A Proposed Algorithm Based on Variance to Effectively Estimate Crack Source Localization in Solids

1. Introduction

A semi-permanent underground high level waste disposal repository can be highly stressed by the high temperature of the used nuclear fuel, underground water, and deep geological conditions [1]. Under such conditions, it is very important to maintain real-time monitoring of the repository’s long-term integrity and evaluate the degree of structural damage to ensure the safety of the disposal system.

To evaluate degree of damage, the most important task is the accurate estimation of crack locations. Time-of-arrival-differences (TOAD) of the crack waves must be accurately measured to precisely calculate crack locations.

Acoustic emission sensors are used to measure crack waves. Recently, the acoustic emission (AE) technique has been widely used for the real-time structural health monitoring of a structure [2,3]. AE has been widely used to evaluate the damage mechanisms of various structures because it is closely related to crack initiation and growth in materials.[4].

A large number of discontinuities formed by blasting and excavation exist around a radioactive waste repository. These discontinuities are the main causes of scattering and dispersion of elastic waves, and interference from reflected waves. In such conditions, the crack wave becomes undetectable because of the noise signal. For this reason, to calculate the exact location of cracks, a method is needed to isolate the crack wave from noise signals.

Traditionally, the estimation of source localization is performed through a triangulation method and a circle intersection technique based on TOAD [5,6]. Recently, a method of calculating TOAD using time-frequency analysis has been studied. Representative time-frequency analysis include Short-time Fourier transform (STFT), wavelet transform and Wigner-Ville distribution. [7,8,9,10,11]. In time-frequency domain, the dispersive characteristics of crack wave are well expressed in homogeneous medium such as metals, but there is a disadvantage that dispersive characteristics do not come out well in a non-homogeneous medium such as rock. Thus, many researches are underway to estimate TOAD and crack location in a rock [12,13].

This paper suggests an algorithm for determining TOADs using moving window, and to calculate a crack location from a crack signal with noise in rocks.

2. Basic Idea

Figure 1 shows the crack wave generated from cracks in rocks. The crack wave starts slightly before 1.3 msec, however a noise signal generated from various causes makes it difficult to accurately determine the arrival time of the crack wave. Given the uncertainty in the arrival time of the crack wave, the estimate of the crack localization contains a significant error. For example, an error of 0.1 msec in the arrival time of a crack wave with a velocity of 5,000 m/s may result in errors of 0.5 meters in the crack localization. To better assess the integrity of structures such as tunnels or other underground facilities, a method is needed to reduce the uncertainty in crack wave arrival time detection, and crack localization.

To more accurately detect the crack wave arrival time from monitoring signals, we first investigated the characteristics of the crack wave and noise signals, employing frequency analysis. Figure 2 shows result of frequency analysis (power spectrum) obtained from crack wave signal. The crack signal obtained by AE sensor is non-stationary signal that has narrow frequency range, while the noise is stationary signal that has broad band as shown in Figure 2. A stationary signal has the characteristic that the ensemble mean and variance value does not change regardless of the signal length. Therefore, if the variance of the crack signal in noisy environment is calculated according to the length change of the signal, the variance of the noise does not change, but only the variance of the crack signal changes as explained in Figure 3.

In this paper, we propose a method of estimating the crack wave arrival time using moving window. The TOAD is estimated by calculating the variance of the measured signal mixed with noise applied to moving window of different sizes as shown in Figure 4. The starting point of the crack signal is the point at which the variances of measured signal change as window size changes.

3. Background Theory for Finding TOAD

As mentioned above, the algorithm determines the variance from the window covering a certain range of signals. Then, an identical method is applied with windows of varying sizes. The theoretical formulas for the proposed method in this chapter are as follows.

The windowed signal $f_k(\tau , T)$ of a crack signal $f\left(t\right)$ can be expressed by the following equation:

$$\begin{gathered} f_k\left(\tau ,T\right)={\displaystyle \frac{1}{M\cdot T}}{\int }_{k\tau }^{k\tau +T}W(t-k\tau )\cdot f\left(t\right)dt \\ \mathrm{M}={\int }_0^{T}W\left(t\right)dt \end{gathered}$$

(1)

where,

$$k=\mathrm{0,1},2,\dots ,N-{\displaystyle \frac{T}{∆t}}$$

where, W(t) is the window function, T is the size of the moving window, N is the number of window data, and $∆t$ is the sampling time.

The variance is as follows:

$${\sigma }^2=E\left[x^2\right]-(E{\left[x\right])}^2$$

(2)

If the windowed signal is substituted for Eq. (2), the variance of windowed signal can be seen as follows:

$$VS\left(k,T\right)={\int }_0^T{f}_k^2\left(\tau ,T\right){\displaystyle \frac{dT}{T}}-{\left\{{\int }_0^T{f}_k\left(\tau ,T\right){\displaystyle \frac{dT}{T}}\right\}}^2$$

(3)

And Eq. (3) can be written as Eq. (4).

$$\begin{gathered} VS\left(k,T\right)={\displaystyle \frac{1}{M^2{T}^4}}\left[T{\int }_0^T{\left\{{\int }_{kT}^{k\tau +T}W\left(t-k\tau \right)f\left(t\right)dt\right\}}^{2}d\tau \right. \\ \left.-{\left\{{\int }_0^T{\int }_{kt}^{k\tau +T}W\left(t-k\tau \right)f\left(t\right)dtd\tau \right\}}^2\right] \end{gathered}$$

(4)

As shown in Figure 1, a crack signal generated in rock can be expressed as the sum of the background noise and the p-wave signal. If the background noise is assumed to be white noise and its mean is zero, then Eq. (4) can be defined as Eq. (5).

$${VS}_n\left(k,T\right)={\sigma }^2$$

(5)

As shown in Eq. (5), the noise signal has a constant value, regardless of the size (T) of the window.

To verify the behavior of the p-wave in Eq. (4), the p-wave is considered to be a sine-wave with the frequency $\mathsf{\omega }$ , as follows:

$$f\left(t\right)=sin\omega t$$

(6)

If Eq. (6) is substituted into Eq. (4), as follows:

$${VS}_p\left(k,T\right)={\displaystyle \frac{1}{4M^2{T}^4{\omega }^2}}(2{\omega }^2-\omega Tsin2\omega T-2cos2\omega T+8cos\omega T-6)$$

(7)

As can be seen from Eq. (7), now that the p-wave is a function of the size of the window, it can be confirmed that the variance varies according to the size of the window.

Therefore, as can be confirmed from Eq. (5) and Eq. (7), even if a p-wave exists in the noise, if the variance is calculated using the proposed method with the moving window, the noise signal does not change, but the variance of the p-wave changes.

The proposed method calculates the variance while changing the size of the window. Here, determining the window size is the most important factor for ascertaining the arrival time of the p-wave. In this chapter, we will describe how to determine the size of the window.

Figure 5 (b) shows the results of the variance with the size of the window when the p-wave is assumed to be a sine-wave. As can be seen from the graph, it can be confirmed that the change in variance is largest within one wavelength of the p-wave. The size of the window has to be determined within a p-wave wavelength, because the proposed method determines the arrival time of the p-wave by observing the change in variance according to the size of the window.

4. Validation Using an Experiment

To verify the proposed method for estimating the arrival time and location of crack signal, experiment was carried out using 6 AE sensor on the rock specimens. The AE signal was generated using a pencil lead break and the location of source points and sensors is shown in Table 3 and Figure 7.

As shown in Figure 8, the AE signal was measured using AE-300, which is a program developed by Rectuson & Fuji ceramics, and the AE-603 SW-GA sensor, which has a resonance frequency band in the range of 60 kHz ± 20%. The attachment of the sensor significantly affects the accuracy of the results. For this reason the AE sensors were fixed firmly to the surface of the rock using a high-degree vacuum adhesive. The amplitudes of the free amplifier and the main amplifier were amplified to 40 dB and 20 dB, respectively, because the crack initiation signal has a weak amplitude.

Figure 9 shows the AE signals from each AE sensor when the pencil lead break point was (x,y,z)=(125 mm, 0 mm, 50 mm). The AE signals from Ch.3~Ch.6 arrive earlier than the AE signals from Ch.1~Ch.2. This is because the pencil lead break point is farthest from sensor 1 and sensor 2, and between sensor 3 ~ sensor 6 (Figure 7). Similarly, Figure 10 and Figure 11 show the results from each AE sensor when the pencil lead break point was (x,y,z)=(0 mm, 0 mm, 50 mm), and (x,y,z)=(-125 mm, 0 mm, 50 mm).

In Figure 9, Figure 10 and Figure 11, the defined initiating point (arrival time) has a high probability of error depending on the analyzer. As shown in Figure 12, Figure 13 and Figure 14, it is arrival time when the signal variance initiates to vary according to moving window size which was defined as a result of calculation of raw AE signal, with equation (4) (the proposed moving window method). The estimated arrival time delay obtained using the proposed method is presented in Table 4.

Figure 15, Figure 16 and Figure 17 show the estimated source location results derived by the substitution of Table 1 and Table 2 in equation 11 (Table 1: sensor location; Table 2: arrival time delay calculated for each noise source.). Colors represent the variance in AE wave velocity in Figure 15, Figure 16 and Figure 17. The location of the source is where velocity variance is the least. This is because the velocity variance becomes 0 where the scan source accords with the true source. In Table 3, we present the estimated results of the source location obtained using the above method. As shown in Table 3, the maximum error is 1.5 mm, so the proposed algorithm for arrival time delay and location estimation is valid.

The earlier result from the experiment using a rock specimen had a high signal to noise ratio, and could not be used to verify detection capability, the biggest advantage of the proposed method. Instead, the experiment was carried out using artificial noise.

We supposed that the noise was white noise with a variance size of 340. The signal to noise ratios in sensors 1~6 were respectively 0.025, 0.026, 0.015, 0.016, 0.016, 0.012. These values are too low to define the initial point of the crack signal as directly as Figure 18. However, as shown in Figure 19, we defined the initial point of the crack signal using the moving window method, and estimation was possible based on that initial point, as accurately as Figure 20. This means the proposed method is useful even when the signal to noise ratio is low.

5. Verification Test at In-Situ Test

A field demonstration test was carried out in KURT (KAERI Underground Research Tunnel) in KAERI (Korea Atomic Energy Research Institute). (Figure 21)

As shown in Figure 22, the wall of the tunnel consists of rocks and is finished with shotcrete, and the ground is concrete. The entire interior of the real tunnel is concrete; this research experiment was carried out on the ground. There were 8 AE sensors placed in the tunnel, as shown in Figure 23. The distance between sensors was 0.5 m, and they were arranged in two lines. We excited several points (Figure 23(a)) using an impact hammer (Figure 23(b)) because we could not generate real cracks in the ground at several points. Exciting point 2, in particular, was used to investigate whether the sensors could calculate the crack location, even when the excitation occurred at the edge of the AE sensor array.

Figure 24 shows the results of the calculation with the moving window, and the signals from each sensor, when exciting point 1 was excited with an impact hammer. Figure 25 shows the estimated impact location using the calculated time delay from Figure 24. As shown in the result, the estimated location of excitation (-0.267 m, 0.248 m) was highly accurate, considering the true location of excitation (-0.25 m, 0.25 m). Figure 26 and Figure 27 show the estimated impact location results for each impact using the above method (presented in Table 5).

6. Conclusions

Crack signals from rock and concrete structures occur in conjunction with background noise from various causes. As a result, errors occur when estimating the crack location because the crack wave arrival time cannot be accurately determined. In this paper, by analyzing the crack signal and background noise characteristics, we were able to propose a concept of a ‘moving window’ to accurately determine the crack wave arrival time in a noisy environment. Experiments on a rock specimen and tunnel concrete showed that the proposed method is good at estimating crack locations. It is expected that the proposed method can be applied to the ‘Structural health monitoring’ of large structures, to find defects and weakness such as cracks.