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The present work concerns the estimation of the probability density function (p.d.f.) of measured data in the Lamb wave-based damage detection. Although there was a number of research work which focused on the consensus algorithm of combining all the results of individual sensors, the p.d.f. of measured data, which was the fundamental part of the probability-based method, was still given by experience in existing work. Based on the analysis about the noise-induced errors in measured data, it was learned that the type of distribution was related with the level of noise. In the case of weak noise, the p.d.f. of measured data could be considered as the normal distribution. The empirical methods could give satisfied estimating results. However, in the case of strong noise, the p.d.f. was complex and did not belong to any type of common distribution function. Nonparametric methods, therefore, were needed. As the most popular nonparametric method, kernel density estimation was introduced. In order to demonstrate the performance of the kernel density estimation methods, a numerical model was built to generate the signals of Lamb waves. Three levels of white Gaussian noise were intentionally added into the simulated signals. The estimation results showed that the nonparametric methods outperformed the empirical methods in terms of accuracy.

Structural health monitoring (SHM) is an emerging technology that merges with a variety of techniques related to diagnostics and prognostics. Monitoring the status of structural health can improve the safety and maintainability of critical structures in many fields, such as civil engineering, aerospace, and military industry. An ideal SHM system includes several subsystems in which the damage detection methodology is the key part. Therefore, numerous damage-detection methods have been researched in years [

The portion of the SHM process that has received the least attention in the technical literature is the development of statistical models for discrimination between features from the undamaged and damaged structures. The algorithms, which analyze statistical distributions of the measured or derived features to enhance the damage identification process, have been developed [

Elementary parametric estimation method has been adopted under the assumption that the p.d.f. of the measured data is normal distribution [

Since the type of p.d.f. of measured data from field experiments is varied and can hardly be predicted, more robust approach methods should be considered. The nonparametric statistic methods can give the parameters of distribution and do not rely on assumptions that the data are drawn from a given probability distribution. Therefore, introducing the nonparametric statistic methods is crucial in Lamb wave-based damage detection.

The aim of this paper is to demonstrate the necessity and feasibility of application of kernel density estimation, which is the most popular nonparametric estimation method in Lamb wave-based damage detection. Two kinds of kernel density estimation methods, the one based on the Gaussian approximation and the one based on the smoothing properties of linear diffusion processes, were briefly introduced in this paper. The signals of Lamb waves with different levels of white Gaussian noise were acquired by using numerical simulation. The framework of applying nonparametric estimation method in Lamb wave-based damage detection was demonstrated by using the simulated signals. The characteristics of noise-induced error in the arriving time of damage-scattered Lamb waves, which is the index used to locate damage, was analyzed. Based on this analysis, the outcomes of two kinds of kernel density estimation method as well as the parametric estimation methods were compared. The results show that the nonparametric methods outperform the parametric method in terms of accuracy and reliability.

Lamb waves are a kind of elastic waves propagates in thin plate and shell structure. With a high susceptibility to interference on a propagation path, for example, damage or a boundary, Lamb waves can travel over a long distance even in materials with a high attenuation ratio, and thus a broad area can be quickly examined [

Lamb waves are made up of a superposition of longitudinal and shear modes, and its propagation characteristics vary with entry angle, excitation, and structural geometry. A Lamb mode can be either symmetric or antisymmetric, formulated by

Lamb waves can be actively excited by a variety of means, such as ultrasonic probe, laser, interdigital transducer, and piezoelectric element. The piezoelectric element can also be used as sensor to collect signals of Lamb waves perfectly. The piezoelectric element is particularly suitable for integration into a host structure as an in situ generator/sensor, for their neglectable mass/volume, easy integration, excellent mechanical strength, wide-frequency responses, low power consumption and acoustic impedance, as well as low cost. Applications of piezoelectric element in Lamb wave-based damage detection are numerous.

Lamb mode selection is an important part for damage detection. The basic symmetric mode,

The algorithms for Lamb wave-based damage identification can be roughly divided into two categories. The first category is the algorithms that identify and locate damage by observing the damage-reflected Lamb waves, such as Time-of-Flight (ToF) method [

For the algorithms that focus on the damage-reflected waves, the arriving time of the Lamb waves is the key index used to locate damage. Since the signal of Lamb waves is wave packet in the form, several methods have been developed to measure the arriving time of Lamb waves, such as threshold method, correlation method, wavelet method [

ToF, defined as the time lag from the moment when a sensor catches the damage reflected signal to the moment when the same sensor catches the incident signal, was widely used to locate damage [

Consider a sensor network consisting of _{0} mode and the incipient

Because there are two unknown damage parameters, (

Damage localization using ToF method in a plate. (a) Locus based on accurate ToF value, (b) Locus based on ToF with error.

There is a prerequisite in the traditional approach. That is all of the measured ToF values

The concept of probability-based approach was introduced by Zhao et al. [

The main frame of data fusion-based method can be divided into two steps.

The inspection area of the structure was evenly meshed. For a certain measured ToF, each mesh node will be evaluated about its possibility for the presence of damage by using a probability density function.

All evaluated results for each measured ToF were combined to give the detection result in a matrix form. Each element of the matrix represents the probability of the presence of damage for one mesh node.

The detection result in matrix form can be illustrated in an image shown in Figure

Damage localization result of probability-based method.

It is obvious that the p.d.f. of damage occurrence is the key part of probability based method. Su et al. [

Satisfied results have been obtained by using this kind of p.d.f. But it should be noticed that the standard variance

The concept of probability-based approach was also adopted in some other Lamb wave-based damage detection methods rather than ToF method. Wang et al. [

There are mainly two disadvantages in the existing work. First, empirical formula usually are simpler to write down and faster to compute, but it depends heavily on the experimental environment. Any change which is inevitable in experiment may cause a big error in the estimated results. That is, the simplicity of empirical formula makes up for its nonrobustness. Since the data measurement work in the Lamb wave-based damage detection is not time consuming, it is reasonable that the density function should be estimated by using robust statistic method. Second, the p.d.f. used in existing work is the distribution function about the location of damage in the plane

Therefore, probability density estimation methods will be introduced in Section

In statistic, density estimation is the method that estimates the parameters of a distribution based on the observed samples. Depending on whether a priori knowledge about the type of the distribution is required, the density estimation methods can be divided into two categories: parametric estimation and nonparametric estimation.

Parametric estimation mainly includes point estimation and interval estimation. In statistics, point estimation is the use of sample data to calculate a single number of possible values of an unknown population parameter, in contrast to interval estimation, which is an interval. Most commonly used point estimation methods are method of moment estimation, maximum likelihood estimation, and Bayesian estimation. For instance, if it is known that the sample data come from a normal distribution, then the two parameters of normal distribution, expectation and variance, can be calculated by using (

Nonparametric estimation is a method that estimates the parameters of an unknown distribution while does not rely on assumptions about the type of this distribution. Commonly, nonparametric estimation methods include histogram, nonparametric regression, and kernel density estimation, which is the most popular one.

Kernel density estimation is a nonparametric method to estimate the probability density function of a random variable. Kernel density estimation is a fundamental data smoothing problem where inferences about the population are made, based on a finite-data sample. In some fields such as signal processing and econometrics, kernel density estimation was also termed as the Parzen-Rosenblatt window method, after Emanuel Parzen and Murray Rosenblatt, who are usually credited with independently creating this method in its current form [

Let

The bandwidth of the kernel is a free parameter which exhibits a strong influence on the resulting estimate [

If the kernel function is normal and it is assumed that the distribution being estimated is Gaussian, then it can be derived from (

Kernel density estimation is an ongoing research topic in statistics. Botev et al. [

Given

Chaudhuri and Marron [

Because any bounded domain can be mapped onto

It has been proved that the estimator given in (

Therefore, motivated by the idea of acquiring the estimator from the solution of diffusion PDE, Botev proposed that the most general linear time-homogeneous diffusion PDE can be a starting point for the construction of a better kernel density estimator. The simple diffusion model described in (

The solution of (

The novel plug-in bandwidth selection method for the diffusion estimator defined in (

Assuming that

Denote the functional dependence of

It is noticed that the assumption in the

Given

Set

if

Deliver the Gaussian kernel density estimator in (

The above section explains how to estimate the bandwidth

Assuming that

The second one is

Given the data

Let

Estimate

Substitute the estimate of

Deliver the diffusion estimator in (

The flow chart of the entire bandwidth selection algorithm was shown in Figure

Flow chart of kernel density estimation via diffusion.

Feasibility of using the kernel density estimation method to estimate the p.d.f. of experiment results was demonstrated in a thin plate structure via finite-element (FE) simulation. Eight PZT wafers were surface installed at an aluminium plate. The aluminium plate was 600 mm × 600 mm × 1.5 mm in size, supported with all its four edges. The elastic modulus, poission’s ration, and density of the aluminium are 71e9GPa, 0.35, and 2711 Kg/m^{3}, respectively. The thin plate was three dimensionally modeled using eight-node brick solid elements. To ensure simulation precision, the largest dimension of FE elements was less than 1 mm and the plate was divided into multilayer in thickness, guaranteeing that at least ten elements were allocated per wavelength of the incident diagnostic wave, which has been demonstrated sufficiently to portray the characteristics of elastic waves in the thin plate [

Schematic of numerical simulation mode.

Gaussian noise is statistical noise that has its probability density function equal to that of the normal distribution, which is also known as the Gaussian distribution. A special case is white Gaussian noise, in which the values at any pairs of times are statistically independent (and uncorrelated). It is well known that noise comes from many natural sources is Gaussian noise. Therefore, in order to simulate the environment noise, three signal-to-noise ration (SNR) levels (20 dB, 30 dB, and 40 dB) of white Gaussian noise were intentionally added into the numerical simulated Lamb waves signals.

In numerical model, four sensor pairs are used to locate the damage. The sensor pairs are s2-6 formed by sensor 2 and sensor 6; s4-8 by sensor 4 and 8; s3-7 by sensor 3 and 7; s3-5 by sensor 3 and 5. The process of adding three levels white Gaussian noise in the signals captured by the four sensor pairs repeated 30 times. That is, there are 30 ToF results for each sensor pair under each level of noise.

It can be expected in theory that the nonparametric estimation methods should have a better performance than parametric estimation method when deal with the distribution without a priori knowledge about its type. The advantage of kernel density estimation method will be demonstrated in this paper by estimating

ToF is given by comparing the arriving time of incident waves and damage-scattered waves. Since the incident waves is strong, the errors in arriving time of incident waves can be neglected. Without loss of generality, the errors in ToF was considered to be caused entirely by the errors in the arriving time of damage-scattered waves.

As mentioned in Section

The signals of Lamb waves received by sensor 8.

It is obvious that

Parametric estimation method, the kernel density estimation based on the Gaussian approximation, and the adaptive kernel density estimation via diffusion were used to estimate

The estimation results for the signal with 40 dB SNR noise was shown in Figure

p.d.f. estimated results for samples from s4-8 with 40 dB noise.

p.d.f. estimated results for refined samples from s4-8 with 40 dB noise.

p.d.f. estimated results for samples from s4-8 with 30 dB noise.

p.d.f. estimated results for samples from s4-8 with 20 dB noise.

The fact that only 4 samples were affected by both

Lilliefors test was adopted to check whether the refined samples came from a normal distribution. In statistics, the Lilliefors test, named after Hubert Lilliefors, was an adaptation of the Kolmogorov-Smirnov test [

The calculated value from the Lilliefors test was 0.1373, which was less than the critical value 0.1699 corresponding to 5% significance level. The null hypothesis that the refined data came from a normally distributed population was accepted. It explained why the empirical formula given in the previous work was a normal distribution type and why the damage detection results based on the empirical formula was satisfied. Since the noise in previous work [

The estimation results for the signals with 30 dB SNR noise were shown in Figure

The estimation results for the signals with 20 dB SNR noise were shown in Figure

The damage localization under 20 dB noise environment was selected as the example to show that an accurate estimation was important for the localization result. The p.d.f. estimation results given by three kinds of density estimation methods introduced in Section

Damage localization result based on parametric estimate method (partial view).

Damage localization result based on kernel density estimation with Gaussian approximation (partial view).

Damage localization result based on kernel density estimation via diffusion (partial view).

The characteristics of noise-induced error in ToF data measured by using threshold method were analyzed.

The empirical formula method and the parametric estimation method presented in existing work had the same assumption that the experimental data came from a normal distribution. This assumption had been verified by real experiments and numerical simulation. The results in this paper revealed that the type of distribution of ToF data was related to the noise level. The empirical formula method and the parametric estimation method were developed in laboratory environment where the noise was weak. It had also been proved in this paper that the ToF data measured from high SNR signal (SNR > 40 dB) were distributed normally. Therefore, the density estimation method with the normality assumption presented in existing work can work well in laboratory environment.

However, the signals of field experiment usually contained much more strong noise. The results in this paper showed that even for the signal with 40 dB SNR, the distribution of measured ToF data were not normal distribution. In this case, nonparametric estimation method must be emplyed to estimate the p.d.f. correctly. Further, investigating about the signals with 30 dB and 20 dB noise showed that, with the increasing noise, only the kernel density estimation via diffusion, which is purely data driven, can give a satisfied estimating result.

The damage localization under 20 dB noise environment had been carried out. Parametric estimation method with the normality assumption, the kernel density estimation based on the Gaussian approximation and the kernel density estimation via diffusion were adopted to estimate the p.d.f. of measured data. Three different p.d.f. were obtained by employing the above-motioned three kinds of density estimation methods. By using each p.d.f, a damage location result can be calculated. Through comparing the three results of damage location, it can be seen that an accurate estimation of p.d.f. has a direct effect on the accuracy of the results. Applying kernel density estimation in Lamb wave-based damage detection was necessary.

The noise studied in this paper was the white Gaussian noise. The noise in the real field experiment was much more complex. Further study was needed to reveal the characteristic of errors in ToF data caused by noise in field experiment. However, the complex nature of noise in field experiment could not be a trouble for the application of kernel density estimation method, instead, it could be a reason to apply this method. It had been proved that when deal with simple noise, the kernel density estimation method introduced in this paper performed better, in comparison with empirical methods. Since the kernel density estimation method did not rely on any assumption about the distribution to be estimated, it could be expected that the kernel density estimation method could demonstrate a greater advantage in a complex noise environment.

This work was financially supported by National Natural Science Foundation of China under grant no. 50905141, the Program for New Century Excellent Talents in University of China, and the NPU Foundation for Fundamental Research under grant no. NPU-FFR-JC20110258.