## 2 Materials and Methods
### Simulation Setup and Study Organization  
In this work, Matlab (The MathWorks, Natick, MA, USA) simulations with Field II [26; 27] were carried out by modeling a 64-element phased array probe with a 340 \(\upmu\)m pitch. The central working frequency was 2 MHz, and a 2-cycle Hanning-weighted sinusoidal burst was used as the excitation signal. The transmit focus was set at a 70 mm depth and dynamic focusing was implemented in reception. A 100 MHz sampling frequency was considered.  
First, the analytical study on the spatial coherence trend was performed by simulating a numerical uniform speckle phantom, made of 140,000 scattering points with a Gaussian amplitude distribution, randomly placed in a 100 \(\times\) 1 \(\times\) 70 mm\({}^{3}\) volume centered around \((x,y,z)=(0,0,65)\) mm (i.e., \(>\)10 scatterers per resolution cell).  
In B-mode imaging with standard dynamic RX focusing, the spatial coherence is known to decrease before and after the fixed TX focus [20; 23]; thus, in this work, both the normalized covariance \(C_{N}(l,t)\) and total spatial correlation \(SC(t)\) were evaluated using Equations (1) and (2), and averaging the values obtained in a small 5 \(\times\) 5 mm\({}^{2}\) area centered at \((x,z)=(0,70)\) mm, i.e., over the TX focus. Three different scenarios were analyzed (Table 1):  
1. the number of MLT beams varies (i.e., \(N_{TX}=1/4/6/8/12\)), but the total image sector is fixed (\(\theta_{SECT}=90^{\circ}\)), as well as the number of lines (192); consequently, the angle among the TX beams (\(\theta_{TX}=\theta_{SECT}/N_{TX}\)) changes together with the number of beams (usually, this is the classic MLT implementation);the number of MLT beams varies (i.e., \(N_{TX}\) = 1/4/6/8/12), but the same angle (\(\theta_{\rm TX}\)) among the beams is used in all configurations; in particular, this angle was set to be equal to the one that would be obtained applying 12-MLT to scan a 90\({}^{\circ}\) sector (i.e., \(\theta_{12}\)). Thus, in this case, the total image sector also changes in the different MLT configurations;
3. the number of beams is fixed (i.e., \(N_{TX}\) = 4), while the angle among them changes, as it would do in 4/6/8/12-MLT when a 90\({}^{\circ}\) sector is acquired. Thus, also here, the total image sector changes in each case.  
These three scenarios were specifically chosen to evaluate the influence of the number of simultaneously transmitted beams or of their angular spacing, or of both factors, on the SC trend.  
The second part of this study aimed at testing the performance of short-lag F-DMAS beamforming in MLT imaging. Results were also compared to those of DAS with Tukey apodization in RX. In both cases, Tukey apodization was also applied in TX, in order to reduce TX cross-talk [22].  
Image quality was evaluated in terms of lateral resolution (at \(-6\) dB), contrast ratio (CR), contrast-to-noise ratio (CNR), and speckle signal-to-noise ratio (sSNR) [8]:  
\[CR=20 log_{10}\left(\frac{\mu_{cyst}}{\mu_{bck}}\right), \tag{9}\]  
\[CNR=\frac{\left|\mu_{cyst}-\mu_{bck}\right|}{\sqrt{\sigma_{cyst}^{2}+\sigma_{ bckt}^{2}}}, \tag{10}\]  
\[sSNR=\frac{\mu_{bck}}{\sigma_{bck}}. \tag{11}\]  
Point spread function (PSF) simulations were carried out considering a series of vertically aligned point scatterers along the \(x\) = 0 mm axis, from \(z\) = 10 mm to 90 mm with a 20 mm step; in addition, resolution measurements were made at the focal depth on the PSF at \((x,z)\) = (0, 70) mm. CR, CNR, and sSNR were instead evaluated by simulating a numerical phantom of 140,000 points (i.e., \(>\)10 scatterers per resolution cell), with a 10-mm-diameter anechoic cyst centered at \((x,\,z)\) = (0, 70) mm, in a 100 \(\times\) 1 \(\times\) 70 mm\({}^{3}\) uniform tissue background starting at \(z\) = 30 mm. Measurements were performed considering two 6-mm-diameter circular areas inside and outside the cyst.