## 3 Results and Discussions
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **Fitting Method** & **SSE** & **R-Square** & **RMSE** \\ \hline \multirow{4}{*}{1} & 2-order polynomial & 99.88 & 0.9394 & 0.2551 \\  & 3-order polynomial & 99.07 & 0.9399 & 0.2544 \\  & 4-order polynomial & 57.67 & 0.965 & 0.1944 \\  & 5-order polynomial & 55.95 & 0.9656 & 0.1919 \\  & 3-order Zernike polynomial & 99.84 & 0.9395 & 0.2553 \\ \hline \multirow{4}{*}{2} & 2-order polynomial & 183.5 & 0.8763 & 0.2328 \\  & 3-order polynomial & 177.8 & 0.8802 & 0.2293 \\  & 4-order polynomial & 170.1 & 0.8854 & 0.2245 \\  & 5-order polynomial & 166.2 & 0.888 & 0.2221 \\  & 3-order Zernike polynomial & 180.5 & 0.8783 & 0.231 \\ \hline \multirow{4}{*}{3} & 2-order polynomial & 158.2 & 0.9337 & 0.2464 \\  & 3-order polynomial & 156.2 & 0.9345 & 0.2451 \\ \cline{1-1}  & 4-order polynomial & 68.16 & 0.9714 & 0.162 \\ \cline{1-1}  & 5-order polynomial & 65.72 & 0.9724 & 0.1593 \\ \cline{1-1}  & 3-order Zernike polynomial & 156.6 & 0.9343 & 0.2454 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fitting error: SSE (sum of squares for error), R-square, and RMSE (root mean squared error). Figure 7: The input signals of the acoustic sources A, B, and C.  
The wavefronts for Model 1, Model 2, and Model 3 at different times, shown in Figure 8, are generally consistent with the sound field distributions simulated by FEM. Four probes were placed at Points 1, 2, 3, and 4 (at 1, 1.5, 2, and 2.5 mm to both \(x\)-axis and \(y\)-axis), the same as in the FEM simulation. Data collected by these probes for Model 1, Model 2, and Model 3 are plotted in Figure 9. Sensor data of Model 1 (red curve) only has one peak, probe data of Model 2 (green curve) has three peaks with similar amplitude, while probe data of Model 3 (blue curve) has two peaks and the second peak is stronger than the first peak. Figure 8: Wavefront at different time. (**a**) Model 1, (**b**) Model 2, and (**c**) Model 3 at 0.6, 0.8, 1, 1.2, and 1.4 \(\upmu\)s (from top to bottom). Figure 10 shows normalized FEM simulation results and k-wave simulation results separately displayed on the same graph. The simulation results of the two methods have the same trend, but there are slight differences in details, k-wave simulations are simply based on a superposition of sound waves, while FEM simulates the complete process of sound waves generated by thermal expansion. There are also some differences between the different methods. In Model 1, Model 2, and Model 3, the arrival time of sound waves at detection points in FEM is earlier than that in k-wave. This is because the starting point of sound wave propagation in FEM is at the boundary of the heat source (strain caused by temperature gradient), while the starting point of sound wave propagation in the k-wave simulation is at the center of the sound source. The propagation distance of photo-generated acoustic waves in FEM is reduced by half the heat source radius compared to k-wave, which leads to the sound waves in the FEM simulation at the detection point arriving earlier. In addition to the main wave, there are waves that are generated by multi-point excitation in Models 2 and 3, but these fail to fully participate in the superposition. The k-wave approach was only used as a comparative simulation to verify the suitability of the model. The actual photoacoustic process is closer to FEM simulation. Therefore, only the signal differentiation problem in the FEM model is considered here. Figure 10 shows that the excited sound wave has only a single peak in the negative direction, which can be used as the main peak of detection. In the FEM simulation, there is slight vibration after the sound wave passes. This is because of the coarsening of edge mesh and the inhomogeneity of mesh division due to the compromise of calculation time. This kind of slight vibration can be eliminated by refining the mesh and decreasing the time step. Figure 9: Sound pressure measured by sensors that were placed at (**a**) Point 1, (**b**) Point 2, (**c**) Point 3, and (**d**) Point 4.