The inverse split-step parabolic equation method is derived. Spatial localization of single and multiple low-frequency harmonic point sources in a deep ocean is demonstrated by using simulated pressure field measurements from a vertical array. In addition, the method is applied to a source extended in range and depth. This method is solved in a single iteration, unlike some inverse techniques that vary parameters.

The coherent pressure field of an acoustic harmonic point source is gradually transformed into an incoherent pressure field as a function of propagation distance through the oceanic internal wave field. A receiver located in the weak scattering region would measure a fluctuating acoustic pressure with a Gaussianly distributed logarithmic amplitude and phase. The consequence is a biased estimate for the conventional maximum likelihood estimate of pressure or intensity and subsequently a biased detection criteria. It is proposed that detection performance may be improved in this case by logarithmically transforming the pressure prior to detection processing. An analysis and discussion will be presented using the Garrett and Munk internal wave field model. [Work supported by NUSC IR/IED.]

The existing theory of acoustic propagation through an oceanic internal wave field with a Garrett and Munk spectrum is modified and, by numerical computation, is shown to be consistent. The fractional sound speed computation is rederived to satisfy the Garrett and Munk spectrum and used to compute a stochastic simulation of the internal wave sound speed fluctuation field. The Garrett and Munk spectrum in (ω, j) space has been normalized by 4π, and the acoustic scattering is redefined to accommodate scattering from the internal wave phase fronts as in an acoustic phase grating. These modifications are then used to compute the coherent acoustic intensity by two methods: a first-order multiple scatter approximation and a stochastic simulation. Also, the Rytov approximation is shown to be equivalent to the first-order multiple scatter approximation in the form of the stochastic parabolic equation method in the unsaturated region. The computational results show agreement in the weak scattering region using typical deep ocean values. The stochastic simulation method is accurate in the saturated and unsaturated regions; however, the method requires long computer execution times. Phase front fragments propagating along rays with sound speeds reduced by the stochastic internal wave field are used to discuss the computational results.

The Garrett and Munk internal wave field is a second-order process. It is also stationary, homogeneous, and anisotropic. Acoustic fields propagating through the internal waves are scattered by the internal wave sound speed fluctuations. The differential scattering cross section is derived employing the Born approximation and the “n frequency shell” condition. For comparison, the internal wave sound speed fluctuation is approximated by an anisotropic Gaussian correlation function and the resulting differential scattering cross section is evaluated for typical horizontal and vertical correlation lengths. Both differential scattering cross sections are compared as a function of scattering angle for selected angles of incidence and acoustic frequencies.

The canonical deep ocean sound channel is parametrized by the sound-speed minimum, buoyancy frequency scale, and adiabatic fractional sound-speed gradient. Variations in the sound-speed parameters depend on spatial and temporal variations in temperature and salinity. These variations are described by probability density functions. The sound channel near the sound-speed minimum is approximated by a Hirsch sound-speed profile, and the low-frequency acoustic field is derived. Analytic field density functions for received frequency, intensity, wave vector, and arrival time are derived by the application of the transformation of random variables, assuming the sound-speed parameters are uncorrelated Gaussian random variables. The resulting field density functions are non Gaussian. Estimates for the sound-speed parameters and associated variances are used to evaluate the field density functions.

The acoustic field of the fluctuating parabolic waveguide was expressed as an eigenmode expansion of the Green function. The field was assumed to be negligibly influenced by the boundary conditions and the source/observer locations were on the waveguide axis. Variations in pressure, arrival time, and angle for each mode, m, and source frequency, ω, were examined for fluctuations in axis velocity, C0, and waveguide curvature parameter, a. The results showed the measurable quantities to be approximately ten orders of magnitude more sensitive to fluctuations in the waveguide curvature parameter than to fluctuations in axis velocity. The arrival time asymptotic behavior was tA/t0 ∼ O(exp(−ωm2/4ω2)) for large source frequencies, ω ≫ ωm = (2m + 1)aC02, and fluctuations in the waveguide curvature parameter resulted in negligible variation in arrival time. For source frequencies near the mode-dependent cutoff frequency, ωm, arrival times showed extreme sensitivity to fluctuations in the waveguide curvature parameter. Mode-dependent arrival angles were least sensitive to fluctuations in the curvature parameter near grazing angles of π/4 radians. Fluctuations in pressure exhibited similar sensitivity to the waveguide parameters and intrinsic high-order moments.