## 2 Principles and Algorithms  
\(R(f,\,z)\) represents the intensity of the backscattered radio frequency (RF) signal that is obtained by the PMUT. It can be represented by Equation (1) in the frequency domain:  
\[R(f,\,z)=G(f)\cdot A(f,\,z)\cdot B(f) \tag{1}\]  
where \(f\) is the wave frequency, \(z\) denotes the depth of the ROI from the transducer, \(G(f)\) is the transmit pulse, \(A(f,\,z)\) is the attenuation, and \(B(f)\) is the backscattering. It is generally accepted that \(G(f)\) can be modeled using a Gaussian function, and the Gaussian-shaped envelope is denoted by Equation (2) in the frequency domain:\[G(f)=exp\left\{-\frac{\left(f-f_{c}\right)^{2}}{2\sigma^{2}}\right\} \tag{2}\]  
where \(f_{c}\) denotes the center frequency and \(\sigma^{2}\) represents the variance of the transmit pulse. _A(f, z)_ denotes the accumulated attenuation of the ultrasound in soft tissue. In a medium with a uniform attenuation property, the cumulative attenuation can be expressed as:  
\[A(f,z)=exp[-4\alpha(f)z] \tag{3}\]  
where \(\alpha(f)\) is the coefficient of attenuation expressed in Np/cm. \(\alpha(f)\) is commonly assumed to be linearly proportional to the frequency \(f\) and can be therefore written as \(\alpha(f)=\beta_{N}f\)[16]. Equation (3) can then be expressed as Equation (4):  
\[A(f,z)=exp[-4\beta_{N}fz] \tag{4}\]  
where \(\beta_{N}\) denotes the attenuation coefficient in Np/MHz/cm. _B(f)_ in Equation (1) represents the backscattered echo signals, and it is modeled as a random process as a result of the abundant random scattering in soft tissue. _B(f)_ is given by an exponential form that is expanded by the Taylor series, as denoted by Equation (5). _B(f)_ is then used to derive a close-form expression for the spectral shift in the frequency domain [17]:  
\[ \begin{split} B(f)=f^{n}=exp\left\{n\cdot\log(f)\right\}\\ =exp\left\{n\cdot\log(f_{c})+n\cdot\log\left(1+\frac{f-f_{c}}{f_ {c}}\right)\right\}\\ \approx f_{c}\cdot exp\left\{n\left(\frac{f-f_{c}}{f_{c}} \right)-\frac{(f-f_{c})^{2}}{2f^{2}}\right\}\\ \propto exp\left\{-\frac{n\left(f^{2}-4f_{c}f\right)}{2f_{c}^{2 }}\right\}\end{split}  \tag{5}\]  
Higher-order terms in the Taylor series expansion are ignored in Equation (5). For human tissue, the typical values of \(n\) range from 1 to 2. _S(f_o)_ denotes the cross-correlation function between two power spectra at different depths and it can be expressed as Equation (6):  
\[S(f_{o})=\int_{-\infty}^{\infty}R(f+f_{o},z_{1})R(f,z_{2})df \tag{6}\]  
where \(z_{1}\) and \(z_{2}\) (\(z_{1}<z_{2}\)) denote two different depths relative to the PMUT, and \(f_{o}\) represents the spectral shift. Since \(z_{1}\) and \(z_{2}\) are close, we can assume that the attenuation coefficients at depths \(z_{1}\) and \(z_{2}\) are similar. The cross-correlation algorithm is adopted, and the spectral shift between the two power spectra is given by Equation (7):  
\[f_{o(max)}=-\frac{4\sigma^{2}\beta(z_{1}-z_{2})}{1+\frac{uz^{2}}{f_{c}^{2}}} \tag{7}\]  
We can assume that the variance of the Gaussian emission pulse remains constant along the depth, and the relationship between the downshift slope of the center frequency (versus depth) and the attenuation coefficient (expressed in dB/MHz/cm) is given by Equation (8):  
\[\beta=-\frac{8.686}{4\sigma^{2}}.\frac{df_{c}(z)}{dz} \tag{8}\]  
where \(z\) denotes the depth from the PMUT and \(\sigma^{2}\) represents the variance of the transmit pulse. \(f_{c}(z)\) represents the center frequency of the power spectrum at depth \(z\). The method description and precise derivation can be found in the work of [PERSON] and [PERSON] [11; 17; 21]. The amplitude of ultrasonic waves in biological tissues decreases exponentially, as described by Equation (9) [18]:  
\[u=u_{o}e^{-\beta_{N}fz} \tag{9}\]  
where \(u_{o}\) is the amplitude of impulse voltage corresponding to the initial sound pressure \(p_{0}\), and \(u\) is the voltage value at \(z\) under the sound pressure \(p\).