## 3 Results  
### Numerical Simulations  
Our numerical study shows that the AP estimated in each patch (see the scatterers and patches in Figure 3c) varies significantly (see Figure 4a,b). For the flat bone model, the AP significantly varies even if the aberrator is laterally invariant (the plate has a constant thickness and is parallel to the probe array) because the depth and lateral coordinate of the targets are not constant. A shallower target depth (patch #1) leads to a stronger aberration magnitude. An asymmetrical AP is obtained if the target is far from the bisector of the array. The irregular APs in Figure 4b and larger phase shift compared to Figure 4a come from the irregularities of the realistic bone model in its near and far surfaces and its asymmetric shape. Note that the aberration profiles in patches 1 and 2 are not accurately estimated for elements 75 to 96 (see Figure 4b,d) because the phase jump between the beamsm reference signal and the recorded wavefronts are larger than \(2\pi\) and thus not properly captured with the beamsm channel correlation method [40]. Both the PS and GB techniques improve the FWHM compared to the conventional method (compare Figure 5b,d with Figure 5a). For the flat bone and realistic bone models, Table 1 indicates that the PS technique improves the FWHM by 0.18 mm and 0.14 mm compared to the conventional reconstruction, respectively. The GB reconstruction further improves the FWHM compared to the conventional PS method by about 0.08 mm and 0.25 mm, respectively. The most significant improvement obtained by the GB method is the reduced background clutter (compare Figure 5d,b; see Figure 5f for more detail). Table 1 indicates that the GB method improves SCR by 4.9 dB and 2.4 dB compared to the PS technique for the flat and realistic bone model, respectively. The PS method improves the SCR by 4 dB and 4.3 dB compared to the conventional method for the flat and realistic bone model, respectively. The GB reconstruction technique detects the scatterers at their correct location (compare the location of the scatterers with the red dots in Figure 5d,g for more details). As Table 1 indicates, the GB technique reduces the localization axial error by 1.2-1.3 mm and the localization lateral error by about 0.3 mm compared to conventional reconstruction. In contrast, the PS method does not significantly reduce the localization axial/lateral error compared to the conventional method. Figure 4: The aberration profile in five isoplanatic patches (indicated by P) when the aberration/background model is **(a)** flat/homogeneous and **(b)** realistic/heterogeneous. The extracted synthetic-transmit-focused RF data in different isoplanatic patches for the **(c)** flat bone and **(d)** realistic bone model; the background was homogeneous. Figure 5: The reconstructed numerical images using (**a**) conventional beamforming, (**b**,**c**) phase-screen modeling, and (**d**) geometry-based reconstruction techniques with a flat bone model (Figure 3a). The aberration profile (AP) was estimated using the dataset generated with a (**b**) homogeneous and (**c**) heterogeneous background wave-speed model. The red dots indicate the true location of the scatterers. The blue triangles in (**b**) indicate the region that was used as background for quantifying the signal-to-clutter ratio calculation in Table 1. The yellow boxes in (**c**) indicate the regions at which the scatterers are completely missed. The lateral FWHM (**e**) and signal-to-clutter ratio (**f**) and localization error (**g**) obtained with all the methods for each scatterer.