Seismic motion field stochastic simulation method based on spatial correlation

Abstract

The invention relates to the technical field of seismology, in particular to a seismic oscillation field stochastic simulation method based on spatial correlation, which is composed of a single-field-point seismic oscillation model and a spatial correlation model, and specifically comprises the following steps: step 1, building a single-field-point seismic oscillation time-frequency power spectrum model; 2, building a prediction equation of time-frequency power spectrum parameters; step 3, converting a time-frequency power spectrum to a time history; 4, building a spatial correlation model of the time-frequency power spectrum parameters; according to the method, the simulation speed of the seismic motion field is greatly improved, meanwhile, spatial correlation of model parameters is introduced, correlation of time-frequency characteristics between field points in space is well represented, and the seismic motion field simulated through the method can conform to physical cognition.

Description

Technical Field

[0001] The invention relates to the technical field of seismology, and in particular to a random simulation method of seismic motion fields based on spatial correlation. Background Art

[0002] In performance-based seismic design and urban seismic resilience assessment, seismic motion fields associated with various earthquake scenarios are required as input. Due to the limited historical records, engineers usually select records from locations other than the location of interest, and then scale and modify them in time and frequency to match the required earthquake scenario. Although this method is convenient, the result is that the original physical characteristics of the seismic motion are lost. Another method is to build a random seismic motion model based on the relevant knowledge of wave propagation, which regards the seismic motion as a non-stationary random process. There are two types of such methods, namely source-based random models and field-based random models. Source-based random models require a detailed source model and a propagation path in a specific area, which is inconvenient for engineering applications. Field-based random models are based on statistical regression to discover the relationship between seismic scenario parameters and the time-frequency characteristics of seismic motion, parameterize ground motion, and combine the uncertainty of model parameters to capture the natural variability of earthquake ground motion. At the same time, they rely on fewer known inputs, such as magnitude, fault distance, and site shear wave velocity. They are more convenient to use and have received widespread attention in recent years.

[0003] Researchers have developed a variety of field point-based models using different time-frequency analysis methods, such as modulated white noise models, models based on wavelet packet transform, bimodal KT spectrum models, and time-frequency spectrum models based on S transform, etc. Although these models can effectively express the time-frequency characteristics of earthquake motion, they are all developed for a single field point and do not consider the correlation between the time-frequency characteristics of earthquake motion between points in space. Summary of the invention

[0004] The purpose of this application is to provide a random simulation method of seismic field based on spatial correlation, aiming to solve the problems in the prior art.

[0005] The present invention provides a method for random simulation of seismic field based on spatial correlation, which specifically includes the following steps:

[0006] Step 1: Build a single-field point ground motion time-frequency power spectrum model;

[0007] The earthquake motion is regarded as a random process, and the time-frequency power spectrum is used to describe the change of earthquake motion energy with time and frequency. A simplified model of the time-frequency power spectrum is proposed.

[0008] Step 2: Build the prediction equation of time-frequency power spectrum parameters;

[0009] The log-normal distribution probability density function is used to describe the change of normalized energy in the time-frequency power spectrum in time and frequency. Based on historical records, empirical prediction equations between five time-frequency power spectrum parameters and earthquake scenarios are established, and the total residuals, intra-event residuals, and inter-event residuals of the five time-frequency power spectrum parameters are calculated; the earthquake scenarios include magnitude, fault distance, and site shear wave velocity;

[0010] Step 3: Transformation of time-frequency power spectrum to time history;

[0011] The time-frequency power spectrum parameters predicted in step 2 are brought into the simplified time-frequency power spectrum model in step 1 to obtain the time-frequency power spectrum of a certain point. The time-frequency power spectrum is then converted into a seismic time history using the spectral representation method to obtain the simulated seismic motion of a single field point.

[0012] Step 4: Build a spatial correlation model of time-frequency power spectrum parameters;

[0013] Based on the observation records of multiple field points of historical major earthquakes, a spatial correlation model of the intra-event residuals of the time-frequency power spectrum parameters was constructed. The semivariogram was used to describe the spatial correlation between the time-frequency power spectrum parameters of two field points, and an exponential function was selected.

[0014] Furthermore, the seismic time-frequency power spectrum model in step 1 is as follows:

[0015]

[0016] Where: S(f,t): time-frequency power spectrum, f and t represent frequency and time respectively;

[0017] E T : total seismic energy;

[0018] A 2 (t): Temporal variation of normalized seismic energy;

[0019] The variation of normalized energy of ground motion in frequency;

[0020] Furthermore, the change of the normalized seismic energy in time and frequency in step 2 is as follows:

[0021]

[0022]

[0023] Where: μ t : corresponds to the mean value of the probability density function of energy variation over time;

[0024] σ t: corresponds to the standard deviation of the probability density function of energy variation over time;

[0025] μ f :: corresponds to the mean of the probability density function of energy variation over frequency;

[0026] σ f : corresponds to the standard deviation of the probability density function of energy variation over frequency;

[0027] Furthermore, the empirical prediction equation in step 2 is as follows:

[0028]

[0029] Where: Z j : The jth model parameter mapped to the normal space; α j,1 : The first prediction equation coefficient corresponding to the jth model parameter, there are 8 coefficients in the prediction equation; M: moment magnitude; R: fault distance; V: site shear wave velocity; ε j : Total residual.

[0030] Furthermore, when constructing the empirical prediction equation for the time-frequency power spectrum parameters in step 2, it is necessary to identify the values ​​and distribution of the five time-frequency power spectrum parameters of the historical earthquake records, and map the five parameters to the normal space based on the inverse standard normal cumulative probability distribution function.

[0031] Furthermore, the historical earthquake records in step 2 are selected from the NGA-West2 database; the selected historical earthquakes are all of moment magnitude greater than 4.5, each earthquake has at least 5 stations recording three parts of earthquake records, and the selected earthquakes are all strike-slip earthquakes.

[0032] Furthermore, the earthquake records selected in the step 2 are subject to multiple preprocessing operations, including rotating the seismic motions in the horizontal directions to the directions perpendicular to the fault and parallel to the fault, and only considering the seismic motions in the direction perpendicular to the fault; all records are truncated to retain seismic motions within the energy range of 0.1% to 99.9%.

[0033] Further, the expression of the spectrum representation in step 3 is as follows:

[0034]

[0035] Where: x(t): time history obtained by spectral representation; S XX (f n ,t,β): predicted time-frequency power spectrum; Δf: frequency interval; f n : The frequency at the nth frequency point; φ n : Random phase, uniformly sampled between 0 and 2π.

[0036] Furthermore, the observation records of multiple sites of historical major earthquakes in step 4 are six major earthquake events with multi-site observation records selected from the NGA-West2 database; and there must be at least 30 pairs of station records at each site distance.

[0037] The beneficial effects of the present invention are as follows: based on the current random seismic motion model, the present invention proposes a simplified model of the time-frequency power spectrum, which greatly improves the speed of seismic motion field simulation, and at the same time introduces the spatial correlation of model parameters, which well characterizes the correlation of time-frequency characteristics between field points in space, so that the seismic motion field simulated by the present invention can be consistent with physical cognition. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 This is a comparison chart between the true and predicted values ​​of the five parameters of the time-frequency power spectrum.

[0039] Figure 2 This is a graph showing how the semivariogram of the five parameters of the time-frequency power spectrum changes with distance.

[0040] Figure 3 This is a comparison chart between real earthquake records and three simulated earthquake motions.

[0041] Figure 4 The pseudo acceleration response spectrum calculated from real records, the BSSA14 attenuation relationship, and the comparison chart of 500 samples simulated by the present invention are shown.

[0042] Figure 5 Comparison of real and simulated pseudo-acceleration response spectra for all stations with periods of 0.1 s, 0.5 s, 1 s, and 2 s. DETAILED DESCRIPTION

[0043] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. 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 creative work are within the scope of protection of the present invention.

[0044] A random simulation method of seismic motion field based on spatial correlation consists of two parts: a single-field seismic motion model and a spatial correlation model. The single-field seismic motion model is a multi-parameter seismic time-frequency power spectrum model, which describes the change of seismic energy in time and frequency based on the probability density function of log-normal distribution. The time-frequency power spectrum is controlled by 5 parameters. Based on the historical earthquake database, an empirical prediction equation between these 5 parameters and earthquake scenario parameters (magnitude, fault distance and site shear wave velocity) is established. On the basis of the single-field seismic motion model, the spatial correlation of the intra-event residuals of the time-frequency power spectrum parameters is introduced, and a spatial correlation model is established based on the multi-field records of historical large earthquakes. By combining the single-field seismic motion model with the spatial correlation model, the time-frequency power spectrum considering the spatial correlation at each field point in the entire field can be obtained, and then the time history of each field point is obtained by using the spectrum representation method, and then the entire seismic motion field is obtained.

[0045] The specific steps include:

[0046] Step 1: Build a single-field point ground motion time-frequency power spectrum model;

[0047] The earthquake motion is regarded as a random process, and the time-frequency power spectrum is used to describe the change of earthquake motion energy with time and frequency. A simplified model of the time-frequency power spectrum is proposed.

[0048] Step 2: Build the prediction equation of time-frequency power spectrum parameters;

[0049] The log-normal distribution probability density function is used to describe the change of normalized energy in the time-frequency power spectrum in time and frequency. The change of energy in time and frequency is characterized by the log-normal distribution probability density function, which is controlled by the mean and standard deviation. Based on historical records, an empirical prediction equation between five time-frequency power spectrum parameters and earthquake scenarios is established; and the total residuals, intra-event residuals, and inter-event residuals of the five time-frequency power spectrum parameters are calculated; the earthquake scenarios include magnitude, fault distance, and site shear wave velocity;

[0050] Step 3: Transformation of time-frequency power spectrum to time history;

[0051] The time-frequency power spectrum parameters predicted in step 2 are brought into the simplified time-frequency power spectrum model in step 1 to obtain the time-frequency power spectrum of a certain point. The time-frequency power spectrum is then converted into a seismic time history using the spectral representation method to obtain the simulated seismic motion of a single field point.

[0052] Step 4: Build a spatial correlation model of time-frequency power spectrum parameters:

[0053] Based on the observation records of multiple field points of historical large earthquakes, a spatial correlation model of the intra-event residuals of the time-frequency power spectrum parameters is constructed. The semivariogram is used to describe the spatial correlation between the time-frequency power spectrum parameters of two field points, and the form of an exponential function is selected. The spatial correlation of the intra-event residuals of the time-frequency power spectrum parameters is measured by the semivariogram in geostatistics. The semivariogram is a function related to distance and has a complementary relationship with the correlation, that is, the greater the distance between the two sites, the larger the semivariogram and the smaller the correlation. After obtaining the spatial correlation model of the intra-event residuals of the time-frequency power spectrum parameters, the correlation matrix of the entire field can be calculated, and then the covariance matrix can be obtained. Based on the joint normal distribution sampling, the time-frequency power spectrum parameters of each field point can be obtained.

[0054] The form of the ground motion time-frequency power spectrum model in step 1 is as follows:

[0055]

[0056] In the above formula: S(f,t): time-frequency power spectrum, f and t represent frequency and time respectively;

[0057] E T : total seismic energy;

[0058] A 2 (t): Temporal variation of normalized seismic energy;

[0059] The variation of normalized energy of ground motion in frequency;

[0060] The change of the normalized seismic energy in time and frequency in step 2 is as follows:

[0061]

[0062]

[0063] Where: μ t : corresponds to the mean value of the probability density function of energy variation over time;

[0064] σ t : corresponds to the standard deviation of the probability density function of energy variation over time;

[0065] μ f :: corresponds to the mean of the probability density function of energy variation over frequency;

[0066] σ f : corresponds to the standard deviation of the probability density function of energy variation over frequency;

[0067] The empirical prediction equation in step 2 is as follows:

[0068]

[0069] In the above formula: Z j : The jth model parameter mapped to the normal space; α j,1 : The first prediction equation coefficient corresponding to the jth model parameter, there are 8 coefficients in the prediction equation; M: moment magnitude; R: fault distance; V: site shear wave velocity; ε j : Total residual.

[0070] When constructing the empirical prediction equation of the time-frequency power spectrum parameters in the step 2, it is necessary to identify the values ​​and distribution of the five time-frequency power spectrum parameters of the historical earthquake records, and map the five parameters to the normal space based on the inverse standard normal cumulative probability distribution function.

[0071] The historical earthquake records in step 2 are selected from the NGA-West2 database, and a total of 3004 earthquake motions are selected; the selection principle is that the moment magnitude is greater than 4.5, each earthquake has at least 5 stations recording three parts of earthquake records, and the selected earthquakes are all strike-slip earthquakes.

[0072] The earthquake records selected in the step 2 are subject to multiple preprocessing operations; the specific operations are: 1. The seismic motions in the horizontal directions are rotated to the directions perpendicular to the fault and parallel to the fault, and only the seismic motions in the direction perpendicular to the fault are considered; 2. In order to minimize data redundancy and ensure the accuracy of power spectrum calculation, all records are truncated, and seismic motions within the energy range of 0.1% to 99.9% are retained, in order to facilitate the prediction of the time envelope function.

[0073] The expression of the spectrum representation in step 3 is as follows:

[0074]

[0075] In the above formula: x(t): time course obtained by spectral representation; S XX (f n ,t,β): predicted time-frequency power spectrum; Δf: frequency interval; f n : The frequency at the nth frequency point; φ n : Random phase, uniformly sampled between 0 and 2π.

[0076] The multi-site observation records of historical major earthquakes in step 4 are 6 major earthquake events with multi-site observation records selected from the NGA-West2 database; and there must be at least 30 pairs of station records at each site distance, and the site distance is 2 kilometers.

[0077] The following is a description with reference to specific embodiments.

[0078] Step 1: Build a single-field point ground motion time-frequency power spectrum model:

[0079] Considering the need to quickly restore the seismic field after an earthquake, a simplified model of the time-frequency power spectrum of ground motion at a single field point is proposed. The changes in the time and frequency of ground motion energy are considered independently, and five parameters are used to describe the time-frequency characteristics of ground motion: total energy, mean of time function, standard deviation of time function, mean of frequency function, and standard deviation of frequency function. The changes in energy in time and frequency are normalized;

[0080] Step 2: Build the prediction equation of time-frequency power spectrum parameters:

[0081] The energy variation function in time and frequency is expressed by the probability density function of lognormal distribution, which is controlled by two parameters: mean and standard deviation. 3004 earthquake motions were selected from the NGA-West2 database. The earthquake moment magnitude was greater than 4.5, and each earthquake had at least 5 three-component earthquake motion records. The 5 time-frequency power spectrum parameters of each earthquake in the database were identified. The mean and standard deviation of the time function were related to the basic characteristics of the earthquake motion (90% of the significant duration and 45% of the total energy accumulation time). The mean and standard deviation of the frequency function were identified by the moment estimation method using the correspondence between the first-order moment and the second-order moment. The data distribution of the identified 5 time-frequency parameters was fitted, and the 5 time-frequency parameters were mapped to the normal space using the inverse standard normal cumulative probability distribution function.

[0082] The relationship between the five time-frequency parameters and the magnitude, fault distance, and site shear wave velocity was analyzed to build an empirical prediction equation. The Adam optimizer in machine learning was used to optimize the prediction equation coefficients, and the total residual, intra-event residual, and inter-event residual were calculated. The results are as follows: Figure 1 As shown;

[0083] Step 3: Convert the time-frequency power spectrum into time history:

[0084] After obtaining the predicted time-frequency power spectrum, the time-frequency power spectrum is inversely transformed into the ground motion time history using the spectral representation method to obtain the simulated ground motion at a single field point;

[0085] Step 4: Build a spatial correlation model of time-frequency power spectrum parameters:

[0086] Six large earthquake events with multi-site observation records were selected from the NGA-West2 database to provide data for the construction of the spatial correlation model. The semivariogram in geostatistics is used to estimate the correlation between spatial position-related variables. The semivariogram is a function of the field point distance and is complementary to the correlation. The farther the site distance, the larger the semivariogram and the smaller the correlation. The semivariogram uses an exponential function. The semivariogram of the five time-frequency power spectrum parameters is regressed for each earthquake, and the regression results of the six earthquakes are averaged to obtain the overall spatial correlation model, as shown in Figure 2. Figure 2 As shown;

[0087] Combine the above steps to simulate the seismic field:

[0088] Given the earthquake scenario parameters: moment magnitude, fault distance of each field point and site shear wave velocity, five time-frequency power spectrum parameters of each field point can be obtained based on the prediction equation. In the prediction equation calculation, random sampling is used for the residual term, the residual within the event is adopted by the joint normal distribution, the covariance is calculated based on the spatial correlation model, and the correlation of the residual between events is not considered. The five predicted time-frequency power spectrum parameters are brought into the time-frequency power spectrum model, and the spectral representation method is used to convert the time-frequency power spectrum into a time history, thereby obtaining the entire seismic field.

[0089] The specific form of the prediction equation selected in this embodiment is as follows:

[0090]

[0091] Where: Z j : The jth model parameter mapped to the normal space; α j,1 : The first prediction equation coefficient corresponding to the jth model parameter, there are 8 coefficients in the prediction equation; M: moment magnitude; R: fault distance; V: site shear wave velocity; ε j : Total residual.

[0092] Based on the current random simulation method of seismic motion, the present invention proposes a simplified time-frequency power spectrum model, introduces the spatial correlation of time-frequency power spectrum parameters, and establishes a random simulation method of seismic motion field based on spatial correlation. The time-frequency power spectrum is controlled by only 5 parameters, which greatly improves the speed of seismic motion field simulation. For the simulation of a 200km×200km free surface field, the grid size is 1km, and there are a total of 40401 field points. It only takes 34.66 seconds to simulate once using the random simulation method of seismic motion field proposed by the present invention. At the same time, the empirical prediction equation has well established the relationship between the seismic time-frequency parameters and the earthquake scenario parameters, such as Figures 3 to 5 As shown in the figure, in the simulation of real earthquakes, the simulation results are also in good agreement with the actual records.

[0093] It will be apparent to those skilled in the art that the invention is not limited to the details of the exemplary embodiments described above and that the invention can be implemented in other specific forms without departing from the spirit or essential features of the invention. Therefore, the embodiments should be considered exemplary and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description, and it is intended that all variations falling within the meaning and scope of the equivalent elements of the claims be included in the invention. Any reference numeral in a claim should not be considered as limiting the claim to which it relates.

Claims

1. A random simulation method of seismic field based on spatial correlation, characterized in that: The specific steps include: Step 1: Build a single-field point ground motion time-frequency power spectrum model; The earthquake motion is regarded as a random process, and the time-frequency power spectrum is used to describe the change of earthquake motion energy with time and frequency. A simplified model of the time-frequency power spectrum is proposed. Step 2: Build the prediction equation of time-frequency power spectrum parameters; The log-normal distribution probability density function is used to describe the change of normalized energy in the time-frequency power spectrum in time and frequency. Based on historical records, empirical prediction equations between five time-frequency power spectrum parameters and earthquake scenarios are established, and the total residuals, intra-event residuals, and inter-event residuals of the five time-frequency power spectrum parameters are calculated; the earthquake scenarios include magnitude, fault distance, and site shear wave velocity; Step 3: Transformation of time-frequency power spectrum to time history; The time-frequency power spectrum parameters predicted in step 2 are brought into the simplified time-frequency power spectrum model in step 1 to obtain the time-frequency power spectrum of a certain point. The time-frequency power spectrum is then converted into a seismic time history using the spectral representation method to obtain the simulated seismic motion of a single field point. Step 4: Build a spatial correlation model of time-frequency power spectrum parameters; Based on the observation records of multiple field points of historical major earthquakes, a spatial correlation model of the intra-event residuals of the time-frequency power spectrum parameters was constructed. The semivariogram was used to describe the spatial correlation between the time-frequency power spectrum parameters of two field points, and an exponential function was selected.

2. The random simulation method of earthquake motion field based on spatial correlation according to claim 1 is characterized in that: The seismic time-frequency power spectrum model in step 1 is as follows: Where: S(f,t): time-frequency power spectrum, f and t represent frequency and time respectively; E T : total seismic energy; A 2 (t): Temporal variation of normalized seismic energy; :The change of normalized energy of earthquake motion in frequency.

3. The random simulation method of earthquake motion field based on spatial correlation according to claim 1 is characterized in that: The change of the normalized seismic energy in time and frequency in step 2 is as follows: Where: μ t : corresponds to the mean value of the probability density function of energy variation over time; σ t : corresponds to the standard deviation of the probability density function of energy variation over time; μ f :: corresponds to the mean of the probability density function of energy variation over frequency; σ f : corresponds to the standard deviation of the probability density function of the energy variation over frequency.

4. The random simulation method of earthquake motion field based on spatial correlation according to claim 1 is characterized in that: The empirical prediction equation in step 2 is as follows: Where: Z j : The jth model parameter mapped to the normal space; α j,1 : The first prediction equation coefficient corresponding to the jth model parameter, there are 8 coefficients in the prediction equation; M: moment magnitude; R: fault distance; V: site shear wave velocity; ε j : Total residual.

5. The random simulation method of earthquake motion field based on spatial correlation according to claim 1 is characterized in that: When constructing the empirical prediction equation of the time-frequency power spectrum parameters in the step 2, it is necessary to identify the values ​​and distribution of the five time-frequency power spectrum parameters of the historical earthquake records, and map the five parameters to the normal space based on the inverse standard normal cumulative probability distribution function.

6. The random simulation method of earthquake field based on spatial correlation according to claim 1 is characterized in that: The historical earthquake records in step 2 are selected from the NGA-West2 database; the selected historical earthquakes are all of moment magnitude greater than 4.5, each earthquake has at least 5 stations recording three parts of earthquake records, and the selected earthquakes are all strike-slip earthquakes.

7. The random simulation method of earthquake field based on spatial correlation according to claim 6 is characterized in that: The selected earthquake records are subject to multiple preprocessing operations, including rotating the seismic motions in the horizontal directions to the directions perpendicular to the fault and parallel to the fault, and only considering the seismic motions in the direction perpendicular to the fault; all records are truncated to retain seismic motions within the energy range of 0.1% to 99.9%.

8. The random simulation method of earthquake motion field based on spatial correlation according to claim 1 is characterized in that: The expression of the spectrum representation in step 3 is as follows: Where: x(t): time history obtained by spectral representation; S XX (f n ,t,β): predicted time-frequency power spectrum; Δf: frequency interval; f n : The frequency at the nth frequency point; φ n : Random phase, uniformly sampled between 0 and 2π.

9. The random simulation method of earthquake motion field based on spatial correlation according to claim 1 is characterized in that: The observation records of multiple points of historical earthquakes in step 4 are: Six large earthquake events with multi-station observation records were selected from the NGA-West2 database; And there must be at least 30 pairs of station records at each station distance.

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