## 2 Numerical Experiment Design
### Effective Bulk Modulus and Effective Density Calculation  
Effective bulk modulus and Effective density in scanned imaging were calculated as [15]:  
\[\rho=c^{-1}\;Z_{0}\left(\frac{-1-\frac{p_{1}}{p_{e}-p_{0}}-\sqrt{ \frac{4\frac{p_{1}}{p_{e}-p_{0}}+1}{p_{e}-p_{0}}}}{\frac{p_{1}}{p_{e}-p_{0}}-2} \right)\text{, }\frac{Z}{Z_{0}}>1\text{,}\] \[\rho=c^{-1}\;Z_{0}\left(\frac{\left(1-\frac{p_{1}}{p_{e}-p_{0}} \right)+\sqrt{1-4\frac{p_{1}}{p_{e}-p_{0}}}}{\frac{p_{1}}{p_{e}-p_{0}}+2} \right)\text{, }\frac{1}{3}<\frac{Z_{1}}{Z_{0}}<1\text{.} \tag{1}\] \[\rho=c^{-1}\;Z_{0}\left(\frac{\left(1-\frac{p_{1}}{p_{e}-p_{0}} \right)-\sqrt{1-4\frac{p_{1}}{p_{e}-p_{0}}}}{\frac{p_{1}}{p_{e}-p_{0}}+2} \right)\text{, }0<\frac{Z_{1}}{Z_{0}}\leq\frac{1}{3}\text{.}\]  
\[K=c\;Z_{0}\left(\frac{-1-\frac{p_{1}}{p_{e}-p_{0}}-\sqrt{\frac{p _{1}}{p_{e}-p_{0}}+1}}{\frac{p_{1}}{p_{e}-p_{0}}-2}\right)\text{, }\frac{Z}{Z_{0}}>1\text{,}\] \[K=c\;Z_{0}\left(\frac{\left(1-\frac{p_{1}}{p_{e}-p_{0}}\right)+ \sqrt{1-4\frac{p_{1}}{p_{e}-p_{0}}}}{\frac{p_{1}}{p_{e}-p_{0}}+2}\right)\text{, }\frac{1}{3}<\frac{Z_{1}}{Z_{0}}<1\text{.} \tag{2}\] \[K=c\;Z_{0}\left(\frac{\left(1-\frac{p_{1}}{p_{e}-p_{0}}\right)- \sqrt{1-4\frac{p_{1}}{p_{e}-p_{0}}}}{\frac{p_{1}}{p_{e}-p_{0}}+2}\right)\text{, }0<\frac{Z_{1}}{Z_{0}}\leq\frac{1}{3}\text{.}\]  
In the above equations, \(p_{e}\), \(p_{0}\), and \(p_{1}\) are, respectively, the highest amplitude of the source pulse from the probe, the reflection of the wavefront back from the front interface of the sample layer, and the second echo back from the interface between the target sample layer and next material layer separately. \(p_{e}\) was the maximum amplitude value of the emission source. It was set in the software to 1 \(\upmu\)Pa on the absolute scale. \(p_{0}\) and \(p_{1}\) were the maximum absolute values of the detected reflection signal amplitudes obtained from the probe's upper surface. The values were averaged from the ten linear distributed arrays  
Figure 1: Material model of the layers used for simulations.  
on the probe line laterally. \(c\) is indicated the sound velocity from the time of flight in the sample layer at the measured location, described as \(c=2d/(t_{1}-t_{0})\), where \(t_{1}\) and \(t_{0}\), are the first peak of the first and second echoes. \(d\) is the thickness of the target layer. Effective density \(\rho\) is calculated from \(\rho=Z/c\), where \(Z\) is the acoustic impedance of the sample at the scanned location. The baseline impedance value was referred to in the previous layer is \(Z_{0}=\rho_{0}c_{0}\).  
For the multiple layers of various materials:  
\[p_{k}=\frac{Z_{n}}{Z_{0}}(p_{e}-\text{sig}(Z_{1}-Z_{0})|p_{0}|)\cdot\left(\prod _{i=2}^{k}t_{i-1,i}\right)r_{k-1,k}\left(\prod_{i=1}^{k}t_{i,i-1}\right),k=1,2, \cdots,n. \tag{3}\]  
where the transmission and the reflection coefficients are \(t_{i-1,i}=(2Z_{n})/(Z_{n-1}+Z_{n})\) and \(r_{k-1,k}=(Z_{n}-Z_{n-1})/(Z_{n}+Z_{n-1})\) the reflection coefficient at the interface between layer \((n-1)\) and \(n\). The reflection coefficient of the interface between the last layer and ambient material is expressed as \(r_{k-1,0}=(Z_{0}-Z_{n-1})/(Z_{n-1}+Z_{0})\). \(n\) numbers of \(Z_{n}\) values are obtained by solving \(n\) numbers of Equation (2) for the \(n\) numbers of layers in the samples. The effective density values are expressed as \(\rho_{n}=Z_{n}c_{n}^{-1}\).