## 1 Introduction  
The current standard method for bone strength assessment and fracture risk prediction is based on areal bone mineral density (_aBMD_) measured by dual-energy absorptiometry (DXA) [1]. Although _aBMD_ is an important biomarker of bone quality, additional bone factors, including macro- and micro-structural bone parameters, as well as viscoelastic properties, are known to determine individual bone strength; therefore, to quantify these parameters for bone assessment, quantitative ultrasound (QUS) methods have been introduced as nonionizing alternatives. Early bone QUS technologies used dedicated hardware to measure acoustic properties, such as the speed of sound (SOS) and broadband ultrasoundattenuation (_BUIA_), at anatomical sites that contain mostly trabecular bone, such as the heel [2]. More recent QUS devices are aimed at imaging bone by using dedicated hardware electronics and ultrasound probes. An example of this by [PERSON] et al. developed ultrasonic image reconstruction methods to image the cortical diaphysis of long bones using quantitative ultrasonic tomography [3; 4]. Another tomographic approach to image long bones is based on full-waveform inversion [5]. Additionally, [PERSON] et al. used Split-Step Fourier imaging to image bone fractures and to monitor bone healing [6]. Furthermore, a Born-based inversion method has been implemented on an ultrasonic wavefield imaging technique to reconstruct internal structures of long bones [7]. Limitations of these studies were that either the speed of sound or the thickness needed to be assumed a priori. Axial transmission devices can retrieve cortical parameters (i.e., porosity, thickness, and speed of sound), by measuring the propagating velocity of dispersive guided waves [8; 9; 10; 11; 12]; however, this technique is challenged by large soft tissue thickness, irregular bone shapes, and it does not provide direct image guidance. A few recent technologies utilize sophisticated array-based pulse-echo imaging technology to estimate _BMD_ in trabecular bones at major fracture sites (i.e., spine and proximal femur [13]), or to measure structural and material properties in the cortical bone (i.e., tibia and radius) [14; 15]. Most medical ultrasound scanners on the market implement the standard delay-and-sum (DAS) beamforming method to reconstruct the brightness mode (hereinafter called B-mode) images. This technique uses a transducer array to transmit and receive focused ultrasound signals inside the body. Conventionally, the reconstruction of B-mode images using DAS is done by adding time specific delays to the individual ultrasound signals which are received at each element of the receiver array before summing all signals to create a beamformed received signal; therefore, the sensitivity of the beamformed signal can be maximized to a certain depth and direction. In medical ultrasound scanners, transmit and receive focusing is performed by assuming a constant speed of sound of soft tissue (1540 m/s) along the entire sound propagation path. This approximation provides satisfactory image quality for most soft tissues, because the true velocities only vary within 10% when compared with the assumed value [16]; however, this is not the case for mineralized tissues, such as cortical bones. The radial speed of sound in cortical bone is between the range of 2800 to 3500 m/s [17], which results in a substantial refraction at the soft tissue and cortical bone interface. In case of a wrong assumption on the constant sound velocity, the delay estimation, which is necessary to focus on a particular image location, is incorrect, subsequently leading to a phase-distorted DAS signal. As a result, in a conventional B-mode image reconstructed by medical ultrasound scanners, the internal bone structures appear blurred or cannot be reconstructed at all. Aside from radiofrequency echographic multispectrometry (REMS) technology [18], this conventional DAS beamforming is currently used for bone strength assessment and fracture risk prediction.