Imaging the Earth Using Time-Domain Surface-Wave Measurements: Evaluation and Correction of the Finite-Frequency Phase Shift

Surface waves propagating from earthquakes, active sources or within the ambient noise wavefield are widely used to image Earth structure at various scales, from centimeters to hundreds of kilometers. The accuracy of surface-wave, phase-velocity measurements is essential for the accuracy of the Earth models they constrain. Here, we identify a finite-frequency phase shift in the phase travel time that causes systematic errors in time-domain, phase-velocity measurements. The phase shift arises from the approximation of monochromatic surface waves with narrow-band filtered surface waves. We derive an explicit formula of the finite-frequency phase shift and present a numerical method for its evaluation and for the correction of the measurements. Applications to high-frequency and long-period examples show that the phase shift is typically around pi/60-pi/16 for the common settings of ambient-noise imaging studies, which translates to 0.2%-0.8% phase-velocity measurement errors. The finite-frequency phase shift depends on the (a) second derivative of the wavenumber with respect to frequency; (b) width of the narrow-band filter; (c) epicentral or interstation distance; and (d) center frequency of the filter. In conversion to phase velocity, the last two factors cancel out. Frequency-domain methods for phase-velocity measurements have the advantage of not producing the finite-frequency phase shift. Both time- and frequency-domain measurements, however, can be impacted by a break-down of the far-field approximation (near-field phase shift), which our calculations also show. Our method offers an effective means of improving the accuracy of the widely used time-domain, phase-velocity measurements via the evaluation of and corrections for the finite-frequency phase shift.