Imaging the Earth using time-domain surface-wave measurements:
Evaluation and correction of the finite-frequency phase shift
Abstract
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 π/60-π/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 (1)
second derivative of the wavenumber with respect to frequency; (2) width
of the narrow-band filter; (3) epicentral or interstation distance; (4)
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.