## 2 Cross-Correlation Based Photoacoustic Velocity Measurement in Time-Domain  
When a pair of pulsed lasers irradiates an object with a time delay \(t_{p}\), a pair of PA signals is generated, denoted as \(p_{1}\) and \(p_{2}\). In the scenario where the object remains stationary, only the original time delay \(t_{p}\) exists between the arrival time of the PA signal pair. Moreover, if no additional time shift \(t_{s}\) appears, the time delay \(\tau\) between the arrival time of the PA signal pair is expressed as \(\tau=t_{p}\). However, if the object is in motion, as illustrated in Figure 1, a time shift \(t_{s}\) appears between the arrival time of the PA signal pair. The first PA signal \(p_{1}\) is generated by the first pulse irradiation. The second PA signal \(p_{2}\) arises from the object during motion after \(t_{p}\) seconds. Though \(p_{1}\) reaches UST promptly at time \(t=0\) s, \(p_{2}\) arrives at the UST \(t_{p}\pm t_{s}\) seconds later due to the object's motion. To determine the time shift \(t_{s}\) between a PA signal pair generated solely by an object's movement, CC analysis was performed on the PA signal pairs \(p_{1}\) and \(p_{2}\). CC analysis calculates the correlation between signals with a varying time delay and expresses it as a correlation coefficient. We assumed that the signals were correlated, and the correlation coefficient demonstrates the highest correlation at a certain time delay. The time shift \(t_{s}\) can be ascertained by identifying the peak correlation coefficient. This time shift represents the time difference between the arrival of the extracted PA signals at the UST. Utilizing this information, the propagation distance \(d\) of the PA wave is subsequently calculated as  
\[d=t_{s}c. \tag{1}\]  
where \(c\) represents the speed of sound in the medium. When considering the angle between the UST and the object as \(\theta\), the propagation distance of the PA waves to the receiver is expressed as \(d\)=\(l\cos\theta\), where \(l\) corresponds to the actual distance the object has moved. By dividing the displacement \(d\) by the time shift \(t_{s}\), the velocity \(v\) of the object is computed as follows:  
\[v=\frac{l}{t_{p}}=\frac{d}{t_{p}\cos\theta}=\frac{t_{s}c}{t_{p}\cos\theta}. \tag{2}\]  
This is analogous to the classical equation of the Doppler frequency derived for a moving object, with the key distinction being that the velocity is now proportional to the time shift rather than the frequency shift [19; 21]. The proposed method enables the estimation of both the object's moving velocity and direction of movement using the time shift \(t_{s}\) obtained from the CC analysis of the PA signal pair. A CC function was employed to determine the time shift \(t_{s}\) between the two PA signals, \(p_{1}\) and \(p_{2}\). By taking the complex conjugate of \(p_{1}\) as \(p_{1}^{*}\), the continuous function for CC is defined as follows:  
\[(p_{1}\star p_{2})(t)=\int_{-\infty}^{\infty}p_{1}^{*}(\tau)p_{2}(t-\tau)d\tau. \tag{3}\]  
For discrete signals \(p_{kn}\), where \(k=1,2\), and the number of samples is \(n=0,\ldots,N-1\) with \(N\) the total number of samples and a time separation of \(\Delta t\), the relationship between the continuous signal \(p_{k}(t)\) and the discrete signals \(p_{kn}\) is defined as \(p_{kn}\equiv p_{k}(t)|_{t=n\Delta t}\). The  
Figure 1: Principle of the velocity measurement using photoacoustic (PA) signals. PA signals are generated by irradiating an object with a pair of laser pulses separated by a pre-selected time delay \(t_{p}\). When the time \(t\) = 0 s, the first pulse generates PA signal 1. After \(t_{p}\) seconds, the second pulse irradiates the object, generating PA signal 2. During the \(t_{p}\) seconds, the object undergoes movements, causing a time-shift \(t_{s}\) in the arrival time of the PA signal pairs. CC results can be interpreted as the deterministic correlation between two deterministic signals. Then, for these discrete samples of length \(N\), the CC function is defined as follows:  
\[\hat{R}_{p_{1}p_{2}}(m)=