Disaggregation of probabilistic seismic hazard and construction of conditional spectrum for China

Abstract

For mainland China, the primary obstacle in conditional spectrum (CS) based ground motion selection work is that the corresponding seismic hazard disaggregation results were not released for the China national standard GB 18306–2015 “Seismic Ground Motion Parameter Zonation Map”, which refers to the fifth-generation seismic hazard map. Therefore, this study firstly constructed a probabilistic seismic hazard map for mainland China using the three level seismicity source models as applied to produce the fifth-generation seismic hazard map. The derived peak ground acceleration (PGA) values in our seismic hazard map were basically consistent with the PGA range bins in fifth-generation seismic hazard map for most of the 34 principal cities considered. Then, a three-dimensional disaggregation scheme was performed for PGA and 5%-damped spectral acceleration (Sa) corresponding to mean return periods of 475 and 2475 years. The results clearly identified the potential source areas and corresponding fault zones that dominate the hazard of the target cities regarding different intensity measures. Based on the magnitude-longitude-latitude disaggregation results of three example cities: Xichang, Kunming, and Xi’an, approximate and exact CS were established with/without considering multiple casual earthquakes and possible strike directions of the potential source areas. The mean exact CS lies between the results of approximate CS using long and short axis GMMs. The conditional standard deviation of exact CS is approximately 1.1–1.5 times larger than the approximate CS for the periods away from the conditional period. For three example cities, hazard consistency of the spectral accelerations of the ground motion realizations were validated for matching exact CS and approximate CS. The results suggested that the geometric mean approximate CS without inflation of standard deviation is a practical choice when the target exceedance rate was not too low. Moreover, for the 34 studied cities, we tabulated the uniform hazard curve and disaggregation results for PGA and Sa values (0.2, 0.3, 0.5, 0.7, 1.0, 1.5, and 2.0 s) at mean return period of 475 and 2475 years (https://github.com/JIKUN1990/China-Seismic-Hazard-Deaggregation-34cities).

Appendix

Formula derivation for Eqs. (9) and (10)

For kth seismic source grid point, $f_{k} (x)$ is the conditional probability density function(PDF) of ln Sa(T_i) given ln Sa(T*). Supposing that there are two possible strike directions: strike direction α and strike direction β. The corresponding conditional PDFs were defined as $f_{k1}^{{}} (x)$ and $f_{k2} (x)$ respectively.

$$f_{k1} (x) = \frac{1}{{\sigma_{k1} \sqrt {2\pi } }}e^{{ - \frac{{(x - \mu_{k1} )^{2} }}{{2\sigma_{k1}^{2} }}}}$$

$$f_{k2} (x) = \frac{1}{{\sigma_{k2} \sqrt {2\pi } }}e^{{ - \frac{{(x - \mu_{k2} )^{2} }}{{2\sigma_{k2}^{2} }}}}$$

in which, $\mu_{{\ln Sa_{k1} (T_{i} )|\ln Sa(T^*)}}$ is denoted as $\mu_{k1}$, $\mu_{{\ln Sa_{k2} (T_{i} )|\ln Sa(T^*)}}$ is denoted as $\mu_{k2}$, defination of them could be found in Eqs. (6) and (7). $\sigma = \sigma_{k1} = \sigma_{k2} = \sigma_{{\ln Sa_{k} (Ti)|\ln Sa(T^*)}}$, which is defined in Eq. (8).

g(x) is The conditional PDF of $\ln Sa(Ti)|\ln Sa(T^*)$. It could be obtained via the total probability theorem form as follows.

$$\begin{aligned} g(x) & = \sum\limits_{k = 1}^{N} {w{}_{k}(f_{k1} (x)P_{k1} + } f_{k2} (x)P_{k2} ) \\ & = \sum\limits_{k = 1}^{{N_{{}} }} {w{}_{k}\left( {P_{k1} \times \frac{1}{{\sigma \sqrt {2\pi } }}e^{{ - \frac{{(x - \mu_{k1} )^{2} }}{{2\sigma^{2} }}}} + P_{k2} \times \frac{1}{{\sigma \sqrt {2\pi } }}e^{{ - \frac{{(x - \mu_{k2} )^{2} }}{{2\sigma^{2} }}}} } \right)} \\ & = \sum\limits_{k = 1}^{N} {\frac{{w{}_{k}}}{{\sigma \sqrt {2\pi } }}\left( {P_{k1} \times e^{{ - \frac{{(x - \mu_{k1} )^{2} }}{{2\sigma^{2} }}}} + P_{k2} \times e^{{ - \frac{{(x - \mu_{k2} )^{2} }}{{2\sigma^{2} }}}} } \right)} \\ \end{aligned}$$

Then $\mu_{{\ln Sa(T_{i} )|\ln Sa(T^*)}}$ could be derived as:

$$\begin{aligned} \mu_{{\ln Sa(T_{i} )|\ln Sa(T*)}} & = \int_{ - \infty }^{ + \infty } {g(x)xdx} \\ & = \sum\limits_{k = 1}^{N} {w_{k} \int_{ - \infty }^{ + \infty } {x\frac{1}{{\sigma \sqrt {2\pi } }}\left( {P_{k1} \times e^{{ - \frac{{(x - \mu_{k1} )^{2} }}{{2\sigma^{2} }}}} + P_{k2} \times e^{{ - \frac{{(x - \mu_{k2} )^{2} }}{{2\sigma^{2} }}}} } \right)dx} } \\ & = \sum\limits_{k = 1}^{N} {w_{k} \left( {\int_{ - \infty }^{ + \infty } {x\frac{1}{{\sigma \sqrt {2\pi } }}\left( {P_{k1} \times e^{{ - \frac{{(x - \mu_{k1} )^{2} }}{{2\sigma^{2} }}}} } \right)dx} + \int_{ - \infty }^{ + \infty } {x\frac{1}{{\sigma \sqrt {2\pi } }}\left( {P_{k2} \times e^{{ - \frac{{(x - \mu_{k2} )^{2} }}{{2\sigma^{2} }}}} } \right)dx} } \right)} \\ & = \sum\limits_{k = 1}^{N} {w_{k} \left[ {P_{k1} \int_{ - \infty }^{ + \infty } {x\frac{1}{{\sigma \sqrt {2\pi } }}\left( {e^{{ - \frac{{(x - \mu_{k1} )^{2} }}{{2\sigma^{2} }}}} } \right)dx} + P_{k2} \int_{ - \infty }^{ + \infty } {x\frac{1}{{\sigma \sqrt {2\pi } }}\left( {e^{{ - \frac{{(x - \mu_{k2} )^{2} }}{{2\sigma^{2} }}}} } \right)dx} } \right]} \\ & = \sum\limits_{k = 1}^{N} {w_{k} (P_{k1} \times \mu_{k1} + P_{k2} \times \mu_{k2} )} \\ \end{aligned}$$

$\sigma_{\ln Sa(T_{i})|\ln Sa(T^*)}$ could be derived as:

$$\begin{aligned} \sigma^{2}_{\ln Sa(T_{i})|\ln Sa(T^*)} & = E(x^{2} ) - E(x)^{2} \\ & = \sum\limits_{k = 1}^{N} {w_{k} } [P_{k1} \times (\sigma^{2} + \mu_{k1}^{2} ) + P_{k2} \times (\sigma^{2} + \mu_{k2}^{2} )] - E(x)^{2} \\ & = \sum\limits_{k = 1}^{N} {w_{k} } [P_{k1} \times (\sigma^{2} + \mu_{k1}^{2} ) + P_{k2} \times (\sigma^{2} + \mu_{k2}^{2} )] - \sum\limits_{k = 1}^{N} {w_{k} } (P_{k1} + P_{k2} )E(x)^{2} \\ & = \sum\limits_{k = 1}^{N} {w_{k} } [\sigma^{2} + P_{k1} (\sigma^{2} + \mu_{k1}^{2} - E(x)^{2} ) + P_{k2} (\sigma^{2} + \mu_{k2}^{2} - E(x)^{2} )] \\ \end{aligned}$$

in which

$$\begin{aligned} E(x^{2} ) & = \int_{ - \infty }^{ + \infty } {g(x)x^{2} dx} \\ & = \sum\limits_{k = 1}^{N} {w_{k} \int_{ - \infty }^{ + \infty } {\frac{1}{{\sigma \sqrt {2\pi } }}\left( {P_{k1} \times e^{{ - \frac{{(x - \mu_{k1} )^{2} }}{{2\sigma^{2} }}}} + P_{k2} \times e^{{ - \frac{{(x - \mu_{k2} )^{2} }}{{2\sigma^{2} }}}} } \right)x^{2} dx} } \\ & = \sum\limits_{k = 1}^{N} {w_{k} \left( {P_{k1} \times \int_{ - \infty }^{ + \infty } {\frac{1}{{\sigma \sqrt {2\pi } }}\left( {e^{{ - \frac{{(x - \mu_{k1} )^{2} }}{{2\sigma^{2} }}}} } \right)x^{2} dx} + P_{k2} \times \int_{ - \infty }^{ + \infty } {\frac{1}{{\sigma \sqrt {2\pi } }}\left( {e^{{ - \frac{{(x - \mu_{k2} )^{2} }}{{2\sigma^{2} }}}} } \right)x^{2} dx} } \right)} \\ & = \sum\limits_{k = 1}^{N} {w_{k} [P_{k1} \times (\sigma^{2} + \mu_{k1}^{2} ) + P_{k2} \times (\sigma^{2} + \mu_{k2}^{2} )]} \\ \end{aligned}$$