Low strain pile testing based on synchrosqueezing wavelet transformation analysis

2.1. Low strain testing

Generally, the length of a pile is greater than its width. In low strain conditions, a pile can be simplified to a one-dimensional elastic body. When the impact load is on top of the pile, the pile body produces stress waves (Fig. 1) and its propagation characteristics conform to a one-dimensional wave equation [32]:

where $u$ is displacement, $t$ is time, and $c$ is the elastic wave velocity.

Fig. 1Low strain testing sketch: A – sensor, T – pile body, S – soil, H – force-hammer; E – signal analysis system

During low strain testing of pile after power hammer excitation, a stress wave is generated and spreads downward in the pile. When the stress wave encounters the wave impedance surface, transmitted and reflected waves appear. The transmission and reflection of energy depend on the wave impedance of two different media, a condition that complies with the law of energy conservation. Wave’s propagation characteristics can be expressed as [33]:

where $v$, $\sigma$ are vibration velocity and stress at the interface of the particle respectively; $i$, $r$, $t$ are incident waves, reflected waves and refracted waves respectively.

In accordance with Eq. (2), vibration velocity and section stress of the reflected wave can be preliminarily expressed as:

From Eq. (3), the velocity and stress at the particle interface are associated with the reflection coefficient $F$ that is determined by the wave impedance $z$ that relies on the pile concrete cross-sectional area, velocity, and density. When the interface impedance indicates a great difference in the pile, the reflected signal detected is significant. Thus, the distance $H$ between the top and the bottom of the pile can be obtained by:

where $c$ is the elastic wave velocity, $\mathrm{\Delta }t$ is the time difference between incident and reflected wave on the pile top.

The wave impedance of internal pile defects or interfaces is different from that of uniform concrete in the pile body. As a result, speed response amplitude and phase changes occur when elastic wave propagation takes place in the pile body. Analysis of the amplitude and speed phase can determine the position of defects and the integrity of the pile body, among other features, for evaluating the pile quality.

2.2. SST theory

The SST algorithm is based on wavelet analysis theory. In this study, $f\left(t\right)$ is processed by using the following wavelet transform [34]:

where $a$ is the scale factor, $b$ is the time shift, and ${\phi }^{\mathrm{*}}\left(t\right)$ is the complex conjugate of the mother wavelet.

In the synchrosqueezing phase, where $\left(a,b\right)\to \left({\omega }_x\left(a,b\right),b\right)$, the energy is converted from a time-scale plane to a time-frequency plane. The instantaneous frequency of $f\left(t\right)$ can be preliminarily estimated as [35]:

where $\omega \left(a,b\right)$ is instantaneous frequency.

Then, the synchrosqueezing value of wavelet coefficients is obtained by determining the wavelet coefficients in the central frequency domain:

where $T_f\left(\omega ,b\right)$ is the wavelet synchrosqueezing coefficient.

The time-domain signal can be reconstructed by using SST back-transformation, which is expressed as:

where $C_{\phi }=1/2\left({\int }_0^{+\mathrm{\infty }}\overline{\widehat{\phi }\left(\zeta \right)}d\varsigma /\varsigma \right).$

When the signal is discrete, the wavelet synchrosqueezing coefficients are expressed as:

where ${\omega }_l$ is the central frequency in [${\omega }_l-∆\omega /2$, ${\omega }_l+∆\omega /2$].

In the same way, the time-domain discrete signal can be reconstructed by using SST back-transformation, which is expressed as:

where $C_{\psi }=1/2\left({\int }_0^{+\mathrm{\infty }}\overline{\widehat{\phi }\left(\zeta \right)}d\varsigma /\varsigma \right)$, $\widehat{\phi }\left(\varsigma \right)=(1/\sqrt{2}){\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\phi e^{-i\varsigma t}dt$.

3.1. Signal recognition of the pile bottom reflection

The low strain detection signal in Fig. 3 is transformed with the SST. The instantaneous frequency spectrum is shown in Fig. 4. The results of the analysis show that the signal frequency range is 0 to 6000 MHz. The signal energy in Fig. 4 is concentrated at the low frequency 500 MHz, intermediate frequency 2000 MHz, and high frequency 4000 MHz. For the three characteristic frequencies, the energy is most concentrated near the intermediate frequency 2000 MHz.

Fig. 4Instantaneous frequency spectrum of low strain testing signal with SST

Three frequency bands were taken at 400-600 Hz, 1800-2000 Hz, and 4000-4200 Hz for SST inverse transformation. The signals within the three frequency bands can be reconstructed as shown in Fig. 5.

Fig. 5 shows the reconstructed signal diagram. In Fig. 5(b), the low frequency band signal shows that no reaction interface is evident in the pile and the reflected signal is mainly from the background trend. In Fig. 5(c), the signal energy is concentrated and its peak overlaps with the original signal peak. With higher resolution, the part presents reactions at each interface in the pile. In Fig. 5(d), the high-frequency part shows that different interfaces in the pile have different reactions.

Considering the high-frequency portion as a major research target, we compare it with the continuous wavelet transformation, as shown in Fig. 6. Fig. 6(a) shows several reflection peaks as different reflective interfaces that form multiple peaks about the original signal. If the pile length is known in advance, then the pile bottom reflection peak becomes difficult to recognize. The reflection time of the pile bottom must be determined. SST can be adopted for signal processing and for obtaining time-frequency characteristics. Fig. 6(b) shows the time-frequency contour of the signal after the SST transformation. In the figure, two red ellipse regions show two noticeable features at 4000 Hz; these features correspond to the reflection times of the top and bottom of the pile. Fig. 6(c) shows the scale and wavelet coefficients of a continuous wavelet transform. Fig. 6(c) shows multiple peaks of reactions, but no apparent characteristics are indicated with regard to the reflected signal of the pile bottom.

Fig. 5Reconstructed signal in three frequency bands with SST: a) original signal, b) 400-600 Hz, c) 1800-2000 Hz, d) 4000-4200 Hz

Fig. 6a) Low strain testing signal, b) SST transform, c) wavelet transform

The time-domain signal is reconstructed by using SST back-transformation. In Fig. 7, the reflected signal is reconstructed from 4000 Hz to 4200 Hz, as in Fig. 5(d). The reflected signal of the pile bottom is strengthened, whereas the other reflected signal is suppressed. Reflected wave arrival time from the pile bottom is 3.35 ms, as green horizontal line marked in Fig. 7. The velocity is 1593 m/s and the initial wave time is 0.82 ms. In accordance with Eq. (4), we calculate the pile length as 2 m. The result is consistent with the actual pile length.

Fig. 7Reconstructed low strain testing signal

3.2. Signal and noise separation

With a low strain testing a signal against a strong noise background, the reflection time of wave impedance interface formation becomes difficult to identify. Fig. 8(a) shows the testing signal amplitude in the time domain against a strong noise background. The SST is used for signal and noise separation. In general, the containing noise signal is de-noising data minus the original noiseless data, and the ratio of signal to noise energy is regarded as the signal to noise ratio (SNR). We can use this parameter to measure the de-noising effect. The formula is [36]:

where $S_{NR}$ is SNR, ${‖\cdot ‖}_2^2$ is square norm, $M$ is original noiseless data, $S$ is containing noise data. In Fig. 7(a), $S_{NR}=\mathrm{}$1.249.

Fig. 8a) Noise interference, b) SST de-noising

The signal in Fig. 8(a) was processed using low-pass filtering SST. Fig. 6(b) shows the original signal s₀ and the processed signal $s_1$; these signals have corresponding reflection peaks. The SNR increased from 1.249 (0.964 dB) to 10.796 (10.333 dB), and the SST de-noising effect was considered significant.

To further study the effects of SST de-noising under different noise levels, we added different levels of random noises to the Fig. 3 data and used the SST method for de-noising, comparing the results with those from EMD and Wavelet threshold. In the process, the EMD method abandons high-frequency components and IMF1 (intrinsic mode function 1) and IMF2 (intrinsic mode function 2), and an adaptive threshold is adopted in Matlab wavelet de-noising. De-noised data can be obtained by all three methods, and SNR is calculated by Eq. (11) in the five noise level. Table 1 shows the effects of the three de-noising methods. From the SNR results in Table 1, the de-noising performance of the SST method is significantly better than that of the other two methods, and EMD and Wavelet threshold have similar effects.

According to Table 1, the increase in the times of SNR can be obtained with the three de-noising methods, as shown in Fig. 9. In Fig. 9, the minimum increase in the times of SNR with SST is 6 and the highest value is more than 10. The result shows that the SST has good ability to remove noise. Even in a strong noise background, the SST can significantly improve SNR, and its performance is better than that of the other two methods.

Fig. 9Increase in times with low strain testing signal SNR under different noise levels

3.3. Effect analysis of wavelet basis function

The SST is based on conventional wavelet transformation, during which the wavelet function selection is involved. To study the effect of the wavelet basic function on the results, we selected three types of wavelet (Morlet, Mexican, Bump). Fig. 10 shows the findings of the basic function of the three types of wavelet. In Fig. 10(a)-(c), the squeezing wavelet coefficients in the time-frequency domains are in good agreement with each other. In the SST process for pile with a low strain detection signal, the wavelet basis function has less affects result.

In our study of the de-noising effect of SST with different basis functions, the three wavelet functions are still the research object. In Fig. 11(a), a low strain test signal is under a strong noise background ($S_{NR}=\text{0.822}$). We tested the function of the three wavelets for low strain test signal de-noising. The result is shown in Fig. 11(b).

Fig. 10SST with different wavelet basis functions: a) Morlet wavelet, b) Mexican wavelet, c) Bump wavelet

Fig. 11a) Low strain test signal containing strong noise, b) De-noising results with three types of wavelet basis function

Fig. 11(b) shows that the results from the three different wavelet basis functions are highly consistent with the original signal $s_0$. The selection of wavelet basis function has little influence on the de-noising effect. From Fig. 11(b), the SNR of the three wavelet basis functions can be obtained according to Eq. (11), as shown in Fig. 12.

As evident in Fig. 12, the change in magnitude of the SNR is at the same level with the three de-noising methods. The original signal SNR increases dramatically. These findings further confirm that the SST has strong de-noising performance and the wavelet basis function selection is not greatly influenced.

Fig. 12SNR with SSTs of different wavelet basis functions