## Appendix C Phase Aberration Correction (PAC) Methods
### PAC III: Cross-Correlation (CC)  
The cross-correlation method was used to correct for small fluctuations in travel times caused by intracortical pores to determine \(F_{z,B}\) after ACF correction. The shift in the time-of-flight of the backside signals \(\Delta TOF_{B,\mathrm{R}xi}\) from each receiving channel were estimated with respect to the time-of-flight of the reference channel \(Rx_{\mathit{R}\mathit{R}\mathit{R}}\). For a focus depth smaller than the depth of the backside surface, backside echoes were not in phase. The \(\Delta TOF_{B,\mathrm{R}xi}\) showed a concave shape with negative curvature (Figure A3e for focal depth of 23 mm and 24 mm). When focus positions converged towards confocal backside focus position \(F_{z,B}\), the negative curvature of the concave shape of \(\Delta TOF_{B,\mathrm{R}xi}\) decreased. At \(F_{z,B}\) the backside signals were in phase by means of \(\Delta TOF_{B,Rxi}\) = 0. For a focus depth larger than \(F_{z,B}\), the reflected backside signals were defocused and \(\Delta TOF_{B,Rxi}\) transitioned to a convex shape with increasing positive curvature towards larger focus depths (Figure A3e for focal depths of 25 mm and 26 mm). On the retrieved \(\Delta TOF_{B,Rxi}\) a second order fit was performed to estimate the curvature parameter \(p_{1}\) (Figure A3f) using the following equation. \[\Delta TOF_{B,\ Rx_{i}}\ =\ p_{1}\cdot\left(R_{x_{i}}\ -\ R_{x_{ref}}\right)^{2}\ +\ p_{2}. \tag{9}\]  
The parameter \(p_{2}\) represents the value of \(\Delta TOF_{B,Rxrf}\) at the reference channel, which was not used for further analysis. The parameter \(p_{1}\) represents the curvature of the second order fit. The change of the curvature of \(\Delta TOF_{B,Rxi}\) from negative values for focus depth smaller than the confocal focus depth towards positive values for focus depth larger than the confocal focus depth, showed a linear dependence of \(p_{1}\) over the focus depth. The focus position where \(p_{1}\) remained zero was defined as \(F_{z,B}\) position. It was determined by a linear fit, \(p_{1}\) = \(m\cdot F_{z}\) + \(n\), from \(\pm 2\) focus position around the focus depth, where \(p_{1}\) had the smallest distance to zero (Figure A3e). Instead of using the amplitude of \(H_{B}(F_{z})\) for focus shift \(\Delta F_{z}\) between confocal frontside and backside bone reflections, the zero-crossing value of \(p_{1}\) (Figure A3f) was used for \(F_{z,B}\) to estimate \(\Delta F_{z}\) for _Ct.Th_ and _Ct.v\({}_{11}\)_ calculation. \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Model** & \multicolumn{2}{c}{**Cr.\(\
u_{11}\)Ref [m/s]**} \\ \hline \hline \multicolumn{2}{c}{_r40 dx0 Po2_} & \multicolumn{2}{c}{3428.6} \\ \multicolumn{2}{c}{_r40 dx0 Po4_} & \multicolumn{2}{c}{3321.8} \\ \multicolumn{2}{c}{_r40 dx0 Po6_} & \multicolumn{2}{c}{3189.4} \\ \multicolumn{2}{c}{_r40 dx0 Po8_} & \multicolumn{2}{c}{3127.0} \\ \multicolumn{2}{c}{_r40 dx0 Po10_} & \multicolumn{2}{c}{3038.0} \\ \multicolumn{2}{c}{_r40 dx0 Po12_} & \multicolumn{2}{c}{2953.8} \\ \multicolumn{2}{c}{_r40 dx0 Po14_} & \multicolumn{2}{c}{2848.7} \\ \multicolumn{2}{c}{_r40 dx0 Po16_} & \multicolumn{2}{c}{2774.6} \\ \multicolumn{2}{c}{_r40 dx0 Po18_} & \multicolumn{2}{c}{2704.2} \\ \multicolumn{2}{c}{_r40 dx0 Po20_} & \multicolumn{2}{c}{2681.6} \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of _Ct.Th\({}^{\text{MF}}\)_, _Ct.\(\
u_{11}\)\({}^{\text{MF}}\)_ and relative errors (RE) for each PAC method.