Local accuracy of cross-term corrections of three-dimensional parabolic-equation models

This paper deals with accuracy assessment for cross-derivative (depth and azimuth) term corrections to a three-dimensional (3D) parabolic equation (PE). The focus is on local errors, involving comparison of the two sides of the PE at insertion of (scaled) Helmholtz-equation reference solutions. For media with a particular type of lateral sound-speed variation, mode expansion together with wavenumber integration to compute modal expansion coefficients produces very accurate Helmholtz solutions for the field and its spatial derivatives. There are explicit expressions for the wavenumber integrands in terms of Airy and exponential functions, and accuracy improvements by PE cross-derivative terms are easy to assess. For a relevant example, with 3D effects similar to those in a 3D Acoustical Society of America wedge benchmark, inclusion of a leading-order as well as an additional, higher-order, cross-derivative term in the PE is favorable. The additional term provides a fourth-order accurate approximation of the PE square-root operator. A fifth-order accurate Padé approximation yields further improvement, approaching the numerical accuracy limit set by the PE method itself. The adiabatic approximation is exact for the particular media under study, but the local PE errors are similar for related 3D wedge examples with mode coupling.