## 1 Introduction
There have been efforts to overcome this false assumption of a constant speed of sound in cortical bone. [PERSON] et al. [14] proposed the first in vivo image reconstruction of cortical bone using a conventional medical ultrasound scanner and seismic image reconstruction. This reconstruction method provides local estimations of _Ct.Th_ and anisotropic sound velocity. Consequently, the need for further methods brought about the multi-focus (MF) imaging technique, that was developed by our group to measure cortical thickness (_Ct.Th_) and the compressional sound velocity propagating in the radial bone direction (_Ct.v1_) [19]. Our method aims at imaging cortical bone at the central anteromedial tibia. This anatomical site is of clinical interest, as it is easy to access, and is composed of a thick and regular cortical bone shell. Alterations, such as reduced cortical thickness and the occurrence of large intracortical pores, have shown to be associated with reduced hip strength [20] and increased fracture risk [21]. The ultrasonic speed of sound in cortical bone has been proposed as a biomarker of bone quality since the 1970s [22], and is related to bone density and elastic constants, which are correlated to bone quality and fracture risk [23]. The MF method is based upon the consecutive focusing of ultrasound waves at varying depths, followed by the retrieval of focus locations and pulse-travel times of signals reflected from the periosteal (frontside) and endosteal (backside) cortical bone interfaces using conventional DAS beamforming (Figure 1). So far, the MF method has been validated on plates with constant thickness, positioned parallel to the probe array; however, it is important to extend this, as typical human cortical bones exhibit curvatures at their periosteal and endosteal interfaces. These curvatures introduce a distortion of the propagating wavefronts and the round-trip travel time, resulting in phase distorted beamformed signals. The objective of this research is to analyze the effect of the phase aberration caused by (a) bone curvature, (b) bone tilt with respect to the beam axis, and (c) material inhomogeneities due to the presence of cortical pores on the estimations of _Ct.Th_ and _Ct.v\({}_{11}\)_. The phase aberration from bone surface curvature leads to a different round-trip travel time when compared with a flat plate bone model (surface time shift, ST), and was corrected using the concept of refraction compensation proposed by [PERSON] et al. [24]. Bone tilt, with respect to the beam axis of the transducer array, shows orientation dependence of the received echoes compared with a flat bone interface. Additional time shifts caused by the orientation dependence of received echoes were corrected using autocorrelation analysis (ACF). Differences in the round travel time of the received echoes based on the interaction of ultrasound wave refractions with cortical pores were determined using cross correlation analysis (CC). We show the need to incorporate three phase-aberration correction (PAC) methods for non-plate shaped bone structures by means of numerical finite-difference time-domain (FDTD) simulation models, as well as measurements on a human tibia bone. Precision and accuracy values of estimated _Ct.Th_ and _Ct.v\({}_{11}\)_ with and without corrections were compared.