## 1 Introduction  
An ultrasonic coda wave is the tail part of a wave signal corresponding to the diffuse wave field. Because a coda wave is scattered and reflected multiple times in a medium, it has a high sensitivity to minor changes in material, either caused by a uniform property change or localized scatters. The time-lapse changes in the coda wave are typically calculated using coda wave interferometry (CWI) [1, 2, 3] by cross-correlating the disturbed and reference signals. The CWI analysis evaluates the relative wave-velocity change \(dV/V\) between two coda wave signals by measuring the relative time shift \(dt/t\), which can be estimated in two ways: doublet method [1] and stretching method [4]. The doublet method measures time shifts \(dt\) of many short segments at times \(t\) and then evaluates the relative time shift \(dt/t\). The stretching method assumes a uniform change of wave velocity in the medium so that the perturbed signal is a stretched or compressed version of the reference signal. [PERSON] et al. [5] measured temperature-induced velocity change in concrete and found that the stretching method gives stable results even for signals with a low signal-to-noise ratio (S/N). Because this study will analyze short time windows in a signal, only the stretching method will be used in the following analysis. In the stretching analysis, the perturbed signal \(S_{1}\) is interpolated at times \(t(1+\epsilon)\) with a stretching factor \(\epsilon\) to obtain \(S_{1}^{\prime}\), and then compared with the reference signal \(S_{0}\) by calculating the cross-correlation coefficient (CC):\[CC(\varepsilon)=\frac{\int_{t_{1}}^{t_{2}}S_{1}[t(1+\varepsilon)]S_{0}[t]dt}{\sqrt{ \int_{t_{1}}^{t_{2}}S_{1}^{2}[t(1+\varepsilon)]dt\int_{t_{1}}^{t_{2}}S_{0}^{2}[t ]dt}} \tag{1}\]  
where [\(t_{1}\), \(t_{2}\)] is the selected time window for calculation. The stretching parameter \(\varepsilon_{\text{max}}\) that maximizes the cross-correlation coefficient is regarded as the relative time shift \(-dt/t\), and then the relative wave-velocity change \(dV/V=-dt/t=\varepsilon_{\text{max}}\)[1; 4]. Figure 1 illustrates the stretching technique on two ultrasonic waveforms. According to [PERSON]'s research [6], for a Poisson medium, the CWI analysis result is primarily affected by the wave velocity variation of S waves. Therefore, in the case of homogeneous velocity changes, the CWI analysis result is close to the S wave-velocity change. On the other hand, if the velocity change is inhomogeneous, the CWI analysis result may depend on the selected time window positions of the signal. This difference has been explained theoretically by many researchers in their studies [7; 8; 9; 10], and the time-dependency of sensitivity kernels may be used to image local changes in media. The heterogeneous structure of concrete promotes rapid formation of a diffuse field. Therefore, CWI has been used to investigate small velocity changes induced by the acoustic-toelastic effect or nonlinear ultrasonic waves in concrete. [PERSON] and [PERSON] [11] reported the CWI technique could give a resolution of \(2\times 10^{-5}\) relative wave-velocity change in concrete induced by uniaxial stress. [PERSON] et al. [12] extracted the third-order elastic constants in concrete by measuring \(dV/V\) in different propagation and polarization directions under uniaxial stress. CWI also shows advantages in nondestructive evaluation (NDE) for very slow and weak changes in the material. For example, [PERSON] et al. [13] used the CWI method to monitor the self-healing process of internal cracks in biomimetic mortar due to the biomineralization effect. Their experimental results clearly showed that the biomimetic samples cured with nutrition spray demonstrated a higher velocity increase than other test groups. Recently, [PERSON] and [PERSON] [14] developed a nonlinear thermal modulation ultrasonic test for concrete damage evaluation by measuring the sensitivity of relative velocity changes to temperature variations.