## 2 Materials and Methods
### Numerical Ultrasound Propagation Model  
Ultrasound wave propagation in bone and water was simulated using a 2D finite-difference time-domain (FDTD) code (Simsonic, www.simsonic.fr, accessed on 10 March 2022) [25]. The simulation model considers elastic wave propagation including mode conversion, multiple scattering, frequency-independent absorption, refraction, and diffraction. A convergence study, as described in [19], provided stable results at grid sizes of 7 \(\upmu\)m and time steps of 0.93 ns. Table 2 shows the material properties used for the models in this study. Material properties were used from an ex vivo study [26] and a previous acoustic microscopy study in a human femoral cortical bone [27]. All bone models were simulated as hollow cylinders immersed in water. The cylinders were defined by an outer curvature radius \(r\) and a wall thickness \(d\). All bone models were placed 15 mm below a linear array with 64 transmitter and receiver elements (element and pitch sizes: 0.3 mm); therefore, the models assumed the sound propagation in the transverse image plane (i.e., perpendicular to the bone's long axis, at the antero-medial midshaft of a tibia, where the outer bone surface is flat or slightly curved and the sound velocity of the tissue matrix can be assumed to be isotropic in the simulation plane). The transducer elements emitted broadband pulses with a center frequency of 4 MHz and a \(-\)6-dB bandwidth of 60%. Phase delays were applied to focus the transmit beam consecutively, at depths ranging from 13 mm to 40 mm, with an increment of 1 mm. The signals received by all elements were captured and downsampled to a sampling rate of 80 MHz for further processing. The sufficient aperture size of 64 was chosen based on a side study, which can be found in Appendix A.  
\begin{table}
\begin{tabular}{c c} \hline
**Abbreviation** & **Description** \\ \hline _Cl.Th_ & Cortical thickness \\ _Ct.v11_ & Cortical compressional sound velocity propagating in the radial bone direction \\ _vH2O_ & Speed of sound in water \\ _dx_ & Lateral shift of center of mass of curved bone plate model relative to beam axis \\ \(r\) & Bone plate curvature radius \\ _Ct.Po_ & Cortical porosity \\ _F_\({}_{z}\) & Focus depth in \(z\)-direction \\ _H_F(\(F_{z}\)) & Amplitude of Hilbert-transformed envelope signal of beamformed frontside reflection at focus depth \(F_{z}\) \\ _H_B(\(F_{z}\)) & Amplitude of Hilbert-transformed envelope signal of beamformed backside reflection at focus depth \(F_{z}\) \\ _FB_ & Front- and backside reflection \\ _ATOF_ & Shift in time-of-flight between peak position of _H_F(\(F_{z}\)) and _H_B(\(F_{z}\)) \\ _\(\Delta F_{z}\)_ & Shift in focus depth between peak position of _H_F(\(F_{z}\)) and _H_B(\(F_{z}\)) \\ _F_\({}_{z,B}\) & Confocal focus depth position of backside reflection \\ _\(\theta\)_ & Semi-aperture angle of transmit and receive beams \\ _k_\({}_{\mathit{eff}}\) & Correction factor _k_\({}_{\mathit{eff}}\) for effective aperture _k_\({}_{\mathit{eff}}\)\(\theta\) \\ _\(\theta_{\mathit{crit}}\)_ & Critical angle based on [PERSON]’s law \\ _\(\Delta\theta\)_ & Difference of the semi-aperture angle to the critical angle \\ _Tx_\({}_{\mathit{tr}}\)_, _Rx_\({}_{i}\) & Transmit or receive channel number \\ _Rx_\({}_{\mathit{rf}}\) & Reference receive channel with maximum amplitude at envelope signal of pre-beamformed backside reflection \\ _V_\({}_{\mathit{gb}}\) & Gated pre-beamformed backside reflection signals \\ _V_\({}_{\mathit{ACF}}\) & Signal after using autocorrelation function (ACF) \\ _l_\({}_{\mathit{ACF}}\) \(l\) & Magnitude of the ACF signal \\ _\({}_{\mathit{\alpha}}\)_\({}_{\mathit{ACF}}\)_ & Inclination angle of the fitted ellipsoid on _V_\({}_{\mathit{ACF}}\) to the major semi-axis \\ _M_\({}_{\mathit{ACF}}\) & Time shift correction based on _R_\({}_{\mathit{ACF}}\) \\ \hline \end{tabular}
\end{table}
Table 1: List of abbreviations.