Derivation and application of extended parabolic wave theories. I. The factorized Helmholtz equation

Abstract

The reduced scalar Helmholtz equation for a transversely inhomogeneous half-space supplemented with an outgoing radiation condition and an appropriate boundary condition on the initial-value plane defines a direct acoustic propagation model. This elliptic formulation admits a factorization and is subsequently equivalent to a first-order Weyl pseudodifferential equation which is recognized as an extended parabolic propagation model. Perturbation treatments of the appropriate Weyl composition equation result in a systematic development of approximate wave theories which extend the narrow-angle, weak-inhomogeneity, and weak-gradient ordinary parabolic (Schrödinger) approximation. The analysis further provides for the formulation and exact solution of a multidimensional nonlinear inverse problem appropriate for ocean acoustic and seismic studies. The wave theories foreshadow computational algorithms, the inclusion of range-dependent effects, and the extension to (1) the vector formulation appropriate for elastic media and (2) the bilinear formulation appropriate for acoustic field coherence.

The One-Way Wave Equation: A Full-Waveform Tool for Modeling Seismic Body Wave Phenomena

“…However, the left factor serves as a 'corrector' term by annihilating this error. Therefore, an exact plane-wave solution to equation (26) for forward propagating waves is given byũ(x 1 ) = G exp [iωP 1 x 1 ]c, where the three-vector c represents the initial amplitudes of all three forward propagating body-waves at x 1 = 0.…”

“…This 6×6 matrix representation is referred to as a 'displacement-stress' formulation and is convenient when dealing with boundary-value (i.e., reflection/transmission) problems [e.g., 15; 64]. The formulation of [63] uses an operator-splitting approach similar to that of [26] and was subsequently reduced to the acoustic wave equation [65; 66]. In this approach, the forward and reverse wave coupling is approximated using an iterative process based on a generalization of the Born method (which they refer to as a Bremmer or Neumann series summation).…”

“…However, this operator approach makes the explicit assumption of weak heterogeneity to obtain a 'simple' expression for the operator-root in terms of a perturbation series. A significant step forward was realized by [26] who applied a similar operatorsplitting approach to that of [55], but sought a more detailed and widely applicable expression for the operator-root. This permits a generalization of the one-way wave equation technique and introduces a new class of propagation algorithms.…”

Phase Space Methods and Path Integration: A Microscopic Approach to Direct and Inverse Wave Propagation.

“…While functions of a finite set of comuting self-adjoint operators can be defined through spectral theory, functions of noncomouting operators are represented by pseudo-differential operators [2,5]. The formal wave equation (1) is now written explicitly as a Weyl pseudo-differential equation In the form…”

“…Consistent with taking the square root of the indefinite Helmholtz operator, the corresponding symbols, generally, have both real and imaginary parts characterized by oscillatory behavior (4,6], as illustrated in Figure 2. Nonuniform and uniform perturbation solutions corresponding to definite physical limits (frequency, propagation angle, field strength, field gradient) recover several known approximate wave theories (ordinary parabolic, rangerefraction parabolic, Grandvuillemin-extended parabolic, half-space Born, Thomson-Chapman, rational linear) and systematically lead to several new full-wave, wide-angle approximations [2][3][4]6]. …”