Determination of GMPE functional form for an active region with limited strong motion data: application to the Himalayan region

Abstract

Advancement in the seismic networks results in formulation of different functional forms for developing any new ground motion prediction equation (GMPE) for a region. Till date, various guidelines and tools are available for selecting a suitable GMPE for any seismic study area. However, these methods are efficient in quantifying the GMPE but not for determining a proper functional form and capturing the epistemic uncertainty associated with selection of GMPE. In this study, the compatibility of the recent available functional forms for the active region is tested for distance and magnitude scaling. Analysis is carried out by determining the residuals using the recorded and the predicted spectral acceleration values at different periods. Mixed effect regressions are performed on the calculated residuals for determining the intra- and interevent residuals. Additionally, spatial correlation is used in mixed effect regression by changing its likelihood function. Distance scaling and magnitude scaling are respectively examined by studying the trends of intraevent residuals with distance and the trend of the event term with magnitude. Further, these trends are statistically studied for a respective functional form of a ground motion. Additionally, genetic algorithm and Monte Carlo method are used respectively for calculating the hinge point and standard error for magnitude and distance scaling for a newly determined functional form. The whole procedure is applied and tested for the available strong motion data for the Himalayan region. The functional form used for testing are five Himalayan GMPEs, five GMPEs developed under NGA-West 2 project, two from Pan-European, and one from Japan region. It is observed that bilinear functional form with magnitude and distance hinged at 6.5 M _w and 300 km respectively is suitable for the Himalayan region. Finally, a new regression coefficient for peak ground acceleration for a suitable functional form that governs the attenuation characteristic of the Himalayan region is derived.

Appendices

Appendix 1

Different GMPEs mentioned in Tables 1 and 2 are ranked based on the LLH procedure. The recorded data from 2015 Nepal earthquake is used in ranking the GMPEs using LLH. Table 4 provides the ranking of GMPEs along with LLH score for peak ground acceleration.

Table 4 Ranking of GMPEs based on LLH criteria for peak ground acceleration

Appendix 2

The steps used in the mixed effect algorithm are as follows:

• I. Estimate the model coefficients, i.e., θ using a fixed-effects regression algorithm assuming random effect term, η equals zero.

• II. Using θ, solve the variance of the residuals, σ ² and τ ², by maximizing the likelihood function described as follows:

$$ {\displaystyle \begin{array}{l}\kern3em C\left({\varepsilon}_{ij}^{(t)},{\varepsilon}_{ij^{\hbox{'}}}^{(t)}\right)=C\left({\varepsilon}_{ij}+\eta, {\varepsilon}_{ij^{\hbox{'}}}+\eta \right)\\ {}\kern8em =\rho \left({d}_{jj^{\hbox{'}}}\right){\sigma}^2+{\tau}^2\forall i,j,{j}^{\hbox{'}}\\ {}C\left({\varepsilon}_{ij}^{(t)},{\varepsilon}_{ij^{\hbox{'}}}^{(t)}\right)=0\forall j,j,i\ne {i}^{\hbox{'}}\end{array}} $$

Parameter $\rho \left({d}_{j{j}^{\prime }}\right)$ denotes the spatial correlation between intraevent residuals at two sites j and j ^′ as a function of ${d}_{j{j}^{\prime }}$, the separation distance between j and j ^′. The detailed discussion about determining $\rho \left({d}_{j{j}^{\prime }}\right)$ is given in Jayaram and Baker (2010).

• III. Given θ, σ ², and τ ², η _i can be calculated as follows:

$$ {\eta}_i=\frac{1_{n_i,1}^{\prime }{C}_c^{-1}{\varepsilon}_i^{(t)}}{\frac{1}{\tau^2}+{1}_{n_i,1}^{\prime }{C}_c^{-1}{1}_{n_i,1}} $$

C _c is defined as conditional covariance metric(s) for the total residual, ${\varepsilon}_i^{(t)}=\left[{\varepsilon}_{i1}^{(t)},{\varepsilon}_{i2}^{(t)},\dots, {\varepsilon}_{i{n}_i}^{(t)}\right]$, which is the collection of total residuals at all the sites during an earthquake i. ${1}_{n_i,1}^{\prime }$ is the transpose of the column metric(s) of ones of length n _i . The above equation is valid only if the interevent residual follows a normal distribution and the intraevent residuals at multiple sites during a given earthquake jointly follow a multivariate normal distribution

• IV. Given η _i , estimate the new coefficients (θ) using a fixed effects regression algorithm for ln(Y _ij ) − η _i .

• V. Repeat steps 2, 3, and 4 until the likelihood in step 2 is maximized and estimates for the coefficient convergence.

For more detail regarding the algorithm, refer to Jayaram and Baker (2008) and Jayaram and Baker (2010).