The dynamically correct Poynting vector formulation for acoustic media with application in calculating multidirectional propagation vectors to produce angle gathers from reverse time migration

Abstract

The Poynting vector (PV) has been widely used to calculate propagation vectors of a pressure field (PF) in acoustic media. The most widely used acoustic PV formula is the negative of a product of the time and space derivatives. These two derivatives result in a phase shift between the PF and its PV; particularly, for a PF at a local magnitude peak, the PV modulus is zero and thus the propagation direction there is undefined. This “zero-modulus” issue is not consistent with the physical definition of the PV, which is the directional energy flux density of a PF because this definition indicates that the variation of the PV modulus should be consistent with the PF magnitude. This PV is only considered as kinematically correct and defined as K-PV. We derive the dynamically correct PV (D-PV) formula for acoustic media, which is the negative of the product of the reciprocal of the density, the PF itself, and a factor that is obtained by applying a time integration and a space derivative to the PF. There are two derivations. One uses the slowness vector, and the other is by simplifying the elastic PV. This D-PV does not suffer from the zero-modulus problem, and we also use it to update the multidirectional PV (MPV), which produces a D-MPV. Two strategies are provided to reduce the computational complexity of the time integration in the D-PV formula. Because the MPV already involves Fourier transforms between the time and frequency domains (which facilitates implementation of the time integration), its updated version causes only a very minor increase in the computational complexity of the original one. Numerical examples indicate that the D-PV provides more reliable propagation vectors than the K-PV, and the D-MPV provides more accurate angle-domain common-image gathers from reverse time migration of acoustic media than the K-MPV.