# Ultrasonic Pulse-Echo Signals for Quantitative Assessment of Reinforced Concrete Anomalies
## 3 Formulation for Accurate Assessment of Void Defects
Some generalized white noise signals exhibit a flat spectrum, and their statistical properties are determined solely by a scalar parameter H, known as the Hurst exponent. For a signal \(x(t)\) exhibiting a zero-mean Gaussian stationary process, its autocorrelation function is expressed as follows: \[r_{H}[k]=\frac{\sigma^{2}}{2}\Big{(}|k-1|^{2H}-2|k|^{2H}+|k+1|^{2H}\Big{)}\] (9) where \(\sigma\) is the process variance of signal \(x(t)\), H is the Hurst exponent, and k is the correlation lag. Notably, when H equals 0.5, the process is classified as uncorrelated white noise, whereas for other H values, it is labeled as colored Gaussian noise. Moreover, when a generalized white noise signal is subjected to EMD, it acts as a dyadic filter bank. 8. It is important to note that the log-variance of the IMFs follows a simple linear model, which the Hurst exponent of the process ultimately governs. \[\log_{2}V_{H}[k]=\log_{2}V_{H}[2]+2(H-1)(k-2)\log_{2}\rho_{H}\] (10) (9) For k \(\geq\) 2 and \(\rho_{H}\approx\) 2, the energy of each of the IMFs E\({}_{k}\) can be parameterized as a function of the first IMF energy and Hurst exponent parameters [62]: \[E_{k}=\bigg{(}\frac{E_{1}}{\beta_{H}}\bigg{)}\rho_{H}{}^{-2(1-H)^{k}},k\geq 2\] (11) \[E_{1}=\frac{1}{N}{\sum_{n=1}^{N}}(\text{IMF}_{1})^{2}\] (12)
1. This particular model can execute denoising BMD-based techniques. The process entails breaking down the noisy signal into IMFs and gauging their energy levels about the estimated noise-only IMF energies derived from Equation (10). From there, the signal reconstruction is accomplished by adding up the IMFs whose energy levels deviate from the expected noise model. 1. Peak detection techniques are typically employed to estimate TOF, thereby differentiating between the reflection signal from the front surface and the reflection signal from the defect. Despite the denoising process, the defect echo signals may still exhibit dispersion and weakness, requiring specialized methods for identification and estimation. Techniques such as filtering, cross-correlation, envelop moment analysis, and matching pursuit decomposition with dispersion compensation are necessary for accurate defect detection in such scenarios. The envelope of an echo signal constitutes a vital characteristic that can be employed to extract information regarding the location of the echo waveform. The Hilbert transform facilitates the computation of instantaneous features of a time series, including the envelope amplitude and instantaneous frequency. The instantaneous envelope denotes the amplitude of the complex Hilbert transform, which can be utilized to identify the dynamic characteristics of non-linear systems by improving the accuracy of envelope detection through the use of local maxima interpolation. The Hilbert transform involves casting the actual signal a(t) into an analytical signal (i.e., complex envelope) \(\widetilde{\text{a}}\)(t). The \(\tilde{\text{a}}\)(t) is a complex-valued time-domain signal, where the original actual signal a(t) is the genuine part, and the Hilbert transformed signal \(\tilde{\text{a}}\)(t) is the imaginary part (Equation (13)). By leveraging this approach, one can obtain a complex-valued time-domain signal that provides a more comprehensive representation of the underlying physical phenomenon. \[\widetilde{\text{a}}\text{(t)}=\text{a(t)}+j\tilde{\text{a}}\text{(t)}\] (13)
* In Equation (14), \(\text{j}\) is the imaginary number, while H[\(\cdot\)] denotes the Hilbert transform operation. In the time domain, the Hilbert transform is defined  
Figure 5 shows the A-scans for the cases measured on the concrete deck without void defect (a) and with void defect (b) below the UPE device. In Figure 5b, the peak at about 80 microseconds is slightly smaller than the last peak at about 60 microseconds; it could be interpreted as the reflection echo peak from a decayed surface wave or the reflected echo from a defect void.