Probabilistic seismic hazard analysis based on monthly maximum PGA distribution in three cities in southwest China

2.1. Historical earthquake events

The data of historical earthquake events are recorded in historical earthquake catalog as surface wave magnitude ($Ms$) of 3 cities in southwest China, i.e., Chongqing, Chengdu, and Kunming. Historical earthquake events from 1920 to 2020 with $Ms$ greater than or equal to 3 and epicentral distances within 400 km are selected for analysis. The temporal and spatial distribution of earthquake events are shown in Fig. 2 and 3 respectively.

Fig. 1Technical route of probabilistic seismic hazard analysis based on PGA

Fig. 2Temporal distribution of earthquake events from 1920 to 2020

a) Chongqing

b) Chengdu

c) Kunming

Fig. 3Spatial distribution of earthquake events with epicentral distances within 400 km

a) Chongqing

b) Chengdu

c) Kunming

2.2. Attenuation relation of seismic parameters

PGA data can be back-calculated from $Ms$ data recorded in the earthquake catalog using the attenuation relation of seismic parameters. Generally, it is believed that the relations between seismic motion parameters and magnitude satisfies following function [5]:

where, $b_0$ denotes regression parameter, $\epsilon$ denotes random error, $R$ denotes epicentral distance, $M$ denotes magnitude of earthquakes. In this paper, PGA data is calculated based on the attenuation relation proposed by Tatsuo [6]:

where, $a$, $b$, $c$, $d$, $e$ denotes regression parameters. The empirical formula proposed by Wason [7] can be used for conversion between $Ms$ and $Mw$:

2.3. Monthly maximum PGA distribution

In order to fit the better model of PGA distribution, logarithmic calculation for PGA ($ln$ PGA) is taken. PGA distribution is studied using an equal time step method with a time step of 1 month, i.e., monthly maximum PGA values are statistically analyzed. The temporal distribution of the monthly maximum $ln$ PGA values of historical earthquake events from 1920 to 2020 is shown in Fig. 4. In Fig. 5, it can be considered that ln PGAs at the three cities satisfy with normal distributions respectively.

Fig. 4Temporal distribution of monthly maximum ln PGAs

a) Chongqing

b) Chengdu

c) Kunming

Fig. 5Comparisons between ln PGA distribution and normal distribution

a) Chongqing

b) Chengdu

c) Kunming

Then, the Kolmogorov Smirnov test (K-S test) on the fitting results is performed. The K-S test can be used to test whether a set of random numbers with an unknown distribution comes from a specific continuous distribution function $F_0$, and KS-statistic is:

where, $k$ denotes the $k$th quantile data in the sample, and $n$ denotes the sample size [8]. Under the condition of a significant level $\alpha =$ 0.05, all the sample data of three cities satisfy with normal distribution.

2.4. Frequency-PGA relation

Generally, there is an empirical formula [9] for the relation between the number of earthquakes ($N$) and the magnitude ($M$):

Similarly, it can be seen that the relation between $ln\left(lnN\right)$ and PGA satisfies a linear relation in Fig. 6, namely:

where, $p_1$, $p_2$ denote regression parameters. And the rationality of the regression results of $N$-PGA relations can be tested by coefficient of determination ($R^2$), which reflects the percentage of the fluctuation of the dependent variable that can be described by the independent variable [10]:

where, SSR and SSE denote sum of squares for regression and sum of squares for error, respectively. The fitting results for all three cities have 0 $<R^2<$ 1 and are close to 1, demonstrating good regression results.

Fig. 6ln(lnN)-PGA relations

a) Chongqing

b) Chengdu

c) Kunming

After getting $N$ in regression, the frequency ($v$) within time $t$ can be obtained:

where, $v_i$ can be $v_0$ or $v_u$, respectively denote the calculated result of $N_i$ corresponding to PGA = 0 or PGA = $e^u$. $u$ denotes the mean value in $ln$ PGA Normal Distribution, which is also the mode of $ln$ PGA.

2.5. Poisson model

It is generally believed that the probability of $N$ earthquakes occurring within time ($t$) satisfies the Poisson model [2]:

where, $v$ denotes average frequency within $t$. And average frequency ($v$) of PGA $>x$ can be calculated from Eq. (11):

where $P(PGA>x|u,\delta )$ and $v_i$ can be calculated from the results of $ln$ PGA distributions and frequency-PGA relation, respectively. Probability of seismic event PGA $>x$ occurring within $t$, that is, the exceedance probability for each given PGA ($Pr$), can be calculated from Eq. (12):

And recurrence interval ($T$) can be calculated from Eq. (13) [3]:

2.6. Year exceedance probability and recurrence interval

The parameters calculated from regression are shown in Table 1. The annual exceedance probability v.s. PGA curves and recurrence interval v.s. PGA curves of the three cities can be obtained by these parameters and Eqs. (12) and (13). The PGA corresponding to the annual exceedance probability Pr of 0.0021 (i.e. 475 y of recurrence interval $T$) of Chongqing, Chengdu and Kunming are 48.08 gal, 103.0 gal and 116.0 gal respectively.

Compared with the values of SIZM (in Table 2), the calculation results of Chongqing and Chengdu are close to those of the Seismic Zoning Map, while the calculation results of Kunming are relatively small. This may be due to the too large epicentral distance threshold selected, leading to the excessive consideration of the proportion of small earthquakes in the surrounding area. The influence of the epicentral distance threshold will be discussed in the following text. Meanwhile, the results calculated using $v_0$ are slightly greater than the results calculated by $v_u$, and the results calculated using $v_0$ are closer to the results of SIZM of China.