the study of the Earth’s shallow structure.

Fig. 5

Normalized, real part of rigidity kernels for period of a 12.8 s, and b 6.1 s, for the vertical component at station TYS (F-net). Crosses on the top map denote the locations of the FAMT solution (SRC) and the station (TYS), respectively. The horizontal slices of the three-dimensional kernels at depths of 2, 11, 21, and 30 km are displayed. The kernels are smoothed using a running average with a cell of $11.7 \times 11.7$ km horizontally and 6.3 km vertically, normalized with the maximum amplitude for each period, and clipped at a normalized amplitude of $\pm \;0.5$