On the onset ofPPand other mini-max phases

Abstract

S U M M A R Y Theory predicts that pulses that propagate on stationary non-minimum time paths, such as the P P path, undergo phase shifts such that the onset is emergent and the observed pulse is non-causal relative to the arrival time predicted by ray theory. The phase shift is n / 2 radians, at least at high frequencies, so that the observed pulses should approximate to the Hilbert transform of the pulse recorded over a minimum time path. Although onsets of phases that follow non-minimum time paths (usually called mini-max phases) are predicted to be emergent, those observed often seem clear and no more difficult to read than those of arrivals that follow minimum time paths. An example of a P P seismogram which is clearly emergent is shown here. Simulations obtained by Hilbert transforming the impulse responses of conventional long-period and short-period seismographs convolved with attenuation operators, show that the first half-cycle of mini-max phases which should have the emergent onset will usually be of low amplitude relative to the second half-cycle. This suggests that on observed seismograms the first half-cycle of mini-max phases may be obscured by earlier arrivals. The second half-cycle will then be taken as the first motion and could appear to have a well-defined onset. The effects of phase shifts that result in emergent onsets can be corrected for to some extent, and ideally onset times of mini-max phases would be read from such corrected records. However, onset times of P P read directly from conventional seismograms are reported in bulletins and these times have been assumed by some to be reliable enough to be used to determine anomalies in wavespeed in the upper mantle in the vicinity of the reflection point. Perhaps surprisingly the assumption appears to be justified in that the apparent onset time of a mini-max phase on a conventional short-period seismograph may be little different from the onset time read after correction for the n / 2 phase shifts.Key words: Hilbert transform, mini-max phases, P P onsets.By considering the radiation from a point disturbance in a uniform compressible liquid sphere, Jeffreys & Lapwood (1957) show that P P travels by a stationary non-minimum time path (commonly referred to as a mini-max path),

Northern Chile earthquake of 1980 May 26: evidence of unilateral fracture

“…51) now shows a prominent pulse at the onset. Douglas, Sheehan & Stewart (1992) show that this pulse can be derived from the complex WRA P by applying an attenuation operator with a t* of l.Os, where t* is the ratio of traveltime to specific quality factor. The surface reflection p P P is weak at WRA.…”