Geophys. J . Inr. (1 992) 108, 394-398

RESEARCH N O T E

On the onset of PP and other mini-max phases

A. Douglas,’ A. F. Sheehan2* and R. C. Stewart’? 1 MOD(PE), Blacknest, Brimpton, Reading, RG7 4RS, UK

Department of Geology, University of Reading, Reading, RG6 2AB, U K

Accepted 1991 June 24. Received 1991 June 12; in original form 1990 December 5

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), whereas p P , which like PP is a reflection from the free

* Now at: Department of Earth, Atmospheric & Planetary Sciences, Massachusetts Institute of Technology, Cambridge; Massachusetts 02139, USA. t Now at: Camborne School of Mines, Geothermal Energy Project, Rosemanowes Quarry, Herniss, Penryn, Cornwall, TRlO 9DU, UK.

surface, travels by a true minimum-time path. Jeffreys & Lapwood (1957) show that ‘if p P is a step-function of time, P P has a logarithmic infinity, rising from a gradual beginning’. This is because PP touches a caustic surface, and the distorting effect, at least at high frequencies, is a n / 2 phase shift for each component of the spectrum. The main effect of this phase shift should be to give PP (and other phases that follow mini-max paths) what is usually described as an emergent onset compared to P and p P .

The term ‘emergent onset’ is also applied to P onsets when such onsets are difficult to pick because the signal emerges gradually from the background noise. For such P

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On the onset of mini-max phases

Table 1. ISC parameters of 1980 May 26 earthquake. Date: 1980, May 26 Origin Time: 18-41-38.7 Latitude: 19.425 Longitude: 69.33"W Region: Northern Chile Focal Depth (pP-P): 109 km Magnitude: mb=6

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signals however there must nevertheless always be an onset marking the first arrival of energy from a disturbance and in the absence of noise this could be read to any desired accuracy given unlimited magnification of the signal. The same is not true of the PP onset however. Increasing the magnification on a seismogram free from noise and other interfering arrivals would simply result in the apparent onset of PP moving earlier and earlier until it coincides with P. The wavefront of PP is in fact P .

Several studies of mini-max phases show that such phases are Hilbert transforms of pulses that have followed minimum time paths. Thus Lynnes & Ruff (1985) show earthquake seismograms where the pulse shape of P can be closely matched by Hilbert transforming PP. Choy & Richards (1975) show the same effect for SSH (like PP expected to have undergone a n / 2 phase shift) which can be closely matched by Hilbert transforming SS which has not been phase shifted. What is not clearly shown by these studies is the predicted emergent nature of the mini-max phases. For example, the apparent onset of SS before and after Hilbert transforming often appears to be equally clear (see for example fig. 5 of Choy & Richards 1975) and Lynnes & Ruff (1985) show an example of PP (their fig. 4) with an apparent onset that is not emergent. In this note we show an example of a PP phase with a clearly emergent onset and comment briefly on why such onsets are rarely noticed. The onsets of mini-max phases often seem to be no more difficult to read than phases that follow minimum time paths.

0 - V Figure 1. Broad-band seismograms for the north Chile earthquake of 1980 May 26 derived from recordings made at Warramunga, Australia. (a) P K f K P seismogram after Hilbert transforming and inverting. (b) PP seismogram. (c) Seismogram (a) after convolution with an attenuation operator with f* = 1.0s. (d) PKIKP seismogram. (e) PP seismogram after Hilbert transforming. (f) Seismogram (d) after convolution with an attenuation operator with t' = 1.0s.

Figure 1 shows broad-band (BB) PKIKP and PP seismograms for an earthquake in north Chile (Table 1). These seismograms have been derived by filtering the short-period (SP) seismograms recorded at Warramunga, Australia (WRA) at an epicentral distance of 135"; the bounce point for PP being in the South Pacific at 60.2"s 147.0"W. The relative magnification of the SP and BB systems is shown in Fig. 2. The method used to derive the BB seismograms is described by Douglas, Marshall & Young (1987). Examples of BB seismograms for earthquakes derived from SP recordings are shown by Douglas, Stewart & Richardson (1984) and Douglas, Richardson & Hutchins (1990). At a distance of 135" the Jeffreys-Bullen tables (Jeffreys & Bullen 1988) predict that PKIKP will be the only P K P phase. However, A d a m & Randall (1964) and Bolt (1968) present evidence of an earlier arrival preceding P K I K P by about 10s. Cleary & Haddon (1972) suggest that these precursors to PKIKP are generated by scattering of PKP-waves at the core-mantle boundary. The precursors can be seen on the WRA SP seismogram of the north Chile earthquake (see Fig. 4). These arrivals are of low amplitude compared to PKIKP itself and the effect of these precursors on the BB seismograms is assumed here to be negligible.

The take-off angles of PKIKP and PP at the source differ by about 20" and studies of the focal mechanism show that

FREQUENCY, H r

Figure 2. Relative magnification of the short-period and broad- band seismographs as a function of frequency. --- Broad-band seismograph. - Short-period seismograph.

396 A . Douglas, A . F. Sheehan and R. C. Stewart

the ray paths for both phases leave the source in the same negative lobe of the radiation pattern (Sheehan 1985). It is clear from Fig. l(b) that the observed P P has a positive first motion that, as predicted by theory, emerges gradually from the noise. Making the reasonable assumption that P K I K P (Fig. Id) is a good approximation to the source pulse radiated on the P P path, then Hilbert transforming and inverting PKIKP (to allow for the change in polarity of P P on reflection at the free surface) should produce a seismogram that mimics the initial deflection of the PP seismogram and this is seen to be so (Fig. la) . Conversely Hilbert transforming P P (Fig. l e ) should correct for the effects of the caustic and the change of polarity at the free surface at the bounce point and show a pulse similar to the observed PKIKP (Fig. Id) and this is also seen to be so. The similarity between the Hilbert transform of P K I K P and the observed P P (and observed P K I K P and the Hilbert transform of P P ) , is further increased when the P K I K P seismogram is convolved with an attenuation operator t o allow for the expected greater anelastic attenuation of the PP path compared to the PKIKP path. Fig. l(c) shows the effect of applying the attenuation operator of Carpenter (1966) to the Hilbert transform of P K I K P (Fig. l a ) and Fig. l(f) the effects of the operator on P K I K P (Fig. Id). The amplitude response of the operator has the form exp ( - w t * / 2 ) where w is angular frequency and t* is the ratio of the traveltime to the average quality factor on the

path. Here a t* of 1.0s has been used. The similarity between Figs l (b) and (c) and Figs l (e ) and (f) confirms that the observed BB P P is indeed the Hilbert transform of the source pulse and has the predicted emergent onset.

Minimax phases are never first arrivals so there will almost always be interference from other arrivals, either specific phases or their codas, with traveltimes similar to those of the minimax phases. Further, layering near the bounce point for such phases as P P and SS may also produce interfering arrivals. In addition it seems the form of the minimax pulses is such that interference could easily result, as is shown below, in the onset appearing t o be clear and impulsive. Perhaps then it is not surprising that emergent onsets are difficult to observe.

Figure 3 shows seismograph impulse responses and their Hilbert transforms each convolved with the attenuation operator of Carpenter (1966). Also shown is the Hilbert transform of the seismograph step response again convolved with an attenuation operator. The Hilbert transform of the impulse response should approximate to the form of the minimax phase expected from sources that radiate pulses of short duration. For sources radiating pulses of long duration the onset of mini-max phases can be investigated by examining the step response.

Taking t* for P P paths to be around 2.0 s then the form of the pulse expected from an impulse source in a half-space as recorded on a World Wide Standard Seismograph Network

b WWSSN LP t*=5s

I.- 1:̂

c WWSSN LP t*=lOs + T r

0 55 SP 0 10s LP and BE Lu

Figure 3. Seismograph impulse responses, their Hilbert transforms and the Hilbert transforms of the step responses each convolved with an attenuation operator. (a) Responses for the WWSSN LP seismograph convolved with an operator with t * = 2.0 s. (b) Responses for the WWSSN LP seismograph convolved with an operator with r * = 5.0 s. (c) Responses for the WWSSN LP seismograph convolved with an operator with r * = 10.0 s. (d) Responses for the broad-band seismograph convolved with an operator with t* = 2.0 s. (e) Responses for the short-period seismograph convolved with an operator with t * = 2.0 s. The impulse response is the top trace of each group of three; the Hilbert transform the middle trace, and the Hilbert transform of the step the bottom trace. AB indicates section of the response that might be taken for leading edge of the pulse in the presence of interfering arrivals.

On the onset of mini-mar phases 397

Long-Period system (WWSSN LP) is as shown in Fig. 3(a). It can be seen from these pulses that given interfering arrivals (or noise) of about a quarter the amplitude of the first peak then the emergent character of the onset could be obscured. The apparent onset could then appear to be as impulsive as P. This might explain the apparently impulsive onsets shown by some PP pulses such as those on the JCT and BHP seismograms shown by Lynnes & Ruff (1985). In the presence of interfering arrivals with amplitudes similar to that of the first peak of the Hilbert transformed responses, AB (Fig. 3a) could appear to be the leading edge of the response giving again the appearance of a well-defined onset. The PP arrival on the LON seismogram shows such an apparently step-like onset.

SH seismograms are usually simpler than those of P and so interfering arrivals would be expected to be relatively less prominent on SSH than on P P seismograms. However, t* for SSH is likely to be large (between 5 and 10 s) compared to that for PP. The effect of such large attenuation effects is to reduce the first motion of the Hilbert transformed responses relative to second motion as shown in Figs 3(b) and (c). This is likely again to make observation of the emergent nature of the onset difficult, even when interfering arrivals have relatively small amplitude. Then AB may again appear to be the leading edge of the response. This appears to be true of the SSH (and sSSH) pulses shown in fig. 5 of Choy & Richards (1975).

From the above discussion it is clear that it should be no surprise that the emergent onsets of minimax phases are difficult to observe. Obviously emergent onsets are most likely to be seen at distance ranges where interfering arrivals will usually be small. This is so for P P and SS for example recorded at distances that lie in the core shadow zone for P and S. In fact PP on the OGD seismogram shown by Lynnes & Ruff (1985) is recorded in just this distance range (A = 116.9"), the P and its coda is relatively small and close examination shows that the onset of PP does appear to be emergent.

Comparing Fig. l(b) with Fig. 3(d) shows that the observed P P does have the form of the Hilbert transform of the BB impulse response convolved with an attenuation operator. The main reason the emergent onset of the BB PP seismogram (Fig. lb) can be observed is that there are no interfering arrivals of significant amplitude. (Presumably the arrival observed is true P P , i.e. the reflection from the sea bed at the bounce point, not that from the surface of the ocean.) However there is another reason why the emergent onset is clear which is that the maximum peak-to-peak amplitude of the arrival is from first peak to first trough and the arrival can be displayed so that this amplitude fills the available range. Had there been larger amplitude oscilla- tions following the first trough [as there is on the OGD seismogram of Lynns & Ruff (1985)] then such an arrival would normally be scaled down for display to accommodate the large amplitudes and the fact that the pulse had an emergent onset might well then have been overlooked.

Like other seismic phases, minimax phases are used for deriving estimates of source pulses and, from their traveltimes, upper mantle structure in the vicinity of the mid-ray reflection point. Now that much seismological data is available in digital form it is easy to allow for the n/2 phase shifts. This has been done for the source studies by

Lynnes & Ruff (1985) and for traveltime studies by, for example, Girardin (1980), Stark & Forsyth (1983) and Kuo, Forsyth & Wysession (1987).

Times measured from records after compensating for the n / 2 phase shift are presumably reliable estimates of the traveltimes for the geometric ray path. However the onset times of PP read directly from seismograms are regularly reported in bulletins and are assumed to be reliable enough to be used (without compensation for non-causality) to determine anomalies in wavespeed in the upper mantle in the vicinity of the reflection point (Stewart 1976, 1980; Stewart & Keen 1978; Dorbath & Dorbath 1981; Darragh 1985). Perhaps surprisingly this assumption appears to be justified. This can be seen from Fig. 3(e) which shows for the WWSSN SP system the impulse response and its Hilbert transform, and the Hilbert transform of the step response each convolved with the attenuation operator with t* = 2.0 s. Although the Hilbert transformed responses are non-causal it seems unlikely that this would be recognized by anyone observing such an arrival and an onset time would be picked either just before peak A or just after depending on whether the peak was obscured or not by other arrivals. Whichever arrival time was picked it is unlikely it would differ by more than a few tenths of a second from the time that would be obtained by correcting for non-causality. This conclusion is supported by the onsets shown by the SP PKIKP and PP seismograms of the north Chile earthquake and their Hilbert transforms (Fig. 4). Inspection of the seismograms shows that although none of the arrivals has a clear onset an observer would find it no more difficult to define an apparent arrival time when the

0 10 201

Figure 4. Short-period seismograms recorded at Warramunga, Australia from the north Chile earthquake of 1980 May 26. Also shown are their Hilbert transforms. (a) PKIKP as recorded. (b) Seismogram (a) after Hilbert transforming. (c) PP as recorded. (d) Seismogram (c) after Hilbert transforming. The vertical lines mark onset an observer might pick.

398

arrival is expected to be non-causal (Figs 4b and c) than when it is not. Times that an observer might pick as marking onsets are shown on the figure. The differences between the times taken from the non-causal seismograms and those from the causal seismograms are less than 0.5 s. Studies such as those listed above that use PP times from bulletins to determine upper mantle structure in the vicinity of the reflection point, find discrepancies between observed and computed times of f 5 s or more. From the results presented here it is clear that such discrepancies are much larger than can be accounted for by non-causality of PP.

A. Douglas, A . F. Sheehan and R . C. Stewart

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