## 2 Materials and Methods
TTM is an automatic method developed for DOP measurement and based on the gray scale mapping function (GSMF) determination [18] without a threshold subjectively selected by the technician. In particular, after the ROI selection in the diagnostic image, the gray levels were computed as a function of depth, averaging adjacent columns into the ROI. The mean depth profile \(g(z)\) obtained, was then interpolated with a non-linear curve, according to the following logistic function (\(\zeta\), \(\
u\) and \(\xi\) are constant):  
\[b(z)=\frac{\zeta}{1+e^{-\
u z+\xi}}, \tag{4}\]Following Equation (4), the DOP was evaluated as the depth at which \(b\)(z) intersects the tangent threshold \(T_{th}\), estimated on \(b\)(z) as follows:  
\[T_{th}=K_{max}\cdot\frac{\Delta g_{min}}{L_{fs}}, \tag{5}\]  
where \(K_{max}\) is assumed as the maximum system sensitivity and depends both on some characteristics of the medium (i.e., the attenuation constant and a factor related to the absorption in the medium) and some parameters related to US scanner settings (i.e., the bandwidth central frequency, the electronic noise amplitude, and the echo signal amplitude at \(z=0\)) as reported in [18]. The main advantage of the AdSTM, as for the TTM, is the automatic determination of the adaptive threshold based on the human eye sensitivity, as defined in Equations (1) and (5), respectively. The arbitrary threshold from the operator is no longer required. On the other hand, the main disadvantage of the TTM is the slow GSMF determination and the high acquisition time. As in [22], the phantom clips were acquired on a multi-purpose, multi-tissue US phantom (CIRS, Model 040 GSE). The device [26] is embedded with test objects of known characteristics and it was designed to provide tissue-mimicking properties under B-mode ultrasound. Such phantoms provide the possibility of testing the US probe on two different zones with attenuation coefficients of 0.70 and 0.50 dB-cm\({}^{-1}\)-MHz\({}^{-1}\). The main characteristics of CIRS 040 GSE are reported in Table 1.  
Four real-time diagnostic US systems for general purpose imaging (intermediate technology level), indicated as A, B, C, and D, each one of them equipped with three US probes (linear, convex, and vector array, respectively), were tested by means of the phantom zone with 0.70 dB-cm\({}^{-1}\)-MHz\({}^{-1}\) of attenuation, paying attention to display the speckle background only. All the probes were set at their own central frequency among the suitable range (nominal frequency). Moreover, scanning settings (e.g., overall gain, sensitivity time control (STC), and dynamic range) were preserved throughout the clip acquisitions because of their influence on SNR and DOP. In order to compare the different diagnostic systems tested in this work, the phantom and in-air clip acquisitions were carried out according to two different configuration settings (sets 1 and 2 in Table 2), which corresponded to the raw scanner working condition (QA protocols to provide a comparison of the outcomes from different scanners at similar working conditions) and the default preset working condition (clinical preset) provided by the US specialist, respectively. In addition, the ROI was always automatically selected in the same area of the diagnostic image according to the probe model. The influence of the ROI selection was evaluated, in terms of repeatability uncertainty, with changes in ROI size and position by means of Monte Carlo simulations (see the Monte Carlo simulation section). Moreover, a tissue-mimicking phantom with uniform speckle was used in our tests to reduce the aforementioned error component as much as possible.