Multi-mode ultrasonic lamb wave imaging spatial resolution quantification method

Abstract

The invention discloses a multi-modal ultrasonic Lamb wave imaging spatial resolution quantification method, and relates to the technical field of ultrasonic Lamb wave imaging, and the method specifically comprises the steps: 1, employing an array transducer to carry out point-by-point excitation and receiving on the basis of a full-focus method data collection mode, collecting complete full-matrix data, building a multi-modal Lamb wave propagation model, and carrying out the calculation of the multi-modal Lamb wave propagation model; the method comprises the following steps: step 1, constructing a corresponding Fisher information matrix based on partial derivative information of each channel in combination with observation noise variance, and step 2, carrying out CRLB numerical calculation by adopting an MATLAB platform based on a derivation result of the Fisher information matrix. Physical relations among the defect imaging precision, the wavelength and the modal characteristics are established theoretically, and a quantitative basis is provided for multi-modal selection and system design.

Description

Technical Field

[0001] The present invention belongs to the technical field of ultrasonic Lamb wave imaging, and particularly relates to a method for quantifying the spatial resolution of multimodal ultrasonic Lamb wave imaging. Background Technique

[0002] With the development of structural health monitoring technology (SHM), Lamb waves have been widely used in non-destructive testing of plate-like structures in fields such as aerospace, rail transit, and civil engineering due to their characteristics of long propagation distance, high sensitivity, and sensitivity to surface and near-surface defects. However, the propagation of Lamb waves is accompanied by significant dispersion characteristics and modal diversity, and its propagation behavior is significantly affected by factors such as excitation frequency, structural thickness, and material parameters. In a complex environment with coexisting multimodes, the interpretation of signals and the imaging quality highly depend on the effective discrimination and reasonable utilization of each mode.

[0003] Currently, the methods for imaging resolution analysis mainly focus on empirical parameter optimization, image quality index evaluation, or numerical simulation means, such as transverse intensity curve (TIC) measurement, simulation comparison experiments, etc. These methods have defects such as certain subjectivity or strong dependence on specific models, lack accuracy evaluation indicators with theoretical universality, and are difficult to be extended to complex structures, arbitrary modal combinations, or different imaging system design scenarios.

[0004] In the field of statistical signal processing, the Cramer-Rao lower bound (CRLB), as the theoretical lower limit for measuring the minimum variance of parameter estimation, has long been used for performance analysis in scenarios such as radar, multi-antenna arrays, and sound source localization. Its advantage lies in that it can calculate the lowest variance that any unbiased estimator can achieve through the Fisher information matrix (FIM) according to the given signal model and parameter relationship. Therefore, there is an urgent need for a method for quantifying the spatial resolution of multimodal ultrasonic Lamb wave imaging to obtain the theoretical limit of spatial resolution by getting rid of the dependence on specific algorithms and establishing the physical relationship between modal characteristics and imaging capabilities. Summary of the Invention

[0005] The purpose of the present invention is to provide a method for quantifying the spatial resolution of multimodal ultrasonic Lamb wave imaging to solve the technical problem of the lack of a theoretical evaluation standard for spatial resolution in the prior art.

[0006] To achieve the above purpose, the present invention adopts the following technical solutions: A method for quantifying the spatial resolution of multimodal ultrasonic Lamb wave imaging, where CRLB represents the Cramer-Rao lower bound, includes: Step 1: Based on the full-focusing method data acquisition method, use an array transducer for point-by-point excitation and reception, collect complete full matrix data, and establish a multimodal Lamb wave propagation model; Step 2: Based on the partial derivative information of each channel and combined with the observation noise variance, construct the corresponding Fisher information matrix; Step 3: Based on the derivation result of the Fisher information matrix, use the MATLAB platform to perform numerical calculation of the CRLB.

[0007] Furthermore, use an array transducer for point-by-point excitation and reception to collect complete full matrix data. The specific method is as follows: Record the time-domain signals of all transmit-receive channels, perform three-dimensional fast Fourier transform on the full matrix data, convert the time-domain signals to the frequency-wavenumber domain, obtain the energy distribution characteristics of the signals in three-dimensional space, identify the wavenumber ranges and energy concentration regions of different Lamb wave modes through frequency-wavenumber spectrum analysis, use the Hilbert-Huang transform, extract the intrinsic mode functions of the signals through empirical mode decomposition, combine instantaneous frequency analysis to separate the time-frequency components of the aliased modes, design matching window functions or perform band-pass filtering operations according to the wavenumber distribution ranges of different modes, and use Hilbert envelope detection or short-time energy integration methods to extract the amplitude information of each mode, determine the Lamb wave dispersion curve, and determine the wavenumber according to the excitation frequency and group velocity in the Lamb wave dispersion curve.

[0008] Furthermore, establish a multi-modal Lamb wave propagation model. The specific method is as follows: Obtain the time-domain signal expression of the multi-modal Lamb wave by linearly superposing each modal component to construct a signal model. The signal component of each mode is a function of the defect position and is represented by the formula where S represents the signal model, i represents the i-th receiving array, j represents the j-th Lamb wave mode, N represents the total number of receiving arrays, M represents the total number of Lamb wave modes, represents the out-of-plane amplitude of the j-th Lamb wave mode at the receiving element i, represents the wavenumber of the j-th Lamb wave mode, represents the spatial coordinate of the receiving element i, represents the spatial coordinate of the defect point, represents the phase of the j-th Lamb wave mode, represents Gaussian white noise with a variance of

[0009] Furthermore, based on the partial derivative information of each channel and combined with the observation noise variance, construct the corresponding Fisher information matrix. The specific method is as follows: ​For the observed signals of each array element channel, the partial derivatives with respect to the defect spatial position parameters are obtained respectively. According to the chain rule, the partial derivative of the signal intensity with respect to the position parameters is expanded into the joint contribution of the amplitude term and the phase term. Based on the partial derivative information of each channel and combined with the observation noise variance, the corresponding Fisher information matrix is constructed. Specifically, each element of the Fisher information matrix is composed of the outer product superposition of the partial derivatives of each channel. Among them, the diagonal elements measure the sensitivity to a single position parameter, and the non-diagonal elements reflect the correlation between the position parameters. The theoretical lower limit of the defect position estimation error is deduced through the Fisher information matrix. The Fisher information matrix provides a quantitative reference standard for the optimization of different modal combinations, array configurations, and excitation frequency selections. By systematically analyzing the FIM, the differences in the contributions of different Lamb wave modes to the spatial resolution are revealed.

[0010] Furthermore, the partial derivative of the signal intensity with respect to the position parameters is expanded into the joint contribution of the amplitude term and the phase term. The specific method is as follows: Using the formula and the formula , the partial derivative of the signal intensity with respect to the position parameters is expanded into the joint contribution of the amplitude term and the phase term, where represents the signal intensity of a certain Lamb wave mode at time t, t represents time, A represents the amplitude of the corresponding Lamb wave mode, f represents the excitation frequency, that is, the frequency of the transmitted signal, represents the phase of the corresponding Lamb wave mode, x represents the abscissa, and y represents the ordinate.

[0011] Furthermore, combined with the observation noise variance, the corresponding Fisher information matrix is constructed. The specific method is as follows: There are M modes of Lamb waves, and the size of the Fisher information matrix (FIM) is (2M + 3) × (2M + 3). Using to represent the Fisher information matrix, where represents the signal model, j represents the jth mode, represents the auxiliary variable of the jth mode, representing the partial differentials with respect to the amplitude and phase respectively, represents the spatial coordinates of the defect point source, represents the observation noise variance, represents the partial derivative of the signal model with respect to the auxiliary variable of the jth mode.

[0012] Furthermore, the partial derivative of the signal model with respect to the auxiliary variable of the jth mode. The specific method is as follows: Using the formula to represent the partial derivative of the signal model with respect to the auxiliary variable of the jth mode, where represents the signal model, j represents the jth mode, The auxiliary variable representing the j-th mode, and M represents the total number of Lamb wave modes. Denotes the characteristic factor, and is represented by the formula Denotes.

[0013] Furthermore, based on the derivation result of the Fisher information matrix, the numerical calculation of the CRLB is carried out using the MATLAB platform. The specific method is as follows: The numerical calculation of the CRLB is carried out using the MATLAB platform. Physical parameters are set according to the actual detection scenario, including plate geometric parameters, excitation frequency range and step size, wave speeds of each mode, wave numbers, and amplitude attenuation laws. A complete input data set is constructed, and the inverse matrix of the Fisher information matrix is obtained through matrix operations. The diagonal elements corresponding to the defect spatial position parameters are extracted to obtain its theoretical minimum variance, that is, the CRLB value.

[0014] In summary, due to the adoption of the above technical solutions, the beneficial effects of the present invention are as follows: 1. By adopting the full focusing method data acquisition method, the present invention uses a two-dimensional array transducer to excite and receive point by point to collect complete full matrix data. This comprehensive data acquisition method can obtain richer information, covering the time-domain signals of all transmit-receive channels. By performing a three-dimensional fast Fourier transform on the full matrix data, the time-domain signals are converted to the frequency-wave number-space domain to obtain the multi-dimensional energy distribution characteristics of the signals, and it is possible to more intuitively understand the propagation of Lamb waves in different dimensions. 2. By decomposing the partial derivative of the signal intensity with respect to the position parameter into the joint contributions of the amplitude term and the phase term, the present invention can more carefully analyze the influencing factors of the signal intensity change with the defect position, providing a more accurate theoretical basis for accurately evaluating the defect position. The Fisher information matrix constructed based on the partial derivative information of each channel and the observation noise variance can deduce the theoretical lower limit of the defect position estimation error under the given imaging conditions. This lower limit provides an objective standard for evaluating the performance of the detection system, enabling researchers and engineers to understand the minimum error range that can be achieved for defect position estimation under the current detection conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0015] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0016] Figure 1 Shows a step diagram of a method for quantifying the spatial resolution of multi-modal ultrasonic Lamb wave imaging. Figure 2 Shows a method step diagram for constructing the Fisher information matrix; Figure 3 Is a result comparison diagram for implementing TFM imaging of the present invention; Figure 4 Is a result comparison diagram for implementing 3D-FFT of the present invention; Figure 5 Is a schematic diagram of the CRLB result of the present invention; Figure 6 Is a quantitative comparison diagram of the TIC results after implementing TFM imaging of the present invention. Detailed implementation manners

[0017] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0018] As Figure 1 , Figure 2 , Figure 3 , Figure 4 , Figure 5 And Figure 6 shown, a method for quantifying the spatial resolution of multimodal ultrasonic Lamb wave imaging specifically includes the following steps: Step 1: Based on the full focus method data acquisition method, use an array transducer for point-by-point excitation and reception, collect complete full matrix data, and establish a multimodal Lamb wave propagation model.

[0019] A two-dimensional array transducer is adopted, with the element pitch set to half of the Lamb wave wavelength. An A*A array of elements is used to cover the data detection area. A laser rangefinder is used to measure and record the actual spatial coordinates of each element. Each element is sequentially activated as a transmitter, and the remaining elements synchronously receive the echo signals. Through point-by-point excitation and reception by the array transducer, complete full matrix (FMC) data is collected, including recording the time-domain signals of all transmit-receive channels. Three-dimensional fast Fourier transform (3D-FFT) is performed on the full matrix data (FMC) to convert the time-domain signals to the frequency-wavenumber-space domain, and the multi-dimensional energy distribution characteristics of the signals are obtained. Through frequency-wavenumber spectrum analysis and in combination with predefined Lamb wave dispersion curves, the wavenumber-frequency mapping relationships of different modes (such as S0, A0, A1) and their energy concentration regions are identified. For the aliased modes in the frequency-wavenumber domain, the Hilbert-Huang transform (HHT) is further adopted. The intrinsic mode functions (IMFs) of the signals are extracted through empirical mode decomposition (EMD), and the transient interference components are separated in combination with instantaneous frequency analysis. Based on the dynamically generated wavenumber-frequency relationship, an adaptive window function (such as a Gaussian window, a Hanning window) or a band-pass filter is designed. For example, low-pass filtering is implemented for the S0 mode (low wavenumber region at the current frequency), and band-pass filtering is performed on the A0 mode (medium to high wavenumber region) to suppress cross-talk between modes. Finally, the amplitude information of each mode is extracted using Hilbert envelope detection or short-time energy integration methods. Combining the group velocity and excitation frequency in the known dispersion curve, the wavenumber calculation results are verified and corrected; By linearly superposing the modal components, the time-domain signal expression of the multi-modal Lamb wave is obtained, and a signal model is constructed. The signal component of each mode is a function of the defect location. The specific formula of the signal expression is as follows: ; where S represents the signal model, i represents the i-th receiving array, j represents the j-th Lamb wave mode, N represents the total number of receiving arrays, M represents the total number of Lamb wave modes, represents the out-of-plane amplitude of the j-th Lamb wave mode at the receiving element i, represents the wavenumber of the j-th Lamb wave mode, represents the spatial coordinates of the receiving element i, defining the array layout and the signal acquisition position, represents the spatial coordinates of the defect point, represents the phase of the j-th Lamb wave mode, represents Gaussian white noise with a variance of .

[0020] Step 2: Based on the partial derivative information of each channel and in combination with the observation noise variance, construct the corresponding Fisher information matrix.

[0021] Based on the existing measurement model, the partial derivatives of the observed signals of each array element channel with respect to the defect spatial position parameters are obtained respectively, and the chain rule is used for expansion to obtain the sensitivity expression of the signal intensity to the position change. The existing measurement model refers to the measurement model established based on the data acquisition principle of the total focusing method (TFM) and combined with the propagation characteristics of multi-modal Lamb waves. According to the chain rule, the partial derivative of the signal intensity with respect to the position parameter is expanded into the joint contribution of the amplitude term and the phase term. The specific formula is as follows: ; ; Among them, represents the signal intensity of a certain Lamb wave mode (such as S0, A0) at time t, where t represents time, A represents the amplitude of the corresponding Lamb wave mode, which is obtained by extracting the amplitudes of different modes through wavenumber analysis and mode separation means, f represents the excitation frequency, that is, the frequency of the transmitted signal, represents the phase of the corresponding Lamb wave mode, which is determined according to the position of the sensor in the array and the aperture ratio and can be obtained through geometric calculation of the wavefront propagation path. x represents the abscissa and y represents the ordinate.

[0022] Subsequently, based on the partial derivative information of each channel and combined with the observation noise variance, the corresponding Fisher information matrix is constructed. Specifically, each element of the Fisher information matrix is composed of the outer product superposition of the partial derivatives of each channel. Among them, the diagonal elements measure the sensitivity to a single position parameter, and the off-diagonal elements reflect the correlation between position parameters. The noise variance is usually set as a constant according to the Gaussian white noise assumption. If there are M modes of Lamb waves, the size of the Fisher information matrix (FIM) is (2M + 3) × (2M + 3). The Fisher information matrix is as follows: ; Among them, represents the signal model, and j represents the jth mode, represents the auxiliary variable of the jth mode (representing the partial differentials with respect to the amplitude and phase respectively), represents the spatial coordinates of the defect point source, represents the observation noise variance.

[0023] Furthermore, the specific calculation formula of the signal model with respect to the auxiliary variable of the jth mode is as follows: ; Among them, represents the signal model, and j represents the jth mode, represents the auxiliary variable of the jth mode (representing the partial differentials with respect to the amplitude and phase respectively), M represents the total number of Lamb wave modes, Denote the characteristic factor, and use the formula It is shown that the theoretical lower limit of the defect position estimation error under given imaging conditions is derived through the Fisher information matrix. The Fisher information matrix provides a quantitative reference standard for the optimization of different modal combinations, array configurations, and excitation frequency selections. By systematically analyzing the FIM, the differences in the contributions of different Lamb wave modes to the spatial resolution are revealed.

[0024] Step 3: Based on the derivation result of the Fisher information matrix, use the MATLAB platform to perform numerical calculation of the CRLB.

[0025] Based on the derivation result of the foregoing Fisher information matrix, use the MATLAB platform to perform numerical calculation of the CRLB. Set the physical parameters according to the actual detection scenario, including the geometric parameters of the plate (thickness, material density, elastic modulus), the excitation frequency range and step size, the wave velocities, wave numbers, and amplitude attenuation laws of each mode, construct a complete input data set, obtain the inverse matrix of the Fisher information matrix through matrix operations, extract the diagonal elements corresponding to the defect spatial position parameters, and obtain its theoretical minimum variance, that is, the CRLB value.

[0026] In the actual calculation process, in order to accurately reflect the physical characteristics of the imaging system, it is necessary to update the propagation velocity and wave number parameters in real time according to the dispersion relations of each Lamb wave mode at different excitation frequencies, so as to ensure that the CRLB result is highly consistent with the actual detection conditions. By changing the input parameters such as frequency, modal type, and array aperture, the minimum variance change curve under the corresponding imaging conditions can be systematically generated.

[0027] As described above, only the preferred specific embodiments of the present invention are provided, but the protection scope of the present invention is not limited thereto. Any person skilled in the art within the technical scope disclosed by the present invention, according to the technical solution of the present invention and its inventive concept, makes equivalent substitutions or changes, and should be covered by the protection scope of the present invention.

[0028] The preferred embodiments of the present invention disclosed above are only used to help explain the present invention. The preferred embodiments do not describe all the details in detail, nor do they limit the present invention to only the specific embodiments. Obviously, many modifications and changes can be made according to the content of this specification. This specification selects and specifically describes these embodiments to better explain the principle and practical application of the present invention, so that those skilled in the art can well understand and utilize the present invention. The present invention is only limited by the claims and their full scope and equivalents.

Claims

1. A method for quantifying the spatial resolution of multimodal ultrasonic Lamb wave imaging, characterized in that, Including: Step 1: Based on the full-focusing method data acquisition method, an array transducer is used for point-by-point excitation and reception to collect complete full matrix data, and a multi-modal Lamb wave propagation model is established; Step 2: Based on the partial derivative information of each channel and combined with the observation noise variance, a corresponding Fisher information matrix is constructed; Step 3: Based on the derivation result of the Fisher information matrix, the numerical calculation of the CRLB is performed using the MATLAB platform.

2. The multimodal ultrasonic Lamb wave imaging spatial resolution quantization method according to claim 1, wherein An array transducer is used for point-by-point excitation and reception to collect complete full matrix data. The specific method is as follows: Record the time-domain signals of all transmit-receive channels, perform a three-dimensional fast Fourier transform on the full matrix data to convert the time-domain signals to the frequency-wavenumber domain, obtain the energy distribution characteristics of the signals in the three-dimensional space, identify the wavenumber range and energy concentration region of different Lamb wave modes through frequency-wavenumber spectrum analysis, use the Hilbert-Huang transform, extract the intrinsic mode functions of the signals through empirical mode decomposition, combine with instantaneous frequency analysis to separate the time-frequency components of the aliased modes, design matching window functions or perform band-pass filtering operations according to the wavenumber distribution ranges of different modes, use Hilbert envelope detection or short-time energy integration methods to extract the amplitude information of each mode, determine the Lamb wave dispersion curve, and determine the wavenumber according to the excitation frequency and group velocity in the Lamb wave dispersion curve.

3. A method for quantifying the spatial resolution of multimodal ultrasonic Lamb wave imaging according to claim 1, characterized in that, Establish a multi-modal Lamb wave propagation model. The specific method is as follows: The time-domain signal expression of the multi-modal Lamb wave is obtained by linearly superposing each modal component, and a signal model is constructed. The signal component of each mode is a function of the defect position, and the formula is used to represent the signal expression. Here, S represents the signal model, i represents the i-th receiving array, j represents the j-th Lamb wave mode, N represents the total number of receiving arrays, and M represents the total number of Lamb wave modes. represents the out-of-plane amplitude of the j-th Lamb wave mode at the receiving element i. represents the wave number of the j-th Lamb wave mode. represents the spatial coordinate of the receiving element i. represents the spatial coordinate of the defect point. represents the phase of the j-th Lamb wave mode. represents white Gaussian noise with a variance of .

4. A method for quantifying the spatial resolution of multimodal ultrasonic Lamb wave imaging according to claim 1, characterized in that, Based on the partial derivative information of each channel and combined with the observation noise variance, construct a corresponding Fisher information matrix. The specific method is as follows: Calculate the partial derivatives of the observed signals of each element channel with respect to the defect spatial position parameters respectively. According to the chain rule, expand the partial derivatives of the signal intensity with respect to the position parameters into the joint contributions of the amplitude term and the phase term. Based on the partial derivative information of each channel and combined with the observation noise variance, construct a corresponding Fisher information matrix. Specifically, each element of the Fisher information matrix is composed of the outer product superposition of the partial derivatives of each channel. Among them, the diagonal elements measure the sensitivity to a single position parameter, and the non-diagonal elements reflect the correlation between the position parameters. The theoretical lower limit of the defect position estimation error under the given imaging conditions is derived through the Fisher information matrix. The Fisher information matrix provides a quantitative reference standard for the optimization of different mode combinations, array configurations, and excitation frequency selections. By systematically analyzing the FIM, the differences in the contributions of different Lamb wave modes to the spatial resolution are revealed.

5. A method for quantifying the spatial resolution of multimodal ultrasonic Lamb wave imaging according to claim 4, characterized in that, Expand the partial derivatives of the signal intensity with respect to the position parameters into the joint contributions of the amplitude term and the phase term. The specific method is as follows: Using the formula and the formula , the partial derivative of the signal intensity with respect to the position parameter is expanded into the joint contribution of the amplitude term and the phase term, where represents the signal intensity of a certain Lamb wave mode at time t, t represents time, A represents the amplitude of the corresponding Lamb wave mode, f represents the excitation frequency, that is, the frequency of the transmitted signal represents the phase of the corresponding Lamb wave mode, x represents the abscissa, and y represents the ordinate.

6. A method for quantifying the spatial resolution of multimodal ultrasonic Lamb wave imaging according to claim 4, characterized in that, Combined with the observation noise variance, construct a corresponding Fisher information matrix. The specific method is as follows: There are Lamb waves with M modes, and the size of the Fisher information matrix (FIM) is (2M + 3) × (2M + 3). Using to represent the Fisher information matrix, where represents the signal model, j represents the j-th mode, represents the auxiliary variable of the j-th mode, and represents the partial differentials with respect to the amplitude and phase respectively, represents the spatial coordinates of the defect point source, represents the observation noise variance, represents the partial derivative of the signal model with respect to the auxiliary variable of the j-th mode.

7. A method for quantifying the spatial resolution of multimodal ultrasonic Lamb wave imaging according to claim 4, characterized in that The partial derivative of the signal model with respect to the auxiliary variable of the j-th mode. The specific method is as follows: Using the formula represents the partial derivative of the signal model with respect to the auxiliary variable of the j-th mode, where represents the signal model, j represents the j-th mode, represents the auxiliary variable of the j-th mode, M represents the total number of Lamb wave modes, represents the characteristic factor, and is represented by the formula as shown.

8. A method for quantifying the spatial resolution of multimodal ultrasonic Lamb wave imaging according to claim 1, characterized in that, Based on the derivation result of the Fisher information matrix, perform the numerical calculation of the CRLB using the MATLAB platform. The specific method is as follows: Use the MATLAB platform to perform numerical calculations of the CRLB. Set the physical parameters according to the actual detection scenario, including the geometric parameters of the plate, the excitation frequency range and step size, the wave velocities of each mode, the wave numbers, and the amplitude attenuation law. Construct a complete input data set, obtain the inverse matrix of the Fisher information matrix through matrix operations, extract the diagonal elements corresponding to the defect spatial position parameters, and obtain its theoretical minimum variance, that is, the CRLB value.

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