## 3 Sample, Experimental Devices, and Protocols
### Transmitted and Received Signals  
Figure 5 shows the synoptic view of the data acquisition and the electro-acoustic devices. The electro-acoustic devices and the transducers were analyzed as continuous linear and stationary causal filters. Let us consider an object to be imaged. Whatever the transducer used, the recorded \(RF\)-signal, corresponding to the response returned by the object at the rotation angle \(\phi\) (i.e., the backpropagation), can be written as a function of time \(t\):  
\[s^{\phi}(t)=(r^{\phi}\otimes g^{\phi})(t)\otimes x(t) \tag{29}\]  
where \(r^{\phi}(t)\) is the impulse response of the mechanical apparatus including the mechanical offset of the bench depending of the angle of rotation \(\phi\). \(g^{\phi}(t)\) is the response of an object that is, according to the selected configuration, the hydrophone (Configuration A); the reference plate (\(g^{0}(t)=p(t)\), independent of the angle of rotation \(\phi\) (Configuration B); the copper thread (\(g^{0}(t)=w(t)\), independent of the angle of rotation \(\phi\) (Configuration C); and the object to be imaged (\(g^{\phi}(\phi,t)=m^{\phi}(t)\)) (Configuration D). x(t) is the response of the electro-acoustic chain, and can be written:  
\[x(t)=h_{T}^{*}(t)\otimes e(t) \tag{30}\]  
Figure 5: Synoptic view of the data acquisition and electro-acoustic setup. \(e(t)\) is the electric input conveyed to the transducer via the generators. \(h_{T}^{*}\) are the impulse responses of the electro-acoustic devices, including the transmitter-receiver devices and the transducers themselves. Pulse-mode method: USCT was first realized using an ultrasonic pulse-mode method. In this case, the electric input signal \(e(t)\) was an electronic pulse signal delivered by the pulse-receiver generator, and assimilated to a Dirac delta function (\(e(t\ )\approx\ \delta(t)\)). The recorded \(RF\)-signal can then be written:  
\[[s^{\Phi}(t)]^{pulse}=(r^{\Phi}\otimes g^{\Phi})(t)\otimes h_{T}^{*}(t) \tag{31}\]  
The time and frequency (\(-6\) dB) parameters for each transducer (f\({}_{0}\) = 500 kHz, 1 MHz, 2.25 MHz) are presented in Table 2. They were recorded using the hydrophone at a distance of 150 mm in the Fresnel zone (Configuration A). The nominal frequency was the frequency determined for the maximum amplitude of the spectrum modulus. The bandwidth was calculated for a threshold at \(-6\) dB. The area under the curve (AUC) was the integral of the spectrum in the bandwidth. The pulse duration was the inverse of the bandwidth, and the axial resolution was the product of the ultrasonic wave velocity and the pulse duration. WCE-mode method: In a second phase, the WCE-mode method was used. The method requires a calibration of the electro-acoustic response x(t), which must correspond to a wavelet function \(\psi_{j}(t)\), as described in Section 2.2:  
\[\mathrm{x(t)}=\psi_{j}(t)=h_{T}^{*}(t)\otimes\mathrm{e(t)} \tag{32}\]  
The recorded \(RF\)-signal can then be written:  
\[[s^{\Phi}(t)]^{WCE}=(r^{\Phi}\otimes g^{\Phi})(t)\otimes\psi_{j}(t) \tag{33}\]  
The dilatation scale \(j\) was adapted to the experimental electro-acoustic configuration for the sampling frequency \(F_{e}\) = 500 MHz, and a number of samples \(N\) = 65,536, and was equal to \(-9\) for f\({}_{0}\) = 500 kHz, \(-8\) for f\({}_{0}\) = 1 MHz, and \(-7\) for f\({}_{0}\) = 2.25 MHz. The time and frequency (theoretical and \(-6\) dB) parameters of the theoretical analyzing wavelets are presented in Table 3. \begin{table}
\begin{tabular}{c c c c} \hline
**Parameter (\(-6\) dB)** & **500 kHz** & **1 MHz** & **2.25 MHz** \\ \hline Nominal frequency & 477 kHz & 858 kHz & 2 MHz \\ Broadband width & 591 kHz & 900 kHz & 1 MHz \\ Bandwidth & (229–820) kHz & (0.5–1.3) MHz & (1.5–2.6) MHz \\ AUC (normalized) & 0.394 & 0.459 & 0.76 \\ Pulse duration (\(t_{pulse}\)) & 1.7 μs & 1.2 μs & 1 μs \\ Axial resolution & 2.5 mm & 1.8 mm & 1.5 mm \\ \hline \end{tabular}
\end{table}