Ground Motion Prediction Equation based on Morikawa and Fujiwara (2013)

December, 2023
Nobuyuki Morikawa and Hiroyuki Fujiwara
National Research Institute for Earth Science and Disaster Resilience

The National Research Institute for Earth Science and Disaster Resilience (NIED) has developed seismic hazard assessments and maps of peak acceleration with respect to the response spectrum (Fujiwara et al., 2023). In addition, the Headquarters for Earthquake Research Promotion released a Probabilistic Seismic Hazard Analysis of Response Spectra (provisional edition) in 2022 (Earthquake Research Committee, Subcommittee for Strong Ground Motion Evaluation, 2022). The application of ground motion prediction equation in these assessments is explained here.

The following references should be cited when conducting seismic hazard assessments according to the following explanations.

Download the set of coefficients for the ground motion prediction equation (CSV format)

The ground motion prediction equation is based on Morikawa and Fujiwara (2013). An improved version of the correction terms for site amplification (Morikawa and Fujiwara, 2014) and an additional correction term for intraplate earthquakes shallower than about 60 km in depth that occur in the Philippine Sea plate (Morikawa and Fujiwara, 2015) are applied to the original equation by Morikawa and Fujiwara (2013).

The ground motion prediction equation is expressed as follows, which applies a quadratic magnitude model.

Peak acceleration, peak velocity, 5% damped acceleration response spectrum (A)

\begin{align} {\log_{10} A =}{a \cdot \left( Mw^{'} - 16 \right)^2 + b_{k} \cdot X + c_{k} - \log_{10}\left( X + d \cdot 10^{0.5Mw^{'}} \right)} + G_{d} + G_{s} + AI + PH + \sigma \tag{1} \\ \text{Where } Mw^{'} = \min_{}(Mw,8.2) \end{align}

In the following, the ground motion prediction equation based on Morikawa and Fujiwara (2013), shown in the above equation, is divided into the basic equation, correction terms for deep sedimentary layers, shallow soft soils, anomalous seismic intensity distribution, and for intraplate earthquakes on the Philippine Sea Plate.

(1) Basic equation of Morikawa and Fujiwara (2013)

The basic equation of the ground motion prediction equation by Morikawa and Fujiwara (2013) is expressed as follows.

Peak acceleration, peak velocity, 5% damped acceleration response spectrum (A)

\begin{align} {\log_{10} A =}{a \cdot \left( Mw^{'} - 16 \right)^2 + b_{k} \cdot X + c_{k} - \log_{10}\left( X + d \cdot 10^{0.5Mw^{'}} \right)} \tag{2} \\ \text{Where } Mw^{'} = \min_{}(Mw,8.2) \end{align} \begin{align} Mw &: \text{Moment magnitude} \\ X &: \text{Fault distance in } {[}km{]} \\ a, b_k, c_k ,d &: \text{Regression coefficients} \end{align}

*) The subscript \(k\) indicate the earthquake type, which can be divided into the following categories; \begin{align} 1 &: \text{Shallow crustal earthquakes} \\ 2 &: \text{Subduction zone inter-plate earthquakes} \\ 3 &: \text{Subduction-zone intra-plate earthquakes} \end{align}

This basic equation is applicable to all earthquakes and all evaluation sites, and is extrapolated to apply even when the shortest fault distance is longer than 200 km, although it is constructed based on earthquake records within 200 km in Morikawa and Fujiwara (2013).

(2) Correction term for deep sedimentary layers

The correction term for deep sedimentary layers is common to all seismic intensity indices and is expressed by the following equation.

\[ {G_{d} = p_{d} \cdot \log_{10}}\left\{ \frac{\text{max}\left( D_{l\ \rm min},D_{1400} \right)}{300} \right\} \tag{3} \] \begin{align} D_{1400} &: \text{Depth to the layer whose } V_{s}=1400m/s \text{ at the evaluation site in } {[}m{]} \\ p_{d}, D_{l\ \rm min} &: \text{Regression coefficients} \end{align}

This correction term for deep sedimentary layers is also applied for all earthquakes and all evaluation sites, as in the basic equation. The subsurface structure model is J-SHIS V3.2 (Fujiwara et al., 2023), which is used in the National Seismic Hazard Maps for Japan, 2020 edition. However, for the Ogasawara Islands and other islands that do not have a subsurface structure model, the value of \(G_d\) is uniformly set to 0 (zero). For locations where there is no layer with \(V_{s}\) = 1400 m/s, the top depth of the underlying layer is used under the same conditions as when the regression equation was created.

(3) Correction term for shallow soft soils

The correction term for shallow soft soils is the same for all seismic intensity indices as the correction term for deep sedimentary layers, and is expressed by the following equation.

\[ {G_{s} = p_{s} \cdot \log_{10}}\left\{ \frac{\text{min}\left( V_{S\rm max},AVS30 \right)}{350} \right\} \tag{4} \] \begin{align} AVS30 &: \text{Average S-wave velocity up to 30 m depth at the evaluation site in } {[}m/s{]} \\ p_{s}, V_{S\rm max} &: \text{Regression coefficients} \end{align}

Note: Fujiwara et al. (2023) incorrectly stated that equation (4) above is correct.

For the evaluation on an engineering bedrock (\(AVS30\)=400m/s), the following equation is used:

\[ {G_{s} = p_{s} \cdot \log_{10}}\left\{ \frac{\text{400}}{350} \right\} \cong p_{s} \times 0.058 \tag{5} \]

(4) Correction term for anomalous seismic intensity distribution

The correction term for the seismic anomalous seismic intensity distribution is common to all seismic intensity indices and is expressed by the following equation.

\[ AI = \gamma \cdot X_{vf} \cdot (H - 30) \tag{6} \] \begin{align} X_{vf} : & \text{Distance to the volcanic front at the evaluation site in } {[}km{]}\\ &\text{In the case of southwestern Japan, }X_{vf} = \min_{}(X_{vf}, 75) \\ H : & \text{Hypocenter depth in }{[}km{]} \\ \gamma : & \text{Regression coefficients }(\gamma_{\rm NEJapan}: \text{Northeastern Japan,} \gamma_{\rm SWJapan}: \text{Southwestern Japan}) \end{align}

For volcanic front locations, we use Morikawa et al. (2006) for northeastern Japan and Earthquake Research Committee (2009) and Fujiwara et al. (2009) for southwestern Japan. For northeastern Japan, however, we refer to Catalog of Quaternary Volcanoes in Japan, which extends to the Izu and Ogasawara Islands (Fujiwara et al., 2009). For northeastern Japan (Pacific Plate) earthquakes, all earthquakes deeper than 30 km are applied to evaluation sites north of 36°N (actually, they are varied in the range of 36.5°N~35.5°N). For earthquakes in southwestern Japan (Philippine Sea Plate), earthquakes in the region from Kyushu to the Ryukyu Islands at depths of 60 km or greater are applied to evaluation sites west of 136.9°E. The "depth of hypocenter" is the center depth of the fault plane.

(5) Correction term for intraplate earthquakes of the Philippine Sea Plate

For intra-plate earthquakes in the Philippine Sea Plate, we use the correction term \(PH\) by Morikawa and Fujiwara (2015) is used.

(6) Stabdard deviation

For the standard deviation, we tentatively use the same one as the National Seismic Hazard Maps for Japan, including the peak acceleration. For shallow crustal earthquakes such as those on active faults, the standard deviation depends on the fault distance (\(X\))

\[\sigma = \left\{ \begin{matrix} 0.23 & X \leqq 20km \\ 0.23 - 0.03 \cdot \frac{\log_{10}\left( X/ 20 \right)}{\log_{10}\left( 30/ 20 \right)} & 20km < X \leqq 30km \\ 0.20 & 30km < X \\ \end{matrix} \right.\ \tag{7}\]

For subduction earthquakes, we use the amplitude-dependent standard deviation

\[ \sigma = \left\{ \begin{matrix} 0.20 & PV \leqq 25cm/s \\ 0.20 - 0.05 \cdot \frac{PV - 25}{25} & 25cm/s < PV \leqq 50cm/s \\ 0.15 & 50cm/s < PV \\ \end{matrix} \right.\ \tag{8}\]

Where \(PV\) is the peak velocity on stiff soil (\(V_{s}\) = 600 m/s), which is obtained by Si and Midorikawa (1999)'s ground motion prediction equation.