Two algorithms to stabilize multidirectional Poynting vectors for calculating angle gathers from reverse time migration

Abstract

Angle-domain common-image gathers (ADCIGs) obtained from reverse time migration are important for velocity and reflectivity inversion. Using the Poynting vector (PV) is an efficient way to calculate ADCIGs, but it suffers from inaccuracy and instability. A well-known reason is that a PV can give only one direction per grid point per time step, and thus it cannot calculate the individual directions of overlapping wavefields. This problem can be addressed by using a multidirectional PV (MPV), which decomposes the wavefields into several “approximate” directions and then calculates PVs for each decomposed wavefield. However, the MPV still suffers from another instability problem. The PV is the product of the time and space derivatives of the wavefield, and so it will be zero when the magnitude of the wavefield is at a local peak, which means that the directions are undefined. This leads to unstable points when the wavefields are close to a local magnitude peak, and it thus reduces the quality of the ADCIGs. We have developed two methods to stabilize the MPVs. The first method makes use of the property that the seismic wavelet has a short time duration, during which the propagation direction is stable. Thus, for each point in a decomposed wavefield, a time shift is used to locate the optimal PV during a short time duration, and the optimal location coincides with the local maximum magnitude of the time derivative. Therefore, there is a time shift between the wavefield and its corresponding PV. The second method combines the existing optical flow (OF) with the multidirectional scheme to produce a multidirectional OF (MOF). The MOF is iterative, and thus it has greater computational complexity. Numerical examples show that the time-shift MPV and MOF give more accurate ADCIGs than those using MPV only.