Fitting method of spatial correlation model of ground motion considering the influence of elevation

Abstract

The present invention relates to a method for fitting a seismic motion spatial correlation model that considers the influence of elevation. The method comprises the following steps: obtaining data required for fitting the seismic motion spatial correlation, including residuals within earthquake events, path effect terms, and site effect terms, to form a data set; the path effect terms include site spacing and azimuth; the site effect terms include shear wave velocity and elevation; selecting an appropriate function form to consider the correlation between each dependent variable and the inter-event residuals; using the seismic motion spatial correlation model to calculate the correlation matrix of the intra-event residual field, and using the maximum likelihood parameter estimation method to calculate the model parameter values ​​that minimize the negative log-likelihood value of the intra-event residual field, thereby obtaining the seismic motion spatial correlation. The seismic motion spatial correlation model that considers the influence of elevation, fitted by the method of the present invention, has higher accuracy and better stability, and can be applied to fields such as regional infrastructure earthquake risk assessment.

Description

Technical Field

[0001] The invention belongs to the technical field of earthquake engineering, and in particular to a method for fitting a seismic motion spatial correlation model taking elevation influence into consideration. Background Art

[0002] As earthquake hazard and risk assessments shift from single structures to larger clusters of buildings and infrastructure, models of spatial correlation of ground motion play a key role in regional infrastructure earthquake risk assessment. The excitations of a seismic event are spatially correlated, and this localized spatial correlation can increase the likelihood of simultaneous damage to many structures during a single earthquake. However, accurate quantification of earthquake damage is only possible when the correlation between ground motion intensities at different locations is known.

[0003] Currently, scholars mostly use an isotropic spatial correlation model, in which the correlation between sites decays exponentially with increasing distance between them. However, this approach suffers from significant variability in spatial correlation models for different earthquakes. Some researchers have taken site and path effects into account to optimize existing spatial correlation models. Given that the spatial correlation of ground motion is affected by elevation, with stronger correlation between sites at similar elevations, existing spatial correlation models are inaccurate in areas with large elevation differences. Therefore, a method for fitting a ground motion spatial correlation model that considers the influence of elevation has been proposed.

[0004] Chinese patent publication number CN111458744A, entitled "Spatial Rotational Earthquake Motion Simulation Method," discloses decomposing actual earthquake motion into a first horizontal acceleration, a second horizontal acceleration, and a vertical acceleration. The Fourier spectra of the rocking component and the torsional component are obtained based on linear elasticity theory and fast Fourier transform results. Based on random vibration theory, the Fourier spectra of the rocking component and the torsional component are transformed to obtain a spatially accounted rotational power spectrum. Non-stationary rotational and translational earthquake motions are then derived through LDLT decomposition and intensity envelope function analysis. This method simplifies the propagation direction of earthquake waves to horizontal and vertical, without considering the influence of other elevation factors, including azimuth angle.

[0005] A Chinese patent, publication number CN116558753A, titled "A Method for Simulating Long-Period Earthquake Motions," discloses separating actual long-period earthquake motion records based on earthquake motion parameters to obtain surface and body wave components. Multivariate empirical mode decomposition is then performed on the body wave components to obtain intrinsic mode function components. Time-frequency parameters are estimated for the surface and eigenmode function components, and random phase angles are introduced. These components are then combined and superimposed to generate long-period earthquake motion simulation records. However, this method is primarily applicable to low-frequency, long-period earthquake motions, which have significant surface wave secondary effects. Therefore, seismic motions can be simulated using the separated surface waves. This method is not applicable to non-long-period earthquake motions and lacks universal applicability. Summary of the Invention

[0006] The purpose of the present invention is to solve the problem that the existing seismic motion spatial correlation model does not consider the influence of elevation parameters, and the seismic motion field fitting is not accurate enough in areas with large drop. A seismic motion spatial correlation model fitting method that considers the influence of elevation is proposed.

[0007] The present invention provides a method for fitting a spatial correlation model of ground motion that takes into account the influence of elevation, including the following steps:

[0008] Step 1: Establish a seismic motion spatial correlation fitting dataset:

[0009] Based on real historical earthquake information and the global geographic information database, the data required for fitting the spatial correlation of seismic motions are obtained, including residuals within earthquake events, path effect terms, and site effect terms, which form a data set; the path effect term includes site spacing and direction angle; the site effect term includes shear wave velocity and elevation;

[0010] Step 2: Determine the spatial correlation function form of each dependent variable:

[0011] Considering the correlation between each dependent variable and the residual between events in step 1, select appropriate functional forms respectively; among them, Formula 1 is used to express the relationship that the residual correlation decays with the increase of the site spacing:

[0012]

[0013] Where, d E is the site spacing, l E and γ E are model parameters;

[0014] The correlation between the direction angle and the residual is expressed by formula 2:

[0015]

[0016] Where, d A Refers to the difference in earthquake direction angle at the site, l Aare model parameters;

[0017] The relationship between the site shear wave velocity Vs30 and the residual correlation is expressed in Formula 3:

[0018] ρ S =exp(-d S / l S ) Formula 3

[0019] Where, d S Refers to the site Vs30 difference, l S are model parameters;

[0020] The relationship between site elevation and residual correlation is expressed in the form of Formula 4:

[0021] ρ H =exp(-d H / l H ) Formula 4

[0022] Where, d H Refers to the site elevation difference, l H are model parameters;

[0023] Step 3: Construction of spatial correlation model of earthquake motion:

[0024] Considering all dependent variables in step 2, the spatial correlation model of ground motion is constructed as shown in Formula 5:

[0025] ρ EASH =(w1·ρ E (d E )+(1-w1)·ρ H (d H ))·(w2·ρ A (d A )+(1-w2)·ρ S (d S )) Formula 5

[0026] Where w1 and w2 are weight parameters;

[0027] Step 4: Maximum likelihood parameter estimation:

[0028] The correlation matrix of the residual field within the event is calculated using the spatial correlation model of earthquake motion in step 3. The model parameter value that minimizes the negative log-likelihood value of the residual field within the event is calculated using the maximum likelihood parameter estimation method. The spatial correlation of earthquake motion is obtained, and the random residual vector z that obeys the multivariate normal distribution has a negative log-likelihood value as shown in Formula 6:

[0029]

[0030] Where n is the dimension of the residual vector z, and R is the covariance matrix of the residual vector z;

[0031] Step 5: Model effect comparison:

[0032] Using the seismic motion spatial correlation model obtained in step 4, the maximum log-likelihood value of the model is calculated and compared with the traditional isotropic spatial correlation model and the optimization model proposed by scholars.

[0033] The initial parameters of the model parameters in step 2 are: l E =50,γ E =1,l A =30, l S =80,l H =100.

[0034] In step 3, the initial values ​​of the weight parameters are w1=0.5 and w2=0.5.

[0035] In the minimization of the negative log-likelihood value described in step 4, since earthquake events are not unique, the negative log-likelihood function of each earthquake event needs to be added together, and the model parameters are optimized by minimizing the overall negative log-likelihood function.

[0036] The model involved in the comparison in step 5 is the traditional isotropic spatial correlation E model, as shown in formula 7:

[0037]

[0038] And the EAS model proposed by scholars is shown in Formula 8:

[0039] ρ EAS =ρ E (d E )·(wρ A (d A )+(1-w)ρ S (d S )). Formula 8

[0040] Formula 9 is used as the comparison model:

[0041]

[0042] Where, LL BL is the log-likelihood mean of the comparison model, LL M is the mean log-likelihood of the EASH model.

[0043] The present invention has the following beneficial effects: Based on spatial correlation analysis in seismic statistics, the present invention proposes a method for fitting a seismic motion spatial correlation model that considers the influence of elevation. The method uses maximum likelihood parameter estimation to fit the spatial correlation model parameters, thereby obtaining the final seismic motion spatial correlation model. Comparison with existing model methods shows that the EASH model has a larger log-likelihood mean and a smaller log-likelihood standard deviation. Therefore, the seismic motion spatial correlation model fitted by the present method that considers the influence of elevation is more accurate and stable.

[0044] The present invention proposes a method for fitting a spatial correlation model of earthquake motion taking into account the influence of elevation, which can be applied to fields such as regional infrastructure earthquake risk assessment. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 Flow chart of the method of the present invention.

[0046] Figure 2 This is a diagram of the DI value of the EASH model of the present invention relative to the traditional model.

[0047] Figure 3 This is a diagram of the DI value of the EASH model of the present invention relative to the EAS model. DETAILED DESCRIPTION

[0048] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0049] like Figure 1 As shown, the method for fitting the earthquake motion spatial correlation model considering the influence of elevation of the present invention is implemented by the following steps:

[0050] Step 1: Establish a seismic motion spatial correlation fitting dataset:

[0051] Based on real historical earthquake information and the global geographic information database, the data required for fitting the spatial correlation of seismic motion is obtained, including residuals within earthquake events, path effect terms (site spacing, azimuth angle), and site effect terms (shear wave velocity, elevation), to form a data set; in this embodiment, historical earthquake information of the United States, Japan, and Taiwan, China was selected.

[0052] Step 2: Determine the spatial correlation function form of each dependent variable:

[0053] Considering the correlation between each dependent variable (site spacing, azimuth, shear wave velocity, elevation) and the residual between events, appropriate functional forms are selected. Formula 1 is used to express the relationship that the residual correlation decays with the increase of site spacing:

[0054]

[0055] Where, d E is the site spacing, l E and γ E is the model parameter. In this embodiment, l E =50,γ E =1.

[0056] The correlation between the direction angle and the residual is expressed by formula 2:

[0057]

[0058] Where, d A Refers to the difference in earthquake direction angle at the site, l A is the model parameter. A =30.

[0059] The relationship between the site shear wave velocity Vs30 and the residual correlation is expressed in Formula 3:

[0060] ρ S =exp(-d S / l S ) Formula 3

[0061] Where, d S Refers to the site Vs30 difference, l S is the model parameter. S = 80. The relationship between site elevation and residual correlation is expressed in Formula 4:

[0062] ρ H =exp(-d H / l H )Formula 4

[0063] Where, d H Refers to the site elevation difference, l H is the model parameter. H =100.

[0064] Step 3: Construction of spatial correlation model of earthquake motion:

[0065] Considering all the dependent variables in step 2, the earthquake motion spatial correlation model (EASH) is constructed as shown in Formula 5:

[0066] ρ EASH =(w1·ρE (d E )+(1-w1)·ρ H (d H ))·(w2·ρ A (d A )+(1-w2)·ρ S (d S Formula 5

[0067] Wherein, w1 and w2 are weight parameters. In this embodiment, the initial values ​​of the weight parameters are w1=0.5 and w2=0.5.

[0068] Step 4: Maximum likelihood parameter estimation:

[0069] The correlation matrix of the residual field within the event is calculated using the spatial correlation model of earthquake motion in step 3. The model parameter value that minimizes the negative log-likelihood value of the residual field within the event is calculated using the maximum likelihood parameter estimation method. The spatial correlation of earthquake motion is obtained, and the random residual vector z that obeys the multivariate normal distribution has a negative log-likelihood value as shown in Formula 6:

[0070]

[0071] Where n is the dimension of the residual vector z, and R is the covariance matrix of the residual vector z.

[0072] Since earthquake events are not unique, the negative log-likelihood function of each earthquake event needs to be added together, and the model parameters are optimized by minimizing the overall negative log-likelihood function.

[0073] Step 5: Model effect comparison:

[0074] Using the earthquake motion spatial correlation model EASH obtained in step 4, the maximum log-likelihood value of the model is calculated and compared with the traditional isotropic spatial correlation model and the optimization model proposed by scholars.

[0075] The model involved in the comparison is the traditional isotropic spatial correlation E model, as shown in Formula 7:

[0076]

[0077] And the EAS model proposed by scholars is shown in Formula 8:

[0078] ρ EAS =ρ E (d E )·(wρ A (d A )+(1-w)ρ S (d S Formula 8

[0079] Formula 9 is used as the comparison model:

[0080]

[0081] Where, LL BL is the log-likelihood mean of the comparison model, LL M is the mean log-likelihood of the EASH model.

[0082] The results of spatial correlation model parameters for different period fitting are shown in Table 1. The mean and standard deviation of log likelihood of traditional model, EAS model and EASH model are shown in Table 2. The DI value of EASH model relative to traditional E model is shown in Table 2. Figure 2 As shown, the DI value of the EASH model relative to the EAS model is as follows Figure 3 Through comparative analysis of the calculation results, it is found that the EASH model has a larger mean value of log-likelihood and a smaller standard deviation of log-likelihood. Therefore, the spatial correlation model of earthquake motion considering the influence of elevation fitted in this embodiment has higher accuracy and better stability.

[0083] Table 1 shows the results of the spatial correlation model coefficients of the earthquake motion fitted in this embodiment.

[0084]

[0085]

[0086] Table 2 compares the spatial correlation model of earthquake motion fitted by the present invention with the traditional model and the EAS model.

[0087]

[0088] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above and that the present invention can be implemented in other specific forms without departing from the spirit or essential characteristics of the present invention. Therefore, the embodiments should be considered in all respects as exemplary and non-restrictive.

Claims

1. A method for fitting a spatial correlation model of ground motion considering the influence of elevation, characterized in that: The following steps are involved: Step 1: Establish a seismic motion spatial correlation fitting dataset: Based on real historical earthquake information and the global geographic information database, the data required for fitting the spatial correlation of seismic motions are obtained, including residuals within earthquake events, path effect terms, and site effect terms, which form a data set; the path effect term includes site spacing and direction angle; the site effect term includes shear wave velocity and elevation; Step 2: Determine the spatial correlation function form of each dependent variable: Considering the correlation between each dependent variable and the residual between events in step 1, select appropriate functional forms respectively; among them, Formula 1 is used to express the relationship that the residual correlation decays with the increase of the site spacing: Where, d E is the site spacing, l E and γ E are model parameters; The correlation between the direction angle and the residual is expressed by formula 2: Where, d A Refers to the difference in earthquake direction angle at the site, l A are model parameters; The relationship between the site shear wave velocity Vs30 and the residual correlation is expressed in Formula 3: ρ S =exp(-d S / l S ) Formula 3 Where, d S Refers to the site Vs30 difference, l S are model parameters; The relationship between site elevation and residual correlation is expressed in the form of Formula 4: ρ H =exp(-d H / l H ) Formula 4 Where, d H Refers to the site elevation difference, l H are model parameters; Step 3: Construction of spatial correlation model of earthquake motion: Considering all dependent variables in step 2, the spatial correlation model of ground motion is constructed as shown in Formula 5: ρ EASH = (w1·ρ E (d E )) + (1 - w1)·ρ H (d H ))·(w2·ρ A (d A )) + (1 - w2)·ρ S (d S ) Formula Five Where w1 and w2 are weight parameters; Step 4: Maximum likelihood parameter estimation: The correlation matrix of the residual field within the event is calculated using the spatial correlation model of earthquake motion in step 3. The model parameter value that minimizes the negative log-likelihood value of the residual field within the event is calculated using the maximum likelihood parameter estimation method. The spatial correlation of earthquake motion is obtained, and the random residual vector z that obeys the multivariate normal distribution has a negative log-likelihood value as shown in Formula 6: Where n is the dimension of the residual vector z, and R is the covariance matrix of the residual vector z; Step 5: Model effect comparison: Using the seismic motion spatial correlation model obtained in step 4, the maximum log-likelihood value of the model is calculated and compared with the traditional isotropic spatial correlation model and the optimization model proposed by scholars.

2. The method for fitting a spatial correlation model of ground motion considering the influence of elevation according to claim 1, characterized in that: The initial parameters of the model parameters in step 2 are: E =50,γ E =1,l A =30, l S =80,l H =100.

3. The method for fitting a spatial correlation model of ground motion considering the influence of elevation according to claim 1, characterized in that: In step 3, the initial values ​​of the weight parameters are w1=0.5 and w2=0.

5.

4. The method for fitting a spatial correlation model of ground motion considering the influence of elevation according to claim 1, characterized in that: In the step 4, the negative log-likelihood value is minimized. Since earthquake events are not unique, the negative log-likelihood function of each earthquake event needs to be added together, and the model parameters are optimized by minimizing the overall negative log-likelihood function.

5. The method for fitting a spatial correlation model of ground motion considering the influence of elevation according to claim 1, characterized in that: The model involved in the comparison in step 5 is the traditional isotropic spatial correlation E model, as shown in formula 7: And the EAS model proposed by scholars is shown in Formula 8: ρ EAS = ρ E (d E )·(wρ A (d A )+(1 - w)ρ S (d S )) Formula VIII.

6. The method for fitting a spatial correlation model of ground motion considering the influence of elevation according to claim 5, characterized in that: Formula 9 is used as the comparison model: Where, LL BL is the log-likelihood mean of the comparison model, LL M is the mean log-likelihood of the EASH model.

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