Surface consistent finite frequency phase corrections

Static time-delay corrections are frequency independent and ignore velocity variations away from the assumed vertical ray path through the subsurface. There is therefore a clear potential for improvement if the finite frequency nature of wave propagation can be properly accounted for. Such a method is presented here based on the Born approximation, the assumption of surface consistency and the misfit of instantaneous phase. The concept of instantaneous phase lends itself very well for sweep-like signals, hence these are the focus of this study. Analytical sensitivity kernels are derived that accurately predict frequency-dependent phase shifts due to P-wave anomalies in the near surface. They are quick to compute and robust near the source and receivers. An additional correction is presented that re-introduces the nonlinear relation between model perturbation and phase delay, which becomes relevant for stronger velocity anomalies. The phase shift as function of frequency is a slowly varying signal, its computation therefore does not require fine sampling even for broad-band sweeps. The kernels reveal interesting features of the sensitivity of seismic arrivals to the near surface: small anomalies can have a relative large impact resulting from the medium field term that is dominant near the source and receivers. Furthermore, even simple velocity anomalies can produce a distinct frequency-dependent phase behaviour. Unlike statics, the predicted phase corrections are smooth in space. Verification with spectral element simulations shows an excellent match for the predicted phase shifts over the entire seismic frequency band. Applying the phase shift to the reference sweep corrects for wavelet distortion, making the technique akin to surface consistent deconvolution, even though no division in the spectral domain is involved. As long as multiple scattering is mild, surface consistent finite frequency phase corrections outperform traditional statics for moderately large velocity contrasts, also in the case of a free surface with topography. The kernels could also be used to update the velocity model, if one has knowledge of (wave propagation related only) surface consistent phase operators.