Random medium modeling and underwater acoustic field simulation

The von Karman random medium model is established based on the small scale random medium theory in this paper. Combined with the finite element theory, the numerical simulation of the sound field under this model is realized by using the spectral-element method, and the characteristics of the sound field are compared with those under the uniform medium underwater environment. The results show that the correlation length reflects the variation of the heterogeneous size, and the scattering wave generated by random disturbance may provide a more reasonable explanation for the amplitude and energy anomalies in some trial data.


Introduction
In recent years, with the deep researching on seismic data people found that seismic data also carry the effect of non-deterministic (or statistical) heterogeneity in the earth's interior, that is, self-organization.Random media can characterise the self-organisation of the Earth's internal media, reflecting the smallscale non-homogeneity and statistical features attached to the large-scale background field.This smallscale inhomogeneity is also important in explaining the propagation of seismic coda wave, and smallscale perturbations have a greater influence on the characteristics of the backscattered wave field than the large-scale background field.Therefore, we should use effective medium theory to correctly model the waveforms associated with small-scale inhomogeneities.The structure of such a large number of randomly distributed small-scale anomalies, which cannot be easily described by conventional methods, is conveniently described by statistical methods, such as the random medium model.
Non-homogeneous media can be characterised by the autocorrelation function of the velocity perturbations and the corresponding power spectral density function.This correlation function was first proposed by von Karman [1] to characterise the random velocity field of turbulent media.Since then, it has been frequently used in the statistical literature, in the study of turbulence problems [2] and in the study of stochastic media (wave scattering) [3] .The von Karman function is explicitly defined as belonging to the class of continuous correlation functions.A large amount of work has been carried out by scholars for the study of stochastic medium modeling.Klimes.L [4] summarised and reviewed the most used types of correlation functions and explained the physical significance of their parameters; Anne Obermann [5] et al. proposed the spatial sensitivity problem of coda wave based measurements, and investigated the depth sensitivity of the coda wave to the local velocity perturbation by using the 2D numerical wave field simulation.
The research on non-uniform medium model is mainly concentrated in the field of seismology, with few applications in underwater acoustic engineering.In order to make the applicability of the ocean sound field model closer to the complex and real marine structure, especially the complex seafloor topography and the random change of geoacoustic parameters, it is significant to establish the seafloor non-uniform medium model.

Von Karman type random medium modeling
According to the theory of random media, a small-scale random perturbation in a random inhomogeneous medium can be described by an autocorrelation function.The power spectrum of the random perturbation ()  r is the Fourier transform ()  k of the autocorrelation function ()  r .According to the spectral expansion theory of stochastic processes, the known power spectral function ()  k generates the random power spectral function ()  k , and the random perturbation ()  r described by ()  k can be obtained by performing a Fourier inverse transformation on it.Finally, the non-uniform medium model with the specified mean, variance, and autocorrelation function can be generated through the normalisation process.
According to the von Karman autocorrelation function, the steps to build a 2-D random medium model with zero-mean, specified variance, horizontal correlation length a and vertical correlation length b are shown as Figure 1 [6] :  Comparing Figure 2a and 2c, Figure 2b, and 2d, respectively, the sizes of nonhomogeneous bodies are obviously different, and therefore the correlation lengths reflect the change in the size of the nonhomogeneous bodies.It can also be seen that when the transverse correlation length and the longitudinal correlation length are different, the non-homogeneity exhibits the property of extending like a certain direction.Therefore, when studying the seafloor non-homogeneous medium and modeling the non-homogeneous medium, it is important to choose a random model that satisfies the statistical properties of the seafloor medium.

Numerical simulation of sound field
Based on completing the modeling of von Karman random medium, a seawater-random medium with semi-infinite elastic bottom parametric environment model (the random medium model) is set up as shown in Figure 3a.In the random medium correlation length a=b=10m, h=0.3, In order to analyse the effect of random medium on the underwater acoustic, a homogeneous medium model, shown in Figure 3b, is designed relative to Figure 3a.Parameters of the two models are shown in Table 1.Table 1.Parameters of the two models.In our work, the spectral element method [7] is adapted to complete the numerical simulation of the sound field under the above models.The Ricker wavelet with a center frequency at 30 Hz was employed in the simulation, and the source located at zs = 100 m, the horizontal line array arranged on the seafloor with 20m spacing between array elements.The time domain waveform;

Conclusion
In this paper, a von Karman-type random medium model is established according to the random medium theory, and by comparing the differences in sound field characteristics between the random medium model and the uniform medium model, the following understanding is preliminarily obtained: 1) The correlation length reflects the change in the size of the non-homogeneous body, and when the transverse correlation length and the longitudinal correlation length are different, the non-homogeneity exhibits the property of extending in a certain direction.Therefore, a stochastic model that satisfies the statistical properties of the seafloor media should be selected when modeling non-homogeneous media.
2) Scatter wave generated by random perturbations reduce the reflected energy from the seafloor and are distributed over all moments after the direct wave signal.This may provide a more reasonable explanation for the amplitude and energy anomalies obtained in trial data.

Figure 1 .
Figure 1.Flow chart describing how the 2-D von Karman random medium model are built.K is a constant of proportionality and takes values in the range 0.3-0.8.The two-dimensional von Karman autocorrelation function expressed as: ( ) ( ) ( ) ( ) 1 2

Figure 2 1 Figure 2 .
Figure 2 shows a two-dimensional von Karman-type random medium model (200m × 200m) built according to the above steps.Taking the longitudinal waves of the ocean floor as an example, and the average velocity 0 =2000m/s p v

Figure 3 .
Figure 3. Parametric environment models.(a) The seawater-random medium with semi-infinite elastic bottom parametric environment model; (b) The homogeneous medium model.In the random medium correlation length a=b=10m, h=0.3,

Figure 4 .
Figure 4.The snapshots of the sound field at the time of interested.Scatter wave generated by random perturbations can be clearly seen in Figure 4 and 5. Label (a), (c) and (b), (d) correspond to the random medium model and homogeneous medium model, respectively, in Figure 4 and 5. Between the direct wave and the sea-surface reflection signal, the time-domain waveform exhibits a smooth straight line in the uniform medium model, but the scatter wave in the random medium model, and the subsequent multiple reflections are buried in the scattering wave which is almost indistinguishable (Figure 5c).