# Ground-motion models

## Definitions

In this section we review ground motion models that are appropriate for the prediction of strong ground motions at the sites. Ground-motion models, also called ground motion prediction equations (GMPE) are most commonly derived from strong ground motion recordings. For a given event of magnitude $$m^*$$ located at some distance $$r^*$$, the strong motion intensity $$i^*=f(m^*,r^*,{\theta})$$ can be estimated in terms of its conditional mean $$\eta_{(I|M,R)}$$ and its error term $$\varepsilon_I$$. The expected seismic intensity at the project site is a conditional random variable $$I/M,R$$ and the ground-motion models provide an estimate of the median and the error term of ground-motion intensity in terms of earthquake magnitude, distance and site and geologic conditions. The conditional mean $$\eta_{(I|M,R)}=f(m^*,r^*,{\theta})+\varepsilon$$ is an empirical regression model of earthquake magnitude $$m^*$$, source-to-site distance $$r^*$$ and geological/geotechnical site conditions {$$\theta$$}. The error term $$\varepsilon$$, is a random variable with a log-normal distribution with mean $$\mathrm{E}[\varepsilon]=0$$ and variance $$\mathrm{Var}[\varepsilon] \approx \sigma_{lnI|M,R}^2$$

Most GMPEs are defined in terms of finite fault distance metrics assuming a planar rupture geometry. The trace of the rupture is defined as the projection of the top edge of the rupture on the ground surface. The rupture plane, trace and the surface projection allow to define four finite fault distance metrics: $$R_{RUP}$$, $$R_{JB}$$ ,$$R_{X}$$ and $$R_{Y0}$$. The rupture distance $$R_{RUP}$$ is the distance from the site to the closest point of the rupture plane. The Joyner-Boore distance $$R_{JB}$$ is the closest distance between the site and the surface projection of the rupture plane and the auxiliary parameters $$R_X$$, $$R_{Y0}$$ and $$Z_{TOR}$$ are the coordinates from the site to the closest point of the top of the rupture.

Figure 8 shows these distances between a site and the rupture plane. A full description of these finite fault metrics can be found in Kaklamanos et al..40

## Logic Tree

A minimum number of independent GMPEs are selected to capture the expected range of possible ground-motions in the target region. They are weighted in a logic tree to address their relative confidence based on the available information. The chosen set represents the composite distribution of epistemic uncertainty in the ground motion model for the site.

A region-specific ground-motion model is one of the weighted branches of the modeled logic tree (under certain guidelines).42 One or more of these models are complemented with GMPEs developed for similar seismic regions or suited for global analysis.

The lack of strong movement records in West Africa does not allow for a specific ground motion model at the time of writing, and the standard practice consists in select ground-motion models suitable for for similar tectonic regimes43

Table 3 summarizes the addressed tectonic region and rupture, distances and site context required for each model. The site parameters $$V_{s30}$$ and $$Z_{1.0}$$ are the average shear-wave velocity in the top 30 m and the depth to engineering rock ($$V_s=1000\,\mathrm{m/s}$$) respectively.

 Model Code Tectonic Rupture context Distance context Site context Abrahamson et al. 2014 ASK14 ASC Mw, rake, Ztor, dip, width Rrup, Rjb, Rx, Ry0 Vs30, z1.0 Akkar et al. 2014 ASB14 ASC Mw, rake Rjb Vs30 Chiou Youngs 2014 CY14 ASC Mw, rake, Ztor, dip Rrup, Rjb, Rx Vs30, z1.0 Atkinson Boore 2011 AB11 SCC Mw Rrup Vs30 Pezeshk et al. 2011 PZT11 SCC Mw Rrup 760 or 1500 m/s Toro et al. 2002 TAS02 SCC Mw Rjb 800 or 1828 m/s Silva et al. 2002 SGD02 SCC Mw Rjb 760 m/s

Kadiri and Kijko44 compared the results of a seismic hazard under a stable continental (SCC) and a shallow crustal seismicity (ASC). They concluded that the activity in Western Africa corresponds more to a stable continental activity. Therefore, we adopted a branch configuration with a higher share (64%) of this tectonic type (Figure 9).

1. “Estimating Unknown Input Parameters When Implementing the NGA Ground-Motion Prediction Equations in Engineering Practice.”↩︎

2. Kaklamanos, Baise, and Boore.↩︎

3. J. Bommer and F. Scherbaum, “The Use and Misuse of Logic Trees in Probabilistic Seismic Hazard Analysis,” Earthquake Spectra 24 (2008): 997–1009.↩︎

4. Jonathan Stewart et al., “GEM-PEER Task 3 Project: Selection of a Global Set of Ground Motion Prediction Equations” (Pacific Earthquake Enginnering Research Center, 2013).↩︎

5. “Seismicity and Seismic Hazard Assessment in West Africa.”↩︎