Higher Approximations to Study Statistical Characteristics of Waves in Multiscale Inhomogeneous Media

Abstract

The previously obtained integral field representation in the form of double weighted Fourier transform (DWFT) describes effects of inhomogeneities with different scales. The first DWFT approximation describing the first-order effects does not account for incident wave distortions. However, in inhomogeneous media the multiscale second-order effects can also take place when largescale inhomogeneities distort the field structure of the wave incident on small-scale inhomogeneities. The paper presents the results of the use of DWFT to derive formulas for wave statistical moments with respect to the first-and second-order effects. It is shown that, for narrow-band signals, the second-order effects do not have a significant influence on the frequency correlation. We can neglect the contribution of the second-order effects to the spatial intensity correlation when thickness of the inhomogeneous layer is small, but these effects become noticeable as the layer thickness increases. Accounting for the second-order effects enabled us to get a spatial intensity correlation function, which at large distances goes to the results obtained earlier by the path integral method. This proves that the incident wave distortion effects act on the intensity fluctuations of a wave propagating in a multiscale randomly inhomogeneous medium.

Some Possibilities of Spatial Signal Processing in an Inhomogeneous Medium Based on DWFT

“…Our 3-D space can be considered as a surface in this expanded space. Kravtsov and Tinin (2000), Tinin and Kravtsov (2008), Kravtsov et al (2011), Tinin (2018a this surface select so that eikonals of partial waves are independent of the coordinates ρ,ρ 0 . This yields…”

“…Since the phase screen approximation describes strong amplitude fluctuations, the DWFT approximation can be used to describe wave propagation under such conditions. Moreover, the consideration of the second eikonal approximation allows us to use 6 for describing intensity fluctuations at large distances (Tinin, , 2018a, which has been carried out by the path integral method before (Klyatskin, 1980). Thus, the integral representation of the field as DWFT 6 has a combination of the following properties (Tinin, 2018b):…”

“…Note that in partial waves in these representations there remains the dependence on source and observer coordinates. Kravtsov and Tinin (2000), Tinin and Kravtsov (2008), Kravtsov et al (2011), Tinin (2018a, instead of finding the solution of wave equation 2 in the small-angle approximation for the field, find the solution for V…”

“…holds, we can compute integrals in 6 using the stationary phase method and derive the GO approximation (Tinin, 2018a;Tinin & Kravtsov, 2008):…”

“…Using DWFT wavefield expression 6, obtain for the two-frequency two-position coherence function (Tinin, 2017(Tinin, , 2018a(Tinin, , 2018b Figure 5. Increments of phase (a) and amplitude (b) after spatial processing 21 when a Gaussian irregularity is at the origin for z b = 0m (blue line), z b = 20km (red line), and z b = 50km (green line).…”