I. The Caracas, Venezuela Earthquake of 1967: A Multiple Source Event. II. Seismic Waves at the Epicenter's Antipode Citation Rial M., José Antonio (1979) I. The Caracas, Venezuela Earthquake of 1967: A Multiple Source Event. II. Seismic Waves at the Epicenter's Antipode. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/725f-v881. https://resolver.caltech.edu/CaltechTHESIS:11112019-154510111 Abstract This dissertation consists of two unrelated subjects. Although both deal with the construction of synthetic seismograms, the first (Chapter I) is the detailed study of a complicated earthquake source, whereas the second (Chapter II) deals with the structure of the earth's interior as seen from the antipode of a seismic wave source. The two can be classified as problems involving forward inversion of seismological data. There is, however, a significant common ground in the spirit of the approach in the sense that seismic wave interference phenomena, present in both, are used as allies in reducing the inherent non-uniqueness of the inversion process. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Geophysics) Degree Grantor: California Institute of Technology Division: Geological and Planetary Sciences Major Option: Geophysics Thesis Availability: Public (worldwide access) Research Advisor(s): Helmberger, Donald V. Group: Seismological Laboratory Thesis Committee: Unknown, Unknown Defense Date: 18 December 1978 Funders: Funding Agency Grant Number NSF EAR76-06619 NSF EAR78-14786 Record Number: CaltechTHESIS:11112019-154510111 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:11112019-154510111 DOI: 10.7907/725f-v881 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 13564 Collection: CaltechTHESIS Deposited By: Mel Ray Deposited On: 12 Nov 2019 00:02 Last Modified: 11 Dec 2024 20:22 Thesis Files Preview PDF - Final Version See Usage Policy. 30MB Repository Staff Only: item control page

Jose Antonio Rial M. Copyright €) 1979 I. THE CARACAS, VENEZUELA EARTHQUAKE OF 1967: A MULTIPLE SOURCE EVENT II. SEISMIC WAVES AT THE EPICENTER'S ANTIPODE Thesis by Jose Antonio Rial M. In Partial Fulfilln~nt of the Requirements For the Degree of Doctor of Philosophy California Inst'itute of Technology Pasadena, California 1979 (Submitted December 18, 1978) -ii- ACKNOWLEDGMENTS I owe the Consejo de Desarrollo Cientffico y Humanfstico of the Universidad Central de Venezuela, Caracas, for all five years I have spent doing my graduate work. I have neveY' been professionally so satisfied as during my stay at Caltech's Seismolog ·ical Laboratory. D. Helmberger has been my guide and dear friend, for whom I feel a profound admiration and respect, not only for his scientific stature, but for his even taller humanism. Seismo Lab people were most exceedingly helpful at all times; I am grateful to all of them. I must acknowledge the inspiring discussions with D. Harkrider, D. Anderson, H. Kanamori, L. Ruff, V. Cormier, their advice and encouragement. I deeply regret having to leave, and deep inside I am sure that this is not goodbye. This research was partially supported by National Science Foundation grants EAR76 - 06619 and EAR78-14786. -iii- GENERAL INTRODUCTION This dissertation consists of two unrelated subjects. Although both de a·1 with the canst ruction of synthetic seismograms, the first (Chapter I) is the detailed study of a complicated earthquake source, whereas the second (Chapter II) deals with the structure of the earth S 1 interior as seen from the antipode of a seismic wave source. The two can be classified as prob .lems involving forward inversion of seismological data. There is, however, a significant common ground in the spirit of the approach in the sen s e that seismic wave interference phenomena, present in both, are used as allies in reducing the inherent non-uniqueness of the inversion process. -ivTABLE OF CONTENTS Acknowledgments ii General Introduction Table of Contents CHAPTER I: iv THE CARACAS VENEZUELA EARTHQUAKE OF 1967: A MULTIPLE SOURCE EVENT Abstract vi I. l INTRODUCTION I. 2 I~ETHOD I.3 BODY WAVES 5 P-Waves 5 Body Wave Synthetic Seismograms 7 The Three Sources 8 SH-Waves 14 SURFACE WAVES 22 Synthetic Seismograms 22 Di recti vi ty Function 25 THE ~~ODEL 29 Tectonic Implications 31 Holocene Geology of the Caracas Region 33 I.4 I. 5 OF ANALYSIS References CHAPTER II: 35 SEISmC \~AVES AT THE EPICENTER 1 S ANTIPODE Abstract II.l 3 39 INTRODUCTION I I. 2 DATA SET II.3 THEORETICAL DEVELOPMENT 41 46 49 -v- II.4 The ~1ethod 53 The Langer Approximation 58 The Focusing Effect at the Antipode 61 The Response Integrals 64 PKP Potential 64 PKP at the Antipode 73 SYNTHETIC SEISMOGRAMS 76 Results 77 Synthetic and Observed Seismograms 85 I I. 5 SURFACE WAVES AT THE ANTIPODE 11.6 92 Introduction 92 DISCUSSION AND CONCLUSIONS 99 References 103 APPENDIX A ·1Ol APPENDIX B 111 APPENDIX C 116 -vi- CHAPTER I The Caracas, Venezuela earthquake of 1967: A multiple source event. -viiABSTRACT A general study of Caribbean plate tectonics is first focused on the determination of fault parameters and source processes of the Caracas (Venezuela) earthquake of July 29, 1967 (mb == 6.5, t\ = 6.7) . Syn- thetic seismograms which closely reproduce the observed P-, SH-and Love \-Jave sei smograrns were generated using genera 1i zed ray and mode theory, respectively. The results indicate a compli cated faulting proc- ess, consisting of at least three separated sources aligned along a NlOW-trending 11 en echelon 11 vertical left-lateral stri ke-slip system of three faults that ruptured from north to south, at three discrete places with an extreme separation of 90 km. The process of rupture progressed southwards with a mean velocity of 3 km/sec. The focal depths of the individual sources varied between 8 km and 27.5 km. The total disloca- tion was calculated as 120 em along the direction NlOW and the total average moment as 4 x lo 26 dyne-cm. The multiple character of the event severely constrains the number of suitable source models that can be infel~red, thus facilitating the process of inversion. Tectonic implications are briefly discussed and local geology is successfully invoked to support the source model. -1I. 1 INTRODUCTION Studies on Caribbean seismotectonics are not numerous, and in general are limited to a descriptive account of seismicity, fault plane solutions or seismic-wave transmission [Sykes and Ewing, 1965; Molnar and Oliver, 1969; ~1olnar and Sykes, 1969; Papazachos, 1964; Tarr, 1968; Rial, 1976a,b]. In the last twenty years several major earthquakes, with surface wave magnitudes 6.5 and larger, have occurred along the eastern Caribbean seismic loop from S. Domingo to Colombia (Figure 1.1). It is the purpose of this and subsequent studies to investigate in detail the focal processes, source parameters and te ctonic significance of selected events of this group in the hope of obtaining a clearer picture of the seismic energy release style that characterizes the Carribean plate boundaries. In this paper one seismic event of critical importance is analyzed in detai"l, the Caracas (Venezuela) earthquake of July 29, 1967 (mb =6.5, H s = 6. 7) which occurred on the proposed southern boundary of the plate in a region where great earthquakes of magnitude over 8 have occurred in the past [Gutenberg and Richter, 1954] and in which seismic hazards are continually increasing because of grovling population and industrial expans ion. The present analysis of this event is based primarily on teleseismi c records of 1ong- peri od body and 1ong- and short - period surface waves . No near-field data, including accurate aftershock locations, are available. The only seismological station in the region (CAR) recorded most aftershocks on a short-period vertical instrument, the only one left in -2- I COCOS PLATE Fig. 1.1 ( 5) Focal mechanisms of large earthquakes along the Caribbean plate boundaries (shaded areas are compressional first motions). 1: June 11, 1971 H = 12h 56m t11s = 6.5 2: October 8, 1974 H = 09h 50m Ms = 7.1 3: December 24, 1967 H = 20h 03m Ms = 6.7 4: December 25, 1969 H = 2lh 32m Ms = 7.2 5: June 12, 1974 H = l6h 25m lvls = 6.5 6: July 30, 1967 H = OOh OOm Ms • 6.7, the Caracas earthquake 7: July 29, 1967 H = lOh 24m Ms = 6.2 8 : J u1y 13, 1974 H = 01h 18m ~1s = 7. 2 -3- working condition after the main shock, whose epicenter was located 60 km to the NW of the station under the ocean [Fiedler, 1968]. The fault plane solution for the Carac·as event (Figure !.2) based on first P-vJave arrivals shows a typical double couple strike-slip radiation pattern vrith tvJo nodal planes oriented NlOW and N80E v.Jith vertical and nearly vertical dips. The closeness of the epicenter to the major east-west tectonic 1i neaments of north em Venezue 1a made Molnar and Sykes [1969] suggest the N80E plane as representing the right-lateral earthquake fault. The evidence here presented, however, shows that the rupture occurred along the NlOW left-lateral strike-slip fault that ruptured from northwest to southeast in at least three separated places. The rup- ture process spread southward with a mean velocity of 3 km/s. 1.2 METHOD OF ANALYSIS The determination of the source parameters for the Caracas earthquake was carried out by fitting the observed P-, SH-, and Love wave (Gl) seismograms with computer synthesized ones corresponding to physically possible models. The Cagniard-deHoop formalism and generalized ray theory [Helmberger, 1974; Langston and Helmberger, 1975] are used in the construction of the body wave synthetics; the method of Harkrider [1964] is used in the generation of short-period surface wave synthetics. Finally, the directivity function defined by Ben-Menahem [1961] is calculated and compared with that observed for G2 waves recorded at stations sy~netri­ cally ·located with respect to the epicenter. One major problem in the construction of synthetic seismograms is uniqueness. No matter how perfect the fits one can accomplish by -4- Event Carquake, 07/29/67 N 0 0 E 0 .... AZ1 = N 11.5° W AZ2 =N 70° E Fig. 1.2 Dip1 =88° + 3° Dip2 =80° +5° Fault-plane solution for the Caracas 1967 event from P-wave first arrivals . Open circles are dilatation, filled circles are compression. -5patiently and diligently modifying the initial source model on a trial and error basis, the final model is good because it fits the data, not because it rep1·esents the physics of the source process uniquely. It is therefore quite important to look for characteristics in the whole radiated wave field that make the number of physically plausible models decrease to a minimum. Unfortunately, there are no general r·ules to search for uniqueness other than to hope that the source process itself contains easily identifiable (unique) characteristics. In the process of fitting the generated seismograms many parameters are involved, thus it is also desirable to check the resulting models using P-, S-, and surface \vaves as independent constraints. In this study long - period body vJaves and short- and long-period surface waves recorded at ~IWSSN stations have been used in the analysis. When necessary, individual phases have been digitized and rotated to separate the components of the ground rnoti on . I. 3 BODY WAVES P-Waves One important feature of the Caracas earthquake is that its P-wave train indicates a multiple source (see Figure 1.3). A c·lose \vaveform inspection shows that for many stations the first P onset appears identical to a synthetic P-wave for a strike-slip event of focal depth hetvJeen 12 and 14 km as recorded by a WWSSN station at teleseismic range as shown by Helmberger [1974]. The three initial phases (first 10 s) can thus be identified as the sequence P, pP, sP. The remarkable fact is that the trio repeats itself after about from the initial P onset, and 16 s - 6- p TRI t::.=76.282° Az =45.461° Mg=6000 KON t::.=74.299° Az =30.938° Mg= 1500 A TU t::. =84.396° Az =52.561° Mg= 1500 PTO t::. =59.650° Az =48.477° Mg=l500 VAL t::.=61.66° Az =36.23° Mg= 1500 ST J 6 =38.794° Az =15.882° Mg =l200 at 30 sec ----..1 1 min . 1----- Fig. 1.3 P-waves of the Caracas earthquake at several WWSSN stations -7possibly again 17 s after the latter. This is most clearly seen at sta- tions STJ, KON, and VAL (Figure 1.3). After several records around the epicenter are compared, it is concluded that at least three similar sources, separated in space and time, must account for the waveforms seen . . It will be shmvn that this multiplicity is an important factor ·in the previously mentioned search for uniqueness. Body Wave Synt~eti c Seismograms By using generalized ray theory and the Cagni ard-deHoop forrnul ati on, the construction of synthetic seismograms is s tra·i ghtforward [see Langston and Helmberger, 1975] . In the present case of a triple source three seismograms lagged in space and time are constructed, each with its independent focal parameters, and then summed in the time domain. Thus for P-waves the final seismogram W(t) of the multiple shock can be represented by the expression ~H t) 3 = L (.l)i ( t) H( t- Ti ) ( 1) i=l where each w;(t) is the total vertical response at the far field re- ceiver for each double couple point source, origin time lags, Ti are the corresponding H is the Heaviside step function, and + (R n ~) sp ns n.H( t- t 2 ) ]*s ( t) *I ( t )·*0 ( t) 1 \.Vhere the asterisks are convolution operators. ( 2) The terms in brackets in (2) represent, respectively, the direct P-wave, the pP phase, and sP phase contributions to the seismogram. These terms can be visualized as the -8- surface-source interaction near the epicenter. Rpz is the receiver function (P-wave, z component) defined in Helmberger [1974]. S(t) is the far-field source time function, taken here as a trapezoid of unit area for each event, and defined by three parameters o1 , o2 , o 3 which repre- sent the rise time, rupture time, and the roll-off time histories at the source as seen at the receiver. Each trapezoid then represents the source directivity for each individual event. This directivity thus de- fined does not appreciably change with azimuth [Burdick and Mellman, 1976], making it necessary to appropriately modify the o•s in the case of observed azimuthally dependent directivity. Q(t) and I(t) are the Q operator [Futterman, 1962] and the inst rument response respectively, Rpp and Rsp are the reflection coefficients for pP and sP respectively.., t ., and t2 are the corresponding lags with respect to the direct P-wave, and H is the Heaviside step function. p is the usual ray parameter and nv = (l/i-p 2 )112 where v is either P- or S-wave velocity. Here ; and . s-2 are the time derivatives of the P- and SV-wave potentials, respectively. For S··\11/ave seismograms equations (1) and (2) are simply transformed by substituting in the required S-wave potentials, coefficients~ time responses, and time lags. The Three Sources The focal mechanism for the first event is known (Figure !.2). Two three-point source models representing three consecutive seismic events separated in space and time and oriented along each of the two nodal planes were tested. In each case a pr·e liminary timing of the individual events was determined from stations located along great circles perpendicular. to each nodal plane. One of the two sets of event-times represents -9- the absolute timing of the events and therefore is the most important constraint at this stage in resolving the earthquake fault plane from the nodal plane. Of the tvw tested models, each with a different set of event times as in ·i tial conditions, only one v-1ill hopefully satisfy them and at the same time will closely reproduce the sixteen observed seismograms. It is indeed assumed throughout this stage that the three events are aligned along or are closely parallel to one of the two nodal planes on Figure 1.2. There are perhaps several justifications to assume this, but the most obvious one stems from the observed similarity in waveforms from one event to the other, independent of azimuth and distance. This coherence suggests common focal geometry and this in turn suggests common tectonic origin. The near source structure was assumed as a simple layer over halfspace, as suggested by Burdick and Langston [1977] for the Caracas region. The crustal layer is 33 km thick and a sharp transition in P- and S-wave velocities exists between crust and mantle (6.28 km/s/8.1 km/s for P-waves 3. 70 km/sec I 4. 68 km/s for S-waves). The best fits thus obtained are shown in Figure I.4 with the follow; ng source parameters: I) Far Field Source Time Hjstory (Trapezoid) 0 3 ( s) Strength Source 4.0 2. 0 2. l. Source 2 4.0 2.0 2. 2. Source 3 3.0 2.0 2. 2. -10- f-------1 1 min 1 T 1 em P- Wave \~ Syn. =Synthetic Obs. " Observed Fig. 1.4 Comparison of observed and synthetic P-wave seismograms at WWSSN stations around the epicenter. -11- II) Time of Occurrence and Relative Locations (X is measured in km from the first source along the fault trace. Y is measured in km perpendicularly. South is negative, East is positive.) Time (s ) X ( km) Y ( km) Focal Depth (km) Source 1 0. 0. 0. 13. Source 2 12. -52. 25. 10. Source 3 32. -90. 40. 25 III) Fault Plane Solutions (Azimuth is measured clockwise from North to the fault trace such as to define the sense of the movement of the hanging block, i.e., the reference direction of rake. See Langston and Helmberger [1975].) Fault Az·imuth ( deg) Dip ( deg) Rake ( deg) Source 350 87. 0. Source 2 350 87. 0. Source 3 340 89. 0. The stability of these results is good, in that small changes in fault dip, timing, or relative epicenter location (a few degrees, seconds, kilometers) can make the difference between a good fit (all stations) and a partial fit (somf.~ stations only). However, there certainly may be other comb·i nations of parameters that, without substantially changing the picture, could produce equally good fits to the data. Our task is to make use of as many independent data as possible in order to reduce the -12number of other combinations. Figure 4 shows that the above set of parameters reproduces fai r ly well the observed seismograms. Unfortunately, the P-waves at South American stations are not available in the distance range required (30° ~ ~ ~ 90°) for P-wave arrival times coincided with the stations change of record period. Thus the model obtained may also be biased, especially latitude-wise. The best fits (Figure 1.4) require that: ( 1) The first and last events have an extreme separation of about 97 km. (2) The three shocks occurred on three parallel faults of a seemingly 11 en echelon 11 left latera·! strike-slip system oriented NlOW, (SlOE) with the northernmost event occu r ring first. (3) The sequence of events propagated south with a mean velocity close to 3 km/sec; the overall process of fracture is completed in 37 s . In propagating south the focal depth first decreased and then in- (4) creased. The three sources had comparable strength, the second and third (5) events being double in amplitude relative to the first. A total average moment, M0 , of 5. 0 x l0 26 dyne-em is obtained by comparing sP amplitudes of synthetic and observed seismograms. The indi vi dua 1 events have average moments of 1. 2, 2. 0, and 2 x 10 26 dyne-em. T/Q a = 1.1 was used, where Tis P-wave travel time and Q is the path . average qua 1ity factO I" for P-~'laves. a Figure I. 5 shows s chemati ca lly the source model and the generalized rays included in the calculation of the synthetic seismograms. Each source generates one direct ray and two reflected rays with similar teleseismic travel times . No crustal -13- CD®@ ,.-A-.. r-"'---. ,.-A-.. p sP W(t) = B Fig. !.5 Interpretation of the focal process that generated the seismograms on Figures 1.3 and 1.4 -14- reverberations v1ere considered. The sequence of the three events was indeed such that the three trios of phases did not interfere stronqly, thus providing the clue to the multiplicity of the event, facilitating the individual identification of phases. It shall be shown that stronger interference occurs, hovJever, for the slower SH-waves, and this in turn helps considerably in the assessment of a more unique solution. In this regard P- and S-waves complement each other nicely in the process of inversion. SH-Waves Starting with the model already described for P-waves, SH-vJave synthetic seismograms for ten \~\~SSN stations suitably located around the epicenter were generated. Only a few small changes in the parameters were necessary to obtain the rather remarkable fits shown in Figure 1.6. This time two South American stations, ARE and LPA, provided excellent constraints, thus restricting the number of physically plausible models. It should be clear from Figure 1.6 that the spatia-temporal sequence of events needed to fit all ten stations must now account for the rather obvious north-south asymmett·y in wave forms observed when seismograms from CMC, SFA, or STJ, say, are compared with those recorded at LPA or ARE. Such asymmetry is best seen in detail in Figure I. 7 for stations 01C and LPA, which are 1ocated within 5 degrees of the pas tul ated di rec- tion of rupture propagation, on opposite sides of it. A simple calcula- tion can be performed to determine the relative distance bet1t1een events 1 and 2 from the relative arrivals of event 2 at CMC and LPA with respect to event 1. It is easy to show that this distance, liL~ may be determined -15- jt:_::: I Syn. ~ h - J\v.vv~~ l v ~ G.DH t::. = 59.0° Az =5 .6° Obs~SFA t::.=3JC Az= 355.8° r-syn ~~0~ :::~ FSJ ___/\ ~ ~ ~ Syo J\ A L - ,...VAVJAV -Syn. ~3• ~ Obs. Obs. SCH STJ A•39• Az=0.83° ~~~~. Az=15.JC Syn . Obs -~Syn PTO ~;~~28~0 ~ Obs. l. t-------1 1 crn 1rnin T SH-Wave UNM Syn. =Synthetic 1'1=31.8° Az=289.4° -"·~Syn. ~Obs. LPA Fig. I.6 !::, =27.3° t, = 47° Az= 188 .1° Az =169.7° Obs. = Observed t-------1 1min Comparison of observed and synthetic SH-wave seismograms at WWSSN stations around the epicenter. Notice the directivity effect between CMC, STJ to the north, and ARE, LPA to the south of the epicenter. -16- Observed SH CD®® r'---..,-A---. ~ CMC - - /::.=65 .2° Az =342° j_ 1 em T LPA /::.=47° Az =169.2° ~1 min --I '--v-...A.y--"--v----J CD®® Relative location to CMC N ~ t ~~ 1 ~~2 0 30 LLJ....J km ~,3 ~ to LPA F·ig. I. 7 Comparison of observed seismograms at stations O~C and LPA. Notice the effect of source locations on the relative arrivals of events 1, 2, and 3 at LPA. -17from the equation ( 3) where t 2 and t 1 are the arrival times of events 2 and 1, respectively, and Pct~C and PLPA are the S-wave ray parameters at CMC and LPA, respectively. It is assumed that the two stations are just along the direction of the rupture, which progresses horizontally. With 62t = 8 s obtained directly from the seismograms (Figure 1.7) and PCMC = 0.10, PLPA = 0.12, vJe get 6L = 36.4 km That is, the second event occurred 36.4 km to the southeast of the first, along the great circle between CMC and LPA. This resu"lt does not depend on the absolute times at which each event happened so it provides a very effective constraint on the basis of which several parameters can be fixed. In generating the SH-wave seismograms, however, this fact had been ignored; a new model was obtained by just adjusting the model previously determ·ined for P-waves to fit the SH seismograms. The corrected new parameters were found to be: IV) Corrected Source Parameters Using SH-Wave Seismograms Time (s) X (km) y (km) Foe a1 Depth (km) Source 0. 0. 0. l 3. 5 Source 2 15. -30. 20. 8.0 Source 3 33. -75. 50. 27 . 5 -18- The total average moment, M0 for SH-waves is 3.0 x 10 26 ctyne-cm. The average moments for the individual events are 0.64, 1.02, and 1.3 x 10 26 ctyne-cm. This model is df=picted in F·igure I. 7. the same as before. Other parameters remain essentially The most significant changes are that the second and third sources are now set to the north of their previous locations, making the extreme distance between sources 1 and 3 decrease to 90 km. This is indeed a direct result of the constraints imposed by the seismograms at LPA and ARE. For this model L'il. of equation (3) is seen to be 36.0 km, which agrees quite well with the above estimate. The changes in focal depth reflect the higher sensitivity of S-waves to this parameter. None of these changes affects substantially the fHs shown previously on Figure I. 4. In Figure I.8 it is shown how the synthetic seismograms are constructed from three distinct point sources and how the interferf~nce of the ·ir phases (Sand sS) produces the final synthetic waveform. In contrast with the P-wave seismograms, the SH slower velocity makes it possible for reflected and direct phases from two consecutive events to strongly interfere, thus producing a unique pattern which is azimuth dependent. This fact makes itself evident in the process of generating the synthetic seismograms, since parameters related to the spatia-temporal sequence of events become crucially important. As at each individual station a quite unique interference pattern appears;, the closer all these patterns can be reproduced synthetically the closer the final model will be to uniqueness . At this stage it is necessary to point out the fact that at station LPA a rather conflicting phase (dotted in Figure 1.9) arrives just after the third event. This wave, however, is not an SH-wave. The particle -19- -y- v--· y- One source Synthetic v--(1) Two sources Syn . *- ~ 1vJv- +-- ~ )r- ~ CMC STJ SCH ARE t.. = 65 .1° 6 = 39° Az = 15 .r !:.. =43° Az =0 .83° 6=27 .3° Az= 188. 1° Az = 342° Three sources Syn (3) ObsSCH ~t:.l:. STJ CMC I \v,~) \ j_ 1----l 1 rnin \\\ 1 em T t::.. ARE F·i g. I. 8 (3) \ Relativ~ location of stat1ons and sources Synthetic SH waveforms for 1, 2, and 3 sources compared to selected observed seismograms . Notice the variation in interference patterns that develop in moving from northern stations CMC to STJ. -20- motion for this phase describes a prograde vertical ellipse, characteristic of an S-coupled PL-wave. As the phase arrives at LPA along a nongreat- circle path, its effect cannot be separated by rotation of the horizontal components, so "it appears on the rotated SH seismograms of Figure I.9. The fact that a similar phase is not observed at ARE supports this interpretation. The SH-'t'lave seismograms have produced complementary evidence to the process of inversion. that must be resolved. Nevertheless, there are ambiguities in the model Firsts it is necessary to prove that the observed wave foi1Tis could have not been the result of radiation from a single fault, with a bilateral component of rupture. This indeed would produce a com- plex, multiple-like time function due to the extra stopping phase, thus creating the illusion of multiplicity. Various reasonable bilateral- rupture models on single faults \'Jere tried, No one could reproduce the wave fonns as v1ell as the multiple event model (see Figure 1.9). Second, it is appropriate to further establish that the choice of fault planes in the direction Nl 01~ is the correct one. In Figure I. 9 a refined mode 1 of the source that includes three propagating ruptures along NlOW striking faults is compared with the observed seismograms at 01C and LPA. Here it is obvious that the relative duration of the first event at the two stations is \'Jell reproduced by the synthetics. The observed difference in time duration can thus be explained as a Doppler-type effect caused by a signal that propagates toward one of the stations (LPA) and away from the other. The small bilateral component shovm in the model is necessary, although it may not be indispensible to achieve the best fit. -21- Syn. I I I I 20 40 60 80 Time sec CMC Obs. j_ 1 ern SH f-1 min --l T Syn. .,., ..., .. ' , ... \ \ I _ ~,_; LPA Obs. I 0 i I I 20 41 60 Time sec SH Fig. 1.9 One fault source models and three fault source models for stations CMC and LPA. The dotted part of the signal at LPA is an S-coupled PL phase, not an SH phase. -22The source model is now complete enough to establ·ish the sense and direction of the rupture propagation and the fault orientation for the first event. The location and time of arrival of the second shock are reasonably well established by independent methods, and its wave form is satisfactorily reproduced in Figure 1.9 by considering a 25 km long unilateral fracture propagating Sl0°E. As for the third event, its location and time of arrival are reasonably well established and its wave form reproduced satisfactorily by a 22 km long unilateral fracture also propagating Sl0°E. Table 1 summarizes the best model obtained in generatinq the body wave synthetics for the Caracas 1967 earthquake. (See also Figure 1.10.) !.4 SURFACE WAVES Synthetic Seismograms A check on the longer period, average characteristics of the model proposed can be accomplished by generating surface waves with predominant periods between 15 and 90s using the appropriate set of source parameters. For this purpose five records of Love (Gl) waves were selected on the basis of quality and path simplicity. The comparison between synthetic and observed seismograms is shown in Figure I.ll, along v-1ith the calculated values of the moment M in each ca s e. 0 For the construction of the synthetic seismograms the theor·y given in Harkrider [1964] Y./as used as it is novJ implemented on an interactive routine procedure for a NOVA 1200 minicomputer at the Seismological Laboratory, Caltech . The source time function was assumed triangular with an area equivalent to the sum of the three individual trapezoids prev·iously described. To account for -23- TABLE 1 CARACAS 1967 EARTHQUAKE SOURCE PARAMETERS Epicenter: 11°.0 N, 67°.25 L~* l. Relative location (X, Y in km) 2. Re 1ati ve separat·i on From Source in km From Source 1 in sec 3. 4. Fault parameters Strike ( o) Dip ( o) Rake ( 0 ) Length (km) Width (krn) Focal depth (km) Rupture parameters Ri s e t i me ( s e c ) Rupture vel. (km/sec) Sense of rupture t1ode 5. Source 1 Source 2 Source 3 0. ' 0 . ' -30. ' 20. -75.' 50. 0. 0. 36. 15. 90. 33. 350 350 350 87 87 87 0. 0. 0. 33. 25. 10. 8. 22. 10. 27.5 2.5 2.5 3.0 3.0 SlOE, NlOW bilateral SlOE unilateral SlOE unilateral 1.0 0.6 2.0 1.02 2.0 10. 13.5 2.5 (t) 3.5 Avg. Moment (M 0 ) (x l0 26 dyne-cm) P-wave S-wave (t) 0.8 at LPA and ARE (*) Fiedler [ .1968] 1. 34 -24- 67°W ', / ' 'IO / oo /?"J-- l I I I / CARIBBEAN SEA--- --- ---· -- ----l ----,., / / / / / I o 10 20 30 40 50 km VENEZUEL.A Fig. I. 10 The thre_e events in the 1oca l setting. The dotted 1 i nes on land represent the crest lines of the Coastal Ranges near Caracas. -25propagation effects different rise times were used (0.8 s for LPA, 2.5 s for the other stations). The earth structure between epicenter and station was selected from among appropriate models. As every path selected can be considered as pure-path, either oceanic (for stations GDH, KEV, NUR) or continental (LPA), a simple 40-layer ocean [Anderson, 1964] and an average continental [Gutenberg, 1951] earth models were used. The synthetic wave forms thus calculated reproduce quite accurately the observed ones. The values of M0 are in good agreement with those obtained using SH waves (Figure I. 11). Di recti vi ty Function The directivity function for the Caracas event was calculated from the spectral ratios of the long period G2 surface waves recorded at a pair of stations located symmetr-ically with respect to the epicenter. cedure has been described elsewhere [Kanamori, 1970a]. The pro- Here it consists basically in the evaluation of the amplitude spectral ratio of equalized surface waves recorded at stations RES (Canada) and ARE (Peru). The spectra are determined from the G2 Love wave windowed between group velocities 4.7 and 4.2 km/s. The seismograms are digitized every 2 s. algorithm is applied to obtain the spectral densities. The usual FFT The spectral den- sity ratios are equalized using the values of the attenuation coefficients from Kanamori [l970c]. The theoretical directivity function \'Jas calculated using Ren-f~P.nahem [1961] equations. For equalized seismic signals to an identical distance this is: I Dl = o(e 0 ) c/vR+cos 6 = c/vR-cos 80 0 sin~ (c/vR-cos 80 ) sin Tr~f (c/vR+cos 80 ) (4) -26- ,...----- - --- - - - - - - - - - - - - - - - - - - - -·- - - - - - - . GDH KEV !J. =6578.6 krn Az=5.6° !J. =9060.6 km Az=20.52° M0 =2 .81xl0 26 !J. =9083 .1 km Az=29 .9 5° M0 =2.64xl0 26 f---1 1min 1 1 em T LOVE WAVES LPA !J. =5140 km Az =1 69.r !J. = 689 1 krn Az=54 .1° - - - - -·------------ - Fig. 1.11 Co mparison between synthetic and ob s erved Love wave seismograms. Paths to GDH, KEV, NUR, and t·1AL at e oceanic. Path to LPA is contin ental. -27- IDI is the directivity function magnitude, c and vR are the phase vel oc- ity cor responding to the frequency f wave component and the rupture veolcity, respectively. 8 0 is the angle with which the wave leaves the source, measured from the rupture direction, and Lis the fault length. From (4) it is easy to predict, on the basis of the previous analysis, the shape of the observed directivity function. In fact, with 8 0 = 0' we shall have peaks in the directivity function for the ratio G (w) {RES} I G2(w){ARE} when 2 2:!:f. (C/V R+l) = n1T c W·i th n = 1, n = l ,2 ( 5) L= 90 km (the distance between the first and third events), c ~ 4. 5 km/sec vie get vR = 3. 0 km/sec and T = 1/f ::e 50 s. In Figure I. 14 the observed directivity function is shown, and in fact a peak exists at T = 52 s. Working backwards from this value we obtain a fault length of 93.7 km when vR = 3 km/s . Taking into account the averaging properties of 1ong period radiation, the above numbers are remarkably close to both the distance betv.Jeen the extreme events (90 km) or the total length of ruptured faults (80 km) given by the model in Figure 1.11. A second peak on Figure 1.12 at about 100 s could be ex- plained by rupture along a 130 km long fault and a rupture velocity of 2.1 km/s. l~ithout more information than that of Figure I.l2, one t.,rould have chosen to take the mean valuesll5 km and 2.5 km/s as the source parameters. The directivity function method has received sever·e criticism, especially when only one station is used and successive order surface v.Javes are analyzed. From the results here reported it seems that the -2R- 1.5 c 0 -+- u c ·:::::1 1.0 > +- u QJ a 05 · .~ •• o~~--~~~--~----~--L__.~~--~~-~~~~ 30 Fig. I. 12 40 50 60 Period, sec 80 100 120 140 Di recti vi ty function for Love \'iaves (G2) recorded at RES {Canada) and ARE (Peru). -29- stability of the method increases v1hen two stations symmetrically located with respect to the epicenter are used and the same order wave is analyzed in both. \>Je can conclude that the directivity function thus obtained may be useful and render accurate information, as long as other independent methods are used to check its results. The ~lode 1 Table and Figure !.10 summarize the findings of this study. The focal process is best described by considering the action of three different sources separated in space and time; occurring in succession from northwest to southeast on three seemingly 11 en echelon 11 left-lateral strikeslip faults. The relative locations of the three souY'ces are well resolved, es·· pecially for the first two. The third epicenter is obtained mainly on the basis of the constraints imposed by stations north and south of the epicenters, and is also reliably located. The relative location and separation figures (Table 1) are thus based upon direct seismogram measurements and Pand SH-wave synthetics. Errors are probably wi hti n 10% of those figures. Fault parameters strike, dip, rake and focal depth are fairly well established from the initial trials based upon the factual knowledge of the first event's fault-plane solution (Figure 1.2) and the observed P-wave azimuth independent similarity in 1vave forms among the three events (see Figures 1.3 and !.4). These geometrical parameters by themselves are sufficient to fit closely the v1ave forms of each individual event. A number of models \>.Jere tr-ied in \'Jhich the fault geometry of either the second or third sources were solely varied thus making it possible to estimate that errors -30- in these parameters are within 1% of the figures in Table l. This and the above error estimates simply represent the particular parameter interval within which the synthetic seismograms remain stable. These rather detai 1ed results are indeed the fortuitous consequence of the source character, and especially of the resultant interference of SH phases which strongly constrains the model, large deviations from the model drastically changing the resulting synthetic wave form. Other parameters such as rise time, rupture velocity ' and fault dimensions were last to be included in order to refine the original point-source trapezoidal far-field time function. Besides providing a closer fit to the detailed time duration of each event, a more realistic model that includes length and width of the faults is obtained. The 11 en echelon 11 character of the three faults (Figure I.lO) is needed to fit the data. ~~odels vJith faults oriented otherwise (including the direction of the line joining the three sources) do not successfully fit a11 seismograms. As mentioned before, the P-wave seismograms show that there is little difference in the focal mechanisms of the three shocks besides the focal depth . The relative location of the individual events and their relative amplitudes become the crucial parametets where the SH seismograms are to be fitted. In Figure 1.8 the delicate azimuthal de- pendence of SH wave forms at stations CMC, SCH, and STJ is shown. This is fairly reproduced by the synthetics only by considering the 11 en echelon 11 faulting of Figure 1.10, as suggested originally by the P-wave synthetics. The best fit for the P-wave seismograms was found for a model with NlOW trending fault planes while the three epicenters are on a N34W trending -31line. The SH seismograms which are indeed more sensitive to azimuthal changes in the radiation pattern and relative locations and include stations that give better latitudinal control require that the epicenters be located on a N41W trending line. The difference in trending line with the P-wave model is of 7 degrees,which is not sufficient to drastically change the P wave forms. A compromise so 1uti on is depicted in Figure I.lO on which the three events are located on a N37W trending line and locations of the second and third events are closer to those which best fit the SH seismograms. With the location of the first event taken from Fiedler [1968] the modeled faults align along the well-known fault system of Tacagua on northwest Ca~acas, whose topographical expression is a short, narrmv valley that sharply cuts the east-west trending Coastal Range (highest point 9072 feet) north of Caracas and suggestively displaces it from its west continuation by as much as 10 km in a left-lateral sense (see Figure 1.10). Tectonic Implications The 1967 event is the only major, destructive earthquake occurring in the Caracas region since 1900 (October 29th). Instrumentally detected events have been routinely reported since 1963, when the WWSSN station CAR local bulletins report a substantial number of events between magnitudes 2.5 and 4 that occurred mainly within 20 and 60 km of the station. Precise epicenter locations, however, cannot be obtained due to the absence of other seismological stations in the region. Tectonically, the area offshore north-central Venezuela is of critical importance in the understanding of the present and past movements of -32- the Caribbean plate. It has long been said that the southern boundary of the Caribbean plate runs along the inferred east-\•Jest right-lateral strike-slip San Sebastian fault, which presumably runs parallel to the coast 20 to 50 miles offshore. The major release of se·ismic energy in the 1ast sixty-seven years, the Caracas 1967 earthquake, occur-s nevertheless along a direction almost perpendicular, which suggests that although caused by a similar state of stress, (1) the Caracas 1967 event does not represent the current state of plate displacements, or (2) the recent plate displacements are northwest-southeast in this locality. There exist several reports on the structure of the southern Caribbean basin and the Venezuelan borderland that point to the fact that this region is dominated by northwest-trending strike-slip faults that presumably cut the San Sebas ti an-El Pilar east-west trending fault system thus excluding its forming a continuous plate boundary [Peter, 1972; Bell, 1972; Rial, 1973]. It is also \.<Jell known that north of the Caracas region the island chain of the Netherlands Antilles is heavily cut by northwest trending structures [Silver et al., 1975]. With the focal parameters determined in this paper, and by using the well-known relation ~1 0 = )150, where M0 is the total average moment over the fault surface S caused by the average dislocation D, and 11 dyne cm- 2 , the total displacement induced by the Car·acas J.1 = 3 x l0 earthquake is calculated to be 120 em along the direction NlOW. This gives an estimate of 1.8 em/year for the local rate of tectonically accumulated displacement since the 1900 shock. -33- Holocene Geology of the Caracas Region Several recent studies on Holocene sediments and structural patterns as well as seismic morphogenesis in North-Central Venezuela and the Caracas area point out quite explicitly that the modern regime of tectonic deformation in the area is associated with NW-SE faulting and differential vertical movements of large blocks of the Caribbean coastal range. Picard and Goddard [1975] in a detailed study of Central Venezuela coastal sediments and geomorphology conclude that the coastal region can be divided into two major segments, a v/estern segment of subsided coast line with cliffs submerged as much as 600 feet, and an eastern segment associated vdth a wide continental platform vJhich has emergent morphology. The region in which these two segments are in contact coincides with the NW-trending Tacagua system of faults near Caracas. these co as tl i ne It is suggestive that verti ca 1 movements occur in such a way that caul d 1·1ell be explained in part by pivoted left lateral displacements along the NW-SE fault sys tern. In an excellent study of seismic morphogenesis and recent tectonics of Caracas. Singer [1977] concludes that the Holocene tectonic movements in the area occur along "en echelon" strike-slip fractures oriented Nl0°-l5°W which disrupt and cut the east-west trending faults north and south of the Co as tal Range, and that the present and recent sei srni city of Caracas could well be related to the activity of the NW-SE fault system. It is indeed quite difficult to establish along which of the faults in the NW system the Caracas 1967 earthquake occurred. There are several candidates, all of them showing indications of recent activity, and some -34- even aligned along the zones most heavily damaged during the 1967 earthquake such as the 11 Humbo 1dt fault, 11 a verti ca 1 in OW-trending topographically well-defined fault whose inferred trace crosses the 1967 heavily damaged sites of Los Palos Gran des (Caracas) and the town of Caraballeda on the coast. According to Singer (1977) large, chaotic deposits of identifiably avalanche-~ypeinduced by large seismic events in the years 1100 to 1500 are found within the alluvial sediments brought dm'ln by the i•Quebrada Chacaito, 11 the steep canyoned creek that runs along the Humboldt fault trace. In a geo·log·ical time scale, however, the Caracas earthquake might well represent a minor episode in the secular displacement of the Caribbean plate. It is easy to show that a large east-west trending fault 11 locked 11 in the vicinity of Caracas can alter the local shear stress pattern so that the present eastward movement of the Caribbean plate induces displacement along the northwest left lateral faults. If the Sebastian fault is secularly active and is presently locked, the results of this study may indicate that the Caracas event represents a minor break and that future large earthquakes in the region might well occur along the east-west fault zone. -35REFERENCES Anderson, D. L. , Universal dispersion tables. 1. Love waves across oceans and continents on a spherical earth, Bull . Seism. Soc. Am., 54 681-726, 1964. Bell, J . S., Geotectonic evolution of the southern Caribbean area, Geol.:_ Soc. Amer. t·1em., 132, 369-386, 1972. Ben - Menahem, A., Radiation of seismic surface waves from finite moving sources, Bull. Se·i sm. Soc. of Am., il_, 401, 1961. Burdick, L. J., and C. A. Langston, Modeling crustal structure through the use of converted phases in teleseismic body wave forms, Bull. Seism. Soc. of Am., §1_, 677-692, 1977 Burdick, L. J., and G. t~ell m an, Inversion of the body waves from the Borrego Mountain earthquake to the source mechanism~ Bull. Sei sm. Soc. of Am., 66, 1485-1499, 1976. Fiedler, G., Estudio sismol6g·ico de la region de Caracas con relaci6n al terremoto del 29-7--1967, Bo1. 1.~1.M.E., Univers·idad Central, Caracas anoY!_, Nos. 23-24,1968. Futterman, W. I., Dispersive body waves, J . Geophys. Res . , 67,5279-5291, 1962. Gutenberg, B., ed., Internal Constitution of the Ea rth, Dover Publishers, 1951. Gutenberg, B., and C. Richter, Seismicity of the Earth and Associated Phenomena, 2nd ed., Princeton University Press, 1954. Harkrider, D., Surface waves in multilayered elastic media. I. Rayleigh and Love waves from buried sources in a multilayered elastic halfspace, Bull. Sei sm. Soc . of Am., ~. 627-679, 1964. He1mberger, D. V., Generalized ray theory for shear dislocations, Bull. Seism. Soc. of Am., §!!_, 45-64, 1974. -36Jordan, T. H., The present-day motions of the Caribbean plate, Jour. Geophys. Res., 80, 4433-4439, 1975. Kanamori, H., Synthesis of long-period surface waves and its application to earthquake source studies--Kurile Is. earthquake of October 13, 1963, Journ. Geophys. Res., ]2, 5011,5027, 1970a. Kanamori, H., The Alaska earthquake of 1964: Radiation of long period surface waves and source mechanism, Journ. Geophys. Res., J!i, 50295040, 1970b. Kanamori, H., Velocity and Q of mantle \vaves, Phys. Earth. Planet. Int., 1_, 1970c. Langston, C. A., and D. Helmberger, A procedure for modeling sha11mv dislocation sources, Geophys. J., R.A.S . , 42, 117-130, 1975. Molnar, P., and J. Oliver, Lateral variations of attenuation in the upper mantle and discontinuities in the lithosphere, J. Geophys. Res., 74, 2648-2682, 1969. Molnar, P., and L. Sykes, Tectonics of the Caribbean and Middle America regions from focal mechanisms and seismicity, Bull. Geol. Soc. of Am., 80,1639-1670,1969. Papazachos, B. C., Dispersion of Rayleigh waves in the Gulf of Mexico and Caribbean Sea, Bull. Seism. Soc. of Am., ~' 909-926, 1964. Peter, G., Geologic structure offshore north-central Venezuela, Trans. Caribbean Geol. Conf. 6th, 283-294, 1972. Picard, X. and D. Goddard, Geomorfologia y sedimentaci6n de la Costa entre Cabo Codera y Puerto Cabello, Bol. AVGMP, Caracas,]~, l, 39-106, 1975. Rial, J. A., La fosa tect6nica de Tacagua y su relaci6n con la sismicidad de la region de Caracas, Bol. Asoc. Ven. Geol., Caracas,~. 3, 49-54' 1973. Rial, J. A., Implications of the focal processes of northern Venezuela earthquakes in Caribbean tectonics, EOS, ~. 954, 1976a (abstract). -37Rial, J. A. Seismic-wave transmission across the Caribbean plate: High attenuation on concave side of Lesser Antilles island arc, Bull. Seism. Soc. of Am.,~. 1905-1920, 1976b. Silver, E. A., J. E. Case, and J. H. MacGillary, Geophysical study of the Venezuelan Borderland, Geol. Soc. Amer. Bull ., 86, 213-226, 1975. Singer, A., Tect6nica reciente, morfogenesis sismica y riesgo geol6gico en el graben de Caracas, Venezuela, Div. Geotecnia, Ministerio de Minas y Energia, Caracas, 1977. Sykes, L. R., and M. Ewing, The seismicity of the Caribbean region, J. Geophys. Res. , ZQ_, 5065-5074, 1965. Tarr, A. C., The dispersion of Rayleigh waves in the crust and upper mantle under the Western North Atlantic Ocean, Gulf of Mexico and Caribbean Sea~ Ph.D. thesis (unpublished), University of Pittsburgh, 1968. -38- CHAPTER II Seismic waves at the epicentre 1 s antipode -39- ABSTRACT The antipodal region (178° ~ 6 ~ 180°) of a seismic wave source is investigated in detail and shown to provide a new set of remarkable data to use in the exploration of the ear-th•s interior. Body and surface waves converge individually at antipodal distances after having sampled laterally the totality of the planet. The waves are focused and strongly amplified up to one order of magnitude with respect to the normal phase recorded two or more degrees away. The delicate interference patterns thus formed yield information on departures from lateral homogeneity and sphericity of the core and mantle, the structure of the inner core, global dissipation characteristics of the upper mantle, and provide strong constraints on earth models. Synthetic sei srnograms have been generated that closely reproduce the phases Pdiff' PKIKP, PKIIKP, PKP(BC), PKP, and PP observed at WWSSN long period instruments located within 5 degrees from the antipode of the New Zealand (Inangahua) earthquake of t~ay 23, 1968. Preliminary results in- dicate that the lower mantl e and upper core are laterally homogeneous as seen by 15 s waves but the core-mantle boundary region is probably 1atera l"ly inhomogeneous. The inner core-outer core boundary appears to be a sharp transition with a P-wave velocity jump of the order of 0.8 km/s. The resolution of the long period data is poor, but the poten- tial richness of the method when better data sets are available strongly motivated the investigation . Analysis of surface Rayleigh waves at a single antipodal station indicates that the average dissipation of seismic energy in the upper mantle is higher than previously thought. -40- Ray theory underlies the computation of body wave seismograms. The mathematical approach and necessary corrections to geometrical optical solutions are thoroughly explained. The amplification phenomena and interference patterns produced by focusing of rays is successfully described by the uniform asymptotic forms of the Legendre function in terms of Bessel functions of order zero and one: Langer• s approximation of the WKBJ radial eigenfunctions correctly quantifies the interaction of finite frequency rays with curved boundaries at grazing incidence. Suggested future lines of research using antipodal observations include monitoring of inner core phases, study of focal processes of large earthquakes and the exploration of planetary interiors. -41- II.l INTRODUCTION Most of the current and past research on the structure and compo- sition of the earth's interior is based upon observations of seismic waves at the surface and their interpretation in terms of appropriate models. The investigation requires different combinations of source- receiver locations according to what particular region is to be explored. All such possible combinations have already been used except that in which the receiver is located at or near the source's antipode [Rial, 1977, 1978]. It is the purpose of this work to show that the antipodal region (6 ~ 180°) of a seismic wave source can be used to extract valuable information about the earth's interior. Due to the sphet'i cally symmetric seismic velocity stl~ucture of the earth and the existence of a low velocity fluid core, many different families of seism·ic rays are focused at the antipode of a source creating the seismic analogue of a convergent optical lens ·image (Figure II.l). The unique ability of antipodal seismic stations to sample an infinite number of rays makes it possible to discuss the homogeneity or inhomogeneity of portions of the earth which would otherwise require a large number of observations. The antipode acts as a collector of seismic body and surface wave infor·mation, thus providing the investigator with a global view of the earth concentrated in a very reduced al~ea. Body waves such as PKP, PP, PcPPKP, PPP, SS, SKS, etc. arriving at the antipode produce delicate interference patterns whose stabi 1i ty, form, and extent strongly depend on the existence of deviations from a perfect radial symmetry of the structure sampled by the rays. Most of the P waves arriving at the -42- 175° Source~~-7--~~~~--~~~~~-4~~~~---L-~-L~IS~ Figure II.l The focusing of rays PP and PKP? (branch AR of PKP) at the antipode. Dashed lines rep~esent selected wavefronts with propagation time in minutes. -43antipode strongly interact with the outer and inner boundaries of the core, and some sample the structure in the neighborhood of the earth's center (a region accessible only from antipodal observations). Further- more, multiple surface waves from large earthquakes recorded at a single antipodal station can be used to determine azimuthally-independent, global averages of the anaelastic dissipation of seismic energy as a function of depth in the mantle . Observations show that most body wave phases are strongly amplified up to one order of magnitude with respect to the normal phase recorded two or mor·e degrees away from the antipode. Since the signal/noise ratio is enhanced in the same proportion, the seismograms provide clear and strong arrivals of otherwise weak signals, and perhaps of other seismic phases not yet observed elsewhere. These phenomena have received little attention in the seismologi cal litet·ature since it is quite a fortuitous occurrence that a seismological station is located vdthin one or two degrees from the antipode of a regular size earthquake with axisymmetric radiation pattern. Brief accounts on the poss ·i ble focusing of seismic energy at antipodal distances are found in Gutenberg and Richter [1934], although no attempt is made to analyze the phenomena, nor are observations reported. Other indirect references are found in Bolt and O'Neill [1964] who suggested the possibility, confirmed in the present work, that very weak signals like PKIIKP can be amplified strongly enough at the antipode to be observable. Burridge [1963] computed time domain antipodal amplitudes of body waves reflected from the surface of a homogeneous solid sphere and Alterman and Loewenthal [1972] discussed briefly the ampl ·ification effect -44- for simplified homogeneous earth models. Antipodal radio waves have, by contrast, received repeated attention since the times of Van der Pol and Bremmer (1937), but in seismology they have even been regarded as unimportant (Chapman, 1969). In the present work only WWSSN records at few locations around the antipode of three earthquakes were available. The study deals mainly with body and surface waves recorded on the long period (T = 100 s, 9 T = 15 s) instruments, so the resolution is limited and the conclusions s drawn regarded as preliminary. Short period signals were subjected to some scrutiny, but the highly complex nature of the earthquakes• sources impeded a similar treatment. The ideal sources for experiments of this nature are, of course, underground nuclear explosions and the ideal recei ver is an array of seismometers centered at the proper ant ·ipode. The simplest way to show how antipodal observations may be used is by comparing and matching observed seismograms with theoretically generated ones. In this dissertation synthetic seismograms of body \'laves are computed in two stages. First, since only natural earthquake records are available, it becomes necessary to determine the source geometry and time functions of the events to be studied. To this end we apply the Cagnard-deHoop method as developed by Helmberger (1968), Langston and Helmberger (1975), and Langston (1978) to reproduce teleseism·i c P waves recorded by the WWSS network at distances between 30 and 90 degrees from the epicenter. The source time function is modeled as that corresponding to a propagating finite dislocation imbedded in a layered medium containing a free surface. Second, following results of Friedman ( 1951) for the eigenfunctions of the radial form of the Helmholtz -45- equation, Langer (1949) uniform asymptotic expansions of Bessel functions of large order near turning points, and accounting fol~ the focusing effect at the antipode by appropriate representations of the Legendre functions, we calculate the Green•s functions for reali s tic earth models with radially varying elastic and anaelastic properties. Computationally we use the method developed by Richa rds (1970, 1973, 1976) and Cormier and Richards (1977), to generate impulse responses that include the effects of frequency dependent transmission - reflectionconversion coefficients. The theoretical development is a high fre- quency approximation that allows the displacement equations to uncouple into P, SV, and SH independent displacement potentials that solve canonical forms of the Helmholtz equation. The convolution of the source time functions with the impulse responses results in the synthetic seismogram. The se 1ect ion of Richards • method as opposed to that of Gil bert and Helmberger (1972) is due to the fact that the Cagniard-deHoop scheme as it is deve -1 oped presently for a spherically 1aye red earth does not easily lend itself to the computation of the effect of frequency dependent transmission coefficients nor to the antipodal focusing. Surface wave synthetic seismograms can be computed using t :1 e method developed by Saito (1967) and Kanamori and Cipar (1974), after a suitable representation of the Legendre polynomials in terms of Bessel functions is made to account for the antipodal focusing (Rial and Chael, 1978, in preparation). The order of subjects to be presented is as follovJs. In Section II.2 we discuss the observed seismograms; the emphasis is on the first twelve minutes of long period records that include Pdiff' PKIKP, PV-IIKP, -46- PKP, PP, PcPPKP, and PPP. In Section II.3 we describe the basic theory underlying the computation of the synthetic seismograms in a rather heuristic manner, which relies more on ·physical motivation than on mathematical formality. None of the results are new to mathematics, and with . the exception of the focusing effect treatment at the antipode, they are not new to seismology. The approach used in developing the theory is meant to be uniform, in the sense that wave propagation problems are solved by the introduction of solutions to the wave equation in terms of infinite series in the inverse powers of the frequency. Such series may have restricted domains of validity, so when a solution ·is required out of those domains, a new, more general series is developed from mathematical and physical arguments and successfully applied. In Section II.4 synthetic and observed seisrnogt·ams are compared, results are discussed, and in II.5 we show how surface waves recorded at antipodal distances are used to determine Q for the mantle. II.2 DATA SET The Inangahua earthquake of May 23, 1968 (Ms = 7.1) that occurred in New Zealand with epicentral coordinates 41.72S, 172.03E and a focal depth of 21 km (ISC) is one of the three shallow seismic events recorded by an adequate number of stations situated within 5 degrees of the epicenter1s antipode in the last 20 years. On the basis of record quality and symmetry of the fault plane solution this event ~"as selected as the subject of this research. The shock was recorded at MAL(6 = 175°), TOL(6 = 176 . 5°), and PTO(l1179.25°) as shown in the long period vertical seismograms of Figs. II.2 and II.3. Figure II.2 shmt.Js the remarkable -47- ANTIPODAL FOCUSING NEW ZEALAND May 23, 1968 PTO L1 = 179 25° PKIKP N PKP 1 1em T 1 min. --l I- PcPPKP pp MAL PKIKP PcPr~<P ppp ~=~ Figure 11.2 The focusing effect strongly amplifies the signals up to one order of magnitude with respect to the normal phase recorded 2 or more degrees away. (Inangahua earthquake of May 1968). I~WSSN long period vertical. -48- PKP 1 MIN /:;:, pp PKIKP (deg) PTO 179.25° I I I I I I I I I I I I I I I TOL 176.50° I I I I I I I I I I MAL 175.00° pp ~ESK 166.03° Figure 11.3 The Inangahua event at four antipodal stations. Note the arrival between PKIKP and PKP. WWSSN long period vertical. -49- focusing effect that occurs as a consequence of the constructive interference of coherent seismic pulses arriving almost simultaneously to the receiver. All but the direct PKIKP phases are strongly amplified. The first arrival at both stations is the P-wave diffracted by the core, and the first optical arrival is the direct P wave transmitted close to the axis source-receiver. Figure II.3 shows the Inangahua event as recorded at four stations close to the antipode. The ·interference of PKP is nicely demonstrated and the focusing effect clearly seen. Between PKIKP and PKP a rather unexpected phase, which also shows amplification, is recorded. Its travel time coincides approximately with PKIIKP and its subsidiary branches (Fig. II.4), that is, multiple reflections from the inner side of the inner core. It is shown below that this phase can be generated synthetically by the interference of PKIIKP and PKP(BC), the long period diffracted energy off the tip of the BC branch of PKP. In this dissertation synthetic seismograms have been generated for the first seven minutes of the record, that is, including up to PP, at d·istances ranging from!::.. = 175° to the antipode itself. In the analysis of surface waves the data set consisted of the vertical component of the long period instrument at PTO which includes multiple Rayleigh wave arrivals up to R6. 11.3 THEORETICAL DEVELOPMENT The propagation of seismic waves through the earth is one of the basic problems of theoretical seismology. Scholte (1956), following closely the mathematical theory of high frequency radio waves propagating around the earth derived analogous results for several distinct seismic -50- PKIIKP Figure I I. 4 The PKII KP rays converge at the antipode after being reflected from the inner side of the inner core. -51body wave phases in a homogeneous two-layered earth with a fluid core. Although most of Scholte's work is limited to the geometrical optics of body waves, some of his results include the field at the PKP caustic and diffracted P in the shadow zone. The mathematical theory hinges on tvJO basic tools of the analysis, the Watson transformation ( ~iatson, 1918; Van der Pol and Bremmer, 1937) and the Debye (rainbow) expansion of the incident field into an infinite series of interactions with the core boundary, analogous to the multiple internal reflection treatment of the Fabry-Perot interferometer. In the homogeneous earth case the \~atson transform is applied to each individual term of the Debye expansion. The asymptotic behavior of each term is usually dominated by contributions of two types: (1) saddle point contributions associated with geometrical optic rays, corresponding to the first term of the WKBJ solutions to both the radial and tangential forms of the wave equation in spherical coordinates of cylindrical symmetry, and (2) residue series contributions, associated to surface waves diffracted along major interfaces of · the medium. Contributions from higher order saddle points associated to caustics, cusps and focusing can also be evaluated by suitable manipulations of the integral responses. The general case of high frequency seismic waves propagating through an inhomogeneous (radially varying) earth has been developed only recently in the seismological literature, although the basic equ ations and mathematical theory had long been available from the works of Karal and Keller (1959), Babich and Al ekseyev (1958), Chekin (1959), Friedman (1951), Bremmer (1949), Langer (1931, 1934, 1949), Budden (1961), etc. The availability of a 1arge amount of seismic wave data and high- -52speed computers in the last decade has been a decisive incentive for seismologists to pursue the computation of synthetic seismograms for realistic earth models as a means of modeling the interior of the earth on the basis of observations at the surface. Chapman (1969), Richards (1970), r-uchs and Muller (1971), Gilbert and Helmberger (1972), Richards (1973, 1976), Cormier and Richards (1977), and others have developed such theoretical-computational techniques. The present work deals basically with antipodal observations of body waves and the construction of appropriate synthetic seismograms for realistic earth models. For this purpose two major features of anti- podal rays must be brought into consideration: (1) Most of the observed phases graze the boundaries of the outer and inner cores, and (2) most of them are focused in the antipodal region. In order to correctly describe these effects in the theory, it is necessary to use high frequency approximations to the solutions of the elastodynamic equations that remain valid for grazing incidence and give finite amplitudes at foca 1 points of rays. The mathemat i ca 1 theory to be used has been ap- plied by Richards (1973, 1976) following results of Friedman (1951) and Langer (1949) and will be slightly modified to account for the focusing effect. The computational method consists essentially in numerical inte- gration along suitable contours on the complex ray parameter plane to obtain the frequency domain response of a realistic earth model to a po·int source whose time history may be represented by a Dirac delta function. After applying the inverse Fourier transformation a Green •s function in the time domain is obtained for each particular seismic ray. For a given seismic event the source time function is modeled by -53using Cagniard-deHoop first motion approximation technique as explained by Langston and Helmberger (1975); the convolution of these two functions results in the synthetic seismogram. It is desirable for antipodal observations that the radiation pattern of the event to synthesize be axisymmetric. If this is not the case, an expression for it in terms of ray parameter must be included in the response integral before performing the contour integration. The Method: The physical problem to solve is that of point source generated high frequency elast ·ic waves, propagating in a perfectly elastic*, radially inhomogeneous earth, which after interacting with major internal discontinuities at grazing or nearly grazing incidence, converge and focus at t he antipode of the source point. The mathemati- cal description of high frequency electromagnetic waves in radially varying inhomogeneous media has been given in detail by Friedman (1951) and applied to body wave propagation in the earth by Richards (1973) . l~e earth is assumed to be a sphere centered at the origin of a system of spherical polar coordinates r,e,¢ where r is the distance from the center, e the colatitude, and ¢ the longitude. The differential equations satisfied by the wave displacement potentials generated at a point source of coordinates (b,O,O) are ( 1) *l~e shall assume perfect elasticity at this moment for simplicity•s sake. -54- where¢ and~ are azimuthally independent scalar functions that represent the P-SV displacement potentials for the high frequency uncoupled wave motions. w is the angular frequency, a (r) and S(r) are the P- and S-wave velocities in the medium. ¢and~ are related to the elas- tic wave displacement vector u by the expression ~(r,esu.l) l ~ = 172 [grad¢+ curl(O,O,- ~ ae)] + e1(;) p where p is the density at the source. The solution ¢(r,w,e) to the first of equations (1) is given by a series expansion of the form (Friedman, 1951) 00 ¢(r,w.e) = L (n+~) un(r) P (cos e) n=O ( 2) n where un(r) satisfies the radial form of the wave equation i ( ru ) ( ) r-b) _-=-'n_ + [K2(r) _ n n+l ](ru ) = - -6 (-2TI-r d~ ~ n K( r) ( 3) = w/ a ( r) The radiation condition applies on both ¢ and un(r), such that lim r + co rj~c£. ar ~ iK o¢1 == 0 where K ·js an assumed constant value of K(r) at r = oo. 0 polynomial Pn (cos G) satisfies the equation The Legendre -55- Sln1 8 d ( s1n . 8 d8 d p -ae-n) + n( n+ l)P n = 0 (4) Our basic interest here resides on the high frequency representations of the solutions to equations (3) and (4). As mentioned above, body wave r-ays arriving at antipodal distances have their turning points in the neighborhood of the major discontinuities of the earth, the outer and inner boundaries of the liquid core. In order to solve for the di sp 1acements associ a ted w"ith those rays, boundary cond·i ti ons must be satisfied at those discontinuities. It will be shown that the ampli- tudes associated with the geometrical optical rays represented by the vJKBJ solution of equation (3) are unbounded at the turning point, and thus a better approximation must be sought. In much the same way, the asymptotic (WKBJ) forms of the Legendre polynomials result in unbounded amplitudes at the antipode (8= n) which analogously is a turning point for the high frequency latitudinal component of the wave displacement. In order to solve equation (3) we must determine the solution of d 2 ~ + w2[ dr 2 ~ = r 1 _ n (n+ 1)]~ = 0 c/ (r) w r 22 (5) R( r) We can obtain an asymptotic form of the eigenfunctions ~n(r), valid for w lar·ge by assuming an infinite series solution in the inverse po'.'Jers of frequency, so it wi 11 be useful at high fr·equ enci es. In fact, the leading term is just the solution which one would construct according to the principles of geometrical optics, and the subsequent terms are corrections to it. To this end we follow Karal and Keller (1959) and use the following ansatz: -56- ~ (r) ~ n eiwS(r) I (iw)-j AJ.(r) (6) . 0 J= (For a time-harmonic solution we append the factor e-i wt to (6)). expansion (6) is called a ray series. functions to be determined. The S(r) and Aj(Y') are w-independent S(r) is the phase function, and the Aj 1 S are the amplitude coefficients of the ray series. Inserting (6) into (5) we have, i wS 11 ( r) I (i w)- j A. ( r) - w2 [ S 1 ( r) i Ij (i f j AJ. ( r) + i wS 1 ( r ) I ("i wf j A~ ( r ) + i w S 1 ( r ) I (i wf j A~ ( r ) j J j J. + I (i w) - j A1! ( r) + w2 [ j J uJ j J 1 - n~ n21)] I 2 a (r) a r j (i wf j A. ( r) = 0 J where A~. ( r ) J dA. ( r) =___,Jd':--r By retaining the highest power of w we find that (7) S(r) where rp is that value of r for which the integrand vanishes. The fol- 1ow ·i ng recurrence re 1ati ons are found: 2S 1 ( r ) A1• ( r ) + S 11 ( r ) A. ( r ) = - ( A. J where we define A_ J 1 = 0. J- 1 Thus, we have the result A.(r) = [-1-- (n+l)nfl/4{b.- ]_ Jr J 2 2 J 2 r a 2(r) wr j = 0, 1 , · · · , A~ l ::: 0 ) 11 , j = 0 , 1 , · • · p [-1 - !!_(n±.l)_fl/4(A. )lldrr) J-1 a2 w2r2 -57- Considering only the term with j=O, we get and S(r) is given by (7). b 0 is a constant to be determined. Substitut- ing these results back into (6) we obtain g~ b R(r) r( 1 0 [+ _ !iin+l) l/4 exp - ri(r) (8) 2 2 ) wr In a homogeneous mecli um a ( r) = a = a functions, H(i)(kr) with i n = 5 and ( 8) must have the fonn of Hankel 1 representing outgoing waves and i = 2 ingoing waves (from the center of the sphere.) We use the conventional Four·ier transform pair given by I 00 F( w) = f(t) ei wt dt - oo 00 f ( t) - 2 7T I F( ul ) e -iwt ctcu Then, by using the Debye asymptotic forms (Watson, 1966) for H~i)(kr) we find that e ±in/4 w We see that (8) coinci des with Richards' (1973) radial e i genfunctions 1 for wr > {n+ ) a (r). 2 The approximation (8) neglects terms of order 1/w. It is easy to see the failure of expression (8) when 2 n ( n+ 1 ) . l/ a = = p2 /r2 whel~e p 1s the ray parameter ~vhose value at the 2 2 to r turning point of the ray is precisely rp/a(rp) . -58- The Langer Approximation. The behavior of solutions like (8) near the turning point is easy to infer from (5). If we represent equa- tion (5) as \)J 11 + is 2 (r)t)J = 0, where w is large and positive and S = S(r·) is such that s2 (r) is a monotonically increasing function of rand has a zero (turning point) at r = rp, we have that r r l/;(r) = ~- exp[i w J S(r')dr'] + l/~ exp(-iw J S(r')dr'] 17 S (r) r S (r) p rp for r > rp (above the turn ·ing point), and r r C D t)J(r) = - exp[w Jp (-s 2 ) l/2 dr'] + exp[-w Jp (-s 2 ) l/2 dr'] 2 2174 114 ( -s ) ( -s ) r r for r < rp (below the turning point). These forms show that the solution is oscinatory above the turning point and exponential below it (A,B,C,D are integration constants). This behavior suggests the possibility of representing the solution of (5) in terms of functions that reproduce the oscil"latory and exponential charac t ers and are bounded at the turning point. The suggestion by Rayleigh (1912) to use such functions was formally developed by Langer (1931) who shows that associated with a given equation of the type (5) there may be a certain ''related'' eq uation whose solutions asymptotically reproduce those given above. The Airy functions A; (z) have the adequate behavior, and in order t o represent solutions of (5) it ·is easy to shov1 that their argument must be proportional to - w213 (see, fo r instance, Olver, 1974). Operating as in the previous case we try a new ansatz for equation (5) in the form -59- ~ ~ W[-w213s(r)] fj=O w- 2j BJ.(r) + w- 4/ 3 W[-<}/ 3 S(r)] where W- tW = 0 Bi(t). Ij=O 2 CJ. (r) w- j (9) ·is the equation satisfied by the Airy functions Ai(t), It is easy to show that the first series on the R.H.S. of (9) is sufficient by itself as a solution of equations of the form (5). The more general form (9) is needed for equations of more general type (Olver, "1974). Substituting the first series of (9) into (5) we obtain, after retaining terms in powers of w2 : -S(Sj )2 + [-\-- n~n;l)] = 0 a ( 10) w r and with terms of order w21 3 , S11 B + 2S 1 B 1 0 0 = 0 Conditions (10) and (11) are sufficient for our purposes. get S ( r) and from ( 11 ) B = _ __..-:1---:;--:= o [s•(r)]l/2 Therefore, we can write ( 11 ) From (10) we -60- l Jr [ {2 rR( r) 1 _(n+1)n]lI 2 }l 16 "7 r a 2r.2 w dr = l)Jn ( r) ~ K -_J:...----~--.}_7) _ !!_{.Q_+l) Jl I 4 [ L 2 2 . a Ai ( -w213 S( r)) ( 12) w r where Ai is the Airy funct ·ion and K is a constant to be determined. Using the identity wh e re H is the Hankel function of order 113, and making 113 we have Thence (12) is given by R(r) rv 112 112 Kl;; exp( ±2Trii3)(I) 2 r[-1 _ lI4 a 2 !7] ( 13) r · n( +l) 2 (n+ll2) where v~e made the usual change val1d for n large,-;- rv p = 2 w To determine a (r) = a 5 K explicitly we compare (13) with the solution of (5) for = constant and obtain ·K _ 112(3)116 -516 -n12i - as w e 2 and finally, 2 -61- '\ rr 1/2 1 ~;; 1/2 Q-112 e±i rr/6 H( 1)I ( 2) (ws) R(r) rv (--) 1/3 2 rw 1/2 ( 14) r Q - [z-- P2,I r 2 J1/2 1 with a s= I Q dr 1 rp Equation (14) is identical to equation (12) of Richards (1976). The Langer approximation of the radial eigenfunctions includes terms of order down tow2/3 . By using the uniform asymptotic expansion (14) in the description of the radial functions we obtain a bounded value of the amplitude at the turning point and correct for the fact that the transmission, reflection~ and conversion coefficients at grazing incidence become fre- quency dependent functions. The Focusing Effect at the Antipode: As mentioned before, the WKBJ solution to the Legendre equation fails at the antipode of the source (rr=8), which in analogy to the previous considerations represents a turning point for the waves propagating in the 8-direction. The breakdown of the WKBJ solution comes from the simple geometrical fact that a spherical Legendre polynomial is represented by a linear cosine function . Legendre•s equation can be written for n-+ co and m = 0 as dP ( 1. 8 n ) l )2 p -s-,.--.n---::-8 ere s n de - + ( n + 2n =0 d ( 15) Following the same method as in previous pages, we use the ansatz : Pv _ to find 112 (cos e) rv e i vS( 8) co I (i v f j AJ. ( e) , j=O v = n + l /2 -62- 2 Pn(cos e) rv (nn sin e) 1/2 cos{(n + l/2)8- -£} (16) o< e< n Equation (16) can of course be determined by the saddle point approximation of the integral representation of Pn(cos e) (Copson, 1965). The failure of (16) at 8 = 0 or TI requires a search for a related 11 11 equation of (15) and the introduction of a more general ansatz. F1~om (15), when e + 0 and n + oo such that ne remains bounded, we gett with n = ne dP n 1 d n dn (n drl) + pn = 0 ( 17) Since Pn is finite at e = 0, the only permissible solution of (17) is the Bessel function J 0 . Since it is evident that equation (15) is notal- tered by changing e by n-e = e•, we can, in the spirit of Langer's work, assume the following form of Pn(cos e•) v -2j ( 18) with .. +lj + J = 0 Jo t o o . - dJ 0 Jo = dt It is found by inspection that the t\vo series in (18) are necessary to obtain uncoupled recurrence relations for the Aj and Bj coefficients. By inserting (18) into (15) and proceeding as before, we find s (8 I ) : ±8 I e• Ao = (sin e' ) 1/2 -63- and retaining terms down to v0 e• B = ( .e• )l/2 {b + l J (sin t;, )l/2 (All t o s 1 n e' o 2 t;, o + co s A•)d} o t;, ( 19) l·lore general express ·ions can of course be obtai ned, but for the pul~pose of this vwrk, and without loss of generality, we get, with e• = n-e The asymptotic expansion (20) was obtained by Szeg~ (1934), and is uniform in an interval containing the antipode and g·i ves a finite result for v ~ oo and e• ~ 0. The focusing effect created by rays converging to a point is correctly described by the Bessel function J [ve•] in a 0 J (v e •) first approximation as it is evident that the term* (e• cot e•-1 )~.-tends to zero very rapid .ly in the neighborhood of the focal point. The zeros of the Bessel function represent, for a given value of the frequency wand fixed ray parameter, the spectral ho.les induced by destructive interference of rays arriving at a point with phases differing by n, thus forming alternate ''1 i ght and shadov/ 11 annular regions around the antipode of high and low intensity of the wave field. These annuli are controlled in extent and number by the argument v (n- e ) of the Bessel function of order zero. It is instr·uctive to observe that in the frame of reference of the antipode the function J [ v (1r -e )] becomes J (kr), 0 0 where now r is the horizontal distance measured from the antipode, and k is the horizontal wave number, since for high frequencies the expression (n+ l/2) = wp is valid with p = R sin ¢ (pis the usual ray vo -64- parameter, R is the earth•s radius, ¢ the angle of incidence at the antipodal receiver, and V the seismic v-1ave velocity at the surface.) 0 Physically, the Bessel function represents standing waves generated by the interference of converging axisymmetric travelling waves. Compu- tationally, expressions (16) and (20) have been compared within the ranges of distance and frequency adequate to the present study. For 178° ..s_ 8 .s. '1 80°, * the asymptotic form ( 20) is used in full, whereas for 8 < 178°, representation ( zn) is substituted by ( 16). At 8 = 178° both representations are identical for all practical purposes. The Response Integra "!2_. Even though the synthetic seismograms shown below have been generated for an inhomogeneous earth with radially varying elastic constants, it is instructive to start the description of the theoretical manipulations for the simpler case of a tvJO-layered homogeneous perfectly elastic earth with a fluid core . In order to reduce complexity we shall describe the mathematical representation of PKP only. PKP Potential. For a homogeneous earth, it is shown in Appendix A that the displacement potential associated \-Jith the PKP ray can be represented by 00 I n=O (n+ 1/2) g(n) Pn(cos e) (21) where (22) * Heretofore defined as the antipodal region . -65- after the general solution of (1) for a homogeneous mantle and core has been resolved into an infinite series (Debye or rainbow expansion). The PKP ray corresponds to the second term of such series, the rays PKnP with n > 1 correspond to the subsequent terms and the direct and reflected rays (P and PeP) are identified with the first term of the rainbow expansion. The caustic of PKKP at about 240° from the source is in fact the seismological analogue of the l~ainbow produced by light interacting with suspended raindrops. In (22) T and T are the (optical) plane wave transm·ission 12 21 coefficients for waves entering and leaving the core. z: n(j)(ka) are . s pheri ca 1 Besse 1 functions that represent outgoing waves ( i = 1) or ingoing waves (i = 2) from the center of the earth. k and k are the 1 2 local wave numbers ·in the core and mantle respectively, the core radius is a, b the source radius, and r the distance measured from the origin e is the epicentral distance measured from the source of coordinates. axis. The Watson transformation (vJatson, 1918) permits (22) to be rearranged into a contour integral, giving 1 J \I d\1 ( ¢PKP = 2f cos \11T g(v-1/2) Pv-l/ 2 -cos 8) (23) c where Cis the contour indicated in Figure 11.5, taken around the zeros of cos \11T on the positive real axis, in the clockwise direction, on the complex v-plane. Following Vander Pol and Bremmer (1937), expression (23) is equivalent to -66- Im 11 Re 11 c Figure 11.5 Contour C in the complex v- plane. -67- ¢PKP = 2Tii I Residues + i oo J cosvTI v dv [( + ·1 9 v -1/2)- g( -v -1/2)] Pv -l/ 2 (-cos G) 2i (24) 0 where I Residues is a general term detailed in Appendix B. If g( v -1/2) is an even function of v , (24) gives a non-optical field that represents only diffracted waves around the core. For waves like PeP observed in the shadov1 region g(v-1/2) is in fact an even function and its contribution is entirely determined by the residue series of (23) evaluated at the poles of g( v -1/2). In the lit zone, however, since g( v -l/2) is even, the original contour C is deformed to run across the saddle points of the integrand, whereas the Legendre function is split into terms representing travelling waves. On the other hand, for PKP we have, since . e iTIV (l~atson, 1966) that equation (24) gives ¢PKP = 2Tii I Residues + 00 + -!. J \) dv ,_,i TIV pv -l/ ( - cos 8'J g ( v -· l /2 ) 2 1 0 (see Appendix B), where we have used (25) -68- g( v -1/2)- g(- v- 1/2) = -2i eirrv sin vn g( v -l/2) 00 1 I (- ·l )m exp[i (2m+l )vrr] TC"Os vi = m=O Im v > 0 and Im v > 0 tan ( nv) 'u i and a new contour deformation (Appendix B). tains the optical rays -.,..,e sought. The integral in (25) con- In order to obtain their contributions the contour must be deformed to one r say, which runs across the PKP saddle points on the real v axis. by tJ cpPKP = Thence, the ray contributions are given (26) v d\J einv g(v -l/2) P,.> - 112 (- cos El ) r It is instructive to see that the factor e i 7fV Pv-l/ 2 ( -cos e) represents waves arriving at the receiver from the t\'10 opposite directions on the sphere. In fact, eirrv p .,(-cos e) ·- ei rrv [-i einv Qv(l_)l/2(cos e) v .. 1I L + i e-irrv Q( 2 ) v- 1/2 (Nussenzvei g, Using the asymptotic expansions of the Legendre (cos e)] (27) ·1965) functi ens of the second kind g~ e±i (ve- n/4) Qv - l/2(cos El) rv l/2 (2 nv sin e ) (Abramowitz and Stegun, 1965) -69- we see that ei1T\) pv - l/2(-cos 8) = ---"------.--,.,.. [eiv(2Tr-8)- i n/4 (2nv sin e) 11 2 +eiv8+i rr/4] The two terms in brackets represent travelling waves propagating in the negative and positive directions of e respectively, with a constant phase difference of n/2 betvJeen them due to the passage through tne antipode of one of them (Fig. 11.6). At and near the antipode the uniform asymptotic expansion (20) must be used to give <Ppl(p(antipode) rv t (si~· 8 ,) 112 I v dv einv 9(v- 1/2) {J (\>8') 0 I' J. (v8') 1 - ~ (e'cot 8 -1) ve' } , I 8' = n-e (28) Expression (28) is quite general and can be used for most antipodal phases with the appropriate form of g(,l -1/2). We also see from (26) and (27) that if we wish to evaluate the direct ray of PKP at distances not near the antipode, we get 2 ¢PKP(direct) 'VI v dv g(v-1/2) Q~ }1/2(cos e) r that is, -70- 8v + -rr/4 Receiver Mantle Figure 11.6 Two PKP rays propagating in opposite directions. -71- r x (2) Qv-l/2(co~ 8) (29) with G(v-l/2) Explicit expressions for r and T are given by Scho1te (1956), Chapman 12 21 (1969), Richards (1970,1973), etc. To compute the radial PKP displace- ments in (29) we first substitute all transcendental functions by their \~KBJ asymptotic forms, compute ~ 12 ~ and retain the highest powers of p w, substitute v = wp and obtain /rKP(direct) ' -' e -in/4 a ( w )l/2 J pl/2 T T eiw J ( r,p ) rrs s 2np sin 8 12 21 r• ( 30) which for simplicity Y.Je have evaluated when the source and receiver are at the same radius rs. 2 I - :z) (l a Using the fact that l/2 _ ( 2 2 -l(a) x -a ) - n cos x dx - X we get: J(r,p) ( 31 ) ;~ow r• is a scaled do\'m contour in the comp .lex p-plane, w is real, and r 0 and rk are the turning points of the PKP ray extension into the core I -72- and the core leg respectively, i.e., those points at which the respective integrands vanish. The term v~ithin brackets in (31) 9ives the phase along the straight ray in the mantle leg and the next integral gives the phase along the straight core leg. Comparing (30) and (31) with equation (16) of Richards (1973) we see that the only difference between the two express ions (accounting for the fact that Richards' expression is for PKKP and not for PKP) is that for the inhomogeneous case the phase J(r,p) contains integrals in v·thich the vtave velocity a is not a constant, but an analytic function of the radius. Otherwise, (30) above and (16) of Richards (1973) are identical. Therefore, it ap~ears that the generalization from the homogeneous case to the inhomogeneous case can be made by careful inspection of the integral responses. This fact is not surprising since the expres- sions above are geometrical optics approximations. Therefore, in pass- ing from the homogeneous to the inhomogeneous case, for high enough frequencies we just substitute in (29) say, the WKBJ Hankel functions for their inhomogeneous counterparts given by (8). Equation (30) however will fail to give the right displacement if the rays graze at the core boundary, since in that case t .1e product of plane \'lave coefficients The remedy is to incorporate in T r the general12 21 ized cosines of incidence that are computed from Langer's uniform (T T ) is zero. 12 21 asymptotic radial eigenfunctions given by (14). Thus the transmission coefficients will account for tunneling and the frequency dependent ·interaction with the boundary. Since r·epl~esentation (14) is uniformly asymptotic and reproduces (8) far from the turning points, the most general expression for the PKP displacement should be obtained by -73- replacing in (29) all Hankel functions with the corresponding radial eigenfunctions given in (14); hm'lever, mantle and core rays far from their interaction with discontinuities can safely be represented by forms ( 8). PKP at the Antipode. ing the previous reasoning Starting from expression (28) and follow ~~e easily find, at and near the antipode, 81 1 2 ) 1 to 1): (using only the first term of (20) and approximating ( . s1 n 8 ( 32) and (see Appendix B) r A J(r,p) = frp cos a dr + f' cos a dr + TIP rp where now TPK and TKP are generalized transmission coefficients, (2) (2) (l) (l) ) mwp-l/Z(rm) and kwp-l/ 2 (rm) are the radial eigenfunctions (14 in the mantle and core, respectively (analogous to G(\J-l/2) in (29)) with (1) and (2) referring to outgoing and ingoing waves respect'ively. transmission coefficients are given by The -74- where c(l) and c( 2 ) are the generalized cosines defined by Richards (1976), the suffix m refers to the bottom of the mantle, c to the top of the core, and cos i X = (l (x = m,c) , s = p/r r = rm Equation (32) is given by Rial (1978), with Ls there being an appropriate source strength factor. Finally, the time- domain synthetic se ·ismoqram for PKP for a point source delta function is given by co urPKP (e,r,t) = il J w Re iwJ -iwt dw ( 33) 1 2 e -in/2 p (cos i. ) 1 f(p,w, e ) cos 1 s ( 34) [,g(p,w,S) e A dp e } { 0 with a a g(p,w,e) = __s__ (~) yrs ap 1/2 In the actual computations the velocity profile a= a(r) is described as a complex polynomial in r whose imaginary part takes into account the anaelastic dissipation of the earth. Equation (33) may ·include the source strength, the radiation pattern as a function of p and the source time function either as a convo·lution integral with (33) or as a frequency domain function multiplying g(p, w,e ). For long enough periods the inner core may have an effect on the PKP energy even for antipodal distances. must be multiplied by In that case, f(p,w,e) in (32) -75- ( 35) \JI where R.1 is the generalized reflection coefficient from the outside of the inner core. (The inclusion of (35) into (32) improves gence of the integrand for the long period waves.) the conver- More direct interac- tions with the inner core, as in the case of PKIIKP and subsequent branches are computed by including the factor k(2) T T ~J.L2 KI I K[ ,, (1 ) (r i )J[i(l) )l kwp- 1I 2 ( r i ) · wp-1 /2 . ( 2) 1 (r i /("I_~~) ( 36) wp- 1/2 ( r i ) with 1\ where R1 is the generalized reflection coefficient at the underside of the inner core, and the radial eigenfunctions have obvious meaning. is the inner core radius. r. 1 TKI and TIK are the transmission coefficients in and out of the inner core. (See Cormier and Richards (1977) for a detailed des cri pti on. "' 1\ Explicit expressions for TKI, TIK' RI' and RI are given in their appendix.) Introduction of (35) or (36) in (32) involves careful definition of the general complex contour r' because new strings of poles are now present. Analytically it is a rather difficult problem to determine ac- curately the position of poles in the complex p-plane associated with all the earth's discontinuities. It is simpler indeed to numerically -76- compute the integrand in (32) along appropriate contours and check for convergence by suitably plotting the values of its real and imaginary parts. All the computational work here reported has been checked for convergence in such a fashion. 11.4 SYNTHETIC SEISMOGRAMS Computation of synthetic seismograms is s trai gh tfon•ta rd with the use of expressions of the form ( 32). For each parti cu·l ar body wave ray to be generated a typical procedure consists of (a) selection of a suitable integr·ation contour on the complex ray parameter plane in agreement with the gu·idelines of the previous sect·ion (see Appendix B), (b) checking for integrand convergence, stability and adequate sampling rates within the selected frequency and epicentral distance ranges (the same integration contour is in general valid for a substantially broad range of epicentral distances), (c) inversion of the resulting response to the time domain, (d) convolution with the instrument time response. Procedures (a) through (d) provide synthetic seismograms for a point source with a Dirac delta as time function. As explained in the previous section, severa 1 a 1ternati ves are pass ib 1e when an earthquake source is considered. The general procedure calls for the inclusion of source parameters (radiation pattern, far-field source time function) into the integrands at some adequate stage of the computation. An important factor to consider is, of course, the earth model to be used. We have found that for the range of frequencies considered (0.00781 Hz to 0.5 Hz) smooth earth models PEM (Dziewonski et al., 1975) -77- and l066B are sufficiently accurate. An extension of model MM8 (Anderson and Hart, 1978) for Qa and Q0~ that includes Qa ~ 10000 for the outer core and 500 fo r the inner core was highly adequate. These models are not desirable, however, if higher frequency signals are to be generated. In fact they may not be accurate enough in the high frequency end of the above mentioned window. Nevertheless, the earthquake source time function of the event selected for this study is peaked near 0.08 Hz, thus the resulting synthetics are hardly sensitive to changes in the high frequency end of the spectrum. This indeed results in lack of model discriminating power for frequencies higher than perhaps 0.15 Hz. greate r resolution. The results presented here do not claim This is not a problem of the analytic method, of course, but of the scarcity of analyzable data. It is very encourag- ing that even with such limited resolution the antipodal observations allow the investigator to draw significant conclusions about the earth; a consequence of extremely delicate interaction of the 1arge number of rays that converge at the antipode after· havi nq thoroughly sampled the earth's core and mantle. As will be seen later, the long time function and the relatively large magnitude of the Inangahua earthquake provide, by cont r ast, an excellent data set to determine world average values of Q for mantle Rayleigh \'>/aves . Results. Synthetic seismograms were generated at epicentral dis- tances ranging between 175 ° and 180° for the phases Pdiff' PKIKP, PKP(BC), PKII KP, PKP, and PP. This corresponds to a 7- minute time win- dow that starts with the first arrival of seismic energy at the receiver -78- (Pdiff). Figure 11.7 shows examples of the computed amplitude spectra for PKP and PP at distances of 175°, 176.5°, 178.34°, 179.25°, and 180°; all except the last selected according to the data available. (The sharp terminations on both sides of the frequency \'lindows ar·e artificially induced to indicate their size.) Except for the antipode signals all others show the effect of interfering signals in the form of spectral holes separated in frequency by an amount inversely proportional to the differential arrival times of the interfering pulses. effect of the event Since neither the far field time function, or the instrument are in- cluded, the resultant amplitude spectra must resemble that of either pure or interfering delta functions, the effect of attenuation reflected on the negative average slope of all curves. The focusing effect on the spectral content is clearly evident, especially for distances within two degrees of the antipode. The amplification is seen to be frequency dependent, reaching a maximum of about one order of magnitude at about 0.5 Hz. At the lowest frequencies spectral amplitudes are comparable. The clear difference in overall values of spectral amplitudes between PP and PKP is a consequence of the diverging character of the spreading coefficient of PKP in the distance range. More subtle differences in the frequency content of PKP and PP can better be noticed at t:. = 180°: A relatively stronger amplitude decrease with frequency for PP beyond 0, 15 Hz may be caused by the extra path of this phase in the upper· mantle, and the rapid relative decrease of the amplitude from that point to the low frequency end is suggestively in agreement v.Jith the expected tunneling of the long period PP energy into the core, which eventually becomes part of the PKP phase. -79- -7- PKP 0... E <r CP 0 __j - II · __L__j__ _ 008 0.90 172 2.54 3.36 4.18 l I I 5.00 -II 008 0.90 Frequency (xiO), cps _L _ 172 2.54 3.36 4.18 J 5.00 Figure 11.7 Synthetic spectra for PKP and PP vertical components at several antipodal distances . (Attenuation is included . The source is a delta function . The effect of the instrument is not included.) -80- The focusing effect alone appears then as the major contributor to the overall changes in spectral amplitude with distance from the antipode for both PKP and PP. As shown below, this seems also to be the case for Pdiff' PKP(BC) and PKIIKP. A perhaps more descriptive account of the focusing effect as a function of distance and frequency is shown in Figure 11.8. Here the amplitude spectra correspond to the approxi- mate (saddle point) evaluation of the PP phase for a simple homogeneous mantle and core earth model without attenuation. The functional form of the amplitude in that case is proportional to the product of a Bessel function of zero order times the square root of the inverse of the wavelength (Rial, 1978). Figures 11.9 and II. 10 show the basic results of this study in the form of time-domain Green's functions for a point source in a PEM earth model with attenuation. As discussed before, phases that have interacted with the inner core-~uter core boundary region have been included. It is instructive to examine Figure 11.9 closely. The most obvious feature is the remarkab 1e effect of the interfering phases PP and PKP arriving along opposite directions first as two least-time rays contained in the great circle plane of incidence, and finally at 6 = 180° the focusing of an infinity of rays an·iving from every direction around the receiver, producing a clear ten-fold amplification. As discussed above, PKP amplitudes are consistently lower than those of PP. the direct wave, is of course not amplified in the process. PKIKP, An interest- ing resuH is that in fact the inner core phases PKIIKP and PKP(BC) show substantially large amplitudes within one degree of the antipode (see Figure 11.10) suggesting that at antipodal distances their effect is -81- 6 r---,----, PP SPECTRAL AMPLITUDES 5 - T0 = 5 sec TN =65 sec 6 Ti =5 sec Q) -g4 :9c:: Q_ E 0 Q) > 3 0 Q) a:: 2 o~L~~~~-~~~~·~~~~~~~~~~~~~~~==~r~~ 174 176° 178° 180° 178° 176° 174° Antipode Figure 11.8 Theoretical variation of the PP spectral amplitudes with distance from the antipode as a function of period. -82- ~ ~= 175° _) --~ 6= 176.5° ~ - ~178.34° _1__ _A___~--1 \ -· 6=179.25° 1 Green S Functions. pp I I 1 min PKIKP I L~ PKP __) I --~ L 1'1 =180° - · - - --- -- - POIFF. PKP(BC) PK IIKP Figure I I. 9 Green functions fo r the vertical components of Pd'ff' PKIIKP, PkP(BC), PKP and PP at ant i podal distance~. Note the ~trong arrivals between PKIKP and PKP due to inner core phases. -83- sec 1.114 1.714 pp PKIKP PKP I PKIIKP+ PKP(BC) l ··. 6 =179.25 Green's function Figure II.lO Detail of F-igure 11.9 for 6 = 179.25°. Note the phase relationships between rays arriving in opposite directions. -84- large enough to provide valuable information on the constitution and structure of the inner cor~-outer core boundary region. Their ampli- tudes and waveforms are highly sensitive to seismic velocity gradients above and bel ow the top of the inner core, the velocity and density contrasts at the boundary, and the sphericity of the inner cor·e. The study of these phases by themselves and relative to the direct ray PKIKP will constitute a highly reward·ing research when seismograms of these phases for highly impulsive earthquake sources or nuclear detonations are recorded at antipodal seismic arrays. It is instructive to observe in Figures 11.9 and 11.10 the relative phase shifts of the different waveforms. PKIKP is, as expected, a simple pulse that resembles a Dirac delta function and retains the source original phase. The two PKP arrivals, however, show the waveform dis- tortion produced when the rays touch internal caustics. The first PKP arrival corresponds to the direct ray travelling the shorter path. Its waveform shows a n/2 phase shift advance re 1ati ve to the source, as a consequence of its crossing of the caustic near the top of the outer core, a result of the rapid P-wave velocity increase belm•J the coremantle boundary. The second PKP arrival has crossed the axial caustic before reaching the receiver and therefore shows an extra n/2 phase advance which results in an inverted version of the direct PKIKP waveform. The same r·elationship can be observed to exist between the first and second arrivals of PP with an extra phase advance of n, due to the surface bounce. This results in a wave form that is the inverted version of the PKP pair. Since the synthetics in Figure II.9 are computed for a surface source, the focal point of the rays is itself at the surface, -85- thus at e = 180° an extra phase advance of rr/4 occurs for all focused rays (see Appendix C). Synthetic and Observed Seis~rams. In order to be able to com- pare with the observed seismograms, the Green's functions are convolved with the far-field time function of the Inangahua shock. The event is found to be a sequence of three thrust fault shocks of almost identical fault plane geometry, at depths that vary between 30 km and 10 km. The second event occurs 4 s after the first and the third 20 s after the second shock. Some of its P waveforms are shown in Figure II.ll. The source time function is obtained by a procedure entirely similar to that described ·in Chapter I of this dissertation. Figure II.12 shows that the model satisfactorily reproduces the observed record at PTO. arrival is the diffracted P wave around the outer core. The first This phase is amplified and presents an interesting experimental demonstration of the well known Poisson spot effect predicted by physical optics (Sommerfeld, 1964). Besides this anecdotal feature, the diffracted P wave can be of importance in recovering properties of the average sei sm·i c ve 1ocity s tructure near the base of the mantle. Although this wave is predominantly long period, it is sensitive to changes in the P-wave velocity gradient in the neighborhood of the core-mantle boundary. The interference pattern of this phase recorded by an antipodal array would provide an excellent data set to study lateral deviations from homogeneity in the velocity structure by applying simple optical arguments that relate the form, dimensions, and symmetry of Fraunhofer interference patterns to the s tructure and geometry of the object causing the diffracted field. -86- ~~ deg. Mag. 36.31 375 SPA 48.48 375 LPA - - 91.04 750 ESK~ 166.03 1500 MAL 175.00 1500 TOL 176.50 1500 PTO 179.25 1500 SBA-- ~ ---1 1- 60 sec Figure II.ll -___j First arrivals of P waves of the Inangahua earthquake. (WI~SSN, long per·i od vertical). · -87- 1.654 PTO .6 =179.25° Obs. Syn. NEW ZEALAND May 23, 1968 i 1em T Figure 11.12 1--------4 1min Comparison of observed and synthetic seismograms at =- 179.25°. The synthetic includes Pdiff• PKIKP, PKIIKP, PKP(BC), PKP and PP. !'::. -88- PEM model. Inst. * ott. * source. PKP(BC) 8.= 180o PKIIKP /4)~~ PDIFF. r--- I ~ -~ ~~ '-------- PKIKP PKP pp f----j 1min L . __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ ___. Figure 11.13 Synthetic seismograms obtained for various antipodal distances. -89- Although in Figure II.l2 the synthetic seismogram reproduces the P(diff) waveform satisfactorily, its amplitude is less than the observed. l~e used model PEM in this case and although model 10668 was also tested, it did not improve what is shown. The lack of a more extensive set of data does not allow us to draw any conclusions at this stage. As seen in the Green's function (Figure II. 10) the arrival after PKIKP in the synthetic of Figure II.l2 has been generated by the interference of PKIIKP and PKP(BC); this latter phase is the long period diffraction that occurs in the shadow zone of the totally reflected rays In the travel time curves it off the outer surface of the inner core. corresponds to the prolongation of the BC branch of PKP off the tip of the triplication. PKIIKP is generated as a sum of multiply reflected rays at the inner s ·ide of the inner core boundary. v It is an interfer- ~ ence head wave (Cerveny and Ravindra~l971Lalso known as a 11 whispering gallery" series. The interaction of the PKP(BC) and PKIIKP whose arrival times are sim"ilar for most earth models, creates a waveform similar to what is observed. The observed phase is, however, only partially reproduced by the synthetics at 175° and 176.5° . (Figure II.l4) . as PKIIKP+ PKP(BC) is at best uncerta ·in. The identification Effects of earth structure near the source cannot be ruled out completely, although it is not evident from our analysis of all recorded seismograms for this event that such interpretation can be given to it. Two earth models were tested in generating these two phases, (Dziewonski et al., 1975) and 10668. Pn~ The first contains a 0.83 km/s jump in P-wave velocity and of 0.6 g/cm 3 in density, and successfully -90- PKIKP PKP --- Obs. TOL L\= 176.5° Syn. T 1em 1 min _L PKIKP PKP Obs. MAL L\= 175.0° Syn. Figure 11.14 Comparison of observed and synthetic seismograms at ~ = 175° and 176.5°. Observe the long period coda in the record at TOL, not reproduced by the synthetic. -91reproduces the observed. The 10668 model includes discontinuities 20 to 25% less pronounced, and does reproduce the v1aveforms, but the amplitudes are too small and the amplification develops poorly. This assertion favors an inner core with sharp P and density contrasts as long as our identification of the phases is correct. The motivation, however, is to show that antipodal observations within the time arrival window of PKI I KP can offer· conclusive evidence about the structure of the inner core, if known sources are available. Again, it is the highly de 1i cate interference phenomena that occur now between tvw different seismic waves that reduces the inherent non-uniqueness of the inversion of seismic wave data. The rays PKP and PP are satisfactorily reproduced in waveform, absolute and relative amplitudes for the station PTO at 179.25°. As the predominant periods of both phases are centered near 15 s, it can be concluded that there are no major lateral heterogeneities in the lower mantle and upper core of the earth with characteristic dimensions longer than perhaps 200 km. Otherwise, the interference of PKP and PP could not have been so closely reproduced. Lateral inhomo- geneities in the elastic constants with such dimensions would have caused local focusing and/or defocusing of rays result"ing in different interference phenomena than those observed. Relative amplitudes betv.reen PP and PKP, relative and absolute phase content, and overall waveforms can be easily altered if the individual rays do not converge coherently, as they appear to do judging by the similarity between the observed and the synthetics. At shorter distances (8 = 176.5°, l75°),however, the ob- served interference cannot be modeled as successfully~ mainly because the short and 'long paths · ofthe rays now interact differently with the -92- core boundary and thus the interference becomes more complex, and in fact more sensitive to even smaller lateral inhomogeneities . PKP at TOL ( e = 176.5°) shm'ls a 1ong period coda that cannot be reproduced by the synthetics. Th ·i s effect could be caused by gu ·i ded seismic energy travelling short paths along the base of the mantle, originated by the ray that travels 183.5° before arriving at the receiver. This ray stays in contact with the core-mantle boundary longer than possibly any other ray. A less pronounced coda can also be observed at MAL ( e = 175°). No evidence of it is clear at PTO. These features do not allm'l any other conclusion but the reinforcement of the view that antipodal observations using seismic arrays are potentially useful in the investigation of the structure of the core-mantle boundary. Figure 11.15 is shown for close comparison of the observed PKP at 179.25° and the predicted at 180°. FY'om this figure it is seen that there is some precursory energy to PKP that could not be reproduced. The nature of this precursor is not known. II. 5 SURFACE ~·JAVES AT THE ANTIPODE Introduction. The anaelasti city of the earth's upper mantle controls the amplitude decay of long period surface waves in the period range from 100 to 350 s. If Q is assumed independent of frequency within this range it is possible to obtain an estimate of its value as a function of depth in the mantle from the study of mantle Love and Rayleigh \o>Javes by computi'ng the relative amplitude decays of surface wave groups that have travelled a number of times around the earth. If the seismological station is located at the antipode (or at the source) of a large earthquake with an appropriate radiation pattern, multiole -93- T 1 em 1------i 1 min ..l PKP PKP ~=179.25° 1::.=180.0° Obs. Figure 11.15 Syn. The phase PKP observed at 179.25° as it would have be~n recorded at 180°. -94- arrivals of surface waves can be used to explore large portions of the mantle with a minimum effort. The Inangahua earthquake recorded at PTO ( ~ = 179 . 25°) provides an excellent example of this situation. Figure II.l6 sho1>Js the l ocati ons of the pair epicenter7'station for this event and the approximate regions sampled by Rayleigh waves. Information from over half of the earth is focused at the antipode of this event where pairs of Rayleigh wave trains R R , R R , R5R6 , etc. 12 34 arrive in succession after having travelled 1/2, 3/2, 5/2 times around the earth. The vertical component of these arrivals at a single seis- mological station is all that is needed to obtain a world average value of Q for Rayleigh \'laves ~iithin a frequency range for which the individual groups can be considered to be at their source's antipode. In the frequency domain, a single surface travelling wave group an~iving at a seismological station can be schematically represented by the expression x1 = e -ike e-ye vJhere k is the I>Jave number, e the epicentral d-istance andy is given by y = w/2QU where w is the angular frequency and U the group velocity. At the anti- pode, successive wave trains can be represented by e with -ik(n+2nTI) n = 1,2,3,···. e - 2QU (n+ 2nTI) By taking the spectral amp-litude ratios of two successive groups one obtains -95- Figure 11.16 Global projection of the Rayleigh wave radiation pattern of the Inangahua earthquake. Dashed lines on regions sampled by the waves. -96- x1(w)l x (w) w u 1T = eQ l 2 and thus x1(w)l 1og x TWT l 2 = 2TIY The R.H.S. of the last expression is obtained experimentally, so that if U(w) is known, Q can be determined for each value of w. The group velocities were obtained by a bandpass filtering technique (Kanamori, "1970). In this method the group velocity is determined from the arrival time of a wave train on a sharply bandpass filtered record at prescribed frequencies. The results are shown in Figure 1!.17 and compared to those obtained by f'1ills and Hales (1977). The compari- son with the work of the above mentioned authors is significant, since both results are supposed to represent global averages. The agreement of both results fortifies our belief that the method is viable and potentially useful. Figure I I. 18 shows the y va 1ues obtai ned by the spectral ratios of successive wave train arrivals (R R )/(R R ) and 1 2 3 4 (R R )/(R R ). The results are stable, showing only minor deviations 1 2 5 6 from a well defined trend. The agreement with Mills and Hales 1 results ·is only partial, however, and shows maximum discrepancies for periods less than 200 s. The vertical bars in t1i'lls and Hales 1 results are an indication of the general scattering of values that is obtained where dissimilar great circle paths are investigated. In our case, however, this circumstance is avoided, and the results represent a time average of the dissipation character of the mantle in the regions depicted in -97- 4.0 --r-~- 3.9 Rayleig;~ ~ave~ <1> 3.8 -l E ~ ! ..--.. ! u ~ ..._.. - 0 >. 3 .7 0 ! 0 0 0 ·u ! 0 0 0 0 ~ 3.6 t 00 Q_ ! 0 o 0o o ! 0 fo <b !oo ::l o 0 I 0 f>o 0 ~ <..9 3.5 0 3.4 Antipodal station ! Mills and Hales ( 1977) 3.3 120 140 160 180 200 220 Period, 240 260 280 300 320 sec Figure 11.17 Dispersion curve for Rayleigh waves obtained by the analysis of a single antipodal record. -98- Rayleigh Wave Attenuation 7 X o R1 R2 I R3R4 R1 R2 I R5 R6 J Antipodal station ! Mills and Hales ( 1977) ~4 3 2 ~ ~ § ~~ I _ _.~..__.__ _ l _ _ _ - - l_ _-L..__---l..._ _ 120 160 140 180 g _L___._ _l__L ____J__~ 200 220 240 Period, sec 260 280 300 320 Figure II . 18 Values of y as a function of period obtained by the analysis of a single antipodal record. -99Figure II.l6. For periods shorter than 200 s our values indicate that attenuation in the upper mantle above 200 km is consistently higher than l~i lls and Hales • results suggest. crepancy is clear. The explanation of this dis- Ant-ipodal observations in this case allow us to sample uniformly a region 240° wide around the epicenter, whereas the method of the above mentioned authors samples a region not \vi der than 140°. This simply implies that the waves arriving at the antipode sample the low velocity layer more extensively in a region of the earth which, as shown in Figure 11.16, contains a fair proportion of continental,oceanicandtectonic areas. However, if the effect of lateral inhornogene iti es along the wave paths is si gni fi cant within the frequency band, the resu"lts in Figure II.l8 might be slightly biased toward low values of Q. No attempt has been made to correct for those effects here, although a preliminary quantification shows that in the \'JOrst case they should indeed be minor. 11 . 6 DISCUSSION AND CONCLUSIONS Antipodal monitoring of seismic \'laves provides a globa ·l vie\v of the earth's interior concentrated in a very reduced area. The unique abi.lity of antipodal stations to sample an infinity of rays \'Jhose paths cover the entire earth laterally allows the investigator to discuss the homogeneity or inhomogeneity of portions of the earth which would otherwise require a large number of observations. The study of these 11 Seismic images 11 generated at the antipode can be compared to the determination of aberr·ations of optical lenses by the analysis of the images they produce. At their focal points seismic signals generate delicate inter- ference patterns of Ft' aunhofer type whose form, extent and regularity -100- strongly depend on the coherency of the interfering signals . At an antipodal array of seismic stations lateral inhomogeneities associated with a given family of rays will be mapped as deformed~ asymmetric interference patterns of anomalous amplitudes, thus providing direct evidence of localized anomalies. Antipodal observations unfortunately require source-receiver combinations not frequently found. Nevertheless, Fiji-Kermadec and In- donesian earthquake active zones are within antipodal distance from Algeria and Northern South America, where an appropriate density of receivers could offer an excellent prospect. The ideal experiment, designed to collect the maximum amount of rigidly controlled data waul d require an array of broadband seismometers ·1ocated w"ithin b.'lo degrees of the antipode of a large underground nuclear detonation. Several conclusions, some of preliminary character, can be derived from the results presented in this study. Seismic waves at the antipode of their source are amplified as much as ten times with respect to the normal phase observed two or more degrees m-.Jay, in agreement with theoretical calculations. Since the signal/noise ratio is enhanced in the same proportion, very weak waves interacting \'Jith the inner core like PKIIKP and PKP(BC) are rendered visible. If our identification of inner core phases is correct~ the inner core-outer core boundary must be sharply defined, with a P wave velocity contrast of the order of 0.8 km/s. The true structure of the boundary cannot be resolved, however, until appropriate, short-period sources are available. The lower mantle and upper core of the eat~th are homogeneous laterally as seen by waves as short as 15 s in period. The core-mantle -101- boundary region, however, is probably 1atera lly inhomogeneous. In the construction of synthetic seismograms the strong amplification due to the focusing of rays is successfully represented mathematically by the high frequency, uniform asymptotic representation of the Legendre function in terms of Bessel functions of order zero and one. Langer's approximation to the WKBJ solutions of the wave equation successfully quantifies the tunneling effect of long period P waves interacting with the core/mantle boundary. The analy$iS of surface Rayleigh waves at a single antipodal station provided a global, stable average of Q values for the upper mantle that indicates higher dissipation between periods of 120 and 200 s than previously thought. Antipodal observations have a much wider potential use than what could be concluded from an analysis of the results here presented. In fact, perhaps the most exciting prospect of antipodal observations is that they could produce unique constraints on the structure, velocity, and attenuation of the inner core. Previous investigators have studied the inner core from generally weak phases (recorded between 110° and the PKP-B caustic) whose paths interact near critical angle with its surface and do not prov·i de unambiguous results . The inner core can be explored thoroughly from about 155° to the antipode by noting the behavior of PKIKP and the development of the interference between PKIIKP and PKP(BC). Source parameters of large earthquakes can be constrained by antipodal observations. The direct P arrival of shallow focus events is not -102- contaminated by converted phases like sP; local heterogeneities near the source affect only marginally rays travelling at zero pa rameter; near receiver effects are equally minimized and very large first arrivals are indeed smallest at the antipode. These facts suggest that modelling of large earthquake s with complicated source time functions is greatly simplified and possibly less ambiguous at the antipode than anywhere else . Furthermore, it is easy to show that surface waves generated by a point source double couple produce zero amplitude on the horizontal component of an antipodal (or a near-source) receiver, so that any horizontal wave movement that is detected in the long period surface waves must represent the effect of source finiteness. At this point it is natural to conclude by saying that antipodal observations are simple, economical and powerful enough to be applied to the exploration of planetary interiors when the technology is adequate. As a common corollary to both chapters of this dissertation, we infer that interference phenomena associated with seismic waves help to decrease the inherent non-uniqueness of certain inversion problems in seismology. -103- REFERENCES Abramowitz, H. and I. A. Stegun, Handbook of Mathematical Functions, U.S. National Bureau of Standards, 1965. Alterman, Z. and D. Loewenthal, Computer generated seismograms, in Methods of Computational Physics, ed. B. Bolt, v. 12, Academic Press , 19 72. Anderson, D. L. and R. S. Hart, Attenuation models of the earth, ~· Earth. p·Janet. Int. }§_, 289-306, 1978. Ansell, J. H., On the scattering of SH waves from a point source by a sphere, Geophys. J. Roy. Astr. Soc. 54, 349-387, 1978. Babich, V. ~~. and A. S. Alekseyev, A ray method of computing wavefru nt intensities, Bull. (lzv) Acad. Sci. USSR, Geophys. Ser. no. l, 9-15, 1958. Bolt, B. A. and M. E. a• Neill, Times and amplitudes of the phases PKiKP and PKIIKP, Geophys. J. Roy. Astr, Soc. 2_, 223··231, 1964. Born, M. and E. ~Jolf, Pt~inciples of Optics, Pergamon Press, New York, 1959. Bremmer, H., Terrestrial Radio t~aves, Elsevier Pub. Co., 1949. Brune, J. N., J. E. Nafe, and L. E. Alsop, The polar phase shift of surface waves on a sphere, Bull. Seism. Soc. Am._§], 247-257, 1961. Budden, K. G., Radio Waves in the Ionosphere, Cambridge University Press, 1961. Burridge, R., The reflection of a pulse in a solid sphere, Proc. Roy. Soc. London, series fi 276, 367-400, 1963 . .J , Cerveny, V. and R. Ra vindra, Theory of Seismic Head Waves, University of Toronto Press, 1971. Chapman, C. H., Seismic wave diffraction theory, PhD thesis, University of Cambridge, 1969. Chekin, B. S., Reflection .and refraction of seismic waves at a weak interface, Bull. (Izv.l_Acad. Sci. USSR, Geophys. Ser. no. l, 9-13, 1959. -104Choy, G. L. and P. G. Richards, Pulse distortion and Hilbert transformation in multiple reflected and refracted body waves, Bull. Seism. Soc. Am. ~. 55-70, 1975. Clebsch, A., uber die Reflexion and einer Kugelflache, Crelle's Journ. fUr Math h§l_, S. 195, 1863. Copson, E. T., Asymptotic Expansions, Cambridge Tracts in Math. and Math. Phys. No. 55, Cambridge University Press, 1965. Cormier, V. and P. Richards, Full wave theory applied to a d·i scontinuous velocity increase; The inner core boundary, J. Geophys. 11, 3-31' 1977. Duwalo, G. and J. A. Jacobs, Effects of a liquid core on the propagation of seismic waves, Can. Journ. Phys. 37, No.2, 109-128, 1959. Dziewonski, A.M., A. L. Hales, and E. R. Lapwood, Parametrically simple earth models consistent with geophysical data, Phys. Earth. Planet. Int. lQ., 12-48, 1975. Friedman, B., Propagation in a non-homogeneous atmosphere, in The Theor_y of Electromagnetic Waves, Morris Kline, ed., Dover Publications, New York, 1951. Fuchs, K., and G. Muller, Computation of synthetic seismograms with the reflectivity method and comparison with observation, Geophys. J. Roy. Astr. Soc. ~. 417-433, 1971. Gilbert, F. and D. V. Helmberger, Generalized ray theory fot a layered sphere, Geophys. J. Roy. Astr. Soc. ~. 57-80, 1972. Gutenberg, B. and C. Richter, On seismic waves, Gerlands Beitr. Geo~~· 43, 56-133, 1934. Helmberger, D. V., The crust-mantle transition in the Berling Sea, Bu1._L_ Seisrn. Soc. Am. 58, 179-214, 1968. Hill, D. P., Phase shift and pulse distortion in body waves due to internal caustics, Bull. Seism. Soc. Am. 64, 1733-1742. Kanamori, H., Velocity and Q of mantle waves, Phys. Earth. Planet. Int. £, 259-275, 1970. Kanarnori, H. and J. J. Ci par, Foca 1 process of the gr·eat Chi ·1ean earthquake, May 22, 1960, Phys. Earth. Planet. Int. i• 128-136, 1934. -105- Karal, F. C. and J. B. Keller, Elastic wave propagation in homogeneous and inhomogeneous media, J. Acoust. Soc. Amer. ll• 694-705, 1959. Langer, R. E., On the asymptotic solutions of ordinary differential equations, with an application to the Bessel functions of large order, Trans. Amer. t~ath. Soc. ~' 23-64, 1931. Langer, R. E., On the asymptotic solutions of differential equations, with an application to the Bessel functions of large complex order, Trans. Am. t~ath. Soc. 34, 447-480, 1932. Langer , R. E., The asymptotic solutions of ordinary differential equations of the second order, with special reference to the Stokes phenomenon, Bull . Amer. Math. Soc. 40, 545-582, 1934 . Langer, R. E., The asymptotic solutions of ordinary linear differential equations of the second or·der, with special reference to a turning point, Trans. Amer. ~1ath. Soc. §]_, 461-490, 1949. Langston, C. A. and D. Helmberger, A procedure for modeling shallow dislocation sources , Geophys. J. Roy. Astr. Soc . 42, 11'7-130, 1975. Langston, C. A., The February 9, 1971 San Fernando Earthquake: A study of source finiteness in teleseismic body \'laves, submitted to BulL Seism. Soc. Am. Mills, J. M. and A. L. Hales, Great circle RayleiQh wave attenuation and group velocity, Part I: Observations for per·iods between 150 and 600 s for seven great-circle paths, Phys. Earth Planet. Int. J...!, 109-119, 1977 . Nussenzveig, H. M., High-frequency scattering by an impenetrable sphere, Annals of Physics 34, 23- 95, 1965. Olver, F.l~.J., 11 Asymptotics and Special Functions 11 , Academic Press, New York, 1974. Rayleigh, L., On the propagation of waves through a stratified medi urn, with spe cial reference to the question of reflection, Proc . Roy. Soc. London, Ser. A §_~, 207-226, 1912. -106- Rial, J. A., The focusing of seismic waves at the epicenter's anti- pode (abstract), EOS 58, No. 12,1191,1977. Rial , J. A. , On the focusing of seismic body waves at the epicenter's antipode, Geophys. J. Roy. Astr. Soc . .§i, 1978. Richards, P. G., "A contribution to the theory of high frequency elastic waves with applications to the shadm'l boundary of the Earth's core," PhD Thesis, California Institute of Technology, 1970. Richards, P. G., Calculation of body waves for caustics and tunne"lling in core phases, Geophys. J. Roy. Astr. Soc. 35, 243-264, 1973. Richards, P. G., On the adequacy of plane-\tlave reflection -t ransmission coefficients in the analysis of seismic body waves, Bul~ei~m. Soc. Am. 66, 701-718, 1976. Saito, M. , Excitation of free oscillations and surface waves by a point source in a vertically heterogeneous earth, J. Geophys. Res. ~, 3689-3699' 196 7. Scholte, J.G.J., On seismic waves in a spherical earth, Meded. Verh. K. ned. t4et. Inst. ~, 1-55, 1956. Sommerfeld, A., "Lectures on Theoretical Physics", Optics, Vol. IV, Academic Press, 1964. Szego, Von G., Uber einige asymptotische entwickengen der Legendreschen funktionen, Proc. Land. fqath. Soc. Series ? 36, 427- 450, 1934. Vander Pol, B. and H. Bremmer, The diffraction of electromagnetic waves from an electrical source round a finitely conducting sphere, with applications to radiotelegraphy and the theory of the rainbow, Parts I and II, Phil . Mag. 24, 141-190, 825-864, 1937. Watson, G. N., The diffraction of electric waves by the earth, Proc. Roy. Soc. A95, 83-99, 1918. Watson, G. N., "A Treatise on the Theory of Bessel Functions", Cambridge University Press, 1966. -107- APPENDIX A A point source of i rrot ational character in an in f inite elastic solid is represented by the Green function solution of the Helmholtz equation for the displacement potentials as r > b r < b where R is the distance from source to rec~iver along the minimum-time ray, and the expansion in an infinite series using spherical coordinates (r, e ,~) is due to Clebsch (1863), jn and hn are spherical Bessel and Hankel functions, respectively, k1 is the wave number w/a 1, b is the radius to the source, and e is the colatitude of the receiver measured from the source axis. a 1 is the P wave velocity in the solid and it is assumed that the source is ~ - independent. The secondary field <Ps produced by scattering of cp PRH~ from a fluid sphere of radius a imbedded in the medium and centered at the origin is possibly given by ¢sec = I (2n+l) anhn (k 1r) Pn (cos e) X = I (2n+l) cnhn( ksr) Pn(cos e) cp • = I · with r > a (2n+ 1) bnj n( k2r) Pn(cos e) (A. 1 ) (fJ.• 2) r < a k2 - w/ a2 a2 = P-v.Jave velocity in the fluid ks :: (u/ S f3 = S-wave ve 1ocity in the solid (A.3) -108- All expressions satisfy radiation conditions at infinity. The boundary conditions at r = a are used to determine the constants an,bn,cn. We require continuity of nor·mal stresses and displacements and discontinuity of shear stresses. l~ith a time-harmon ·ic factor e-i (Jlt we identify h~l) with outgoing waves and h~ 2 ) with incoming waves (from the center of the sphere). The splitting of the scattered field into an infinity of individual optical terms each of which represents a specific geometrical ray is a particular case of the rainbow expans·ion, the first term of which contains the analogue of the rays P direct and PeP (S and ScS), the second term the PKP (or SKS, PKS or SKP) rays, and subsequent terms contain PKjP, etc. for j ; 1. For the PKP case we proceed as follows: The ingoing elementary spherical wave h~ 2 )(k r) Pn(cos e) incident at the boundary r =a gives 1 rise to a transmitted wave that is, an ingoing wave within the fluid sphere. T12 is the optical transmission coefficient equal to (¢;/<P~)r=a where the superscripts t and i stand for transmitted and incident wave, respectively. The ratio h~ 2 )(k 1 a)/h~ 2 )(k 2 a) represents conversion of the wave that now propagates in a different medi urn. is transformed by the outgoing wave 11 The i ngoi ng e1ementary ~'lave total reflection 11 at the origin to produce at r =a -109- h( 2 )(k a) 1 T { 12 h 2)(k a) n 2 h ( 1 ) ( k r) P (cos 8) 2 n n and at r = a it is transmitted again through the boundary and becomes (A.4) The original incoming potential is given by the sum of elementary waves I (n+l)2 h(l)(k b) h( 2 )(k r) P (cos e) n 1 n 1 n n=O so that oo 1 (1) <PPKP = L (n + 2) hn (klb) Tl2 T2l n=O {h~2) (klaD~~l) (k2a~ ~ -,~-h~c..'(k a) 2 x h~ 11 (k a) 1 h~l)(k 1 r) P11 (cos 8) (A.5) It is simple to prove that in order to obtain the representations for subsequent terms of this group, PKjP wHh j > l, ~tie s·imply multiply (A.5) by where R is the reflection coefficient at the underside of the boundary 22 r = a. Explicit expressions for T ,T ,R are g·iven by Scholte (1956), 12 21 22 Duwalo and Jacobs (1959) among others. The 11 total reflection 11 at the origin invoked above stems from the fact that the field within the sphere -110- must be represented in terms of jn(k 2r) functions to avoid the singularity of either h(l) or h( 2 ) at r = 0. n n -111- APPENDIX B The contour C in Figure B.l can be analyzed as follows. correspond to poles of the integrand in (23) . v~ start near the real axis where The two series vn and lvl = k 1a and lvl = k2 a, respectively. The two contours c3 and c4 run along the imaginary axis. depending on the behavior of the integrand at The x•s (However, Ivi -r oo , they may be at an angle with the imaginary axis, running symmetric with respect to the origin; see for instance, Ansell, 1978). From Figure B.l we see that since I I - - c, + 2wi Dl 02 c3 c2 v• n c4 Thence icc ::: c J = 0 (Friedman, 1951) I Res vn J - - I + 21ri I Res J J= I+ J = J c1 c2 0 I + 2ni[I Res vn + I Res v~] c4 ioo Ic I · cos v dv [g(v-l/2)-g( -v- l/2)]P vnv·- l/2 (-cos e) l = 2T 0 + 2ni[I Res vn + I Res v ~] where we changed v by -v along c4. A new contour analysis can be made in order to be able to cross the saddle points of the integrand on the real axis. -112- Im v I -.......;, ---.... - . . . . . o1 '-. ao '~ X )( x\ X \ X x lin \ .: c )( 1 ... \ \ ' ·- - - - Rev Fiq . B. l lmv D -- ......................... X ~X ~ X~ f X )(X s )( oo \ \\ ,../ \ Fig. 8.2 -113- From Figure B.2, J = -J - 2ni I ~es vn , where t•l A Res vn refer to the s residues picked up by the integrand of 1 J COS v dvVTI [ g(V- l/2) - g(-v- l/2) ] pV _ '1/2 (-COS e) 2f so 00 1 J = 2f c J COS v dvVTI [ g(V -l/2) - g(-v- l/2)] pV _ l/2(-COS e) 0 A + 2ni[I Res vn + I Res v~] - 2ni I Res vn In the inhomogeneous case one can start from the general series solution vvi th (2) ±in/ 4 ( ) 1/2 r r(l) (r) "' _e- - - _a_a~s-;-;;-_ e +iw J -=n wr [ ] 1I 4 rn [ ] = [1 _ i (n +2 21/2) 2 -] wr see Richards (1973), and obtain after applying the Watson transform ¢ = with t I\) ~~l_)l/2(r) fv-l/2(rs) pv-l/2(-cos e)sec VTI dv c The contour C can now be deformed as before and by making the substitution -114- s 1. n vn E;v(1)_ 112 ( r s ) one finds that the cont r ibutions for antipodal rays are given in general by ¢ rv tJ v dv ei1TV E;~~)l/2 E;~l_)l/2(rs) Jo[v('rr-8)] r• from which we easily obtain equation (32). Typical contours used in the computation of PKP and PKIKP are shown in Figure 8.3. The x•s represent the string of poles associated with the core-mantle boundary. · - 115- Imp >< )( X X X PKP )( )( PKIKP Figure B. 3 Typ·ical complex contours used in the comput ations. the complex ray parameter. p is -116- APPENDIX C - The phase shift at the antipode Brune et al (196.1) have shown how waves propagating over the surface of a sphere undergo a phase shift of ±n/2 for every Cl'ossing of the antipode or the source regions. This argument can be used in an analogous manner for body waves that converge to the antipodal region of a sphere because they also display cylindrical symmetry and the axial caustic is a line, not a point. In Figure C.l this situation is illustrated by using generic rays which approach the antipode of a homo~ geneous sphere. ~~e arbitrarily select the phase of the least time ray as be·ing zero (paths S) so that those rays that arrive along the longel' paths L have crossed the axial caustic and suffered a n/2 phase shift that d·istorts the wave form as shown in points C (Hill, 1974). !~hen the antipode A is approached the interference of the rays is seen to amplify the upswing part of the signal that results in the wave form A, whose phase shift with respect to the incident pulse S is now only n/4. Thus, at the antipode itself the phase of the pulse is n/4 shifted with respect to the first (least time) arriving pulse. This situation is seen to occur for PKP and PP waves in F·igure II.9. Thel' t"'!, the first arr·ival (S) of PKP is a Hilbert transform of the oY'ig·inal phase. At the antipode the phase of PKP is then~+*= 34rr . Similarly, since the phase of the first PP arrival is 3n/2, its phase at the antipode is 3n 1r 3n y+ 4= 7T + -·;r 1I 1 ' • I- rr/4 + -+-t- · 0 7T/2 S L C B 0 rr/2 A + ~c B s Caustic Pulse Distortion ,;"11. -. ('~. 1 T Phase 0 rr 14 rr/2 37T/4 7T