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A pkp study of the earth's core using the warramunga seismic array Bertrand, Aimee Elizabeth Surrendra 1972

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A PKP STUDY OF THE EARTH'S CORE USING THE WARRAMUNGA SEISMIC ARRAY by AIMEE ELIZABETH SURRENDRA BERTRAND B. Sc. Un iver s i t y of the West Indies, 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of GEOPHYSICS We accept th i s thes i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 197 2 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver 8, Canada i ABSTRACT PKP core phases recorded at the U.K.A.E.A. - type seismic array, WRA, in Northern T e r r i t o r y , A u s t r a l i a over the d i s tance range 113° to 17&° are used to determine a ve loc i t y -dep th model for the earth's core . Paper recordings played out at high speed (kO mm/sec) from analog magnetic tape with amplitude gain contro l and narrow bandpass f i l t e r s provided the data ava i l ab le fo r ana l y s i s . Such data enabled p r e c i s i o n measurement of r e l a t i v e times between seismic t races . Four d i s t i n c t t rave l time branches were observed. Two of these are the well-known PKP„^ and PKPp.,- branches which are t rans -mit ted through the outer and inner cores, r e s p e c t i v e l y . The two additional branches are forerunners to the DF branch at d i s tances a less than 143 . In recent years, the ex is tence of one of these branches designated PKP , has been accepted to account for ob-served precursors . However, the ex is tence of a d i s t i n c t second branch, designated PKP_ , has been a subject fo r debate. Since the precursor branches provide information about the " t r a n s i t i o n reg ion " between the f l u i d outer core and the s o l i d inner core, the ident -i f i c a t i o n in th i s study of two d i s t i n c t sets of forerunners is of some importance. Travelt ime and t rave l t ime gradient ( dT /dA ) measure-ments o f each phase observed on the seismograms were made. The dT/dA values were measured by a least-squares technique. These measurements were s t rong ly perturbed by s t ructure beneath the array and i t was necessary to co r rec t for the e f f e c t by an emp-i r i c a l approach. The corrected dT/dA values were smoothed by the method of summary values and, for comparison, by a polynomial - r e g r e s s i o n technique. The smoothed dT/dA values for a l l four i i branches were inverted by the Herglotz-Weichert method in order to obta in a ve l oc i t y -dep th model fo r the e a r t h ' s core . The f i n a l v e l o c i t y model, UBC1, determined in th i s manner was the one which gave the- best f i t to a l l observat ions in th i s study. Near a depth of 4000 km., UBC1 required a -s l ight red -uct ion in the J e f f r e y s - B u l l e n (1940) v e l o c i t i e s in order to ob-t a i n t rave l times that agreed with the observat ions fo r the outer core . The model exh ib i t s three v e l o c i t y d i s c o n t i n u i t i e s at depths °f 4393 km , 4810 km , and 5120 km. The magnitude of the v e l o c i t y increases at the d i s c o n t i n u i t i e s are 0.10, 0.24 and 0.92 km/sec, r e s p e c t i v e l y . These d i s c o n t i n u i t i e s def ine two s h e l l s surrounding the inner core . In the outermost s h e l l , 417 km th i ck , the v e l o c i t y gradient is near zero in magnitude. In the second s h e l l , 3Q0 km th ick , the v e l o c i t y gradient is s l i g h t l y negat ive. i i i ACKNOWLEDGEMENTS I greatly appreciate the advice and encouragement given to me by my supervisor Dr. R. M. Clowes. Special thanks should be given to Dr. J.R. Cleary, Hr- H.G.Doyle and the Department of Geophysics and Geochemistry, Australian National University for t h e i r h o s p i t a l i t y to Dr. Cloves while he obtained the data used i n this study. The help of Dr. C. Wright i n providing many of the computer programs used i n t h i s study i s deeply appreciated. F i n a n c i a l assistance f o r t h i s project was provided by a University of the West Indies Postgraduate Scholarship, a National Research Council Bursary, and National Research Council operating and computing grants numbered A-7707 and C-1257, respectively of Dr. R. M. Clowes. iv TABLE OF CONTENTS Page 1. INTRODUCTION 1 1.1 The Project 1 1.2 Previous Work on Core Structure 2 1.2.1 Discovery of Central and Inner Core 2 1 .2.2 Ve loc i t y Solut ions for Ear th ' s Core 4 1.3 Seismic V e l o c i t i e s and Physical Propert ies of the Core 14 1.3.1 Central Desity of the Core 14 — 1 . 3 . 2 . S o l i d i t y of the Inner Core 15 1.4 The Warramunga Seismic Array 17 1.5 Use of an Array in Determining Structure 19 2 . MEASUREMENT AND ANALYSIS 21 2.1 Determination of dT/dA and Azimuth From a Seismic Array 21 2.1.1 Some Poss ib le Methods 21 2 . 1 . 2 Deta i l s of the Least Squares Method 23 2.2 Measurement Techniques 26 2 .2 .1 Data Ava i l ab le for the Study 26 2 . 2 . 2 Measurement of dT/dA and Azimuth 29 2 . 2 . 3 Sources of Systematic Error in dT/dA and Azimuth Measurements 34 2.3 Smoothing of Corrected dT/dA Values 52 2 .3 .1 Smoothing of dT/dA by the Method of Summary Values 52 2 . 3 . 2 Smoothing of dT/dA by UBC TRIP 63 2 . 4 Inversion of Smoothed dT/dA Data Using the Herg lotz -Wiechert Integral 66 2 . 4 . 1 . Descr ip t ion of the Herglotz-Wiechert Technique 66 V TABLE OF CONTENTS (cont 'd) Pagj. 2.4.2 App i i c a t i on of the Herglotz-Wiechert Method to (1T/4A Values for the Ea r th ' s Core 70 3. DISCUSSION 73 3.1 General Observations 73 3.2 S p e c i f i c Observations 76 3.2.1 I d e n t i f i c a t i o n of Phases % 3.2.2. Turning Points and End Points for the T r a v e l -time Branches 77 3.2.3 Amplitude Observations 80 3.2.4 A Record Sect ion for the Core Phases 82 3.3 Discuss ion of the Analyzed Data 83 3.3.1 Corrected dT/dA Measurements 83 3.3.2 Travel time Measurements 87 3.4 V e l o c i t y Model UBC1 93 4. CONCLUSION 100 BIBLIOGRAPHY 103 APPENDICES APPENDIX 1. Seismic Ray Theory 107 APPENDIX 11. L i s t of Earthquakes Used in Study 116 APPENDIX 111. E l l i p t i c i t y and Focal Depth Correct ions 120 to Measured Travel times APPENDIX IV. Method of Summary Values 123 APPENDIX Va UBC TRIP - Tr iangular Regression Package 126 Vb L i s t of dT/dA Smoothed by UBC TRIP 129 v i LIST OF FIGURES Page FIGURES 1.1 V e l o c i t y models of Gutenberg and R i c h t e r 3 1.2 B o l t ' s 1964 T2 v e l o c i t y model 7 1.3 Adams and Randall 1964 v e l o c i t y model 9 1.4 Buchbinder's 1971 v e l o c i t y model 12 1.5 Warramunga Array, A u s t r a l i a 16 2.1 Diagram to i l l u s t r a t e a l e a s t squares technique of determining dT/dA and azimuth 22 2.2 Example of data used i n the study 27 2.3 Matching waveform technique used to determine r e l a t i v e a r r i v a l times at each seismometer 30 2.4 Measured t r a v e l times 33 2.5 Reduced t r a v e l times 34 2.6 R e f r a c t i o n of a planar s e i s m i c beam i n c i d e n t on a d i p p i n g i n t e r f a c e between a ha If-space and the o v e r l y i n g l a y e r . 41 2.7 Measured dT/dA 49 2.8 dT/dA c o r r e c t e d f o r l o c a l s t r u c t u r e beneath the a r r a y 50 2.9 d T / d A smoothed by the method of summary values 62 2.10 Comparison of dT/dA smoothed by polynomial r e g r e s s i o n and method of summary values 65 3.1 Traveltime - d i s t a n c e p l o t of seismograms 81 3.2 F i t of smoothed dT/dA curves to c o r r e c t e d data 84 3.3 F i t of smoothed t r a v e l t i m e curves to data . 85 3.4 F i t of smoothed reduced t r a v e l t i m e curves to data 86 3.5 Traveltime r e s i d u a l s (0 - C) 89 3.6 Traveltime d i f f e r e n c e s 90 v i i LIST OF FIGURES (cont 'd) 3.7 The v e l o c i t y model UBC1 for the e a r t h ' s core 32 A-1 Seismic ray theory 108 A-2 Seismic ray theory for d iscont inuous increase in v e l o c i t y at a boundary. 1 1 2 A-3 Seismic ray theory fo r d i scont inous decrease in v e l o c i t y 114 at a boundary. v i i i LIST OF TABLES Page TABLE 1.1 Coordinates of the ind iv idua l seismometers of the Warramunga array 18 2.1(a) Summary points for the DF branch 56 (b) Summary points for the GH branch 57 (c) Summary points for the IJ branch 58 (d) Summary points fo r the AB branch 59 2.2 L i s t of dl/dH smoothed by method of summary values 60 2.3 Equations for dT/dA smoothed by polynomial regress ion Sk 3.1 Equations f o r t rave l times smoothed by polynomial reg -res s ion 88 3.2(a) Pos i t i on of t rave l t ime cusps in v e l o c i t y model UBC1 96 (b) Ve loc i t y -depth r e l a t i o n s h i p for v e l o c i t y model UBC1 96 3.3 Travel t imes for model UBC1 97 CHAPTER I  INTRODUCTION 1.1 The Project The core of the earth is less a cce s s i b l e than most of our p lanetary neighbors and i t s p roper t ie s are almost as d i f f i c u l t to determine. Phys ical c h a r a c t e r i s t i c s of the e a r t h ' s core can only be in fer red from measurements made near the sur face . Seismologtcal i nves t i ga t ions have proved most success -f u l in th i s regard. The ana l y s i s of se ismic body waves which have penetrated the e a r t h ' s core (e.g. PKP, SKS and other phases) i s one such method which has provided d e t a i l e d informat ion concerning the phys ica l p roper t i e s of the core. The present study uses PKP phases recorded at the Warramunga se ismic array in Northern T e r r i t o r y A u s t r a l i a as a means of d e r i v i n g a v e l o c i t y - d e p t h model for the e a r t h ' s core . Emphasis is placed on the c o n t r o v e r s i a l ' t r a n s i -t i on reg ion 1 - that reg ion , a few hundred k i lometres t h i ck , which separates the ' f l u i d ' outer core from the ' s o l i d ' inner core . The PKP phases which t rave l through the e a r t h ' s inner core are o f ten designated PKIKP. The precursors or forerunners to the PKIKP phases provide information on the nature of the t r a n s i t i o n reg ion . These precursors are normally too small to be r e a d i l y i d e n t i f i e d on ind i v idua l seismograms, except for large earthquakes. The s i g n i f i c a n c e of the array study is that these precursors , in a d d i t i o n to the major PKIKP phases, can be i d e n t i f i e d , not jus t by the i r t rave l times, but even more d e f i n i t i v e l y by the i r apparent v e l o c i t i e s or t rave l t ime gradients (dT/dh). The remainder of th i s chapter is devoted to a b r i e f d i s cu s s i on of the work done in the past regarding the s t ruc tu re of the e a r t h ' s core . The r e l a t i o n s h i p of seismic v e l o c i t i e s with the physical p roper t ie s of the core are b r i e f l y descr ibed. The Warramunga seismic array is descr ibed in som? d e t a i l , and mention is made of the use of an array in determining s t ruc tu re . Chapter 2 is devoted to d e t a i l s of the measurement and ana lys i s techniques used, and the r e s u l t s obta ined. Chapter 3 concerns a d i scus s ion of the r e s u l t s . A b r i e f d i g re s s ion on the bas ic seismic ray theory involved in th i s study is presented in Appendix I. 1.2 Previous work on core s t ruc ture 1.2.1 Discovery of the Central and Inner Core At the timrn of th i s century, observat ional se i smolog is ts noted that up to d i s tances of about 100°, both P and S waves were recorded with mo or less uniform amplitudes; beyond 100°, the amplitudes decayed r ap id l y with complete e x t i n c t i o n of short period waves at d i s tances greater than 105°. Short period P waves appeared cons i s t en t l y and s t rong ly again at d i s tances greater than about 143°, but at these d i s tances , were delayed some two minutes compared with the extension of the t rave l time curve for d i s tances less than 105°. The corresponding short period S waves d id not reappear. The explanat ion suggested (Gutenberg, 1914) was that at a depth of about 2900 km, there was a sharp d i s c o n t i n u i t y at which the v e l o c i t y of P waves f e l l by approximately 40%. This d i s c o n t i n u i t y marked the boundary between s o l i d materia l (mantle) and l i q u i d materia l (s ince S waves were not transmitted throught i t ) . Thus, the ex is tence of a l i q u i d core of the earth was pos tu la ted. It was not long before more sen s i t i ve seismographs showed that the P wave t rave l time curve, which extended from 143° to 180°, was assoc iatednear i t s beginning with small amplitude P-wave onsets, extend-ing as far back as 110°. The immediate explanat ion of these observat ions was that the waves were motion which was d i f f r a c t e d at a cusp in the t rave l t ime curve near 143°. A competing hypothesis, suggesting the ex i s tenc of an inner core in which the P v e l o c i t y was greater than in the outer core, was advanced in 1936 by the Danish se i smolog i s t I. Lehmann. This hypothesis «4i -3-Q OJ In > t 0 3 u 110 o 3 z o 6-W.TEH oefto— R I C H T E R I'O 6-5 01 . 1 * 1 p . . Velocity models of Jeffreys and Gutenberg-Richter T E F F R E V S DISTANCE I,lb. Traveltime Curves C»U W a t K & - — R I C H T K R OlSTONCE I.Ic. Ray paths through core corresponding:'to travelti, i ;e curves Figure I.I Velocity Models of Jeffreys and Gutenberg-Richter, -im-proved c o n s i s t e n t with observat ions of the t ravel t ime curves at a l l d i s t a n c e s , It was accepted inS+ead,' of the d i f f r a c t i o n hypothes is , a f ter Je f f reys (1939) showed by an extension o f A i r y ' s theory of d i f f r a c t i o n near a c a u s t i c , that for waves with per iods of approximately 1 s e c , d i f f r a c t i o n . c o u l d not occur more than 3° from the cusp at 1 4 3 ° . With the d iscovery of the inner c o r e , a l l the main boundaries w i th in the ear th had apparent ly been loca ted , and the bulk of P and S wave observat ions could be expla ined by the geometry of a t r i p a r t i t e earth con-s i s t i n g of a mantle, inner and outer c o r e . Thenext step was to obta in v e l o c i t y -depth s o l u t i o n s cons is ten t with observat ions for a t r i p a r t i t e e a r t h , and in p a r t i c u l a r for the core of the e a r t h . 1.2.2 V e l o c i t y So lu t ions for E a r t h ' s Core Gutenberg and Richter (1937) favored a v e l o c i t y d i s t r i b u t i o n in which the v e l o c i t y of P waves increased with depth down to the mantle-core boundary at about 2900 km, where i t dropped sharply from 13-7 km/sec. to approximately 8 km/sec. The v e l o c i t y then increased s lowly to 10.2km/sec at a depth of 4850 km, thereupon inc reas ing r a p i d l y andcont inuously to 11.4 km/ sec .over a depth of 300 km, and f i n a l l y decreas ing slowly to 11.3 km/sec, at the e a r t h ' s cen te r . They therefore postulated a t r a n s i t i o n region 300 km th ick to e x i s t between the inner and outer c o r e , with the inner core boundary at a depth of 5150 km. J e f f r e y s (1939) model d i f f e r e d from that of Gutenberg and Richter in that i t showed a v e l o c i t y decrease from a depth of 4980 km to 5120 km, fo l lowed by a sharp increase of 1 :8 km/sec. at the boundary of the inner c o r e . The v e l o c i t y decrease in the zone surrounding the inner core was needed to exp la in h is observed t ravel t imes. The two models are shown in F igure 1.1a, and the form of the corresponding t rave l t ime curve is shewn in F igure 1.1b. Note the d i f f e r e n c e in the lengths of the branch BC, The ray which leaves the focus at the shal lowest angle necessary to enter the core corresponds to the point A on the curve , and s u c c e s s i v e l y steeper rays trace the t rave l t ime curve ABCDEF. Rays corresponding to the branches AB and BC - 5 -penetrate only the outer core, the cusp at B being a d i r e c t consequence of the discont inuous decrease in v e l o c i t y at the mantle-core boundary and the subse-quent, P v e l o c i t y v a r i a t i o n in the outer core, and is assoc iated with a caus t i c ( i . e . large ampl i tudes) . The branch CD of the travel t ime curve represents rays which undergo tota l interna l r e f l e c t i o n at the boundary of the inner core. Rays that represent the branch DF penetrate the inner core; the point F thus corresponds to a ray t r a v e l l i n g d i a m e t r i c a l l y through the ea r th . The paths of these rays are i l l u s t r a t e d in Figure 1.1c, in which the points of emergence of rays that correspond to the points in Figure 1.1b are shown by the same l e t t e r s . The hypothesis of a s ing le v e l o c i t y jump in the t r a n s i t i o n region between the inner and outer cores proved inadequate, however, with the d i s -covery (Gutenberg, 1957 . 1958) of short period precursors to the DF branch in the approximate d is tance range \25°<c&<- 1^3° . These precursors a r r i ved at the s ta t ions up to 15 seconds e a r l i e r than the DF1 a r r i v a l s . Gutenberg on (1957) suggested, on the basis of exper iments^viscos i ty by Kuhn and Vielhauer ( 1 9 5 3 ) . tha t the PKP wave pulse may become dispersed into a frequency - depend-ent wave t r a i n in a t r a n s i t i o n zone between a l i q u i d outer core and a more viscuous inner core. However, h is suggestion became unacceptable when no gradual increase in wave periods could be observed in the time in terva l from the beginning of the short period precursors to the a r r i v a l of the DF phases. Another poss ib le mechanism for the e a r l y PKP onsets was discussed by Knopoff and MacDonald (1958), in a study of the e f fec t s of a postulated magnetic f i e l d in the inner core. They pred ic ted that a toro ida l f i e l d of about 5 x 10^ oersted was required to s p l i t PKIKP pulses into two components d i f f e r i n g in transmiss ion time by 12 sees, at A =140°, and that the pulse separat ion would be independent of frequency. They observed however, that a f i e l d of that strength was u n l i k e l y , s ince the seismological evidence for the e q u a l i t y of the travel t imes of the ax ia l and equator ia l PKIKP waves, with -6-allowance for e l l i p t i c i t y , placed an upper l i m i t of order 5 x 10"' oersted on the magnitude of the magnetic f i e l d , provided the f i e l d was t o r o i d a l . Bolt (1959, 1962, 1964) suggested that the precursors to the PKIKP waves (DF branch) could be caused by an add i t i ona l d i scont inuous increase in v e l o c i t y at a shallower level than the prev ious l y accepted boundary of the inner core . The replacement of a s ing le v e l o c i t y d i s c o n t i n u i t y by two ad-jacent jumps has the fo l l ow ing consequences for the t rave l time curve: l e t the jumps occur at r a d i i Rj and R2 in s ide the core of radius R c = 3,4-73 km where R 2 <C Rj R c . As shown in Figure 1.2a, the fami ly of rays through the mantle and outer core which are def ined by decreas ing valves of ray parameter p = d T / d £ , f i r s t generate the locus ABC as p rev ious l y desc r ibed . At C, a ray becomes tangentia1 to the spher ica l surface of radius R, and f o r steeper angles of incidence (smaller p) the rays are t o t a l l y i n t e r n a l l y r e f l e c t e d to generate the receding branch QG. The v e l o c i t y cont ras t or magnitude of the v e l o c i t y jump at R f determines the c r i t i c a l angle for to ta l in terna l r e f l e c t i o n , and hence determines the point at which r e f r a c t i o n beneath R] commences - i . e . i t determines the p o s i t i o n of the point G. The rays r e f r ac ted normally in the she l l of the core between r a d i i R, and R2 (R|> R2) generate the advancing branch GH. At H, a ray once again becomes tangent ia l to the spher ica l surface of radius R 2 , and the t o t a l l y r e -f l e c t e d rays generate the receding branch HD. As before, the magnitude of the v e l o c i t y jump at R determines the po s i t i on of the cusp at D, where r e f r a c -t ion beneath R 2 begins. The rays r e f r ac ted beneath R 2 (that i s , through the inner core) generate the advancing branch DF which has monotonica l ly decreas ing gradient p. The main point of the double-jump hypothesis is immediately apparent - i t produces an add i t i ona l sec t ion of the PKP curve - CGH - to which the observed precursors to PKIKP (DF branch) may be re fe r red (cf. Figure 1.1 for one v e l o c i t y jump). By extens ion of the preceding arguments to a t r i p l e v e l o c i t y jump hypothes is , i t can be seen that the m»TEL TIME VELOCITY (KM/SEC) " f -L--8-r e s u l t s would be the add i t i on of two branches of precursors to the DF branch. For any hypothesis to be u s e f u l , i t must be shown to lead to com-p le te t rave l t ime curves which do not v i o l a t e accepted cons t ra in t s such as those provided by ray theory. For example, ray theory demands that the receding branches CG and HD generated by t o t a l l y r e f l e c t e d raysi. must be concave^ upwards, while the advancing branches generated by re f rac ted rays must be concave downwards. The branches CG and GH must extend p r a c t i c a l l y s t r a i gh t from C and G r e s p e c t i v e l y , with the main curvature near G and H. Since the v e l o c i t y increases d i scont inuous ly at R] and R 2 , i t is expected that the only cusp to be assoc iated with large amplitudes would be the one at B. These cons t ra in t s are discussed more f u l l y in Appendix I. Empir ica l cons t ra in t s are provided by observed travel times fo r the ind iv idua l branches. Using the v e l o c i t y tables of Je f f reys (1939) as h is s t a r t i n g po int , Bolt found that he could r e t a i n the caus t i c at B, and at the same time e l im inate the negative v e l o c i t y gradient of J e f f r e y s ' s o lu t i on only by a reduct ion of J e f f r e y s ' v e l o c i t y ju s t beneath the region producing the c a u s t i c . His f i n a l procedure was to take the v e l o c i t y as constant at 10*03 km/sec. from th i s region to the f i r s t d i s c o n t i n u i t y ( i . e . from 0.52 R c to 0.48 R c or depths of 4,564 km to 4 ,602 km), assume the v e l o c i t i e s in the she l l bounded by Rj and R£ and in the inner core to be constant, and to c a l c u l a t e so lut ions for Rj and R 2 by bu i l d i n g up times for the three superimposed s h e l l s . Three so lu t i on s , T|, T 2 and T3 were determined, of which his preferred s o l u t i o n , T 9 , is i l l u s t r a t e d in Figure 1.2b. The corresponding v e l o c i t y model is shown in F igure 1.2c and the r e s u l t i n g dT/^A V s A curve in 1.2d. Bolt concluded that " the re is in the core a d i s c r e t e she l l with thickness of order 420 kms, and with a mean P v e l o c i t y near 10.31 km/sec. This she l l surrounds the inner core having mean radius 1,220 kms and mean P v e l o c i t y 11.22 km/sec. The mater ia l in the intermediate she l l is not l i k e l y to have marked r i g i d i t y . 20m 3 0 i 20 i» 30s 11* CDs l « m . 3 0 s rt W "~ F s A $ < I.'Jb. A-R Traveltime Ourvss, .a 10 o s i ; i ; r ~ i i i K, t i i i J i i i i ——. , s y s I / J i i / i " ' Si «o 1 , 1 Ro I.5c. A-R Velocity Model. R ,V , -1978 km(0.57Ru) 1 1 9.87 -10.00 km/sec R ,V , -1579 km(0.4555*) & 9.98 -10.41 km/sec R ,v -1249 k a f O . ^ Z B c ) 0 10.15 -IT.16 km/sec 144.1 3.501 130.0 3.455 156.4 2.764 125.0 2.650 160.5 2.161 IIO.O 1.956 I.3a. V e l o c i t y Solution Cusp T B I8m 37.2s I I8n; 48.1S J 20m 13. 2s G ian 46.9s H 20m 17.4s D ISm 33.2s ~i • o — — D — t i 1 \ 1 F \ I M fi» *4* t i 0 I.3d. A~R dt/dA Model. ISO Figure 1.5. Mans aid Randall I964 Velocity Kodel -10-w The inner core is 1 ike1y to be sol id . Jul ian • , -fit oi (1972) have s ince reported that they may have observed the phase PKJKP on seisinograms recorded at the LASA ar ray in Montana. The PKJKP phase is one which t r ave l s through the inner core as an S wave and elsewhere as a P wave. A l iquid inner core would prevent the convers ion from P to S. Adams and Randa l l ' s (1963) ana l y s i s of seismograms w r i t t e n from certain New Zealand earthquakes in the range 148°.<- A £~ 156° confirmed the existence of the l a t t e r part of the GH branch introduced by Bo l t , and ob-servations publ i shed by Nguyen-Hai .(1961) are now known to belong to th i s branch. In 1964, Adams and Randall used t rave l t ime data from 24 earthquakes and one s e r i e s of nuclear exp lo s ions , together with observat ions publ ished by other workers, to determine yet an add i t i ona l branch to the t rave l t ime curve der ived by Bo l t . Th i s add i t i ona l branch, IJ in the i r no t a t i on , was introduced to e x p l a i n the e a r l i e s t precursors observed, and necess i ta ted the i n t roduc t i on of an add i t i ona l v e l o c i t y d i s c o n t i n u i t y at a rad ius r = 0.57 R c (depth of 4,390 km) where R c is the radius of the e a r t h ' s co re . This d i s c o n t i n u i t y is above the level of r = 0.54 R c, which is the depth to which rays would have to penetrate the core before a c a u s t i c would be formed at B, as in B o l t ' s model and consequently the i r model does not e x h i b i t th i s feature. The second d i s c o n t i n u i t y was placed at a depth of 5,112 km. Their results were obtained by d i f f e r e n t i a t i n g the t rave l t ime data to obta in a P - ^ ' r e l a t i o n s h i p . The Herg lotz-Wiechert inversion technique, to be descr ibed l a t e r , was then used to obta in a v e l o c i t y - d e p t h model for the core . The d e t a i l s o f t he i r f i n a l model are shown in Figure 1.3a with the corresponding traveltime curve, v e l o c i t y model, and P-A curves in F igures 1.3b, 1.3c, and 1.3d r e s p e c t i v e l y . Thus Adams and Randall (1964) suggested the ex i s tence of two s h e l l s , each between 300 and 400 km th i ck , surrounding the inner core , with a small negative v e l o c i t y gradient in each s h e l l requ i red to exp la in the observed t rave l t imes. Hannon and Kovach (1966) v e l o c i t y f i l t e r e d data from several events in the range 130°to 160°. The data were recorded at the extenaed Tonto Forest Array in Ar izona. The v e l o c i t y f i l t e r i n g technique, in the form of time de lay ing and summation across the large ar ray , in conjunct ion with conventional amplitude and t rave l time a n a l y s i s , y i e l d e d evidence of two short period precursors , IJ and GH, to the DF a r r i v a l at d i s tances .1 ess than 140°. At leas t one of these precursors became an intermediate a r r i v a l to DF and AB at d i s tances greater than 145°. Their r e su l t s supported the model of Adams and Randal 1 (1964). From an ana ly s i s of t rave l t ime and amplitude d i s tance data of the World Wide Standardized Seismograph Network (WWSSN), Ergin(1967) found evidenc for the ex i s tence of the GH and IJ branches of Bolt and Adams and Randal l , and a t t r i b u t e d the branches to two discont inuous increases in v e l o c i t y at depths of 4,500 km and 4,685 km r e s p e c t i v e l y . Delays in PKIKP t rave l t ime observed at 153° and 162° were in terpreted as the r e s u l t of two low-ve loc i ty layers w i th in the inner core. Erg in obtained two AB branches instead of one, as in previous models, the second branch being 3-6 seconds la ter than the f i r s and in terpreted as the r e s u l t of a decrease in v e l o c i t y at a depth of 4,015 km To obta in AB t rave l t ime near 180° and beyond, he found that lower v e l o c i t y was needed at the base of the mantle, and that th i s reduct ion in v e l o c i t y resu l ted in the propagation of d i r e c t P up to 130° and beyond. Based on amplitude v a r i a t i o n s , the beginning of the PKIKP branch (that i s , the point D in the previous models) was put at around A =125°> a n d waves observed along the o o DF branch at d i s tances shorter than 125 , down to 106 were interpreted as ord inary r e f l e c t i o n s from the core boundary, which were not qui te ext inguished at the c r i t i c a l angle. Bolt (1968) found no evidence fo r the delays at 153° and 162° reported by Egin (1967). A group of small DF a r r i v a l s about 1 second e a r l i e r o o than expected in the range 1 5 6 ^ A ^ 160 seemed to ind i ca te e i ther that the curvature in that range had been overest imated, or the presence of a s l i g h t d i s c o n t i n u i t y in the inner core near a depth of 5,520 km. Gogna (1968) analysed t rave l times of about 3000 observations from 35 e a r t h -9.0 4000 4500 ' 5000 DEPTH (km) RG - }V &, - 1 8 2 1 km(0.524Rc) 9.99 -10.00 km/sec RpV,, -1521 km(0.437Rc) 10.15-10.17 km/sec R D , V 0 , -1226 km(0.353Rj 10.26-10.83 km/see Cusp A DT/D/1 (Peg) (Sec/Deg) & 140.0 3.18 J I42.O 2.61 H I5<5.0 2.08 D 120.0 I.21 I.4b. Buchbinder's Velocity f.'odel. I.4a. Buchbinder's Velocity Solution Figure 1.4. Buchbinder's 1971 Velocity Iylodel. - 1 3 -quakes and confirmed, once more, the ex is tence of the GH branch of o o precursors as postulated by Bo l t . On h i s model, GH extended from 130 to 153 > but the p o s s i b i l i t y of i t extending beyond 153° was not ru led out. In con-t r a s t , he obtained no evidence in support of the add i t i ona l IJ branch as suggested by Adams and Randal l . Buchbinder (1970 examined trave l times and amplitudes of PKP and higher mu l t i p le K phases from a worldwide d i s t r i b u t i o n of short period seismograms. In h is r e s u l t i n g v e l o c i t y model, the v e l o c i t y at the bottom of the mantle is 13.44 km/sec. r e s u l t i n g in a negative v e l o c i t y gradient,{he v e l o c i t y decreasing from 13.6 km/sec. at a depth of 2,700 km to 13.44 km/sec. at the core-mantle boundary, placed at a depth of 2,892 km. The caus t i c B is placed at A =143°. The observed t rave l times and amplitudes of PKP were in terpreted to give the turn ing points D and G near 120° and 140° r e s p e c t i v e l y rather than at 110° and 125° as in previous models. Together with the t rave l times and amplitudes of the higher mu l t i p l e K phases these re su l ted in a v e l o c i t y d i s c o n t i n u i t y of 0.01 km/sec. at a depth of 4,550 km to account fo r the e a r l i e s t forerunners of PKP,and a second d i s c o n t i n u i t y of 0.02 km/sec at a depth of 4,850 km to account fo r the la ter forerunners . Since he could not reso lve the forerunner data into branches, he was not able to def ine the d i s c o n t i n u i t y p r e c i s e l y , but instead produced the two branches of forerunners to i l l u s t r a t e the poss ib le range of t rave l t imes and amplitudes. The top of the inner core boundary was placed at a depth of 5,145 km + 10 km, and represented a d i s c o n t i n u i t y of 0.576 km/sec. These re su l t s are l i s t e d in Figure 1.4a. Figures 1.4b, c and d show Buchbinders v e l o c i t y model, the reduced t rave l t imes , and the P - A curves correspo nding to h is model. The branches CGG1 and U J 1 of Buchbinder 's model correspond to the IJ and GH branches, r e s p e c t i v e l y , of the model of Adams and Randall (1964). A s i g n i f i c a n t d i f f e r e n c e is that on Buchbinder 1 s model, the sect ions GG1 and J J 1 of the respect i ve branches -14-represent phases p a r t i a l l y r e f l e c t e d o f f the corresponding boundar ies , whi l s t they represent the t ransmit ted phases on the Adams and Randall mode I. 1.3 Seismic V e l o c i t i e s and Physica l Proper t ies of the Core. 1.3.1 Central Density of the Core The v e l o c i t i e s of P and S body waves, o t and ^ r e s p e c t i v e l y , are given by the equations (Bu l len , 19(53). foe2 = k + 4/3 A. f^ 2 ~<a where k is the bulk modulus or incompress ib i1 i t y of the medium p. is the dens i t y , and /X is the corresponding r i g i d i t y . From these e q u a t i o n s , we can obta in the quant i ty 0 = k/p = oC 2 - V3/>0 2 The v a r i a t i o n of d e n s i t y p with depth for a non-homogenous reg ion is given by (Bu l len , 1963). where g is the g r a v i t a t i o n a l fo rce per uni t mass and & = dk/cfp - c f dtf/riz. 0) Here the quant i ty p denotes h y d r o s t a t i c pressure . For a chemica l l y homogenous medium, 0 is u n i t y . For values greater than u n i t y , 9 g ives a measure of the a d d i t i o n c f heavy mater ia l in excess of pure ly h y d r o s t a t i c compression of a homogenous medium. The J e f f r e y s ' model for the c o r e , with i t s large negat ive v e l o c i t y gradient in the t r a n s i s t i o n reg ion y i e l d s a value of dfifdz - ~ '5 x !0 e r r / s e c 2 Assuming g r a v i t y at a depth of 5000 km to be approximately 600 c . p / sec , equat ion (1) becomes 8 = dk/dp + 25 Reasonable values for dk/dp in the c o r e , based on the theory of s o l i d s t a i r , range from 3 to 5 u n i t s . Consequent ly , wi th a v e l o c i t y decrease in the t r a n s i s t i o n r e g i o n , as in the J e f f r e y ' s model, Q becomes approximately 30 u n i t s . Th is value for 8 is not only greater than at any other reg ion -15-in the ea r th , but implies the admixture of a cons iderable proport ion of mater-i a l which is denser than inthe outer core. This e n t a i l s , in turn , a large dens i ty increase at the inner core boundary, and a centra l den i s ty of about 3 20g/cm . On the other hand, Bo l t ' s - (1964) T 2 model has zero v e l o c i t y gradient in the t r a n s i t i o n reg ion, and 0 is then of the order of 5 un i t s , implying 3 a cent ra l dens i ty of about 15 gm/cm . This lower value of cent ra l dens i ty is more compatible with re su l t s of recent laboratory experiments using t rans ient shock wave techniques to in fer the e l a s t i c propert ies of i ron at 4 m e q a b a r s These r e s u l t s ind icate a maximum t e r r e s t r i a l centra l dens i ty of the order of 13g/cm3. The e f f e c t of po s i t i ve v e l o c i t y gradients in the t r a n s i t i o n reg ion, such as those in Bmchbinder 1s (1971) model would be to reduce the index 6 even f u r t h e r , and to br ing the cent ra l dens i ty even lower than in the Bolt case. The e f f e c t of combined po s i t i ve and negative v e l o c i t y grad ients , as in the Adams and Randall (1964) model, would, prov id ing the negative gradient is reasonably smal l , r e s u l t in a centra l dens i ty for the core s l i g h t l y higher than that pred icted in the Bolt case. It is apparent, then, that core v e l o c i t y models do provide empir ica l r e s t r a i n t s on acceptable dens i ty p r o f i l e s with the core , and v i ce versa. 1 .3.2 S o l i d i t y o f the Inner Core That the outer core is in a f l u i d s tate has been es tab l i shed by the f a i l u r e to observe transmitted S waves in the appropr iate d i s tance range. The v e l o c i t y of S waves is a d i r e c t func t ion of the r i g i d i t y of the medium, which, for a l i q u i d , has the value zero. At the boundaries of the t r a n s i t i o n region in the Bolt (1964) model, the v e l o c i t y of P waves, oc , increases by 3% and 9%, corresponding to 6% 2 and 18% increases in a • If the t r a n s i t i o n region were in a l i q u i d s ta te , the v e l o c i t y of P waves through i t would be given by (since /X. = 0) k - pc* 2 The jump in oc of 6% at the f i r s t boundary requires a corresponding increase Figure 1.5. The Warramunga seismic array* The inset shows the location of the array i n the stable shield area. - 1 7 -in k of about 6%, s ince the dens i ty p is not l i k e l y to increase by any large amount. Bul len (1963) has ind icated that an increase in k of th i s magnitude is u n l i k e l y , and consequently the equation has to be modif ied to include the r i g i d i t y term JJ^ i . e . k + V3 IL= f o £ 2 With th i s mod i f i c a t i on , some of the increase in ot at the boundary can be taken up by the r i g i d i t y term, r e s u l t i n g in a smaller increase in k at the boundary. This impl ies that the materia l in the t r a n s i t i o n zone has s i g n i f i c a n t r i g i d i t y , and is therefore not completely l i q u i d as in the 2 outer core. S i m i l a r l y , the larger jump in ot of 18% at the boundary of the inner core requi res an even larger increase in r i g i d i t y in the inner core , i nd i ca t i n g the inner core to be completely s o l i d . Bul len (1963i p.243) advances fur ther arguments in favor of a s o l i d inner core. That the inner core is s o l i d has been confirmed recen t l y by observat ion of the PKJKP phase by Ju l i an « t a l ( l 9 7 2 ) 1.4 The Warramunga Seismic Array The Warramunga seismic array (WRA) s i tuated near Tennant Creek in Northern T e r r i t o r y , A u s t r a l i a is one of four seismic arrays of the United Kingdom Atomic Energy Author i ty (U.K.A.E.A.) type which were deployed as a means of de tec t i ng and i d e n t i f y i n g underground nuclear e x p l o s i i o n s . The other three arrays are located in Canada (YKA), Southern India (GBA) and in B r a z i l . Another B r i t i s h designed array (EKA), smaller than any of the o thers , is located in Scot land. WRA began operat ion in 1965. It is s i tua ted on gran i te outcrops in the s tab le Precambrian sh i e l d area, approximately 500 km from the nearest po'mt of the Au s t r a l i an coast (see inset , F igure 1 . 5 ) . It is a medium aperture array c o n s i s t i n g of twenty shor t -per iod v e r t i c a l component Willmore Mk II seismometers with a natural period of one second and a damping f ac to r of 0.6. The seismometers are arranged in two l ines approx-Table 1.1 Coordinates of the Individual Seismometers of the Warramunga Seismic Array Seismometer Latitude E Longitude S Cartesian Coordinates X , km Y, km. Elevation, • feet • E l 134.34776 19,96097 -0.310 -1.476 1294.85 B2 134.35250 19.94417 0.183 0.373 1284.27 133 134.356S1 19.92453 0. 638 2. 558 1265.89 134 134.3G050 19.90497 1.025 4.724 1252.65 B5 134.36954 19.87971 1.762 7. 519 1218.72 336 134.36352 19.S5645 1.863 10.095 1200.07 B7 134,3S0S0 19.84227 3.150 11.662 • 1203.09 B3 134.3S213 19.81541 3.291 14/635 1176.96 B9 134.38530 19.79286 3. 623 • 17.132 1184.94 BIO 134.39418 19.76862 4.554 19.816 3188.50 . Rl 134.340S5 19,94411 -1.033 0.391 1281.07 R2 134.36555 19.94S90 1.552 -0.140 1265.19 R3 134.33S31 19.95000 3.934. -0.264 1236.49 R4 134.40302 19.95181 5.998' -0.465 . .1228.55 R5* • 134.42764 19.95949 8.052 -1.314 1217.83 R6 134.45578 19.95525 10.998 -0.846 1155.42 ' R7 134.47742 19.95667 13.263 -1.004 1167.15 R8 134.50191 19.95726 15.826 -1.074 1212.53 R9 134.51631 19.95912 17.334 -1.2S1 1190.GS RIO 134.54195 19.96109 20.016 -1.502 11G1.54 * moved to new site 18 September, 1968. R2 and R6 have subsequently been moved to new sites. The geographic coordinates of the origin of the cartesian coordinate system are 19.94777 S, 134.350S1 E . - 1 9 -imately at r i gh t angles to each other (see Figure 1 .5), each l i ne being 2 2 . 5 km long and c o n s i s t i n g of ten seismometers equa l l y spaced. The geographic co -ord inates of the ind iv idua l seismometers and the Cartes ian coordinates r e -l a t i v e to the point of i n t e r s e c t i o n of the two arms of the array are l i s t e d in Table 1.1. The seismometers are operated at a displacement magn i f i ca t ion of about 250,000. The s i gna l s from each seismometer are telemetered to a centra l r e -cording s t a t i o n where they are recorded s imultaneously with a d i g i t a l time code on 24-track FM magnetic tape. The p r e c i s i o n t iming and FM telemetry system are descr ibed in d e t a i l by Truscot t ( 1 9 6 4 ) , and Keen et a l . ( 1 9 6 5 ) . A l l data are forwarded to the U.K.A.E.A. center in England. A dup l i ca te recorder is operated by the Au s t r a l i an National U n i v e r s i -ty. As a means of improving i t s e f f i c i e n c y , a small array c o n s i s t i n g of two l i n e s , each of length 2 . 5 k i lometres and conta in ing f i v e v e r t i c a l component seismometers, was i n s t a l l e d in 1967 , p a r a l l e l to the main arms of the ar ray . This small array or ' c l u s t e r ' d i s c r im ina tes against random noise by d i r e c t summation, and t r i ggers the tape record ing system only when the conelator output exceeds a s p e c i f i e d t r i gger l e v e l . The mode of operat ion is s i m i l a r to that of the 2k element c l u s te r in use at YKA and is descr ibed in Whiteway (1965). 1.5 Use of an Array in Determining Structure Before the development of seismic a r rays , the t rave l times of P and S waves provided the most r e l i a b l e means for i nve s t i g a t i ng the s t ruc ture of the e a r t h ' s i n t e r i o r . D i f f e r e n t i a t i o n of the smoothed t rave l t ime-d i s t ance ( T - A) curves y ie lded the f i r s t d e r i v a t i v e , dT/dA , as a funct ion of d i s t ance , A This d e r i v a t i v e was then used in the Herglotz-Wiechert in tegra l (to be descr ibed l a te r ) in order to determine a v e l o c i t y - d e p t h p r o f i l e fo r the region under con s i de ra t i on . Extensive use of th is method has y ie lded ve l o c i t y - dep th p r o f i l e s that are in reasonable agreement except in the upper 1000 kms of -20-of the mantle and the anomalous t r a n s i t i o n region of the core . However, an array can be used to measure dT/dA d i r e c t l y , and is capable of supply ing more f i n e - s t r u c t u r e d e t a i l than the conventional t rave l t ime method. A i so , the s ignal enhancing a b i l i t y of an array makes poss ib le the detec t ion and i d e n t i f i c a t i o n of phases too small to be detected on a s i ng le seismcgram. These propert ies of an array are u t i l i z e d f u l l y in th i s study. Past use of seismic arrays in the i nves t i ga t i on of the s t ruc ture of the e a r t h ' s i n t e r i o r has been conf ined almost e x c l u s i v e l y to s tudies of the e a r t h ' s mantle. Such studies were pioneered by Niazi and Anderson (1965) who invest igated the upper mantle s t ruc ture beneath western North America by measurement of P-wave t rave l t ime gradients at the Tonto Forest Array (TFSO) in centra l Ar izona. The f i r s t s tud ies of th i s nature at WRA were made by Cleary et al (1968) who concluded, among other things, that the dT/dA measurements were s t rong ly perturbed by va r i a t i on s in s t ruc ture underneath the a r ray . Wright (1968) using dT/dA measurements at WRA from 30 earthquakes in the Marianas Islands and surrounding regions found evidence for a low v e l o c i t y layer at a depth c lose to 800 km in the mantle. His complete work on the lower mantle using WRA data is contained in h is doctora l thes i s (Wright 1970). As the present study is of s im i l a r type to that made by Wright, many of the techniques of measurement and ana lys i s devised and used by him have been adopted for use in th is study. These w i l l be f u l l y descr ibed in Chapter 2. -21-CHAPTER 2 MEASUREMENT AND ANALYSIS 2.1 Determination of dT/dA and azimuth from a se i s m i c a r r a y .  2.1.1 Some P o s s i b l e Methods ' In order .to o b t a i n a v e l o c i t y d i s t r i b u t i o n f o r the e a r t h ' s core, knowledge i s r e q u i r e d of the v a r i a t i o n of the t r a v e l t i m e g r a d i e n t , dT/dA , w i t h depth i n the core. T h i s i s obtained by determining dT/dA for those s e i s m i c waves which are known to pass through the earth's core, a r r i v i n g at the su r f a c e a t d i s t a n c e s i n the range 110° to 180° for PKP. Both dT/dA and the azimuth of a r r i v a l a t a se i s m i c a r r a y can be estimated from the a r r i v a l times a t the i n d i v i d u a l seismometers In at l e a s t t h ree ways: (1) A C o r r e l a t i o n Method: The a p p r o p r i a t e time delays corresponding to tu n i n g to a p a r t i c u l a r azimuth and dT/dA f o r each of the two l i n e s are i n s e r t e d for each seismometer of the a r r a y , and a summed output determined f o r each l i n e . The normalised output amplitudes of the summed l i n e s are c r o s s - c o r r e l a t e d assuming a s i n u s o i d a l s i g n a l and the c o r r e l a t o r output, determined as a f u n c t i o n of azimuth and dT/dA , givesmaximum value at the s i g n a l azimuth and v e l o c i t y , ( B i r t i l l and Wh?teway, 1965). This method was explored i n some d e t a i l by Wright (1970) who concluded that i t gave i n s u f f i c i e n t l y high p r e c i s i o n to be of any use i n h i s work on mantle P-wave v e l o c i t i e s . Even higher p r e c i s i o n i s r e q u i r e d when d e a l i n g w i t h the earth's core, s i n c e the dT/dA values f o r the core are at most h a l f as large as the dT/dA values f o r the mantle. A new c o r r e l a t i o n technique which may give more r e l i a b l e azimuth and dT/dA measurements has been d e s c r i b e d by Muirhead(1968). -- 2 2 -Figure 2.1 Diagram to illustrate a least-squares method of estimating dT/dA and azimuth. - 2 3 -(2) A Fourier Transform Technique: dA /dT and azimuth values are determined at a number of di f ferent frequencies by measurement of differences in phase angle of Fourier spectral densit ies across a fixed array (SSna et a l , 1964). Again, however, this method does not appear to be precise enough for use in deep earth -investigations. (3) A Least-Squares Technique: The best straight l ine is f i t t ed to the set of arr ival times at the individual seismometers by the method of least squares. The individual arr ival times are functions of the azimuth of a r r i v a l , the arr iva l time at the or ig in of the array and the traveltime gradient or apparent ve loc i ty . Consequently their least-squares f i t results in the best possible estimate of a l l three parameters, for the particular event being analysed. Since this method was used in obtaining the dT/dA measurements in this study, i t wi l l now be described in more d e t a i l . 2.1.2 Details of the Least Squares Method The least squares method has been described by Kelly (1964) and Otsuka (1966). Let the or ig in 0 in Figure 2.1 be the point of intersection of the two arms, of WRA, and take a cartesian coordinate system with the y and x axes pointing north and east respectively, in the direct ion of the two arms of the array. For a medium aperture array such as WRA, and at the distances in question, the wave front can be assumed to be plane provided the structure of the crust and upper mantle in the v i c i n i t y of the array is reasonably simple. Consider a seismometer at Sj , distance R; from Oj and let £S;Oy by flj. Suppose a P wave arr ival crosses the array from azimuth 0 with apparent -24-surface v e l o c i t y V, and l e t the a r r i v a l time at Sj be T- subject to an e r ror e ; . Let the a r r i v a l time at the o r i g i n of the array be To. If can be seen from F igure 2 .1 that * t = To-T; - V - ' ^ R i C o s <f> Gs&L + RL'Sin<f>Sm®i) = To-Ji - V ' Qiji Cos $ -h Xi Sin (fi) where y t- = R £ Cos G ; . and Xi=Ri£in&c . Fur ther m o r e } if w e let P= S i n ^ / v and a = Cos tf/v ? +ben e-t = To - T i - (*j,-a+ xiP) and i f N is the number of seismometers in the array, J * N £ CTO-T; - C * . P + ^ Q ) ) 2 To obta in best est imates of the three unknowns P, Q,} and To, and hence best estimates of azimuth 0, apparent v e l o c i t y V, and t rave l t ime T o at the o r i g i n of the a r ray , the least squares cond i t i on that the sum> of the e r r o r s , or f (P^ Q., To) , be a mimimum is used. Thus This gives r i s e to the three normal equat ions: o r , i f W e let; x,'tj; = Cxup ? ^ ^ i 2 - CxxU e +c ? thfi oju/rhons carv be written*, Qxv]p + cvynd •+ C Y T J - Tvlx = © -25-The v a l u e s of x j and y j , the c a r t e s i a n c o o r d i n a t e s of the i n d i v i d u a l seismometers,are known and the values of T; are determined by measure-ment of the a r r i v a l times at each of the seismometers. Consequently the three normal equations can be solved to give the best estimates of P, Q, and To. From these, we o b t a i n : w=(p 2 + n 2 ) " 2 -x * ' Z. apparent v e l o c i t y of P wave dT/dA = r d / V a t the deepest po i n t of the r aY ( ro ' s t n e m e a n r a d i u s of the earth) and 0 = Tan"' (P/Q) = azimuth of a r r i v a l a t the a r r a y . A knowledge of To, the a r r i v a l time at the o r i g i n of the a r r a y , and the N.O.A.A. (National Oceanic and Atmospheric A d m i n i s t r a t i o n of the U.S.) origin time of the event enables the t r a v e l t i m e t o be determined. Knowledge of the e r r o r s i n the c a l c u l a t e d q u a n t i t i e s , p a r t i c u l a r l y i n the dT/dA v a l u e s , are e s s e n t i a l i f s t a t i s t i c a l s i g n i f i c a n c e t e s t s are to be a p p l i e d i n the smoothing of the data a t a l a t e r stage. On the assumption that the e r r o r s e; i n the measured onset times are i n -dependent Gaussion v a r i a b l e s , each w i t h mean zero and v a r i a n c e Kelly (1964) showed that the root-mean-square e r r o r s i n Vj,dT/dA and 0 are given by:-(&vMr.ffl.s. = (S(dT/dA)/ dT/d/s)r m_s - «- v / ( N D ) % a r x Cos2? -2G>vCr,i j) Sin <?5 Cos <f> + Var tj S m 2 ^ ) 2 -tU) = V / C N O ) ! (Varx Smz$ + 2 G>v(*, y) S)n$G>s $ ' r. n>'S- ' i (in rod .cms) -f Var y Cos* <f>) X a. u>Ur* D= Var ac Vary - (.Coy C*»y) ) ' Vary = VN I C y t - y ) z -26-The e s t i m a t i o n of the q u a n t i t y tr , poses a s i i g h t problem. C o r b i s h l e y (1970) approached the problem by e s t i m a t i n g a random readi n g e r r o r f o r a number of events o c c u r r i n g i n a small r e g i o n of the e a r t h . Wright (1970) p r e f e r r e d to c a l c u l a t e a value of ~ v f o r each set of r e l a t i v e onset times on the grounds t h a t the e r r o r i n each onset depends on four main f a c t o r s : (1) The waveform o f the event, s i n c e low frequency events give l e s s c l e a r l y d e f i n e d peaks and zer o s . (2) The s i g n a l - t o - n o i s e r a t i o s i n c e random bu r s t s o f noise cause spurious changes i n waveform from one seismometer to the next. (3) The instrumental constants of the seismometers which vary s l i g h t l y from seismometer to seismometer and (4) r a p i d v a r i a t i o n i n l o c a l s t r u c t u r e , an azimuth-and-distance-dependent f e a t u r e . Each value o f r e s i d u a l €.i i s c a l c u l a t e d a f t e r P, Q. and To have been determined. Assuming that the r e s i d u a l s have zero mean and v a r i a n c e o"^-c h a r a c t e r i s t i c of that p a r t i c u l a r set of a r r i v a l times, then «• = . £ « i / N - 3 The value of or thus determined c o n t a i n s not only random re a d i n g e r r o r s , but a l s o systematic e r r o r s due to p o s s i b l e d i f f e r e n c e s i n instrumental constants of the i n d i v i d u a l seismometers and r a p i d v a r i a t i o n s i n l o c a l s t r u c t u r e . 2.2 Measurement Techniques 2.2.1. Data A v a i l a b l e f o r the study As mentioned p r e v i o u s l y , data acquired by the A u s t r a l i a n N a t i o n a l U n i v e r s i t y f o r WRA are recorded on 24 Channel FM analog magnetic tape. At the time the data used i n t h i s study were acquired , -28-a system of d i g i t a l convers ion and ana ly s i s had not been i n s t i t u t e d . Consequently, the usual procedure was to play back the data onto chart paper. F i r s t l y , the seismic s i gna l s on tape were f i l t e r e d using narrow bandpass f i l t e r s (0.4 to 2.0Hz) having i den t i ca l c h a r a c t e r i s t i c s . A 16-channel chart recorder was the only one ava i l ab l e at the time, and the twenty seismic traces were therefore played out in a two-step procedure. The s i gna l s from one l i n e of the array, together with the s i gna l s from the end seismometers of the other l i n e and a t iming s ignal were reproduced in one pass. The l i ne pos i t ions were reversed and the procedure repeated. It was o f ten noted that the amplitude of the seismic traces var ied cons iderab ly over the time range of i n t e r e s t . This necess i tated d u p l i c a t i o n of c e r t a i n parts of the records at higher or lower gain set t ings in order to obta in amplitudes best su i ted fo r a n a l y s i s . The playback speed (40 mm/sec) used in obta in ing the paper records enabled high re so lu t i on t iming to be achieved. An example of the paper seismograms is shown in Figure 2.2. Note the c lear waveforms which enabled p rec i s i on measurement of the r e l a t i v e d i s tances between peaks or troughs of a given cyc le across the array. Seismograms from the 115 earthquakes used in th i s study were reproduced in the above manner. They represented data over the d i s tance range of 113° to 176° although th i s range is not uniformly covered due to the loca t ion of a c t i ve seismic zones with respect to WRA. The major i ty of events occurred over the d is tances of 130° to 150°. Among other things, th i s uneven spread of the data l a te r resu l ted in some d i f f i c u l t y in smoothing the measured dT/dA va lues . The sparseness of data in the ranges 120° to 130° and 150° to 160° led to d i f f i c u l t y in determining p r e c i s e l y the end points of the -29-precursor branches of the traveltime curve, a point which wi l l be discussed la ter . A l i s t of the earthquakes used in this study is given in Appendix II. 2.2.2 Measurement of dT/dA and Azimuth If precision in measurement of dT/dA comparable to precision obtained from an array such as LASA is to be obtained on a medium aperture array such as WRA, onset times must be measured with a standard de-v iat ion of between 0.01 and 0 .03 seconds. The measurement of P onset times with such accuracy is extremely d i f f i c u l t , i f not impossible. Measurement, instead, of re lat ive P times obtained by matching P waves recorded at each seismometer can y ie ld accuracy of the required degree. Since the method was f i r s t described by Evernden (1953), the technique of measuring re lat ive onset times by matching waveforms has been commonly used in the calculat ion of phase ve loc i t ies of surface waves across t r ipar t i te arrays. Matching surface waves with periods of 20 seconds or larger to an accuracy of 0.2 seconds is comparable to matching P waves with periods, of about 1 second to an accuracy of 0.01 sec. With the chart speed of 40mm/sec used for the paper records, waveforms could be matched to an accuracy of ± 0 . 5 mm, which corresponds to an average timing error of 0.012 sec. Thus the precision in timing measurement was sat isfactory. The technique used in matching the waveforms is as follows: (1) The paper speeds on the two lines of the array, the Red and Blue l i n e s , were determined by averaging at least f ive sets of measurements of the distance between 5-sec. intervals in the region of measurement of dT/dA , for each l ine . The two values thus obtained were further averaged to give an overall value for the particular event. It was -30-c » lure 2.3.. Matching technique for determining; relative arrival times at each seismometer. From the record at pit R4(a), a family of curves with different ampli-tudes is constructed (b), and matched with tho record at tho other pits (c), (d). -31-found in p rac t i ce that th is o v e r a l l speed was wi th in 0.10 mm/sec of the speed of e i the r l i n e , thus e l i m i n a t i n g the necess i ty of a c o r r e c t i o n term. (2) Para l lax co r rec t i ons in m i l l imet res were determined fo r each trace by observing the d i s tance dev ia t ions in the onset of square wave pulses app l ied s imultaneously to a l l traces at the s t a r t of the record. The cor rec t ions were never more than 1 mm with the vast major i ty center ing around 0.5 mm. (3) A t race was se lec ted from both l i nes of the array, which appeared to be the leas t contaminated by noise or other i n t e r -ferences (Figure 2 .3a) . One cyc le of the phase whose dT/dA value was to be measured, was traced onto transparent paper. The v e r t i c a l l i ne corresponding to a chosen reference time mark was drawn on the transparent paper, as was the hor izonta l l i ne i nd i ca t i n g the ze ro -si gnoJ level of the t race . In order to compensate for v a r i a t i on s in amplitude between traces for a p a r t i c u l a r event, the se lected curve was used to cons t ruct , on the same transparent paper, a se r ie s of curves of d i f f e r e n t amplitudes, as shown in F igure 2.3b. (k) Using th i s transparent paper as an over l ay , the curves on i t were matched with each seismometer output for that event, and the pos i t i on of the f i r s t zero of each seismometer output was determined r e l a t i v e to the reference time mark. |n add i t i on the r e l a t i v e times of the f i r s t peak and f i r s t trough of the appropr iate cyc le for each trace were measured. Thus three sets of r e l a t i v e onset times were determined for the p a r t i c u l a r phase of the p a r t i c u l a r event. In some cases, onl-yone or two sets of r e l a t i v e times could be determined, depending on the c l a r i t y of the waveform. -32-The reason for determining more than one set of onset times, and hence dT/dA and azimuth, for a p a r t i c u l a r phase is to compensate to some extent fo r the changes in waveform fromone seismometer to another. These changes may be due to rap id va r i a t i on s in loca l s t ruc ture beneath the array or even to mu l t ip le a r r i v a l s caused by s t ruc ture at the deepest point of the ray. Random noise e f f e c t s could a l so play an important part in causing changes of waveform for the precursors to the PKIKP phases, which were of ten f a r smaller than PKIKP in the e a r l y parts of the appropr iate d is tance ranges. Each set of a r r i v a l times, and consequently each dT/dA and azimuth determin-a t i on for a p a r t i c u l a r phase, was regarded as independent. By means of a computer program (Wright, 1970) the a r r i v a l times were used to c a l c u l a t e a least square value of dT/dA , azimuth 0 and a r r i v a l time To at the o r i g i n of the array. Residuals of the measured times r e l a t i v e to the i r least squares f i t , were determined. If a seismometer res idua l was found to exceed 0.02 seconds, the corresponding onset time was d iscarded and the e n t i r e procedure repeated (insing the remaining times. In poorer f i t s from lower q u a l i t y data, a r e j e c t i o n value of a 0.04 seconds was used. This r e j e c t i o n procedure ensured that the standard dev iat ions of the onset times were kept less than 0.03 seconds, and consequently ensured the high p rec i s i on of th i s method of determining dT/dA and azimuth. The root-mean-square er ror s in the least squares dT/dA and azimuth determinat ion were a l so ca l cu l a ted by the program to f a c i l i t a t e future weighting of the data. Figure 2.4. Measured Traveltimes o 5 S o I REDUCED TRAVEL TIME a - D F VS DELTA g-l a - GH z - U x - AB B B J 1 I* 1 1 I : I I 1 1 1 1 T 1 1 1 1 110.0 115.0 120.3 125.3 130.3 135.0 140.0 145.0 150.0 155.0 160.0 • 165.0 170.0 175.0 183.3 DELTA f DEG) Figure 2.5. Reduced Traveltimes. - 3 5 -This matching waveform technique and subsequent c a l -culations were carried out for any coherent signal observed on the seisinograms within the time range of interest. It was often nec-essary, in the early parts jf the appropriate distances ranges, to resort to the high amplitude data in order to measure the precursors to PKIKP. If a' particular phase was identif ied as belonging to one of the time travel branches on the basis of i ts dT/dA value and a rough estimate of i ts trowel .-hune i , the actual travel-time was determined by measuring the onset time of a clear wave and sub-tracting from i t the or igin time as given by N.O.A.A. It was often d i f f i c u l t to pick the onset of the phase on the sei sinogram accur-ately, because of the background noise and in some cases where a l l phases arrived closely together, because of interference from preceding phases, for example in the range 140°- 145*. This resulted in quite a large scatter of travel-times and, as wi l l be seen later , in the residuals with respect to the smoothed travel -times. To ensure uniformity, e l l i p t i c i t y and, focal depth cor-rections as described in Apperidix III were applied to the travel-times. The 115 events analysed in the above manner yielded a total of 574 d T / d A and 188 traveltime measurements for a l l PKP phases. Figures 2.4 and 2.5 show the traveltime and reduced traveltime measurements. These wil l be discussed further in Chapter 3. The dT/dA measurements are shown in Figure 2.7. The large scatter exhibited by the dT/dA values is mainly a con-sequence of systematic errors in their measurement. These errors now wil l be discussed in more d e t a i l . -36-2.2.3. Sources of Systematic Error in dT/dA and Azimuth Measurements ' Systematic e r ro r s in dT/dA and azimuth measurements can be a t t r i b u t e d to four main sources: (1) The e f f e c t of d i f f e rences in seismometer constants (2) D i f ferences in e l eva t i on of the ind iv idua l seismometers (3) El 1 i p t i c i t y of the earth and (4) The e f f e c t of s t ruc ture beneath the array. (1) The e f f e c t of d i f f e rences in the seismometer constants. . Muir-head (1968) invest igated the problem of phase s h i f t s due to small d i f f e r e n c e s is seismometer constants in a medium aperture "array, and showed that fo r an instrument with a natural frequency of 0 .9 Hz record ing a sinusoidal motion of a period of 1 s e c , the f i r s t zero of cross -over point of the recorded wave would be 0.04 sec. l a ter than i f the instrument had a natural frequency of 1.1Hz. He a l s o showed that the e f f e c t of d i f f e rence s in the damping constants of the seismometers was of the second order and could be s a fe l y neg lected. The seismometer c a l i b r a t i o n scheme in operat ion fo r U.K.A.E.A. arrays is descr ibed by Keen et al (1965). The natural f requencies of the instruments at WRA show small d i f f e rences from one seismometer to another, with a maximum d i f f e rence of about 0.18 Hz between R6 and B9. In a d d i t i o n , the frequencies vary s l i g h t l y from day to day, again with a maximum v a r i a t i o n of about 0.20 H? from 0 .9 to 1.1 Hz. In p r i n c i p l e , i t is poss ib le to remove such e r r o r s . Wright (1970) invest igated th i s problem f u l l y , and ob-tained co r rec t i on s to d T / d A with a maximum valve of about 0.05 sec/ deg - . He concluded that " the e f f e c t of d i f f e rences in the natural periods of the seismometers on the onset times i s , in almost a l l instances, too small to produce apprec iab le scat ter of systematic e r ro r s in the values of dT /dA and azimuth". He invest igated - 3 7 -parametergof the order of 8 - 10 szc/deq in magnitude, and consequent-ly a c o r r e c t i o n of 0*05 Sec \def^ to these values is almost n e g l i g i b l e . In the case of the e a r t h ' s core, however, where the PKIKP branch has a value of d T / d A ranging from about 2.0 Sec/deg -to 0.0 sec/cteg at 180°, a c o r r e c t i o n of that magnitude i s more s i g n i f i c a n t . Never-the le s s , in the face of co r rec t ions for loca l s t ruc ture (to be descr ibed l a te r ) which amounted in some instances to as much as 0.9 Sec/detj, the c o r r e c t i o n for d i f f e rence s in seismometer constants seemed i n s i g n i f i c a n t , and was consequently ignored in th i s study. (2) D i f ferences in E levat ion of Individual Seismometers. The d i f f e r e n c e in e leva t i on of the highest and lowest seismometers of the array is less than 0.05 km. At the large d i s tances involved, the systematic d i f f e r e n c e in a r r i v a l time due to th i s d i f f e r e n c e in e l eva t i on is less than 0.01 sec. assuming a P wave v e l o c i t y of 6km/sec at the e a r t h ' s sur face, and can consequently be ignored. (3) E l l i p t i c i t y of the Earth. The e l l i p t i c i t y c o r r e c t i o n to t rave l t imes is given by the fo l l ow ing equation (see Appendix III). 6T - -f ( A X h o + h.) It is a l so poss ib le to co r rec t the d T / d A values for e l l i p t i c i t y by c a l c u l a t i n g ST for a number of imaginary epicentres at- regular i n te rva l s of d i s tance and azimuth, and d i f f e r e n t i a t i n g the r e s u l t i n g tables to obta in S( d T / d A ). - the e l l i p t i c i t y c o r r e c t i o n for the dT/dA measurements. Wright (1970) c a r r i e d out th i s procedure and obtained maximum cor rec t ions of about 0.024 sec/olecj . On the basis of larger co r rec t i ons to be determined such as those involved in c o r r e c t i n g for s t ructure under the array and in r e s t r a i n i n g the smoothed dT/dA values to an acceptable t rave l time .curve , he p re f fe r red to ignore the e l l i p t i c i t y co r rec t ions to d T / d A . The -38-same procedure is adopted in th i s study for s im i l a r reasons. (4) E f f e c t of Structure Underneath the Array. By f a r the most ser ious source of systematic e r ro r s in dT/dA and azimuth values determined'at WRA, is that due to the s t ruc tu re -o f the crus t and upper mantle in the v i c i n i t y of the array. One s t r ingent requirement of an ideal arrays ite is that the geology should be reasonably homogenous and uncomplicated by major e l a s t i c d i s c o n t i n u i t i e s . I n i t i a l l y , i t was expected that the WRA s i tuated in the middle of the Aus t r a l i an stable sh ie ld f u l f i l l e d these requirements. However, when the array records of the nuclear exp los ion tongshot were analysed towards the end of 1966, they y ie lded a value of dT/dA which was about 11% higher than the value expected on the basis o f the J e f f r e y - B u l l e n tab le s . Studies by C leary and Hales (1966) and Carder et al (1966) s t rong ly suggested that at the d i s tance in quest ion, the J-B tab les , and hence the expected dT/dA val ue was in error by less than 1%. This provided the f i r s t h in t of an anomalous s t ruc ture under the array, and the large d iscrepancy between the measured and expected dT/dA values was a t t r i b u t e d to th i s s t ruc tu re . From the re su l t s of the seismic ex-peri ment WRAMP, Underwood (1967) i n fe r red the presence of a near-surface dippi'ng s t ruc ture underneath the array. Unfortunate ly, he used too l i t t l e data to enable a r e a l l y d e t a i l e d model of the reg ion beneath the:array to be determined. As more d T / d A and azimuth measurements accumulated, i t became evident that the s t ructure beneath the array could not be approximated by a s ing le plane d ipp ing i n t e r f a c e , and in f a c t , appeared to diverge cons iderab ly from that simple approximation. Owing to the f a c t that d e t a i l s of the s t ruc ture underneath -39-the array are not yet a v a i l a b l e , and that the s t ructure appears to be oneof great complexity, an empir ica l method of c o r r e c t i n g the dT/dA and azimuth measurements fo r s t ruc ture must be used. Wright (1970) devised such a method of co r rec t i on based e n t i r e l y on N i a z i ' s (1966) theory on the e f f e c t of a d ipp ing layer measure-ments of dT /dA and azimuth. The method involves s e l e c t i n g a r e a l i s t i c model of the c rus t and ad jus t ing the d ip angle and d ip d i r e c t i o n , keeping the P wave v e l o c i t i e s constant, u n t i l the apparent v e l o c i t y and azimuth ca l cu l a ted by N i a z i ' s theory agreed with the measured values of dT/dA and azimuth. The weakness of th i s method l i e s in the f a c t that any number of models f i t the data, and any s t ruc ture with a s p e c i f i c v e l o c i t y contrast could be replaced by an equiva lent s t ruc ture with a d i f f e r e n t v e l o c i t y contrast and d ip angle. If, however, c e r t a i n l i m i t s based on observat iona l evidence are placed on parameters such as the dip angle of the s t ruc tu re , and expected apparent v e l o c i t y , the method is capable of y i e l d i n g f a i r l y accurate r e s u l t s . Niazi (1966) has shown how data from earthquakes at almost equal d is tances and at opposite azimuths from the array can be used to separate the e f f e c t s of a s t ruc ture c o n s i s t i n g of a s ing le plane d ipp ing in ter face from e r ro r s in a given dT /dA curve. C leary, Wright and Muirhead (1968) used th i s technique to work out a simple model of the crust beneath the array, and determined the d ipp ing in ter face to have a d ip angle around 6 - 7°, d ip d i r e c t i o n of around 230°, and a v e l o c i t y r a t i o of 0.7. The corresponding parameters deduced by Underwood (1967) were a d ip angle of 5°,dip d i r e c t i o n of 200° and v e l o c i t y r a t i o of about 0.9. Wright (1970), us ing the same method as in 0 1968, estimated a d ipp ing in ter face with d ip angle of about 6.5 to o e o 7 , and d ip d i r e c t i o n 215 - 235 , with a v e l o c i t y contrast of 0.7 to give the best co r rec t i ons fo r dj"/dA and azimuth. Consequently, -40-the predominant evidence i n d i c a t e s t h a t the d i p angle of the i n c l i n e d plane assumed to be r e s p o n s i b l e f o r the large dT/dA and azimuth anomalies i s about 6° or 7°, with a v e l o c i t y c o n t r a s t of approximately 0.7. The o d i p d i r e c t i o n i s somewhat more f l e x i b l e , but seems to be near 220 . It should be noted that the above s t r u c t u r e s used to c o r r e c t the dT/dA and azimuth measurements are a l l s i n g l e d i p p i n g plane s t r u c t u r e s , whereas, i t appears that the s t r u c t u r e underneath the array i s f a r more complicated and may c o n s i s t of a number of d i f f e r e n t d i p p i n g planes. However, an important f a c t , noted by Wright (1970) i s that the e f f e c t of several plane d i p p i n g i n t e r f a c e s on dT/dA and azimuth measurements i s i n d i s t i n g u i s h a b l e from that of a s i n g l e i n t e r f a c e , and consequently, the assumption that the s t r u c t u r e underneath the array could be approximated by a s i n g l e plane d i p p i n g i n t e r f a c e i s j u s t i f i e d . The e f f e c t of a s i n g l e iplane d i p p i n g i n t e r f a c e i s to i n t r o d u c e approximately s i n u s o i d a l v a r i a t i o n s i n azimuth and dT/dAas a f u n c t i o n of azimuth t h a t are 90° out of phase. The dT/dA anomaly i s l e a s t i n the d i p d i r e c t i o n and the azimuth anomaly changes from negative to p o s i t i v e . A c l o s e look a t the dT/dA and azimuth measurements made i n t h i s study reveal not only s t a r t l i n g d i s c r e p a n c i e s between expected and observed values but a l s o great f l u c t u a t i o n s in the amount of the discrepancy (Figure 2.7). Quite o f - t e n i t was p o s s i b l e to c o r r e l a t e the magnitude of the anomaly w i t h azimuth. Although the azimuth range was not uniformly covered, the r e s i d u a l s of the measured dT/dA values with respect to the expected values e x h i b i t a q u a s i - s i n u s o i d a l v a r i a t i o n . The azimuth r e s i d u a l s a l s o i n d i c a t e some degree of s i n u s o i d a l v a r i a t i o n but changed more r a p i d l y than do the dT/dA r e s i d u a l s and are not c o i n c i d e n t w i t h them. On t h i s Eigure .2.6 Refraction of a planar seismic beam incident on a dipping interface between a half-space and the overlying layer. . -42-bas i s , the assumption of a s ing le plane d ipp ing in ter face underneath the array as used in the empir ica l method of c o r r e c t i o n , seems to be a j u s t i f i a b l e one. For purposes of ana lys i s a computer program wrj t ten by C. Wright was used to work out the e f f e c t on dT/dA and azimuth, of any combination of d ipp ing in ter faces for a seismic event whose azimuth and d i s tance are accura te ly known. The program i s based e n t i r e l y on N i a z i ' s theory, which w i l l be b r i e f l y ou t l i ned below. The geometry of a th in planar beam of seismic waves inc ident on a d ipp ing in ter face from beneath is shown in Figure 2 .6 . The i n te r sec t i on of the d ipp ing i n te r face with an imaginary hor izonta l plane ( i . e . the d i r e c t i o n of s t r i k e ) i s taken as the x ax i s . OP and OQ, are the in te r sec t i ons of the v e r t i c a l plane conta in ing the inc ident beam with the imaginary hor izonta l plane and the d ipp ing i n te r f a ce , r e s p e c t i v e l y . The d ip angle is denoted by symbol £ , and the angle between the seismic beam and v e r t i c a l axis OZ, i . e . the true angle of inc idence, is denoted by i . The angle 0 between OP and the x -ax i s i s the true azimuth from which the beam has been inc ident on the i n t e r -f a c e . i Q i s the angle of incidence at the f r e e sur face. The r e -f r a c t i o n of the rays takes place according to S n e l l ' s law with i 1 and not i being the angle of inc idence, where i 1 is the angle subtended between the ray and the normal to the in ter face ON. Consequently, the angle of r e f r a c t i o n w i l l be r ' , which is d i f f e r e n t from r = i Q . In the process of r e f r a c t i o n , in genera], the rays do not remain in the v e r t i c a l plane, but appear to be a r r i v i n g at the surface from a new azimuth, the apparent azimuth 0 ' , the corresponding apparent d ip being 6*' . Niazi (1966) has shown that the d i r e c t i o n -43-cos ines of the beam reaching the surfaces are given by (. = (-m0 +n tf) /<x. m= (Cbsr'-rvO>sS)/Sin S r\ - Cos r'Gos S ± ocS'tnS Sin r' where OL , 0 t Y are the d i r e c t i o n cosines of the l i n e perpendicular to the apparent plane of inc idence, i . e . to both the inc ident rays 2 2. 2. a n d ON, and ot 4/3 + = * If V j and are the wave v e l o c i t i e s in the upper and lower medium r e s p e c t i v e l y , then since Cos 'i' = S i n t S m ^ Sin£ + COS*-CDSS oi - Tan S Cot t - S m ^ / R 0 = Cos 4/^ * - - T a n £ Ccstf/R _L where K = C ' + T a n 2 ^ Cot2"- + T a n ^ C o S 2 ^ - 2.Sin y4 T a n 6" 6^ i ) 2 " The apparent angles of inc idence, apparent azimuth and new apparent v e l o c i t y of the rays can then be obtained from the d i r e c t i o n cosines I , m, n as f o l l ows : Cos t c - n $ =Tan~' (w/0--44-The actual procedure was as fo l l ows . A d ipp ing in ter face between the c rus t and mantle was taken as having a v e l o c i t y r a t i o of 0 . 7 . Taking a f i xed d ip d i r e c t i o n of 10° and a f i x e d d ip angle of 0 . 5 , the azimuth 0 was var ied at i n t e r v a l s of 2 0 ° from 0 ° to 3 6 0 ° . It was assumed from the J-B tab les that the expected values of dT/dA for the DF branch lay in the range of 2 . 0 2 2 sec/ deg. at 113° to 0 . 3 sec/deg. at 176 (where 113° to 176° was the d is tance range of the DF data ) , corresponding to expected apparent v e l o c i t i e s of 55 km/sec to about 380 km/sec. Consequently, keeping the dip d i r e c t i o n and angle f i xed at the values chosen, four quant i t i e s were ca l cu l a ted at each value of azimuth from 0 ° to 3 6 0 ° at 2 0 ° i n t e r v a l s . These four values were: (1) The apparent azimuth 0 , and consequently the azimuth anomaly. (2) The measured apparent v e l o c i t y V ' , and consequently (3) The measured dT/dA obtained from V ' . (k) The r a t i o of measured ve loc i ty /expected v e l o c i t y where the expected v e l o c i t i e s used var ied from 55 km/sec - 100 km/sec in 2 km/sec steps from 100 km/sec - 200 km in 10 km/sec steps and from 200 - 380 km/sec in 20 km/sec steps. Thus the e n t i r e range of expected apparent v e l o c i t i e s corresponding to the e n t i r e .range of expected dT/dA values for DF was covered. The d ip d i r e c t i o n was o . 0 o then var ied from 10 to 360 at 10 i n t e r v a l s , and keeping the dip angle f i x e d at 0 . 5 , the e n t i r e procedure was repeated at each new d ip d i r e c t i o n . The d ip angle was increased by 0 . 5 ° from 0 . 5 ° to 1 0 . 0 ° , and at each new d ip angle va lue, the c a l cu l a t i on s were again performed fo r d ip d i r e c t i o n s from 10° to 3 6 0 ° as before. Thus, an e n t i r e set of tables was generated to enable cor rec t ions to be app l ied to the dT/dA and azimuth measurements f o r the DF bnanch. -45-The co r rec t i on s were made as f o l l ows : The true azimuth : 0, and measured dT/dA and azimuth, and therefore the azimuth anomaly of the event were noted. The tables were checked for a point where at the same true azimuth 0 the corresponding dT/dA value and azimuth anomaly in the tables agreed as c l o s e l y as poss ib le with the measured dT/dA and azimuth anomaly. The dip d i r e c t i o n and dip angle so s p e c i f i e d were then assumed to be the parameters of the plane d ipp ing in ter face respons ib le for th i s p a r t i c u l a r set of anomalies in dT/dA and azimuth measurement. In th i s manner, a d ip d i r e c t i o n and d ip angle were worked out for each dT/dA and azimuth determinat ion for the DF branch. On the assumption that the same method of c o r r e c t i n g for loca l s t ruc ture could be appl ied to the other phases as w e l l , the e n t i r e procedure was repeated f o r the measured values of the GH, IJ and AB t rave l t ime branches. For the GH and IJ data, expected apparent v e l o c i t i e s of 30 to 37 km/sec corresponding to dT/dA values of about 3.6 to 3 .0 sec/deg were used, r e s p e c t i v e l y , at approximately 2 km/sec i n t e r v a l s . The range of dT/dA values for IJ branch was taken onthe basis of the measured dT/dA values which seemed to have the i r major i ty of values in th i s range and the co r re s -p o n d i n g range of expected dT/dA values per the Adams and Randall (1964) model. The range of expected dT/dA values for the GH branch was estimated in the same manner as in the IJ case, but i t was extended to a maximum value of about 2 .9 sec/deg in order to cover the range of expected dT/dA values in the Bolt (1968) model as w e l l . For the AB co r rec t i on s expected apparent v e l o c i t i e s of 25-31 km/sec correspond-ing to dT/dA values ofa bout 4 .4 to 3 .6 sec/deg were used, again at roughly 2 km/sec i n t e r v a l s . Thus, a d ip d i r e c t i o n and d ip angle was worked out for each dT/dA and azimuth determinat ion for the &H, IJ and -46-AB branches. In e f f e c t , a l l that was done was to take a measured d T / d A and azimuth value that had been assigned to a p a r t i c u l a r branch, p lace l i m i t s on the expected true value of dT/dA for that branch, and search the tables for a combination of d ip angle and d ip direction,until the azimuth anomaly and dT/dA of the measured data agreed with that of the tab les , for the p a r t i c u l a r 0 (true azimuth) in quest ion. A c e r t a i n amount of ambiguity is involved in th i s method of c o r r e c t i o n . For example, i t is qu i te poss ib le for a measured dT/dA and azimuth va lue, assigned to the DF branch, to be corrected to an expected dT/dA va lue, l y i n g somewhere on the GH or the IJ branches, and for a measured IJ value to be corrected to an expected value l y i n g somewhere on the DF branch. The two r e s t r a i n i n g f ac to r s used to remove th is ambiguity were (1) As prev ious ly mentioned, a f i x e d range of expected dT/dA va lues, based on the d i s t r i b u t i o n of the measured values and on expected values taken from e x i s t i n g models fo r the appropr iate d is tance range, was assigned to co r rec t a p a r t i c u l a r branch. These ranges d id not over lap for the d i f f e r e n t branches. (2) Evidence has shown that the d ip angle of the d ipp ing in ter face respons ib le for the anomaly is c lose to 7°> and consequently the d ip angles obtained in the cor rec t ions should a l l be c lose to, or around th i s va lue. An important f a c t in connection with th i s is that the larger the anomaly in the measurement, the larger was the d ip angle of the plane needed to co r rec t th i s anomaly. Consequently, i f a measured IJ value was des i red to be corrected to some expected value l y i n g , fo r example on the GH or AB branches, the d ip angle needed to e f f e c t th i s c o r r e c t i o n would be much larger than that needed to co r rec t the measured value to some point on the expected IJ branch. In p a r t i c u l a r , i t was observed that the measured 61/dA and azimuth -47-values for the e a r l y part of the IJ branch could be cor rected to be on the GH branch, thus r e s u l t i n g in a t rave l t ime model s im i l a r to B o l t ' s (1964) model. On the basis of the larger d ip angles, greater than 1 0 ° , needed for th i s c o r r e c t i o n , and s ince the phase IJ was C l e a r l y observed on array seismograms in that d i s tance range (Bolt d id not observe th i s phase at a l l ) , the c o r r e c t i o n to B o l t ' s model was r e j e c t e d . The d ip angles of the c o r r e c t i n g planes were found to be centered around the value of 6 ° - 7°. although in some cases the d ip angle reached as low as 0 . 5 ° . In even fewer cases, i t reached as o ° high as 10 . The d ip d i r e c t i o n s were concentrated around - j 100 and 2 0 0 ° but a few d ip d i r e c t i o n s as low as 1 0 ° on as high as 3 5 0 ° were obta ined. Often the three measurements in any one event would have, c l o s e l y s im i l a r d ip angles and dip d i r e c t i o n s , as would most events around the same azimuth 0, but i t was poss ib le though unusual to have a l l three measurements in any one event with va s t l y d i f f e r i n g d ip angles and dip d i r e c t i o n s . This d i s p a r i t y was due to the changes in the magnitude of the anomalies for the three measurements. In turn these could be due to systematic changes in waveform from one seismometer to another as a, r e s u l t of random noise e f f e c t s , extremely rap id va r i a t i on s in the loca l s t ruc tu re , irregularities in the source reg ion or the deepest point of the ray path,or more than one phase a r r i v i n g at near ly the same time. The appropr iate d ip vectors were d iv ided into groups so that each group contained dip vectors of approximately the same va lue, and the d i s tance and azimuth ranges in each group were not too large. The -48-data fo r the DF branch was d iv ided into 25 groups, the IJ into 15. the GH into 19, and the AB into 2 1 . The d i s tance range var ied from 5° to 1 0 ° and the azimuth range from 3° to 1 0 ° . The purpose of the d i v i s i o n into groups was to obta in an average di-p vector fo r each group under the assumption that a l l the events in the p a r t i c u l a r group were a f fec ted by th i s 'average' s t ruc ture determined, and that the f i n e s t ruc ture v a r i a t i o n in each group was merely a consequence of the empir ica l method of c o r r e c t i o n , and somewhat d ivergent from the degree of phys ica l r e a l i t y obta inable by th i s method. The averaging of the d ip vectors was approached as a problem Q f the s t a t i s t i c s of a spher ica l d i s t r i b u t i o n . This approach is commonly adopted i * paleomagnetic studies and is descr ibed by Fisher (1953) and Watson (1956) . The dip d i r e c t i o n L and d ip angle A ° f t n e mean.dip vector are givenby: L = T a n " ' ( 2u>;nu Li) A = Sin"' ( ^ u t n ; / R ; where m;, n\ and I } are the d i r e c t i o n cosines of the d ip vectors in the d i s t r i b u t i o n , R is the length of the i r re su l tant vector and wj is the weight of the ind iv idua l vec tor s . The weights are normalized so that the i r sum is equal to the number of observat ions N in the group. A measure of the d i s t r i b u t i o n of the ind iv idua l vectors with respect to the mean vector is given by the quant i ty K = N - I / N - a The higher the value of K the more c l o s e l y are the ind iv idua l vectors grouped around the i r mean. LL) in A - DF Q - GH 2 - IJ X - RB A A DT/D-DELTF) VS DELTA X (UNCORRECTED) X Z j z i 1 x 1 z , z a * A 21 X I* z • 2<%, X x* 4» -1 135.0 110.0 1)5.0 120.0 125.0 130.0 140.0 145.0 DELTA(DEG) 150.0 -1 155.0 F i g u r e 2.7. M e a s u r e d DT/DA 1E0.0 165.0 170.0 175.0 iao UJ • O Q A - DF • - GH Z - IJ X - RB A i 1 ^ A * DT/D-DELTR VS DELTA (CORRECTED) z z z , o 4 110.0 115.0 120.0 125.0 130.0 135.0 140.0 145.0 DELTA(PEG) 150.0 I 155.0 160.0 -1 165.0 I 170.0 I 175.0 ieo.0 Figure 2.8. DT/DA corrected for l o c a l structure -51-The measured azimuth, apparent v e l o c i t y and d ip vector for each observat ion in a p a r t i c u l a r group were used to c a l c u l a t e the angle of incidence i Q of the waves at the e a r t h ' s surface as descr ibed in- N i a z i ' s theory. The d i r e c t i o n cosines LL n't were estimated for each vector from i t s ' angle of inc idence, and the quan t i t i e s L, A and K were ca l cu l a ted for the e n t i r e group; The observat ions were rearranged i f necessary to obta in s a t i s f a c t o r y estimates of K. The weights W| were taken as the square of the rec ip roca l of the root-mean square error on the corresponding dT/dA measurement. The averaged d ip vectors obtained in the manner were used to prepare tables as descr ibed prev ious ly and the f i n a l corrected dT/dA values were estimated from these tab les . It should be mentioned at th i s po int that the s t a t i s t i c a l approach used for the problem of averaging the d ip vectors is not s t r i c t l y the co r rec t one s ince the d ip angle i t s e l f is known with greater p r e c i s i o n than the d ip d i r e c t i o n , and the p r o b a b i l i t y d i s t r i b u t i o n is somewhat a n i s o t r o p i c . A co r rec t s t a t i s t i c a l model would be one intermediate between a spher ica l and a c i r c u l a r d i s t r i b u t i o n but the procedure then becomes somewhat too complex to be j u s t i f i e d by the nature of the problem. The dT/dA values as o r i g i n a l l y measured are shown in F igure 2.7. They e x h i b i t cons iderable scat ter and f i t t i n g a smooth curve through the values for each branch would be extremely d i f f i c u l t . The same dT/dA measurements corrected fo r s t ruc ture in the manner j u s t descr ibed are d i sp layed in Figure 2.8. The improvement is s e l f -evident and the scat ter in the observat ions has been reduced s u f f i c i e n t l y that smoothing techniques could be appl ied meaningful ly. -52-2.3 Smoothing of Corrected dT/dA Values In the reduction of random errors by smoothing techniques, i t is highly desirable that the errors be reduced as far as possible without losing valuable information, and that the magnitude of the reduction be estimated. Three possible techniques of smoothing in common use are: (1) To subtract a twelfth of the fourth difference from, or, add a quarter of the second difference to each observed value except the two end points. (Jeffreys, 1937). This method has two severe disadvantages. F i r s t l y , the magnitude of the reduction cannot be properly estimated since the errors in neighbouring points tend to be correlated, and secondly, i t does not remove as much of the error as the uncertainties would allow (Jeffreys, 1934) nor does i t give any control over the amount of smoothing to be applied. (2) To.fit a simple function such as a polynomial or a series of polynomials to the data by least squares. In this way, it is possible to get as much smoothing as desired or j u s t i f i e d , but the standard errors in the coefficients of the function are not a convenient measure of the reduction of the error unless the coefficients themselves have some physical significance. A variation of this technique was used for comparison as discussed in Section 2.3.2. (3) The method of summary values devised by Jeffreys (1937, 1961) in which the amount of smoothing can be controlled, no valuable information is lost as ascertained by a significance test, and the magnitude of the reduction can be estimated. This method is superior to the other two and was adopted for smoothing purposes in this study. Appendix IV contains a description of this method. 2.3.1 Smoothing of dT/dA by the Method of Summary Values The method of summary values involves f i t t i n g a linear and quadratic curve to each of several groups of the data. The curvature -53-in each group is estimated beforehand to be reasonably sma l l . From an appropr iate c a l c u l a t i o n (Appendix IV) the coordinates of i n te r sec t i on of the l i near and quadrat ic forms can be determined and are known as the summary po int s . They best represent the data in the appropr iate range. The summary points determined for the several groups are in terpo la ted to give the complete smoothed curve. A computer program based on the theory of Appendix IV was wr i t ten by C. Wright and was used in th i s study. The corrected dT/dA values f o r any one branch were d iv ided into groups in which the d i s tance ranges were smal l , and the curvature a l so estimated to be smal l . The program was used to c a l c u l a t e the summary points and the i r corresponding uncer ta in t i e s for each range of the branch. The summary points were o then in terpo la ted at 0 .01 i n te rva l s by means of d iv ided d i f f e r e n c e s , the s i z e of the in terva l being chosen so as to f a c i l i t a t e comparison of the ca l cu l a ted values with the measured values in which the d i s tance o coordinates were given to 0 .01 . The res idua l s between the ca l cu l a ted and measured dT/dA values were determi ned^and in order to determine whether adequate smoothing had taken place a ch i - squared tes t was 2 performed on these r e s i d u a l s . The "x. value for each res idua l was taken as the product of the weight for the corresponding dT/dA measurement and the square of the r e s i d u a l , the total-x? for each range being the sum of the ind iv idua l x? of the points in that range. An ove ra l l the sum of the "x? for each of the ranges in the p a r t i c u l a r branch was ca l cu l a ted a l so .For adequate smoothing, and assuming the error to be wholly random, must be in the range V + JlV where V is the -54-number of degrees of freedom. Thus *v is equal to the number of f i x e d parameters less the number of adjustab le parameters. In th i s case "V is equal to the number of observat ions used, less the number of summary points for the case being considered ( i . e . for one range with M po in t s , the number of summary points is 2 , and V = M - 2 , so that the x- 1 for that range must be wi th in ( M - 2) + •j[2(M-2) . If there are N ranges in the e n t i r e branch, then the number of summary points is 2N, and i f the e n t i r e number of observat ions in the branch is M 0 , then the ove ra l l x 2 must be in the range ( M 0 - 2 N ± J 2 (M 0 - Z N ) If x i s too large ( i . e . i f the res idua l s are too large) then the summary points do not represent the data in the range as accura te ly as they cou ld , the curvature is being underestimated, and s i g n i f i c a n t features are being l o s t . To co r rec t th i s problem, the number of ranges should be increased so that the summary points f a l l c lo ser to the 2 data in any given range and the res idua l s are thereby reduced. If -x, i s too smal l , i n s u f f i c i e n t smoothing is taking place and poss ib ly meaningless d e t a i l i s being reta ined in the curve. To co r rec t th i s problem, the number of ranges should be increased, so that the re s idua l s in any given range would increase a l so . In p r a c t i c e , owing to the heavy concentrat ion of data in some distanceranges, and r e l a t i v e lack in the other reg ions, some d i f f i c u l t y was encountered in attempting to obta in a s a t i s f a c t o r y value of T& for each of the ranges, and at the same time a s a t i s f a c t o r y 2 overa l l •% for the e n t i r e branch. A f ter somewhat exhaustive attempts 2 to obta in a s a t i s f a c t o r y so lu t i on with regard to the - X , for each range, 2 the ove ra l l oo for the branch and the re ten t ion of reasonably small d i s tance ranges, the f o l l ow ing procedure was adopted: that s o l u t i on - 55 -f o r which the maximum number of ranges gave s a t i s f a c t o r y values of x z and ihe o v e r a l l value of x,2 f o r the e n t i r e branch remained s a t i s f a c t o r y was allowed to stand. The r e s u l t of th i s procedure is thatthe larger part of the branches adequately smoothed but there were poss ib ly one or two regions on i t that were e i ther undersmoothed or oversmoothed. Since at th i s stage the data was being smoothed to determine a c o r r e c t i o n term corresponding to cons t ra in ing the data to an acceptable t rave l t ime curve, th i s lack of adequate smoothing over the e n t i r e branch is not too important(Wright 1970). In order to cons t ra in the smoothed data to an acceptable t rave l t ime curve and consequently to reduce any remaining systematic e r ro r s even f u r t h e r , the area under each smooth curve ( i . e . in each range) was evaluated by Simpson's ru le of in tegra t ion and subtracted from the t rave l t ime d i f f e r e n c e between the end points of the range. If T| and T^ are the t rave l times at the end points of the range of in teg ra t ion A t and A a r e s p e c t i v e l y , and i f c , a n d « r z are the respect i ve standard er ror s in T, and T z and i f , a l so , the area under the smooth curve in that range is A with a standard errore^then the c o r r e c t i o n C is given by: c = ((Tz ± <%) - On i °0 - O* ± c r 3 ) ) / a 2 - A , ^ = C Ta - T , - R ) / A 2 - A , ± 6*72 + ^ Z ^ 2 ) % 2 - A . where the second term on the r i ght is the error in the c o r r e c t i o n term C. Th i s c o r r e c t i o n C was ca l cu l a ted for each of the ranges in the d i f f e r e n t branches, and app l ied to each of the dT/dA values in these ranges, a procedure which merely 'involved changing the basel ine of the dT/dA values in the ind iv idua l ranges. The trave l t ime curves to which the smoothed dT/dA data were re s t ra ined are the J-B tables for the DF and AB branches (which branches for comparison were a l so res t ra ined to the Bolt (1968) tables) and the Adams and Randall (196^) - 5 6 -TABLE 2.1(a) SUMMARY POINTS FOR 3 14 DATA Pi RESTRAINED INITIAL SUMMARY POINTS: a dT/dA o (Dnfi) (SEC/DEC) (SEC/DEG) 115.45 1.956 0.018 118.98 1.946 0.017 122.10 ' 1.858 0.030 123.68 1.857 0.033 125.55 1.890 0.016 131.26 1.862 Q.017 134.06 1.886 0.010 138.08 1.867 0.010 143.06 1.748 0.012 165 .22 0 .896 -0 .016 DP BRANCH II NTS TO J . B . TRAVEL-TIME CURVE X 2 v C (SEC/DEC) 49.32 45 +0.000 21.02 18 +0.083 38.06 35 +0.034 48.41 78 -0.039 148.11 128 +0.023 Total x 2 = 304.92 on 304 degrees of freedom FINAL SUMMARY POINTS: A dT/dA o (DEG) (SEC/PEG) (SEC/DEG) 115.30 1.962 0.016 118.78 1.9S2 0.013 122.56 1.940 0.016 125.38 1.924 0.016 131.58 1.895 0.011 134.93 1.846 0.010 138.04 1.828 0.010 147.81 1.706 0.015 155.34 1.432 0.019 165.32 0.919 0.016 49.48 45 i 29.64 32 j 81.53 77 106.27 108 46.09 42 Total x 2 = 313.01 on 304 decrees of freedom. o = ST1). ERROR ON dT/dA VALUE OP SUMMARY POINT v = NO. OP DEGREES OF FREEDOM C = CORRECTION FOR RESTRAINING DATA TO TP.AVF.LT] ME CURVI: -57-TABI.E 2.1(b) SUMMARY POINTS FOR Gil BRANCH 3 05 DATA POINTS (a) RESTRAINED TO A-R TRAVELTJMIi CUR Vii I N I T I A L SUMMARY POINTS: A (DEC) 136.08 141.77 dT/dA (SEC/DEG) 2.653 2 .624 (SEC/DEG) 0.020 0 .021 69.36 84 (SEC/DEG) -0 .008 143.97 LSI.S3 2 .605 2 .501 0 .019 0 .037 19.21 17 -0 .008 T o t a l v 2 = 88.57 on 101 d e c r e e s o f freedom FINAL SUMMARY POINTS: A (DEG) 136.20 141.93 dT/dA (SEC/DEG) 2 .645 2.613 . (SEC/DEG) 0.018 0 .017 79.42 84 14 3.7 5 151.94 2 .601 2 .494 0.015 0.024 15.21 17 (158.0 2 .300 T o t a l x 2 = 94.63 on 101 degrees o f freedom. *Taken from A-R c u r v e i n o r d e r t o e x t e n d b r a n c h beyond 152° (b) RESTRAINED TO BOLT TRAVELTIME CURVE FINAL SUMMARY POINTS: A (DEG) dT/dA (SEC/DEG) C= 0.087 (SEC/DEG) v 2 (SEC/DEG) 136.19 141.91 2 .740 2 .704 0.019 0.020 77 .59 84 143.75 151 .91 2 .692 2.588 0.020 0.021 16.1! 17 (156 .00 2 .100 T o t a l x 2 = 93.74 on 101 degrees o f freedom. * Taken from B o l t ' s c u r v e i n o r d e r t o e x t e n d b r a n c h beyond 152° -58-TABLE 2.1(c) SUMMARY POINTS FOR I J BRANCH 81 DATA POINTS RESTRAINED TO A+R TRAVEL-TIME CURVE .INITIAL SUMMARY POINTS: A (DEG) 135.68 140.46 146.62 150.82 dT/dA (SEC/DEG) 3.455 3.317 (SEC/DEG) 0.032 0 .015 3.195 3.054 0.017 0 .030 47.42 19 .54 T o t a l x 2 ~ 66.96 on 77 degrees of freedom, 60 17 (SEC/DEG) +0.024 +0.003 FINAL SUMMARY POINTS: A dT/dA a x 2 (DEG) (SEC/DEG) (SEC/DEG) 135.68 3.459 0.030 ,140.46 3.320 0.013 146.6S 3.198 0.016 150.87 3.056 0.021 T o t a l x 2 = 75.84 on 77 degrees of freedom. 57.23 60 18.61 17 -59-TABLF. 2.1(d) SUMMARY POINTS FOR AB BRANCH 70 DATA POINTS (a) RESTRAINED TO J.B. INITI A L SUMMARY POINTS: A dT/dA a X 2 v C (DEG) (SEC/DEG) (SEC/DEG) (SEC/DEG) 146.01 3.670 0.017 149.896 4.010 0.024 *1S7.642 4.279 0.034 161.599 4.416 0.032 165.716 4.488 0.026 173.441 4.495 0.008 33.01 37 +0.145 3.56 12 29.73 21 -0.093 T o t a l x 2 = 66.30 on 70 degrees o f freedom. * Not used i n i n t e r p o l a t i o n because o f u n s a t i s f a c t o r y b e h a v i o u r o f c u r v e i n t h a t r e g i o n . FINAL SUMMARY POINTS: A dT/dA o (DEG) (SEC/DEG) (SEC/DEG) 146.01 3.814 0.015 149.896 4.165 0.020 161 .599 4 .319 0 .030 165.716 4.395 0.028 35.67 37 27.94 26 173.441 4.419 0.033 T o t a l x 2 = 63.61 on 63 degrees o f freedom. (b) RESTRAINED TO BOLT FINAL SUMMARY POINTS: C = -0.055 sec/dog and -0.081 sec/deg r e s p e c t i v e l y A dT/dA o x 2 v (DEG) (SEC/DEG) (SEC/DEG) 146.01 3.625 0.013 149.89 3.968 0.020 161.60 4.335 0.026 165.72 4.406 0.022 36.75 37 24.32 26 173.44 4.409 0.032 T o t a l x 2 = 61.0? on 63 degrees o f freedom. NOTE: (].) A l l v a l u e s o f unsmootlicd /dA above 4.5 r e s t r a i n e d t o 4.5. -60-TABLE 2.2. DT/DA SMOOTHED BY THE METHOD OF SOBS&Rr VALUES DISTANCE i l DT/DA (SEC/DEG) (DEG) ! DF GH IJ AB i I (A-R) (BOLT) (J-B) (BOLT) 1 1 3 . 0 1 . 9 6 2 1 1 4 . 0 1 . 9 6 2 1 1 5 . 0 1 . 9 6 2 1 1 6 . 0 1 . 9 6 2 1 1 7 . 0 1 . 9 5 9 1 1 8 . 0 1 . 9 5 6 1 1 9 . 0 1 . 9 5 2 1 2 0 . 0 1 . 9 4 4 1 2 1 . 0 1 . 9 4 1 1 2 2 . 0 1 . 94 1 1 2 3 . 0 1 . 94 1 1 2 4 . 0 1 . 9 3 9 1 2 5 . 0 1 . 9 3 1 1 2 6 . 0 1 . 9 2 1 1 2 7 . 0 1 . 9 1 5 1 2 8 . 0 1 . 9 1 0 1 2 9 . 0 1 . 9 0 6 1 3 0 . 0 1 . 9 0 2 1 3 1 . 0 1 . 8 9 7 1 3 2 . 0 1 . 8 8 2 2 . 6 5 1 2 . 7 4 7 1 3 3 . 0 1 . 8 7 8 2 . 6 4 9 2 . 7 4 5 1 3 4 . 0 1 . 8 6 5 2 . 6 4 9 2 . 7 4 4 3 . 4 8 4 1 3 5 . 0 1 . 8 5 4 2 . 6 4 8 2 . 7 4 3 3 . 464 1 3 6 . 0 1- 8 5 2 2 . 6 4 6 2 . 7 4 1 3 . 4 4 4 1 3 7 . 0 1 . 8 4 9 2 . 6 4 3 2 . 7 38 3 . 4 0 8 1 3 8 . 0 1 . 8 2 9 2 . 6 3 9 2 . 7 3 4 3 . 3 7 8 1 3 9 . 0 1 . 8 2 4 2 . 6 3 4 2 . 7 2 9 3 . 3 5 1 1 4 0 . 0 1 . 8 1 8 2 . 6 2 9 2 . 7 2 4 3 . 3 2 8 14 1 . 0 1 . 8 0 8 2 . 6 2 2 2 . 7 17 3 . 3 0 7 1 t 2 . 0 1 . 7 9 4 2 . 6 1 5 2 . 7 1 0 3 . 2 8 8 1 4 3 . 0 1 . 7 7 3 2 . 6 0 6 2 . 7 0 1 3 . 2 7 0 1 4 4 . 0 1 . 7 5 6 2 . 5 9 7 2 . 6 9 2 3 . 2 5 2 1 4 5 . 0 1 . 7 3 8 2 . 5 8 7 2 . 6 8 2 3 . 2 3 2 3 . 6 0 0 3 . 5 0 0 1 4 6 . 0 1 . 7 1 8 2 . 5 7 6 2 . 6 7 1 3 . 2 1 1 3 . 8 2 0 3 . 6 10 1 4 7 . 0 1 . 6 9 7 2 . 5 6 4 2 . 6 59 3 . 187 3 . 9 2 0 3 . 7 2 0 1 4 8 . 0 1 . 6 7 3 2 . 551 2 . 6 49 3 . 159 4 . 0 1 0 3 . 8 1 0 1 4 9 . 0 1 . 6 4 8 2 . 5 3 7 2 . 6 3 2 3 . 128 4 . 0 9 0 3 . 8 9 0 1 5 0 . 0 1 . 6 2 0 2 . 5 2 3 2 . 5 8 0 3 . 0 9 0 4 . 1 7 0 3 . 9 7 0 1 5 1 . 0 1 . 5 9 1 2 . 5 0 7 2 . 5 5 0 3 . 0 4 7 4 . 2 0 0 4 . 100 1 5 2 . 0 1 . 5 5 9 2 . 491 2 . 5 0 0 3 . 0 3 0 4 . 2 0 0 4 . 2 0 0 1 5 3 . 0 1 . 5 2 6 2 . 4 7 3 2 . 4 5 0 4 . 2 0 0 4 . 3 0 0 1 5 4 . 0 1 . 4 8 9 2 . 4 5 5 2 . 4 0 0 4 . 2 0 0 4 . 300 1 5 5 . 0 1 . 4 5 1 2 . 4 3 6 2 . 3 0 5 4 . 3 0 0 4 . 4 0 0 1 5 6 . 0 1 . 4 1 1 2 . 4 1 5 2 . 2 1 0 4 . 3 0 0 4 . 4 0 0 1 5 7 . 0 1 . 3 6 8 2 . 394 4 . 3 0 0 4 . 4 0 0 1 5 8 . 0 1 . 3 2 3 2 . 3 0 3 4 . 3 0 0 4 . 4 0 0 1 5 9 . 0 1 . 2 7 5 4 . 3 0 0 4 . 4 0 0 1 6 0 . 0 1 . 2 2 5 4 . 3 0 0 4 . 4 0 0 1 6 1 . 0 1 . 1 7 2 4 . 3 2 0 4 . 4 1 0 1 6 2 . 0 1 . 1 1 6 4 . 3 4 0 4 . 4 2 0 1 6 3 . 0 1- 0 5 8 4 . 3 6 0 4 . 4 30 1 6 4 . 0 Q . 9 97 ' 4 . 3 70 4 . 4 3 5 1 6 5 . 0 Q . 9 3 3 4 . 3 9 0 4 . 4 4 0 1 6 6 . 0 0 . 7 1 5 4 . 4 0 0 4 . 4 4 6 1 6 7 . 0 0 . 6 2 3 4 . 4 1 0 4 . 4 5 0 1 6 8 . 0 0 . 6 05 4 . 4 2 0 4 . 4 5 7 1 6 9 . 0 0 . 5 9 8 4 . 4 2 0 4 . 4 5 8 1 7 0 . 0 0 . 5 5 4 4 . 4 2 0 4 . 4 5 8 1 7 5 . 0 0 . 3 0 0 4 . 4 2 0 4 . 4 5 9 1 8 0 . Q 0 . 0 0 0 4 . 4 2 0 4 . 4 5 9 -61-tables for the IJ and GH branches, th i s l a t t e r branch a l so being re s t ra ined to the Bolt (1968) tables for the sake of i n t e r e s t . Since none of these tables provided standard er ror s in the i r t ravel times, i t was not poss ib le to estimate the error on the c o r r e c t i o n C fo r any of the ranges. With the appropr iate c o r r e c t i o n C appl ied to each group in the p a r t i c u l a r branch, the corrected dT/dA values of the e n t i r e branch were once again smoothed by the method of summary values to obta in the o new summary po int s . These were in terpo la ted at 1 i n te rva l s to give the f i n a l smoothed dT/dA va lues. The i n i t i a l and f i n a l summary points used in obta in ing the f i n a l smoothed dT/dA values fo r the respect ive branches are Listed in Tables 2.1(a) to 2.1(d) i n c l u s i v e . Note that the GH branch has been re s t ra ined to the t rave l t ime curves of both Adams and Randall and Bo l t . The AB branch was re s t ra ined to both J-B and Bo l t . Owing to lack of data in the range 150°to 160° for AB, i t was not poss ib le to obta in s a t i s f a c t o r y summary points for th i s reg ion, so the procedure adopted was to smooth the data in the ranges 145 - )50° and 160 - 175° separate ly , and to f i l l in the gap so created with the values of J-B and Bolt r e s p e c t i v e l y . In a d d i t i o n , values of dT/dA for the AB branch that were greater than 4.5 sec/deg were re s t ra ined to th i s va lue, as i t is the maximum given for P CP phases in the 1968 tab les . The f i n a l smoothed dT/dA values are l i s t e d in Table 2.2 Figure 2.9 shows these values res t ra ined to the d i f f e r e n t t rave l t ime branches. From the f i g u r e , i t is seen that the GH branch res t ra ined DT/DA SMOOTHED B7 METHOD OF SUMMARY VALUES in A rrjEG) Figure 2.9 DT/DA smoothed by the method of Summary Values -63-to B o l t ' s data has greater curvature near 150° than the one res t ra ined to that of Adams and Randal l . The AB branch re s t ra ined to the J-B t rave l times has greater curvature near i t s beginning than the one re s t r a ined to Bo l t ' s times. A l so , i t i s s h i f t ed to a lower* level near the l a t t e r part, of the range. On the basis of the best f i t to the data, the dT/dA curve for the GH branch re s t ra ined to Adams and Randa l l ' s t rave l times and the dT/dA curve for the AB branch re s t ra ined to the J-B t rave l times were se lected for the purpose of i nver s i on . The curves as shown for branches IJ and DF were used in the invers ion procedure. 2.3.2 Smoothing of dT/dA data by UBC Tr iangu lar Regression Package - TRIP  Mainly fo r the sake of comparison, the dT/dA values corrected fo r loca l s t ruc ture were a l so smoothed by polynomial regress ion using the U.B.C. computer program U.B.C. TRIP (1972). The method is b r i e f l y descr ibed in Appendix V.a. The regress ion equations determined as g i v ing the best f i t to the data are l i s t e d in Table 23. The independent va r i ab le s fo r the DF, GH, IJ and AB branches were taken as ( A - 1 1 3 ° ) , ( A -132°) ( A - 1 3 4 ° ) and ( A -145°) r e spec t i ve l y where the numerical quant i t i e s are the turn ing points of the respect i ve branches. The turn ing points were taken as that d i s tance , truncated to a whole number, at which the ind iv idua l phaseswere f i r s t observed. Smoothed curves at 1° in te rva l s f o r the d i f f e r e n t branches were generated from the appropr iate regress ion equat ions. The ca l cu l a ted valves are l i s t e d in Appendix Vb, and are shown in Figure 2.10. For comparison, the smoothed curve obtained by the method of summary values i s shown in dashed l i n e s . The lesser -64-TABLE 2 .3: RECRESS 1 ON EQUATIONS FOR SMOOTHED dT/dA DAT A DF BRANCH 314 data points R 2 = 0.9643 ]NDEPENPEN'T VAR1ARLF. COEFFICIENT STD . F.RROR F - PROB TOLERANCE CONSTANT 1.9S87 0.0180 (A-113°) -0.1384/10 0.2528/10 3 0.0000 1.0000 (A-113°) 2 0.1095/10 2 0 . 1384 /10'' 0.0000 1.0000 (A-113°) 3 -0.3608/10'' 0 .1146/10= 0.0000 1.0000 (4-113°)* 0.2432/10 6 0 . 5876/10' 0.0001 1.0000 P = d T / d A = 1.9587 - 0.01384 (A-113) + 0.1095 10 -0.0360 / V 1 1 3 V + 0.0024 /A -113V 10 / V 10 / A - U 3 \ 2 GH BRANCH 106 data points R 2 = 0.7573 INDEPENDENT VARIABLE COEFFICIENT CONSTANT (A-132") ( A-132") 2 (4-132°)3 2 .GS78 -0.1694/10 0.1326/10 2 -0 .4839/10" STD.F.RROR F-PROB TOLERANCE 0 .01990 0.S931/10 3 0.0000 1.0000 0.84 86/10'* 0.0000 1.0000 0.1301/101 0.0004 1.0000 P - 2.6878 - 0 .01694 (A- 132 ) + 0 . 1326 /'A-132V- -0.0484 /A-132V- -0 .04 4 /A-1321 : v io / v io y I J BRANCH R2 = 0.8156 81 data points INDEPENDENT.VARIABLE COEFFICIENT STD.ERROR F-PROB. TOLERANCE CONSTANT (4-134°) (A-134°) 2 3.4556 0.0255 -0.1833/10 0.1376/10 2 0.0000 1.0000 -0.3761/10 3 0.307G/10 3 0.0223 1.0000 3.4556 - 0 .0183 (A-134) 0.03761 AB BRANCH R2= 0.9756 INDEPENDENT VARIABLE COEFFICIENT CONSTANT (A* 145°) (A-145°) 2 ( A - 1 4 5 0 ) 3 3. 5480 0.1172 -O.S9M/10 2 0.1203/10 3 70 data points STD.ERROR F-PROB TOLERANCE 0.0194 0.8280/10 3 0.0000 1.0000 0.2310/10 3 0.0000 1.0000 0.4134/10'' 0.0050 1.0000 P » 3.5480 + 0.11 7?. (A-Hi;; - 0.5964 ^£_-JJJ:^7 * 0.1203 ^ A -1 4 r,j 3 DT/DA SMOOTHED BY POLYNOMIAL REGRESSION ANO METHOD OF SUMMARY VALUES POLVtvlCrtlAL E.£(JRlrS5 l o M ' S U M M A R Y V A L U E S . C D O L U C O < D G i no.o n 115.J -1 120.0 I 125.0 130.0 -1 135.0 140.0 DELTA (DEG) 145.0 ~1 150.0 155.0 I 160.0 - 1 1BS.0 " I 170.0 - | 175.0 160.0 F i g u r e 2.10 Comparison o f DT/DA smoothed by P o l y n o m i a l R e g r e s s i o n and method o f Summary V a l u e s . -66 -curvature near the s t a r t of the AB branch f i t t e d by polynomial regress ion can be a t t r i b u t e d to the lack of data in the range 150° to 160°. Thus e x c e l l e n t agreement between the sets of curves wee-i obta ined. However, the smoothed dT/dA values in terpo la ted from the summary points were used for the invers ion process. 2.4 Inversion of Smoothed dT/dA values using the Herglotz-Wiechert Integral 2.4.1 Descr ip t ion of the Herglotz-Wiechert invers ion technique As was shown in Appendix I for a spher ica l and symmetrical earth model, the ray parameter p is given by p = In Sm I /v = dT/dA Al so , the in tegra l s for the distance A , 2 and t rave l t ime T l 2 . for a ray of parameter p terminat ing at leve l s r j and r£ where n > r2> were der i ved : i&,2 = P I r - ' C r f - p 1 ) " * dr (0 • ± T « = I tfr-'Cf-p^dr - (Z) where i \ = r/v fo r convenience. These in tegra l s may be solved e i ther by numerical approximation or by d i r e c t in tegra t ion when the v e l o c i t y func t i on is su i t ab l y de f ined . Consider the l a t t e r case for a m u l t i -layered earth where the v e l o c i t y in each layer is descr ibed by a power law of the form v = ar* 5, with a and b constants. Note that b is i den t i ca l with ^ of Appendix I. Thus v(r) i s continuous but dv/dr is in general d iscont inuous at each tabulated po int . If v =^ ) 2. r ' > 1 2" in the layer bounded by r a d i i r^ and r^, s ub s t i t u t i on in equations (1) and (2) y i e l d s (Engdahl et a l , 1968) - 6 7 -i A , z - O-^T'CCos'Xf/ri,) - COS^'CP/IZ)) .... (3) i T, 2 = 0 -b -^'a -^p^ - C<-P^ -00 where r\t and t\z a r e t n e values of n at r] and r2 r e s p e c t i v e l y . For the ray which reaches i t ' s deepest point w i th in the l ayer , the value of P i s equal to r[ , so that (3) and (k) become A,2 = 2 0-b.i)"'<:os"'Cp/ri,) - - f t ) T « = 2 0-b,x)- ,(nf- P 2) i C« The radius of penetrat ion rp for th i s p a r t i c u l a r ray is determined from the v e l o c i t y r e l a t i o n s h i p and is e a s i l y seen to be given by Using equations (3) to (6), the t rave l t ime and d i s tance of any ray or ray segment can be determined by bu i l d i ng up the times and d i s tances over the proper number of l ayers , each layer being represented by new values of a, b, and T^. The values of a and b are ca l cu l a ted from a given v e l o c i t y d i s t r i b u t i o n for that layer bounded by r a d i i r j and ( r l ? r 2 ) where b l z = In (V,/v-z) / J n ( r . / r a ) <8) a ( i = v , r , - b * (?) and v] and V2 are the given values of v at r] and r2 r e s p e c t i v e l y . The theory embodied in equations (3) to (9) has several important a p p l i c a t i o n s . It can be used in ray t r ac ing to inves t i ga te complexity in t rave l t ime curves caused by regions of high v e l o c i t y gradients or low v e l o c i t y layers or to co r rec t ep i cen t ra l d is tances f o r foca l depth (see Sect ion 2.2.1). Of importance to th i s study, however, i s the f a c t that i t can be used to der ive T - A and ?-A curves f o r a s p e c i f i e d v e l o c i t y d i s t r i b u t i o n , thus enabl ing models to - 6 8 -be generated for comparison with the observat ional data . As w e l l , the equations can be used to s t r i p the earth to any required depth h by c a l c u l a t i n g the con t r i bu t i on *Ck i ;Lof the region down to h to the to ta l d i s tance A in quest ion. The adjusted d i s tance or the d i s tance corresponding to the s t r ipped earth is given by ( A - A,2. ) and the adjusted t rave l t ime by ( T - T ( i ). Equations (3) to (9) formed the basis of the computer program (Wright, 1970) wr i t ten to inver t the dT/dA values by the Herg lotz -Wiechert method and so generate a corresponding v e l o c i t y d i s t r i b u t i o n . Provided that TK= r/v is a monotonic decreas ing f u c t i o n of depth beneath the sur face , equation (1) can be solved to y i e l d (Bul len, 1 9 6 3 ) . A, la ro/r, ~ TT ' / C o s h 'Cp/p.)ciA . (iq) o where r e is the mean radius of the ea r th , r, i s that radius where the v e l o c i t y of the wave is V, , P, = T\t i s the parameter of a ray that penetrates to a depth rQ-r, and reaches the surface at a d i s tance A| from the source, and V, = r, / p ( . For convenience (10) can be wr i t ten At ± r, = r D expC-TT'f tn(p/ P l + C(P/P.)Z-l)2)d:A) 0 0 To evaluate v, , dT/dA as a func t ion of d i s tance from 0 to A•, is required so that r, can be ca l cu l a ted from ( 1 1 ) . V = H / p i can then be est imated. If a d i s c o n t i n u i t y e x i s t s , t h e earth must be s t r ipped of the region down to the d i s c o n t i n u i t y before the c a l c u l a t i o n s can proceed for the region beneath the d i s c o n t i n u i t y . If h is the depth to the d i s c o n t i n u i t y and dT/dA values are known from the observat iona l data over the d i s tance range A% ^° ^ 1 > - 6 9 -some e x i s t i n g model is used to f i l l in the d i s tance range from 0° to A^. Then the cont r ibu t ions to the d i s tances in the range A 2 . *° due to that region to depth h, for values of dT/dA between p ( A 2 ) and p^A,) are ca l cu l a ted by summation over the appropr iate number of l ayer s . The cont r ibu t ions are then subtracted fromthe appropriate distanoes between A2 +0, A1 so that a dT/dA curve c h a r a c t e r i s t i c of a s t r ipped earth of radius ro - h is obta ined. The computations then proceed as before, using equation (11) with the d i f f e r e n c e that r Q i s now replaced by r Q - h, and the d i s tance range of i n teg ra t ion is that of the s t r ipped d is tances corresponding to A2. to A-l . The f i r s t s t r ipped d i s tance , i . e . corresponding to A 2 . should be equal to 0, meaning that the depth h to the d i s c o n t i n u i t y used in the s t r i p p i n g technique is the maximum depth at ta ined by the ray path of parameter pC^Az) . The s t r ipped distances increase from A z t o A l . . The computer program mentioned prev ious ly enabled the earth to be s t r ipped to any appropr iate d i s c o n t i n u i t y by eva luat ing (3) from an assumed v e l o c i t y - d e p t h model above the d i s c o n t i n u i t y . The quant i ty p in equation (3) r e fe r s to the observat iona l dT/dA values to be inverted,f\j and TI2 represent the dT/dA va lues , ca l cu l a ted from the v e l o c i t y model assumed above the d i s c o n t i n u i t y for the corresponding v e l o c i t y values V] and Thus a table of dT/dA values versus s t r ipped d is tance is obta ined. It is in terpo la ted to give values at 1° d i s tance i n t e r v a l s , beginning at the f i r s t s t r ipped d i s tance of 0 ° . This is to f a c i l i t a t e eva luat ion of the radius of penetrat ion r j in equation (11) by means of Simpson's ru le of i n teg ra t i on , r^ is evaluated for every t h i r d point in the table ( i . e . the range of in tegra t ion is from 0° to 2 ° , then 0° to 4 ° , then 0° to 6 ° e t c . to obta in values of r ] , at 2 ° , 4 ° , 6 ° , e tc . ) S i n § e v = r/p, and both r and p are known, the v e l o c i t y v fo r the appropr iate depth could -70-be c a l c u l a t e d . Consequently, a v e l o c i t y versus depth table can be der ived from the measured dT/dA values in conjunct ion with an assumed v e l o c i t y model above the d i s c o n t i n u i t y . F i n a l l y , equations (3) and (4) are used to c a l c u l a t e the true d i s tances and t rave l times at dT/dA i n te rva l s of 0.02 sec/deg. for both the assumed and the ca l cu l a ted v e l o c i t y d i s t r i b u t i o n . These values together with the c o r r e s p o n d i n g dT/dA va lues, radius and depth of penetrat ion as well as the v e l o c i t y values at each dT/dA in terva l of 0.02 sec/deg. were the f i n a l output of the computer program. Close examination of th i s table enabled assessment of any adjustments in v e l o c i t y and d i s con t i nu i t y -dep th values which might be necessary to ensure a better f i t to the data. The e n t i r e procedure can be repeated for add i t i ona l d i s c o n t i n u i t i e s . 2.4.2 App l i c a t i on of the Herglotz-Wiechert method to dT/dA values fo r the e a r t h ' s core.  The boundary between the e a r t h ' s mantle and core is d iscontinuous with a sharp decrease in compressional wave v e l o c i t y from about 13.6 km/sec to 8.1 km/sec. (J-B va lues ) . Consequently, i f the smoothed dT/dA for the AB branch are to be inverted to obta in a v e l o c i t y d i s t r i b u t i o n for the outer core, the earth must be s t r ipped to the mantle-core boundary. For t h i s purpose, the v e l o c i t y d i s t r i b u t i o n corresponding to the J-B values was assumed to the mantle-core boundary at a depth of 2894 km. This d i s t r i b u t i o n and the smoothed dT/dA values fo r the AB branch were used to determine the ve l oc i t y -dep th d i s t r i b u t i o n for the outer core. Because of the large d i s -continuous decrease in v e l o c i t y at the core-mantle boundary, some d i f f i c u l t y was encountered in obta in ing a s a t i s f a c t o r y ve loc i t y -dep th d i s t r i b u t i o n , and an a l t e r n a t i v e but by no means poorer,procedure was adopted. The outer core -71-was assigned an i n i t i a l ve l oc i t y -dep th d i s t r i b u t i o n corresponding to the J-B t ab le s . From th i s i n i t i a l model the corresponding dT/dA and trave l t imes values were c a l c u l a t e d . The model was adjusted u n t i l a s a t i s f a c t o r y f i t of the ca l cu l a ted and the measured value was obta ined. The smoothed dT/dA data fo r the U branch were used to determine the depth to the f i r s t d i s c o n t i n u i t y in the t r a n s i t i o n reg ion . This was done by using the ve l oc i t y -dep th d i s t r i b u t i o n determined fo r the AB branch, together w i th i the J-B d i s t r i b u t i o n abouve the mantle-core boundary, to s t r i p the earth to the depth where i t was assumed that the d i s c o n t i n u i t y producing the receding Bl branch was s i t u a t i o n . This depth was adjusted u n t i l the s t r ipped d i s tance for the f i r s t IJ data point was as near 0 ° as pos s ib le . This ensured that the advancing IJ branch would s t a r t at the des i red d i s tance of about 1 3 4 ° . If th i s f i r s t s t r ipped d i s tance was p o s i t i v e , i t meant that the depth to the d i s c o n t i n u i t y chosen was too shal low, with the r e s u l t that the advancing IJ branch would s t a r t e a r l i e r than des i red , while the receding Bl branch would be correspondingly too long. Likewise, a negative s t r ipped d is tance meaning that the d i s c o n t i n u i t y was too deep would r e s u l t in a receding branch too short , and a l a ter turning point fo r the advancing branch. The depth to the d i s c o n t i n u i t y was adjusted to give s a t i s f a c t o r y re su l t s regarding the turning po int of the advancing branch and the length of the receding branch. The corresponding ve l oc i t y -dep th values and travel t imes for the IJ branch were ca l cu l a ted as p rev ious l y descr ibed. They were used i n conjunct ion wi th the ca l cu l a ted v e l o c i t i e s above the d i s c o n t i n u i t y ( i . e . the AB v e l o c i t i e s and JB model) to determine a s a t i s f a c t o r y turning point f o r the GH branch using the smoothed dT/dA data . for GH. This was repeated for the DF branch, f i n e adjustments of ca l cu l a ted v e l o c i t i e s were necessary to obta in the best poss ib le f i t with the ca l cu l a ted trave l t imes while at the same time ensuring that the -72-turn ing points fo r the var ious branches remained s a t i s f a c t o r y , and that the measured dT/dA values required no s i g n i f i c a n t adjustment. In th i s manner, the measured dT/dA values for a l l four observed t rave l t ime branches were inverted to obta in a v e l o c i t y model for the e a r t h ' s core . This model i s d iscussed fur ther in Chapter 3, along with a general d i scuss ion on the re su l t s of the measurement and ana ly s i s procedures used in th i s study. - 7 3 -CHAPTER 3 DISCUSSION 3.1 General Observations During the measurement and ana lys i s procedures descr ibed in Chapter 2, a number of pecu l i a r observations were made. Genera l ly , there has been no documentation in the publ ished l i t e r a t u r e of s im i l a r observat ions. Because such observat ions may be re levant to other workers in the f i e l d , th i s sec t ion is devoted to a d i s cus s ion of them. While ana lys ing the DF branch in the range 113° to 120°, a phase with a s l i g h t l y lower tiT/dA value and higher amplitude than the phase i d e n t i f i e d as PKP 0 F was con s i s t en t l y observed less than one second a f te r the DF phase a r r i v e d . This second a r r i v a l with d i f f e r e n t apparent v e l o c i t y may simply be a d i f f r a c t e d e f f e c t due to i r r e g u l a r i t y in s t ructure at the source or at some point in the path of the array. Rapid l a te ra l v a r i a t i o n in e l a s t i c parameters near the array s i t e may a l so be a poss ib le exp lanat ion. The f i r s t a r r i v a l with higher dT/dA value and e a r l i e r t rave l t ime may even suggest the presence of an add i t i ona l d i s c o n t i n u i t y qu i te near to the inner core boundary, the second a r r i v a l in th i s case being the true DF phase. More data is needed before such a p o s s i b l i l i t y could be inves t i ga ted. That th i s anemalous observat ion may be az imuthal ly dependent is suggested by the f a c t that the events in question a r r i ved from azimuths near 300° and 6 0 ° , and at d i f f e r e n t d i s tances . o o In the range 120 to 130 , s im i l a r e f f e c t s were noted some-what s p o r a d i c a l l y , with great v a r i a t i o n in the amplitude of the f i r s t a r r i v a l with respect to the second. Of p a r t i c u l a r i n te res t in th i s range, however, is that at d i s tances of 120.6°, 1 2 1 . 7 ° and 121.8°, - 7 4 -the second a r r i v a l , approximately 1.5 seconds a f te r the DF a r r i v a l had higher dT/dA value and amplitude than the DF phase. These may well be observat ions of the receding branch HD, and the i r amplitudes with respect to the amplitudes of the Df phase agree with Buchbinder 's (1971) p r e d i c t i o n s . Four events in p a r t i c u l a r deserve spec ia l mention. Those are events numbers 25, 26, 29, and 30 in the d i s tance range 125° to 1 3 0 ° , a l l a r r i v i n g from azimuths near 1 5 0 ° . These were the f i r s t events over the e n t i r e range to show any s ign of precursors , but ana ly s i s of the precursors y ie lded puzz l ing r e s u l t s . For the f i r s t three events, precursors were observed j u s t 3 to h seconds before the a r r i v a l of the DF phase. Ana lys i s of the i r dT/dA values y ie lded re su l t s vary ing from 1.0 to 1.5 sec/deg for the f i r s t two events and 1.7 to 2 .0 sec/deg for the t h i r d . The four th event showedaprecursor only 1 to 2 seconds before DF, and with s im i l a r values as the above mentioned precursors . The dT/dA value for PKP 0 F was low for a l l four events - approximately 1ik to 1.5 sec/deg - and th i s was a gen-era l observat ion for a l l events in the azimuth range 100° to 1 6 0 ° . A l l four events showed a second set of presursors some 5 to 7 seconds before PKP^. Their dT/dA values were, on the average, s l i g h t l y higher than the DF va lues, and on the basis that the DF values were low, they could have been, in the extreme, taken as belonging to the GH t rave l t ime branch. This was not done, however, because the i r t rave l t imes , according to any e x i s t i n g models were too slow to belong to th i s branch. The foca l depths of the events ru led out the poss-i b i l i t y of them being pPKP^H a r r i v a l s . A t h i r d set of precursors was noted on a l l four events approximately 8 to 9 seconds ( and in one case 15 seconds) before DF, but the i r dT/dA values were too low to be meaningful. These observat ions could be an azimuthal - 7 5 -o e f f e c t as the events a l l occur at azimuths near 150 , but at d i f f e r e n t d i s t ances . No s a t i s f a c t o r y explanat ion for the precursors can be g iven. At least ten events in the range 134° to 139° showed pre-cursors which were unusual. Some of these a r r i ved approximately 4 seconds before P K P D F with d T / d A values of about 3 to 4 sec/deg while others a r r i ved only 1 to 2 seconds before DF with dT/dA values of 2 .6 sec/deg. Yet others had dT/dA values of about 3.3 to 3 .7 sec/deg but a r r i ved only one second before DF. (The above mentioned values are the dT/dA values uncorrected for loca l s t ruc tu re . Such a c o r r e c t i o n , on the basis of the PKPQF dT/dA va lue, would r e s u l t in about a 5% increase in the i r magnitude). The p o s s i b i l i t y of these phases being pPKP,-j. and pPKP & H were ru led out on the basis of foca l depth of the events in quest ion. It may be poss ib le that they are due to some source of mu l t ip le a r r i v a l s deep in the crus t (as for example f a u l t i n g near the base of the crust (Wright, 1970)). On the basis of t r a v e l -times and dT/dA values i t is l i k e l y that some of them are observat ions of the receding branches 61 andJG-. In the d is tance range 1 3 1 . 6 ° to 1 3 8 . 9 ° , some 20 events showed phases a r r i v i n g 2 to 4 seconds a f te r the DF phase. In the major i ty of cases, these 2nd a r r i v a l s had s l i g h t l y higher dT/dA values and amplitudes than the DF phase. They could probably be observat ions of the receding brand HD, but i f Buchbinder (1971) is c o r r e c t , the ir amplitudes at the d i s tances in quest ion should be lower than that of the DF a r r i v a l s . This lower amplitude behaviour was observed in only four of the 20 events. The high-amplitude data of 9 of these events a l so showed precursors some 2 to 4 seconds before the DF a r r i v a l , with dT/dA values approximately equal to or only s l i g h t l y lower than the dT/dA val ues for the DF phase. Wright (personal communication, 1972) has made s im i l a r observat ions of the precursors - 7 6 -over the same d is tance range from seismograms recorded at the Y e l l -owknife seismic array. However, no s a t i s f a c t o r y explanat ion fo r these phases can be made at the present time. 3.2 S p e c i f i c Observations Concerning PKP Phases  3.2.1 I d e n t i f i c a t i o n of Phases Since u l t imate l y the model generated in th i s study depends upon the i d e n t i f i c a t i o n of the ind iv idua l phases, a few comments on th i s subject would seem to be appropr ia te. The DF phase was e a s i l y i d e n t i f i e d up to d i s tances of about 140° as being the most prominent a r r i v a l on the record . This is in add i t i on to i d e n t i f i c a t i o n on the basis of dT/dA and t rave l t ime measurements. At d is tances of about \k0° to about 146°, i d e n t i f i c a t i o n became somewhat dtifficultdue to the near-simultaneous a r r i v a l s of a l l the phases. Beyond 146° , i t was e a s i l y i d e n t i f i e d on the basis of t ravel t ime and apparent v e l o c i t y . I d e n t i f i c a t i o n of the AB phase l ikewise proved easy in the d i s tance range 150° to 176° because of i t s d T / d A values and longer t r a v e l -time which e f f e c t i v e l y separated i t from the other phases. Even in the ea r l y part of i t s range from about 145 ° to 150°, no problem was encountered in separat ing PKP f l S from PKP.j , both of which tend to have s im i l a r values of dT/dA near 145°. The AB phase always a r r i ved a f te r the IJ phase and i t s uncorrected dT/dA value was a l -ways s l i g h t l y higher than that of the IJ phase. In the range 132° to 138°, separat ion of the IJ and GH phases proved easy on the basis of the i r dT/dA and t rave l t ime meas-urements. However, in the range 138° to 142° where the time sep-a ra t i on of the two phases narrowed g reat ly and the true dT/dA values were sometimes masked by loca l s t ruc tu ra l e f f e c t s , i d e n t i f i c a t i o n became more d i f f i c u l t and was o f ten made a f ter c lose examination of the dT/dA values obtained using r e j e c t i o n leve l s of both 0.02 and 0.04 - 7 7 -seconds on the seismometer r e s i dua l s . The greatest weight was a t t -ached to the 0.02 sec. r e j e c t i o n measurement s ince th i s is the higher p r e c i s i o n measurement. In one or two instances, the i d e n t i f i c a t i o n was made s o l e l y on a t rave l t ime basis i f i t was not poss ib le to o d i s c r im ina te from one dT/dA value to another. Near 145 , the GH branch was masked by the IJ branch. In the l a t t e r part of the range o beyond 145 , the two phases could once more be e a s i l y separated on the basis of the i r dT/dA and t rave l t ime measurements. On the basis of the measured dT/dA values and travel times, l im i ted amplitude information (which w i l l be discussed in Sect ion 3.2.3) and an o v e r a l l de ta i l ed ana lys i s of a l l seismograms, i t is be l ieved that the PKP phases used in th i s study have been c o r r e c t l y i d e n t i f i e d . 3.2.2 Turning Points and End Points for the Travelt ime Branches At th i s po in t , i t should be noted that i t was not poss ib le to determine p r e c i s e l y the turn ing point of the branch DF due to lack . 0 of data in the immediate region before 113.6 . As prev ious ly mentioned, the turn ing point was taken as the -truncated value of 113.6 for use in the invers ion techniques. This is not unreasonable as Adams and Randall (1964) place i t at 110°, Bolt (1964) at 111.3°, Shufbet (1967) near 110°, and Ergin (1968) near 114°. Buchbinder (1971) placed o the turn ing point near 120 on the basis of amplitude data. o The GH phase was f i r s t i d e n t i f i e d at a d i s tance of 132.2 . The dT/dA values (uncorrected) fo r th i s phase were lower than the expected value by approximately 23%, but i t a r r i ved at the expected time per the Adams and Randall model. Since the DF dT/dA values on the same event were a l so 23% lower than expected, the phase was accepted as belonging to the GH branch. The same app l ie s to the o second observat ion of GH at a d i s tance of 134.8 . The f i r s t c lear - 7 8 -o and strong GH phase was observed at a d i s tance of 134.8 . Thus, the c turn ing point for the GH branch was se lected as being 132.0 . Poss ib le IJ precursors occur r ing approximately 12 to 13 o seconds before DF could be barely recognized at d i s tances of 132.0 , 133.11 , 134.5 , and 134.7 » but were too smal l , even on high ampl i -tude data , to be analysed. The f i r s t i d e n t i f i c a t i o n of PKP { J was a l s o made on the record at a d i s tance of 134 .8° . Although i t s ampl i -tude was s im i l a r to that of the background no i se, dT/dA measurements could be made. Thus, i t is l i k e l y that the IJ branch has i t s turn -ing point in the range of 132° to 135°. For the purpose of i nver s ion , o i t was se lec ted as being 134 . F a i l u r e to observe e i ther the GH or IJ phases at shorter d is tances may be due, apart from the obvious explanat ion that the phases simply do not ex i s t in th i s part of the range, to several other reasons. For example, a f te r 1967» the array was t r iggered by an a r r i v a l at the c l u s te r with a 13 second delay loop for record -ing, i f GH or IJ were present fo r d i s tances less than 132° their amplitudes were i n s u f f i c i e n t to t r i g ge r . the array, but the DF phase would t r igger i t . Unfortunately at these d i s tances , the 13 seconds would not neces sa r i l y be s u f f i c i e n t to enable record ing of the ear -l i e s t precursons. This could have an important bearing in determining the turn ing points of the branches f o r , according to Buchbinder (1971)» the turning point is that d is tance at which the amplitudes increase at greater d istances but decrease r ap id l y at smaller d istances, Any phases observed before the turning point are a t t r i bu ted to those phases p a r t i a l l y r e f l e c t e d from the d i s c o n t i n u i t y and p a r t i a l l y transmitted through i t . Consequently, i f the GH and IJ phases ex i s ted in the e a r l y part of the range, the i r amplitudes are so smal l , that by Buchbinder 's c r i t e r i o n they do not in any case belong to the t o t a l l y - 7 9 -transmitted advancing branches which were determined in th i s study. Another poss ib le reason for f a i l u r e to observe these phases, but which e s s e n t i a l l y leads to the same conc lus ion as above, is that the high-amplitude data was at times f a r too noisy for any coherent s igna l s to be picked out. In other cases, there was a lack of high-amplitude data a l l together. In Sect ion 3.1, four events o o which showed precursors in the range 125 to 130 were d i scussed. None of these precursors appeared even remotely to belong to e i ther of the IJ or GH branches. This f a c t suggests that these phases do not e x i s t in the range. Owing to the sparseness of data in the range 150° to 160° i t was not poss ib le to def ine abso lute ly the end points of the IJ and GH branches. The la s t c lear observat ion of IJ occurred at a d i s tance of 150.89° . Events at d is tances 152.7° and 157°showed no evidence of the phase so that i t seems l i k e l y that the end point of the branch is somewhere near 151°. This cannot be said with any c e r t a i n t y because of lack of data in th i s reg ion. The las t c l ea r observat ion of GH a l so occurred at 150.89°. The event at 157.2° y ie lded a phase at the expected time of GH, but with dT/dA values lower than the expected value by about 25%. Since the DF d T / d A value was a l so low by approximately the same amount, the phase ob-served at 157.2° was accepted as belonging to the GH branch. Thus, the end point for the branch had to be greater than th i s d i s tance and was chosen to be 158°. The turning point for the AB branch was assumed to be some-where in the range 143° to 145°. Strong a r r i v a l s of a phase that looked l i k e AB, were recorded at d i s tances of 142.31° and 142.59° but i nc lu s i on of these values in the smoothing of the AB data resu l ted in an undesirable f l a t t e n i n g out of the AB branch. -80-These a r r i v a l s pos s ib l y belong to the receding Bl branch. Be-cause of th i s ambiguity, they were not included in the dT/dA data f o r the AB branch. For purposes of invers ion , the turn ing point for th i s branch was se lec ted as be ing ' a t 145°. The phase was observed o out to 175 , the greatest d i s tance f o r any event analysed in th i s study. 3.2.3 Amplitude Observations Although absolute amplitudes from one event to another could not be compared because of lack of information as to the gain used on each seismogram, i t was poss ib le to compare • the r e l -a t i ve amplitudes of the d i f f e r e n t phases for any one p a r t i c u l a r event. This showed that at d i s tances less than 142°, the largest phase on the seismogram is the DF phase, fo l lowed in magnitude by the GH, then the IJ phases. The GH amplitude var ied from about £ to 1/3 that of DF near 135°, to about 4/5 that of DF near \kQ°. The IJ amplitude var ied from about 1/10 to 1/4 that of DF near 135° to 5- to 3/4 that of DF near 140°. Thus the var iance in amplitude of the IJ phase was more pronounced than the GH phase in the d i s tance range less than 142°, although i t always rema i ned sjnal ler than' the GH phase at these d i s tances . At d i s tances greater than-142° PKP I 7 became more prominent than PKP & H , but both were s t i l l smal ler than the DF phase. Then, at about 145 , IJ exceeded DF in ampl i -tude whi le GH was approximately equal to or only s l i g h t l y larger than DF in amplitude. IJ continued to be the most prominent phase for the res t of i t s d i s tance range a t t a i n i n g a maximum amplitude 10 times that of DF. The GH amplitude usua l l y lay j u s t beneath the IJ ampl i -tude. The AB phase, enter ing at about 145° d id so with amplitude greater than the DF amplitude but by 160°, i t s ' amplitude had f a l l e n to approximately the same or s l i g h t l y less than the DF amplitude. Of p a r t i c u l a r i n te re s t i s the f a c t that while Adams and Randall (1964) Figure 3.1 A Record Section for the Phases DF,GH,IJ,AB.The IJ Phase is shown darkened. -82-i nd i ca te that the GH phase is the most prominent on the record at i o o d i s tances between 145 and 153 , in th i s study the IJ phase is the most prominent at these d i s tances . Close examination of the behav-iour of the theore t i ca l amplitudes determined by Buchbinder (1970 i nd ica te good agreement with the observations j u s t d i scussed. 3.2 .4 A Record Sect ion for Core Phases. F igure 3.1 is a t ravel t ime - d i s tance sect ion for s e i s -mograms in the range where the precursor branches GH and IJ were present. In compi l ing th i s record s ec t i on , only 5 of the poss ib le 20 traces have been reproduced for the sake of c l a r i t y . The phases are i d e n t i f i e d by the i r dT/dA values on the record . In p a r t i c u l a r , the IJ phase is shown darkened throughout the record. This phase has not e a s i l y been observed in the past, and i t s c l ea r observat ion is one of the s i g n i f i c a n t r e su l t s of th i s study. Many of the observat ions d iscussed in the previous three sect ions are exempl i f ied in th i s f i g u r e . The f i r s t seismogram at • o 134.80 is the one on which the IJ phase was f i r s t observed, and the GH phase was c lea r and prominent. The amplitude of the IJ branch with respect to the other branches can be seen to increase u n t i l at a d i s tance of about 146° i t exceeds the DF phase in amplitude. It becomes the most prominent phase, fo l lowed by GH, for the res t of i t s d i s tance range. The decrease in time separat ion between the phases near 143° can be c l e a r l y observed. By 146.17°, the IJ (and poss ib ly GH) phase becomes a l a ter a r r i v a l to the DF phase. At 147.6 the IJ phase is seen a r r i v i n g a f te r the GH phase. The AB phase is f i r s t observed at 1 4 5 . 2 4 ° . The decrease in slope of the t r a v e l -time branches from IJ to DF can be c l e a r l y observed. -83-3.3 D iscuss ion of the Analysed Data  3.3.1 Corrected dT/d.A Measurements. The corrected dT/dA measurements smoothed by the method of Summary values and by a polynomial regress ion technique were shown in Figures 2.9 and 2.10. The e f f e c t of r e s t r a i n i n g the AB curve to both the Bolt t rave l t ime tables and the JB t rave l t imes ind icated that the r e s t r a i n t to the J-B times gave a better f i t to the corrected dT/dA data p a r t i c u l a r l y in the e a r l y part of the AB range where the curvature is Steep. Even so, the best f i t to the data in th i s region would have been one with greater curvature than that given by the J-B r e s t r a i n t but th i s necess i tated having summary points at almost the same d i s tance but with widely d i f f e r i n g dT/dA va lues . This could not be obtained with the d i s t r i b u t i o n of a v a i l a b l e data for the reg ion. The AB curve smoothed by polynomial regress ion l i e s between the two curves smoothed by summary va lues , but in the e a r l y part of the range i t tends to lean heav i l y in favour of the J-B res t ra ined curve. Owing to the lack of data in the range 150° to 160° the curve here has s l i g h t l y less curvature than would be expected. For the same reason, the curves smoothed by summary values in th i s region behaved somewhat e r r a t i c a l l y and were simply replaced by the expected J-B va lues . The IJ branch smoothed by both methods agree reasonably w e l l , as does the DF branch. The GH branch re s t ra ined to the Adams and Randall t rave l t ime curve and smoothed by the summary values method agrees almost exac t l y with that obtained by the polynomial regress ion method. Owing to the f a c t that, the Bolt r e s t r a i n t for th i s branch was truncated at 156° instead of 158° (Bolt places h is GH end point at 156°) and that h is t rave l t imes a l l l i e beneath the Adams and Randall t rave l t imes for the same d i s tances , the curve res t ra ined to his model was sh i f t ed to higher dT/dA values and at ID UJ O <_> _ Ujru I m a a A - DF D - GH Z - IJ X - RB DT/D-DELTR VS DELTR (CORRECTED) B m A i oo -1 170.0 ~T~— 175.0 110.0 115.0 120.0 -1 125.0 -I 130.0 13S.D -| 1 140.0 145.0 DELTA(DEG) 150.0 -1 155.0 -1 160.0 165.0 180.0 Figure 3.2 F i t of smoothed DT/DAcurves on to corrected data. TRAVEL TIME VS DELTA A - DF Q - GH z - IJ x - RB CJ-UJ IT. " I 125.0 I 130.0 I 135.0 "T 150.0 —I 175.0 I BO 1JD.D 115. D ^ 120.0 140.0 145.0 DELTA(DEG) 155.0 160.0 165.0 170.0 Figure 3.3 F i t of smoothed traveltime curves on to data. C - DF D - GH z - IJ x - HB REDUCED TRAVEL TIME VS DELTA -1 150.0 -I 160.0 I 165.0 T 170.0 T 175.0 UO.O -1 1 115.0 120.0 "~l 125.0 - | 130.3 I 135.0 140.J 145.0 DELTAfDEG) 155.0 iao Figure 3.4 F i t of smoothed reduced traveltime curves on to data. the same time f e l l o f f more sharply near 156 . The smoothed dT/dA values used in the actual invers ion procedure were the AB values re s t ra ined to J -B, the IJ and DF data, and the GH data re s t ra ined to the Adams and Randall model. The f i t of these curves to the corrected dT/dA measurements is shown is F igure 3.2. As determined by the y. 2 tests (Section 2.3.t)for the f i n a l summary points used in d e r i v i n g these curves, the f i t is good for a l l the branches observed in th i s study. 3.3.2 T rave l - t ime Measurements The t rave l t ime measurements, a f ter co r rec t i on s for e l l i p -t i c i t y and foca l depth had been app l i ed , were smoothed using the U.B.C. TRIP program as descr ibed in Appendix Va. The independent va r i ab le s were the same as those used in the smoothing of the data, and the r e s u l t i n g regress ion equations are l i s t e d in Table 3.1. These equations were used to c a l cu l a te the t rave l times at 1° i n t e r v a l s . The corresponding t rave l times and reduced trave l t ime curves are superimposed on the data in Figures 3.3 and 3.4 r e s p e c t i v e l y . That the f i t is a good one is evident from the parameter R (which is d iscussed in Appendix Va) for the d i f f e r e n t branches as shown in Table 3.1. B a s i c a l l y , the c lo ser R is to the value un i ty , the better is the f i t of the regress ion equation to the data. R2" has the lowest value of 0.986 for the GH branch. It should be.noted that no cor rec t ions for s t ruc ture be-neath the array were appl ied to the measured t rave l times on the assumption that at the large d i s tances involved, such cor rec t ions would be n e g l i g i b l e . The res idua l s (0 - C), between the smoothed and measured times are shown in Figure 3.5. A p o s i t i v e res idua l impl ies a late a r r i v a l with respect to the smoothed times and v i ce versa . The -88-TARi.'.-: J . I REGRESSION I:QIJATIO.\'S I:OR SMOOTHED TRAVELTIME DATA DP BRANCH 105 data points r.2= 0.99S5 INDEPENDENT VARIARIE coi;n ; icii:MT STD.ERROR F-•I'ROi! TOLIiRANCI: CONSTANT 1119.4653 0.3729 0. .0000 1.0000 (A-113) 1.7656 0. .0000 1.0000 (A-113)1 0.1222/10 0. 3536/103. 0 . ,0000 1.0000 (4-113)' -0.2681/10 3 0.2779/10" 0. .0000 1.0000 T». 1119.4653+1.76S6(A -113) +1.222 / 'A-113.0\' - 0, .Z6S1 f A-U3)3 < 10 J V 1 0 / CU DRANCII 34 data points R2 = 0.9S63 INDEPENDENT VARIABLE COEFFICIENT STD.ERROR F-PROB TOLERANCE CONSTANT 1141.2908 1.1931 (A-132") 3.2137 0.0566 0.0000 1.0000 (A-132°) 2 -0.2175/10 0.7953/10* 0.0000 1.0000 T -= 1141 .2908 + 3.2137 (A-132) - 2 .175/^132^ \ 10 / IJ BRANCH 26 data points R2 = 0.9915 INDEPENDENT VAR I AH J.E COEFFICIENT STD.ERROR F-PROB TOLERANCE •CONSTANT 1140.2857 1.1746 (A-134°) 3.7487 0.0655 0.0000 1.0000 (A-134") 2 -0.19410 0.0142 0.01808 1.0000 T «= 1140 .2857 + 3 .7487 (A-134) -1.9410 ^A-134 J AE JiP.ANCII 23 data points R2 •= 0.9979 INDEPENDENT VARJAI'.LE COEFFICIENT STD.ERROR P-PROR TOLERANCE CONSTANT 1180.8818 0.7517 (A-J4S") 3.8276 0.0432 0.0000 1.0000 (A-145") 3 0.13923/10 0.6171/102 0.0338 1.0000 T = 1130.8818 * 3 . 8 2 7 6 fA-145 ) + 1 . 3 9 2 3 / A O -1J.Y V. 10 / -89-T R A V E L - T I M E RESIDUALS 00 AB -2C — 0< IJ •20J— • . 0 tf*-ui • 0 • • • £ 20J— Ct OC GH -20 — • 9 • 2 0 -0C * — r r 1 _a_ -20 • • • ; . • • » 110 120 130 140 150 160 I70 DELTA COEGJ Figure 3.5 Traveltime Residuals (0-C). -90-T R A V E L - T i r i E . D I F F E R E N C E S AB BRANCH J- B - C f l l C . BOLT—CflLC IJ BRRNCH fl.R - C P L C . G H BRUNCH B O L T — CRLC. OF BRANCH J . 8 — CflLC. eon - cfluc. 120* 140* D I S T A N C E rDE&) 160' Figure 3.6 Traveltime Differences -91-re s idua l s show the leas t scat ter for the DF branch. Thisaa natural consequence of the f a c t that the unsmoothed times for the branch c l o s e l y def ined the regress ion curve. The larger scat ter in the unsmoothed IJ and GH times (Figure 3.3) are d u p l i c a t e d r by the i r corresponding r e s i d u a l s . The scat ter of the re s idua l s for the AB branch may be more a consequence of the sparseness of the data points over the large d i s tance range of the branch, so that the regress ion curve could not be p r e c i s e l y def ined by the data. A l so the onset of th i s phase is o f ten d i f f i c u l t to p ick . The t rave l t ime d i f f e rences of the smoothed times with respect to var ious t rave l t ime models a lready in ex i s tence are shown in Figure 3.6. The smoothed AB and DF times agree more c l o s e l y with the corresponding J-B va lues, but the smoothed AB times may ind icate that lesser curvature is required in the J-B curve near 155°, so that the new times are about 1 second l a ter than the present J-B times. It should be remembered, however, that the smoothed AB t rave l t ime curve may not be an adequate representat ion of the true s i t u a t i o n owing to the lack of data over the range. The IJ branch requi res greater curvature than the Adams and Randall model near i t s beginning and lesser curvature near i t s end po int . The GH branch is s h i f t ed to lower t rave l t ime than those def ined by the Adams and Randall model, and to higher times with respect to the Bolt model. It requires greater overa l l curvature than that demanded by both models. Again, the behaviour of the smoothed travel times of the precursor branches near the i r end points may simply be a consequence of the leas t squares f i t in which too large emphasis is placed on values near the end points o f the branches. Obviously, any error in these end points would tend to bias the r e s u l t i n g smoothed times near these regions in a Figure 3.7 Velocity Model UBC1. for Earth's Core. -93-. corresponding manner. This f a c t is important when i t comes to ad-j u s t i n g the f i n a l v e l o c i t y model in order to obta in agreement between the ca l cu l a ted and the smoothed times. On account of th i s b ia s ing of times near the end points of the precursor branches, exact agreement of the model times and smoothed times in these regions is not to be e x c l u s i v e l y demanded. It is of more importance, however, that the agreement be wi th in the l i m i t s imposed by the scat ter of the t r a v e l -time res idua l s themselves, for the respect ive branches. 3.4 Ve l oc i t y Model UBC1 F igure 3.7 shows the v e l o c i t y model, UBC1, which represents the best f i t to a l l data analysed in this study. Of a l l the models descr ibed in Sect ion 1.2, i t most c l o s e l y resembles that suggested by Adams and Randall (1964). The bas ic d i f f e r e n c e in the two models l i e s in the l oca t ion of the turn ing points for the d i f f e r e n t branches and in the magnitude of the change in dT/dA along the receding branches. From these two d i f f e rence s a r i s e the other d i s s i m i l a r i t i e s : depth to the c o r r -esponding d i s c i n t i n u i t i e s , magnitude of the v e l o c i t y jumps at these d i c o n t i n u i t i e s , and t rave l time gradients in the two s h e l l s def ined by the d i s c o n t i n u i t i e s . In the process of f i t t i n g the AB branch to the data, i t was found that the J-B values of v e l o c i t y fo r the outer core had to be reduced s l i g h t l y from a depth of about 4000 km downwards, in order to get agreement between the ca l cu l a ted and measured t rave l times and dTjdA va lues . Too great a reduct ion at th i s depth however, resu l ted in an undesired fea ture , the formation of a c au s t i c in the ea r l y part of the AB branch and t rave l times that were too slow. Thus, a good r e s t r a i n t was put on the magnitude of the v e l o c i t y reduct ion. -94-The f i r s t velocity discontinuity in UBC1 giving r ise to the receding branch Bl of the traveltime curve is at a depth of 4393 km (0.569-Rc). The end point of the branch AB is at 144.4°. The turning point of the advancing branch IJ is located at 134.2° and the magnitude of the jump at this discontinuity is 0.108 km/sec. The value of dT/dA near B is 3.52 sec/deg and near I it is 3.48 o sec/deg. The turning point I in UBC1 is 4.2 later than that in the A & R model and this results in the s l ight ly deeper discontinuity for the UBC1 model. The change in dT/dA along the receding branch Bl for UBC1 is s l igh t ly less than that in the A & R model, and th is , together with the later turning point I for UBC1, results in a smaller velocity jump. The depth to the f i r s t discontinuity, in model UBC1 is well above the level 0.54 Rc (depth 4497 km) to which rays would have to penetrate in order to form a caustic at a distance of about o 143 . It should be noted that the large amplitude arr iva ls near this distance (see Figure 3.1) were associated with arr ivals of the IJ phase, rather than as the result of a caustic at B. The second discontinuity giving r ise to the receding JG branch.occuned at a depth of 4810 km. in UBC1 . The magnitude of the velocity jump at the discontinuit ies 0.237 km/sec. The end point of the IJ branch occurs at 131.9° with dT/dA value of 2.66 sec/deg. The later starting point for G, and the smaller change in dT/dA a-long JG for model UBC1 results in a smaller velocity jump for this model than in the A & R model. The effect of using the higher dT/dA values for the GH branch,due to restraining the smoothed data to the Bolt traveltime curve rather than the A & R curve, was to raise the discontinuity to about 4765 km and to cause a s l ight ly higher velocity jump of 0.287 km/sec at - 9 5 -the d i s c o n t i n u i t y . The change in dT/dA along JG remained the same as before, but the turn ing points J and G were s l i g h t l y e a r l i e r and l a te r r e s p e c t i v e l y . However, UBC1 was generated from the d T / d A measurements res t ra ined to the A & R curve. The v e l o c i t y gradient in the f i r s t she l l def ined by the f i r s t two d i s c o n t i n u i t i e s of UBC1 is i n i t i a l l y p o s i t i v e then s l i g h t l y less than zero and the thickness of the she l l is 417 kms. Adams and Randall required a more negative g r ad ien t . i n the region on account of the i r larger v e l o c i t y jump at the d i s c o n t i n u i t y . The th i r d d i s c o n t i n u i t y at the inner core boundary, pro-ducing the receding branch HD, occurs at exact l y the same depth as that obtained by Je f f rey s (1939) - 5120 km. This impl ies an inner core of radius 1251 km. The magnitude of the v e l o c i t y jump at the d i s c o n t i n u i t y is 0.92 km/sec while the end point H or branch GH is located at 158.6 with dT/dA value of 2.16 sec/deg. The turn ing point D is located at 113 .2° with dT/dA value of 1.96 sec/deg. The second and th i r d v e l o c i t y d i s c o n t i n u i t i e s def ine a second she l l surrounding the inner core. This she l l is 310 km. th ick and has s l i g h t l y negative gradient. This negative gradient can only be e l iminated by p lac ing the turn ing point D at a l a ter d i s t -ance and the end point H at at an e a r l i e r d i s tance . The depth to the second d i s c o n t i n u i t y would increase and the v e l o c i t y jump at the d i s c o n t i n u i t y would decrease, enabl ing the gradient in the second she l l to be non-negative. Such requirements cannot be met by the data. The v e l o c i t y d i s t r i b u t i o n in the inner core c l o s e l y fo l lows that of Buchbinder (1971). A d i r e c t consequence of the smaller negative v e l o c i t y gradients and smaller v e l o c i t y jumps in the t r a n s i t i o n region of -96-TABLE 3.2a POSITION OF CUSPS FOR MODEL UBC1* CUSP T DT/DA (DEC) BIN SEC (SEC/DES) B 1111.1 19 38. 7 3. 52 I 131.2 19 2.2 3.U8 J 152.6 20 2. 3 2. 71 S 131.9 19 1.1 2.66 H 158.6 20 12. 1 2. 16 0 113.2 18 39.4 1.96 RESULTS OBTAINED BY RESTRAINING GH DATA TO BOLT'S TRAVELTIME CURVE: J 151.6 19 58.9 2. 82 G 132.3 19 1.8 2.7U H 158.9 20 11.9 2. 16 TABLE 3.2b VELOCITY-DEPTH VALUES FOR UBC1. DEPTH VELOCITY DEPTH VELOCITY DEPTH VELOCITY (KM) (KM/SEC) (KB) (Ktt/S EC) (KM) (SH/3EC) 2 8 9 ' ! 1 3 . 6 7 0 0 3 6 5 9 9 . 1 1 0 0 7 8 9 4 8 . 1000 3 7 2 8 9 . 2 0 0 0 2 9 6 1 8 . 1 8 0 0 3 7 9 3 9 . 2 8 0 0 3 0 3 3 8 . 2 6 0 0 3 8 6 8 9 . 3 7 0 0 3 1 0 3 8 . 3 5 0 0 3 9 3 7 9 . 4 8 0 0 3 1 7 2 8 . 1 1 0 0 1 0 7 6 9 . 5 0 0 0 3 2 1 2 8 . 5 3 0 0 1 1 4 6 9 . 5 5 0 0 3 3 1 1 8 . 6 3 0 0 1 2 1 5 9 . 6 2 0 0 3 3 8 1 8 . 7 1 0 0 1 3 9 3 9. 8000 3 1 5 0 8 . 8 3 0 0 1 3 9 3 9 . 9 080 3 5 2 0 8 . 9 3 0 0 4 1 8 8 1 0 . 0 1 0 5 3 5 8 9 9 . 0 3 Q 0 1 5 0 0 1 0 . 0 0 0 0 4 8 1 0 1 0 . 0 0 0 0 1 4 8 1 0 1 0 . 2 3 7 0 4 8 2 9 1 0 . 2 6 9 3 4 9 0 0 1 0 . 2 4 4 0 5 0 0 0 1 0 . 2 0 7 0 5 1 2 0 1 0 . 1 6 0 0 | 5 1 2 0 1 1 . 0 3 0 0 ! 5 4 0 0 1 1 . 1 3 0 0 5 6 0 0 1 1 . 2 4 0 0 5 8 0 0 1 1 . 2 6 0 0 6 2 0 0 1 1 . 2 8 0 0 6 3 7 1 1 1 . 2 8 0 0 RESOLTS OBTMNED BY RESTRAINING GH DATA TO BOLT'S TRAVELTIME CURVE 4 5 0 0 1 0 . 0 0 0 0 4 7 6 5 1 0 . ooooi 4 7 6 5 1 0 . 2 8 7 0 J 4 8 2 9 1 0 . 2 6 9 8 TABLE 3.3 TRAVELTIMES FOR MODEL 'J BC 1. (SURFACE FOCUS) DISTANCE DF GH IJ AB (DEG) MIN SEC MIN SEC BIN SEC MIN SEC 130.0 19 12.0 131.0 13.9 132.0 15.8 19 4.4 133.0 17.7 7. 1 134.0 19.5 9.7 19 1.6 135.0 21.4 12.1 5. 1 136.0 23.2 15. 1 8.5 137.0 25. 0 17.7 11.9 130.0 26.9 20.3 15.3 139.0 28.7 22.9 18.7 140.0 30.5 25.5 22.0 141.0 32.3 28. 1 25.6 142.0 34.0 30.7 28.9 143.0 35.7 33.3 32. 3 144.0 37.4 35.9 35.6 145.0 39.2 38.5 38.8 19 40.4 146.0 4 0.9 41.1 42.0 44.2 147.0 42.6 43.7 45. 2 48. 1 148.0 44.2 46.3 48.4 52.0 149.0 45.3 48.8 51.5 56. 1 150.0 17.4 51.3 54.6 20 0.2 151.0 48.9 53.8 57.7 4.4 152.0 50.4 56.3 20 0.6 8.6 153.0 51.8 58.7 12.8 T54.0 53.2 20 1.2 17.0 155.0 54.5 3.6 21.2 156.0 55.8 6.0 25.5 157.0 57.2 8.4 29.8 158.0 58.5 10.7 34. 1 -98 -model UBC1 is that they would cause the central density of the earth's core to have a lower value than\ that necessitated by the Adams and Randall model (see Section 1.3.1) without greatly affecting the r i g i d i t i e s of the transit ion region and the inner core (Section 1.3.2.). The position of the traveltime cusps and the velocity -depth relationship for UBC1 are shown in Tables 3.2a and 3.2b respectively. The uncertainties in the depths to the discontinuit ies are estimated at being of the order of 10 km on the basis of ob-taining satisfactory turning points for the various branches. The traveltimes of model UBC1 l isted in Table 3.3 agree with the smoothed traveltimes determined in this study within the l imits imposed by the scatter of the residuals. The greatest discrepancy was obtained for the early part of the GH branch. It appeared that the smoothed times were too early by about two seconds in this v i c i n i t y . This was undoubtedly the result of a biasing of the regression curve towards early times owing to the f i r s t measured arr ival time cbeing early, possibly as a result of epicentre mislocation. The model times for the GH branch tend to be shifted downwards by an average of about 1 second with respect to the Adams and Randall times, and upwards by about 3 to 5 seconds with respect to Bolt 's traveltimes in the latter and ear l ier part of the branch respectively. The model times for the IJ branch agree almost exactly with the model times of Adams and Randall, the only difference being that o the UBC1 times are about 0.2 seconds earl ier up to about 146 , and 0.1 seconds later beyond th is . The model times for the AB and DF branches agree with those of the J-B traveltime tables. It must be emphasized that model UBC1 is that model which - 9 9 -gave the best f i t to the d l /d f l and t rave l t ime measurements made in th i s p a r t i c u l a r study. A more e f f i c i e n t d i s t r i b u t i o n of data than that used in the study could r e su l t in small mod i f i ca t ions (such as changes in l oca t ion of the turn ing points of the branches, with r e -s u l t i n g e f f e c t s ) to th i s model. -100-CHAPTER k CONCLUSION Measurements of t rave l t ime and t rave l t ime gradients dT/dA > have been made at the Warramunga seismic array for PKP phases from 115 earthquakes over the d i s tance range 113° to 176°. The data have been used to obta in a ve l oc i t y -dep th model fo r the e a r t h ' s core. The dT/dA measurements at WRA were s t rong ly perturbed by loca l s t ruc ture beneath the array and an empir ica l method of c o r r e c t i n g fo r this e f f e c t was employed. The corrected dT/dA values were smoothed by the method of summary values and a polynomial r e -gress ion technique. Both methods gave re su l t s which were in reason-ably good agreement. The t rave l t ime gradient measurements smoothed by the summary value method were the ones a c t u a l l y used in i nve r t i ng the data to obta in a v e l o c i t y model s ince they had been re s t ra ined to acceptable t rave l t ime curves in order to remove any random e r r o r . The smoothed dT/dA values were inverted by the Herg lo tz -Wiechert method to der ive a ve loc i t y -dep th model for the e a r t h ' s core. This model, UBC1 , was the one which gave the best f i t to the meas-ured dT/dfl and t rave l t ime data. The s i g n i f i c a n c e of th i s study is that i t revealed c l e a r l y the ex is tence of two precursor branches - IJ and GH - to the main DF t rave l t ime branch for the e a r t h ' s core . Prev ious ly , only one of these branches, the GH one, had been widely accepted. Th i s s ing le pre-cursor branch resu l ted in a t r a n s i t i o n region for the e a r t h ' s core conta in ing two v e l o c i t y d i s c o n t i n u i t i e s and one " s h e l l " surrounding the inner core . On the other hand, the ex is tence of two precursor branches, GH as well as IJ, gives r i s e to three v e l o c i t y d i s c o n t i n u i t i e s -101-and two s h e l l s surrounding the inner core . This add i t i ona l branch, PKP,j , accounting for the e a r l i e s t precursors to PKP D F at d i s tances less than 145°, has been c o n s i s t e n t l y observed in th i s study. The corrsequences of i t s observat ion are manifested in the form of a t r i p l e - d i s c o n t i n u i t y , doub le - she l l t r a n s i t i o n region for model UBC1. The f i r s t v e l o c i t y d i s c o n t i n u i t y of 0.11 km/sec occurs at a depth of 4393 km. The region between the core-mantle boundary at a depth of 2894 km. and th i s d i s c o n t i n u i t y generates the AB t r a v e l -time branch fo r the e a r t h ' s core. This branch has i t s ' turn ing point at 144 .4° and i n i t i a l s lope of 3.52 sec/deg. The second d i s c o n t i n u i t y with an assoc iated v e l o c i t y jump of 0.24 km/sec occurs at a depth of 4810 km. The' region between the f i r s t and second d i s c o n t i n u i t i e s , a she l l approximately 417 km. th ick generates the e a r l i e s t precursor branch IJ. This branch begins at 134 .2° with an i n i t i a l slope of 3.48 sec/deg and terminates at 152.6° with a slope of 2.74 sec/deg. The th i rd v e l o c i t y d i s c o n t i n u i t y of magnitude 0.92 km/sec occurs at the inner core boundary at a depth of 5120 km. The region between the second and th i r d d i s c o n t i n u i t i e s , a she l l approximately 300 km th i ck , generates the second precursor branch GH. This branch has i t s turn ing point at 131.9°with an i n i t i a l s lope of 2.66 sec/deg o at that point and extends to 158.6 with a s lope of 2.16 sec/deg. The region beneath the t h i r d d i s c o n t i n u i t y , i . e . the inner core of radius 1251 km., generates the main t rave l t ime branch, DF, o for the e a r t h ' s core . The turning point of th i s branch is 113.2 and i t s ' i n i t i a l s lope is 1.96 sec/deg. The branch terminates at 180.0° with zero s lope. The consequences of a doub le - she l l t r a n s i t i o n region would -102-be manifested in other core phases as well. Thus, phases other than PKP (such as SKP,PKS,SKS) should be closely examined for cor-roborative evidence of a triple-discontinuity transition region. Such a region would have the same effect, producing two additional branches to the traveltime curve, on a l l core phases although the magnitude of the effect would vary. Some problems could arise in such a study. The phases SKS, PKS, and SKP never occur as f i r s t arrivals and the f i r s t two require horizontal component seismometers for optimum recording. Also, since the periods of these phases are considerably longer than that for PKP, identification and sub-sequent measurement of the precursors could be more d i f f i c u l t . Amplitude measurements in addition to dT/dfl and travel-time studies similar to those made in this project would be most useful in defining closely the turning points of the traveltime branches. This could have important consequences on the final core model derived. While i t is not expected that the proposed velocity model, UBC1, for the earth's core will be accepted as the ultimate one, major changes to this model are not presently envisaged. The methods used in this study have enabled a more definitive identification of the traveltime branches and i t is upon such i 'identification that any reliable velocity model depends. -103-BIBLIOGRAPHY Adams, R.D. and M.J. Randall, 1963. Observed Tr ip l i ca t ion of PKP, v o l . 200, NO. 4908, 744-745 Adams, R.D. and M.J. Randall, 1964. The f ine structure of the Earth 's core, B . S . S . A . , v o l . 54, NO. 5, 1299-1313 Adams, R. D. and M. J . Randall , 1969. Distance corrections for deep focus earthquakes. Geophys. J . R. a s t r . S o c , v o l . 18 329-330. Arnold, E.P., 1968. Smoothing traveltime tables. B . S . S . A . , v o l . 58, 1345-1351 B f r t f l l , J.W. and F .E . Whiteway, 1965. The application of phased arrays to the analysis of seismic body waves. P h i l . Trans. Roy. Soc. Lond. A, v o l . 258, 421-493 Bolt, B .A. , 1959. Traveltimes of PKP up to 145 . Geophys. J .R. as t r . Soc, v o l . 2, 190-198 Bolt, B .A. , 1962 Gutenberg's early PKP observations. Nature v o l . 196, 122-124 Bolt, B .A. , 1964. The veloci ty of seismic waves near the Earth's centre. B . S . S . A . , v o l . 54, 191-208 Bolt, B .A. , 1968. Estimation of PKP traveltimes. B . S . S . A . , v o l . 58, NO. 4, 1305-1324 Cleary, J . R. , C. Wright and K. J . Mulrhead, 1968. The effects of l o c a l structure upon DT/LYimeasurements at the Warramunga seismic array. Geophys. J., v o l . 16, 21-29 Buchbinder, G.R.R., 1971. A veloci ty structure of the Earth's core. B . S . S . A . , v o l . 6l_, NO.2, 429-456 Bullen, K . E . , 1938. E l l i p t i c i t y corrections to waves through the earth ' s central core. M.N.R.A.S., Geophys. Suppl . , vol. 4, 317-331 Bullen, K . E . , 1963. An introduction to the theory of Seismology, 3rd Edi t ion, Camb. Univ. Press, Cambridge Carder, O .S . , D.W. Gordon and J .N. Jordan, 1966. Analysis of surface foci traveltimes. B .S .S .A . , v o l . 54, 815-840 Cleary, J .R , and A . L . Hales, 1966. An analysis of the traveltime of P waves to North American stat ions, in the distance range 32 to 100 . B .S .S .A . , v o l . 56, 467-489 Corbishley, D.J., 1970. Multiple array measurements of the P wave traveltime der ivat ive. Geophys. J . , v o l . 19, 1-14 Engdahl, E . R. , J . Taggart, J . L. Lobdel l , E .P . Arnold and G. E . Clawson, 1968. Computational methods. B. S. S. A . , v o l . £8 1339-1344. -104-BIBLIOGRAPHY (cont 'd) Erg fn, K.,1967. Seismic evidence fo r a new layered s t ruc ture of the Ea r th ' s core . J .G.R., v o l . 72, NO.14, 3669-3687 Evernden, J . F . , 1953. D i rec t i on of approach of Rayleigh waves and re l a ted problems - 1. B.S.S.A., vo l . 43 , 335-374 F i s h e r , R.A., 1953. D i spers ion on a sphere. Proc. Roy. Soc. Lond, A, v o l . 217, 295-305 Gogna, H.L., 1968. Travel t imes of PKP from P a c i f i c earthquakes. Geophs. J . , v o l . _[6, 489-514 Gutenberg, B., 19l4 Ueber Erdbebenwellen, VIIA Nach. Ges. Wiss Goettingen Math. Physik, KI., 166-218. Gutenberg, G., 1957- The Boundary of the Ear th ' s Inner Core. Trans. Amer. Geophys. Union., v o l . 38., 750 Gutenberg, B., 1958. Caust ics Produced by Waves Through the Ear th ' s Core. Geophys. J . , vol^3_, 238 Gutenberg, B. and C F . R i chter , 1937. P' and the Ear th ' s Core. M.N.R.A.S., Geophys. Suppl . , v o l . 4 . , 363 Hannon, W.J. and R.L. Kovach, 1966. V e l o c i t y f i l t e r i n g of seismic core phases. B.S.S.A., v o l . 54, 441-454 J e f f r e y s , H., 1934. On smoothing and d i f f e r e n t i a t i o n of t ab le s . Proc. Camb. P h i l . S o c , v o l . 3 0 , 134-138 J e f f r e y s , H., 1937. On the smoothing of observed data. Proc. Camb. P h i l . Soc., v o l . 33, 444-450 J e f f r e y s . H . , 1939. The times of core waves (second paper). M.N.R.A.S., Geophys. Suppl. , v o l . 4 , 594-615 Jeffreys, H., 196l Theory of probability. Third Edition, O.U. P. 2l4-2l6. r J e f f r e y s , H.,and K.E. Bu l len, 1940, 1967- Seismological Tables B r i t i s h A s soc i a t i on for the Advancement of Sc ience, Gray Milne Trus t , London J u l i a n , B.R., D.Davies and R.M. Sheppard, 1972. PKJKP. Nature, v o l . 235, 317-318 Keen, C . G . , J . Montgomery, W.M.M. Mowat, J . E . Mullard and D.C. P i a t t , 1965. B r i t i s h seismometer array record ing systems, The Radio and E l e c t r o n i c Engineer, Vo l . 3 £ , 297-306 K e l l y , E . J . , 1964. Limited network process ing of se ismic s i gna l s . Massachusetts Inst. Techno1. Group Rept. 44 Knopoff, L. and J .G .F . Macdonald, 1958. Magnetic f i e l d and centra l core of the ea r th . Geophys. J . , v o l . 1, 216 -105-BlBLIOGRAPHY (cont 'd) . Kuhn, W. and S. Vielhauer, 1 9 5 3 - Boziehungen zwischen der Ausbreitungvon Longitudinal und Transversalwellen in relaxierden medien 2. Phyzik. Chem., vol. 202,214. Lehmann, I.,'1936. P', Publ. Bur. Central Seism. Intern. , Ser A, Trav. S c i . , v o l . _I4, 87-115 ' Muirhead, K . J . , 1968. E l im ina t ing f a l s e alarms when de tec t ing seismic events au tomat i ca l l y . Nature, v o l . 217, 533-534 Nguyen-Hai, 1961. Propagation des ondes dans le noyeau t e r r e s t r e d 'apres les seismes profonds de F i d j i . Ann. Geophys., v o l . J 2 , 60-66 N i a z i , M., 1966. Correct ions to apparent azimuths and t rave l t ime gradients fo r a d ipp ing Mohorovicic d i s c o n t i n u i t y , B.S.S.A., v o l . 56, 491-509 N i a z i , M., and D.L. Anderson, 1965. Upper mantle s t ruc ture of western North America from apparent v e l o c i t i e s of P waves. J.G.R., v o l . 70, 4633-4640 Otsuka, M., 1966. Azimuth and slowness anomalies of se ismic waves measured on the centra l C a l i f o r n i a Seismic A r ray . , B.S.S.A., v o l . 56, 223-239 and 655-675 Shima, E., K. McCamy and R.P. Meyer, 1964. A Four ier t r a n s f o r m method of apparent v e l o c i t y measurement. B.S.S.A., v o l . 54, 1843-1854 Shurbet, D.M., 1967. The earthquake P phases which penetrate the e a r t h ' s core. B.S.S.A., v o l . 57, 875-890 Stacey, F.D., 1969. Physics of the e a r t h ' s i n t e r i o r . Wiley, New York T ru sco t t , J .R., 1964. The Eskdalemuir sesmic a r ray . Geophys. J . , v o l . 9, 59-67 Underwood, R., 1967. The seismic network and i t s a p p l i c a t i o n s . Ph.D. Thes i s , Aus t ra l i an National Un i ve r s i t y , Canberra Watson, G.S., 1956. Ana lys i s of s i spe r s i on on a sphere. M.N.R.A.S., Geophys. Suppl. , v o l . 7, 153-159 Whiteway, F .E . , 1965. The record ing and ana ly s i s of se ismic body waves using l inear cross a r rays , The Radio and E l e c t r o n i c Engineer, v o l . 29, 33-46 Wright, C , 1968. Evidence for a low v e l o c i t y layer fo r P waves at a depth c lo se to 800 km. Earth Planet. Scf. Le t te r s , v o l . 5, 35-40 -106-BlBLIOGRAPHY (cont'd) Wright, C , 1970. P wave inves t i ga t ions of the Ea r th ' s Structure using the Warramunga Seismic Array. Ph.D. Thes i s , Aust -r a l i a n National Un i ve r s i t y , Canberra -107-APPENDIX I A Short Review of Seismic Ray Theory If t h e v e l o c i t i e s of e l a s t i c waves were uniform w i t h i n the ea r th , t h e n se i smic rays would be s t r a i gh t l i ne s f o l l o w i n g/Chords a s ' i n F igure A . l a and the t rave l t ime from a sur face focus to an observatory at an angular d i s -tance A from the focus would be T = 2 R / V Sin A / 2 (1) However, observat ions of t rave l t imes ind i ca te that they increase less S t r o n g l y with d i s tance than is ind icated by (1), and the T-A curves are more curved than th i s equat ion would suggest. The v e l o c i t y at depth is t h u s greater than at the surface and se i smic rays are r e f r a c t e d as in F igure A. lb . Bu l len (1963) has developed the theory app l i c ab l e to an ear th in w h i c h the v e l o c i t y increases with depth. F igure A . l b shows the geometry of r a y i n a mult i -1ayered ear th in which the v e l o c i t y v increases with depth f r o m the su r face . It is assumed that the Earth is spher i ca l and completely symmetrical about i t s centre in a l l i t s p rope r t i e s , and d i f f rac t ion e f f e c t s a r e ignored. Apply ing S n e l l ' s law to each of the boundaries A, B i n F igure A -1 b. t h e n Sin t, /v, = Sinf,/vz Sm v-i/Vi = Sinfi/v^ ( , , a) But from the two t r i a n g l e s , ^ = r, Smf, = r z Sin iz From equation {Jjx) . S r t i . / v , - r,S.r.f,/V2 = r 2 Si 'n i z / v z s.rzSinf2fa or more' generally, r Sm i / v = Constant = p (2) where i i s the acute angle between the ray and the radius at any po in t . p i s c a l l e d the parameter of the ray, and is const ant for a l l po ints a long the ray. The value of p d i f f e r s from member to m ember of a given fami ly of rays, and at the deepest point of penetra t ion of any ray, where -108-A-Ifejjath of a seismic ray through A-I(b).Seisa:ic ray i n three-layer a hypothetical uniform earth. earth. Figure A-l Seismic Ray Theory. -109-Sln i " 1 , is equal to the value of r /v at that point Consider two rays of a family of rays with parameters p, and p + 4p7 traveltimes T and T + d f subtending angles of A and A + dA at 0 as shown in Figure A . l c , respect ively . Let PQN be the normal to the ray through Qo- Then Smio « N Q o / P o Q o = v 0 . i . d T / r 0 . ^ d A and hence from equation (2), p = rS.ru/v = dT/dA ( 3 ) Let P be any point of a ray whose parameter is p, with polar coordinates (r,e) as shown in Figure A. Id" and let the arc length P 0P be s . Then, from (2) r /v <*%U=P (4) and substi tut ing for ds from the polar re lat ion (ds) 2 = (dr) 2 + (rdo) 2 equation (4) yields the relat ionship • -I , 2 2 * i , V-do - + pr (n - p ) dr (5| ; where for convenience, the parameter n. = r/v (= p at the deepest point cf the ray) . Integrating (5) between PQ and the deepest point of the ray, we obtain the expression for A , i A = P jr"V-p2)~* dr (6; and the corresponding expression for T where ir -  jVr-'tf-p1)"4 ao (7; At this point, i t is convenient to introduce the important functions ^ Ond ^ where ^ : d l n v / d Inr = r / v dv/dr, and | - 2 / i - £ - 2d\r- r/d. la r\ , H can be shown that the downward curvature of a ray with i ts iowest point at some level r in the earth is giv«in by l/v dv/dif\ and this curvature must be less than the curvature ' ^ r of the level surface at r, i f the ray is to ex /s t This condition implies that <J <• I > olf* § >°) <r dr\/dr >Om . Introducing the functions and £ into equations (6) and (7)j and then d i f ferent ia t ing the resul t ing equations lead to the expressions (Bullen, 1963) - 1 1 0 -where the subscr Ipts 0 re fe r to the values of the parameters at the e a r t h ' s sur face . These equations are fur ther s i m p l i f i e d by in t roduc ing the dimensionless va r i ab le s M = n/n.o , A = P/>\ •> T = t / t t 0 i to become u/to - - f o 0 - x a ; - 4 - t { 0 ( M x - \ z r ± d ^ -x-f-y (ro) - ... IP • * dT/dA «-f 0 A^i -A*)->; {W-A^ j : (H) te Consider the r e l a t i o n s h i p between A and T corresponding to d i f f e r e n t types of v a r i a t i o n of v with r. Normal or ' o rd ina ry ' behaviour of v with r e n t a i l s that v increases s lowly as r decreases, and that the ra te of change of v is changing s lowly. This implies that ^ is negative and moderate in va lue, d^/dr and d|/dr are sma l l , a l s o dr^/dr i s p o s i t i v e and ^ Q is negat ive. From the r e l a t i o n p = £ in i 0 p decreases s t e a d i l y between n Q and zero as i 0 goes from 90° ( i . e . A =T = o) to 0 ° . Since ^ Q is negative and P = f\ 0 when A ' s zero, i t fo l lows that for small values of A , X in equation (10) i s large and p o s i t i v e . Owing to the smallness of d|/dr and the range of in tegra t ion for small A , Y is a l so smal l , so that fo r small A , dA/dh i s large and negat ive, and conse-2 2 quently d T/dA is negative tending to o for small A . For small A , then, equations (10) and (11) can be wr i t ten ' dA/dA = | o ( i - A * ) - ! Ol) dLT(d\ =-| 0 A O - A 1 ) " ^ - - 03) t * | 0 Cos "*A 04) T - f o o-A z ) 4 «f 0 sinCA/f 0 ; 05) The above equations hold for small A for normal behaviour of v, and for rays terminat ing at the surface of the earth r=r Q . For that segment of a ray, taken as l y i ng wholly on one s ide of the r ay ' s lowest point and terminat ing at the leve l s r, and r 2 say, the values of & and T are from (14) and (15). -111-and the con t r i bu t i on s to A , T and d&/ d\ for the pa i r of segments of a whole ray which l i e between l eve l s r 0 , and f\ xx.ce. t^wet\ respectively by * ( C o s " A - C o s " ' ( A / / U t ) ) Consider a reg ion L in which the behavior of v i~ normal being represented by j =|o ? £ = § © °r V ~ V o P C r | c ) extending down to a leve l r j , to be under la in by a layer N extending fo r some d i s tance beneath r^. Suppose the v e l o c i t y increases d i s con t i nuous l y at the leve l r j , so that the v e l o c i t y immediately above r j , va, is less than vb, the v e l o c i t y j u s t beneath r j . Thus /Jib is less than Mo. . Let ^ have the constant value k(j 0 in layer N where k ^ 1, so that J £~ ^ c t • Rays fo r which ,UQ, A £ | l i e e n t i r e l y in L and equat ions (12) -(15) apply for th i s reg ion . From equat ion (12), i t i s seen that d&/dA is i n i t i a l l y negative and f i n i t e as p rev ious l y de sc r i bed , becomes less and less negative as A decreases. The behavior i s shown in s ec t i on AB of the curve in F igure A.2a. Rays fo r which Jx b 6- A 6. U<\ a l s o l i e e n t i r e l y in L but cons i s t of rays i n t e r n a l l y r e f l e c t e d o f f the boundary at r j and for these rays, from equations (18) - (20), we have A = | 0 (Yes- 'A - CCS~'(*/M«)) (Zl) T = $. ( o a t ) From equation (23) at A = ? dt\ J d\ jumps di scont i nuous ly to t oO thereby decreas ing through f i n i t e p o s i t i v e values as }\ decreases , becoming zero at some point given by ( I — A 2"^ ~ ^ A A a "~ A ^ ) For rays whose lowest points are in N, \ A b and the appropr ia te A-2fl>). Figure A-2. Behaviour of dA/dA- A, A -A, and T-A, curves for dircontinous increase in velocity at a boundary,. -113-equat i ons are •' ^ T = f „ ( ( I - A 2 ) 1 - 0"«l- * k ^ ^ ( 2 5 ) As A decreases to the va lue yU|> IM l ayer N, d A / d A from (26) jumps di s c o n t i nuously to -00, and as \ decreases beneath the value ,U.b d A / d A increases through f i n i t e negat ive values r e s u l t i n g i n the s e c t i o n C 0 ° f the dk jd\ c u r v e . Equat ions (14), (15), (21), (22) and (24), (25) show that A a ri'- T a r e cont inuous func t i ons of A tor rays i n L and N. The d A /d\ (ordA/dp)curve i s i l l u s t r a t e d i n F igure A . 2 a , and the c o r r e s -ponding p - A and T - A curves in F igu re s A.2b and A,2c r e s p e c t i v e l y Of importance i s the f a c t that ne i the r of the cusps B and C are a s s o c i a t e d w i t h l a rge ampl i tudes s i n c e d A /dX , and there fore cM j d(L o (where e., the angle of emergence i s g iven by p = T\Cc&e.c> ) at these po in t s are not equal to z e r o . Branch BC corresponds to rays r e f l e c t e d o f f the d i s c o n t i n u i t y ( reced ing branch) w h i l s t AB and CD correspond to rays r e f r a c t e d through l a y e r s L and N r e s p e c t i v e l y . On the branch BC, d.A / d A and the re fore d*T" j d A 2" i s zero at both B and C (where A = iX fc> and AX<x r e s p e c t i v e l y ) and consequent ly BC was v i r t u a l l y s t r a i g h t for some d i s t a n c e from B, any marked cu rva tu re being near C, and branch CD l i k e w i s e runs v i r t u a l l y s t a i g h t for some d i s t a n c e from C, any marked curva tu re being near D. It can be seen that the above d i s c u s s i o n a p p l i e s to t h e . v e l o c i t y d i s c o n t i n u i t i e s :n the t r a n s i t i o n r e g i o n , and imposes the c o n s t r a i n t s on the t r a v e l t i m e curve mentioned in s e c t i o n 1.2. For the case of a d i s c o n t i n u o u s " decrease i n v e l o c i t y , as at the mant le -core boundary, the s i t u a t i o n i s somewhat d i f f e r e n t . In t h i s case , l/fc) Va. and AX b > M a. > a n d w e assume that layer N extends downward far enough for jX to r e t u r n through the va lue XiJa, before the bottom of N i s reached. For rays w i t h lowest p o i n t s in L, equat ions (12) - (15) s t i l l a p p l y . For rays which penetra te N, (24) - (26) -114--115-a p p l y , but sin c e \ 4. Ma. f o r these r a y s , there are no rays w i t h lowest p o i n t s i n the upper part of N f o r which Ma. A ^ • Consequently a t A — Ako, there are d i s c o n t i n u i t i e s b"A and ST in 'A and T g i v e n by s u b t r a c t i n g (14) from (24) and (15) from (25) r e s p e c t i v e l y : SA = k j e Cos Ou / A b ) (27) For rays p e n e t r a t i n g N, dA/dh from equation (26) has the value +00 a t A = Aja and decrease as A decreases from/Ua. Thus £\ and T decrease r a p i d l y as A decreases f r o m / j a , and provided N i s deep enough reach minima given by the value of A s a t i s f y i n g the equ a t i o n . ^ ( | - A 2 ) " ^ k ( . u b 2 - A > ( « a - A T ' This minimum i s the cusp D shown i n Figure A.3c and Since dA /d\ i s zero a t t h i s p o i n t , D corresponds to a c a u s t i c . As before, at po i n t C i s zero, and consequently the branch i s v i r t u a l l y s t r a i g h t some d i s t a n c e from C^any marked curvature being near the c a u s t i c D. The corresponding dA /dA~A a nd A-A curves are shown i n Figure s A.3a and A.3b. The above d i s c u s s i o n s cover the v a r i a t i o n i n v e l o c i t y o b t a i n e d i n the e a r t h from the base of the mantle to the center o f the ear t h ' s c o r e . It i s intended that they provide i n s i g h t i n t o the nature of the t r a v e l t i m e curves generated by v e l o c i t y d i s c o n t i n u i t i e s i n t h i s r e g i o n and the con-s t r a i n t s imposed upon these curves by ray theory. APPENDIX 2 LIST OF F.ARJKQ.UAKES USED' IN STUDY Event N o . Date Or i gi n Time Lat i tude . L o n g ! t u d e Di s t a n c e F o c a l Depth Magni tude A z i muth H r . Hi n. Sec (Degrees) (Degrees) (Degrees ) (KIT • ) (Degree 1 J a n . 1 4 / 6 9 23 2 7.9 13.175N • 29.203W 113.47 . 33 ON 5 5 301 .6 2 O c t . 2 4 / 6 9 8 29 12. 1 33 .292N 11 9 . ' 93W 114.11 10 0 5 1 61 .6 3 0 c t . l 8 G 7 1 11 . i,v 8 7 9 . 8 N 2.4 E 116 .50 33 0 5 7 351 .5 4 June 1 2 / 69 15 13 31 1 34 .405N 25.061 E 116.60 25 OD 5 8 259.2 5 N o v . 2 3 / 6 7 13 42 1 6 80.2 N 1.0 W 116.70 10 0 5 8 3 5 2 . 3 6 D e c . 1 9 / 6 8 16 • 30 0 0 3 7 . 232N 116.477W 1 16.80 0 0 6 3 5 7 . 6 7 S e p t . 1 6 / 6 9 14 30 0 0 3 7 . 3 I 4 N 116.460W 116.82 0 0 6 .2 5 7 . 5 8 O c t . 8 / 6 9 14 30 0 c 3 5 . 257N 116.450W 116.83 0 0 5 5 5 7 - 6 9 . O c t . 2 9 / 6 9 22 1 51 4 37 .143N 116.064W 117.12 0 0 5 7 5 7 . 3 10 S e p t . 6 / 6 8 14 0 0 1 £ 37 .135N 116.047W 117 .13 0.0G 5 6 57.8 11 D e c . 2 5 / 6 8 12 17 20 35.-130N 2 4 . 3 3 0 E 117 .27 40 0 5 .0 2 9 9 . 9 12 N o v : 2 1 / 6 7 17 2 25 C 72 . 7 0 N 8 . 5 E 119.20 51 0 5 5 3 4 3 . 9 13 N o v . 3 0 / 6 7 7 23 51 5 4 ) . 5 0 N 2 0 . 5 E 120 .6 29 0 • 6 .0 307.0 14 A p r ! 1 18/69 17 43 53 7 24.283.M 107 .738W • 120.92 33 0 ' 5 .0 73.1 15 S e p t . 2 6 / 6 7 11 1 1 23 1 3 3 . 6 0 S 7 0 . 5 w 121.70 84 0 5 .8 155 .6 16 J a n . 2 1 / 7 0 17 51 3 8 . 5 7 .017N 104.293W 121.80 33 ON 6 .2 9 4 . 1 17 N o v . I / 6 9 11 8 20 S 23 .148N 107 .934W 122 . 3 6 33 ON 5 .6 7 ^ .7 . 18 D e c .13 / 6 9 21 33 21 c -> 3 2 . 7 0 8 S 69 .972W 122 .71 105 OD • 5 .6 155 . 6 19 O c t . 2 7 / 6 9 8 10 5 8 . 3 44 .923N 17 -232E 122 .91 33 ON 5 .3 311.1 20 . S e p t . 2 0 / 6 9 15 26 41 c. 1 .785N 101.031 w 123.0 33 ON 5 .5 101.2 • 21 A p r . 2 6 / 6 9 6 2 49 C 3 0 . 6 4 5 S 71 .542W 123 .87 33 ON 5 .9 . 153.0 22 N o v . 2 7 / 6 7 5 13 12 6, 30.8 S 7 1 . 0 0 w • 124.0 62 0 5 .4 153.6 23 June 23/69 - 7 8 • 27 7 18 .373N 104.546W 124 . 5 9 36 0 5 .3 8! .0 24 O c t . 7 / 6 7 1 14 4 1 2 9 . 6 S 71.1 V/ 125.0 42 0 5 .3 152 . 8 25 N o v . 1 5 / 6 7 21 31 51 5 2 8 . 7 S 71.2 W 125 . 7 0 15 0 6 .2 152.2 26 N o v . 1 3 / 6 9 7 51 29 c 2 7 . 8 S 71 .65 W 126.29 33 ON 5 .8 151.2 27 J a n . 6 / 6 8 23 27 21 1 <• 2 7 . 8 S 71.1 w 126 . 5 0 33 0 5 .8 151 .7 28 A p r . 17/&9 17 37 ' 22 L 28 .26 S 67./86V/ 127.11 82 0 C .0 154.2 29 S e p t . 2 / 6 9 3 47 9 1 2 7 . 7 4 5 S 66.491W 128 .47 174 00 5 . 5 156.2 30 A u g . 2 0 / 6 7 15 3 36 2 2 5 . 2 S 69.0 W 129 . 70 109 0 5 .6 152.2 31 S e p t . 8 / 6 7 8 59 59 I 2 3 . 4 S 7 0 . 7 W 130.50 33 0 5 .5 149.2 32 S e p t . 2 1 / 6 9 2 0 54 • 23 . 5 5 2 S 68.080W 131.55 120 OD 5 .5 152.1 33 D e c . 2 9 / 6 8 16 29 31 1 23 . 9 7 5 S 66 .673W 131 . 77 205 OG 5 .5 153 . 9 3^ S e p t . 1 3 / 6 9 !0 52 58 C 22 . 883S 63.367W 132.00 106 OD 5 . 4 151 .4 35 May 1 6 / 6 8 22 45 19 22.883S 68 .6 W 132.00 104 0 5 .0 151.1 36 D e c . 2 5 / 6 7 10 41 31 i 2 1 . 5 0 S 7 0 . 4 W • 132.20 53 0 5 . 8 148.2 37 • June 6/69 22 25 37 • 22..511S 63.418W 132 .27 125 OD 5 .0 151.1 APPENDIX 2 L i ST OF EARTHQUAKES USED IN STUDY : n t No, D a t e O r i g ) n T i Pfe La t i :ude L o n g i t u d e D i s tanc.e F o c a l D e p t h Ma oil 1 -tude A>-' <-'.!t Hr . M i n Se (Dt-q " ) (D< g r e o s ) ( D e g r e e s ) ( k m . ) ( D e g r e 38 J u l y 25 / 6 9 6 6 42 4 25 552 S 6 3 . 3 low 131 63 579.OD c 5 158 5 39 O c t 2 0 / 6 9 15 2 0 36 5 17 300N 9 5 . 188W 133 11 8 7 . 0 5 4 84 7 40 S e p t 1 7 / 67 7 5 6 22 .7 17 2 N 9 4 . 1 W 134 10 4 5 . 0 . 5 2 85. 1 41 O c t . 2 2 / 6 9 10 21 52 . 1 18 086 S 71 .542W 134 4 6 23 0 5 4 144 ! 42 J u n e 15/68 5 11 17 . 2 14 4 N 92 9 W 134 6 0 2 5 . 0 5 LK 8 9 . 2 43 J u l y 1 9 / 69 4 54 54 1 17 2 5 4 S 72 519W 134 53 54.OD ' 5 9 1 4 2 . 7 4 4 J u n e 2 2 / 6 9 14 30 10 7 16 9 2 9 N 93 608W 134 52 1 5 1 . 0 5 1 85 6 45 A u g 23 / 6 8 22 36 51 3 21 986 S 6 3 . 5 4 7 W 134 70 3 8 . 0 6 5 129 1 46 S e p t . 3 / 6 7 21 . ~i V 30 8 10 6 S 7 9 . 8 W 134 72 5 0 . 0 5 0 ! 4 0 9 4 7 Nov 7 / 6 8 10 13 39 8 16 388s 73 .460V/ 134 70 5 8 156 3 48 O c t 31/68 9 15 46 9 16 289S 73 .298w 134 89 6 7 OD 5 7 141 0 49 j a n 8 / 6 8 18 44 24 5 18 6 S 69 9 W 134 90 1 1 6 . 0 5 *T 146 6 50 S e p t . 2 5 / 6 8 10 38 38 4 15 571N 92 '638W 135 15 1 3 7 . 8 5 7 8 / 7 51 S e p t . 2 8 / 6 8 13 53 35 3 • 13 160S 76 382W 135 31 70.OG 6 0 134 9 52 S e p t . 1 5 / 6 9 7 14 25 8 18 5 5 2 S 69 .021 V/ 135 4 0 177.OD 5 2 147 6 53 Oc t . 5 / 6 7 9 41 3 ! 4 14 5 S 75 4 w 135 5 0 1 C 0 . 0 5 6 13 7 .2 54 S e p t . 2 7 / 6 ? 6 2 39 5 / 3 S 8 1 . 3 V 135 70 3 7 , 0 5 I 124 . 1 55 A u g 23 / 6 8 23 14 52 7 21 8 1 9 S 6 3 . 5 3 0 W 134 96 541 OD 5 2 156 2 56 J u n e 1 1 / 6 9 21 15 32 6 8 590S. 79 688W 136 10 81 . 0 5 0 12 7 .0 57 A p r i l 2 1 / 6 9 2 19 7 1 14 0 9 8 N 91 .015W 136 33 8 2 . 0 5 5 90 3 58 •J-y.i 12/69 14 12 53 0 14 142S 72 74 7W 136 89 1 1 3 . 0 5 2 139. 7 59 J u l y 2 4 / 6 9 2 59 21 0 11 85 IS 7 5 . 1 3 0 W 137 10 1 . 0 5 9 135 .0 60 O c t . 2 0 / 6 9 18 0 28 7 11 859S 75 164W 137 07 1 2 . 0 5 1 134 9 61 O c t . 6 / 6 9 0 37 59 6 11 737S 7 5 .100W 137 20 3 . 0 5 1 1 3 4 . 9 62 O c t 6/69 6 36 45 2 11 782 S 7 4 . 9 9 9 W 137 23 4 . 0 5 3 1 3 4 . 0 63 N o v . 1 6 / 6 9 10 30 1 7 13 3 5 1 N 8 9 . 6 5 0 N 137 43 79.0 4 9 9 1 . 8 64 S e p t . 6 / 6 8 7 49 4 2 0 5 767S 80.261W 137 4 4 6 6 . CD 5 3 123-3 65 N o v . 1 8 / 6 7 12 IS 55 4 13 37N 89 .07W 138 0 0 7 8 . 0 5 1 92 . 0 66 D e c . 2 7 / 6 6 21 22 1.4 8 13 20N 8 8 . 8 1 W 138 2 0 6 6 . 0 5 5 92 . 4 67 J u l y 18/69 J u n e l 1 / 6 8 23 17 10 6 18 241 S 53 314W 138 2 8 1 9 . O D 5 6 154 .3 68 5 5 2 33 . 5 13 SON 8 8 . 8 1 W 138 4 0 1 9 9 . 0 5 3 91 . 4 69 J u n e 6 / 6 9 1.6 16 1 7 12 46 7N 8 8 . 0 0 0 W 138 75 4 8 . 0 5 0 9 3 . 7 70 N o v . 9 / 6 8 17 1 41 1 37 96 IN 8 8 . 4 6 4 W 138 37 1 9 . 0 5 _> 5 4 . 8 71 J u n e 2 0 / 6 8 15 51 56 5 5 6 S 7 7 . 3 W 139 80 3 3 . 0 5 8 1 2 5 . 9 72 J u n e 2 1 / 6 8 0 26 7 . 8 5 7S 77 3W 139 8 0 2 2 . 0 5 6 126.1 73 J u n e l 9 / 6 8 8 13 35 .0 5 6 S 77 2W 139 . 9 0 2 8 . 0 .6 . 4 1 2 6 . 0 74 O c t . 4 / 6 7 6 2 16 4 10 7N 8 6 . OW 140 10 3 3 . 0 5 3 9 7 . 2 A P P E N D I X 2 L I S T O F E A R T H Q U A K E S U S E D I N S T U D Y E v e n t No. D a t e O r i g i n T i m e L a t i t u d e L o n g i t u d e D i s t a n c e F o c a ! D e p t h M a g n i t u d e A z i m u t h H r . M i n . S e c . ( D e g r e e s . ) ( D e g r e e s ) ( D e g r e e s ) ( k m . ) ( D e g r e e s ) 75 O c t . 3 / 6 7 1 8 1 6 3 . 2 1 0 . 9 N 85 .9W 1 4 0 3 0 21 . 0 5 , 8 9 7 . 0 76 S e p t . 2 0 / 6 9 5. 8 5 7.6 5 8 . 2 9 7 N 3 2 . 1 8 9 W 1 4 0 , 4 6 3 3 . O N 5 r . 0 3 4 8 . 9 77 A u g . 2 7 / 6 7 13 8 5 5 , 9 12,3N 8 6 . 2 W 1 4 0 . 4 0 1 8 3 . 0 5 2 9 4 . 7 78 O c t . 1 5 / 6 7 8 0 5 0 . 3 1 1 . 9 N 8 6 . O W 1 4 0 . 5 0 1 6 2 . 0 6 2 9 4 . 5 79 J u n e 2 4 / 6 9 0 35 5 . 5 1 1 . 6 6 IN 8 5 . 7 1 7 W 1 4 0 . 6 6 100 . O G 5 . . 3 9 5.9 8 0 J u n e l 5 / 6 8 7 8 4 8 . 1 5 . 6 N 82.6w 1 4 0 . 8 0 1 6 , 0 6 . 0 1 0 6 . 6 8 1 O c t . . I I/67 20 2 8 10 .2 10.3S 71 . 2 W 1 4 1 , 4 0 585 . 0 5 , , 0 137.8 8 2 M a y 1 6 / 6 8 8 25 9 . 2 3 . 7 S 76.6W 1 4 1 . 6 0 113 . 0 5 l± 1 2 4 . 3 8 3 J a n . 1 / 7 0 1 43 4 6.7 8 . 5 9 8 N 8 3 . 5 2 5 W 1 4 1 . 6 9 4 8 , 0 5 . 101 . 6 8 4 N o v . 8 / 6 7 3 10 5 3.3 1 6 . 8 N 8 5 . 9 W 1 4 1 . 7 0 28 . 0 5 '4 8 7.9 85 M a y 31 /69 1 1 7 17.1 1 . 7 5 3 S 77 ,744w 1 4 1 . 8 4 1 7 2 . , 0 5 , 1 120,7 8 6 D e c . 1 6 / 6 8 3 7 2 4 , 1 7 - 1 2 3 N 8 2 . 2 4 3 W 1 4 2 . 3 1 16 . 0 5 . 3 1 0 4 . 6 87 M a y 2 8 / 6 9 1 3 30 8 . 9 2 , 0 3 6 s 7 6 . 9 2 7 W 1 4 2 . 3 0 177 . O D 5 . 5 1 2 2 . 0 8 8 A p r i 1 2 5 / 6 9 3 34 17.7 7 . 4 5 0 N 8 2 . 0 7 5 W 1 4 2 . 5 9 2 5 . O G 5 , . 4 1 0 4 , 2 8 9 M a y 5 / 6 9 5 34 2 3.5 3 6 . 0 2 3 N 1 0 . 3 9 0 W 1 4 5 . 2 1 2 9 . 0 5 . 5 3 0 4 . 9 90 S e p t . 2 4 / 6 9 3 58 5 6.5 52 5 5 9 N 3 1 . 8 4 1 W 1 4 5 . 7 2 33. . 0 5 . , 2 3 4 5 . 0 91 D e c . 2 4 / 6 9 5 4 4 4 . 5 3 5 . 9 5 3 N . 1 0 . 3 9 6 W 1 4 5 . 2 4 3 3 . . O N 5 . 1 3 0 4 , 3 9 2 F e b . 2 8 / 6 9 2 4 0 3 2.5 3 6 . 0 0 8 N 1 0 . 5 7 3 W 1 4 5 . 3 6 2 2 . 0 ~J I 3 0 4 . 9 9 3 S e p t . 2 4 / 6 9 4 20 5 2 . 5 5 2 . 4 9 9 W 3 1 . 8 4 9 W 1 4 5 . 7 8 3 3 . . O N 5 . 0 3 4 5.0 94 S e p t . 6 / 6 9 1 4 30 3 9.5 3 6 . 3 4 1 N 1 1 .887W 1 4 6 . 0 5 33. . O N 5 , 1 . 1 3 0 7 . 1 95 N o v . 1 4 / 6 9 6 52 5 . 3 4 . 9 3 4 N 7 6 . 8 4 6 W 1 4 6 . 1 7 53. , C 4. . 7 112.0 96 A u g . 3 1 / 6 8 2 1 ' 4 7 3 3.5 4 . 5 0 5 N 7 6 . 3 5 9 W . 1 4 6 . 3 9 9 7 . c 4 . . 6 113.1 97 S e p t , 2 2 / 6 7 8 8 4 . 3 0 . 7 S 2 0 , 1 W 1 4 7 , 6 0 33. . 0 5 , . 3 233 . 6 98 N o v . 2 1 / 6 7 21 50 2 4 . 3 4 3 . 2 N 2 7 . 8 W 1 4 8 . 3 0 33. . 0 5 , . 0 237.0 99 S e p t . 1 / 6 8 4 48 5 2.2 0 . 9 9 5 S 2 4 . 5 1 2 W 150,63 33, . O N 5 .2 2 2 7.3 100 S e p t . 1 / 6 8 8 1 9 1 5 7.2 0 . .889S 2 4 . 5 1 9 W 150.70 33. . O N 5 . 0 22 7.4 101 O c t . 2 0 / 6 9 13 1 1 3 7 . 0 1 0 . . 7 9 6 N 7 2 . 4 8 0 W 1 5 2 . 6 ! 4 0 . O G 5 . 7 105.4 102 O c t . 2 0 / 6 9 1 3 ! 1 3 3 . 5 10.796K 7 2 . 4 S 0 W 1 5 2 . 6 8 5 5 . 0 5. 1 i C 5 , 4 103 O c t . 1 6 / 6 8 1 55 3 . 2 . 7 1 9 . 1 5 2 N 69.838W 157.19 36. , 0 . . 2 8 7 . 9 1 0 4 S e p t . 2 0 / 6 8 6 0 3 . 5 1 0 . . 7 3 5 N 6 2 . 6 6 3 W 161 . 2 2 1 0 7 . . O D 6 . 2 1 1 6 . 7 105 O c t . 2 2 / 6 9 12 5 2 2 2 . 2 •10.92 I N 6 2 . 5 5 2 W 1 6 1 . 4 1 79. 0 5 . ,4 i 16.4 1 0 6 S e p t . 2 2 / 6 9 13 47 5 2.2 4 . 9 7 7 N 3 2 . 6 2 5 W 1 6 0 . 4 5 • 33, . 0 5 . . 7 222. I 107 S e p t , 3 / 6 8 1 5 37 0 . 2 2 0 . 6 3 1 N 6 2 . 2 3 8 W 1 6 4 . 4 2 3 3 . 0 5 . . 5 8 4 . 6 1 0 8 M a y 1 5 / 6 9 20 43 3 3 - 4 1 6 . 5 7 3 N 6 1 , 3 4 1 W 1 6 4 . 7 7 50. , 0 5 . 7 9 9.5 1 0 9 D e c . 2 4 / 6 7 20 3 10.9 1 7 - 4 N 61.1 w 1 6 5 . 1 0 2 4 , ,0 6 . , 4 9 7.3 1 1 0 D e c , 2 5 / 6 9 22 31 2 . 3 1 6 . 0 8 3 N 5 9 . 7 7 1 W 166.03 8 . C ) 6 . . 0 1 0 3 . 7 1 1 1 D e c . 2 5 / 6 9 21 32 2 7 . - 3 1 5 . 7 7 2 N 5 9 , 6 5 0 W 1 6 6 . 0 4 7.0 6 . 4 1 0 5 . 1 APPENDIX 2 L I S T OF EARTHQUAKES USED IN STUDY E v e n t No, . D a t e Or i g i n T i me L a t i t u d e L o n g ! t u d e D i s t a n c e F o c a l D e p t h Magni i t u d e A z i m u t h H r . Mi n. S e c . ( D e g r e e s ) ( D e g r e e s ) ( D e g r e e s ) ( km. ) ( D e g r e e s ) 112 D e c . 2 6 / 6 9 8 4 6 1 5 . 2 1 5 . 8 3 9 N 5 9 . 5 7 W 1 66 . 1 3 2 2 . 0 5 , . 2 1 0 5 .1 113 D e c . 2 6 / 6 9 2 0 3 2 8 . 8 1 5.791N 5 9 . 5 5 5 W 1 6 6 . 1 3 33.ON 5 . . 4 1 0 4 . 9 1 1 4 Jan.8/68 2 0 2 2 1 5 . 6 8.2 N 38.2 W 1 6 6 . 3 0 33.0 5. . 4 2 1 2 . 7 115 S e p t . 2 4 / 6 9 1 8 3 19.0 1 5 . 2 3 7 N 45.7 7 6 W 175.31 3 3 . O N 5 , . 8 1 7 8 . 5 -120-APPENDIX I 11 ELLIPTICITY AND FOCAL DEPTH CORRECTIONS TO MEASURED TRAVELTIMES Bullen (1939, 1938, 1963) has shown that the el 1 ipt ici ty correction 6T to the travel time •foryeocQntric lat i tudes is given approximately by the formula -ST - f (A)Cho + Hi) .....0) where h e and h, are the differences between the actual and mean radii of the earth at the epicentre and observing stations respectively, and are l isted by Bullen, (1937). d lso - fC A ) i s given by Bullen (1938). The value of h , for WRA is +5.0 km while fY&) ranges from 0.10 to 0-095 for PKP D F and 0.100 to 0.074 for PKP A B-Time corrections for focal depth were determined for earth-quakes with focaj depth less than 200 km. from the tables given by Bolt (1968), and the corresponding distance corrections result ing in a surface focus for the event were estimated from near-earthquake P tables. For focal depths greater than 200 km, the equation developed by Adams and Randall (1969) for deep-focus earthquakes, and derived especial ly for use with PKP phases was used: £ - ( 0 - 0 5 + 0 0 2 . 5 H +..o.-;oo,2 8 h a ) p ••--•CO where S is the correction in degrees, h is the focal depth in units of hundredths of t^he earth's radius, and P is the ray parameter dT/d^ -6 is determined from the above formula and the corresponding time correction found from tables of near-earthquake traveltimes using the appropriate values of 5 and h. -121-APPENDIX III ELL IPTI CITY AND FOCAL DEPTH CORRECTIONS  TO MEASURED TRAVELTIMES (cont.) Thus a l l the events were adjusted to have surface f o c i . A check was made on the distance and time corrections obtained for a l l the earthquakes in the above manner by using a computer program designed to calculate these corrections for ray paths of specified parameter t ravel l ing through the earth. The method used in this program was to calculate the quantities A12 O-^d T l 2 . a s described in Appendix I for radii down to the focal depth given, using an assumed velocity model for the crust and mantle. It was found that good agreement was obtained at most focal depths except for those in the v ic in i ty of about 500 kilometres. In these cases the corrections given by the program were smaller by a maximum of about 0.12 deg. than the value obtained using the Adams and Randall equation. The latter values were assumed to be the correct ones. In either case, at the large distances in question, an error of 0.12 deg. is small, compared to the error in epicentre determiniation which can be as large as 0 .5 ° Al l the epicentre coordinates l isted in Appendix II are the preliminary determinations complied by N.O.A.A. on the basis of the J-B traveltime tables. Recent traveltime studies, for example Cleary and Hales (1966), have revealed systematically shorter traveltimes at teleseismic distances than those of Jeffreys and Bullen. Con--122-APPENDIX I I I ELLIPTICITY AND FOCAL DEPTH CORRECTIONS  TO MEASURED TRAVELTIMES (con't) sequently, s ignif icant systematic errors in epicentre'and focal depth determination may exist but until the traveltime controversy is sa t is fac tor i ly resolved, very l i t t l e can be done about th is . The effect of errors in epicentre location on dT/dA measurements is to increase the scatter of the data especial ly in those regions of the curve where d T/dA is large as in the neighbourhood of a caust ic . Errors in focal depth determinations, part icular ly for deep focus earthquakes, may well produce as much scatter in the dT/dA values as epicentre mislocations. The view is commonly held that focal depth determinations of greater than 60 km are somewhat unreliable. In this study, more than 1/3 of the events had focal depths greater than 60 km, and many of these were restrained in some manner to a particular leve l . However, as discussed in Section 2 .2 .3 , systematic errors in the dT/dA measurements are more signif icant than any others. -123-APPENDIX IV METHOD OF SUMMARY VALUES Jeffreys (1937. 1961) devised the method of summary values as a smoothing technique. The following description follows more closely that of Arnold (1968). Consider a number of observed values of the function p = f (A) for a certain range of the argument in which there is very l i t t l e curvature. If a straight line is f i t ted to the points in this range by the method of least squares, any two points (Pi, D]) and (P£, D2) on this straight l ine can be taken to represent the data, but in general, the errors in P| and P2 wil l not be independent thus making estimation of the magnitude of the reduction of error d i f f i c u l t . Moreover, if in this range the curvature is appreciable, the errors have been overestimated and valuable information may be lost . If curvature is accounted for by passing a quadratic through the data and choosing the points (Pi, Di) and (P2, D2) to be the intersections of the linear and quadratic forms, then curvature may be safely neglected and at the same time the uncertain-ties in P] and P2 are rendered independent. Arnold (loc c i t ) has shown that the ith equation of condition for a quadratic form is 0, - Dz v, - Uz If the weight of one observation is w = where 07 is the standard error in PC , the three normal equations for P i , P2 can be formed by multiplying equation (1) by the respective coef f ic ients . -124-APPENDIX IV  METHOD OF SUMMARY VALUES (con't) The conditions that the errors in P] , P 2 and a be independent, are that the coeff ic ients of the off-diagonal elements of the left-hand side of the equation be zero, i .e . £ u > ; ( A ; - O . X ^ ' - 0 * ) = o _ (2) £ ou; (Ai - O./fAi- Oz) - o . (3) £ UK (fAj - D . f O i ~ Dz) = o -(4) Under these conditions, a cannot contribute to Pj and P 2 and i t may assume any value, but there are three equations and only two unknowns. However, i f (k) is subtracted from (3), i t reduces to (2) so that either (3) or (k) is superfluous, and a unique solution ja r Dj and D2 is possible. To f ind D, and D ,^ the moments are formed: \ = - ; = £ u>; A; j Si - &l ->*-t Equation (2) then becomes, putting dj = D] - /ix and d^ = D ^ - ^ z /Ut + d,d 2 = o (5) and equation (3) reduces to M-3 - JtXx C ^ i + ctz) = O - - - - _ - (6) Substituting for either dj or d 2 from (5) into (6), we have Solution of equation (7) yields the values of D] and D2, and Arnold -125-APPENDIX IN/,  METHOD OF SUMMARY VALUES (con ' t ) has shown how by sub s t i t u t i on of equations (2) and (3) into the normal equat ions, the values of Pj and ?2 ace P, - C D l - O 3 L ^ £ u J c C A i - 0 2 ) p t V 10,-02.1 £ w- (At - 0 2 ) Z J(£»>L(& i - 0??) p _ COi-Oi) JTWt "D,)Pt ± | 0,-021 where the uncer t a in t i e s are independent, and reduce to the square roots of the inverse of the c o e f f i c i e n t s of Pj and ?2 . Pj and ?2 are c a l l e d the summary values and represent p (L\) in the range of argument used. The summary values method of smoothing can a l so be used to f i t a quadrat ic ( Je f f rey s , 1937) so that the er ror s at three points are independent of each other and of a poss ib le cubic term. This method appears to be somewhat of an improvement on the f i r s t method of summary va lues, but i t breaks down i f the data are unevenly d i s t r i b u t e d or are not capable of p rec i se l y de f i n i n g a curvature. -126-APPENDIX Va.  U.B.C. TRIP  TRIANGULAR REGRESSION PACKAGE One method of smoothing data is to f i t a polynomial to the data by the method of least squares. The computer programme, U.B.C. Tr ip , was designed with this (among other things) in mind. A polynomial of degree, say m, is requested, being thought by visual examination to give the best f i t to the data. The Trip program uses the data to calculate the coeff icients of m regression equations, the f i r s t equation being a simple linear equation of the form y=bo + b)x» where y is the dependent and x the independent variable. Successive equations are this linear equation augmented by a term in the next higher power of the independent variable^ up to. the power m. 2 Thus, the second regression equation would be y = b Q + bjx + b 2x and the f inal regression equation would be y = b 0 +• b,x + b 2 x 2 . - -t-bmxm After the coeff icients of each of the equations have been calculated by obtaining the best possible f i t of the equation to the data by the usual least squares technique ( i .e . by calculat ing the equation to minimise the sum of the squares of the distances of the predicted values of y from the observed values ) the program calculates several s ignif icant quantities that give an indication of the quality of the f i t . , One of these quantities is the parameter R , thesquare of the multiple correlat ion coef f ic ient , where R = w i C b/£ " y ) A ^ Oil " and y ; is the ith predicted value of ij, (ij. = b 0 + D,Xj) . y. is the i th observed value of y, wj is the weight of the ith observation, and y is the 2 mean of the observed values of y. R always takes a value of between zero and unity, and i ts significance l ies in the fact that the closer i t is to the value unity, the better does the corresponding regression equation f i t the data, -127-2 s ince the phys ica l meaning of R is that i t represents that proport ion of \he to ta l observed var iance of the dependent v a r i ab l e y which i s accounted for by the regress ion l i n e . In add i t i on , the program" ca l cu l a te s quan t i t i e s known as the F - r a t i o and F-probabi1 i ty for each cf the c o e f f i c i e n t s b j , a l ready in the equation where the F r a t i o is given by F t = ( b ; / S t d . E r r o r In b i ) 2 and the F -probab i l t y is the p r o b a b i l i t y of obta in ing a value of F? greater than or equal to the one c a l c u l a t e d , given that b;=o. If th i s p r o b a b i l i t y i s less than 0.05, i t is concluded that the c o e f f i c i e n t b^  is s i g n i f i c a n t l y d i f f e r e n t from 0, and therefore meaningful.,, The to lerance of each potent ia l independent va r i ab le not yet in the regress ion equation (provided the present equation has not yet a t ta ined the degree m) is a l so c a l c u l a t e d . This quant i ty is the proport ion of the var iance of the independent va r i ab le x which cannot be expla ined by the independent va r i ab le s a lready in the equat ion, and i f th i s to lerance f a l l s beneath 0.001, the next higher power of x is not allowed in the equat ion. In a d d i t i o n , the F-probabi1 i ty of each potent ia l independent va r i ab le x, not yet in the equation (again provided the equation has not yet a t ta ined the maximum degree m) is c a l c u l a t e d , and th i s is simply the F-probabi1 i ty that the c o e f f i c i e n t of the next higher power of x would have i f i t were admitted to the present regress ion equat ion. Unless requested otherwise, the c r i t i c a l value of : F-prob. is assumed to be 0.05, and i f the ca l cu l a ted value of F-prob. is less than 0.05, the next higher power of x would be added to the equation and the e n t i r e process repeated using the new equat ion. If F-prob. is greater than 0.05 however, the next higher power of x is not added to the equation and the -128-c a l c u l a t i o n s stop. In p r a c t i c e , i t was requested that the c r i t i c a l value of F-prob. be 1 .0 , so that a l l the powers of X up to the degree requested, were used to c a l c u l a t e a f i n a l r e g r e s s i o n equation, i r r e g a r d l e s s of whether the c o e f f i c i e n t s of the powers were s i g n i f i c a n t or not as i n d i c a t e d by t h e i r F-prob. v a l u e s . The r e s i d u a l s y j - <y- were c a l c u l a t e d , and a p r i n t e r p l o t of the data together w i t h points p r e d i c t e d by the f i n a l r e g r e s s i o n equation obtained. The values of R 2 and F-prob. f o r each of the c o e f f i c i e n t s i n the f i n a l r e g r e s s i o n equation were then examined, along w i t h the r e s i d u a l s and the p r i n t e r p l o t . If the F-prob. of the l a s t c o e f f i c i e n t i n the equation was found to be greater than 0 . 0 5 , a new poly-nomial f i t of degree m-l was requested. If i t was found to be l e s s than 0 . 0 5 , a polynomial f i t of degree m + 1 was requested. The f i n a l r e g r e s s i o n equation 2 chosen as best r e p r e s e n t i n g the data, was "that one i n which R was the nearest to the value u n i t , the F-prob. of more of the c o e f f i c i e n t s were greater than 0 . 0 5 , the r e s i d u a l s appeared to be evenly d i s t r i b u t e d about a mean val u e , and from v i s u a l examination of the p r i n t e r p l o t , the f i t of the r e g r e s s i o n equation to the data seemed s a t i s f a c t o r y . -129-APPENDIX V b . DT/DA SMOOTHED BY POLYNOMIAL RE3RESSI3N DISTANCE (DEG) 1 i D T / n a (SEC /DEG) j DF GH I J 113.0 1.959 114.0 1.950 115.0 1. 94 5 116.0 1.936 117.0 1.929 118.0 1.922 119.0 1. 918 120.0 1. 914 121.0 1.911 122.0 1.908 123.0 1.906 124.0 1.904 125.0 1.903 126.0 1.901 127.0 1. 900 128.0 1. 898 129.0 1. 896 130.0 1.893 131.0 1. 889 132.0 1. 885 2. 688 133.0 1. 880 2.678 134.0 1. 874 2. 668 3.456 135.0 1. 867 2.658 3.437 136.0 1.859 2. 648 3.417 137.0 1. 849 2.640 3.397 138.0 1. 838 2.628 3.355 140.0 1. 812 2.622 3.3 32 141.0 1.7 97 2. 617 3. 309 142.0 1.780 2.613 3.285 143.0 1.762 2. 607 3. 260 144.0 1. 74 2 2.602 3-235 145.0 1.720 2.595 3.208 3. 548 146.0 1. 696 2.588 3. 181 3. 659 147.0 1.671 2. 579 3. 154 3. 760 148.0 1.644 2.568 3.125 3. 849 149.0 1.615 2. 555 3.096 3. 929 150.0 1. 584 2.540 3.066 4. 000 151.0 1.550 2.523 3.035 4. 063 152.0 1.517 2.502 4. 118 153.0 1.481 2. 479 4. 166 154.0 T.443 2.451 4.208 155.0 1. 403 2. 421 4. 244 156.0 1.361 2.386 4.276 157.0 1. 3 18 2. 347 4. 304 158.0 1.273 2.300 4. 329 159.0 1.226 4. 351 160.0 1. 178 4.371 161.0 1-129 4. 390 162-0 1. 077 4.409 164.0 Q. 971 4.448 165.0 Q- 916 4. 470 166.0 0.859 4.494 167.0 0. 802 4. 501 168.0 0.743 4.501 169.0 0. 684 4.503 170.0 0. 624 4.506 175.0 tt. 3 16 4.509 

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