## Appendix A Effect of Aperture and Semi-Aperture Angle \(\theta\)  
The multi-focus (MF) method was introduced in [19] using a 32 element transducer for in silico validation. The bone plates were placed 4 mm below the linear array transducer. In vivo ultrasound measurements on postmenopausal women demonstrated in the study of [PERSON] et al. [15] showed larger bone to transducer ranges up to 30 mm; therefore, simulation models in this study were performed for a realistic transducer/bone distance of 15 mm. To study the effect of the aperture size on the estimations of cortical thickness (_Ct.Th_) and cortical speed of sound (_Ct.\(\
u_{11}\)_), simulation models were created with different transducer array sizes (varying from 32 to 72 elements in increments of 4 elements). For all models a flat bone plate was placed 15 mm below the transducer.  
The tracked backside echo amplitudes for the bone plate model with aperture from 32- to 72-element are shown in Figure 1.  
The peak of the frontside echo occurred for all models at a focal depth of 15 mm. In contrast, the peak position varied for each aperture size and increased from 23.7 mm for the 32-element aperture to 24.9 mm for the 72-element aperture (Figure A1a). Moreover, the tracked front and backside (FB) amplitudes increased with an increasing aperture element number because more receiving signals were captured for delay and sum beamforming. For frontside and backside echoes, the tracked FB echo amplitudes showed a sharpening of the backside peaks with increasing aperture element number. Figure A1b shows an increase of the confocal focus shift \(\Delta F_{z}\) with increasing aperture element number, but \(\Delta TOF\) remained unchanged. The comparison of the estimated \(\mathit{Ct.Th^{MF}}\) and \(\mathit{Ct.\
u_{11}}^{MF}\) to the reference values in Figure A1c shows that the reference values were reached, both for \(\mathit{Ct.Th^{MF}}\) and \(\mathit{Ct.\
u_{11}}^{MF}\) for 64-, 68-, and 72-element apertures.  
Table A1 summarizes the estimated \(\Delta TOF\) between confocal FB reflection echoes, semi-aperture angle \(\theta\), the critical angle \(\theta_{crit}\) defined by [PERSON]'s law, the effective aperture \(k_{\mathit{eff}}\theta\), and \(\mathit{Ct.Th^{MF}}\) and \(\mathit{Ct.\
u_{11}}^{MF}\). For apertures larger than 44 elements, the difference of the semi-aperture angle to the critical angle \(\Delta\theta=\theta_{crit}-\theta\) was smaller than 10\({}^{\circ}\) and the effective aperture was derived iteratively using [19]. For apertures less than or equal to 44-elements, no effective aperture was derived due to \(\Delta\theta\) being larger than 10. In summary, the comparison of the bone plate model with different aperture element numbers revealed a dependence of the estimated \(\mathit{Ct.Th^{MF}}\) and \(\mathit{Ct.\
u_{11}}^{MF}\) on the semi-aperture angle. The previous study determined the effective aperture \(k_{\mathit{eff}}\theta\) with \(k_{\mathit{eff}}=0.1\cdot\Delta\theta\) for \(\Delta\theta<10^{\circ}\) in five iteration steps (Equation (5) in [19]). Due to the larger element number and distance of the transducer to the bone surface compared with the previous study, an adapted factor of 0.122 was used instead 0.1 for \(k_{\mathit{eff}}\). For \(k_{\mathit{eff}}<0.6\), the iteration resulted in incorrect \(\mathit{Ct.Th^{MF}}\) and \(\mathit{Ct.\
u_{11}}^{MF}\) values; therefore, the factor \(k_{\mathit{eff}}\) was not determined in five iterations as the iterative process was interrupted when \(k_{\mathit{eff}}\) reached values smaller than 0.6. For simulation models with 64-, 68-, and 72 elements the RE of \(\mathit{Ct.Th^{MF}}\) and \(\mathit{Ct.\
u_{11}}^{MF}\) was smaller than 0.5%. As simulation models with an aperture size greater than or equal to 64-elements showed no difference in \(\mathit{Ct.Th^{MF}}\) and \(\mathit{Ct.\
u_{11}}^{MF}\), all further simulations were performed with a 64-element aperture transducer.