The derivative of a waveguide acoustic field with respect to a three-dimensional sound speed perturbation

Abstract

Semianalytic expressions are derived for the first-order derivative of a pressure field in a laterally homogeneous waveguide, with respect to an arbitrary three-dimensional refractive index perturbation in either the water column or ocean bottom. These expressions for the ''environmental derivative,'' derived using an adjoint method, require a three-dimensional spatial correlation between two Green's functions, weighted by an environmental parameter basis function, with the Green's functions expressed in terms of normal modes. When a particular set of orthogonal spatial basis functions is chosen, the three-dimensional spatial integral can be converted into a set of one-dimensional integrations over depth and azimuth. The use of the orthogonal basis permits environmental derivatives to be computed for an arbitrary sound-speed perturbation. To illustrate the formulas, a simple sensitivity study is presented that explores under what circumstances three-dimensional plane-wave and cylindrical perturbations produce non-negligible horizontal refraction effects, for a fixed source/receiver geometry. Other potential applications of these formulas include benchmarking three-dimensional propagation codes, and computing Cramer-Rao bounds for three-dimensional environmental parameter estimates, including internal wave components.

Optimal nonlocal boundary control of the wide-angle parabolic equation for inversion of a waveguide acoustic field

“…10 and the perturbative Green's function method in Ref. 11, the presented optimal boundary control approach belongs to the category of variational methods. In fact, the adjoint expressions ͑Born approximation͒ applied in Refs.…”

“…To solve this boundary controllability problem the continuous adjoint of a Claerbout wide-angle PE system is derived analytically. As an adjoint-based optimization the proposed method is complementary to the adjoint approaches that were introduced recently in ocean acoustic tomography 10,11 and in inverse scattering. 12,13 The latter 13 contains an in-depth comparison of three different adjoint formulations for computing Fréchet derivatives, the convena͒ tional adjoint method, the Lagrange multiplier method and an integral equation formulation which is based on the Lippmann-Schwinger integral equation approach.…”

Information and linearity of time-domain complex demodulated amplitude and phase data in shallow water

“…In contrast to sensitivity kernel analysis, this global optimization procedure does not generally provide a physical picture of the dependence of the observations on perturbations to the medium. More closely related to the sensitivity analysis is the adjoint method, 16,17 which gives the gradients for an iterative linear descent algorithm to fit the observations by perturbing the medium. The gradient for a single observation is equivalent to the sensitivity kernel, but to date, authors have generally focused on estimating the medium, and considered all observations together, limiting physical interpretation of the results, although still requiring linearity to be valid.…”

A variational data assimilation system for the range dependent acoustic model using the representer method: Theoretical derivations

“…The use of variational methods for solving inverse problems in ocean acoustic propagation modeled by the parabolic equation (PE) was investigated in recent years. Elisseeff et al (2002) and Hursky et al (2004) applied adjoint modeling to an acoustic tomography inversion problem; Hermand et al (2006) used an adjoint of a similar PE in a geoacoustic inversion problem involving uncertainties in the sound speed; Badran et al (2008) used adjoint modeling in another geoacoustic inversion for the seabed characterization; Le Gac et al (2004) used a variational approach for geoacoustic inversion using adjoint modeling of a PE approximation model with nonlocal impedance boundary conditions; Thode (2004) and Thode and Kim (2004) used the adjoint model to compute the derivatives of a waveguide field with respect to several parameters, including the sound speed, density, and frequency; Charpentier and Roux (2004) used the adjoint method for the inversion of mode and wavenumber in shallow waters; Meyer and Hermand (2005) used the adjoint method with an optimal control method of nonlocal boundaries applied to the wide-angle PE for inversion of the acoustic field and bottom properties; Li et al (2014) used a variational method for the inversion of an internal waveperturbed sound-speed field via acoustic data assimilation, in the presence of acoustic pressure and sound-speed observations.…”