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Summary Spherical Wave Reflection Coefficient (SWRC), a more precise description of seismic wave propagating in subsurface, is the superposition of plane wave reflection coefficient (PWRC). In this paper, frequency-dependent SWRC is investigated in two semi-infinite acoustic media with a planar interface. The frequency-dependent phenomenon occurs in the case of low-frequency and near-field. In general, the lower the frequency (or the near the propagation distance), the larger the difference between SWRC and PWRC. The amplitude and phase of SWRC reposefully approach to that of PWRC at pre-critical angles and oscillate around the value of PWRC at post-critical angles. Moreover, we project the SWRC on complex plane and find an elliptical distribution at pre-critical angles. Curves fitting using different frequency band illustrates that high-frequency information is redundant and low frequency information is indispensable from the perspective of geometry, especially for the estimation of density parameter.

Full waveform inversion often suffers from the cycle-skipping problem, mainly because of the absence of low frequencies in seismograms, especially for elastic media. To address this issue, we present an elastic reflection full waveform inversion (ERFWI) methodology. However, for ERFWI, high nonlinearity still exists in data residuals related misfit function, when true amplitude migration is not adopted. To further relax the cycle skipping and to meet the requirements of true amplitude migration, we develop a traveltime-based ERFWI method to update the low-wavenumber components of P- and S-velocity models. The traveltime-based ERFWI only eliminates traveltime residuals along the wave-path of “rabbit-ear” to extract the long-wavelength background of the middle and deep parts. Because the traveltime of reflected waves is described correctly, the inversion result using the traveltime-based ERFWI method can be used as a velocity model for prestack depth migration or as an initial model for the conventional FWI to obtain high-resolution result. Applications on synthetic data tests show that our traveltime-based ERFWI can efficiently update the low-wavenumber component of the model, and the combined (traveltime-based ERFWI FWI) method can obtain improved result compared with regular FWI alone.