Calculation of the stable Poynting vector using the first-order temporal derivative of the seismic wavefield

Abstract

The Poynting vector is a powerful tool for calculating the propagation directions of a seismic wavefield, and it has a wide range of applications in reverse time migration. However, an instability issue commonly arises while calculating the Poynting vector. The Poynting vector is a product of the temporal and spatial derivatives of the wavefield. The two derivatives are equal to zero at the local extrema of the seismic wavefield, so the Poynting vector cannot provide the propagation directions at these points. Stabilizing techniques, such as smoothing, optical flow (OF) methods, and time-shifting methods, can be applied to address this issue. However, each of these three types of methods comes with trade-offs. Smoothing is easy to implement but has a low angular resolution, whereas the OF and time-shift techniques have high angular resolutions but are computationally inefficient. We have developed a new method that achieves high resolution and high computational efficiency. Based on the fact that a seismic wavefield and its first-order temporal derivative have the same direction of propagation, and that the unstable points of their Poynting vectors are at different locations, we use the first-order temporal derivative of the seismic wavefield to stabilize its Poynting vector calculation. Our method is nearly as accurate as the OF and time-shift techniques and is more computationally efficient than the smoothing technique. Finally, we use numerical simulations to verify the effectiveness of our method.