## 3 Materials and Methods
### Theoretical Demonstration  
According to the properties of EMD, the ultrasound backscattered data \(s(t)\) can be decomposed into numerous IMFs expressed as follows:  
\[s(t)=s_{1}(t)+s_{2}(t)+\ldots\ldots+s_{n}(t). \tag{10}\]  
We assume that the incident wave of ultrasound \(h(t)\) can be decomposed by the EMD, and then we can obtain the following:  
\[h(t)=h_{1}(t)+h_{2}(t)+\ldots\ldots+h_{n}(t). \tag{11}\]  
According to Equation (2), we can obtain the following:  
\[s_{1}(t)=G\times[h_{1}(t)\otimes z(t)],\,\,s_{2}(t)=G\times[h_{2}(t)\otimes z (t)],\,...,\,\,s_{n}(t)=G\times[h_{n}(t)\otimes z(t)]. \tag{12}\]  
Summing all the signals in Equation (12), we can obtain the following:  
\[\begin{array}{ll}s_{1}(t)+s_{2}(t)+\ldots\ldots s_{n}(t)&=G\times[h_{1}(t) \otimes z(t)]+G\times[h_{2}(t)\otimes z(t)]+...+G\times[h_{n}(t)\otimes z(t)] \\ &=G\times[h_{1}(t)\otimes z(t)+h_{2}(t)\otimes z(t)+...+h_{n}(t)\otimes z(t)] \\ &=G\times[(h_{1}(t)+h_{2}(t)+...+h_{n}(t))\otimes z(t)]\\ &=G\times[h(t)\otimes z(t)]\\ &=s(t)\end{array} \tag{13}\]  
Notably, Equation (13) appears to be an inverse demonstration of Equation (10), indicating the following properties: (i) summing \(s_{1}(t)\), \(s_{2}(t)\), \(\ldots\), \(s_{n}(t)\) can reconstruct the original backscattered signal \(s(t)\); (ii) the incident wave \(h(t)\) can be decomposed into \(h_{1}(t)\), \(h_{2}(t)\), \(\ldots\), \(h_{n}(t)\), which can be further treated as the incident ultrasound waves with different frequency components; and (iii) the IMFs of the backscattered signals [\(s_{1}(t)\), \(s_{2}(t)\), \(\ldots\), \(s_{n}(t)\)] are the echo signals generated by the transmitting incident waves \(h_{1}(t)\), \(h_{2}(t)\), \(\ldots\), \(h_{n}(t)\) into the tissue \(z(t)\) for interactions. In this condition, two arbitrary IMFs of the backscattered signal can be further used for normalization. For ultrasound signals, higher-frequency IMFs have larger signal amplitudes and relevant physical meanings [14].  
In particular, the amplitude of the third IMF starts to be smaller than 10% of that of the original backscattered signals [14], and thus the IMFs from \(C_{3}\) to \(C_{n}\) may not be the primary components of the signals. For the above reasons, an IMF-based echogenicity ratio (IER) was defined using the first and second IMFs of the backscattered signals for signal normalization:  
\[IER=\frac{s_{1}(t)}{s_{2}(t)}=\frac{G\times[h_{1}(t)\otimes z(t)]}{G\times[h_{ 2}(t)\otimes z(t)]}=\frac{h_{1}(t)\otimes z(t)}{h_{2}(t)\otimes z(t)}. \tag{14}\]  
In practice, the IER is calculated using the ratio of the envelope amplitude of the first IMF to that of the second IMF. Ideally, the calculation of the IER allows self-normalization of the signal, which produces a gain-independent index that is associated with \(h(t)\) and \(z(t)\). This suggests the presence of a correlation between the IER and the B-scan intensity.  
To confirm the proposed equations, one-dimensional computer simulations were performed for validations. A 5-MHzGaussian pulse with a bandwidth of 60% was generated as the incident wave \(h(t)\). A computer phantom \(z(t)\) with a concentration of 32 scatterers/mm\({}^{3}\) was constructed by a sequence with randomly positioned delta functions. The sampling rate was 100 MHz. Gain factors \(G\) of 5, 10 and 20 were set. The ultrasound backscattered signals \(s(t)\) were then obtained using Equation (2).