Toward broadband propagation

The goal of using phase-rays as a coordinate system for wavefield extrapolation is to enable a more accurate propagation of broadband wavefields through the subsurface. However, this requires extrapolating wavefields at many frequencies. One interesting observation is that phase-rays exhibit frequency-dependent behavior. Thus, one strategy for wavefield extrapolation is to use phase-ray coordinates that adapt to frequency-dependent illumination.

The proposed approach for frequency-dependent wavefield extrapolation is as follows: i) an initial wavefield extrapolation in Cartesian or polar coordinate at the lowest frequency; ii) phase-ray tracing using the current wavefield solution; iii) extrapolating the next wavefield using the traced phase-rays as the new coordinate system; iv) a mapping of the latest wavefield result from phase-ray to Cartesian coordinates; and v) repeating steps ii), iii) and iv) until wavefields at all frequencies are calculated. The broadband wavefield is then obtained by a summation of wavefields over all extrapolated frequencies. Alternatively, because frequency-dependent wavefield illumination usually varies slowly over individual frequency steps, phase-rays could be traced only periodically to save computational cost.