Limited fault simulation method for generating sub-source random vibration by adopting trigonometric series method

Abstract

The invention relates to a finite fault simulation method for generating sub-source random vibration by adopting a trigonometric series method. The finite fault simulation method comprises the following steps: obtaining a sub-fault acceleration source spectrum, normalizing the sub-fault acceleration source spectrum to obtain a probability density function, and generating a sub-fault random frequency component; and carrying out superposition combination on the generated simple harmonic waves corresponding to the random frequency components of all the sub-faults in a time domain to obtain random vibration of the sub-faults, then obtaining an acceleration Fourier amplitude spectrum of the sub-faults, carrying out inverse Fourier transform to the time domain to obtain an acceleration time history of the sub-faults, and adding delay time to carry out superposition to obtain a seismic oscillation time history. The simulated seismic kinetic energy accords with the target spectrum at the high frequency and the low frequency at the same time, and the simulation accuracy is improved.

Description

Technical Field

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

[0002] The random finite fault simulation method is currently the most important near-field high-frequency ground motion simulation method in the field of earthquake engineering. It is one of the most important data sources for ground motion input for structural seismic design and can provide a basis for post-disaster rescue and reconstruction in target areas. Although the random finite fault method has advantages such as clear physical meaning, few input parameters, and easy valuation, it has the defect of insufficient low-frequency simulation and cannot meet the high-precision simulation requirements of both high and low frequencies.

[0003] Generally speaking, any given earthquake motion history can be regarded as a combination of simple harmonic waves of many different frequencies. The frequency components of earthquake motion include frequency and amplitude. Frequency refers to the frequency of each simple harmonic wave contained in the earthquake motion signal, and amplitude refers to the intensity or amplitude of each frequency in the earthquake motion signal. The frequency component of earthquake motion is one of the important parameters reflecting the characteristics of earthquake motion. The recorded earthquake motion shows different frequency components, and its random frequency components depend on the source mechanism and magnitude of the earthquake, as well as the distance from the epicenter and site conditions. Initially, it was believed that earthquake motion was composed of various periodic components in a fixed proportion, but later it was gradually realized that the frequency composition of earthquake motion was variable and different. Earthquake 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 earthquake motion between large and small earthquakes, and there is no quantitative determination method.

[0004] Therefore, providing a finite fault simulation method that can simultaneously meet the high-precision simulation requirements of high frequencies and low frequencies 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 that uses a trigonometric series method to generate sub-source random vibrations. In order to consider the frequency characteristics of ground motion, a normalized acceleration source spectrum is used as a probability density function to control the random frequency components of the sub-fault, and a method combining the trigonometric series method with the random finite fault method is proposed to synthesize the ground motion acceleration time history. The ground motion simulated by this method can be consistent with the target spectrum at both high and low frequencies, which improves the accuracy of the simulation and can be used as the ground motion input for structural seismic design.

[0006] According to the present application, a finite fault simulation method for generating sub-source random vibration using a trigonometric series method comprises the following steps:

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

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

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

[0010] (5) 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;

[0011] (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;

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

[0013] Among them, 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.

[0014] Among them, in step (3), the normalized probability density function is converted into a cumulative probability value through the calculation of the cumulative distribution function, and then uniformly distributed random numbers are generated, and these random numbers are mapped to the value range of the target distribution using the reverse interpolation method of the cumulative distribution function, thereby generating random samples that conform to the specified probability density function and obtaining the frequency components of the random vibration of the sub-fault.

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

[0016]

[0017] 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;

[0018] in, is the corner frequency of the ijth 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; 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 , f k is the random frequency component of the sub-fault with a set probability density, φ k is a random phase.

[0019] Wherein, the sub-fault acceleration Fourier amplitude spectrum A ij (f) 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 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:

[0023]

[0024] Where R represents the fault distance;

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

[0026]

[0027] 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.

[0028] Among them, 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. BRIEF DESCRIPTION OF THE DRAWINGS

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

[0030] Figure 2 These are the PSA simulation results and observation results of the embodiments of the present application. DETAILED DESCRIPTION

[0031] In order to make the purpose, technical solutions and advantages of the present application clearer, the embodiments of the present application will be described in detail in conjunction with the accompanying drawings. It should be noted that the embodiments and features in the embodiments of the present application can be combined with each other arbitrarily without conflict. The present application proposes a finite fault simulation method that uses a trigonometric series method to generate sub-source random vibrations, which is a simulation method for seismic motion and can be used in specific fields such as seismic structure design and earthquake law research.

[0032] According to the present application, a finite fault simulation method for generating sub-source random vibration using a trigonometric series method comprises the following steps:

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

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

[0035]

[0036] in,

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

[0038] (2) Normalizing the sub-fault acceleration source spectrum to obtain the probability density function;

[0039] First, the source spectrum within the frequency range is calculated to generate the initial probability density function, and then the trapezoidal rule is used to numerically integrate the function within the given frequency range to obtain the total area value. Finally, the value of each probability density function is divided by the total area to achieve normalization so that the total integral of the function is 1, thereby ensuring that it meets the definition of the probability density function.

[0040] (3) generating sub-fault random frequency components according to the obtained probability density function;

[0041] By calculating the cumulative distribution function (CDF), the normalized probability density function is converted into a cumulative probability value, 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, thereby generating random samples that meet the specified probability density function and obtaining the sub-fault random frequency component f k .

[0042] (4) All random frequency components f of the generated sub-faults are k The corresponding simple harmonic The superposition combination is performed in the time domain, where the amplitude of the simple harmonic wave of each frequency is C k is the source spectrum amplitude corresponding to the frequency, the phase is a random phase, and the random vibration x of the sub-fault is obtained. ij (t).

[0043] Random vibration of sub-fault x ij (t) can be expressed as:

[0044]

[0045] 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. After selecting appropriate angles from the source and azimuth, the average value can be 0.55. F represents the free surface amplification factor, which can be 2 for SH waves. V represents the horizontal component coefficient of shear wave energy, which can be ρ and β are the density and shear wave velocity of the medium near the earthquake source, respectively, in g / cm 3 and km / s; is the corner frequency of the ijth sub-fault, where 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 , f k is the random frequency component of the sub-fault with a set probability density, φ k is a random phase.

[0046] (5) The obtained 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 A of the sub-fault acceleration. ij (f), as shown in formula (3).

[0047] Sub-fault acceleration Fourier amplitude spectrum A ij (f):

[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 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).

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

[0052]

[0053] Where R represents the fault distance.

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

[0055]

[0056] 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.

[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 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 as exp(-πκf) using a κ filter, where κ is a parameter determined by distance and site conditions, and describes the effect of energy dissipation from the source to the site on the wave field during wave propagation in the crust.

[0058] (6) The obtained sub-fault acceleration Fourier amplitude spectrum A ij (f) After inverse Fourier transformation to the time domain, the sub-fault acceleration time history a is obtained ij (t), add a certain delay time Δt ij The ground motion time history a(t) is obtained by superposition. Furthermore, a(t) can be converted into a pseudo acceleration response spectrum PSA for earthquake-resistant design services.

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

[0060]

[0061] Where nl and nw are the number of sub-sources along the strike and dip of the fault, respectively, and Δt ij It is the superposition of the time delay of the rupture propagating to the ijth sub-source and the time delay of the seismic wave propagating from the ijth sub-source to the site due to the difference in propagation distance.

[0062] Example

[0063] Based on the actual occurrence of M in a certain place s Taking the 8.0 earthquake as an example, this method is used to simulate the seismic motion of a 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, which is compared with the actual record of the station. Figure 1 and Figure 2 shown. Figure 1 The horizontal axis represents time in seconds, and the vertical axis represents acceleration in cm / s. 2 . Figure 1 The horizontal axis represents time in seconds, and the vertical axis represents PSA in 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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