A scaled tank experiment for layered stochastic rough interface scattering reflection coefficient

The stochastic reflection coefficient of the seabed holds valuable acoustic information about the seabed, making it vital for seabed parameter detection. Therefore, it is crucial to achieve some improved results. In this study, we utilized the small-slope approximation method to obtain the fluctuation characteristics of the reflection coefficient for a layered medium with rough interface. Moreover, we addressed the changes in grazing angle by combining the stochastic rough interface scattering with the layered medium reflection. Initially, we present a simulation result, and subsequently, we conducted a scaled tank experiment to verify the variation characteristics of the reflection coefficient. These findings can serve as a theoretical reference for future seabed exploration.


Introduction
The characteristics of the reflection coefficient in layered medium have attracted significant attention.As the incident angle varies, the reflection coefficient demonstrates intriguing variations.Since the reflection coefficient of the seabed contains extensive seabed parameter information, it facilitates the extraction of seabed parameters through analysis of the angle characteristics of the reflection coefficient in the future.When sound waves penetrate the seabed interface, they exhibit marked reflection characteristics.Due to the high cost of conducting experiments in the sea, we opted for a scaled tank experiment to obtain the reflection features at a reduced expense.
Voronovich [1] proposed the small-slope approximation method, and derived the lowest order smallslope approximation scattering cross section in Biot theory.It obtained the numerical results of backscattering and bistable scattering by using the modified power law spectrum.Chen [2] gave a method to calculate the noise coherent sound field in shallow water, which can express the influence of the coherence of submarine noise field, which is based on the theory of ray acoustics.Robert [3] improved the component description of the seafloor scattering interface and was superior to the perturbation theory.Yang [4] improved the perturbation method and small-slope approximation to theoretically derive and study the wave scattering of layered structures with rough interfaces of any number of layers.It gave us the transfer model between different layers.Tang [5] gave the application of small roughness perturbation theory to range-dependent waveguide reverberation.He established a sound field model of rough interface.Based on perturbation theory, fast numerical calculation method is used to deal with the scattering field of gently varying interface.Jackson [6] established the Earth Acoustic Bottom Interaction Model (GABIM).It gave the bottom backscattering intensity and bottom loss values of the stratified seafloor, which was useful in layered interface.Jackson [7] used the smallslope approximation to give a model on layered medium.The modeling community would be justified in adopting small-slope techniques as a wholesale replacement for perturbation theory methods.It was better and, at worst, equivalent.Olson [8] used small-slope approximation to calculate the effect of seafloor interface roughness on passive estimates of reflection coefficients, including coherent and incoherent contributions.But it still needed real measurements and experiment to verify its result.
In this work, we analyze the reflection mechanism of the undulating seafloor and discuss the physical propagation mechanism of the reflection coefficient.We apply the small-slope approximation to calculate the rough interface, and the detailed formula derivations can be found in Olson [8] .Furthermore, we conduct a scaled tank experiment to validate the simulation results and provide a comprehensive comparison between the simulation and experimental data.The findings indicate that the reflection characteristics of layered interface in deep sea has periodic changes.In a certain range of kh, our experimental results agree with the simulation results.

Basic theory
According to Eckart [9] , the reflection in half space coefficient can be expressed as: The reflection coefficients of the  ℎ interface satisfy 11 ( ) ( ) / ( ) , these functions apply to the small-slope approximation 7 8 : For the general case of any number of layers, the scattering cross section is

The environment of the experiment
As is shown in Figure 1, the depth of water and sound source is 37cm and 27cm, the distance between the sound source and the hydrophone is from 6cm to 66cm, and the moving distance is 1cm every time.
We can easily calculate the incident and reflection angle at every point.The sound source is immovable, while the hydrophone and rough interface are movable.We can easily calculate the incident and reflection angle of the material vary from 16° to 73°, which is on the path of the incident and reflected sound waves.After that, we choose 25 points to show our results.The frequency of the transmitted sound wave is from 80kHz to 160kHz, and the frequency variation interval is 1kHz.The parameters of stochastic rough interface after 3D printing material and Geoacoustic parameters are shown in Table 1 and Table 2.

Data processing
In our tank, the intensity of the direct wave attenuates significantly with the propagation distance and the frequency of the transmitted signal.The results of the signal near 100kHz have a large difference, which is very conducive to experimental verification.Therefore, we choose the results of frequency between 85kHz and 105kHz.Figure 3 shows the schematic diagram of incidence and reflection path of sound waves.According to the propagation path of sound waves, when the receiving hydrophone is close to the sound source.In Figure 4, the waveform reflected through the rough interface can be distinguished from the waveform arrived by other means, and the final reflection coefficient will be close to the exact value.However, when the receiving hydrophone is far away from the sound source, the wave reflected by the material overlaps with the waveform of other paths, resulting in large fluctuations in the final result.The distance divide in 30cm from the receiving hydrophone to the sound source.If the distance is more than 30cm, it is not easy to divide the direct and reflection sound waves.The direct wave and the echo reflected by the material are calculated by the difference of sound path.Then we use the voltage changes and convert to the sound pressure value.We obtained the direct wave and the echo reflected wave by calculating the difference of sound path.Finally, we obtain the reflection coefficient and show it Figure 5.

Simulation
Based on the scaling principle, we use the scaled model to give the result.The incident wave frequency is 2125-2675Hz, the total thickness of the two intermediate layers is 0.81m, the height of the interface fluctuation is about 0.00375m, and the correlation length is 0.0375m.There are fluctuations only on the underwater medium, and the medium interface below is smooth without fluctuations.In the layered medium in the tank, the medium is divided into 4 layers.The top layer is water, and the bottom layer is hard concrete medium, and the middle two layers are rough interface after 3D printing material and plexiglass material.In the simulation, we approximate the middle two layers as the same medium, and take the average of the two layers, because the characteristic impedance () difference of the two layers of medium is very small (less than 5%).The simulation environment is shown in Figure 6.There is no loss in layer 1 (in water).
According to the scaling principle, the parameters used in the simulation expand by 40 times compared to the actual parameters, and the results will be the same.The thickness of layer 2 is set to 0.81m.By using the formula derived above, we can calculate the reflection coefficient from layered stochastic rough interface.Experiment was carried out respectively at 85-105kHz, and the results were shown in the Figure 7.
From Figure 7, we can easily find that there is periodic property of reflection coefficient in layered medium.This is completely different from reflection in half space, which decreases monotonically as the grazing angle increases.The solid lines are experimental results, while the dashed lines are simulation results.We can see the variation of the reflection coefficient in layered medium through Figure 7. Then we can get the properties of the seafloor.We can find that there is layered medium with periodic reflection coefficient and find the best incident angle in our future research.

Analysis of results:
In this work, we use the scaled experiment and simulation operation to simulate the layered stochastic rough interface scattering reflection coefficient.We can see that reflection coefficient is coupled with the product of  in this result.If the product on a certain layer, it will be the same, which means that its characteristic impedance is the same.The effect on the reflection coefficient is also the same.According to Olson [8] , in the submarine parameter inversion, the thickness  and the sound velocity ratio  are the easiest one to be retrieved, but it is difficult to determine accurately in the direct calculation simulation, because a small disturbance can lead to a significant change in the results.The acoustic absorption coefficient  is relatively difficult to invert, but is relatively easy to set in our forward calculation, because the error of the absorption coefficient does not significantly affect the result.It lays a foundation for researching the reflection coefficient of random layered rough interface.

Figure 2
is the picture of rough interface and data acquisition card.

Figure 3 .
Figure 3. Schematic diagram of reflection path of sound wave through material.According to the propagation path of sound waves, when the receiving hydrophone is close to the sound source.In Figure4, the waveform reflected through the rough interface can be distinguished from the waveform arrived by other means, and the final reflection coefficient will be close to the exact value.However, when the receiving hydrophone is far away from the sound source, the wave reflected by the material overlaps with the waveform of other paths, resulting in large fluctuations in the final result.The distance divide in 30cm from the receiving hydrophone to the sound source.If the distance is more than 30cm, it is not easy to divide the direct and reflection sound waves.

Figure 4 .
Figure 4. Time domain diagram of sound wave.We can distinguish different waves that receive by hydrophone.When the distance between sound resource and hydrophone is less than 30cm, we can easily distinguish them easily.

Figure 5 .
Figure 5. Reflection coefficient measurement results of rough interface.

Figure 6 .
Figure 6.The simulation environment The loss parameters in the simulation are 0.02dB/m/kHz in layer 2, and 0.01dB/m/kHz in layer 3.There is no loss in layer 1 (in water).According to the scaling principle, the parameters used in the simulation expand by 40 times compared to the actual parameters, and the results will be the same.The thickness of layer 2 is set to 0.81m.By using the formula derived above, we can calculate the reflection coefficient from layered stochastic rough interface.Experiment was carried out respectively at 85-105kHz, and the results were shown in the Figure7.From Figure7, we can easily find that there is periodic property of reflection coefficient in layered medium.This is completely different from reflection in half space, which decreases monotonically as the grazing angle increases.

Figure 7 .
Figure 7.A comparison of simulation and experiment results.From left to right, from top to bottom, it is 85-105kHz for the experiment and 2125-2625Hz for the simulation.The solid lines are experimental results, while the dashed lines are simulation results.We can see the variation of the reflection coefficient in layered medium through Figure7.Then we can get the properties of the seafloor.We can find that there is layered medium with periodic reflection coefficient and find the best incident angle in our future research.

Table 1 .
Acoustic parameters of rough 3D printing interface.

Table 2 .
Geoacoustic parameters in the experiment.