## 2 Materials and Methods
### Numerical Study  
The k-Wave MATLAB toolbox was used to evaluate the PS and GB reconstruction techniques in a two-dimensional (2D) lossless medium, ignoring generation of shear waves [41]. Our numerical study on the PS method consists of two steps: (1) estimating the AP of each isoplanatic patch using an ideal phantom for PS technique (see the left box in Figure 3) and (2) applying the estimated AP on the RF data generated using a phantom with 30 small scatterers at different lateral and axial locations (see Figure 3f) and analyzing the image. For step 1, two aberrator models and two background wave-speed models provide four scenarios (four datasets): (1) a flat bone (Figure 3a) and a homogeneous background (Figure 3e), (2) the flat bone and a heterogeneous background generating speckle (Figure 3d), (3) a realistic bone (based on the micro CT of a human temporal bone (provided by [PERSON] and [PERSON] [42])) and homogeneous background, and (4) the realistic bone and a heterogeneous-background-generating speckle. The homogeneous background is an ideal situation for the PS method since individual responses of echogenic targets can be clearly identified in the RF data. This resembles a scenario where the echoes from individual contrast agent microbubbles are used as beacon signal. The wave-speed in Figure 3f randomly varies by \(+/-0.5\%\) with a uniform distribution with respect to the homogeneous model and mimics realistic brain tissue [43]. Five isoplanatic patches were defined (Figure 3c) by positioning the scatterers at lateral/axial coordinates of \(-8/35\) mm, \(-4/37.5\) mm, \(0/40\) mm, \(4/42.5\) mm, \(8/45.5\) mm, one in each isoplanatic patch. The numerical phantom used for estimating the AP (Figure 3c) was designed such that the hyperbolic wavefront of a scatterer in an isoplanatic patch (acting as a beacon signal) does not interfere with that of another scatterer in another isoplanatic patch; different axial and lateral locations of the scatterers ensured that. This is a very ideal situation for the PS method and most likely not the case in practice where the backscattered wavefront from contrast agent microbubbles is used for AP estimation, even if a small volume of contrast agent is injected. Figure 3: (**a,b**) The layered medium used in the simulation with a flat bone layer or a realistic temporal bone. (**c**) The numerical phantom used for estimating the aberration profile in each isoplanatic patch (P1–P5); the dashed lines separate isoplanatic patches. (**d**) The heterogeneous and (**e**) homogeneous background wave-speed model used in the simulations conducted for aberration profile estimation. (**f**) The numerical phantom used to generate the dataset needed for analysis of the full-width at half-maximum (FWHM), the signal-to-clutter ratio (SCR), and localization error. (**g**) The experimental setup to image a wire phantom while a bone mimicking plate is positioned in front of a phased array. For step 2 (image analysis), the two aberrator models, a homogeneous background and the phantom shown in Figure 3f, where 30 scatterers were positioned in different axial and lateral locations within each patch, were used to generate a numerical dataset. These two additional datasets were used for quantitative evaluation of both the PS and GB methods. The speed of sound (compressional wave-speed) in the silicone rubber, skull, and soft tissue (cutaneous tissue and brain tissue if homogeneous) was 1000 m/s, 3200 m/s, and 1600 m/s, respectively. The mass density of the soft tissue/silicone rubber and skull was 1000 kg/m\({}^{3}\) and 1900 kg/m\({}^{3}\), respectively. The thickness of the silicone rubber, cutaneous tissue, and flat bone layers was 1.3 mm, 1.7 mm, and 2.5 mm, respectively. The minimum and maximum thickness of the realistic bone layer was 2.38 mm and 1.2 mm, respectively. The properties of a P4-1 phased-array transducer (ATL/Philips, 2.5 MHz, 96 elements, pitch = 0.295 mm) and 5 plane waves from \(-\)15 to \(+\)15 degrees (in cutaneous tissue) were used to generate the numerical datasets.