A method for estimating time-varying coherence of ground motion based on bayesian co-optimization

Abstract

The application provides a time-varying coherence estimation method of ground motion based on Bayesian collaborative optimization. The method constructs double target constraints covering physical consistency and spectrum graph distinguishability through generalized S transform and time-frequency bidirectional exponential smoothing method, cooperatively determines the optimal combination of generalized S transform kernel parameters and exponential smoothing operator by using a Bayesian optimization algorithm, and outputs an optimal parameter set and a corresponding time-varying coherence spectrum graph. Therefore, more real data reflecting the spatial variation characteristics of ground motion can be obtained, and the design requirements under the limit adverse state can be met.

Description

Technical Field

[0001] This application relates to a time-varying coherent estimation method for seismic ground motion based on Bayesian co-optimization, applicable to the technical field of earthquake services. Background Technology

[0002] In the seismic design of large structural systems such as bridges, pipelines, and stadiums, it is essential to fully understand the spatial variation characteristics of ground motion to meet design requirements under the most unfavorable conditions. The coherence function, as a standardized cross-power spectrum, effectively measures the attenuation of spatial similarity of ground motions with frequency and distance, and is a crucial means to achieve this goal. However, traditional methods for calculating the coherence function are based on Fourier transforms and assume that ground motion is a stationary random process. This fails to reflect the evolution of spatial similarity over time, potentially leading to an underestimation of the most unfavorable response of large structures.

[0003] To overcome the limitations of traditional methods, time-frequency analysis techniques have been applied to the calculation of evolutionary power spectra and coherence functions. Among these, the generalized S-transform has attracted considerable attention due to its flexible and adjustable focusing capabilities; however, its calculation results depend on the selection of kernel parameters. Furthermore, to obtain robust time-varying coherence spectrum estimates, the evolutionary power spectrum needs to be smoothed to avoid the anomaly of the coherence spectrum displaying values ​​that are consistently 1. However, existing time-spectrum smoothing schemes lack unified engineering standards for parameter configuration, with values ​​often relying on empirical settings. This leads to problems in time-varying coherence calculations, such as low signal-to-noise ratio, poor smoothing transitions, and mismatch with Fourier coherence, limiting its widespread application in engineering practice.

[0004] Therefore, there is a need in the existing technology to establish a time-varying coherent estimation method that can automatically and collaboratively optimize key parameters, ensuring that the estimation results of ground motions take into account both physical consistency and spectral discernibility. This method is used to obtain the true variation law of the spatial correlation of ground motions, guide the establishment of coherent models and parameter fitting, and thus serve the design and response assessment of multi-point input ground motions in the seismic fortification of large-span structures. This allows for a better understanding of the spatial variation characteristics of ground motions and meets the design requirements under extreme conditions. Summary of the Invention

[0005] This application provides a time-varying coherent estimation method for seismic ground motion based on Bayesian collaborative optimization. This method constructs dual objective constraints covering physical consistency and spectral discernibility through the generalized S-transform and time-frequency bidirectional exponential smoothing method. It uses Bayesian optimization algorithm to collaboratively determine the optimal combination of generalized S-transform kernel parameters and exponential smoothing operator, and outputs the optimal parameter set and its corresponding time-varying coherent spectrum. This can obtain data that more realistically reflects the spatial variation characteristics of seismic ground motion and meet the design requirements under extreme unfavorable conditions.

[0006] This application relates to a time-varying coherent estimation method for seismic ground motion based on Bayesian co-optimization, comprising the following steps:

[0007] (1) Preprocess the two waveform data to be analyzed, set the search range and maximum number of iterations of the parameters, and calculate the reference Fourier coherence spectrum;

[0008] (2) Based on the selected set of kernel parameters, the time-varying autopower spectrum and cross-power spectrum of the two ground motion time histories are calculated using the generalized S-transform;

[0009] (3) The power spectrum is post-processed using time-frequency bidirectional exponential smoothing technology, and then the time-varying coherence function of the ground motion at two points in space is calculated;

[0010] (4) Evaluate the quality of the time-varying coherence spectrum generated by the parameter set, establish the objective function, and provide a search guide for Bayesian optimization;

[0011] (5) Using the existing parameter values ​​and the observation results of the objective function, the distribution of the objective function in the parameter space is approximated by the surrogate model and the next set of candidate parameters is selected.

[0012] (6) Repeat steps (2) to (5) until the number of iterations reaches the preset value, obtain the parameter vector that minimizes the objective function, and output the optimal time-frequency coherence spectrum.

[0013] Preferably, step (1) includes the following steps:

[0014] First, baseline correction and bandpass filtering are performed on the two seismic acceleration time histories to be analyzed to eliminate non-physical drift and retain effective frequency band signals.

[0015] Secondly, set the search range and maximum number of iterations for the parameters;

[0016] Next, the reference Fourier coherence spectrum is calculated to provide a physical consistency evaluation benchmark for subsequent optimization.

[0017] Preferably, in step (3), the time-varying power spectrum is smoothed along the time and frequency directions, and the calculation formula is as follows:

[0018]

[0019] In the formula, The current smoothing value, For the current observation value, This is the result of the previous smoothing process. Smoothing coefficient ( ).

[0020] Preferably, in step (4), the objective function is established. Includes the following two items:

[0021] ① Consistency constraints

[0022] The time average of the spectrum was calculated to match the reference Fourier coherence spectrum, and the difference between the two was measured using the Fréchet distance.

[0023] ② Resolution control item

[0024] Introduction As a resolution control index, the coherence spectrum The time-frequency details and numerical stability are constrained, and are defined as follows:

[0025]

[0026] Preferably, in the iteration of step (6), firstly, a generalized S-transform and coherent spectrum generation are performed based on the current parameter set, and the objective function value is calculated; then, the surrogate model in the Bayesian optimization is updated using the calculation results, and the next set of candidate parameters is intelligently decided by the expectation improvement criterion, and then the sequence is updated; this process is repeated to gradually improve the resolution and physical reliability of the coherent spectrum until the iteration terminates.

[0027] This method achieves autonomous and collaborative optimization of key parameters through Bayesian optimization, resulting in an optimal balance between time-frequency resolution and physical consistency in the final time-varying coherent spectrum. On the one hand, this method can effectively capture the non-stationary evolution characteristics of spatial correlation of seismic motions, providing a more realistic numerical basis for multi-point seismic input design; on the other hand, the generated time-varying coherent spectrum can directly serve the establishment of coherent models and parameter fitting, providing reliable support for response assessment in the seismic design of large structures. Attached Figure Description

[0028] Figure 1 This is a flowchart illustrating the time-varying coherent estimation method for seismic motion described in this application.

[0029] Figure 2 This is a diagram illustrating the descent process of the objective function in an embodiment of this application.

[0030] Figure 3 These are two-dimensional time-frequency coherence diagrams obtained by different schemes in the embodiments of this application.

[0031] Figure 4 This is a comparison of coherence curves after dimensionality reduction for different schemes in the embodiments of this application.

[0032] Figure 5 This is a comparison diagram of the coherence functions of the P-band at the initial stage of different schemes in the embodiments of this application.

[0033] Figure 6This is a comparison diagram of the coherence functions of the S-band at the initial stage of different schemes in the embodiments of this application. Detailed Implementation

[0034] This method transforms the calculation of time-varying coherent spectra into a parameter optimization problem under dual objective constraints: the kernel parameters of the generalized S-transform and the power spectrum smoothing operator. The former controls the intrinsic resolution of the time-varying power spectrum, while the latter affects the signal-to-noise ratio and statistical degrees of freedom of the power spectrum. These two sets of parameters jointly determine the quality of the coherent spectrum. Given the complex nonlinear relationship between these parameters and the coherent function after power spectrum normalization, this method introduces a Bayesian optimization algorithm to perform an efficient search with physical consistency and spectral visibility as joint objectives, thereby obtaining high-quality time-varying coherent spectra.

[0035] Specifically, the time-varying coherent estimation method for seismic ground motion based on Bayesian collaborative optimization (BO-TFCH) proposed in this application includes the following steps:

[0036] (1) Data preprocessing and initial condition setting

[0037] The two waveform data to be analyzed are preprocessed, the search range and maximum number of iterations of the parameters are set, and the reference Fourier coherence spectrum is calculated.

[0038] First, the two ground motion acceleration time histories to be analyzed undergo baseline correction and bandpass filtering preprocessing to eliminate non-physical drift and retain effective frequency band signals. If this method is used to analyze coherent attenuation caused by scattering from the subsurface medium, the time difference caused by traveling wave effects also needs to be eliminated; cross-correlation can be used for estimation and correction. For the acceleration time histories recorded at spatial locations i and j... and The cross-correlation function of the two Defined as:

[0039] (1)

[0040] In the formula, The time delay between the two time periods is taken as... The maximum value corresponding to As the optimal time delay between the two records, the records of the non-reference station are translated and aligned accordingly.

[0041] Secondly, the search range and maximum number of iterations for the parameters are set. This method requires optimization of five parameters: three kernel functions that control the generalized S-transform, and two for exponential smoothing. The specific forms and value ranges of each parameter are shown in step (2). The optimization process uses the maximum number of iterations as the convergence termination condition. For this application scenario, the number of iterations can be set to 50-100.

[0042] Next, the reference Fourier coherence spectrum is calculated to provide a physical consistency evaluation benchmark for subsequent optimization, as shown in the following formula:

[0043] (2)

[0044] In the formula, , and These represent the auto-power spectrum and cross-power spectrum of the two signals, respectively. To ensure the reliability of the coherent estimation, a Hamming window with a bandwidth of approximately 1 Hz is used to smooth the power spectrum, avoiding spurious high values ​​in the calculation results.

[0045] (2) Calculation of time-varying power spectrum based on generalized S-transform

[0046] Based on the selected set of kernel parameters, the time-varying autopower spectrum and cross-power spectrum of the two ground motion time histories are calculated using the generalized S-transform.

[0047] The generalized S-transform is defined as:

[0048] (3)

[0049] In the formula, The time history of ground motion acceleration; The time shift factor represents the time center of the spectral analysis. For frequency; It is a kernel function that varies with frequency, and its form is as follows:

[0050] (4)

[0051] The above formula contains three parameters to be optimized: , , ,parameter Controlling the rate of change of kernel function bandwidth with frequency This determines the extent to which the kernel function expands to both sides. As a constant offset term, it plays a balancing and transitional role between the standard S-transform and the short-time Fourier transform. Substituting equation (4) into equation (3), the complete expression for the generalized S-transform is:

[0052] (5)

[0053] when When, formula (5) is equivalent to the standard S-transform; if The above equation can be simplified to a short-time Fourier transform.

[0054] In one implementation, the search range for the three kernel parameters is preset as follows: .

[0055] Spatial location and The time histories of the ground motion at each location are calculated according to equation (5) to obtain the time spectrum. and Furthermore, the self-power spectrum and cross-power spectrum between the two points are calculated using the following formulas:

[0056] Self-power spectrum

[0057] , (6)

[0058] Cross power spectrum

[0059] (7)

[0060] In the formula, is the energy normalization factor of the kernel function, used to ensure the unbiasedness of power spectrum estimation; the superscript * indicates complex conjugate. and The cross-power spectrum reflects the distribution of ground motion energy in the time and frequency domain between the two measuring points. This describes the correlation between the two in the time-frequency domain, providing input for subsequent calculation of the time-varying coherence function.

[0061] (3) Power spectrum smoothing and calculation of time-varying coherence function

[0062] A time-frequency bidirectional exponential smoothing technique is used to post-process the power spectrum, and then the time-varying coherence function of the ground motion at two points in space is calculated. Preferably, the time-varying power spectrum is smoothed first; taking the time direction as an example, the calculation formula is as follows:

[0063] (8)

[0064] In the formula, The current smoothing value, For the current observation value, This is the result of the previous smoothing process. Smoothing coefficient ( Since the time-varying power spectrum is two-dimensional data, smoothing needs to be performed sequentially along both time and frequency directions: first, fix the frequency. Along the time axis power spectrum Smooth the surface to obtain the desired result. ; and then at a fixed time Along the frequency axis right Smoothing is performed to obtain the final result. .

[0065] This process is uniformly represented as: (9)

[0066] in and For the exponential smoothing operator, the smoothing coefficients are... and The range of values ​​is These two coefficients, along with the generalized S-transform kernel parameters, will be determined through Bayesian co-optimization.

[0067] The above smoothing was performed on the self-power spectrum and cross-power spectrum respectively to obtain... , and Finally, the time-varying coherence function between two points i and j in space is defined as:

[0068] (10)

[0069] The above formula quantitatively describes the correlation evolution of two ground motions in the time-frequency domain, with a value range of (0,1). The absolute value of the resulting two-dimensional matrix... This is the time-varying coherence spectrum.

[0070] (4) Parameter quality evaluation and objective function construction

[0071] Comprehensive evaluation parameter set = The generated time-varying coherence spectrum Quality, establish the objective function J This provides a search wizard for Bayesian optimization. Includes the following two items:

[0072] ① Consistency constraint (Fréchet distance)

[0073] To ensure consistency between time-varying coherence analysis and classical theory, it is necessary to guarantee the summation and averaging of the spectrum along the time domain (time-averaged coherence spectrum): The correlation spectrum should match the reference Fourier coherence spectrum as closely as possible, where T represents the effective duration of the seismic motion. Therefore, in one implementation, the difference between the two is measured using the Fréchet distance:

[0074] (11)

[0075] In the formula, C is the set of legal coupling paths connecting two discrete sequences; The distance is Euclidean. For reference Fourier coherence spectra; The smaller the value, the better the fit between the two coherent curves, and the higher the physical guarantee of the coherent spectrum. The Fréchet distance is suitable for comparing curves with inconsistent numbers of discrete sampling points, and can be used for coherence consistency constraints under different frequency point divisions or sampling conditions.

[0076] ② Resolution control item (Rényi entropy)

[0077] In the parameter optimization process, in addition to ensuring the consistency between the time-averaged coherence and the reference Fourier coherence, it is also necessary to optimize the coherence spectrum. The time-frequency details and numerical stability are constrained. To this end, Rényi entropy is introduced as a resolution control index, defined as:

[0078] (12)

[0079] The integration region in the formula is the domain of the spectrum; It can effectively reveal the complexity of two-dimensional images; the better the time-frequency coherence focusing and the higher the resolution, The smaller; conversely The larger.

[0080] Combining the consistency constraint and the resolution control term, the following objective function is constructed:

[0081] J (13)

[0082] In the formula, The regularization coefficient is used to adjust the consistency constraint term. With resolution control items The influence weights between them. In one implementation, It can be determined by a small number of trial calculations or by taking an empirical value (it is recommended to take a value between 0.01 and 0.1).

[0083] The optimization goal of this method is to find the optimal parameter set. , so that:

[0084] (14)

[0085] in, For the predefined parameter search space, see step (2) for details.

[0086] (5) Parameter optimization based on Bayesian optimization

[0087] Using existing parameter values ​​and observations of the objective function, a surrogate model is used to approximate the distribution of the objective function in the parameter space, and the next set of candidate parameters is selected based on the "expected improvement" criterion. This reduces the need for manual parameter tuning and improves the convergence stability of the optimization iteration.

[0088] For each set of parameter vectors Execute steps (2)-(4) to obtain the objective function. Its relationship with the proxy model The following relationship must be satisfied:

[0089] (15)

[0090] In the formula, This represents the prediction bias of the model. Let the set of parameters that have been tested be... =[ The corresponding observation set is: =[ In one implementation, a Gaussian process is used as a surrogate model for... To model this, its probability density function can be written as:

[0091] β (16)

[0092] , represents the set of hyperparameters for constructing the surrogate model, and K is an n×n positive semidefinite covariance matrix whose elements The Gaussian kernel formed by the i-th and j-th groups of parameters is expressed as:

[0093] (17)

[0094] In the formula, hyperparameters A and L are used to describe the scale and smoothness of the objective function in the parameter space.

[0095] Considering the observation bias term The influence of this can be considered as independent and identically distributed Gaussian noise with variance of... Then the proxy model The likelihood function can be written as:

[0096] (18)

[0097] Multiplying equations (16) and (18) and integrating with respect to f, we obtain the marginal likelihood distribution of the observed data:

[0098] (19)

[0099] Based on Bayesian theory, maximizing the marginal likelihood yields the desired result. The optimal solution. Based on this, the current observation set is introduced. Based on the properties of Gaussian processes, the following joint distribution is constructed:

[0100] (20)

[0101] here, It can be derived through a simple formula. The posterior distribution:

[0102] ν (twenty one)

[0103] In the formula, , Gaussian process regression simplifies complex nonlinear problems, provides an estimate of the uncertainty of prediction results, and is a key step in parameter optimization.

[0104] After obtaining a probabilistic approximation of the current objective function, an evaluation criterion is needed to balance "mining" (exploring unknown areas) and "development" (focusing on areas with relatively good objective values) in order to determine the next set of suitable parameter values. These discrimination criteria are called acquisition functions, with the Expectation Improvement (EI) strategy as a representative example, written as:

[0105] (twenty two)

[0106] It is the test point that minimizes the observed objective function in the current evaluation; , and Is it the proxy model in The mean and variance at the given points are as described by formula (21). It is the corresponding parameter set; and These are the cumulative function and probability density of the standard normal distribution, respectively. The parameter ξ is a bias factor that controls the amount of exploration during optimization; its default value is 0.01 times the standard deviation of the parameters. In each iteration, the point corresponding to the maximum value of the EI function is used as the next set of trial points. The value of is the point that is most likely to cause the largest decrease in formula (13).

[0107] (6) Loop Iteration and Result Output

[0108] Repeat steps (2) to (5) until the number of iterations reaches the preset value, obtain the parameter vector that minimizes the objective function, and output the time-frequency coherence spectrum with optimal resolution and physical consistency.

[0109] In each iteration, first based on the current parameter set Perform the generalized S-transform and coherent spectrum generation (steps 2-3), and calculate the objective function value. (Step 4). Subsequently, the surrogate model in Bayesian optimization is updated using the calculation results, and the next set of candidate parameters is intelligently decided based on the expectation boosting criterion. (Step 5), then update the sequence. and This process is repeated cyclically, gradually improving the resolution and physical reliability of the coherent spectrum until the iteration terminates.

[0110] Complete technical process such as Figure 1 As shown, the time-frequency transformation and exponential smoothing parameter set It can be autonomously optimized under the guidance of dual objectives. The final output is the optimal set of parameters. Time-varying coherent spectra with both high resolution and high reliability were generated, thus enabling adaptive analysis from data to high-quality results.

[0111] Example

[0112] To verify the feasibility of this method, two actual ground motions were selected for testing. The data were obtained from the east-west acceleration time histories of measuring points S01 and S03 (Class I site) of the Zigong mobile array during the 2021 Luxian 6.0 earthquake. The straight-line distance between the two points is 343 meters, which is similar to the span of large structures such as stadiums. The data were processed by a 0.02-25Hz bandpass filter, and the time shift caused by the traveling wave effect was eliminated according to formula (1).

[0113] In the coherence spectrum calculation, the maximum number of iterations for Bayesian optimization is set to 60, and the regularization coefficient is... The value is set to 0.2 to ensure that the Renyi entropy and Ferchet distance have equal constraint weights. The optimization process is as follows: Figure 2 As shown, the pink dashed line and the blue solid line represent the minimum value of the surrogate model and J, respectively. The minimum observed value changes with the number of iterations, and the two almost coincide, proving that Gaussian process regression is robust and reliable in this case, and the obtained surrogate model approximates the true objective function. The final parameter optimization results are shown in Table 1.

[0114] Table 1 Optimal Parameter Results

[0115]

[0116] To further illustrate the effectiveness of this method, it is compared with three existing time-frequency coherence calculation methods, whose parameters are mainly selected empirically:

[0117] Method 1: Wavelet coherence method based on Torrence and Webster. Continuous wavelet transform is used for time-frequency analysis, and the power spectrum is processed through cascaded smoothing of time and scale to calculate time-varying coherence. Here, scale-adaptive Gaussian windows are used for time-domain smoothing, and a box window with a 0.6 octave bandwidth is used for scale filtering.

[0118] Method 2: Two-dimensional box window smoothing method based on S-transform. This method is based on the S-transform, treats the time-frequency power spectrum as a whole, and uses a two-dimensional box window with a size of 0.3s × 0.3Hz for smoothing (this parameter is the recommended value in the literature).

[0119] Method 3: Ellipsoidal window smoothing method based on generalized S-transform. This method first adaptively searches for the kernel parameters of the generalized S-transform to obtain the power spectrum with optimal focusing, and then uses a 2s-axis smoothing method. Two-dimensional smoothing was performed using an ellipsoidal window of 0.2 Hz (this parameter is the recommended value in the literature).

[0120] The time-varying coherence spectra of the two test records were estimated using the three reference methods described above, and then compared with the coherence spectra obtained by this method. The results are as follows. Figure 3 As shown.

[0121] The comparison reveals that Method 1, based on wavelet transform, is relatively insufficient in capturing the coherent characteristics of the initial wave phase above 10 Hz, with striped noise observed in the high-frequency region. Method 2 can reflect the overall trend of coherent structure changes with time and frequency, but it has certain shortcomings in time-frequency focusing and detail representation in some frequency bands, with blocky smoothing traces observed in local areas. Method 3 improves in time-frequency focusing, but when the smoothness in the time direction is high, it may lead to a reduction in mid-to-high frequency detail information. In contrast, this method, through a Bayesian collaborative optimization strategy, presents the evolution characteristics of coherent structure with time and frequency in greater detail while maintaining good visual resolution.

[0122] Figure 4 The time-averaged coherence spectra obtained by each method are presented. It can be seen that Method 1 and Method 2, due to the lack of kernel function and smoothing parameter optimization, exhibit significant differences between their average coherence spectra and the reference Fourier coherence spectrum, with Method 2 showing particularly significant distortion in the low-frequency range. Although Method 3 optimizes the kernel parameters, its average coherence spectrum basically matches the Fourier coherence spectrum, but due to its high smoothing degree, some true information is lost. Our proposed method, based on a Bayesian collaborative optimization strategy, preserves richer fluctuation details in the average coherence spectrum and achieves the highest consistency with the Fourier coherence spectrum, indicating that it can effectively maintain the accuracy and detail integrity of the estimation after dimensionality reduction.

[0123] Table 2 shows the Frechet distance and Rényi entropy of the estimation results of BO-TFCH and other reference methods, which are numerically consistent with the analysis results. The BO-TFCH method has the smallest Frechet distance, and its Rényi entropy is lower than that of methods 1 and 2, only slightly higher than that of method 3, reflecting the trade-off between resolution and detail preservation. Although method 3 has the lowest Rényi entropy, its integrated Frechet distance is higher than that of the BO-TFCH method, indicating a difference in fidelity and potentially affecting its practicality.

[0124] Table 2. Statistics of Indicators with Different Calculation Results

[0125]

[0126] To further verify the effectiveness of time-varying coherence spectra in describing non-stationary spatial characteristics, a piecewise averaging strategy was adopted. The time-averaged coherence curves of the initial phase (P-wave portion) and the main shock phase (S-wave portion) were compared with reference Fourier coherence curves, respectively. The time window for the initial phase was set to 5–11 s, and the time window for the main shock phase was set to 13–25 s. The comparison results are shown in […]. Figure 5 and Figure 6 .

[0127] In the initial stage of the earthquake, the coherence curve obtained by Method 1 is significantly lower than the Fourier coherence spectrum from 5 Hz onwards, indicating that the wavelet method is difficult to accurately reflect the spatial similarity of P-band earthquakes. Other methods based on the S-transform, however, can capture the strong coherence characteristics of the first arrival portion better. Among them, the BO-TFCH method proposed in this patent performs best, with its coherence curve showing the highest agreement with the Fourier coherence spectrum. For the mainshock segment, although all four methods can roughly characterize the coherent structure, the BO-TFCH calculation results still show the highest matching degree with the Fourier coherence spectrum. Other methods, either due to empirically given smoothing parameters or limitations of the transform kernel itself, exhibit varying degrees of distortion in the low-frequency or high-frequency ranges.

[0128] It's important to note that the BO-TFCH method achieves optimal results thanks to both the adaptive search of Bayesian optimization and the fidelity of exponential smoothing. While BO-TFCH is an open technical framework that can be further optimized with better smoothing schemes in the future, exponential smoothing is sufficient for the needs of this application.

[0129] Through the above comparison of practical applications, the time-varying coherent estimation method for seismic ground motion based on Bayesian co-optimization proposed in this application outperforms existing time-varying coherent estimation methods in terms of resolution and accuracy. It can provide a more reliable analysis tool for the seismic design of large-span structures and has significant engineering application value.

Claims

1. A time-varying coherent estimation method for seismic ground motion based on Bayesian co-optimization, characterized in that, Includes the following steps: (1) Preprocess the two waveform data to be analyzed, set the search range and maximum number of iterations of the parameters, and calculate the reference Fourier coherence spectrum; (2) Based on the selected set of kernel parameters, the time-varying autopower spectrum and cross-power spectrum of the two ground motion time histories are calculated using the generalized S-transform; (3) The power spectrum is post-processed using time-frequency bidirectional exponential smoothing technology, and then the time-varying coherence function of the ground motion at two points in space is calculated; (4) Evaluate the quality of the time-varying coherence spectrum generated by the parameter set, establish the objective function, and provide a search guide for Bayesian optimization; (5) Using the existing parameter values ​​and the observation results of the objective function, the distribution of the objective function in the parameter space is approximated by the surrogate model and the next set of candidate parameters is selected. (6) Repeat steps (2) to (5) until the number of iterations reaches the preset value, obtain the parameter vector that minimizes the objective function, and output the optimal time-frequency coherence spectrum.

2. The time-varying coherent estimation method for seismic ground motion according to claim 1, characterized in that, Step (1) includes the following steps: First, baseline correction and bandpass filtering are performed on the two seismic acceleration time histories to be analyzed to eliminate non-physical drift and retain effective frequency band signals. Secondly, set the search range and maximum number of iterations for the parameters; Next, the reference Fourier coherence spectrum is calculated to provide a physical consistency evaluation benchmark for subsequent optimization.

3. The time-varying coherent estimation method for seismic ground motion according to claim 1 or 2, characterized in that, In step (3), the time-varying power spectrum is first subjected to exponential smoothing along the time and frequency directions, and the calculation formula is as follows: In the formula, The current smoothing value, For the current observation value, This is the result of the previous smoothing process. Smoothing coefficient ( ).

4. The time-varying coherent estimation method for seismic ground motion according to claim 1 or 2, characterized in that, In step (4), the objective function is established. Includes the following two items: ① Consistency constraints Calculate the time integral of the spectrum to make it match the reference Fourier coherence spectrum, and measure the difference between the two using the Fréchet distance; ② Resolution control item Introduction As a resolution control index, the coherence spectrum The time-frequency details and numerical stability are constrained, and are defined as follows: 。 5. The time-varying coherent estimation method for seismic ground motion according to claim 4, characterized in that, In the iteration of step (6), firstly, a generalized S-transform and coherent spectrum generation are performed based on the current parameter set, and the objective function value is calculated; then, the surrogate model in Bayesian optimization is updated using the calculation results, and the next set of candidate parameters is intelligently decided by the expectation enhancement criterion, and then the sequence is updated; This process is repeated cyclically, gradually improving the resolution and physical reliability of the coherent spectrum until the iteration terminates.

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