Dam crack depth detection method based on Bayesian reasoning

Abstract

The invention discloses a dam crack depth detection method based on Bayesian reasoning, and relates to the technical field of detection, and the method comprises the steps: S1, collecting an initial dielectric constant, dam body average humidity and reflection main peak travel time; s2, calculating a prior dielectric constant, and executing Bayesian reasoning; s3, generating a posterior distribution function, and obtaining a calibration crack depth from the posterior distribution function; and S4, setting a confidence interval, completing the calibration process when the selected point estimation value is within the range of the confidence interval, and otherwise, returning to the step S2. Compared with the prior art, the method has the advantages that the prior dielectric constant is constructed through the reflection main peak travel time and the initial dielectric constant, the average dam body humidity serves as prior input, the crack depth is calibrated through the Bayesian framework, and the method has the advantage that depth systematic deviation caused by humidity gradient is remarkably reduced; and it is ensured that the crack depth calibration process has high precision in a saturated water environment.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of detection, in particular to a dam crack depth detection method based on Bayesian inference. BACKGROUND

[0002] In water conservancy and hydropower engineering, the concrete dam body is inevitably affected by material shrinkage, temperature stress, uneven foundation settlement and external load, and other factors, and various hidden defects will inevitably occur inside, among which cracks are the most common and the most threatening to structural safety. In the prior art, the commonly used high-precision technology is the geological radar method, which uses a geological radar to emit high-frequency electromagnetic waves into the medium and analyzes the amplitude, phase and waveform of the reflected waves to identify interfaces with different dielectric constants, thereby effectively capturing cavities, loose areas and water-rich cracks.

[0003] In existing water conservancy dams, the dam body that blocks water is not a homogeneous medium. Due to the long-term penetration of reservoir water, a complex humidity field is distributed inside the dam body, which transitions from the upstream to the downstream and from the saturated zone to the unsaturated zone. The relative dielectric constant and conductivity of water are much higher than those of dry concrete. This means that the propagation speed of electromagnetic waves inside the dam body is not constant, but along the propagation path, as the water content of the concrete increases, the dielectric constant increases, resulting in a corresponding decrease in wave speed. Traditional crack depth conversion relies on the assumption of a uniform dielectric constant, and once the dielectric constant is not uniform due to changes in concrete humidity, the depth will be severely distorted. The prior art usually selects an average dielectric constant for the entire detection area to calculate the wave speed, which is an oversimplified and invalid assumption in a dam body with a high humidity gradient. A crack located in the front part of the dam body, before the radar wave reaches its location, has already traversed a region with a continuously changing dielectric constant. Using a single average wave speed will attribute all the travel time differences caused by wave speed changes to depth changes, resulting in a deviation between the calculated depth of the crack and the actual depth. SUMMARY

[0004] The present application provides a dam crack depth detection method based on Bayesian inference, which solves the problem of deviation in detection accuracy caused by the influence of dam body humidity when using existing electromagnetic wave technology to detect dam crack depth.

[0005] The present application is implemented by the following technical solutions: A dam crack depth detection method based on Bayesian inference, the method comprising: Step S1: Collect the initial dielectric constant and average humidity of the target dam, collect the initial crack depth detected by the geological radar detection, and obtain the reflection main peak travel time of the target detection crack; Step S2: calculate the prior dielectric constant using the reflection main peak travel time and the initial dielectric constant, perform Bayesian inference based on the prior dielectric constant, the average dam humidity and the initial crack depth, and perform inference update iteration in the Bayesian inference; Step S3: generate a posterior distribution function in the posterior phase of the Bayesian inference, update the posterior distribution function in each iteration cycle, and obtain the calibrated crack depth from the finally updated posterior distribution function; Step S4: set a confidence interval, calculate the approximate distribution of the calibrated crack depth and select a plurality of point estimates, complete the calibration process when all selected point estimates are within the confidence interval, and return to step S2 when there is a point estimate outside the confidence interval.

[0006] Further, the calculation process of the prior dielectric constant comprises: Let the speed of light be c, the reflection main peak travel time be tb, the initial dielectric constant be ε0, the initial crack depth be D0, and the prior dielectric constant be εp, The calculation formula of the prior dielectric constant εp is: , Wherein, α represents the observation weight, and w represents the amplitude weight, and 0<w≤1.

[0007] Further, the setting content of the amplitude weight comprises: Set the main peak amplitude of the target detected crack reflection wave as Ap, set the maximum amplitude of the reflection wave observed in the target dam as Am, and set the amplitude adjustment coefficient γ, The calculation formula of the amplitude weight w is: Wherein, γ>0.

[0008] Further, the process of the Bayesian inference comprises: Based on the prior dielectric constant, the average dam humidity and the initial crack depth, a likelihood function reflecting the propagation of electromagnetic waves in water-containing concrete is constructed, the prior dielectric constant is input as a dielectric correction parameter into the likelihood function, the average dam humidity is set as a humidity adjustment factor, and the humidity adjustment factor is input as a physical correction parameter into the likelihood function; perform Bayesian inference update based on the corrected likelihood function and the prior distribution, and label the likelihood function after completing the Bayesian inference update as the posterior distribution function of the initial crack depth.

[0009] Further, the process of updating the likelihood function to the posterior distribution function comprises: The update convergence condition is set, and in the Bayesian inference updating process, the posterior distribution obtained by the previous updating is used as the prior distribution of the next updating; the likelihood function combined with the humidity adjustment factor is iteratively updated, and in each updating, the posterior distribution of the likelihood function is calculated using the variational Bayesian method, and the posterior distribution function of the initial crack depth is gradually optimized; when the Bayesian inference updating reaches the update convergence condition, the updating is stopped and the posterior distribution function is output.

[0010] Further, the update convergence condition includes: setting a posterior variance threshold for the Bayesian inference updating process, and calculating a variance value of the posterior distribution of each Bayesian inference updating; when the variance value of the posterior distribution is lower than the posterior variance threshold, the Bayesian inference completes the updating process, at which time the updating is stopped and the posterior distribution function is output.

[0011] Further, the process of calculating the posterior distribution by the variational Bayesian method includes: The evidence lower bound of the variational Bayesian and the gradient step length are set; in each Bayesian inference updating iteration, the posterior distribution obtained by the previous Bayesian inference updating is parameterized as a lognormal distribution, the lognormal distribution is adjusted along the direction of increasing evidence lower bound using the gradient step length, and the lognormal distribution after completing the KL divergence adjustment is output as the posterior distribution result of the current updating iteration.

[0012] Further, the gradient step length is set using the Adam optimizer, and the process is set as: The Adam optimizer is set, and in each Bayesian inference updating iteration, in each round of Bayesian inference updating, the first moment and the second moment of the gradient step length are updated based on the gradient information of the lognormal distribution, the adaptive step length is obtained using the Adam optimizer based on the first moment and the second moment, and after dynamic bias correction of the adaptive step length using the Adam optimizer, the adaptive step length is generated as the gradient step length of the current iteration.

[0013] Further, the process of obtaining the calibrated crack depth from the posterior distribution function is set as: constructing a time series according to the time sequence of the likelihood functions of the Bayesian inference iterations of different orders, assigning an iteration weight to the mean value of the posterior distribution at each time point, and using time series fusion calculation based on the iteration weight, and marking the final result of the fusion calculation as the calibrated crack depth.

[0014] Further, the process of time series fusion calculation is set as: Let the calibrated crack depth be represented as Dc, the iteration weight be represented as β, and the ordinal number of the Bayesian inference updating iteration be represented as i; let the crack depth estimate corresponding to the posterior distribution of each iteration be represented as Dp, Then the calculation formula of the calibrated crack depth of the i-th iteration is: , wherein the iteration weight β is set to a range of 0≤β≤1, and i is set to a natural number greater than 1.

[0015] Further, the crack depth estimation of each iteration is set to the median of the posterior distribution in the iteration.

[0016] Further, the setting process of the confidence interval comprises: constructing a confidence interval of the crack depth based on a posterior distribution parameter of the posterior distribution; generating an approximate distribution of the crack depth from the posterior distribution, and extracting a plurality of point estimates from the approximate distribution; when the plurality of point estimates all fall within the confidence interval, determining that the crack depth has reached a calibration condition, and outputting a calibrated crack depth.

[0017] Further, the posterior distribution parameter comprises a mean parameter and a variance parameter of a lognormal distribution, used to represent a variational posterior distribution of the crack depth.

[0018] Compared with the prior art, the present application has the following advantages and beneficial effects: 1. The prior dielectric constant is constructed by the reflection main peak travel time and the initial dielectric constant, the average humidity of the dam body is taken as the prior input, the initial distribution of the dielectric constant is positively correlated with the humidity change, the solution of the crack depth is no longer dependent on the fixed wave velocity, and thus the systematic deviation of the depth caused by the humidity gradient is significantly reduced; 2. The posterior distribution function of the crack depth is introduced through the Bayesian framework, the depth prediction has the confidence information, when the humidity causes strong scattering or the dielectric changes sharply, the posterior distribution will automatically change, and the potential low reliability problem of the traditional radar depth estimation value caused by only a single detection result is avoided; 3. The posterior distribution and the confidence interval of the crack depth are provided, the uncertainty of the depth result is quantified, the reliability and the interpretability are improved, the iteration update and the self-consistent determination mechanism are adopted, the crack depth calibration process is stable, false jumps are avoided, and the geological radar still maintains the robustness in the local water saturation and the complex dielectric environment. BRIEF DESCRIPTION OF DRAWINGS

[0019] The accompanying drawings, which are included to provide a further understanding of the embodiments of the application and are incorporated in and constitute a part of this application, illustrate embodiments of the application and together with the description serve to explain the principles of the application. In the drawings: Figure 1 The flow chart of the present application. DETAILED DESCRIPTION

[0020] To make the objectives, technical solutions, and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the embodiments and accompanying drawings. The illustrative embodiments and descriptions of the present invention are only used to explain the present invention and are not intended to limit the present invention.

[0021] Example 1, as Figure 1 As shown in the figure, this embodiment is a method for detecting the depth of cracks in a dam based on Bayesian inference. The method includes: Step S1: Collect the initial dielectric constant and average humidity of the target dam, collect the initial crack depth of the cracks detected by ground-penetrating radar, and obtain the travel time of the main peak of the reflection of the cracks detected by the target. Step S2: Calculate the prior dielectric constant using the reflection peak travel time and the initial dielectric constant. Perform Bayesian inference based on the prior dielectric constant, the average humidity of the dam body, and the initial crack depth. Perform inference update iteration in the Bayesian inference. Step S3: Generate the posterior distribution function in the posterior stage of Bayesian inference, update the posterior distribution function in each iteration loop, and obtain the calibration crack depth from the finally updated posterior distribution function. Step S4: Set the confidence interval, calculate the approximate distribution of the calibration crack depth and select several point estimates. When all selected point estimates are within the confidence interval, the calibration process is completed; if there are point estimates outside the confidence interval, return to step S2.

[0022] The initial dielectric constant refers to a basic estimation or measurement value of the current electromagnetic environment of the dam before the Bayesian inference calibration. In specific implementation, the standard dielectric constant can be obtained from the dam operation record material as the initial dielectric constant; the field geological radar calibration can be used to obtain a basic dielectric constant value in the non-complex area of the dam or the part with uniform humidity; the constant moisture content test can be performed on the concrete sample of the same material of the dam to measure the initial dielectric constant by using the dielectric spectrum analysis method. The average humidity of the dam body is a parameter representing the overall water content state of the dam, which is used to adjust the humidity sensitivity of the dielectric constant in the model. In specific implementation, the average humidity of the dam body can be estimated by the data of the humidity meter or osmometer arranged in the dam body, and the overall average value can be obtained by using multiple deep buried humidity meters or osmometers to provide the water content at different depths; the average volume moisture content of the entire dam body at a certain period of time can be obtained by using the finite element derivation result of the dam body temperature and humidity field based on the long-term monitoring of the seepage field and temperature field of the target dam. The initial crack depth is a rough estimated depth value of the crack being detected without calibration or correction. The initial crack depth can be directly scanned and collected by using the geological radar, the high-frequency electromagnetic wave is emitted into the medium by using the geological radar, and the amplitude, phase and waveform of the reflected wave are analyzed to identify the crack interface with different dielectric constants; the depth of the crack is calculated according to the geometric shape and travel time of the reflection interface. The reflection main peak travel time refers to the propagation time of the characteristic point with the maximum amplitude in the reflected wave generated by the crack interface in the propagation process of the electromagnetic wave emitted by the geological radar, which is used to represent the time-domain physical quantity of the position of the reflection interface; the meaning can be represented as the characteristic time of the strongest signal of the crack reflection in the round trip propagation time of the whole process of “emission→propagation→crack reflection→return to receiver”. In specific implementation, the reflection main peak travel time can be directly measured and obtained by using the geological radar emission and echo reception.

[0023] In the Bayesian inference framework, the prior represents the preliminary knowledge or estimate held about an unknown parameter before new data is observed. The meaning of the prior permittivity is that before the deep calculation, the preliminary estimate value of the real permittivity is made based on the existing information, which reflects the reasonable estimate value of the equivalent permittivity that can be used in the target dam. The prior permittivity is calculated using the reflection main peak travel time and the initial permittivity. The main peak travel time based on the geological radar echo is combined with the initial permittivity parameter of the target dam, and the permittivity is calculated by correcting the travel time-permittivity relationship model to obtain the prior permittivity used to start the inference process in the Bayesian inference. The prior permittivity can reflect the real propagation characteristics of electromagnetic waves under the influence of dam humidity, and provide reasonable initial value constraints for subsequent posterior inference. Using the prior permittivity, the average humidity of the dam and the preliminary crack depth, the observed radar travel time data is combined with the forward model, and the unknown target crack depth estimate is continuously corrected and updated by the Bayesian statistical method, so as to obtain a more accurate and reliable crack depth posterior distribution.

[0024] As a feasible enumerated embodiment, the calculation process of the prior permittivity can be set as follows: let the speed of light be represented as c, the reflection main peak travel time be represented as tb, the initial permittivity be represented as ε0, the initial crack depth be represented as D0, and the prior permittivity be represented as εp, The calculation formula of the prior permittivity εp is: .

[0025] The meaning of the prior permittivity calculation formula is that the initial permittivity is added to the average permittivity change in the unit travel time change length of the target crack.

[0026] The part represents the theoretical travel time including the round-trip time, and the conversion is derived from: let the one-way theoretical travel time be represented as t, the electromagnetic wave speed be v1, and the derivation formula can be set by known common sense: , c represents the speed of light, Then the round-trip theoretical travel time can be derived as: .

[0027] The part represents the scale factor for converting the time difference into the permittivity correction amount, which can be seen as: The derivation process of the scale factor is that the scale factor is the proportionality coefficient of the small change mapping, and its physical meaning is the permittivity change amount corresponding to the unit travel time change; it also represents the sensitivity of travel time to permittivity. Therefore, the physical form of the scale factor in this embodiment can be set as the differential formula of dε / dt, which can be represented as: .

[0028] The derivation process is as follows: Given the theoretical time, the equation can be expressed as: , Therefore, we get: Therefore, we can obtain: ; Known chain rule: , Then, by taking the derivatives respectively, we can obtain: , , Combining the results, we get: That is, the scaling factor.

[0029] It should be noted that the above The derivation process includes: Let the initial magnetic permeability of the target dam's sampling environment be denoted as μ0, and let the vacuum permittivity be denoted as εz and the vacuum permeability as μz. Then, according to the existing derivations of Maxwell's equations, we know that... The speed of light in a vacuum environment can be expressed as: , In the air environment of the target dam, since there is an airborne medium, let ε be the general symbol for the dielectric constant of the medium and μ be the general symbol for the magnetic permeability of the medium. The speed of electromagnetic waves can then be expressed as: .

[0030] Let ε = ε0∙εz, let μ = μ0∙μz, The electromagnetic wave velocity v1 is then expressed as: ; Since the target dam's air environment is a non-magnetic medium, μ0 can be approximated as 1. ,Right now .

[0031] The principle behind setting ε = ε0∙εz and μ = μ0∙μz is that the electric displacement in the medium is formed by the superposition of the inherent vacuum polarization response and the additional polarization effect of the medium. The vacuum permittivity εz is a dimensional parameter characterizing the fundamental relationship between the electric field and the electric displacement, and is used as an absolute scale for polarization capability. The ratio of the additional polarization generated by the molecular structure of the medium to the vacuum permittivity εz reflects the degree of polarization enhancement of the medium relative to the vacuum. This ratio is a purely relative amplification factor and does not involve any dimension, therefore it is constructed as a dimensionless relative permittivity (i.e., the initial permittivity ε0). By separating the absolute scale from the relative enhancement part, the dielectric properties can be made both universal and comparable.

[0032] Further, the process of performing Bayesian inference comprises: constructing a prior distribution of crack depth and dielectric constant based on a prior dielectric constant, an average dam body humidity and an initial crack depth; establishing a forward relationship between observation data and the crack depth and dielectric constant to be measured by using a reflection main peak travel time measured by the ground penetrating radar; calculating a posterior distribution of the crack depth and dielectric constant to be measured by a Bayesian formula according to the prior distribution and the observation data; and updating the posterior distribution by iteration so as to converge to a stable distribution consistent with the observation data, and obtaining a calibrated crack depth estimation value from the stable distribution (i.e. the finally generated posterior distribution function), which is the calibrated crack depth. The calibrated crack depth can be obtained by selecting a posterior mean, a posterior median and a maximum posterior value of the posterior distribution function; or by using a weighted mean of the posterior distribution of multiple lines and multiple iteration results.

[0033] The confidence interval can be constructed according to the posterior distribution function obtained by Bayesian inference iteration, such as a 95% confidence interval, which represents a probability of 95% that the real crack depth falls within the interval. In the Bayesian framework, the confidence interval reflects the interval of a certain probability mass in the posterior distribution, for example, the 95% confidence interval means that there is a 95% probability that the crack depth falls within the interval; the probability distribution or approximate distribution of the crack depth generated by the posterior sample reflects the uncertainty. The selected point estimates, which can be the posterior mean, the posterior median, the maximum posterior value and other special quantiles (such as 25% and 75%) in specific implementation, if all the selected point estimates are within the confidence interval, it means that the posterior distribution is stable, the point estimate is consistent with the overall distribution, and the calibration process can be ended; otherwise, it means that the posterior distribution is not stable, and the prior needs to be updated or the Bayesian inference needs to be performed again. If all the point estimates fall within the interval, it means that the point estimate is consistent with the overall posterior distribution and there is no abnormal deviation. If part of the point estimates fall outside the interval, it means that the posterior distribution is skewed, thick-tailed or the iteration result is unstable, and needs to be further corrected. The relationship between the multiple point estimates and the confidence interval is used to judge whether the posterior distribution converges and is reliable, thereby indirectly judging the reliability of the crack depth calibration.

[0034] The process of Bayesian inference comprises: Based on the prior dielectric constant, the average dam body humidity and the initial crack depth, a likelihood function reflecting the propagation of electromagnetic waves in water-containing concrete is constructed, the prior dielectric constant is inputted into the likelihood function as a dielectric correction parameter, the average dam body humidity is set as a humidity adjustment factor, and the humidity adjustment factor is inputted into the likelihood function as a physical correction parameter; Bayesian inference update is performed based on the corrected likelihood function and the prior distribution, and the likelihood function after the Bayesian inference update is marked as the posterior distribution function of the initial crack depth.

[0035] A likelihood function describing the propagation characteristics of electromagnetic waves in water-bearing concrete is constructed, and the humidity adjustment factor is input into the likelihood function as a physical correction parameter. As a specific example, the average humidity of the dam body (i.e., the humidity adjustment factor) is denoted as M, and its form can be specifically set as L(D0|εp,M). The likelihood function reflects the degree of physical matching between the observed data (including the travel time and amplitude of the reflection peak) and the parameters. The humidity adjustment factor is used to correct the influence of wave velocity changes with the humidity gradient, improving the model's adaptability to actual dam conditions. The likelihood function is updated using Bayes' theorem. As a specific example, the Bayes' theorem is denoted as P, and the final output calibrated crack depth (i.e., the value after the initial crack depth update) is denoted as D. The initial crack observation data is denoted as data, which mainly includes the travel time tb of the reflection peak of the target crack measured by ground-penetrating radar and the amplitude of the reflected wave. In practical applications, waveform characteristics, phase information, spectrum, and other data can also be added, all of which can be directly obtained through ground-penetrating radar measurement. The Bayesian formula can be set as: P(D|data)∝L(data|D0,εp,humidity)⋅P0(D), where P0(D) represents the prior distribution set in the initial stage, indicating the prior probability distribution of crack depth before observation data is available. In specific implementations, this can be estimated and set using historical data or engineering experience theory. The symbol ∝ in mathematics and Bayesian formulas represents proportionality, indicating that the posterior distribution is proportional to the likelihood function multiplied by the value of the prior distribution. For example, in the example form of this embodiment, the product L⋅p(D) gives the shape or relative size of the posterior distribution. The posterior distribution also needs to be transformed into the true probability distribution through a normalization constant so that the total area is 1.

[0036] Furthermore, as a feasible implementation, the process of updating the likelihood function to the posterior distribution function includes: Set the update convergence condition. During the Bayesian inference update process, the posterior distribution obtained from the previous update is used as the prior distribution for the next update. Iteratively update the likelihood function combined with the humidity adjustment factor. In each update, the variational Bayesian method is used to calculate the posterior distribution of the likelihood function, and the posterior distribution function of the initial crack depth is gradually optimized. When the Bayesian inference update reaches the update convergence condition, the update stops and the posterior distribution function is output.

[0037] In the Bayesian inference updating process, the posterior distribution function obtained by the previous updating is used as the prior distribution function of the subsequent updating to realize the iterative progressive correction of the prior information. An iterative updating process is performed on the likelihood function combined with the humidity adjustment factor, wherein the posterior distribution of the likelihood function is calculated by using a variational Bayesian method in each updating step to gradually optimize the posterior distribution function corresponding to the initial crack depth. The variational Bayesian method is a mathematical method used to approximately solve complex posterior distributions in Bayesian inference, and is particularly suitable for high-dimensional, nonlinear, or large observation noise problems. In specific implementation, a tractable variational distribution is usually used to approximate the true posterior, and the variational parameters are optimized to make the approximation as close to the true posterior as possible. When the Bayesian inference updating process satisfies the updating convergence condition, the iteration is terminated, and the posterior distribution function of the initial crack depth is finally output as the final result of the calibrated crack depth. Through the above iterative updating process, the Bayesian posterior distribution combined with the humidity adjustment factor and the observation data is optimized, thereby improving the calibration accuracy of the dam crack depth and quantifying the uncertainty of the crack depth. In specific implementation, the convergence condition can be completed by setting a posterior threshold value. When the mean value of the posterior distribution changes by less than the threshold value, the Bayesian inference process is completed. In particular, when the first iteration is performed, the prior probability distribution of the crack depth is pre-set by the system, and in specific implementation, it can be obtained by setting the historical data or engineering experience and theoretical estimation.

[0038] In particular, as a feasible implementation, the updating convergence condition includes: setting a posterior variance threshold value for the Bayesian inference updating process, and calculating the variance value of the posterior distribution of each Bayesian inference updating; when the variance value of the posterior distribution is lower than the posterior variance threshold value, the Bayesian inference completes the updating process, at which time the updating is stopped and the posterior distribution function is output.

[0039] A posterior variance threshold value is pre-set, and the threshold value size can be selected according to engineering requirements and observation data accuracy. The smaller the threshold value, the more concentrated the posterior distribution is required, and the more refined the iteration is. A larger threshold value indicates faster convergence, but higher uncertainty. Since the smaller the posterior variance, the more stable the crack depth estimation is, and the observation and prior information are fully fused, and when the variance is large, the posterior distribution still has large fluctuations, indicating that the depth estimation has uncertainty, therefore, as a preferred embodiment, the posterior variance threshold value can be set to a small value. When the variance value of the posterior distribution is lower than the posterior variance threshold value, the posterior distribution is concentrated enough, and subsequent iterations will not significantly change the crack depth estimation, at which time the iteration is stopped, and the final posterior distribution function is output, realizing accurate, stable, and controllable crack depth calibration.

[0040] Further, as a feasible implementation, the process of calculating the posterior distribution by the variational Bayesian method includes: The evidence lower bound of the variational Bayes and the gradient step size are set; in each Bayesian inference update iteration, the posterior distribution obtained by the previous Bayesian inference update is parameterized as a lognormal distribution, the gradient step size is used to adjust the KL divergence of the lognormal distribution in the direction of increasing the evidence lower bound, and the lognormal distribution after the KL divergence adjustment is output as the posterior distribution result of the current update iteration.

[0041] In the variational Bayes method, the evidence lower bound is used to represent the approximation degree of the approximate posterior distribution to the true posterior distribution in the variational Bayes inference, and its mathematical form is the lower bound value of the true posterior probability. In the calculation process, the variational distribution gradually converges to the true posterior distribution by maximizing the evidence lower bound. Therefore, the evidence lower bound can be used as an optimization target of the variational Bayes update process to guide the iterative update of the posterior distribution. The gradient step size is used to control the adjustment range of the posterior distribution parameter in the variational Bayes update process. The posterior distribution parameter represents the parameter of the corresponding posterior probability distribution shape, which is used to represent the parameterization result of the crack depth probability distribution obtained by the Bayesian inference update in combination with the radar observation data, the prior dielectric constant and the humidity adjustment factor. The posterior distribution parameter can specifically include a mean parameter and a variance parameter used to describe the shape of the variational posterior distribution, which are used to reflect the center position of the crack depth estimation value and the uncertainty degree thereof. After calculating the gradient direction of the evidence lower bound each time, the gradient step size is set to make the variational distribution perform parameter update in the direction of increasing the evidence lower bound. The gradient step size is used to constrain the range of single update, so as to avoid iteration instability caused by too fast update or reduce the convergence efficiency caused by too slow update. In this embodiment, the evidence lower bound and the gradient step size can be set by an empirical rule when being specifically applied. In each Bayesian inference update, the evidence lower bound value corresponding to the lognormal distribution obtained in the previous iteration is determined, the gradient of the lognormal distribution in the direction of increasing the evidence lower bound is calculated, and the parameters of the lognormal distribution are updated according to the set gradient step size, so that the KL divergence between the updated lognormal distribution and the true posterior distribution is reduced. The lognormal distribution after the KL divergence adjustment is determined as the posterior distribution of the current iteration step, which is used for output and as the prior distribution input of the next update iteration.

[0042] More specifically, the process of parameterizing the posterior distribution as a lognormal distribution can be used as an example, and its content can include: the posterior distribution is denoted as pk, and the lognormal distribution is denoted as qk. The obtained posterior distribution definition formula pk(D|data) is defined as a parameterized lognormal distribution definition formula qk(D)=LogNormal(D;μk,σk 2) where μk represents the mean of the posterior distribution obtained from the k-th Bayesian update in the log domain, and σk represents the standard deviation of the posterior distribution after the k-th update in the log domain. The μk is used to represent the central tendency of the posterior estimate of the crack depth in the log space, and the σk is used to measure the dispersion of the crack depth estimate in the log space.

[0043] As a specific application example, the parameterization process includes: from the k-th posterior sampling set {D (s)} S s=1 or the posterior density estimate, the logarithm of the sample is first taken to obtain {logD (S)}; the mean and variance of the logarithmic sample are calculated, i.e. and ; the μk and σk are used to define the lognormal approximation distribution. Where S represents the iteration step index of the variational Bayesian method, i.e. the update order number.

[0044] Further, as a feasible implementation, the Adam optimizer is used to set the gradient step size, and the process is set as: The Adam optimizer is set, and in each Bayesian inference update iteration, in each round of Bayesian inference update, the first moment and the second moment of the gradient step size are updated based on the gradient information of the lognormal distribution, the adaptive step size is obtained based on the first moment and the second moment using the Adam optimizer, and after dynamic bias correction of the adaptive step size using the Adam optimizer, the adaptive step size is generated as the gradient step size of the current iteration.

[0045] The purpose of introducing the Adam optimizer in the variational Bayesian update process is to adaptively adjust the gradient step size to improve the stability and convergence speed of the lognormal distribution parameter update. In each Bayesian inference update iteration, based on the gradient information of the lognormal distribution with respect to the lower bound of the evidence, the Adam optimizer accumulates the first moment (mean estimate) and the second moment (non-central variance estimate) of the current gradient, and updates them exponentially. The Adam optimizer calculates the adaptive step size of the current gradient direction based on the updated first moment and second moment according to its internal rules, so that larger gradients are suppressed and smaller gradients are amplified, thereby obtaining a more stable parameter update amplitude. The adaptive step size is calculated by the Adam optimizer according to the first moment and the second moment, and dynamic bias correction is performed on the adaptive step size; the adaptive step size obtained by the Adam optimizer is used as the gradient step size of the current iteration, which is used to update the parameters of the lognormal distribution.

[0046] In the embodiment, the process of obtaining the calibrated crack depth from the posterior distribution function is set as follows: likelihood functions of Bayesian inference iterations in different orders are arranged in time sequence to form a time series, mean values of the posterior distributions at each time point are assigned iteration weights, fusion calculation is performed based on the iteration weights, and the final result of the fusion calculation is marked as the calibrated crack depth.

[0047] Likelihood functions generated in the process of Bayesian inference iterations in different orders are arranged in iteration time sequence to form a time series, mean values of the posterior distributions obtained in each iteration are extracted, and the mean values are assigned weights according to iteration order or convergence state. The mean value of the iteration is more stable, and the weight is higher. In the process of obtaining the calibrated crack depth from the posterior distribution function, likelihood functions of Bayesian inference iterations in different orders are arranged in time sequence to form a time series, mean values of the posterior distributions at each time point are assigned iteration weights, and fusion calculation is performed on the time series mean values based on the iteration weights. The final result of the fusion calculation is marked as the calibrated crack depth. The time series fusion calculation method can be Kalman filtering method, which uses the posterior mean value of the time series as the observation value, combines the estimated variance and the predicted variance to perform recursive filtering update, and obtains the optimal estimation value after fusion. A single weight weighted average method can also be used to weight and sum the mean values of the posterior distributions at different time points according to the preset iteration weight to obtain the calibrated crack depth.

[0048] Further, as a feasible implementation, the process of time series fusion calculation is set as follows: Let the calibrated crack depth be represented as Dc, the iteration weight be represented as β, and the order of Bayesian inference update iteration be represented as i. Let the crack depth estimate corresponding to the posterior distribution of each iteration be represented as Dp, Then the calculation formula of the calibrated crack depth of the i-th iteration is: , Where the iteration weight β is in the range of 0≤β≤1, and i is a natural number greater than 1.

[0049] The mean value of the posterior distribution generated in each Bayesian inference iteration is fused with the previous round of calibration results according to an iteration weight. The iteration weight controls the contribution proportion of the previous round of calibration results and the current posterior mean value. When the iteration weight tends to or is equal to 0, the current iteration posterior mean value is dominant, and new information is quickly responded to; when the iteration weight tends to or is equal to 1, the historical cumulative result of the previous period is dominant, and the iteration process is more smooth and stable. Adjusting β can balance the demand for historical stability and quick response to new observations. A larger β can reduce the influence of observation noise or prior fluctuations on the calibration depth in a single iteration, delay the update step, make the calibration result smoother, and reduce the risk of over-adjustment; a smaller β allows the calibration crack depth to be quickly adjusted to respond to the latest data, and is suitable for situations where the crack depth changes significantly with the environment or humidity. When β is equal to 1, the posterior estimate of the current iteration is completely ignored, and the calibration depth completely follows the results of the previous round; when β is equal to 0, the calibration crack depth is completely determined by the posterior estimate of the current iteration, and the historical iteration results are not involved in the fusion. In specific implementation, under the condition of ensuring the effect of regular iteration, β can generally not be set to 1 or 0. As a feasible implementation, the crack depth estimate of each iteration is set to the median of the posterior distribution in the iteration. The crack depth estimate corresponding to the posterior distribution obtained in each iteration is set to the median of the posterior distribution, which is used as the input of the time series fusion calculation. Compared with the mean value, the median can more accurately reflect the typical position of the crack depth; under the influence of high humidity gradient or observation noise, the posterior distribution may be skewed or multi-tailed, and the median estimate can suppress the deviation caused by occasional abnormalities.

[0050] Further, as a feasible implementation, the setting process of the confidence interval comprises: Based on the posterior distribution parameter of the posterior distribution, a confidence interval of the crack depth is constructed; an approximate distribution of the crack depth is generated according to the posterior distribution, and a plurality of point estimates are extracted from the approximate distribution; when the plurality of point estimates all fall within the confidence interval, it is determined that the crack depth has reached the calibration condition, and the calibrated crack depth is output.

[0051] According to the posterior distribution completed after the iteration of Bayesian inference, the confidence interval of the crack depth is determined by using the distribution parameters such as special set values (mean values), and the confidence interval can be 95%, 90% or set according to the engineering requirements; an approximate distribution of the crack depth is generated according to the posterior distribution, and a plurality of point estimates are extracted from the approximate distribution; when the plurality of point estimates all fall within the confidence interval range, it is determined that the crack depth has reached the calibration condition, and the calibrated crack depth is output. The confidence interval of the posterior distribution quantifies the reliable range of the crack depth, reflecting the reliability of the observation data and the model in the engineering. Extracting a plurality of point estimates from the approximate distribution and verifying whether they fall within the confidence interval is equivalent to verifying the internal consistency of the posterior distribution and the stability of the iterative calibration. As a feasible implementation manner, the posterior distribution parameters include the mean parameter and the variance parameter of the lognormal distribution, which are used to represent the variational posterior distribution of the crack depth. In variational Bayesian inference, the posterior distribution of the crack depth is approximately represented by a lognormal distribution, and the parameters of the parameterized lognormal distribution constitute the variational distribution parameters, which are used to approximate the real Bayesian posterior distribution, realize the feasible iterative update and confidence interval construction. The posterior distribution parameters include the mean parameter and the variance parameter of the lognormal distribution, which are used to represent the variational posterior distribution of the crack depth, wherein the mean parameter reflects the central tendency of the crack depth, and the variance parameter is used to quantify the uncertainty of the crack depth estimation and provide basic information for subsequent time series fusion, point estimation selection and confidence interval construction.

[0052] In different embodiments of the present technology, the construction of the prior dielectric constant can not be limited to a single calculation model. In this embodiment, the calculation process of the prior dielectric constant includes: Let the speed of light be represented as c, the travel time of the main reflection peak be represented as tb, the initial dielectric constant be represented as ε0, the initial crack depth be represented as D0, and the prior dielectric constant be represented as εp. The calculation formula of the prior dielectric constant εp is: , Wherein, α represents the observation weight, and w represents the amplitude weight, and 0 < w ≤ 1.

[0053] In this embodiment, the meaning of the prior dielectric constant calculation formula is that the main part containing the initial dielectric constant is added to the part calculated based on the travel time of the main reflection peak and the initial crack depth observation to obtain. The (1-α)·ε0 part in the formula ensures that the prior dielectric constant is not completely dependent on the dielectric observation component part , keeps physical rationality. When α is small, the prior permittivity mainly depends on the initial permittivity; when α is large, the dielectric observation component has a greater impact on the prior permittivity. The amplitude weight adjusts the influence of the reflection signal strength on the prior, and when the amplitude of the reflection signal is low, the amplitude weight can reduce its contribution. When the amplitude weight is equal to 1, the observation amplitude is completely reliable, and the contribution of the observation calculated permittivity reaches the maximum state. When the amplitude weight is close to 0, the observation amplitude is very weak, and the prior permittivity is mainly obtained through the initial permittivity combined with the observation weight value. The amplitude weight is not equal to 0 in order to always retain the adjustment and optimization of the initial permittivity by the dielectric observation component; the amplitude weight is not more than 1 in order to prevent excessive dependence on the dielectric observation component, because the target dam body is spacious, the dam humidity and crack uniformity are low, and the humidity gradient is complex, so that excessive influence of non-accurate measurement on the crack depth calibration process can be avoided. Since the amplitude weight is only allocated to the dielectric observation component, when the amplitude weight is between 0 and 1, the sum of the total weights is no longer strictly equal to 1, but less than 1, that is, it becomes a damping type weighted fusion; in the specific implementation process, the adjustment mode of the damping type weighted fusion can reduce the dependence of the dielectric observation component on the shallow crack and weak reflection signal, and gradually increase the contribution of the dielectric observation component to the deep crack and high amplitude signal. The observation weight represents the degree of influence of the information carried by the current reflection travel time on the correction of the prior permittivity. The amplitude weight represents the degree of influence of the main peak amplitude of the target detection crack reflection wave on the correction of the prior permittivity.

[0054] It should be noted that in the formula part represents the dielectric observation component, and this part of the expression refers to the effective permittivity obtained by reflection travel time and initial depth inversion, that is, the local medium refractive property calculated by the dielectric observation component.

[0055] The derivation process of the dielectric observation component is as follows: let the dielectric observation component be represented as εe, and let the electromagnetic wave velocity be represented as 2D0 / tb, The dielectric observation component calculation formula is represented as: .

[0056] Further, as a feasible implementation manner, the setting content of the amplitude weight includes: The main peak amplitude of the target detection crack reflection wave is set as Ap, the maximum amplitude of the reflection wave observed in the target dam is set as Am, and the amplitude adjustment coefficient γ is set, The calculation formula of the amplitude weight w is represented as: , where γ>0.

[0057] The amplitude weight calculation formula means that when the main peak signal is strong, the reflection interface is obvious, and the observation data is highly reliable, at this time the amplitude weight can be increased, the proportion of the dielectric observation component is increased, and the observation calculation part fully participates in the calculation of the prior dielectric constant; when the main peak signal is weak, it may be affected by noise or path attenuation, the observation data is less reliable, at this time the amplitude weight can be reduced, the proportion of the dielectric observation component is reduced, and the influence of weak signal on the prior dielectric constant is reduced. In the case of Ap / Am less than 1, when the amplitude adjustment coefficient is equal to 1, the amplitude weight changes linearly with the amplitude ratio; when the amplitude adjustment coefficient is greater than 1, the sensitivity of the amplitude weight is greater, the change range is increased, the weight value will decrease at a faster speed, and the value after reduction is too low, so that the weight contribution is low; when the amplitude adjustment coefficient is between 0 and 1, the amplitude weight has high tolerance to low amplitude, the function curve is relatively flat in the low amplitude area, the weight decreases at a slower speed, and still retains a certain contribution after reduction.

[0058] In particular, in the case of Ap / Am equal to 1, at this time the main peak of the target crack reflection wave is the maximum amplitude of the reflection wave observed by the target dam; no matter what value the amplitude adjustment coefficient takes, the amplitude weight value is always 1 at this time, and the total weight sum of the prior dielectric constant calculation formula is 1, so that the initial dielectric constant and the dielectric observation component can have the same influence on the prior dielectric constant.

[0059] It should be noted that the calculation process of the prior dielectric constant has been constructed in embodiment 1. In the description, the calculation formula of the prior dielectric constant in embodiment 1 is denoted as calculation formula 1, and the calculation formula of the prior dielectric constant in this embodiment is denoted as calculation formula 2. Although calculation formula 1 and calculation formula 2 adopt different mathematical expressions, they both belong to the prior dielectric constant construction method based on the reflection main peak travel time information to correct the initial dielectric constant. They have the following common points in principle and function: Both are based on the physical function relationship between electromagnetic wave propagation travel time and dielectric constant. Both calculation formulas are based on the physical law that the propagation speed of electromagnetic wave in medium changes with dielectric constant, and through the deterministic correspondence between reflection main peak travel time and dielectric constant, the deviation of measured travel time from initial travel time is used to represent the change trend of dielectric constant, so that the prior dielectric constant can reflect the electromagnetic characteristics in the current crack environment.

[0060] The two calculation formulas are constructed by using the initial dielectric constant and the correction amount based on the travel time deviation. The correction amount based on the travel time deviation is a dielectric constant correction term derived from the travel time deviation, and is used to describe the change in the dielectric constant corresponding to the change in the travel time of the main reflection peak. Both calculation formulas can make the prior dielectric constant close to the actual dielectric characteristics in value by calculating or weighting the correction term. Both calculation formulas use the travel time of the main reflection peak as a directly observable physical quantity, and automatically adjust the prior dielectric constant based on the travel time change without relying on additional structural assumptions, so that the generated prior parameters can provide initial values that meet the actual situation for subsequent variational Bayesian inference, can meet the verification process of Bayesian inference update, and can improve the model convergence speed and inference accuracy.

[0061] In the construction process of the specific calculation formula, the calculation formula 1 uses a first-order linearization to back-propagate the prior dielectric constant, and uses a first-order linear expansion of the travel time deviation, which only contains a first-order deviation term. The calculation formula 2 uses a quadratic term to back-propagate the prior dielectric constant, and obtains the equivalent dielectric constant by squaring the reflection travel time, and the calculation result contains a second-order dimension of the travel time deviation. Both are different approximation methods of the same physical model. The correction amount of the calculation formula 1 is determined by the geometric scale and the initial dielectric constant, and does not have additional weight adjustment capability. The calculation formula 2 contains adjustable weights a and w, so that the output result can be scaled and adjusted according to the target material characteristics, signal-to-noise ratio or experience parameters. Since the travel time of the main reflection peak is a limited small amount in actual use, its change range is restricted by the physical conditions of the concrete dielectric constant and the crack depth, and will not be too large. For example, in the specific implementation of the engineering scene of detecting cracks in the dam by using the ground penetrating radar, the travel time of the main reflection peak is generally not very large. The commonly used frequency band of the ground penetrating radar (GPR) in the concrete dam is 100MHz-500MHz, and the dielectric constant of the concrete is generally ε≈6-10. The typical travel time of the main reflection peak is in the order of tens to two hundred nanoseconds, which is not large. Therefore, when the travel time deviation of the main reflection peak is within the normal range, the difference between the calculation results of the two calculation formulas is in the linear approximation order, and the numerical size is at the same level. The deviation between the calculation results of the two calculation formulas can be limited within an adaptable range.

[0062] Here, more specific implementation examples of the calculation processes of the two calculation formulas are given. The following examples are preferred implementation applications for improving the numerical accuracy of the two calculation formulas, and are not limited to the scenes set by the calculation formulas. The calculation formula 1 is a first-order linear expansion of the travel time deviation, does not contain a weight factor, has low noise sensitivity, and is more suitable for scenes with stable signals, shallow cracks and small travel time deviations. Specifically, it can include: 1. High signal-to-noise ratio, main peak travel time stable detection environment, radar antenna and dam body coupling is good, clear echo, peak stable, linear term can directly reflect the small offset of travel time, get stable dielectric estimation; 2. The scene of shallow crack depth or small change range, the crack depth is in the high resolution interval of radar, the shallow interference is low, the reflection main peak travel time is close to the theoretical travel time, the linear correction model with small deviation has high precision under the condition of small deviation; 3. The scene of uniform dam material structure and weak dielectric gradient, the dam body has no obvious water area mutation, the material dielectric constant distribution is stable, the linear travel time and dielectric relationship can accurately reflect the dielectric disturbance, and the second sensitive term does not need to be introduced.

[0063] The calculation formula 2 adopts a second-order dimensional sensitive structure for travel time, and contains adjustable observation weight and amplitude weight, so it is suitable for the case where noise exists, the signal amplitude difference is large, or the detection scene is complex. Specifically, it can include: 1. The scene of significant reflection signal amplitude difference, for example, the echo intensity difference of cracks with different depths is large, and the main peak energy decays obviously with depth. Adjusting the amplitude weight can weaken the influence of low-quality echoes; 2. The scene of dam material water content change too uneven and refractive index gradient obvious, such dam body has alternating distribution of wet and dry areas, and the electromagnetic wave propagation speed gradient is large. The second travel time term can better capture the change of dielectric constant in the highly nonlinear region; 3. The scene of large travel time error or noise that cannot be ignored, there are obvious environmental interferences such as equipment drift, electromagnetic noise, rough interface diffraction, etc. The second term structure has stronger sensitivity, is suitable for establishing priori with strong constraint in the noise background, can amplify the change trend, and compensate part of the travel time deviation.

[0064] The above specific embodiments further illustrate the purpose, technical solutions and beneficial effects of the present application. It should be understood that the above description is only a specific embodiment of the present application and is not used to limit the protection scope of the present application. Any modification, equivalent replacement, improvement, etc. within the spirit and principles of the present application should be included in the protection scope of the present application.

Claims

1. A method for detecting a crack depth of a dam based on Bayesian inference, characterized by, The method comprises: Step S1: collecting the initial dielectric constant and the average humidity of the dam body of the target dam, collecting the initial crack depth detected by the geological radar detection, and obtaining the reflection main peak travel time of the target detection crack; Step S2: calculating the prior dielectric constant using the reflection main peak travel time and the initial dielectric constant, performing Bayesian inference based on the prior dielectric constant, the average humidity of the dam body and the initial crack depth, and performing inference update iteration in the Bayesian inference; Step S3: generating a posterior distribution function in the posterior stage of the Bayesian inference, updating the posterior distribution function in each iteration cycle, and obtaining the calibrated crack depth from the final updated posterior distribution function; Step S4: setting a confidence interval, calculating the approximate distribution of the calibrated crack depth and selecting a plurality of point estimates, completing the calibration process when all selected point estimates are within the confidence interval, and returning to step S2 when there is a point estimate outside the confidence interval.

2. The method for detecting the depth of dam crack based on Bayesian inference according to claim 1, characterized in that, The calculation process of the prior dielectric constant comprises: Let the speed of light be c, the reflection main peak travel time be tb, the initial dielectric constant be ε0, the initial crack depth be D0, and the prior dielectric constant be εp, The calculation formula of the prior dielectric constant εp is represented as: , Wherein α represents the observation weight, w represents the amplitude weight, and 0<w≤1. 3.The method of claim 2, wherein, The setting content of the amplitude weight comprises: Setting the main peak amplitude of the reflection wave of the target detection crack as Ap, setting the maximum amplitude of the reflection wave observed in the target dam as Am, and setting the amplitude adjustment coefficient γ, The amplitude weight w is calculated as where γ >

0.

4. The method of claim 1, wherein, The process of the Bayesian inference comprises: Based on the prior dielectric constant, the average humidity of the dam body and the initial crack depth, a likelihood function reflecting the propagation of electromagnetic waves in water-containing concrete is constructed, the prior dielectric constant is input into the likelihood function as a dielectric correction parameter, the average humidity of the dam body is set as a humidity adjustment factor, and the humidity adjustment factor is input into the likelihood function as a physical correction parameter; based on the corrected likelihood function and the prior distribution, Bayesian inference update is performed, and the likelihood function after the Bayesian inference update is marked as the posterior distribution function of the initial crack depth.

5. The method of claim 4, wherein, The process of updating the likelihood function to the posterior distribution function comprises: Setting an update convergence condition, in the Bayesian inference update process, the posterior distribution obtained by the previous update is used as the prior distribution of the next update; the likelihood function combined with the humidity adjustment factor is iteratively updated, the posterior distribution of the likelihood function is calculated using the variational Bayesian method in each update, and the posterior distribution function of the initial crack depth is gradually optimized; when the Bayesian inference update reaches the update convergence condition, the update is stopped and the posterior distribution function is output.

6. The method of claim 5, wherein, The update convergence condition comprises: setting a posterior variance threshold for the Bayesian inference update process, and calculating the variance value of the posterior distribution of each Bayesian inference update; when the variance value of the posterior distribution is lower than the posterior variance threshold, the Bayesian inference update process is completed, at which time the update is stopped and the posterior distribution function is output.

7. The method of claim 5, wherein, The process of calculating the posterior distribution by the variational Bayesian method comprises: The evidence lower bound and the gradient step size of the variational Bayes are set; in each Bayesian inference update iteration, the posterior distribution obtained by the previous Bayesian inference update is parameterized as a lognormal distribution, the lognormal distribution is adjusted in the direction of increasing the evidence lower bound by using the gradient step size, and the lognormal distribution after the KL divergence adjustment is output as the posterior distribution result of the current update iteration.

8. The method of claim 7, wherein, The gradient step size is set using the Adam optimizer, and the process setting is as follows: The Adam optimizer is set, and in each Bayesian inference update iteration, the first moment and the second moment of the gradient step size are updated based on the gradient information of the lognormal distribution in each Bayesian inference update, the adaptive step size is obtained based on the first moment and the second moment using the Adam optimizer, and the adaptive step size is generated as the gradient step size of the current iteration after dynamic bias correction of the adaptive step size using the Adam optimizer.

9. The method of claim 4, wherein, The process of obtaining the calibrated crack depth from the posterior distribution function is set as follows: the likelihood functions of Bayesian inference iterations of different orders are constructed into a time series in time sequence, the mean values of the posterior distributions at each time point are assigned iteration weights, and the final result of the fusion calculation is marked as the calibrated crack depth based on the iteration weights using time series fusion calculation.

10. The method of claim 9, wherein, The process of the time series fusion calculation is set as follows: Let the calibrated crack depth be represented as Dc, the iteration weight be represented as β, and the ordinal number of the Bayesian inference update iteration be represented as i; let the crack depth estimate corresponding to the posterior distribution of each iteration be represented as Dp, The calculation formula of the calibration crack depth of the i-th iteration is represented as: , Where the iteration weight β is set to be in the range of 0≤β≤1, and i is set to be a natural number greater than 1.

11. The method of claim 10, wherein, The crack depth estimate of each iteration is set to be the median of the posterior distribution in this iteration.

12. The method of claim 1, wherein, The setting process of the confidence interval includes: The confidence interval of the crack depth is constructed based on the posterior distribution parameter of the posterior distribution; the approximate distribution of the crack depth is generated according to the posterior distribution, and a plurality of point estimates are extracted from the approximate distribution; when the plurality of point estimates all fall within the confidence interval, it is determined that the crack depth has reached the calibration condition, and the calibrated crack depth is output.

13. The method of claim 12, wherein, The posterior distribution parameter includes the mean parameter and the variance parameter of the lognormal distribution, which is used to represent the variational posterior distribution of the crack depth.

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