A method for generating strong non-gaussian seismic ground motion samples

Abstract

This invention discloses a method for generating ground motion samples of strong non-Gaussian earthquakes. The method involves setting a non-stationary marginal cumulative distribution function for each measuring point and a non-Gaussian cross-correlation matrix between measuring points, thereby initializing the underlying non-stationary Gaussian vector process. Based on a non-stationary translational transformation relationship, the Gaussian cross-correlation matrix of the initialized underlying non-stationary Gaussian vector process is iteratively adjusted to obtain the Gaussian cross-correlation matrix corresponding to the target non-Gaussian cross-correlation matrix. The Gaussian cross-correlation matrix corresponding to the target non-Gaussian cross-correlation matrix is ​​then sparsely expanded to generate a set of samples of non-stationary Gaussian vector processes. Based on a non-stationary translational transformation relationship, the samples of the non-stationary Gaussian vector processes are mapped to a set of non-Gaussian, non-stationary earthquake ground motion samples through a nonlinear transformation. This invention overcomes the inherent biases of traditional methods based on Gaussian or stationary assumptions, and can provide more reliable input data for fields such as seismic design and earthquake risk assessment.

Description

Technical Field

[0001] This invention relates to the field of earthquake simulation, and more particularly to a method for generating ground motion samples of strong non-Gaussian earthquakes. Background Technology

[0002] Earthquake ground motion simulation is a core research foundation and a major bottleneck in earthquake engineering, civil engineering, and seismic resistance engineering. With the increasing demands for earthquake risk management, seismic design, and structural reliability assessment, the ability to accurately simulate earthquake ground motion has become a persistent pursuit in these fields. Measured ground motion data show that even over short distances, the amplitude, phase, and spectral characteristics of ground motions can change significantly, substantially affecting the dynamic response of spatially extended structures. However, existing earthquake ground motion simulation methods have significant limitations in describing the complexity of actual seismic signals. Traditional earthquake ground motion models are typically based on Gaussian or stationary process assumptions, such as the Spectral Representation Method (SRM). These methods are relatively simple in mathematical modeling and have certain applicability in some engineering applications. However, actual seismic signals often exhibit significant non-Gaussianity and non-stationarity, meaning their probability distribution deviates from a Gaussian distribution, and their statistical characteristics change dynamically over time. This makes it difficult for simulation methods based on Gaussian processes and stationary assumptions to effectively capture extreme values, heavy-tailed characteristics, and time-varying characteristics in seismic signals, resulting in significant deviations in simulation results when describing key features of strong earthquake events.

[0003] To overcome these problems, researchers have recently attempted various improvement methods, such as wavelet analysis, empirical mode decomposition (EMD), and time series modeling. However, while these methods improve simulation accuracy, they often suffer from high computational costs and insufficient modeling flexibility when dealing with complex seismic signals. Therefore, existing simulation methods struggle to effectively balance computational efficiency and accuracy, particularly lacking efficient means to simulate strongly non-Gaussian and non-stationary processes with high fidelity. Exploring a new method that balances efficiency and physical consistency has become an urgent need in the field. Summary of the Invention

[0004] This invention proposes a numerical method for simulating ground motion during strong non-Gaussian and non-stationary earthquakes, particularly suitable for seismic response analysis of spatially extended structures. The core of this method lies in combining the theory of non-stationary translational vector processes with a sparse decomposition algorithm. Through iterative optimization, it inversely derives the underlying Gaussian cross-correlation matrix from a given target non-Gaussian cross-correlation matrix and marginal cumulative distribution function (CDF), overcoming the challenge of generating high-precision time histories of non-Gaussian and non-stationary ground motions. This method is especially suitable for accurate seismic analysis of spatially extended structures such as long-span bridges and pipelines. Compared to conventional methods, it effectively overcomes the difficulty in accurately simulating the coupling of strong non-Gaussian and non-stationary characteristics, as well as the limitations of traditional spectral representation methods in spatial correlation modeling.

[0005] The technical solution adopted in this invention is: S1. Based on the statistical characteristics of the ground motion of the target earthquake, set the non-stationary edge cumulative distribution function of each measuring point and the target non-Gaussian cross-correlation matrix between measuring points, and then initialize the bottom non-stationary Gaussian vector process; S2. According to the non-stationary translation transformation relationship, the Gaussian cross-correlation matrix of the initialized bottom non-stationary Gaussian vector process is iteratively adjusted to obtain the Gaussian cross-correlation matrix corresponding to the target non-Gaussian cross-correlation matrix. S3. Using the random pre-orthogonal adaptive Fourier decomposition method, the Gaussian cross-correlation matrix corresponding to the target non-Gaussian cross-correlation matrix is ​​sparsely expanded to generate a set of samples of non-stationary Gaussian vector processes. S4. Based on the non-stationary translation transformation relationship, the samples of the non-stationary Gaussian vector process are mapped into a set of non-Gaussian, non-stationary seismic ground motion samples through nonlinear transformation.

[0006] The statistical characteristics of the ground motion of the target earthquake include non-stationary edge distribution characteristics and spatial correlation characteristics between measuring points.

[0007] In step S1, initializing the underlying non-stationary Gaussian vector process includes: setting each element of the initial cross-correlation matrix to the modulus of the target non-Gaussian cross-correlation matrix; and setting each component of the underlying non-stationary Gaussian vector process to a standard normal distribution.

[0008] The non-stationary translation transformation relationship is specifically expressed by the following formula: in, Represents a non-Gaussian process with respect to a vector. Indicates a non-Gaussian distribution. Indicates a Gaussian distribution. This represents a Gaussian vector process, where t represents time. Represents the Gaussian correlation function. This represents the value of the Gaussian random vector process at the first point. This represents the value at the second point in the Gaussian random vector process. Indicates time, Denotes the joint density function, This represents the Gaussian autocorrelation coefficient. Represents the cross-correlation function. This represents the variance at time t. This represents the variance at time s.

[0009] Step S2 specifically involves: S2.1. Based on the non-stationary translation transformation relationship, convert the Gaussian cross-correlation matrix of the current underlying non-stationary Gaussian vector process into the corresponding non-Gaussian cross-correlation matrix. S2.2 Calculate the relative error between the current non-Gaussian cross-correlation matrix and the target non-Gaussian cross-correlation matrix, and judge the relative error: If the relative error is greater than or equal to the preset relative error threshold, the iteration continues; If the relative error is less than the preset relative error threshold, the iteration stops, the current Gaussian cross-correlation matrix is ​​taken as the Gaussian cross-correlation matrix corresponding to the target non-Gaussian cross-correlation matrix, and the process jumps to S3. S2.3. Update the Gaussian cross-correlation matrix of the current underlying non-stationary Gaussian vector process based on the relative error to obtain the updated Gaussian cross-correlation matrix. Then, perform a positive semi-definite matrix projection on the updated Gaussian cross-correlation matrix to obtain a Gaussian cross-correlation matrix that satisfies the covariance matrix constraint condition. This Gaussian cross-correlation matrix is ​​used as the Gaussian cross-correlation matrix of the underlying non-stationary Gaussian vector process in the next iteration. Return to S2.1.

[0010] Step S3 specifically involves: S3.1. Select the optimal basis function from the pre-set overcomplete dictionary according to the random pre-orthogonal maximum selection principle, and perform orthogonalization to generate the standard orthogonal basis; S3.2. Using orthonormal bases, perform sparse expansion on the Gaussian cross-correlation matrix corresponding to the target non-Gaussian cross-correlation matrix to generate samples of the non-stationary Gaussian vector process.

[0011] The non-Gaussian, non-stationary earthquake ground motion samples are used for seismic design and earthquake risk assessment.

[0012] The beneficial effects of this invention are: (1) High-precision simulation of complex statistical characteristics: This invention directly models the strong non-Gaussianity and non-stationarity of ground motion. Through non-stationary translation theory and iterative optimization, it can accurately reproduce key features such as extreme values, heavy-tailed distribution and time-varying spectrum in actual ground motion, overcoming the inherent bias of traditional methods based on Gaussian or stationary assumptions.

[0013] (2) Solved the problem of “reverse transformation”: In response to the “reverse problem” of inferring the underlying Gaussian process from a non-Gaussian target, this invention proposes a systematic iterative optimization and mathematical compatibility judgment mechanism (such as handling correlation overbound and non-positive definite matrices), which effectively solves the problem of “incompatibility” of target statistics that is common in existing methods and improves the robustness and practicality of the method.

[0014] (3) Balancing computational efficiency and model flexibility: By combining the SPOAFD sparse representation method, the computational complexity and data redundancy are significantly reduced while ensuring high-precision decomposition of non-stationary Gaussian processes. Compared with methods such as Empirical Mode Decomposition (EMD) or wavelet analysis, it has a clear mathematical foundation and convergence guarantee, and is more stable and flexible in dealing with high-dimensional vector processes.

[0015] (4) Strong engineering applicability: The entire process (from the definition of statistical objectives to the final time history generation) forms a complete closed-loop solution that can directly serve fields such as seismic design and earthquake risk assessment, providing more reliable seismic motion input data. Attached Figure Description

[0016] Figure 1 This is a comparison chart of the covariance of the present invention and two traditional methods. Detailed Implementation

[0017] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0018] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and do not limit the scope of protection of this invention.

[0019] This embodiment includes the following steps: S1. Based on the statistical characteristics (non-stationary, non-Gaussian) of the ground motion of the target earthquake, set the non-stationary edge cumulative distribution function of each measuring point and the target non-Gaussian cross-correlation matrix between measuring points, and then initialize the underlying non-stationary Gaussian vector process; That is, based on the modulus of the target non-Gaussian cross-correlation matrix, the underlying Gaussian cross-correlation matrix is ​​initialized, and the underlying Gaussian process is set to follow a standard normal distribution.

[0020] S2. According to the non-stationary translation transformation relationship, the Gaussian cross-correlation matrix of the initialized bottom non-stationary Gaussian vector process is iteratively adjusted to obtain the Gaussian cross-correlation matrix corresponding to the target non-Gaussian cross-correlation matrix. The improved Iterative Translation Algorithm (ITAM) calculates the non-Gaussian cross-correlation matrix corresponding to the current Gaussian matrix through a forward translation transformation and compares it with the target matrix. Based on the error, the underlying Gaussian cross-correlation matrix is ​​iteratively updated, and the nearest positive semi-definite matrix projection method is used to ensure its mathematical feasibility until convergence.

[0021] S3. Using the random pre-orthogonal adaptive Fourier decomposition method, the Gaussian cross-correlation matrix corresponding to the target non-Gaussian cross-correlation matrix is ​​sparsely expanded to generate a set of samples of non-stationary Gaussian vector processes. This method utilizes the converged underlying Gaussian cross-correlation matrix and employs the Stochastic Preorthogonal Adaptive Fourier Decomposition (SPOAFD) method to efficiently generate samples of non-stationary Gaussian vector processes. SPOAFD achieves a sparse and high-precision representation of Gaussian processes by adaptively selecting the optimal basis functions.

[0022] S4. Based on the non-stationary translation transformation relationship, the samples of the non-stationary Gaussian vector process generated in S3 are mapped into a set of non-Gaussian, non-stationary seismic ground motion samples through nonlinear transformation.

[0023] That is, by applying the theory of non-stationary translation transformation, the generated non-stationary Gaussian samples are mapped through nonlinear transformation into the final non-Gaussian, non-stationary seismic ground motion samples with specified statistical characteristics.

[0024] The statistical characteristics of the ground motion of the target earthquake include non-stationary edge distribution characteristics and spatial correlation characteristics between measuring points.

[0025] In step S1, initializing the underlying non-stationary Gaussian vector process includes: setting each element of the initial cross-correlation matrix to the modulus of the target non-Gaussian cross-correlation matrix; and setting each component of the underlying non-stationary Gaussian vector process to a standard normal distribution.

[0026] Specifically, problem definition and initialization: defining the objective as a non-stationary, non-Gaussian random vector process. Define the non-stationary marginal cumulative distribution function of each component. and the target non-Gaussian cross-correlation function matrix Initialization of the underlying non-stationary Gaussian vector cross-correlation function matrix To accelerate convergence, its initial value can be set as the modulus of the target non-Gaussian cross-correlation matrix. .set up The marginal distribution is a standard normal distribution (zero mean, unit variance), and this assumption is only used to simplify calculations during the iterative inversion phase.

[0027] The non-stationary translation transformation relationship is specifically expressed by the following formula: in, Represents a non-Gaussian process with respect to a vector. Indicates a non-Gaussian distribution. Indicates a Gaussian distribution. This represents a Gaussian vector process, where t represents time. Represents the Gaussian correlation function. This represents the value of the Gaussian random vector process at the first point. This represents the value at the second point in the Gaussian random vector process. Indicates time, Denotes the joint density function, This represents the Gaussian autocorrelation coefficient. Represents the cross-correlation function. This represents the variance at time t. This represents the variance at time s.

[0028] The specific non-stationary translation transformation relationship is used according to the following steps: Nonlinear transformation from non-stationary Gaussian components The mapping yields: in Translation vector components The inverse function of the defined edge CDF. Then, the non-stationary, non-Gaussian CC matrix. The elements and their corresponding non-stationary Gaussian C-matrix The relationship between elements can be represented as: in, Expressing expectations, This indicates a Gaussian correlation function with normalization; This represents the joint Gaussian probability density function (PDF). Represents the underlying Gaussian components The standard deviation. Describes the inverse cumulative distribution function of the non-Gaussian marginal distribution of the j-th component. The cumulative distribution function of the Gaussian marginal distribution of the j-th component is given. The inverse cumulative distribution function represents the non-Gaussian marginal distribution of the k-th component. : The cumulative distribution function of the Gaussian marginal distribution of the k-th component.

[0029] Step S2 specifically involves: S2.1. Based on the non-stationary translation transformation relationship, convert the Gaussian cross-correlation matrix of the current underlying non-stationary Gaussian vector process into the corresponding non-Gaussian cross-correlation matrix. S2.2 Calculate the relative error between the current non-Gaussian cross-correlation matrix and the target non-Gaussian cross-correlation matrix, and judge the relative error: If the relative error is greater than or equal to the preset relative error threshold, the iteration continues, that is, S2.3 continues to be executed; If the relative error is less than the preset relative error threshold, the iteration stops, the current Gaussian cross-correlation matrix is ​​taken as the Gaussian cross-correlation matrix corresponding to the target non-Gaussian cross-correlation matrix, and the process jumps to S3. S2.3. Update the Gaussian cross-correlation matrix of the current underlying non-stationary Gaussian vector process based on the relative error to obtain the updated Gaussian cross-correlation matrix. Then, perform a positive semi-definite matrix projection on the updated Gaussian cross-correlation matrix to obtain a Gaussian cross-correlation matrix that satisfies the covariance matrix constraint condition. This Gaussian cross-correlation matrix is ​​used as the Gaussian cross-correlation matrix of the underlying non-stationary Gaussian vector process in the next iteration. Return to S2.1.

[0030] Specifically, iteratively inverting the underlying Gaussian cross-correlation matrix An improved iterative translation algorithm (ITAM) is used for iterative optimization, and the process is as follows: (1) In the i-th iteration, the non-stationary translation transformation relationship is used to transform the current bottom-level Gaussian cross-correlation matrix. Convert to the corresponding non-Gaussian cross-correlation matrix .

[0031] (2) Calculate the corresponding non-Gaussian cross-correlation matrix With the target matrix The relative percentage error between the moduli of each element. If the overall error tends to stabilize and is below the preset threshold, the iteration converges and proceeds to step three; otherwise, continue to the next step.

[0032] (3) Based on the error, calculate the Gaussian cross-correlation matrix for the next iteration using a specific update equation (such as adaptive adjustment based on the ratio). .

[0033] (4) For the updated Perform the projection of the nearest positive semidefinite matrix (using the Frobenius norm minimization method) to ensure that it satisfies the mathematical condition (positive semidefiniteness) for a covariance function.

[0034] (5) Repeat steps (1)-(4) until convergence. Finally, the underlying Gaussian cross-correlation matrix, compatible with the non-Gaussian statistical properties of the target, is obtained. .

[0035] In S2, the construction of the underlying Gaussian cross-correlation matrix adopts an improved ITAM iterative inversion method. By iteratively updating the parameters of the Gaussian cross-correlation matrix multiple times, it makes it statistically approximate the target non-Gaussian cross-correlation matrix.

[0036] Step S3 specifically involves: S3.1 Select an overcomplete dictionary that satisfies the boundary conditions, and then adaptively select the optimal basis function from the preset overcomplete dictionary according to the random pre-orthogonal maximum selection principle, and perform orthogonalization processing to generate an orthogonal basis; S3.2. Using the orthonormal basis generated in S3.1, perform a sparse expansion on the Gaussian cross-correlation matrix corresponding to the target non-Gaussian cross-correlation matrix to generate samples of the non-stationary Gaussian vector process.

[0037] Specifically, step S3 treats the underlying Gaussian process Y(t) as a function defined in Bochner space. An overcomplete dictionary (e.g., a parameterized Poisson kernel) satisfying the boundary vanishing condition (BVC) is constructed or selected. The stochastic preorthogonal adaptive Fourier decomposition (SPOAFD) algorithm is applied: optimal basis functions are adaptively selected from the dictionary sequentially using the stochastic preorthogonal maximum selection principle (SPOMSP). After each selection, Gram-Schmidt orthogonalization is performed to generate a set of orthonormal bases. Based on this base, the optimal basis functions are selected according to the following conditions: The Gaussian process with statistical properties is sparsely expanded to generate a series of samples of non-stationary Gaussian vector processes, which is implemented as y_sample(t).

[0038] The Stochastic Preorthogonal Adaptive Fourier Decomposition (SPOAFD) method in S3 can be used for any Its SPOAFD series is well defined and in Converging to That is, the parameters selected through the principle of maximum selection. ,satisfy in } is the core An orthogonal system. In this invention, we employ a parameterized Poisson kernel. The pre-dictionary consists of: Based on this, a corresponding SPOAFD expansion was developed for stochastic processes.

[0039] The non-Gaussian, non-stationary seismic ground motion samples are used as input for seismic design and seismic risk assessment.

[0040] Finally, non-Gaussian ground motion samples are generated through mapping: using the non-stationary translation transformation relationship determined after convergence, each Gaussian sample y_sample(t) generated in step three is transformed through a nonlinear transformation (using the inverse function of the defined edge CDF). The data is mapped to the final non-Gaussian, non-stationary seismic ground motion samples x_sample(t). These samples strictly conform to the target edge distribution and cross-correlation structure specified in step one. Thus, the entire simulation process from the statistical target to high-precision time history samples is completed. Table 1 below shows the actual seismic ground motion data using different methods, demonstrating that the SPOAFD method possesses both fast convergence and optimal error.

[0041] Table 1. Absolute error of covariance for different numbers of terms under different methods In practical engineering applications, the non-Gaussian, non-stationary seismic ground motion samples generated by this invention can be directly used as input ground motion data in structural seismic analysis and seismic risk assessment. The specific application methods are as follows.

[0042] First, based on the site conditions of the target project (such as site type, magnitude, epicentral distance, soil structure, etc.), the non-stationary marginal cumulative distribution function of seismic ground motion at each measuring point is determined to characterize the statistical characteristics of the change of ground motion intensity over time. At the same time, combined with multi-point seismic observation data or engineering experience models, a non-Gaussian cross-correlation matrix between measuring points is constructed to reflect spatial correlation and phase difference characteristics.

[0043] Secondly, a set of non-Gaussian, non-stationary seismic ground motion samples generated using the method of this invention can be used as multi-point seismic inputs and directly applied to the time history analysis models of structural systems such as bridges, super high-rise buildings, underground structures, and lifeline projects, for conducting structural dynamic response analysis, vulnerability analysis, and collapse probability assessment.

[0044] Furthermore, in numerical simulation of earthquake engineering, the ground motion samples generated by this invention can replace traditional artificially synthesized seismic waves based on Gaussian or stationary assumptions, significantly improving the ability of ground motion to fit actual strong earthquake records in terms of peak value, duration, energy accumulation, and spatial correlation, thereby enhancing the reliability of seismic design parameter selection and earthquake risk assessment results.

[0045] Therefore, this invention is not only a method for statistical modeling of seismic motion, but also a method for generating seismic ground motion samples for engineering seismic design and earthquake safety assessment, with clear engineering applications and practical technical effects. The above detailed embodiments illustrate the technical solution and beneficial effects of the present invention. It should be understood that the above description is only the most preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, additions, and equivalent substitutions made within the scope of the principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for generating ground motion samples of strong non-Gaussian earthquakes, characterized in that, The method includes the following steps: S1. Based on the statistical characteristics of the ground motion of the target earthquake, set the non-stationary edge cumulative distribution function of each measuring point and the target non-Gaussian cross-correlation matrix between measuring points, and then initialize the bottom non-stationary Gaussian vector process; S2. According to the non-stationary translation transformation relationship, the Gaussian cross-correlation matrix of the initialized bottom non-stationary Gaussian vector process is iteratively adjusted to obtain the Gaussian cross-correlation matrix corresponding to the target non-Gaussian cross-correlation matrix. S3. Using the random pre-orthogonal adaptive Fourier decomposition method, the Gaussian cross-correlation matrix corresponding to the target non-Gaussian cross-correlation matrix is ​​sparsely expanded to generate a set of samples of non-stationary Gaussian vector processes. S4. Based on the non-stationary translation transformation relationship, the samples of the non-stationary Gaussian vector process are mapped into a set of non-Gaussian, non-stationary seismic ground motion samples through nonlinear transformation.

2. The method for generating ground motion samples of strong non-Gaussian earthquakes according to claim 1, characterized in that: The statistical characteristics of the ground motion of the target earthquake include non-stationary edge distribution characteristics and spatial correlation characteristics between measuring points.

3. The method for generating ground motion samples of strong non-Gaussian earthquakes according to claim 1, characterized in that: In step S1, initializing the underlying non-stationary Gaussian vector process includes: setting each element of the initial cross-correlation matrix to the modulus of the target non-Gaussian cross-correlation matrix; and setting each component of the underlying non-stationary Gaussian vector process to a standard normal distribution.

4. The method for generating ground motion samples of strong non-Gaussian earthquakes according to claim 1, characterized in that: The non-stationary translation transformation relationship is specifically expressed by the following formula: in, Represents a non-Gaussian process with respect to a vector. Indicates a non-Gaussian distribution. Indicates a Gaussian distribution. This represents a Gaussian vector process, where t represents time. Represents the Gaussian correlation function. This represents the value of the Gaussian random vector process at the first point. This represents the value at the second point in the Gaussian random vector process. Indicates time, Denotes the joint density function, This represents the Gaussian autocorrelation coefficient. Represents the cross-correlation function. This represents the variance at time t. This represents the variance at time s.

5. The method for generating ground motion samples of strong non-Gaussian earthquakes according to claim 1, characterized in that: Step S2 specifically involves: S2.

1. Based on the non-stationary translation transformation relationship, convert the Gaussian cross-correlation matrix of the current underlying non-stationary Gaussian vector process into the corresponding non-Gaussian cross-correlation matrix. S2.2 Calculate the relative error between the current non-Gaussian cross-correlation matrix and the target non-Gaussian cross-correlation matrix, and judge the relative error: If the relative error is greater than or equal to the preset relative error threshold, the iteration continues; If the relative error is less than the preset relative error threshold, the iteration stops, the current Gaussian cross-correlation matrix is ​​taken as the Gaussian cross-correlation matrix corresponding to the target non-Gaussian cross-correlation matrix, and the process jumps to S3. S2.

3. Update the Gaussian cross-correlation matrix of the current underlying non-stationary Gaussian vector process based on the relative error to obtain the updated Gaussian cross-correlation matrix. Then, perform a positive semi-definite matrix projection on the updated Gaussian cross-correlation matrix to obtain a Gaussian cross-correlation matrix that satisfies the covariance matrix constraint condition. This Gaussian cross-correlation matrix is ​​used as the Gaussian cross-correlation matrix of the underlying non-stationary Gaussian vector process in the next iteration. Return to S2.

1.

6. The method for generating ground motion samples of strong non-Gaussian earthquakes according to claim 1, characterized in that: Step S3 specifically involves: S3.

1. Select the optimal basis function from the pre-set overcomplete dictionary according to the random pre-orthogonal maximum selection principle, and perform orthogonalization to generate the standard orthogonal basis; S3.

2. Using orthonormal bases, perform sparse expansion on the Gaussian cross-correlation matrix corresponding to the target non-Gaussian cross-correlation matrix to generate samples of the non-stationary Gaussian vector process.

7. The method for generating ground motion samples of strong non-Gaussian earthquakes according to claim 1, characterized in that: The non-Gaussian, non-stationary earthquake ground motion samples are used for seismic design and earthquake risk assessment.

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