For shallow earthquakes, the surface-reflected depth phases (pP and sP) arrive shortly after the primary arrival, and the time separation among the three phases can be used to determine the origin depth of the earthquake. To model the relative arrival times and amplitudes of these phases, and the core reflections and water-column reverberations for a given earthquake, we construct stick seismograms using the IASPEI91 velocity model and the Harvard CMT focal mechanisms at the distances and azimuths of the recording seismometers. While the differing arrival times and amplitudes are features observable in the time series, they also affect the spectrum, and we compute the spectrum for a time window that includes the P wave and subsequent arrivals. We quantify the effects of variations in these properties over the focal sphere in terms of differences in the slope of the log spectrum at different stations. To determine the depth of an earthquake, we compare our observed spectral variations with the predicted spectral variations for earthquakes originating at depths within 30 km of the pde depth and identify the depth with the smallest L1 misfit as the true earthquake depth. We demonstrate the effectiveness of this method by applying it to a group of 35 thrust earthquakes in the Aleutian arc near the Andreanof Islands, but we also describe some complications introduced by strongly directive ruptures, as illustrated for the 1995 Jalisco, Mexico, event.
Online material: Observed and predicted variations in pulse width for Aleutian Island earthquakes.