# Frequency-Dependent Acoustic Reflection for Soil Classification in a Controlled Aquatic Environment
## 2 Materials and Methods--Acoustic Spectrum Characterization
### Experimental Setup  
This computation must be performed independently for each tested frequency, covering 100 kHz to 400 kHz in 10 kHz increments. To measure \(A(f,t)\) in the ranges mentioned above, two transducers were placed facing each other at 20 cm in a water tank, and the readings in the frequency range 100-400 kHz were recorded (see Figure 3 for the schematic diagram showing the experimental setup for determining the energy defined as \(I_{1}\)) [PERSON] et al. [24] presented the methodology in detail and briefly described here. Figures 4 and 5 show the transducer's dimensions and their reception curve. Figure 3: The experimental setup for determining the incident energy \(I_{1}\)[24]. The experimental setup for measuring the energy \(I_{0}\) reflected from soil samples was designed such that the incident energy \(I_{1}\) reaching the soil samples will be identical to the energy measured in the previous study [24]. Two transducers were positioned next to each other, facing down in a water tank at a fixed distance of 20 cm from the soil sample placed at the bottom of the tank. The positioning of the transducers was such that the reflected signal reached the recording transducers separately from the direct arrival and reflections from the boundaries. The position of the transducer in the water was such that the direct arrival at the receiver would be completely isolated in the time domain from any reflection from the boundaries. Also note that the distance between the transducers and the soil samples in terms of the acoustic wavelength is significantly greater than 4.83 cm, corresponding to the far field threshold. The thickness of the samples was such that the whole signal was reflected from the top of the sample before the signal reached the bottom of the sample, such that the reflections of the top and bottom soil boundaries did not overlap. The experimental setup is presented schematically in Figure 6.  
For each soil type, 31 acoustic readings were performed per sample, over 10 samples per soil type, at frequency intervals of 10 kHz within the 100-400 kHz range. The reflected signal was recorded at each frequency, and then the gain function developed by [PERSON], [PERSON], and [PERSON] [24] was applied to obtain \(I_{0}\) (stacked over the 10 samples) and allowing (together with \(I_{1}\)) for subsequent reflection coefficient spectrum calculation. For completeness, the gain function by [PERSON] et al. [24] for determining reflected energy based on the recorded energy is outlined next:  
\[log(I_{0})=C_{1}log(I_{i})+C_{2}log(R_{i})+C_{3}f^{2}(R_{i}-1)+C_{7}+C_{8}I_{i} +C_{9}f+C_{10}f(R_{i}-1) \tag{2}\]  
where  
* \(I_{0}=\) the energy coming out of the ground after performing amplification to \(I_{i}\)  
Figure 4: The dimensions of the transducers (MISTRAS Group, West Windsor Township, NJ, USA). Figure 5: Transducer reception ranges in decibel values as a function of frequency. (MISTRAS Group, NJ, USA). * \(R_{i}\) = distance between the transmitting transducer and the ground, the same as between the ground and the receiving transducer;
* \(I_{i}\) = The energy that reaches the receiving transducer after receiving an acoustic reading from the ground; and
* \(C_{1},C_{2},C_{3},C_{7},C_{8},\ C_{9},\ C_{10}\)= are the constant coefficients given in Table 1, which were determined by [PERSON] et al. [24]. Please note that each value on the right side of Equation (2) is a normalized one by the following equation [25]:  
\[X^{\prime}=\frac{X-\mu}{\sigma} \tag{3}\]  
where  
X is the original value of the feature for a specific training example;  
\(\mu\) is the mean of the feature values over all the training examples;  
\(\sigma\) is the standard deviation of the feature over all the training examples; and  
\(X^{\prime}\) is a specific training example's normalized (standardized) value. The standard deviation and mean values of each feature appear in Table 2 and should be applied whenever using the trained nonlinear regression model by [PERSON] et al. [24] for calculating \(I_{0}\):