## Appendix C Phase Aberration Correction (PAC) Methods
### PAC III: Cross-Correlation (CC)
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Model** & **Correction** & \(\textbf{Ct.Tr}^{MF}\) **[mm]** & \(\textbf{RE}_{\textbf{CL.Tr}}\) **[\%]** & \(\textbf{Ct.v}_{\textbf{11}}^{MF}\) **[m/s]** & \(\textbf{RE}_{\textbf{CL.Vi1}}\) **[\%]** \\ \hline _r40 dx5.11_ & No & 3.61 & 9.84 & 3150 & 10.11 \\  & ST & 3.47 & 13.28 & 3040 & 13.25 \\  & ST + ACF & 3.80 & 4.91 & 3335 & 4.83 \\  & ST + ACF + CC & 3.80 & 4.91 & 3335 & 4.83 \\ \hline _r40 dx6.11_ & No & 3.47 & 13.37 & 3030 & 13.53 \\  & ST & 3.29 & 17.74 & 2895 & 17.39 \\  & ST + ACF & 3.54 & 11.48 & 3110 & 11.25 \\  & ST + ACF + CC & 3.54 & 11.48 & 3110 & 11.25 \\ \hline _r40 dx0 Po2_ & No & 3.60 & 10.02 & 3065 & 10.60 \\  & ST & 3.88 & 3.08 & 3310 & 3.46 \\  & ST + ACF & 3.90 & 2.45 & 3330 & 2.88 \\  & ST + ACF + CC & 3.93 & 1.83 & 3350 & 2.29 \\ \hline _r40 dx0 Po4_ & No & 4.39 & 9.67 & 3640 & 9.58 \\  & ST & 3.82 & 4.57 & 3165 & 4.72 \\  & ST + ACF & 3.81 & 4.47 & 3170 & 4.57 \\  & ST + ACF + CC & 3.76 & 6.08 & 3135 & 5.62 \\ \hline _r40 dx0 Po6_ & No & 3.57 & 10.79 & 2830 & 11.27 \\  & ST & 3.85 & 3.79 & 3050 & 4.37 \\  & ST + ACF & 3.85 & 3.87 & 3055 & 4.21 \\  & ST + ACF + CC & 3.92 & 1.92 & 3120 & 2.18 \\ \hline _r40 dx0 Po8_ & No & 3.54 & 11.49 & 2805 & 10.30 \\  & ST & 3.79 & 5.27 & 3015 & 3.58 \\  & ST + ACF & 3.83 & 4.18 & 3010 & 3.74 \\  & ST + ACF + CC & 3.89 & 2.87 & 3055 & 2.30 \\ \hline _r40 dx0 Po10_ & No & 5.16 & 29.12 & 3950 & 30.02 \\  & ST & 3.98 & 0.48 & 3030 & 0.26 \\  & ST + ACF & 3.95 & 1.24 & 3015 & 0.76 \\  & ST + ACF + CC & 3.98 & 0.48 & 3030 & 0.26 \\ \hline _r40 dx0 Po12_ & No & 5.58 & 39.69 & 4090 & 38.47 \\  & ST & 3.86 & 3.46 & 2835 & 4.02 \\  & ST + ACF & 3.87 & 3.23 & 2830 & 4.19 \\  & ST + ACF + CC & 3.92 & 1.94 & 2870 & 2.84 \\ \hline _r40 dx0 Po14_ & No & 4.78 & 19.54 & 3440 & 20.76 \\  & ST & 3.78 & 5.40 & 2760 & 3.11 \\  & ST + ACF & 3.78 & 5.40 & 2760 & 3.11 \\  & ST + ACF + CC & 3.78 & 5.40 & 2760 & 3.11 \\ \hline _r40 dx0 Po16_ & No & 4.82 & 20.55 & 3870 & 39.48 \\  & ST & 3.79 & 5.36 & 2635 & 5.03 \\  & ST + ACF & 3.79 & 5.14 & 2630 & 5.21 \\  & ST + ACF + CC & 3.90 & 2.58 & 2695 & 2.87 \\ \hline _r40 dx0 Po18_ & No & 3.82 & 4.52 & 2585 & 4.41 \\  & ST & 3.98 & 0.56 & 2680 & 0.89 \\  & ST + ACF & 3.97 & 0.79 & 2685 & 0.71 \\  & ST + ACF + CC & 3.83 & 4.21 & 2640 & 2.37 \\ \hline _r40 dx0 Po20_ & No & 5.65 & 41.24 & 3750 & 39.84 \\  & ST & 2.22 & 44.45 & 2675 & 0.25 \\  & ST + ACF & 3.33 & 16.66 & 2730 & 1.80 \\  & ST + ACF + CC & 3.03 & 24.14 & 2670 & 0.43 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._Figure A4: Model _r40 dx5.11_: (**a**) Comparison of tracked amplitude at each correction step for frontside amplitudes (tracked amplitude after ST, ST + ACF and ST + ACF + CC correction overlap). The reference-tracked amplitude of the plate model was shown by the grey dashed line (**b**) and backside amplitudes (tracked amplitude after ST and ST + ACF correction overlap), respectively. (**c**) Curvature parameter \(p_{1}\), retrieved from second order fit of using CC, as a function of focal depth (black circles). Linear fit (red line) was used to retrieve \(F_{z,B}\) at zero-crossing point. (**d**) Comparison of focus shift \(\Delta F_{z}\) and shift (black crosses) in time-of-flight \(\Delta\)TOF (grey circles) to the reference \(\Delta F_{z}\)\({}^{Ref}\) (dashed black line) and \(\Delta\)TOF\({}^{Ref}\) (dashed grey line) of the plate model. (**e**) Estimated _Ct.Th\({}^{MF}\)_ and _Ct.v\({}_{11}\)\({}^{MF}\)_ after each correction step compared to the reference _Ct.Th\({}^{Ref}\)_ (dashed black line) and _Ct.v\({}_{11}\)\({}^{Ref}\)_ (dashed grey line) value.