A method for joint hazard analysis of mainshocks and aftershocks based on Copula theory and BIC criterion

Abstract

The present invention provides a method for joint hazard analysis of main shocks and aftershocks based on Copula theory and BIC criterion, including: forming a main shock and aftershock database by selecting main shock and aftershock ground motion records; establishing an optimal marginal distribution model of main shock and aftershock parameters based on BIC criterion; selecting an optimal Copula function based on the empirical distributions of main shock and aftershock parameters and BIC criterion; establishing an optimal joint distribution of main shock and aftershock parameters based on Copula theory, and obtaining the Copula joint probability density function of main shock and aftershock parameters; establishing the conditional probability density function of aftershock parameters based on Copula theory; generating the conditional probability surface of aftershock parameters, and establishing the joint hazard surface of main shocks and aftershocks. The present invention takes into account the correlation between main shocks and aftershocks, and uses Copula theory to combine the marginal distributions of main shock parameters and aftershock parameters, so as to comprehensively and quantitatively evaluate the hazards of main shocks and aftershocks at the site.

Description

Technical Field

[0001] The present invention belongs to the technical field of seismic risk and earthquake damage prediction and analysis, and particularly relates to a method for joint hazard analysis of main shocks and aftershocks based on Copula theory and BIC criterion. Background Art

[0002] China is located in the Circum-Pacific Volcanic Seismic Belt. Among the numerous earthquake events that have occurred in China, there are many strong earthquake events, resulting in a large number of casualties and property damage, which has brought a huge impact on the social and economic development of China.

[0003] Existing seismic hazard analyses usually only consider the impact of one earthquake (main shock) and do not consider the potential threat of aftershocks. However, numerous earthquake events have shown that a series of aftershocks often occur after the main shock, including strong aftershocks.

[0004] Seismic hazard can better describe the exceedance probability of different intensity earthquakes occurring at an engineering site within a certain period of time, and can comprehensively consider the activity of potential seismic source areas and their contributions to the target site. However, existing joint hazard analysis methods only consider the action of one earthquake, that is, the main shock, and ignore the potential threat of aftershocks. This research defect has led to insufficient understanding of the hazards of real earthquake events, and therefore it is impossible to comprehensively and truly evaluate the seismic hazard level of engineering sites.

[0005] In view of this, it is necessary to consider aftershock factors in the seismic hazard analysis of engineering sites and comprehensively consider the hazard levels of main shock-aftershock sequences. Therefore, there is an urgent need for a method for joint hazard analysis of main shocks and aftershocks. Summary of the Invention

[0006] The purpose of the present invention is to solve the defect that only the action of one earthquake (main shock) can be considered in existing seismic hazard analyses, and to provide a method that can incorporate aftershock events into the existing seismic hazard analysis process. By using Copula theory, the main shock and aftershock parameters with non-linear correlation are considered simultaneously to form a new joint hazard analysis model that can consider both main shock events and aftershock events, providing a theoretical basis for the hazard analysis of engineering sites and applying it to the safety evaluation of engineering sites in engineering construction.

[0007] A method for joint hazard analysis of main shocks and aftershocks based on Copula theory and BIC criterion includes the following steps:

[0008] Step 1: Based on international and domestic ground motion databases, select main shock and aftershock records to form a main shock-aftershock sequence and establish a main shock-aftershock database;

[0009] Step 2: Based on the BIC criterion, select the probability distribution model that best matches the main shock parameters and aftershock parameters, and establish the optimal marginal distribution probability model of the main shock parameters and aftershock parameters;

[0010] Step 3: Based on the optimal marginal distribution model in Step 2, generate empirical distribution data, and based on the BIC criterion, select the Copula function that best matches the empirical distribution function;

[0011] Step 4: Based on the optimal marginal distribution probability model in Step 2 and the optimal Copula function in Step 3, according to Copula theory, establish the optimal joint distribution probability model of the main shock parameters and aftershock parameters, and obtain the Copula joint probability density function;

[0012] Step 5: Based on the optimal marginal distribution probability model in Step 2 and the joint probability density function in Step 4, according to Copula theory, establish the conditional probability density function of the aftershock parameters;

[0013] Step 6: Based on the conditional probability density function in Step 5, use the direct integration method to generate the conditional probability surface of the aftershock parameters;

[0014] Step 7: Based on the conditional probability surface of the aftershock parameters in Step 6, combined with the known main shock hazard analysis results of the site, use the total probability theorem to obtain the main-aftershock joint hazard surface.

[0015] Furthermore, the main-aftershock earthquake records in Step 1 are taken from the strong earthquake databases of authoritative institutions at home and abroad.

[0016] Furthermore, in Step 2, the optimal marginal distribution is identified based on the BIC criterion.

[0017] Furthermore, Step 3 specifically includes: generating empirical distribution data based on the optimal marginal distribution and identifying the optimal Copula function based on the BIC criterion.

[0018] Furthermore, Step 4 specifically includes: based on the optimal marginal distribution probability model and the optimal Copula function, according to Copula theory, calculate and obtain the joint probability density function of the main shock parameters and aftershock parameters according to the following formula, as shown in the following formula:

[0019] f MS-AS (IM MS ,IM AS )=D(u MS ,u AS )×f MS (IM MS )×f AS (IM AS )

[0020] where, f MS (IM MS ) and f AS (IM AS ) are the marginal probability density functions of the main shock parameters and aftershock parameters respectively, and D(u MS , u AS ) is the probability density function of the optimal Copula function, which can be solved by the following formula:

[0021]

[0022] where, C(u MS , u AS ) is the probability distribution function of the optimal Copula function.

[0023] Furthermore, the specific steps of step 5 include:

[0024] Based on the obtained optimal Copula probability density function and the optimal marginal distribution function of the aftershock parameters, establish the conditional probability density function of the aftershock parameters according to the Copula theory, as shown in the following formula:

[0025] f AS|MS (IM AS |IM MS ) = D(u MS , u AS ) · f AS (IM AS )

[0026] where, fAS|MS(IMAS|IMMS) is the conditional probability density function of the aftershock.

[0027] Furthermore, the specific steps of step 6 include:

[0028] Based on the obtained conditional probability density function of the aftershock parameters, use the direct integration method in mathematics to integrate the conditional probability density function to obtain the conditional probability surface of the aftershock parameters, as shown in the following formula:

[0029] P[IM AS |IM MS = ∫f AS|MS (IM AS |IM MS )dIM AS

[0030] where, P[IM AS |IM MS is the conditional probability surface of the aftershock parameters.

[0031] Furthermore, the specific steps of step 7 include:

[0032] Based on the conditional probability surface of the aftershock parameters obtained, combined with the existing main shock hazard analysis results of the target site, and using the total probability theorem, the combined hazard surface of the main shock and aftershocks is obtained, as shown in the following formula:

[0033] MAR(IM MS ; IM AS ) = MAR(IM MS ) · P[IM AS |IM MS )

[0034] In the formula, MAR(IM MS ; IM AS ) is the combined hazard surface of the main shock and aftershocks, and MAR(IM MS ) is the main shock hazard curve (obtained based on the existing main shock hazard analysis results of the target site).

[0035] Advantages of the present invention: The present invention expands the traditional seismic hazard function from a single seismic event (only considering the main shock event) to two seismic events, can simultaneously consider the main shock event and the aftershock event, expands the traditional single-event seismic vulnerability analysis method, combines the main shock parameters and aftershock parameters using the Copula theory, and can comprehensively and quantitatively evaluate the main shock and aftershock seismic hazard level of the engineering site; solves the problem of non-linear correlation between the main shock parameters and aftershock parameters that cannot be considered in the traditional seismic hazard analysis process, and obtains a three-dimensional surface result. The main shock and aftershock combined hazard analysis method based on the Copula theory of the present invention can be used to establish a hazard model of the engineering site under the combined action of the main shock and aftershocks, evaluate the main shock and aftershock hazard level of the engineering site, provide a theoretical basis for the safety assessment of the engineering site, and can be used for the safety evaluation work of the engineering site in engineering construction. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 is a flowchart of a main shock and aftershock combined hazard analysis method based on the Copula theory and the BIC criterion of the present invention;

[0037] Figure 2 is a magnitude-distance distribution diagram of the main shock and aftershock sequence ground motions selected by the present invention;

[0038] Figure 3 is a diagram of the identification result of the optimal marginal distribution of the main shock parameters and aftershock parameters in an embodiment of the present invention;

[0039] In the figure, (a) is the identification of the main shock parameters and their optimal distribution, and (b) is the identification of the aftershock parameters and their optimal distribution;

[0040] Figure 4 is a diagram of the empirical distribution data of the main shock parameters and aftershock parameters in an embodiment of the present invention;

[0041] Figure 5 is the optimal joint distribution probability model of the main shock parameters and aftershock parameters in the present invention;

[0042] Figure 6 is the conditional probability surface diagram of the aftershock parameters in the present invention;

[0043] Figure 7 is the joint hazard surface diagram of the main shock and aftershocks in the present invention. Detailed implementation manners

[0044] The following combines the accompanying drawings and embodiments to further describe in detail the specific implementation manners of the present invention. The following embodiments are used to illustrate the present invention, but are not used to limit the scope of the present invention.

[0045] As Figure 1 shown, a method for analyzing the joint vulnerability of main shocks and aftershocks based on the Copula theory in this embodiment includes:

[0046] Step S1, based on the international and domestic ground motion databases, select the main shock and aftershock records to form the main shock-aftershock sequence and form the main shock-aftershock database;

[0047] Step S2, based on the BIC criterion, select the probability distribution model that best matches the main shock parameters and aftershock parameters, and establish the optimal marginal distribution probability models of the main shock parameters and aftershock parameters;

[0048] Step S3, based on the optimal marginal distribution model in Step 2, generate empirical distribution data, and based on the BIC criterion, select the Copula function that best matches the empirical distribution function;

[0049] Step S4, based on the optimal marginal distribution probability models in Step 2 and the optimal Copula function in Step 3, according to the Copula theory, establish the optimal joint distribution probability model of the main shock parameters and aftershock parameters, and obtain the Copula joint probability density function;

[0050] Step S5, based on the optimal marginal distribution probability models in Step 2 and the joint probability density function in Step 4, according to the Copula theory, establish the conditional probability density function of the aftershock parameters;

[0051] Step S6, based on the conditional probability density function in Step 5, use the direct integration method to generate the conditional probability surface of the aftershock parameters;

[0052] Step S7, based on the conditional probability surface of the aftershock parameters in Step 6, combined with the known main shock hazard analysis results of the site, use the total probability theorem to obtain the joint hazard surface of the main shock and aftershocks.

[0053] The main-aftershock combined hazard analysis method provided in this embodiment can simultaneously consider the main shock event and the aftershock event, expand the traditional single-event seismic vulnerability analysis method, and use the Copula theory to combine the main shock parameters and the aftershock parameters, so as to comprehensively and quantitatively evaluate the main-aftershock seismic hazard level of the engineering site.

[0054] A main-aftershock combined vulnerability analysis method based on the Copula theory, and the specific steps of this method are as follows:

[0055] (1). Based on the international and domestic ground motion databases, select the main-aftershock sequence ground motion.

[0056] 1. Establishment of the main-aftershock ground motion database: Based on the earthquake record databases of the Pacific Earthquake Engineering Research Center in the United States, the Japan Strong Motion Network Center, and the China National Strong Motion Network Center, collect and sort out the existing main-aftershock records.

[0057] 2. Select the main-aftershock sequence ground motion through specific selection rules. The main-aftershock sequence ground motion selection rules in this example are as follows:

[0058] 1) In this example, only the aftershock with the largest magnitude after the main shock is selected to match the corresponding main shock to form the main-aftershock sequence;

[0059] 2) The main shock and aftershock records should come from the same ground motion station;

[0060] 3) The station should be built on free field;

[0061] 4) All earthquake events should be shallow-focus earthquakes.

[0062] Through the above 4 ground motion selection rules, a total of 662 main-aftershock sequence ground motions that meet the conditions are finally obtained, and a main-aftershock ground motion record database is established based on them. The magnitude-distance distribution diagram of the selected main-aftershock sequence ground motion is as Figure 2 shown.

[0063] (2). Establish the optimal marginal distribution probability model of the main shock parameters and the aftershock parameters.

[0064] 1. In this embodiment, the peak ground acceleration of the main shock and the peak ground acceleration of the aftershock are respectively selected as the main shock parameter and the aftershock parameter, and the normal distribution, the lognormal distribution, and the generalized extreme value distribution are selected as the alternative marginal distribution forms;

[0065] 2. Based on the identification of the optimal marginal distribution by the BIC criterion, and then establish the optimal marginal distributions of the main shock parameters and the aftershock parameters. The cumulative probability distribution histograms of the main shock parameters and the aftershock parameters, the cumulative probability curves of the alternative marginal distributions, and their BIC values are as Figure 3 shown.

[0066] (3). Select the Copula function that best matches the empirical distribution function

[0067] 1. Generate the empirical distribution data of the corresponding main shock parameters and aftershock parameters based on the optimal marginal distributions of the obtained main shock parameters and aftershock parameters, as Figure 4 shown;

[0068] According to the data characteristics of the empirical distribution data, five Copula functions are selected as candidate functions, namely Gaussian copula, Plackett copula, Clayton copula, Frank copula, and Gumbel copula. Based on the BIC criterion, the optimal Copula function is identified. In this embodiment, the BIC values of the five Copula functions are -586, -584, -556, -713, and -439 respectively.

[0069] (4). Based on the Copula theory, establish the optimal joint distribution probability model of the main shock parameters and aftershock parameters:

[0070] 1. Substitute the obtained optimal marginal distribution of the main shock parameters, the optimal marginal distribution of the aftershock parameters, and the optimal Copula function into the following formula to solve the optimal joint distribution probability model of the main shock parameters and aftershock parameters, as Figure 5 shown.

[0071] f MS-AS (IM MS ,IM AS )=D(u MS ,u AS )×f MS (IM MS )×f AS (IM AS )

[0072] In the formula, f MS (IM MS ) and f AS (IM AS ) are the marginal probability density functions of the main shock parameters and aftershock parameters respectively, and D(u MS ,u AS ) is the probability density function of the optimal Copula function.

[0073] 2. Differentiate the probability distribution function of the optimal Copula function to obtain the probability density function of the optimal Copula function, which can be solved by the following formula:

[0074]

[0075] In the formula, C(uMS , u AS ) is the probability distribution function of the optimal Copula function.

[0076] (V). Establish the conditional probability density function of the aftershock parameters by using the obtained optimal Copula probability density function and the optimal marginal distribution function of the aftershock parameters. According to the Copula theory, the conditional probability density function of the aftershock parameters is established as follows:

[0077] f AS|MS (IM AS |IM MS ) = D(u MS , u AS ) · f AS (IM AS )

[0078] In the formula, f AS|MS (IM AS |IM MS ) is the conditional probability density function of the aftershock.

[0079] (VI). Based on the obtained conditional probability density function of the aftershock parameters, use the direct integration method in mathematics to integrate the conditional probability density function as follows, and obtain the conditional probability surface of the aftershock parameters, as Figure 6 shown.

[0080] P[IM AS |IM MS = ∫f AS|MS (IM AS |IM MS )dIM AS

[0081] In the formula, P[IM AS |IM MS is the conditional probability surface of the aftershock parameters.

[0082] (VII). Based on the obtained conditional probability surface of the aftershock parameters, combine the existing main shock hazard analysis results of the target site, substitute them into the total probability formula as follows, and obtain the main - aftershock combined hazard surface, as Figure 7 shown.

[0083] MAR(IM MS ; IM AS ) = MAR(IM MS ) · P[IM AS |IM MS

[0084] In the formula, MAR(IM MS ; IM AS ​) is the combined hazard surface of the main shock and aftershocks, MAR(IM MS ) is the main shock hazard curve (obtained based on the existing main shock hazard analysis results of the target site).

[0085] The main shock-aftershock combined hazard analysis method based on the Copula theory and BIC criterion provided in this embodiment extends the traditional seismic hazard function from a single seismic event (only considering the main shock event) to two seismic events (considering both the main shock event and the aftershock event), solves the problem of the non-linear correlation between the main shock parameters and aftershock parameters that cannot be considered in the traditional seismic hazard analysis process, and obtains a three-dimensional surface result.

[0086] The main shock-aftershock combined hazard analysis method based on the Copula theory provided in this embodiment can be used to establish a hazard model for engineering sites under the combined action of the main shock and aftershocks, evaluate the main shock-aftershock hazard level of engineering sites, provide a theoretical basis for the safety assessment of engineering sites, and can be used for the safety evaluation of engineering sites in engineering construction.

[0087] The above is only the preferred embodiment of the present invention and is not used to limit the present invention. It should be noted that for those of ordinary skill in the art, without departing from the technical principle of the present invention, several improvements and modifications can be made, and these improvements and modifications should also be regarded as the protection scope of the present invention.

Claims

1. A main-aftershock combined hazard analysis method based on Copula theory and BIC criterion, characterized in that It includes the following steps: Step 1: Based on the international and domestic ground motion databases, select the main shock and aftershock records to form the main shock-aftershock sequence and establish the main shock-aftershock database; Step 2: Based on the BIC criterion, select the probability distribution model that best matches the main shock parameters and aftershock parameters, and establish the optimal marginal distribution probability model of the main shock parameters and aftershock parameters; Step 3: Based on the optimal marginal distribution model in Step 2, generate empirical distribution data, and based on the BIC criterion, select the Copula function that best matches the empirical distribution function; Step 4: Based on the optimal marginal distribution probability model in Step 2 and the optimal Copula function in Step 3, according to the Copula theory, establish the optimal joint distribution probability model of the main shock parameters and aftershock parameters, and obtain the Copula joint probability density function; Step 5: Based on the optimal marginal distribution probability model in Step 2 and the joint probability density function in Step 4, according to the Copula theory, establish the conditional probability density function of the aftershock parameters; Step 6: Based on the conditional probability density function in Step 5, use the direct integration method to generate the conditional probability surface of the aftershock parameters; Step 7: Based on the conditional probability surface of the aftershock parameters in Step 6, combined with the known main shock hazard analysis results of the site, use the total probability theorem to obtain the main shock-aftershock joint hazard surface; 2. The main-aftershock combined hazard analysis method based on Copula theory and BIC criterion according to claim 1, characterized in that, The specific content of Step 4 includes: Based on the optimal marginal distribution probability model and the optimal Copula function, according to the Copula theory, calculate and obtain the joint probability density function of the main shock parameters and aftershock parameters according to the following formula, as shown in the following formula: f MS-AS (IM MS ,IM AS ) = D(u MS , u AS ) × f MS (IM MS ) × f AS (IM AS ) where f MS (IM MS ) and f AS (IM AS ) are the marginal probability density functions of the main shock parameters and aftershock parameters respectively, and D(u MS , u AS ) is the probability density function of the optimal Copula function, which can be solved by the following formula: where \(C(u MS , u AS )\) is the probability distribution function of the optimal Copula function.

3. A main-aftershock combined hazard analysis method based on Copula theory and BIC criterion according to claim 1, characterized in that, The specific content of Step 5 includes: Based on the obtained optimal Copula probability density function and the optimal marginal distribution function of the aftershock parameters, establish the conditional probability density function of the aftershock parameters according to the Copula theory, as shown in the following formula: f AS|MS (IM AS |IM MS ) = D(u MS , u AS ) · f AS (IM AS ) where, f AS|MS (IM AS |IM MS ) is the conditional probability density function of aftershocks.

4. A main-aftershock combined hazard analysis method based on Copula theory and BIC criterion according to claim 1, characterized in that, The specific content of Step 6 includes: Based on the obtained conditional probability density function of the aftershock parameters, use the direct integration method in mathematics to integrate the conditional probability density function to obtain the conditional probability surface of the aftershock parameters, as shown in the following formula: P[IM AS |IM MS = ∫f AS|MS (IM AS |IM MS ) dIM AS where P[IM AS |IM MS is the conditional probability surface of the aftershock parameter.

5. A main-aftershock combined hazard analysis method based on Copula theory and BIC criterion according to claim 1, characterized in that The specific content of Step 7 includes: Based on the obtained conditional probability surface of the aftershock parameters, combined with the existing main shock hazard analysis results of the target site, use the total probability theorem to obtain the main shock-aftershock joint hazard surface, as shown in the following formula: MAR(IM MS ; IM AS ) = MAR(IM MS ) · P[IM AS | IM MS ​ where MAR(IM MS ; IM AS ) is the main-aftershock combined hazard surface, and MAR(IM MS ) is the main shock hazard curve.

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