Accuracy and efficiency considerations for wide‐angle wavefield extrapolators and scattering operators

SUMMARY Several observations are made concerning the numerical implementation of wide‐angle one‐way wave equations, using for illustration scalar waves obeying the Helmholtz equation in two space dimensions. This simple case permits clear identification of a sequence of physically motivated approximations of use when the mathematically exact pseudo‐differential operator (PSDO) one‐way method is applied. As intuition suggests, these approximations largely depend on the medium gradients in the direction transverse to the main propagation direction. A key point is that narrow‐angle approximations are to be avoided in the interests of accuracy. Another key consideration stems from the fact that the so‐called ‘standard‐ordering’ PSDO indicates how lateral interpolation of the velocity structure can significantly reduce computational costs associated with the Fourier or plane‐wave synthesis lying at the heart of the calculations. A third important point is that the PSDO theory shows what approximations are necessary in order to generate an exponential one‐way propagator for the laterally varying case, representing the intuitive extension of classical integral‐transform solutions for a laterally homogeneous medium. This exponential propagator permits larger forward stepsizes. Numerical comparisons with Helmholtz (i.e. full) wave‐equation finite‐difference solutions are presented for various canonical problems. These include propagation along an interfacial gradient, the effects of a compact inclusion and the formation of extended transmitted and backscattered wave trains by model roughness. The ideas extend to the 3‐D, generally anisotropic case and to multiple scattering by invariant embedding. It is concluded that the method is very competitive, striking a new balance between simplifying approximations and computational labour. Complicated wave‐scattering effects are retained without the need for expensive global solutions, providing a robust and flexible modelling tool.