## 2 Materials and Methods
### Near-Field Phase-Screen Approach  
The PS correction technique requires a coherent wavefront backscattered from a small volume for estimating the aberration profile (AP) in each isoplanatic patch. Such a coherent wavefront backscattered from a small volume can be obtained if ideal echogenic point reflectors exist (such as microbubbles [37]) throughout the and/or thanks to transmit focusing in the heterogeneous medium [38; 39]. Transmit focusing is obtained by true transmission of focused waves or by synthetic transmit focusing (post-processing) if a set of unfocused waves (plane or diverging waves) are transmitted. In this paper, we evaluate the microbubble scenario (a more ideal situation for the PS method). Following steps are taken to determine the AP of an isoplanatic patch:  
1. An image is reconstructed using a conventional delay-and-sum reconstruction technique (see Figure 2a), i.e., assuming a medium with uniform wave-speed equal to 1600 m/s. Then, we estimate the coordinates of the echogenic scatterers (used to estimate the AP of the isoplanatic patch), indicated by the green dots in Figure 2a. Note that these coordinates are not the true coordinates of the targets (the red dots in Figure 2a) because phase aberration is ignored in conventional image reconstruction. Figure 1: The block diagram showing the processing procedure for the geometry-based and phase-screen aberration correction techniques when an ultrafast imaging sequence (multi-angle plane wave) is used for data acquisition. 2. For synthetic transmit focusing, transmit time delays for the transmit unfocused waves to reach simultaneously the coordinates of the echogenic scatterer found in step 1 are calculated. Then, the RF data acquired separately with each transmit unfocused wave are shifted by the calculated transmit delay. 3. By summing all the shifted RF data, RF data with transmit focusing are synthetically generated. As a result, the wavefront backscattered from the chosen scatterer has a higher amplitude compared to wavefronts backscattered by other scatterers and the unfocused RF data (compare the amplitude of the backscattered wavefront from scatterer #1 and #2 in Figure 2b,c, where the amplitude of scatterer #2 in panel (c) is three times higher than from panel (b)). 4. The receive travel-times between the location of the scatterer and the array elements are calculated by triangulation (the green-dashed line in Figure 2c) assuming a medium with uniform wave-speed equal to 1600 m/s. They are used to define a time mask with which the backscattered wavefront in the RF data after synthetic transmit focusing is extracted (Figure 2d). 5. A-lines of the extracted wavefront in step 4 are summed up over all elements of the array to obtain a reference signal (see Ref. SIG in Figure 2e), also known as beamsum reference signal [40]. Then, the phase shift between each A-line and the reference signal is calculated and unwrapped to compensate for phase jumps (Figure 2f). This results in the AP in the isoplanatic patch. Figure 2: **(a)** The reconstructed image using the conventional delay-and-sum beamformer; the aberrator model and phantom shown in Figure 3a,c were used to generate the numerical RF data. **(b)** The numerical wavefront backscattered from scatterer #2 for a tilted (by 7.5 degrees in cutaneous tissue) transmit plane wave. **(c)** The numerical wavefront after synthetic transmit focusing at the location of scatterer #2 by coherent compounding of the RF data obtained with 5 tilted plane waves (from \(-15\) to \(15\) degree in cutaneous tissue). **(d)** The extracted wavefront using triangulation assuming a uniform medium. **(e)** The reference signal and the A-line received by element 96 of the array. **(f)** The estimated phase shifted and its unwrapped version for each element. **(g)** The estimated phase shift (aberration profile) in nanoseconds for the isoplanatic patch around target #2.