Finite fault simulation method for generating sub-source random vibration using the trigonometric series method

Abstract

This application relates to a finite fault simulation method for generating sub-source random vibration using the trigonometric series method, including the steps of: obtaining the sub-fault acceleration source spectrum, normalizing the sub-fault acceleration source spectrum to obtain the probability density function, and generating the sub-fault random frequency components; superimposing and combining the simple harmonic waves corresponding to all the generated sub-fault random frequency components in the time domain to obtain the random vibration of the sub-fault, then obtaining the Fourier amplitude spectrum of the sub-fault acceleration, and then performing the inverse Fourier transform to the time domain to obtain the sub-fault acceleration time history, and superimposing with an additional delay time to obtain the ground motion time history. The seismic energy simulated by this application is consistent with the target spectrum at both high frequencies and low frequencies, improving the accuracy of the simulation.

Description

Technical Field

[0001] The present application relates to a method for simulating ground motion, and particularly to a finite-fault simulation method for generating random vibrations of sub-sources by using the trigonometric series method, which is applicable to the technical field of earthquake services. Background Art

[0002] The stochastic finite-fault simulation method is currently the most important method for simulating near-field high-frequency ground motion in the field of earthquake engineering, and is one of the most important data sources for the input of ground motion in structural seismic design, which can provide a basis for post-disaster rescue and reconstruction in the target area. Although the stochastic finite-fault method has the advantages of clear physical meaning, few input parameters, and convenient estimation, it has the defect of insufficient low-frequency simulation and cannot meet the high-precision simulation requirements of both high frequency and low frequency at the same time.

[0003] Generally speaking, any given ground motion time history can be regarded as a combination of many simple harmonic waves with different frequencies. The frequency components of ground motion include frequency and amplitude. Frequency refers to the frequencies of the individual simple harmonic waves contained in the ground motion signal, and amplitude refers to the intensity or magnitude of each frequency in the ground motion signal. The frequency components of ground motion are one of the important parameters reflecting the characteristics of ground motion. The recorded ground motion shows different frequency components, and its random frequency components depend on the source mechanism and magnitude, as well as the epicentral distance and site conditions. It was initially thought that ground motion was composed of various periodic components in a fixed ratio, and later it was gradually realized that the frequency composition of ground motion is variable and different. Ground motion can have different frequency compositions. The larger the magnitude, the richer the long-period components. At present, there is only a qualitative understanding of the distribution law of high- and low-frequency components of ground motion between large and small earthquakes, and there is no quantitative determination method.

[0004] Therefore, providing a finite-fault simulation method that can meet the high-precision simulation requirements of both high frequency and low frequency at the same time is an urgent problem to be solved in the prior art. Summary of the Invention

[0005] The present application relates to a finite-fault simulation method for generating random vibrations of sub-sources by using the trigonometric series method. To consider the frequency characteristics of ground motion, the normalized acceleration source spectrum is used as the probability density function to control the random frequency components of sub-faults, and a method of combining the trigonometric series method with the stochastic finite-fault method is proposed to synthesize the ground motion acceleration time history. The ground motion simulated by this method is consistent with the target spectrum at both high frequency and low frequency, improving the simulation accuracy and can be used as the input of ground motion for structural seismic design.

[0006] A finite-fault simulation method for generating random vibrations of sub-sources by using the trigonometric series method according to the present application includes the following steps:

[0007] (1) Obtain the sub-fault acceleration source spectrum according to the input source parameters;

[0008] (2) Normalize the sub-fault acceleration source spectrum to obtain the probability density function;

[0009] (3) Generate the random frequency components of the sub-fault according to the obtained probability density function;

[0010] (5) Superimpose and combine the simple harmonic waves corresponding to all the generated random frequency components of the sub-faults in the time domain to obtain the random vibration of the sub-fault;

[0011] (5) Window the random vibration of the sub-fault and transform it to the frequency domain through Fourier transform. Superimpose the propagation path term and the site term in the frequency domain to obtain the Fourier amplitude spectrum of the sub-fault acceleration;

[0012] (6) The obtained Fourier amplitude spectrum of the sub-fault acceleration is inverse Fourier transformed to the time domain to obtain the sub-fault acceleration time history, and the additional delay time is superimposed to obtain the ground motion time history. Preferably, the ground motion time history can be further converted into a pseudo-acceleration response spectrum.

[0013] Among them, in step (2), first calculate the source spectrum within the frequency range to generate the initial probability density function; then use the trapezoidal rule to numerically integrate this function within the given frequency range to obtain the total area value; finally, divide the value of each probability density function by the total area value to achieve normalization.

[0014] Among them, in step (3), by calculating the cumulative distribution function, convert the normalized probability density function into cumulative probability values, then generate uniformly distributed random numbers, and use the inverse interpolation method of the cumulative distribution function to map these random numbers to the value range of the target distribution to generate random samples that conform to the specified probability density function, and obtain the frequency components of the random vibration of the sub-fault.

[0015] Among them, the random vibration x ij (t) of the sub-fault is expressed as:

[0016]

[0017] In the formula, is the amplitude of the theoretical source acceleration spectrum, where C = R θφ FV / (4πρβ 3 ) is the proportionality coefficient, R θφ represents the source radiation pattern, F represents the free surface amplification effect factor, V represents the horizontal component coefficient of the shear wave energy, ρ and β are the density and shear wave velocity of the medium near the source respectively, and the units are g / cm 3 and km / s respectively;

[0018] Among them, is the corner frequency of the ij-th sub-fault, N is the number of sub-faults on the fault rupture surface, NR (t) is the cumulative slip area of the sub - fault, Δσ is the stress drop; M 0 is the total seismic moment of the earthquake, lgM 0 = 1.5M w + 16.05, where M w is the moment magnitude of the earthquake; ω k = 2πf k , f k is the random frequency component of the sub - fault with a set probability density, φ k is the random phase.

[0019] Among them, the Fourier amplitude spectrum A ij (f) of the sub - fault acceleration is:

[0020]

[0021] In the formula, H ij is the scaling factor to ensure the conservation of high - frequency radiation energy of the sub - source; P(R, f) is the propagation path term, and G(f) is the site term.

[0022] Among them, the propagation path term P(R, f) is expressed as the product of the geometric spreading attenuation Z(R) caused by the diffusion of seismic waves in the rock medium and the anelastic attenuation A(R, f) caused by the inhomogeneity of the rock medium, P(R, f)=Z(R)×A(R, f); the geometric spreading attenuation Z(R) adopts the following three - segment geometric attenuation model:

[0023]

[0024] Among them, R represents the fault distance;

[0025] The anelastic attenuation A(R, f) is expressed as:

[0026]

[0027] Among them, Q(f)=Q 0 f η , Q is the quality factor, Q 0 represents the value of the quality factor at a frequency of 1 Hz, η is the seismic activity factor of the region, which varies with the region, and β is the shear wave velocity.

[0028] Among them, the site term G(f) is expressed as the product of the near - surface site amplification effect and the attenuation of high - frequency energy of seismic waves within the local site; the near - surface site amplification effect is obtained by linearly interpolating between adjacent values to calculate the site amplification value at the frequency f; the attenuation of high - frequency energy of seismic waves within the local site is represented by the κ - filter as exp(-πκf), and κ is a parameter determined by the distance and site conditions. Description of the Drawings

[0029] Figure 1 are the acceleration time history simulation results and the observation results of the embodiments of the present application.

[0030] Figure 2 are the PSA simulation results and the observation results of the embodiments of the present application. Specific Embodiments

[0031] To make the objectives, technical solutions and advantages of the present application clearer and more understandable, the embodiments of the present application will be described in detail below with reference to the accompanying drawings. It should be noted that, without conflict, the embodiments in the present application and the features in the embodiments can be combined arbitrarily with each other. The present application proposes a finite fault simulation method for generating sub-source random vibration by using the trigonometric series method, which belongs to a simulation method of ground motion and can be used in specific fields such as seismic structure design and earthquake law research.

[0032] A finite fault simulation method for generating sub-source random vibration by using the trigonometric series method according to the present application includes the following steps:

[0033] (1) Obtain the sub-fault acceleration source spectrum according to the input source parameters.

[0034] Detect or calculate through seismic detection equipment to obtain the source parameters of the earthquake. When the source parameters of an earthquake are given, such as moment magnitude, number of sub-faults, slip distribution, stress drop, etc., according to the Brune model, the acceleration source spectrum of each sub-fault of this earthquake can be obtained:

[0035]

[0036] Wherein,

[0037] In the formula, C = R θφ FV / (4πρβ 3 ) is the proportionality coefficient, R θφ represents the source radiation pattern. After selecting appropriate off-source angles and azimuth angles, the average value can be taken as 0.55; F represents the free surface amplification effect factor, which can be taken as 2 for SH waves; V represents the horizontal component coefficient of shear wave energy, which can be taken as ρ and β are the density and shear wave velocity of the medium near the source respectively, and the units are g / cm 3 and km / s respectively. M 0ij is the seismic moment of the ij-th sub-fault, slip ij is the slip weight on each sub-fault, nl and nw are the numbers of sub-faults along the strike and dip of the fault respectively; M 0 is the total seismic moment of the earthquake, lgM 0 = 1.5M w + 16.05, where Mw is the moment magnitude of the earthquake; f is the frequency, is the corner frequency of the ij-th subfault, where N is the number of subfaults on the fault rupture surface, N R (t) is the cumulative slip area of the subfault, and Δσ is the stress drop.

[0038] (2) Normalize the subfault acceleration source spectrum to obtain the probability density function;

[0039] First, calculate the source spectrum within the frequency range to generate the initial probability density function, then use the trapezoidal rule to numerically integrate this function within the given frequency range to obtain the total area value. Finally, divide the value of each probability density function by the total area to achieve normalization, making the total integral of the function equal to 1, thus ensuring compliance with the definition of the probability density function.

[0040] (3) Generate the subfault random frequency components based on the obtained probability density function;

[0041] By calculating the cumulative distribution function (CDF), transform the normalized probability density function into cumulative probability values, then generate uniformly distributed random numbers, and use the inverse interpolation method of the cumulative distribution function to map these random numbers to the value range of the target distribution, thereby generating random samples that conform to the specified probability density function to obtain the subfault random frequency component f k .

[0042] (4) Superpose and combine the generated harmonic waves corresponding to all the subfault random frequency components f k in the time domain, where the amplitude C of each frequency harmonic wave is the amplitude of the source spectrum corresponding to this frequency, and the phase is a random phase, to obtain the random vibration x k of the subfault. ij (t).

[0043] The random vibration x ij (t) of the subfault can be expressed as:

[0044]

[0045] In the formula, is the amplitude of the theoretical source acceleration spectrum, where C = R θφ FV / (4πρβ 3 ) is the proportionality coefficient, R θφ represents the source radiation pattern, and after selecting appropriate off-source angles and azimuth angles, the average value can be taken as 0.55; F represents the free surface amplification effect factor, which can be taken as 2 for SH waves, V represents the horizontal component coefficient of the shear wave energy, and can be taken as ρ and β are the density and shear wave velocity of the medium near the source, with units of g / cm3 and km / s; is the corner frequency of the ij-th subfault, where N is the number of subfaults on the fault rupture surface, N R (t) is the cumulative slip area of the subfault, Δσ is the stress drop; M 0 is the total seismic moment of the earthquake, lgM 0 = 1.5M w + 16.05, where M w is the moment magnitude of the earthquake; ω k = 2πf k , f k is the random frequency component of the subfault with a set probability density, φ k is the random phase.

[0046] (5) Window the random vibration of the obtained subfaults, and transform it to the frequency domain through Fourier transform. Superimpose the propagation path term and the site term in the frequency domain to obtain the Fourier amplitude spectrum A ij (f) of the subfault acceleration, as shown in Equation (3).

[0047] The Fourier amplitude spectrum A ij (f) of the subfault acceleration:

[0048]

[0049] In the formula, H ij is the scaling factor to ensure the conservation of high-frequency radiation energy of the sub-source; P(R,f) is the propagation path term, and G(f) is the site term.

[0050] The propagation path term P(R,f) is expressed as the product of the geometric spreading attenuation Z(R) caused by the diffusion of seismic waves in the rock medium and the anelastic attenuation A(R,f) caused by the inhomogeneity of the rock medium, P(R,f) = Z(R) × A(R,f).

[0051] The geometric spreading attenuation Z(R) is often expressed by a piecewise attenuation function, and the following three-segment geometric attenuation model can be adopted:

[0052]

[0053] Among them, R represents the fault distance.

[0054] The anelastic attenuation A(R,f) is expressed as:

[0055]

[0056] Among them, Q(f) = Q 0 f η , Q is the quality factor, Q 0It represents the value of the quality factor at a frequency of 1 Hz. η is the seismic activity factor of the region, which varies with the region. β is the shear wave velocity.

[0057] The site term G(f) is expressed as the product of the near-surface site amplification effect and the attenuation of the high-frequency energy of seismic waves within the local site. Among them, the near-surface site amplification effect is obtained by linearly interpolating between adjacent values to calculate the site amplification value at the frequency f. The attenuation of the high-frequency energy of seismic waves within the local site is represented by the κ filter as exp(-πκf), where κ is a parameter determined by the distance and site conditions, which describes the influence of the energy dissipation of the wave from the source to the site on the wave field during the propagation of the wave in the crust.

[0058] (6) The obtained Fourier amplitude spectrum A ij (f) of the sub-fault acceleration is transformed back to the time domain through the inverse Fourier transform to obtain the sub-fault acceleration time history a ij (t), and a certain delay time Δt ij is added and superimposed to obtain the ground motion time history a(t). Further, a(t) can also be converted into a pseudo-acceleration response spectrum PSA for seismic design services.

[0059] The superimposed ground motion acceleration time history a(t) is as follows:

[0060]

[0061] In the formula, nl and nw are the numbers of sub-sources of the fault along the strike and dip directions respectively, and Δt ij is the superposition of the time delay for the rupture to propagate to the ij-th sub-source and the time delay for the seismic wave to propagate from the ij-th sub-source to the site due to the difference in propagation distance.

[0062] Embodiment

[0063] Taking the actual M s 8.0 earthquake that occurred in a certain place as an example, this method is used to simulate the ground motion of a certain station. To ensure the accuracy of the simulation results, the station is simulated 30 times and the average is taken as the final simulation result, and it is compared with the actual records of the station, as Figure 1 and Figure 2 shown. Figure 1 In, the abscissa represents time, with the unit of s; the ordinate represents acceleration, with the unit of cm / s 2 . Figure 1 In, the abscissa represents time, with the unit of s; the ordinate represents PSA, with the unit of cm / s 2 .

Claims

1. A finite fault simulation method for generating sub-source random vibration using a trigonometric series method, characterized in that: The following steps are involved: (1) Obtain the sub-fault acceleration source spectrum based on the input source parameters; (2) Normalize the sub-fault acceleration source spectrum to obtain the probability density function; (3) generating sub-fault random frequency components according to the obtained probability density function; (4) Superimposing and combining the simple harmonic waves corresponding to the random frequency components of all generated sub-faults in the time domain to obtain the random vibration of the sub-faults; (5) The random vibration of the sub-fault is windowed and transformed into the frequency domain through Fourier transform. The propagation path term and the site term are superimposed in the frequency domain to obtain the Fourier amplitude spectrum of the sub-fault acceleration; (6) The obtained sub-fault acceleration Fourier amplitude spectrum is transformed to the time domain through inverse Fourier transformation to obtain the sub-fault acceleration time history, and the additional delay time is superimposed to obtain the seismic motion time history.

2. The finite fault simulation method according to claim 1, characterized in that: After step (6), the earthquake time history is further converted into a pseudo-acceleration response spectrum.

3. The finite fault simulation method according to claim 1 or 2, characterized in that: In step (2), the source spectrum within the frequency range is first calculated to generate an initial probability density function; then the function is numerically integrated within the given frequency range using the trapezoidal rule to obtain the total area value; finally, the value of each probability density function is divided by the total area value to achieve normalization.

4. The finite fault simulation method according to claim 3, characterized in that: In step (3), the normalized probability density function is converted into a cumulative probability value by calculating the cumulative distribution function, and then uniformly distributed random numbers are generated. These random numbers are mapped to the value range of the target distribution using the reverse interpolation method of the cumulative distribution function, generating random samples that conform to the specified probability density function and obtaining the frequency components of the random vibration of the sub-fault.

5. The finite fault simulation method according to claim 4, characterized in that: Random vibration of sub-fault x ij (t) is expressed as: In the formula, is the amplitude of the theoretical earthquake source acceleration spectrum, where C = R θφ FV(4πρβ 3 ) is the proportionality coefficient, R θφ represents the source radiation pattern, F represents the free surface amplification effect factor, V represents the horizontal component coefficient of shear wave energy, ρ and β represent the density and shear wave velocity of the medium near the source, respectively, in units of g / cm 3 and km / s; in, is the corner frequency of the ijth sub-fault, N is the number of sub-faults on the fault rupture surface, N R (t) is the cumulative slip area of ​​the sub-fault, Δσ is the stress drop; M0 is the total seismic moment of the earthquake, lgM0 = 1.5M w +16.05, of which M w is the moment magnitude of the earthquake; ω k =2πf k , fk is the random frequency component of the sub-fault with a set probability density, and φk is the random phase.

6. The finite fault simulation method according to claim 5, characterized in that: The sub-fault acceleration Fourier amplitude spectrum A ij (f) is: In the formula, H ij is the scaling factor to ensure the conservation of high-frequency radiation energy of the sub-source; P(R, f) is the propagation path term, and G(f) is the site term.

7. The finite fault simulation method according to claim 6, characterized in that: The propagation path term P(R, f) is expressed as the product of the geometric diffusion attenuation Z(R) caused by the diffusion of seismic waves in the rock medium and the anelastic attenuation A(R, f) caused by the heterogeneity of the rock medium, P(R, f) = Z(R) × A(R, f); The geometric diffusion attenuation Z(R) adopts the following three-stage geometric attenuation model: Where R represents the fault distance; The anelastic attenuation A(R, f) is expressed as: Where Q(f) = Q0f η , Q is the quality factor, Q0 represents the value of the quality factor when the frequency is 1 Hz, η is the regional seismic activity factor, which varies with regional differences, and β is the shear wave velocity.

8. The finite fault simulation method according to claim 6 or 7, characterized in that: The site term G(f) is expressed as the product of the near-surface site amplification effect and the attenuation of the high-frequency energy of seismic waves in the local site; the near-surface site amplification effect is obtained by linearly interpolating between adjacent values ​​and calculating the site amplification value at frequency f; the attenuation of the high-frequency energy of seismic waves in the local site is expressed by the κ filter as exp(-πκf), where κ is a parameter determined by distance and site conditions.

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