Post-earthquake time-varying aftershock risk analysis method based on aftershock cluster

Abstract

The invention relates to the technical field of earthquake risk and earthquake damage prediction analysis, in particular to a post-earthquake time-varying aftershock risk analysis method based on an aftershock cluster, and the method comprises the following steps: employing maximum likelihood estimation to recognize parameters of an ETAS model through earthquake directory information; based on an ETAS model, generating a large number of simulated aftershock cluster sequences; converting the magnitude information of the simulated aftershock cluster sequence into seismic oscillation intensity through a seismic oscillation prediction equation; selecting a main aftershock sequence, solving aftershock vulnerability parameters in different damage states by adopting maximum likelihood estimation, and carrying out damage state related aftershock vulnerability analysis; and carrying out post-earthquake time-varying aftershock risk analysis. According to the method, the ETAS model is used for simulating the aftershock activity model of the site, a post-earthquake risk analysis model capable of considering the action of multiple aftershocks along with time after the main earthquake is formed, and the defect that only single main earthquake and single aftershock can be considered in existing earthquake risk analysis is overcome.

Description

Technical Field

[0001] The present invention relates to the technical field of earthquake risk and earthquake damage prediction and analysis, and in particular to a post-earthquake time-varying aftershock risk analysis method based on aftershock clusters. Background Art

[0002] my country, located in the Pacific Ring of Fire, is prone to earthquakes. Historically, earthquakes have caused significant casualties and property losses. In past earthquakes, a strong main shock was often followed by a large number of aftershocks, posing further threats to life and property.

[0003] Extensive historical earthquake data indicates that aftershocks often exhibit distinct "clustering characteristics" in both time and space. This means that multiple aftershocks occur within a short time interval near the fault where the mainshock occurred. Because the time interval between aftershocks and the mainshock is typically short, structures damaged by the mainshock generally have insufficient time to be repaired and reinforced. Consequently, aftershocks can cause significant incremental damage. Structures that experience incremental damage from aftershocks will in turn suffer further incremental damage from subsequent aftershocks. This continuous iteration of incremental structural damage in an aftershock environment causes the structural damage state to continuously evolve over time, significantly impacting the risk assessment of aftershocks.

[0004] Traditional mainshock and aftershock risk analysis only considers the risks associated with a single mainshock and a single aftershock, which presents significant flaws. To rationally account for the cumulative damage effects of aftershock clusters in post-earthquake risk assessment, it is necessary to shift from traditional mainshock and aftershock risk analysis, which considers a single mainshock and a single aftershock, to a post-earthquake time-varying aftershock risk analysis, which considers a single mainshock and multiple aftershocks occurring over time.

[0005] In view of this, it is necessary to consider the impact of aftershock clusters in the earthquake risk analysis of engineering sites and develop a post-earthquake time-varying risk analysis method based on aftershock clusters. Summary of the Invention

[0006] To address the drawback of existing earthquake risk analysis that only a single mainshock and a single aftershock can be considered, the present invention proposes a post-earthquake time-varying aftershock risk analysis method based on aftershock clusters. The ETAS model is used to simulate the aftershock activity model of the site, forming a post-earthquake risk analysis model that can consider the effects of multiple aftershocks over time after the mainshock. This model provides a theoretical basis for risk analysis of engineering sites and can be applied to safety assessments in engineering construction.

[0007] To achieve the above object, the present invention adopts the following technical solutions:

[0008] A post-earthquake time-varying aftershock risk analysis method based on aftershock clustering includes the following steps:

[0009] Step 1: Based on international and domestic earthquake catalog databases, the parameters of the ETAS model are identified using maximum likelihood estimation;

[0010] Step 2: Based on the ETAS model, a large number of simulated aftershock cluster sequences are generated using the inverse transformation method and the acceptance-rejection method;

[0011] Step 3: Based on the simulated aftershock cluster sequence in step 2, the magnitude information of the simulated aftershock cluster sequence is converted into ground motion intensity through the ground motion prediction equation;

[0012] Step 4: Select a mainshock-aftershock sequence from international and domestic earthquake motion databases. Generate a cloud map of mainshock intensity-mainshock damage-aftershock intensity-aftershock damage through nonlinear time history analysis. Group the cloud map by damage state after the mainshock. Use maximum likelihood estimation to solve for aftershock vulnerability parameters under different damage states, and conduct damage state-dependent aftershock vulnerability analysis.

[0013] Step 5: Based on the results of steps 2, 3, and 4, conduct a post-earthquake time-varying aftershock risk analysis.

[0014] Preferably, the step 1 specifically includes:

[0015] The earthquake catalog used was taken from the earthquake catalog database of authoritative institutions at home and abroad. The maximum likelihood estimation method was used to identify the parameters of the ETAS model. The formula is:

[0016]

[0017] Where: θ is the parameter of the ETAS model to be estimated, θ = {μ, K0, α, c, p}, lnL(θ) is the log-likelihood function; λ(t n |H t ) is t n The earthquake occurrence rate at the time, H t is the aftershock cluster sequence within time t after the main shock, n represents the nth earthquake event in the aftershock cluster sequence, N S Indicates that the selected aftershock cluster sequence contains a total of N S An earthquake event.

[0018] Preferably, the step 2 specifically includes:

[0019] Assume that the simulation time period is [0,t end ], the simulated aftershock sequence to be generated is H t , there is always a positive number B, let λ(t|H t )≤B=U<λ(t n |H t )For {t∈[t n ,tn+1 ], n=1…N} are all valid, then when the main shock magnitude is m1, the algorithm for generating the simulated aftershock sequence is as follows:

[0020] 1) Let t = 0, i = 0, H t ={m1,0};

[0021] 2) Let B = λ(t n |H t ), according to the value of B, determine the time interval dt, dt satisfies the exponential distribution dt ~ exp(B);

[0022] 3) Generate a random variable U from the uniform distribution [0,1];

[0023] 4) If U<λ(t+dt|H t+dt ) / B, then let t=t+dt, i=i+1, ti=t+dt, mi=FM -1 -(U), and let H t =H t ∪{m i ,t i Otherwise, simply let t = t + dt; where FM(·) is the magnitude distribution obtained according to the GR relationship, and its expression is:

[0024]

[0025] Where, β = b / log10e, b is the GR relationship parameter;

[0026] 5) Return to step 2 until t>t end ;

[0027] 6) Output simulated aftershock sequence H t , where t = t end .

[0028] Preferably, in step 3, the earthquake magnitude information in the aftershock cluster sequence is converted into earthquake intensity by using an earthquake motion prediction equation.

[0029] Preferably, the step 4 specifically includes:

[0030] The selected main shock and aftershock ground motion records were taken from the ground motion databases of international and domestic authoritative institutions;

[0031] The mainshock and aftershock ground motion records were amplitude modulated as ground motion input, and nonlinear time history analysis was performed in OpenSEEs to obtain a cloud map of mainshock intensity-mainshock damage-aftershock intensity-aftershock damage, and the data were grouped according to the damage status after the mainshock.

[0032] The maximum likelihood estimation is used to perform maximum likelihood estimation on the grouped cloud map to solve the damage state related aftershock vulnerability parameters and draw the damage state related aftershock vulnerability curve. The likelihood function is shown as follows

[0033]

[0034] Where m R and β R is the median and logarithmic standard deviation of the earthquake vulnerability function, N Cloud is the number of sample points in the cloud map; x i For a sample point in the cloud map, if it reaches a certain limit state L S , then the corresponding Bernoulli random variable Y is y i =1, otherwise y i =0, P(D≥L S ,I M =x i ) is the aftershock vulnerability probability matrix of damage state transfer, that is, when the aftershock intensity is I M ,=x i Under the condition of L S probability.

[0035] Preferably, the step 5 specifically includes:

[0036] Based on the obtained simulated aftershock sequence and damage state-related aftershock vulnerability curve, the post-earthquake time-varying aftershock risk analysis is performed as shown in the following formula:

[0037]

[0038] Where N LS is the number of structural damage states, D Si represents the i-th damage state, P(D Si |D Sj ,i=0…N LS ,j=0…i,z n ) represents t n At the moment, the state transition risk probability matrix of the structure can be expressed as

[0039] P(D Si |D Sj ,i=0…N LS ,j=0…i,z n )=P F (D Si |D Sj ,i=0…N LS ,0=1…i,I M,n =x)

[0040] ×P(IM,n =x|m n =m,r n =r)

[0041] Where, I M,n is the ground motion intensity corresponding to the nth earthquake event, r n is the epicenter distance of the nth earthquake event, P F (D Si |D Sj ,i=0…N LS ,j=0…i,I M,n =x) is the aftershock vulnerability probability matrix of damage state transfer, that is, when the aftershock intensity is I M,n =x, in the damaged state D Sj The structure reaches the damaged state D Si The probability of P(I M,n =x|m n =m,r n =r) is the earthquake motion prediction equation.

[0042] Compared with the prior art, the present invention has at least the following beneficial effects:

[0043] The post-earthquake time-varying aftershock risk analysis method based on aftershock clusters provided by the present invention extends the traditional main-aftershock risk analysis of a single mainshock plus a single aftershock to a post-earthquake time-varying aftershock risk analysis that considers the mainshock and the aftershock clusters formed by a large number of aftershocks after the earthquake. A large number of simulated aftershock cluster sequences are generated using the ETAS model. Based on the damage state-related aftershock vulnerability analysis, a comprehensive and quantitative evaluation of the post-earthquake time-varying aftershock risk level of the engineering site can be achieved. The relationship between the post-earthquake risk and time that cannot be considered in the traditional main-aftershock risk analysis process is solved, and a risk analysis result with time as the coordinate system is obtained. The analysis method of the present invention can be used to establish a post-earthquake time-varying risk model for an engineering site in a complex aftershock environment after the mainshock occurs, evaluate the post-earthquake risk level of the engineering site that varies with time, provide a theoretical basis for the safety assessment of the engineering site, and can be used for the safety assessment of the engineering site in engineering construction. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 This is a flow chart of a post-earthquake time-varying aftershock risk analysis method based on aftershock clustering according to the present invention;

[0045] Figure 2 is a distribution map of earthquake catalogs selected in one embodiment of the present invention;

[0046] Figure 3 is a magnitude-time distribution diagram of a simulated aftershock sequence generated in one embodiment of the present invention;

[0047] Figure 4A cloud diagram of mainshock intensity-mainshock damage-aftershock intensity-aftershock damage in one embodiment of the present invention;

[0048] Figure 5 This is a cloud diagram of aftershock intensity-aftershock damage grouped by main shock damage status in one embodiment of the present invention; in the figure, a) D Si =D S0 ; b) D Si =D S1 ; c) D Si =D S2 ;d)D Si =D S3 ;

[0049] Figure 6 is a diagram of aftershock vulnerability curves related to damage states in one embodiment of the present invention; in the figure, a) D Si =D S0 ; b) D Si =D S1 ; c) D Si =D S2 ;d)D Si =D S3 ;

[0050] Figure 7 This is a graph showing the results of a time-varying aftershock risk analysis after an earthquake in one embodiment of the present invention; DETAILED DESCRIPTION

[0051] In order to enable those skilled in the art to better understand the technical solutions in the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments of the present invention.

[0052] Please refer to Figure 1 As shown, this embodiment provides a post-earthquake time-varying aftershock risk analysis method based on aftershock clusters, including:

[0053] Step S1, based on international and domestic earthquake catalog databases, the parameters of the ETAS model are identified using maximum likelihood estimation;

[0054] Step S2, based on the ETAS model parameters obtained in step S1, using the inverse transformation method and the acceptance-rejection method, a large number of simulated aftershock cluster sequences are generated using the ETAS model;

[0055] Step S3, based on the simulated aftershock cluster sequence of step S2, converting the magnitude information of the simulated aftershock cluster sequence into ground motion intensity through a ground motion prediction equation;

[0056] Step S4: Select a mainshock-aftershock sequence from international and domestic earthquake motion databases, generate a cloud map of mainshock intensity-mainshock damage-aftershock intensity-aftershock damage through nonlinear time history analysis, group the cloud map by damage state after the mainshock, and use maximum likelihood estimation to solve for aftershock vulnerability parameters under different damage states, and perform damage state-dependent aftershock vulnerability analysis;

[0057] Step S5, performing a post-earthquake time-varying aftershock risk analysis based on the results of steps S2, S3, and S4;

[0058] The post-earthquake time-varying aftershock risk analysis method provided in this embodiment can consider the aftershock clusters formed by a large number of aftershocks after the main shock occurs, expand the traditional earthquake risk analysis method of a single main shock plus a single aftershock, use the ETAS model to generate a large number of simulated aftershock cluster sequences, and based on the damage state-related aftershock vulnerability analysis, it can achieve a comprehensive quantitative evaluation of the post-earthquake time-varying aftershock risk level of the engineering site.

[0059] The specific steps of this method are:

[0060] (1) Based on international and domestic earthquake catalog databases, the parameters of the ETAS model are identified using maximum likelihood estimation.

[0061] 1. Establishment of earthquake catalog database: Based on the earthquake catalog databases of Southern California Earthquake Data Center, European Union Institute of Ocean and Experimental Geophysics and China National Strong Motion Network Center, we collected and sorted out the existing earthquake catalogs, such as Figure 2 shown.

[0062] 2. Use the maximum likelihood estimation method to identify the parameters of the ETAS model. The formula is:

[0063]

[0064] Where: θ is the parameter of the ETAS model to be estimated, θ={μ,K0,α,c,p}, lnL(θ) is the log-likelihood function. λ(t n |H t ) is t n The earthquake occurrence rate at the time, H t is the aftershock cluster sequence within time t after the main shock, n represents the nth earthquake event in the aftershock cluster sequence, N S Indicates that the selected aftershock cluster sequence contains a total of N S An earthquake event.

[0065] (2) Generate a large number of simulated aftershock cluster sequences.

[0066] 1. This example is based on the obtained ETAS model parameters and assumes that the simulation time period is [0, t end], the simulated aftershock sequence to be generated is H t There is always a positive number B, let λ(t|H t )≤B=U<λ(t n |H t )For {t∈[t n ,t n+1 ], n=1…N} are all valid, then when the main shock magnitude is m1, the algorithm for generating the simulated aftershock sequence is as follows:

[0067] 1) Let t = 0, i = 0, H t ={m1,0};

[0068] 2) Let B = λ(t n |H t ), according to the value of B, determine the time interval dt, dt satisfies the exponential distribution dt ~ exp(B);

[0069] 3) Generate a random variable U from the uniform distribution [0,1];

[0070] 4) If U<λ(t+dt|H t+dt ) / B, then let t=t+dt, i=i+1, t i =t+dt,m i =F M -1 (U), and let H t =H t ∪{m i ,t i Otherwise, just let t = t + dt. M (·) is the magnitude distribution obtained according to the GR relationship, and its expression is:

[0071] 5) Return to step 2 until t>t end ;

[0072] 6) Output simulated aftershock sequence H t , where t = t end .

[0073] Through the above 6 steps, a simulated aftershock sequence is generated, and the magnitude-time distribution diagram is as follows: Figure 3 shown.

[0074] (3) The magnitude information of the simulated aftershock cluster sequence is converted into seismic intensity through the seismic motion prediction equation.

[0075] 1. Based on the obtained simulated aftershock cluster sequence, the magnitude information of the simulated aftershock cluster sequence is converted into ground motion intensity through the ground motion prediction equation;

[0076] Since the ASK14 model can take into account the influence of aftershock earthquake motions, the ASK14 model is selected as the earthquake prediction equation model in this embodiment.

[0077] (4) Aftershock vulnerability analysis related to damage status.

[0078] 1. Establishment of a mainshock and aftershock earthquake motion database: Based on the earthquake record databases of the Pacific Earthquake Engineering Research Center of the United States, the Japan Strong Earthquake Network, and the China National Strong Earthquake Network Center, existing mainshock and aftershock records are collected and organized.

[0079] 2. The main shock and aftershock earthquake motion records are amplitude modulated as earthquake motion input, and nonlinear time history analysis is performed in OpenSEEs to obtain the main shock intensity-main shock damage-aftershock intensity-aftershock damage cloud map, such as Figure 4 As shown in Figure 2, the groups were grouped according to the damage status after the main shock, as shown in Figure 2. Figure 5 As shown;

[0080] 3. Use maximum likelihood estimation to perform maximum likelihood estimation on the grouped cloud map to solve the damage state-related aftershock vulnerability parameters and draw the damage state-related aftershock vulnerability curve. The likelihood function is shown as follows:

[0081]

[0082] 4. In the formula, m R and β R is the median and logarithmic standard deviation of the earthquake vulnerability function, N Cloud is the number of sample points in the cloud map; x i For a sample point in the cloud map, if it reaches a certain limit state L S , then the corresponding Bernoulli random variable Y is y i =1, otherwise y i =0, P(D≥L S ,I M =x i ) is the aftershock vulnerability probability matrix of damage state transfer, that is, when the aftershock intensity is I M ,=x i Under the condition of L S probability.

[0083] 5. Based on the obtained damage state-related aftershock vulnerability parameters, draw the damage state-related aftershock vulnerability curve, such as Figure 6 shown.

[0084] (V) Based on the obtained simulated aftershock sequence and the damage state aftershock vulnerability analysis results, the full probability formula is introduced as shown below to obtain the post-earthquake time-varying aftershock risk analysis results, as shown in the following formula: Figure 7 shown.

[0085]

[0086] Where N LS is the number of structural damage states, D Si represents the i-th damage state, P(D Si |D Sj ,i=0…N LS ,j=0…i,z n ) represents t n At the moment, the state transition risk probability matrix of the structure can be expressed as:

[0087] P(D Si |D Sj ,i=0…N LS ,j=0…i,z n )=

[0088] P F (D Si |D Sj ,i=0…N LS ,0=1…i,I M,n =x)×P(I M,n =x|m n =m,r n =r)

[0089] Where, I M,n is the ground motion intensity corresponding to the nth earthquake event, r n is the epicenter distance of the nth earthquake event, P F (D Si |D Sj ,i=0…N LS ,j=0…i,I M,n =x) is the aftershock vulnerability probability matrix of damage state transfer, that is, when the aftershock intensity is I M,n =x, in the damaged state D Sj The structure reaches the damaged state D Si The probability of P(I M,n =x|m n =m,r n =r) is the earthquake motion prediction equation.

[0090] The post-earthquake time-varying aftershock risk analysis method based on aftershock clusters provided in this embodiment extends the traditional main-aftershock risk analysis of a single mainshock plus a single aftershock to a post-earthquake time-varying aftershock risk analysis that considers the mainshock and a large number of aftershocks formed after the earthquake. It solves the problem of the temporal relationship between post-earthquake risks that cannot be considered in the traditional main-aftershock risk analysis process, and obtains risk analysis results with time as the coordinate system.

[0091] The post-earthquake time-varying aftershock risk analysis method based on aftershock clusters provided in this embodiment can be used to establish a post-earthquake time-varying risk model for an engineering site in a complex aftershock environment after a main shock occurs, evaluate the post-earthquake risk level of the engineering site that changes over time, provide a theoretical basis for the safety assessment of the engineering site, and can be used for the safety evaluation of the engineering site in engineering construction.

[0092] The above is a specific implementation of the embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principles of the present invention. These improvements and modifications should also be regarded as the scope of protection of this application.

Claims

1. A post-earthquake time-varying aftershock risk analysis method based on aftershock clustering, characterized in that: The steps include: Step 1: Based on international and domestic earthquake catalog databases, the parameters of the ETAS model are identified using maximum likelihood estimation; Step 2: Based on the ETAS model, a large number of simulated aftershock cluster sequences are generated using the inverse transformation method and the acceptance-rejection method; Step 3: Based on the simulated aftershock cluster sequence in step 2, the magnitude information of the simulated aftershock cluster sequence is converted into ground motion intensity through the ground motion prediction equation; Step 4: Select a mainshock-aftershock sequence from international and domestic earthquake motion databases. Generate a cloud map of mainshock intensity-mainshock damage-aftershock intensity-aftershock damage through nonlinear time history analysis. Group the cloud map by damage state after the mainshock. Use maximum likelihood estimation to solve for aftershock vulnerability parameters under different damage states, and conduct damage state-dependent aftershock vulnerability analysis. Step 5: Based on the results of steps 2, 3, and 4, conduct a post-earthquake time-varying aftershock risk analysis.

2. The post-earthquake time-varying aftershock risk analysis method based on aftershock clustering according to claim 1, characterized in that: The step 1 specifically includes: The earthquake catalog used was taken from the earthquake catalog database of authoritative institutions at home and abroad. The maximum likelihood estimation method was used to identify the parameters of the ETAS model. The formula is: Where: θ is the parameter of the ETAS model to be estimated, θ = {μ, K0, α, c, p}, lnL(θ) is the log-likelihood function; λ(t n |H t ) is t n The earthquake occurrence rate at the time, H t is the aftershock cluster sequence within time t after the main shock, n represents the nth earthquake event in the aftershock cluster sequence, N S Indicates that the selected aftershock cluster sequence contains a total of N S An earthquake event.

3. The post-earthquake time-varying aftershock risk analysis method based on aftershock clustering according to claim 1 is characterized in that: The step 2 specifically includes: Assume that the simulation time period is [0,t end ], the simulated aftershock sequence to be generated is H t , there is always a positive number B, let λ(t|H t )≤B=U<λ(t n |H t )For {t∈[t n ,t n+1 ], n=1…N} are all valid, then when the main shock magnitude is m1, the algorithm for generating the simulated aftershock sequence is as follows: 1) Let t = 0, i = 0, H t = {m1, 0}; 2) Let B = λ(t n |H t ), according to the value of B, determine the time interval dt, dt satisfies the exponential distribution dt ~ exp(B); 3) Generate a random variable U from the uniform distribution [0,1]; 4) If U<λ(t+dt|H t+dt ) / B, then let t=t+dt, i=i+1, ti=t+dt, mi=FM -1 -(U), and let H t =H t ∪{m i ,t i Otherwise, simply let t = t + dt; where FM(·) is the magnitude distribution obtained according to the GR relationship, and its expression is: Where, β = b / log10e, b is the GR relationship parameter; 5) Return to step 2 until t>t end ; 6) Output simulated aftershock sequence H t , where t = t end .

4. The method for analyzing the risk of aftershocks after an earthquake based on aftershock clustering according to claim 1, characterized in that: In step 3, the earthquake magnitude information in the aftershock cluster sequence is converted into earthquake intensity using the earthquake prediction equation.

5. The method for analyzing the risk of aftershocks after an earthquake based on aftershock clustering according to claim 1, characterized in that: The step 4 specifically includes: The selected main shock and aftershock ground motion records were taken from the ground motion databases of international and domestic authoritative institutions; The mainshock and aftershock ground motion records were amplitude modulated as ground motion input, and nonlinear time history analysis was performed in OpenSEEs to obtain a cloud map of mainshock intensity-mainshock damage-aftershock intensity-aftershock damage, and the data were grouped according to the damage status after the mainshock. The maximum likelihood estimation is used to perform maximum likelihood estimation on the grouped cloud map to solve the damage state related aftershock vulnerability parameters and draw the damage state related aftershock vulnerability curve. The likelihood function is shown as follows Where m R and β R is the median and logarithmic standard deviation of the earthquake vulnerability function, N Cloud is the number of sample points in the cloud map; x i For a sample point in the cloud map, if it reaches a certain limit state L S , then the corresponding Bernoulli random variable Y is y i =1, otherwise y i =0, P(D≥L S ,I M =x i ) is the aftershock vulnerability probability matrix of damage state transfer, that is, when the aftershock intensity is I M ,=x i Under the condition of L S probability.

6. The method for analyzing the risk of aftershocks after an earthquake based on aftershock clustering according to claim 1, characterized in that: The step 5 specifically includes: Based on the obtained simulated aftershock sequence and the aftershock vulnerability curve related to the damage state, the post-earthquake time-varying aftershock risk analysis is performed as shown in the following formula: Where N LS is the number of structural damage states, D Si represents the i-th damage state, P(D Si |D Sj ,i=0…N LS ,j=0…i,z n ) represents t n At the moment, the state transition risk probability matrix of the structure can be expressed as P(D Si |D Sj ,i=0…N LS ,j=0…i,z n )=P F (D Si |D Sj ,i=0…N LS ,0=1…i,I M,n =x) ×P(I M,n =x|m n =m,r n =r) Where, I M,n is the ground motion intensity corresponding to the nth earthquake event, r n is the epicenter distance of the nth earthquake event, P F (D Si |D Sj ,i=0…N LS ,j=0…i,I M,n =x) is the aftershock vulnerability probability matrix of damage state transfer, that is, when the aftershock intensity is I M,n =x, the probability that a structure in the damaged state DSj reaches the damaged state DSi; P(I M,n =x|m n =m,r n =r) is the earthquake motion prediction equation.

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