## 2 Materials and Methods  
The AdSTM is a SNR-based method developed to assess the maximum depth of penetration on the basis of phantom and in-air clips obtained with the transducer on the phantom and held in-air, respectively. The data acquired was processed in MATLAB, obtaining a single image for both phantom and in-air clips (\(I\) and \(I_{a}\)), which were the result of the average of the first 15 frames. Then, the proper diagnostic average images (\(I_{c}\) and \(I_{c,a}\)) were obtained after cropping through automatic masking of \(I\) and \(I_{a}\) to remove all the US settings details. In order to compute the SNR as a function of depth, the signal \(S(z)\) and the noise \(N(z)\) were estimated according to the mean pixel value on a row within a rectangular region of interest (ROI), automatically selected on \(I_{c}\) and \(I_{c,a}\), respectively. An accurate estimation of DOP is closely related to a suitable number of pixels to include in the ROI at each depth. Nevertheless, the main limitation on the number of pixels per depth is due to the phantom section width. Furthermore, other aspects should be taken into account--artifacts at the transducer-phantom (or transducer-air) interface; the possible presence of strong echoes, corresponding to the phantom bottom; and the delimitation of the different attenuation zones. In addition, an excessive pixel number per depth could increase DOP computational cost. Therefore, the minimal pixel number choice (e.g., 30 pixels) is based on a trade-off between an accurate estimation and a limited computational time (about 10 ms). Then, the AdSTM evaluates the corresponding \(SNR(z)\) from the ratio \(S(z)/N(z)\) and computes the adaptive SNR threshold (\(SNR_{th}\)), automatically estimated as:  
\[SNR_{th}=S_{max}\cdot\frac{\Delta g_{min}}{L_{fs}}, \tag{1}\]  
where \(S_{max}\) is the maximum system sensitivity retrieved from the logistic function of the SNR curve [22], \(\Delta g_{min}\) is the minimum gray level difference distinguishable from the human eye [23; 24; 25] and \(L_{fs}\) the luminance full scale expressed in terms of gray levels. In particular, the ratio \(\Delta g_{min}/L_{fs}\) is always between 0 and 1, while \(S_{max}\) was determined according to the following steps:  
1. The non-linear curve fitting \(f(z)\) was derived, from \(SNR(z)\), by the iterative computation of the coefficients \(\beta\), \(\chi\), \(\gamma\), \(\eta\) according to the sigmoidal function: \[f(z)=\frac{\beta}{1+e^{-\chi z+\gamma}}-\eta,\] (2)
2. The first order derivative \(f\) '(\(z\)) was calculated and its minimum value \(z_{min}\) was used to estimate \(S_{max}\) as follows: \[S_{max}=f(z_{min}),\] (3) Finally, the AdSTM retrieved the depth value corresponding to the maximum depth of penetration from the intersection between the threshold \(SNR_{th}\) and the SNR curve. In order to test the AdSTM on different US scanners equipped with different probes, the results were compared with the ones obtained from the TTM, an alternative algorithm developed in [18], and the mean judgement of eight independent observers without medical expertise through an eye-based test of \(I_{c}\) (naked eye test). Medical expertise is not mandatory for quality controls of the US systems by means of ad hoc tissue-mimicking phantoms, as they are usually performed by technicians. The protocol employed in this study for the naked eye test was designed by the authors, on the basis of the American Association of Physicists in Medicine (AAPM) report [2]. Each observer independently performed the test making sure to preserve the same test conditions--a well-lit room, maximum monitor light without glare, and keeping a fixed distance from it. Moreover, through an in-house MATLAB software the observer could indicate with a simple mouse click, his decision on \(I_{c}\). In order to test subjects' inter- and intra-variability, the test was repeated six times providing a randomized image order to the observer, automatically determined by the software.