Traveltime calculations from frequency‐domain downward‐continuation algorithms

Abstract

We present a new, fast 3D traveltime calculation algorithm that employs existing frequency-domain waveequation downward-continuation software. By modifying such software to solve for a few complex (rather than real) frequencies, we are able to calculate not only the first arrival and the approximately most energetic traveltimes at each depth point but also their corresponding amplitudes. We compute traveltimes by either taking the logarithm of displacements obtained by the oneway wave equation at a frequency or calculating derivatives of displacements numerically. Amplitudes are estimated from absolute value of the displacement at a frequency. By using the one-way downgoing wave equation, we also circumvent generating traveltimes corresponding to near-surface upcoming head waves not often needed in migration. We compare the traveltimes computed by our algorithm with those obtained by picking the most energetic arrivals from finite-difference solutions of the one-way wave equation, and show that our traveltime calculation method yields traveltimes comparable to solutions of the one-way wave equation. We illustrate the accuracy of our traveltime algorithm by migrating the 2D IFP Marmousi and the 3D SEG/EAGE salt models.

The One-Way Wave Equation: A Full-Waveform Tool for Modeling Seismic Body Wave Phenomena

“…These starting models are usually obtained from conventional traveltime tomography and so are limited by the asymptotic ray approximation. However, newer methods such as the so-called strongly damped wave equation can be used to compute the first-arrival traveltimes [108] or one-way wave equations to compute the most energetic traveltimes and amplitudes [54]. In theory, the acoustic wide-angle wave equation should be applicable to acoustic full-waveform inversion (and the narrow-angle wave equation for elastic full-waveform inversion) either as a means of generating a starting model or as an approximate elastic wave extrapolator for the iterative forward and reverse propagation steps.…”

“…As discussed earlier, equation (4) is one such improvement, which involves a replacement of the Taylor series expansion of the dispersion relation by a Padé or rational approximation [e.g., 53]. [54] have shown that higher order extensions of the standard parabolic equation can be implemented effectively and produce surprisingly accurate results for the acoustic case.…”

Analysis of the multi-component pseudo-pure-mode qP-wave inversion in vertical transverse isotropic (VTI) media

“…Unwrapping the phase is achieved by taking the derivative of the wavefield with respect to frequency, dividing by the wavefield and finally taking the imaginary part (Shin et al, 2003;Choi and Alkhalifah, 2013). The traveltime sensitivity kernel for a displacement component i and anisotropy parameter p is given by,…”

Unwrapped phase inversion for near surface seismic data

“…To overcome the phase wrapping problem, Shin et al (2003) used the derivative wavefield with respect to the angular frequency. Taking the derivative of the wavefield with respect to the angular frequency makes the phase information appear in the amplitude component as well.…”

“…Because the amplitude component is not limited in range, getting the phase information from the amplitude component overcomes the phase-wrapping problem even at high frequencies. Shin et al (2003) used this algorithm to calculate the first arrival traveltime from the forwardmodeled wavefield in the frequency-domain.…”