## 2 Methods
### Data Processing  
#### 2.4.1 Removal of Digital Noise from Analog Signals Using Wavelet Packet Transform  
Due to limited sampling accuracy, finite element simulated LSPIT data will generate quantization noise in the digitization process. The simulated time domain signal needs to be denoised. [PERSON] and [PERSON] proposed the WPT based on the wavelet transform [42]. Thus, wavelet packet transform is localized in both the time and frequency domains. For non-smooth signal denoising, wavelet packet denoising is better than wavelet transform [43,44]. The reason is that WPT gives a balanced binomial tree structure that can decompose both the low-frequency and high-frequency parts of the upper layer, making the denoised signal closer to the original signal.  
Mathematically, the wavelet packet transform as a function of the time-frequency is defined as [45]:  
\[W_{p,k}^{i}(t)=2^{\frac{p}{2}}W^{i}(2^{p}t-k) \tag{6}\]  
where \(p\) is the scale, \(k\) is the translation parameter, and \(i\) is the number of decomposition layers. \(P=k=0\) wavelet packet function is the scaling function and the mother wavelet function. The expressions are as follows.  
\[W_{0,0}^{0}(t)=W^{0}(t)=\varphi(t) \tag{7}\]  
\[W_{0,0}^{1}(t)=W^{1}(t)=\psi(t) \tag{8}\]  
The following equations represent other wavelet packet functions.  
\[W^{2i}(t)=\sqrt{2}\underset{k}{\sum}h(k)W^{i}(2t-k) \tag{9}\]  
\[W^{2i+1}(t)=\sqrt{2}\underset{k}{\sum}g(k)W^{i}(2t-k) \tag{10}\]  
Figure 7: Comparison of numerical simulation and experimental data.  
where \(h(k)\) is a low-pass filter associated with a predefined scaling function and a mother wavelet function, and \(g(k)\) is a high-pass filter. Using the inner product of the signal can calculate the wavelet packet coefficients, and each wavelet packet function has the following form.  
\[Y_{p,k}^{i}=\left\langle x,W_{p,k}^{i}\right\rangle=\int_{-\infty}^{+\infty}x(t )W_{p,k}^{i}(t)dt \tag{11}\]  
Selecting the corresponding wavelet basis according to the actual situation is necessary. This paper selects sym5, which is similar to the transient vibration waveform of the simulated data, as the wavelet basis function of this paper. The smaller the root mean square error, the better the denoising effect; the more significant the signal-to-noise ratio, the better the noise suppression effect. Therefore, calculating the root means square error (RMSE) and signal-to-noise ratio (SNR) and observing the denoised waveform can determine the optimal decomposition level [46,47]. Taking bulge imperfection-weak concrete pile as an example, Table 4 shows the SNR and RMSE of its wavelet packet denoising; Figure 8 shows the denoising effect at different levels.  
For Layer 1 and Layer 2 decomposition with smaller RMSE and larger SNR, the data has good matching with the defined standard. However, according to the denoising waveform of Figure 8, there is still slight noise after Layer 1 and 2 of denoising. After Layer 4 and 5 of denoising, the waveform has been deformed compared with the original waveform. Therefore, the optimal decomposition level should choose the Layer 3 decomposition with a smooth denoising waveform, high SNR, and small RMSE.