Hamiltonian perturbation theory for acoustic rays in a range-dependent sound channel

Abstract

A perturbation theory for acoustic ray propagation is presented which is based on the analogy between Hamilton’s Principle of Least Action (independent variable: time) and Fermat’s Principle of Least Time (independent variable: space coordinate). In a vertical ocean section with x positive to the right and z positive upward, the Lagrangian variables z and ż (a dot denotes differentiation with respect to x; ż=dz/dx) are first replaced by the canonical variables (z, p) of the corresponding Hamiltonian formulation. For the zeroth-order case, where the sound c(x,z) reduces to the form c0(z), a canonical transformation, z, p → ω0, J0, can be made. The new variables are the analogs of angles-action variables in dynamics, and are constants. The generator for this transformation is just the Eikonal function for zeroth order and, hence, is recognized as the travel time between a source and receiver of sound. Finally, when the x-dependent perturbations are turned on, ω0 and J0 cease to be constants and a second, infinitesimal, canonical transformation, ω0, J0 → ω, J, is made. For range-dependent sound channels, ω=ω(x) and J=J(x). The canonical equations for these variables are exhibited, and how they may be solved by standard perturbation theory techniques is shown. Finally, some examples of the theory as applied to perturbations which vary slowly in x or, alternatively, which vary rapidly in x compared with the basic variation in the range-independent channel are given. The various perturbations to travel time are included in these calculations.

Ray Tracing On A Heterogeneous Sphere By Lie Series

“…The Lie series approach enables us to estimate any quantities-traveltime, geometrical spreading, polarization, etc.-by expanding its perturbation with respect to the parameter E. The extra work one has to accomplish is computing the Lie derivative of this quantity (Miller 1986). We shall concentrate on the two ingredients of synthesizing seismograms: traveltime and geometrical spreading.…”

“…We extend and complete the work of Virieux (1989). These techniques have recently been applied in the problem of oceanographic tomography (Miller 1986;Wunsch 1987). Because the J .…”

Acoustic tomography by Hamiltonian methods including the adiabatic approximation

“…The procedure of computing ri perturbations as averages over unperturbed paths is thus seen as a special limiting case of a general procedure. Miller [1986] treats this case systematically using Lie series I- Nayfeh, 1973;Lichtenberg and Lieberrnan, 1983], and we will therefore not pursue it further here.…”