## 2 Materials and Methods
### Centroid Downshift Method  
Of the various medical imaging modalities, ultrasound pulse-echo systems have been widely utilized for visualizing the human body's internal organs, as well as for evaluating the pathological state of soft tissue. To analyze the status of scanned tissue, several ultrasonic parameters, including speed of sound [23,24], integrated backscatter [25,26] and scatterer size [27], are extracted from RF data. Ultrasound attenuation is another critical parameter for evaluation of soft tissue, used to distinguish malignant from benign tumors, as well as provide fundamental information for quantitative ultrasound (QUS) [28]. To simplify representation in the frequency domain, most attenuation estimation methods for ultrasound pulse-echo systems assume constant sound speed and linear frequency dependence in a region of interest (ROI). After applying the Born approximation, which ignores multiple weak scattering of propagating acoustic waves in soft tissue, the intensity of the reflected RF signal in the frequency domain, \(R(f,z)\), can be represented as a product of pulse characteristics (\(G(f)\)), diffraction effects (\(D(f,z)\)), and backscatterer (\(B(f)\)) and attenuation terms (\(A(f,z)\)) [29]:  
\[R(f,z)\ =\ G(f)\ D(f,z)\ B(f)\ A(f,z), \tag{6}\]  
where z denotes the distance between the face of a transducer and the center of the ROI. Here, \(G(f)\), represents the transmit pulse characteristics which are closely related to the geometrical design and beam shaping parameters of the transducer. Insonifying pulses are typically modeled as Gaussian in the spectral domain, which is an approximation that has shown to be accurate within 10%, from root-mean-square-error analysis [30]. Most array transducers in modern ultrasound systems utilize beam focusing techniques to improve the spatial resolution at the ROI. As a result, the propagating acoustic waves experience the diffraction effects, expressed as \(D(f,z)\), along the depth of the tissue, and reflected RF signals acquired by linear array transducers contain diffraction effects around the pre- and post-focal sections. The spectral properties of the power spectra are also distorted before and/or after the beam focus. [PERSON] et al. [31] showed, using classical sound theory and focused transducer models, that diffraction effects could underestimate (or overestimate) attenuation coefficients at pre- (or post-) focal regions. To compensate for the diffraction effects, we utilize a simple correction method using two reference TM phantoms whose attenuation properties are known a priori [28]. Since the reflected RF signals for reference and sample phantoms acquired with the same transducer settings (i.e., the same focus, TGC, and beam pattern) exhibit identical diffraction effects, these can be eliminated by comparing the changes to the center frequency at the same depth. Because of the large number of randomly distributed small scatterers in the ROI, the frequency-domain characteristics for backscattered echo signals, \(B(f)\), are expressed as a stochastic model [32]. In general, this backscatterer model is typically in a form that considers only nonlinear frequency dependence, which is incorporated as a power of frequency. The cumulative attenuation through the beam path is modeled as being dependent on frequency and the distance. In a medium with uniform attenuation characteristics, \(A(f,z)\), is given as,  
\[A(f,z)\ =\ \exp[-4\alpha(f)z], \tag{7}\]  
where \(z\) is the distance from the transducer to the ROI and \(\alpha(f)\) is the frequency-dependent attenuation expressed in Neper/cm (or dB/cm). Since, \(\alpha(f)\) is known to exhibit linear dependence on frequency [33; 34], this function can be written as \(\alpha(f)=\beta\cdot f\), where \(\beta\) is the attenuation slope, also called the attenuation coefficient, expressed in Neper/cm/MHz (or dB/cm/MHz). In medical ultrasound systems, estimation of the attenuation coefficients is based on the frequency-dependent attenuation in soft tissue.