## 3 Low-Strain Integrity Testing
### The SE Method
The short-time Fourier transform (STFT) employs a sliding temporal window to decompose an extended signal into multiple short-duration quasi-stationary segments. Each segment subsequently undergoes FT, maintaining spectral resolution while enabling time-frequency localization through the window's temporal constraints. However, the STFT maintains constant time-frequency resolution across the entire analysis domain once its window function is fixed [48]. While this represents a significant improvement in dynamic analysis capability for non-stationary signals compared to the conventional FT, the fixed resolution inherently limits its adaptability. Building upon STFT concepts, [PERSON] and [PERSON] developed the wavelet transform (WT) to achieve superior time-frequency analysis [49]. The WT provides superior analytical capability for non-stationary signals. In contrast to the STFT's fixed window function, WT adaptively adjusts its time-frequency resolution through scalable wavelet bases--employing compact windows for high-frequency components and extended windows for low-frequency analysis [50]. This time-frequency adaptive analysis capability has been widely validated in the field of transportation infrastructure diagnostics [45; 51]. Furthermore, WT allows for various time and frequency resolutions to be obtained in different frequency bands, which is helpful in decomposing complex signals generated by multiscale processes [52]. When dealing with complex noise pollution, these noises often overlap with the target signal in frequency. WT can identify certain characteristics related to noise or vibration sources and provide important information about the time-varying characteristics of signals [51]. WT achieves precise noise reduction through threshold processing while retaining the local sudden change characteristics of signals [53]. [PERSON] and [PERSON] et al. [54] first applied the wavelet transform to low-strain PIT, using it to denoise data derived from finite element simulations. [PERSON] [55] established the theoretical framework for applying wavelet analysis to PIT and developed three characteristic indicators to quantify spectral variations in reflected signals for defect classification. The continuous wavelet transform (CWT) further enhances defect characterization through high-resolution time-frequency analysis, enabling more precise identification of both the location and nature of pile defects [56]. [PERSON] et al. [57] demonstrated the effectiveness of combining SE and CWT methods, where SE test signals reveal detailed time-frequency features through CWT analysis, enabling clear visualization of both pile tips and defects. In subsequent work, [PERSON] et al. [58] applied CWT to estimate lengths of long drilled shafts in field tests, achieving measurement errors below 10%. Further advancing the methodology, [PERSON] [41] developed an acoustic echo approach utilizing complex continuous wavelet transform (CCWT). The CCWT approach maintains CWT's high-resolution frequency analysis while offering additional advantages: (1) simplified defect identification through 3D phase spectrograms, and (2) reduced subjectivity in interpretation compared to conventional methods. Cross-domain denoising techniques can significantly enhance PIT outcomes. The Wigner-Ville distribution (WVD), a well-established time-frequency analysis method, has demonstrated effectiveness across multiple disciplines, including speech processing, radar signal analysis, biomedical engineering, and vibration diagnostics [45; 59; 60; 61]. Unlike conventional methods, WVD offers simultaneous time and frequency resolution while capturing complete signal energy distribution characteristics, making it particularly suitable for analyzing non-stationary signals [62]. Despite its widespread application in PIT, the SE method exhibits several limitations. As noted in ASTM D5882, while effective for detecting major defects, the method demonstrates reduced sensitivity to minor defects that may nevertheless significantly compromise structural integrity [13].