A Stochastic Simulation Method for Seismic Ground Motion Field Based on Spatial Correlation

Abstract

This invention relates to the field of seismology, and in particular to a method for stochastic simulation of ground motion fields based on spatial correlation. It consists of two parts: a single-point ground motion model and a spatial correlation model, specifically including the following steps: Step 1: Constructing a time-frequency power spectrum model of a single-point ground motion; Step 2: Constructing prediction equations for the time-frequency power spectrum parameters; Step 3: Transforming the time-frequency power spectrum into a time history; Step 4: Constructing a spatial correlation model of the time-frequency power spectrum parameters. This invention significantly improves the speed of ground motion field simulation, and by introducing spatial correlation of model parameters, it effectively characterizes the correlation of time-frequency characteristics between field points in space, enabling the simulated ground motion field to conform to physical understanding.

Description

Technical Field

[0001] This invention relates to the field of seismology, and in particular to a method for stochastic simulation of seismic ground motion fields based on spatial correlation. Background Technology

[0002] In performance-based earthquake design and urban seismic resilience assessment, ground motion fields associated with various seismic scenarios are required as input. Due to the limited historical records, engineers typically select records from locations outside the site of interest and then scale and modify them in time and frequency to match the desired seismic scenario. While convenient, this method results in the loss of the original physical characteristics of the ground motion. Another approach is to build stochastic ground motion models based on knowledge of wave propagation, treating ground motion as a non-stationary stochastic process. There are two types of such methods: source-based stochastic models and field-point-based stochastic models. Source-based stochastic models require detailed source models and propagation paths specific to a particular region, making them inconvenient for engineering applications. Field-point-based stochastic models use statistical regression to discover the relationship between seismic scenario parameters and the time-frequency characteristics of ground motion, parameterizing ground motion and capturing the natural variability of seismic ground motion by incorporating the uncertainty of model parameters. They rely on fewer known inputs, such as magnitude, fault distance, and site shear wave velocity, making them more convenient to use and gaining widespread attention in recent years.

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

[0004] The purpose of this application is to provide a stochastic simulation method for seismic ground motion fields based on spatial correlation, which aims to solve the problems in the prior art.

[0005] This application provides a method for stochastic simulation of seismic ground motion fields based on spatial correlation, specifically including the following steps:

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

[0007] By treating seismic motion as a stochastic process, and describing the change of seismic motion energy with time and frequency based on the time-frequency power spectrum, a simplified model of the time-frequency power spectrum is proposed.

[0008] Step 2: Construct the prediction equations for time-frequency power spectrum parameters;

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

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

[0011] By substituting the predicted time-frequency power spectrum parameters from step two into the simplified time-frequency power spectrum model from step one, the time-frequency power spectrum at a certain point can be obtained. Then, by using the spectral representation method to convert the time-frequency power spectrum into the ground motion time history, the simulated ground motion at a single field point can be obtained.

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

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

[0014] Furthermore, the time-frequency power spectrum model of the ground motion in step one 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 energy of earthquake motion;

[0018] A 2 (t): The time-varying normalized energy of earthquake motion;

[0019] The frequency variation of the normalized energy of earthquake motion;

[0020] Furthermore, the changes in the normalized energy of the seismic motion in step two over time and frequency are as follows:

[0021]

[0022]

[0023] Where: μ t : The mean of the probability density function corresponding to the change of energy over time;

[0024] σ t: The standard deviation of the probability density function corresponding to the change of energy over time;

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

[0026] σ f : The standard deviation of the probability density function corresponding to the change of energy with frequency;

[0027] Furthermore, the empirical prediction equation in step two takes the following form:

[0028]

[0029] Among them: Z j : The j-th model parameter mapped onto the normal space; α j,1 : Corresponds to the first prediction equation coefficient of the j-th model parameter; there are a total of 8 coefficients in the prediction equation; M: moment magnitude; R: fault distance; V: site shear wave velocity; ε j Total residual.

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

[0031] Furthermore, in step two, the historical earthquake records 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-thirds of the earthquake records, and all selected earthquakes are strike-slip earthquakes.

[0032] Furthermore, the earthquake records selected in step two need to undergo multiple preprocessing operations, including rotating the ground motions in the two horizontal directions to the direction perpendicular to the fault and the direction parallel to the fault, considering only the ground motions in the direction perpendicular to the fault; and truncating all records, retaining ground motions in the energy range of 0.1% to 99.9%.

[0033] Furthermore, the expression for the spectral representation in step three is as follows:

[0034]

[0035] Where: x(t): time history obtained using 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, which follows uniform sampling between 0 and 2π.

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

[0037] The beneficial effects of this invention are: based on the current random seismic model, this invention proposes a simplified model of time-frequency power spectrum, which greatly improves the speed of seismic field simulation. At the same time, it 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 field simulated by this invention can be consistent with physical cognition. Attached Figure Description

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

[0039] Figure 2 The graph shows the semivariogram of the five parameters of the time-frequency power spectrum as a function of distance.

[0040] Figure 3 A comparison chart of actual ground motion records and three simulated ground motions.

[0041] Figure 4 A comparison chart of the calculated pseudo-acceleration response spectrum, BSSA14 decay relationship, and 500 samples simulated in this invention, to accurately record the data.

[0042] Figure 5 Comparison of real and simulated pseudo-acceleration response spectra for all stations with periods of 0.1 seconds, 0.5 seconds, 1 second, and 2 seconds. Detailed Implementation

[0043] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0044] A stochastic simulation method for seismic ground motion fields based on spatial correlation consists of two parts: a single-field ground motion model and a spatial correlation model. The single-field ground motion model is a multi-parameter time-frequency power spectrum model, which describes the changes in ground motion energy over time and frequency based on a log-normal probability density function. The time-frequency power spectrum is controlled by five parameters, and empirical prediction equations between these five parameters and earthquake scenario parameters (magnitude, fault distance, and site shear wave velocity) are established based on a historical earthquake database. Based on the single-field ground 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 built based on multi-field records of historical large earthquakes. By combining the single-field ground motion model with the spatial correlation model, the time-frequency power spectrum considering spatial correlation at each field point in the entire field can be obtained. Then, the time history of each field point is obtained using spectral representation, thus yielding the entire seismic ground motion field.

[0045] Specifically, the following steps are included:

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

[0047] By treating seismic motion as a stochastic process, and describing the change of seismic motion energy with time and frequency based on the time-frequency power spectrum, a simplified model of the time-frequency power spectrum is proposed.

[0048] Step 2: Construct the prediction equations for time-frequency power spectrum parameters;

[0049] The log-normal distribution probability density function is used to describe the changes in normalized energy over time and frequency in the time-frequency power spectrum. These changes are characterized by the log-normal distribution probability density function, which is controlled by the mean and standard deviation. Based on historical records, empirical prediction equations are constructed between five time-frequency power spectrum parameters and earthquake scenarios. The total residual, intra-event residual, and inter-event residual for these five parameters are calculated. The earthquake scenarios include magnitude, fault distance, and site shear wave velocity.

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

[0051] By substituting the predicted time-frequency power spectrum parameters from step two into the simplified time-frequency power spectrum model from step one, the time-frequency power spectrum at a certain point can be obtained. Then, by using the spectral representation method to convert the time-frequency power spectrum into the ground motion time history, the simulated ground motion at a single field point can be obtained.

[0052] Step 4: Construct a spatial correlation model for time-frequency power spectral parameters:

[0053] Based on observational records from multiple sites of historical major earthquakes, a spatial correlation model of the intra-event residuals of time-frequency power spectral parameters was constructed. A semivariogram was used to describe the spatial correlation between the time-frequency power spectral parameters of two sites, employing an exponential function. The spatial correlation of the intra-event residuals of time-frequency power spectral parameters was measured using a geostatistical semivariogram, a distance-dependent function that is complementary to the correlation; that is, the greater the distance between two sites, the larger the semivariogram and the smaller the correlation. After obtaining the spatial correlation model of the intra-event residuals of time-frequency power spectral parameters, the correlation matrix of the entire field can be calculated, leading to the covariance matrix. Based on joint normal distribution sampling, the time-frequency power spectral parameters of each site can then be obtained.

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

[0055]

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

[0057] E T Total energy of earthquake motion;

[0058] A 2 (t): The time-varying normalized energy of earthquake motion;

[0059] The frequency variation of the normalized energy of earthquake motion;

[0060] The changes in time and frequency of the normalized energy of the ground motion in step two are as follows:

[0061]

[0062]

[0063] Where: μ t : The mean of the probability density function corresponding to the change of energy over time;

[0064] σ t : The standard deviation of the probability density function corresponding to the change of energy over time;

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

[0066] σ f : The standard deviation of the probability density function corresponding to the change of energy with frequency;

[0067] The empirical prediction equation in step two is in the following form:

[0068]

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

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

[0071] In step two, the historical earthquake records were selected from the NGA-West2 database, with a total of 3004 ground motions selected. The selection criteria were that the moment magnitude was greater than 4.5, each earthquake had at least 5 stations recording three-thirds of the earthquake data, and all selected earthquakes were strike-slip earthquakes.

[0072] The earthquake records selected in step two require several preprocessing operations; the specific operations are as follows: 1. Rotate the ground motions in the two horizontal directions to the vertical fault direction and the parallel fault direction, and only consider the ground motions in the vertical fault direction; 2. In order to minimize data redundancy and ensure the accuracy of power spectrum calculation, all records are truncated, retaining ground motions in the energy range of 0.1% to 99.9%, for the purpose of facilitating the prediction of the time envelope function.

[0073] The expression for the spectral representation in step three is as follows:

[0074]

[0075] In the above formula: x(t): the time history obtained using 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, which follows uniform sampling between 0 and 2π.

[0076] In step four, the observation records of historical major earthquakes at multiple sites are selected from the NGA-West2 database for six major earthquake events with multi-site observation records; and each site must have at least 30 pairs of station records at a distance of 2 kilometers.

[0077] The following description is based on specific embodiments.

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

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

[0080] Step 2: Construct the prediction equations for time-frequency power spectral parameters:

[0081] The energy variation in time and frequency is expressed using a log-normal probability density function, controlled by the mean and standard deviation. 3004 ground motions were selected from the NGA-West2 database, with selected earthquakes having a moment magnitude greater than 4.5 and each earthquake having at least five three-component ground motion records. Five time-frequency power spectral parameters were identified for each ground motion in the database. The mean and standard deviation of the time function were correlated with the fundamental characteristics of the ground motion (90% significant duration and 45% total energy accumulation time). The mean and standard deviation of the frequency function were identified using the method of moments, utilizing the correspondence between the first and second moments. The data distributions of the five identified time-frequency parameters were fitted, and the inverse standard normal cumulative probability distribution function was used to map the five time-frequency parameters to a normal space.

[0082] The relationships between five time-frequency parameters and magnitude, fault distance, and site shear wave velocity were analyzed to construct an empirical prediction equation. The Adam optimizer from machine learning was used to optimize the coefficients of the prediction equation, 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 a 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: Construct a spatial correlation model for time-frequency power spectral parameters:

[0086] Six major earthquake events with multi-site observation records were selected from the NGA-West2 database to provide data for building a spatial correlation model. The semivariogram, a geostatistical tool, was used to estimate the correlation between spatially related variables. The semivariogram is a function of the distance between sites and is complementary to the correlation; the greater the distance between sites, the larger the semivariogram and the smaller the correlation. An exponential semivariogram was chosen. For each earthquake, the semivariograms of five time-frequency power spectral parameters were regressed. The regression results of the six earthquakes were then averaged to obtain the overall spatial correlation model, as shown below. Figure 2 As shown;

[0087] Combining the above steps, perform a seismic field simulation:

[0088] Given earthquake scenario parameters: moment magnitude, fault distance at each field point, and site shear wave velocity, five time-frequency power spectrum parameters for each field point can be obtained based on the prediction equation. In the prediction equation calculation, the residual terms are randomly sampled, the intra-event residuals are obtained using a joint normal distribution, and the covariance is calculated based on a spatial correlation model; the inter-event residuals are not considered for correlation. The five predicted time-frequency power spectrum parameters are then substituted into the time-frequency power spectrum model, and the time-frequency power spectrum is converted into a time history using spectral representation, thus obtaining the entire seismic ground motion field.

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

[0090]

[0091] Among them: Z j : The j-th model parameter mapped onto the normal space; α j,1 : Corresponds to the first prediction equation coefficient of the j-th model parameter; there are a total of 8 coefficients in the prediction equation; M: moment magnitude; R: fault distance; V: site shear wave velocity; ε j Total residual.

[0092] This invention proposes a simplified time-frequency power spectrum model based on current stochastic seismic motion simulation methods. It introduces the spatial correlation of time-frequency power spectrum parameters and establishes a stochastic seismic motion field simulation method based on this spatial correlation. The time-frequency power spectrum is controlled by only five parameters, significantly improving the speed of seismic motion field simulation. For simulating a 200km × 200km free surface field with a 1km grid size and a total of 40401 field points, the proposed stochastic seismic motion field simulation method requires only 34.66 seconds per simulation. Furthermore, the empirical prediction equations effectively establish the correlation between seismic motion time-frequency parameters and earthquake scenario parameters, such as… Figures 3 to 5 As shown, in the simulation of real earthquakes, the simulation results are in good agreement with the actual records.

[0093] 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 invention can be implemented in other specific forms without departing from its spirit or essential characteristics. Therefore, the embodiments should be considered in all respects as exemplary and non-limiting, and the scope of the invention is defined by the appended claims rather than the foregoing description. Thus, all variations falling within the meaning and scope of equivalents of the claims are intended to be included within the present invention. No reference numerals in the claims should be construed as limiting the scope of the claims.

Claims

1. A method for stochastic simulation of seismic ground motion fields based on spatial correlation, characterized in that, Specifically, the following steps are included: Step 1: Construct a time-frequency power spectrum model for a single-field ground motion; the time-frequency power spectrum model for ground motion is as follows: ； Where: S(f,t): time-frequency power spectrum, f and t represent frequency and time, respectively; E T Total energy of a seismic earthquake; A 2 (t): The time-varying normalized energy of earthquake motion; The frequency variation of the normalized energy of earthquake motion; By treating seismic motion as a stochastic process, and describing the change of seismic motion energy with time and frequency based on the time-frequency power spectrum, a simplified model of the time-frequency power spectrum is proposed. Step 2: Construct the prediction equations for time-frequency power spectrum parameters; The log-normal distribution probability density function is used to describe the changes of normalized energy in time and frequency in the time-frequency power spectrum. Based on historical records, empirical prediction equations are built between five time-frequency power spectrum parameters and earthquake scenarios. The total residual, intra-event residual, and inter-event residual of the five time-frequency power spectrum parameters are calculated. The earthquake scenario includes magnitude, fault distance, and site shear wave velocity. The normalized energy of earthquake motion varies in time and frequency as follows: in: m t : The mean of the probability density function corresponding to the change of energy over time; s s t : The standard deviation of the probability density function corresponding to the change of energy over time; m m f :: This corresponds to the mean of the probability density function of energy changing with frequency; s s f : The standard deviation of the probability density function corresponding to the change of energy with frequency; The empirical prediction equation takes the following form: Where: Z j : The j-th model parameter mapped onto the normal space; a j,1 : The coefficient of the first prediction equation corresponding to the j-th model parameter. There are a total of 8 coefficients in the prediction equation; M: moment magnitude; R: fault distance; V: site shear wave velocity; e j Total residuals; Step 3: Transformation of time-frequency power spectrum into time history; By substituting the predicted time-frequency power spectrum parameters from step two into the simplified time-frequency power spectrum model from step one, the time-frequency power spectrum at a certain point can be obtained. Then, by using the spectral representation method to convert the time-frequency power spectrum into the ground motion time history, the simulated ground motion at a single field point can be obtained. Step 4: Construct a spatial correlation model for time-frequency power spectrum parameters; Based on observation records from multiple fields of historical major earthquakes, a spatial correlation model of intra-event residuals of time-frequency power spectrum parameters was constructed. A semi-variogram function was used to describe the spatial correlation between time-frequency power spectrum parameters of two fields, and an exponential function was selected.

2. The method for stochastic simulation of seismic ground motion fields based on spatial correlation according to claim 1, characterized in that, In step two, when constructing the empirical prediction equation for the time-frequency power spectrum parameters, it is necessary to identify the values ​​and distributions of the five time-frequency power spectrum parameters from historical earthquake records, and map the five parameters to the normal space based on the inverse standard normal cumulative probability distribution function.

3. The method for stochastic simulation of seismic ground motion fields based on spatial correlation according to claim 1, characterized in that, In step two, the historical earthquake records are selected from the NGA-West2 database; all selected historical earthquakes have a moment magnitude greater than 4.5, each earthquake has at least 5 stations recording three-thirds of the earthquake data, and all selected earthquakes are strike-slip earthquakes.

4. The method for stochastic simulation of seismic ground motion fields based on spatial correlation according to claim 3, characterized in that, The selected earthquake records need to undergo several preprocessing operations, including rotating the ground motions in the two horizontal directions to the vertical and parallel fault directions, considering only the ground motions in the vertical fault direction; and truncating all records to retain ground motions in the energy range of 0.1% to 99.9%.

5. The method for stochastic simulation of seismic ground motion fields based on spatial correlation according to claim 1, characterized in that, The expression for the spectral representation in step three is as follows: Where: x(t): the time history obtained using spectral representation; The predicted time-frequency power spectrum; Frequency interval; f n : The frequency at the nth frequency point; : Random phase, which follows uniform sampling between 0 and 2π.

6. The method for stochastic simulation of seismic ground motion fields based on spatial correlation according to claim 1, characterized in that, In step four, the observation records of historical major earthquakes at multiple sites are selected from the NGA-West2 database for six major earthquake events with multi-site observation records; and there must be at least 30 pairs of station records for each site distance.

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