Introduction

It is well known that, as frequency increases over 1 Hz, the spatial distributions of observed maximum P- and S-wave amplitudes during local earthquakes (hereafter, this is called the “apparent radiation pattern”) are gradually distorted from the expected four-lobe amplitude pattern of a double-couple point source (e.g., Liu and Helmberger 1985; Satoh 2002a; Takenaka et al. 2003; Takemura et al. 2009; Sawazaki et al. 2011; Kobayashi et al. 2015). The frequency-dependent characteristics of the observed apparent radiation patterns have been incorporated into various applications such as the predictions of strong ground motions (e.g., Pitarka et al. 2000; Pulido and Kubo 2004), the estimation of high-frequency seismic energy radiation during large earthquakes (e.g., Nakahara 2013), nonvolcanic/volcanic tremors (e.g., Maeda and Obara 2009; Kumagai et al. 2010; Cannata et al. 2013; Yabe and Ide 2014), and landslides (e.g., Ogiso and Yomogida 2015), and the earthquake early warning systems (e.g., Okamoto and Tsuno 2015). Although the frequency–distance change model for S-wave radiation pattern proposed by Satoh (2002b) has been used in some applications, to achieve more accurate estimation and prediction of high-frequency seismic radiation, a precise frequency- and distance-dependent model for the apparent radiation pattern for both P and S waves is required. Relationship of the apparent radiation patterns between P and S waves has been important due to recent development real-time systems, such as urgent earthquake detection and earthquake early warning (e.g., Okamoto and Tsuno 2015).

High-quality seismograms recorded by dense regional seismic networks for various distances and wide dynamic ranges enable us to investigate frequency- and distance-dependent characteristics of the apparent radiation pattern. Takemura et al. (2009, 2015) and Kobayashi et al. (2015) reported that the apparent P- and S-wave radiation patterns are distorted with increasing distance but still preserving the original four-lobe pattern at hypocentral distances less than 40 km even for high frequencies. In this study, we firstly investigated the frequency- and distance-dependent characteristics of the apparent P- and S-wave radiation patterns using dense and large number seismograms. On the basis of observed characteristics, then we propose a frequency- and distance-dependent model of the apparent radiation pattern to predict the spatial distributions of maximum P- and S-wave amplitudes of local earthquakes.

Apparent radiation pattern for crustal earthquakes

We analyzed velocity waveforms recorded by Hi-net (high-sensitivity seismograph network operated by the National Research Institute for Earth Science and Disaster Resilience (NIED); Okada et al. 2004) during 13 earthquakes occurred in the crust of Chugoku region, Japan (Events 1–13 of Fig. 1a and Additional file 1: Table S1). The mechanisms of these earthquakes were characterized by strike-slip faulting and reported consistently by both the moment tensor (MT) solutions of F-net (full range seismograph network by NIED; Fukuyama et al. 1998) and the first-motion focal mechanisms in the unified hypocenter catalog of the Japan Meteorological Agency.

Fig. 1
figure 1

Attenuation for maximum amplitudes of P and S waves. a Map of stations and epicenters, b estimated apparent attenuation for P and S waves as a function of frequency, and c, d amplitude attenuations for P and S waves, respectively. Gray squares and triangles in a denote the Hi-net and F-net stations, respectively. Focal mechanisms of each earthquake are referred from MT solutions of F-net. |F p | and |F S | are the magnitudes of respective P- and S-wave radiation pattern coefficients expected from MT solutions of the F-net in the 1D crustal structure (Ukawa et al. 1984; Aki and Richards 2002)

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In some previous studies, energy partition of S wave in each horizontal component was analyzed in order to eliminate the effects of differences in site amplification and source size. However, Sawazaki et al. (2011) pointed out that spatial distribution of maximum amplitudes and energy partitioning in each component show different frequency-dependent properties. Therefore, on the basis of the method by Kobayashi et al. (2015), we measured coda-normalized maximum P- and S-wave amplitudes (hereafter, these are referred to as the “P-wave amplitude” and “S-wave amplitude,” respectively) from three-component root-mean-square (RMS) envelopes for the following different frequency bands: 0.5–1, 1–2, 2–4, 4–8, and 8–16 Hz.

Additional file 1: Figure S1 shows examples of filtered velocity seismograms and RMS vector envelopes normalized by averaged coda amplitudes to eliminate the effects of differences in site amplification and source size (e.g., Yoshimoto et al. 1993). Since coda normalization technique is applicable in the seismograms with hypocentral distance less than approximately 150 km (e.g., Sato et al. 2012 Ch. 3; Takemoto et al. 2012), we employ the lapse times of 60–70 s for calculating averaged coda amplitudes. The time windows of τ-seconds, which represent the averaged pulse durations of P and S waves measured from the displacement waveforms at four F-net stations (filled triangles in Fig. 1a), were used to measure P- and S-wave amplitudes.

After measuring P- and S-wave amplitudes, we estimated the master attenuation curves of P- and S-wave attenuations by using the following equation:

$$\ln \left( {L_{i} A_{ij}^{\hbox{max} } } \right) = - \frac{{\pi f_{C} }}{{Q_{j} V_{j} }}L_{i} + B\quad \left( {j = P,S} \right),$$
(1)

where A max ij is the P- or S-wave amplitude at a hypocentral distance of L i km, Q j is the quality factor for the apparent P- or S-wave attenuation, f C is the central frequency of each band, V j is the seismic velocity for P or S waves in the upper crust (assuming 6.00 and 3.55 km/s, respectively), and B is a constant. Since hypocentral distance L i is widely used in empirical attenuation functions of ground motions (e.g., Si and Midorikawa 1999; Boatwright 2007; Yabe et al. 2014), we simply assumed geometrical spreading 1/L i of body waves, rather than inverse of exact ray-path length. Figure 1b shows the apparent attenuations for P and S waves estimated by least square fitting.

Figure 1c, d shows the measured P- and S-wave amplitudes and master attenuation curves as a function of the hypocentral distance for frequencies of 0.5–1 and 4–8 Hz. The color scale represents the magnitude of the P- and S-wave radiation pattern coefficients (|F P | and |F S |; Aki and Richards 2002) estimated from MT solutions in the one-dimensional (1D) crustal velocity structure (Ukawa et al. 1984), which is used in the Hi-net routine hypocenter analysis. The S-wave radiation pattern coefficient |F S | was calculated by RMS of SV- and SH-wave radiation pattern coefficients. Wavelengths in Fig. 1c, d (λ P and λ S , respectively) were calculated by using the central frequencies of each band and seismic velocities in the crust. Observed amplitudes are scattered around master attenuation curves, reflecting the effects of non-isotropic source radiation and fluctuation of amplitude due to small-scale velocity inhomogeneity along propagation path (e.g., Hoshiba 2000; Yoshimoto et al. 2015).

The scatter due to non-isotropic source radiation is most evident in the P-wave amplitudes for the lowest frequency (0.5–1 Hz; left side of Fig. 1c). We confirmed that P-wave amplitudes with larger/smaller |F P | values tend to distribute above/below the master attenuation curve, respectively. As the frequency increased (4–8 Hz; right of Fig. 1c), this tendency become unclear, implying that P-wave amplitudes at higher frequencies do not show a clear four-lobe apparent radiation pattern. Although similar behaviors appeared in the S-wave amplitudes (Fig. 1d), the four-lobe patterns become unclear more rapidly compared to the P-wave ones (Fig. 1c).

We calculated the observed “amplitude fluctuation δA j ” by the following equation:

$$\delta A_{j} \left( {L_{i} } \right) = \frac{{A_{j} \left( {L_{i} } \right) - A_{0j} \left( {L_{i} } \right)}}{{A_{0j} \left( {L_{i} } \right)}} \, \left( {j = P,S} \right) ,$$
(2)

where A j (L i ) is the P- and S-wave amplitude at a hypocentral distance of L i and A 0j (L i ) is the prediction from the master attenuation curve (Eq. 1). Theoretical amplitude fluctuations were defined as |F j | fluctuations from the azimuthal average of |F j | at a hypocentral distance of L i .

Figure 2 shows the comparison of azimuthal dependence of the observed and theoretical amplitude fluctuations (filled and open symbols, respectively) at frequencies of 0.5–1 and 4–8 Hz. Despite of the continuous four-lobe azimuthal variations of theoretical amplitude fluctuations, observed fluctuations are scattered around the theoretical ones. As wavelengths decrease, observed amplitude fluctuations are widely scattered and large amplitude fluctuations also appeared in the nodal directions. We confirmed that one of the key parameters for distortions of the observed P- and S-wave apparent radiation patterns is the wavelength of the seismic waves.

Fig. 2
figure 2

Azimuthal dependence in the observed and theoretical fluctuations; a P and b S waves derived from data at hypocentral distances of 30–100 km during Events 1–13. Filled and open symbols represent observed and theoretical fluctuations, respectively

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Frequency and distance dependences in the apparent radiation pattern

To quantify distortion of the apparent radiation pattern from double-couple point source predictions, we simply calculated the cross-correlation coefficient (CCC) between the observation and theoretical amplitude fluctuations using moving hypocentral distance windows (40–70, 50–80, 60–90, 70–100, 80–110, 90–120, and 100–130 km).

Figure 3 shows the estimated CCCs as a function of the normalized hypocentral distance kL, where k is the wave number (k = 2π/λ = 2πf/V) and L is the average hypocentral distance of each distance range. Observed CCCs showed linear decay from 0.75 to 0.25 with increasing log(kL) from 1.64 to 2.85 and no significant differences in decay patterns between P and S waves. These results suggest that major causes of the frequency- and distance-dependent distortion of the apparent radiation pattern are seismic wave scattering and diffraction in the heterogeneous crust and that effects of crustal heterogeneity are not different for both P and S waves. Saturation of CCC decay was found at greater normalized hypocentral distances (log(kL) > 2.85), showing almost dissipation of non-isotropic source radiation effect. Considering typical correlation length a of crustal heterogeneities (e.g., Takemura et al. 2009; Kobayashi et al. 2015), this feature appears in the strong scattering domain of ka–kL diagram (Fig. 13.11 of Aki and Richards 1980), in where conventional ray theory does not stand.

Fig. 3
figure 3

Normalized distance (kL) dependence in CCCs between observed and theoretical fluctuations. Estimation error of CCC was evaluated by using the bootstrap method (Efron and Tibshirani 1986). Averaged CCC and estimation error were calculated for 100 sets of random resampling data

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To characterize observed kL dependence of CCC, a linear fitting approach was applied by using the following equation:

$${\text{CCC}} = rlog\left( {kL} \right) + {\text{CCC}}_{0} .$$
(3)

The values of r and CCC0 were determined as −0.38 ± 0.023 and 1.35 ± 0.053, respectively, by a least squares estimation for the range of log(kL) < 2.85. The observed CCC could be described by using resultant Eq. (3) (blue line in Fig. 3). We here introduce a set of values k 1 L 1, k 2 L 2, and k 3 L 3 from log(k i L i ) = 0.92, 2.85, and 3.55 (i = 1, 2, 3), respectively, from Fig. 3, for the following discussions.

Apparent radiation pattern modeling and amplitude predictions

Pulido and Kubo (2004) proposed a simple linear frequency-dependent model of S-wave radiation pattern coefficients for strong ground motion prediction. On the basis of the observed linear decay of CCCs against log(kL), we revisited their approach and proposed a new model, which includes the frequency- and distance-dependent characteristics of the P- and S-wave apparent radiation pattern. Our frequency- and distance-dependent apparent radiation pattern coefficient R j (j = P, S) at a station with a takeoff angle θ and azimuth \(\phi\) is described by the following equation:

$$R_{j} \left( {\theta ,\phi ,kL} \right) = \left\{ {\begin{array}{*{20}l} {F_{j} \left( {\phi_{S} ,\delta ,\lambda ,\theta ,\phi } \right),\quad {\text{ if }}kL \le k_{1} L_{1} } \hfill \\ {F_{j} \left( {\phi_{S} ,\delta ,\lambda ,\theta ,\phi } \right) - \frac{{log\left( {kL} \right) - log(k_{1} L_{1} )}}{{log(k_{3} L_{3} ) - log(k_{1} L_{1} )}}\left( {F_{j}^{\text{ave}} - F_{j} \left( {\phi_{S} ,\delta ,\lambda ,\theta ,\phi } \right)} \right),\quad {\text{ if }}k_{1} L_{1} \le kL \le k_{2} L_{2} } \hfill \\ {F_{j} \left( {\phi_{S} ,\delta ,\lambda ,\theta ,\phi } \right) - \frac{{log\left( {k_{2} L_{2} } \right) - log(k_{1} L_{1} )}}{{log(k_{3} L_{3} ) - log(k_{1} L_{1} )}}\left( {F_{j}^{\text{ave}} - F_{j} \left( {\phi_{S} ,\delta ,\lambda ,\theta ,\phi } \right)} \right), \quad {\text{ if }}k_{2} L_{2} \le kL} \hfill \\ \end{array} } \right.,$$
(4)

where F j is the radiation pattern coefficient of a double-couple point source in the 1D crustal velocity structure (Ukawa et al. 1984). The focal mechanism is characterized by using angles of strike \(\phi\) S , dip δ, and rake λ. F ave j is the average radiation pattern coefficient for P and S waves of a double-couple point source (Boore and Boatwright 1984). Takeoff angle θ was also evaluated in the 1D structure model (an example shown in Fig. 4d). The modeled apparent radiation pattern coefficient introduced here is identical to the double-couple point source radiation pattern at small normalized hypocentral distances (<k 1 L 1) and diminishes non-isotropy as increasing normalized hypocentral distance. Furthermore, our model includes not only S-wave radiation pattern but also P-wave one, which would be important for the earthquake early warning.

Fig. 4
figure 4

Spatial distributions of modeled radiation pattern coefficients. Frequency ranges are a 0.5–1 and b 4–8 Hz, respectively. Theoretical radiation pattern coefficients in infinite homogeneous medium are also plotted in c. d Assumed one-dimensional velocity structure model (Ukawa et al. 1984) and calculated takeoff angles of Event 11 (depth of 11 km)

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Figure 4a, b shows the spatial distributions of modeled apparent radiation pattern coefficients for frequencies of 0.5–1 and 4–8 Hz, respectively. We also show the spatial distribution of the radiation pattern coefficient for a double-couple point source in a homogeneous medium as a reference (Fig. 4c), where amplitude nodes (R j  = 0.00) clearly exist. The azimuthal difference of modeled apparent radiation pattern coefficients became unclear with increasing distances and wave numbers.

By using the estimated master attenuation curves and modeled apparent radiation pattern coefficients, we here propose a representation for the prediction of coda-normalized maximum amplitudes for both P and S waves [A max j (θ, \(\phi\), kL)] during local crustal earthquakes as follows:

$$\begin{aligned} A_{j}^{\rm max} \left( {\theta ,\phi ,kL} \right) &= \frac{{\delta R_{j} \left( {\theta ,\phi ,kL} \right)}}{{d_{i} }}exp\left[ { - \frac{kL}{{2Q_{j} \left( k \right)}} + B} \right] \, \hfill \\ \delta R_{j} \left( {\theta ,\phi ,kL} \right) &= \frac{{R_{j} \left( {\theta ,\phi ,kL} \right) - R_{0j} \left( {kL} \right)}}{{R_{0j} \left( {kL} \right)}} \, \hfill \\ \end{aligned}$$
(5)

where R 0j (kL) is the azimuthal average of the modeled apparent radiation pattern coefficient.

Figure 5 shows the comparisons between the observed and predicted spatial distributions of P- and S-wave amplitudes during Events 11 and 14 for low (0.5–1 Hz) and high (4–8 Hz) frequencies. The latter event was not used in the analyzes of the apparent radiation patterns in the previous sections, and its source parameters are shown at the bottom of Additional file 1: Table S1. The effects of rupture directivity (e.g., Boatwright 2007; Pacor et al. 2016) and fluctuation of maximum amplitudes (e.g., Hoshiba 2000; Yoshimoto et al. 2015) were not taken into consideration within our method. Actually, for some earthquakes, directivity amplifications were found in the south–southeast direction from the source even for high frequencies (4–8 Hz). Recent observation study by Pacor et al. (2016) demonstrated that some earthquakes with Mw of 3.5–4 show directivity effects and that weakening of directivity starts at a frequency of approximately 10 Hz. Despite this, our predictions reproduced the observed spatial distributions of P- and S-wave amplitudes for both low- and high-frequency bands reasonably. Since seismograms with hypocentral distances less than 150 km were used in our analysis, practical applicability of our method is limited within same distance range. As the wave number (k = 2π/λ = 2πf/V) and distance (L) increased, both the observed and predicted maximum amplitude distributions became gradually distorted from the original four-lobe pattern of the double-couple point source.

Fig. 5
figure 5

Comparison between predicted and observed maximum amplitudes. Spatial distributions of coda-normalized P- and S-wave amplitudes for frequencies of a 0.5–1 and b 4–8 Hz during Events 11 and 14

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Satoh (2014) employed frequency- and distance-dependent S-wave radiation pattern coefficient for the stochastic Green’s function method based on empirical model of Satoh (2002a, b), which showed very weak distance dependency for frequencies of 2–5 Hz and an isotropic radiation pattern for higher (>6 Hz) frequencies irrespective of distance. Satoh (2002b) constructed this model via observed energy partitioning of S waves in each horizontal component to reduce source and site amplification effects, rather than spatial distribution of maximum amplitude. Although our model does not include directivity effects, our model practically succeeds in reproducing observed spatial distribution of maximum amplitude of small-to-moderate local crustal earthquakes compared to Satoh (2002b)’s model (Additional file 1: Figure S2). This difference may be caused by difference in the method for model construction.

Conclusions

We investigated the frequency and distance dependences in the apparent radiation pattern for both P and S waves during local crustal earthquakes. We demonstrated how the four-lobe apparent radiation pattern, which is expected from a double-couple point source, is gradually distorted with increasing frequency and distance. The observed distortions have common decay pattern for P and S waves and could be characterized by the normalized hypocentral distance kL. These results suggest that major causes of frequency- and distance-dependent distortion of the apparent radiation pattern are seismic wave scattering and diffraction in the heterogeneous crust.

The observed frequency and distance dependences in the apparent radiation patterns for both P and S waves could be simply modeled by using a linear function of log(kL). On the basis of this, we proposed a method for prediction of the spatial distributions of maximum P- and S-wave amplitudes. Our method, which incorporates frequency- and distance-dependent characteristics of the observed apparent radiation pattern, successfully reproduced the observed spatial distributions of P- and S-wave amplitudes during small-to-moderate local crustal earthquakes.

Our method could also provide better insights into source rupture process and practical correction for the effects of the apparent radiation pattern. In future study, this would enable us to estimate the radiated source energy precisely and to obtain better insights into high-frequency seismic sources, such as small earthquakes and non-volcanic/volcanic tremors and the other effects, especially rupture directivity and fluctuation of maximum amplitudes, will be taken into consideration within our method.