## 2 Formulation of the Statistical Problem and Some Analytical Results  
In a cylindrical coordinate system (\(r\),\(z\)), we consider a marine waveguide with horizontal boundaries consisting of a water layer and a bottom layer of liquid sediments. Suppose that in the water column with a constant density \(\rho_{0}\), the average (regular) sound velocity \(c_{0}(z)\) experiences weak two-dimensional fluctuations \(\delta c(r\),\(z)=c(r\),\(z)-c_{0}(z)\), \(|\delta c/c_{0}|<<1\). The layer of liquid sediments is homogeneous with constant values of density, sound velocity, and absorption: \(\rho_{1}\), \(c_{1}\), \(\beta_{1}\). The random sound velocity field \(\varepsilon(r\),\(z)=-2\delta c(r\),\(z)/c_{0}\) is characterized by an average zero value \(\langle\varepsilon(r\),\(z)\rangle=0\), and is described by some anisotropic correlation function \(B_{\varepsilon}(r_{1}-r_{2}\),\(z_{1}-z_{2})\equiv\langle\varepsilon(r_{1},z_{1})c(r_{2}\),\(z_{2})\rangle\) (hereinafter, the angle brackets mean averaging over the ensemble of realizations). This function has amplitude \(B_{\varepsilon}(0,0)=\sigma_{\varepsilon}{}^{2}\) (dimensionless intensity of fluctuations) and is characterized by scales of spatial correlation in the horizontal direction \(L_{r}\) and along the depth \(L_{z}\). The specific form of the correlation function does not limit the generality of the research findings, but is usually dictated by the convenience of analytical evaluations [23; 24; 25]. The pressure field \(p(r\),\(z)\) and components of velocity \(\{u,w\}\) of a point source with a frequency \(\omega\), located within a waveguide at \(r=0\), \(z=z_{0}\), satisfy linear equations of acoustics with random coefficients:  
\[ \begin{array}{c}\frac{\partial}{\partial z}p(r,r_{0},z_{0},z_{0})-i\omega \rho(r,z)w(r,r_{0},z,z_{0})=0,\\ \frac{\partial}{\partial r}p(r,r_{0},z,z_{0})-i\omega\rho(r,z)u(r,r_{0},z,z_ {0})=0,\ \ \ \rho[\frac{\partial}{\partial z}w+[\frac{\partial}{ \partial r}+\frac{1}{r}]u-\frac{i\omega\rho}{c^{2}(rz)}]=\frac{i\delta(r-r_{0} )}{\omega r}\delta(z-z_{0}).\end{array}  \tag{1}\]Variations in the density of the medium \(\rho\) involved in the Equations (1) for acoustic frequencies above 1 Hz do not affect the sound propagation [4; 5], therefore in underwater acoustics it is usually assumed in the water layer that \(\rho=\rho_{0}\), and in the bottom we will also assume \(\rho=\rho_{1}\). The pressure field \(p(r,z)\) in the wave zone can be represented by decomposition into local modes of an irregular waveguide (density in the water \(\rho=\rho_{0}=1\) kg/m\({}^{3}\)):  
\[p(r,z)=\sum\limits_{m}G_{m}(r)\varphi_{m}(r,z);\frac{\partial^{2}}{\partial z^{ 2}}\varphi_{m}(r,z)+\left[k^{2}(r,z)-\kappa_{m}^{2}(r)\right]\varphi_{m}(r,z)=0\] (1 \[{}^{\prime}\] )  
Eigenfunctions \(\varphi_{m}\) at the surface (\(z=H\)) and at the bottom (\(z=0\)) of an ocean satisfy the following boundary conditions: \(\varphi_{m}(r,H)=0\), \(\varphi_{m}(r,0)+g_{m}(r)\varphi^{\prime}{}_{m}(r,0)=0\), where \(g_{m}(r)\) characterizes the impedance of the penetrable bottom, and the square of the wave number \(k^{2}(r,z)=k_{0}{}^{2}(z)[1+\varepsilon(r,z)]\) is a random function due to fluctuations in the speed of sound \(\varepsilon(r,z)\), \(k_{0}\)=\(\omega/c_{0}(z)\). In [27; 28; 29], it was shown that in an irregular waveguide ignoring the backscattered field (that is, in the one-way propagation or forward scattering approximation), the amplitudes of modes \(G_{m}(r)\) satisfy the following quadrature representation:  
\[G_{m}(r)=A_{m}(r)exp\left\{\int\limits_{0}^{r}\left[i\kappa_{m}(\xi)-(2a_{m})^ {-1}\sum\limits_{n}a_{n}[V_{mn}(\xi)(\kappa_{m}(\xi)/\kappa_{n}(\xi))-V_{mn}( \xi)]\right]dz\right\} \tag{2}\]  
where \(\kappa_{m}\)\(r>>1\), \(a_{m}=\varphi_{m}(0,z_{0})/2\), \(A_{m}=ia_{m}[2\pi i\kappa_{m}(r)r]^{-1/2}\), \(m=1,2\)\(\ldots\).