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Time domain determination of earthquake fault parameters from short-period P-waves Somerville, Paul Graham 1975

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TIME DOMAIN DETERMINATION OF EARTHQUAKE FAULT PARAMETERS FROM SHORT-PERIOD P-WAVES by PAUL GRAHAM SOMERVILLE B.Sc. U n i v e r s i t y of New England, 1964 Dip.Ed. U n i v e r s i t y of Sydney, 1965 fl.Sc. U n i v e r s i t y of B r i t i s h Columbia, 1971 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of GEOPHYSICS AND ASTRONOMY We accept t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA December, 1975 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia , I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e fo r reference and study. I f u r t h e r agree t h a t permiss ion for e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed wi thout my w r i t ten pe rm i ss i on . Department of The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date ABSTRACT Sourca parameters of two shallow earthquakes have been determined by the time-domain a n a l y s i s of s h o r t - p e r i o d t e l e s e i s m i c r e c o r d i n g s . For each event, the e f f e c t of the r e c e i v e r c r u s t was deconvolved from a s e t of g l o b a l l y d i s t r i b u t e d r e c o r d i n g s using the homomorphic method. The r e s u l t i n g seismograins were compared with the form of the e l a s t i c wave r a d i a t i o n computed from Savage's model of r a d i a l l y s p reading r u p t u r e on a plane e l l i p t i c a l f a u l t s u r f a c e . T h i s time-domain approach has p e r m i t t e d the d e t e r m i n a t i o n of s e v e r a l k i n e m a t i c parameters p e r t a i n i n g to the dynamics of rupture that are not o r d i n a r i l y e v a l u a t e d from s p e c t r a l a n a l y s i s . These parameters are rupture v e l o c i t y , the d i r e c t i o n o f f u r t h e s t r u p t u r e propagation, and the d u r a t i o n of a ramp d i s l o c a t i o n time f u n c t i o n which was p r e s c r i b e d to be the same everywhere on the f a u l t s u r f a c e . A g e n e r a l l i n e a r i n v e r s e scheme has been a p p l i e d t o i n v e s t i g a t e how w e l l and i n what manner the parameters of the source model are determined by the o b s e r v a t i o n s . T h i s a n a l y s i s y i e l d s best f i t t i n g models, the range of a c c e p t a b l e parameter v a l u e s , and the d i s t r i b u t i o n of i n f o r m a t i o n c o n c e r n i n g s p e c i f i c parameters t h a t i s contained i n s p e c i f i c o b s e r v a t i o n s . A c o n s i s t e n t d i s c r e p a n c y between the observed and model seismograms d u r i n g the f i r s t h a l f - c y c l e of motion i s a t t r i b u t e d to the i n c o r r e c t p r e s c r i p t i o n of the d i s l o c a t i o n time f u n c t i o n . I t i s suggested t h a t a space-dependent f u n c t i o n determined t h e o r e t i c a l l y by Kostrov i n 1964 would tend to remove t h i s d i s c r e p a n c y . i i TABLE OF CONTENTS Chapter 1. I n t r o d u c t i o n i ..1 Chapter 2. Source Theory 5 2.1. The Knopoff-de Hoop Rep r e s e n t a t i o n Theorem .........5 2.2. Representing A F a u l t By A Somigliana D i s l o c a t i o n ...7 2.3. Displacement S o l u t i o n s From Savage's F a u l t Model ...9 2.4. Seismic Moment 15 2.5. E s t i m a t i n g Source Parameters 18 2.6. P r e s c r i b i n g The D i s l o c a t i o n F u n c t i o n 22 2.7. The Form Of The R a d i a t i o n From Savage's Model 26 2.8. The Spectrum Of The R a d i a t i o n From Savage's Model ..29 2.9. Choosing Model Parameters For I n v e r s i o n 31 2.10. A S t a t i o n ' s P o s i t i o n In F a u l t S u r f a c e C o o r d i n a t e s .33 Chapter 3. Seismogram A n a l y s i s 36 3.1. Choosing Earthguake Events ..........36 3.2. D i g i t i s i n g The Seismograms ..40 3.3. Array Beamforming . t....... 42 -3.4. Homomorphic Deconvolution 44 3.5. A t t e n u a t i o n .......54 3.6. Ray Geometry C a l c u l a t i o n s 57 3.7. Nodal Plane S o l u t i o n s 59 3.8. Computing S p e c t r a ..61 Chapter 4. I n v e r s i o n Theory 62 4.1. Formulating The Problem 62 4.2. G e n e r a l i s e d E i g e n v e c t o r A n a l y s i s .....64 4.3. O v e r c o n s t r a i n e d And Underdetermined Systems .66 4.4. The General I n v e r s e ..68 4.5. The P a r t i c u l a r S o l u t i o n 70 i i i 4.6. Weighting The Parameters 71 4.7. Choosing Between Variance And R e s o l u t i o n 75 4.8. Extremal I n v e r s i o n 76 4.9. Handling A Non-linear I n v e r s i o n Problem 80 Chapter 5. I n v e r s i o n P r a c t i c e ..82 5.1. Representing The Seismogram By Measurements 82 5.2. A s s i g n i n g Standard D e v i a t i o n s To The O b s e r v a t i o n s ..83 5.3. F i n d i n g A S t a r t i n g Model 85 5.4. Computing The P a r t i a l D e r i v a t i v e s 87 5.5. Least Squares I n v e r s i o n 87 5.6. Extremal I n v e r s i o n 93 5.7. V E i g e n v e c t o r A n a l y s i s Of The Model Parameters 96 5.8. U E i g e n v e c t o r A n a l y s i s Of Information D i s t r i b u t i o n .99 5.9. C o n c l u s i o n s 102 Chapter 6. Source Parameters 104 Chapter 7. C o n c l u s i o n s 115 References .....117 i v LIST OF TABLES 2.1. Computing F a u l t Surface C o o r d i n a t e s Of A Ray 34 3.1. Hypocentral And Nodal Plane Data For The Events ...... 38 3.2. Geometrical Values At The S t a t i o n s .57 5.1. Seismogram Measurements ......82 5.2. Observed Values And T h e i r Standard D e v i a t i o n s 83 5.3. F a u l t Model Parameters 85 5.4. D e s c r i p t i o n Of I n v e r s i o n I t e r a t i o n s ..................92 5.5. Extremal Parameter Values .93 5.6. J o i n t E x t r e m i s a t i o n Of Two Rat I s . parameters ........ 94 5.7. V E i g e n v e c t o r s 96 5.8. Measures Of The D e v i a t i o n Of The Parameters 96 5.9. Diagonal Of The U.tf Matrix ..98 5.10. U.U* Diagonal Summed Across S t a t i o n s 100 5.11. D E i g e n v e c t o r s Summed Across S t a t i o n s 101 6.1. Moment C a l c u l a t i o n s .....110 6.2. Source Parameters 111 V LIST OF FIGURES 2.1. C r o s s S e c t i o n Of The Source Region 7 2.2. Rupture On Savage's F a u l t Model 9 2.3. F a u l t Surface Coordinate System .........9 2.4. P o i n t D i s l o c a t i o n And Double Couple E q u i v a l e n t 15 2.5. Contours Of Egual D i s l o c a t i o n On The F a u l t .25 2.6. S u c c e s s i v e Apparent Rupture F r o n t s 25 2.7. Displacement P u l s e s From A F a u l t Model 28 2.8. C o n v o l u t i o n a l Model Of A Displacement Pulse ..29 2.9. Spectrum Of A Displacement Pulse ..30 2.10. O r i e n t a t i o n Angles Of The F a u l t Surface 31 2.11. C o o r d i n a t e s Of A Ray Departing From A Focus 33 2.12. C o o r d i n a t e s Of A Ray A r r i v i n g At A S t a t i o n 33 2.13. S p h e r i c a l Trigonometry At The E p i c e n t r e ..35 3.1. Displacement S e n s i t i v i t y Of The Seismographs 36 3.2. The G a u r i b i d a n u r And H e r r i n C r u s t a l Models .39 3.3. Geometry For Array Beamforming 43 3.4. Deconvolution Of The Rat I s . seismogram At AKU ..".....49 3.5. D e c o n v o l u t i o n Of The Broach Seismograms At SNG 51 3.6. Deconvolution Of The Rat I s . seismograms 53 3.7. Deconvolution Of The Broach Seismograms ...53 3.8. Matching An A t t e n u a t o r To The Rat I s . AKU Wavelet ....56 3.9. Nodal Plane S o l u t i o n Of The Rat I s . earthquake .......59 3.10. Nodal Plane S o l u t i o n Of The Broach Earthquake ..59 5.1. Seismogram Measurements 82 5.2. Model Seismograms For S u c c e s s i v e I t e r a t i o n s .......... 92 6.1. Comparison Between Observed And Model Spectra 105 6.2. Comparison Between Two C o n t r a s t i n g S p e c t r a ........... 105 v i ACKNOWLEDGMENTS I t i s a plea s u r e t o acknowledge the c o n t r i b u t i o n s t h a t many i n d i v i d u a l s have made towards t h i s r e s e a r c h . I p a r t i c u l a r l y wish to thank Dr Robert E l l i s f o r h i s encouragement and support d u r i n g the e a r l y stages of the p r o j e c t , and Dr Ralph Wiggins, who so f r e e l y o f f e r e d h i s co n c e p t u a l and computational s k i l l s d u r i n g the e v e n t u a l r e s o l u t i o n of the r e s e a r c h problem. I a l s o wish t o pay t r i b u t e to the i n s p i r i n g example s e t by my s e n i o r c o l l e a g u e O l i v e r Jensen i n h i s approach t o s c i e n c e . Throughout t h i s r e s e a r c h , the e n q u i r i e s and requests which I d i r e c t e d toward the i n t e r n a t i o n a l s e i s m o l o g i c a l community were answered with u n f a i l i n g l y generous a s s i s t a n c e . I owe a s p e c i a l debt of g r a t i t u d e t o Dr H. I. S. Thirlaway and h i s s t a f f at U.K.A.E.A., Bl a c k n e s t f o r the generous help and h o s p i t a l i t y which I r e c e i v e d while c o l l e c t i n g seismograms t h e r e . F i n a l l y , I wish to thank the f a c u l t y members and my f e l l o w graduate students f o r the i n s p i r a t i o n and companionship which they provided. For the d u r a t i o n of t h i s r e s e a r c h , I was supported by a D.B.C. Graduate F e l l o w s h i p . 1 CHAPTER 1. INTRODUCTION The o b j e c t i v e o f t h i s t h e s i s i s t o i n v e s t i g a t e whether a shear d i s l o c a t i o n model of earthquake rupture can d e s c r i b e the t e l e s e i s m i c a l l y r e c o r d e d s h o r t - p e r i o d r a d i a t i o n of earthquakes having s m a l l source dimensions. The comparison between observed motions and those c a l c u l a t e d from the model i s made i n the time domain. I t i s of p a r t i c u l a r i n t e r e s t t o determine whether apparent d e p a r t u r e s from s p h e r i c a l l y symmetric r a d i a t i o n can be accounted f o r by a model of propagating r u p t u r e on a f a u l t s u r f a c e . I f so, then a d e s c r i p t i o n of the shape of the f a u l t s u r f a c e , and of the e v o l u t i o n of r u p t u r e on. i t , might be p o s s i b l e . T h i s kind o f i n f o r m a t i o n i s not o b t a i n a b l e from the a n a l y s i s of the amplitude spectrum by means of the commonly used model of Brune (1970) , s i n c e that model assumes an i n f i n i t e r u p t u r e v e l o c i t y . Indeed, f o r any model, s p e c t r a l a n a l y s i s i s d e f i c i e n t u n l e s s the phase spectrum i s a l s o c o n s i d e r e d . Time domain a n a l y s i s , on the other hand, d i r e c t l y uses a l l of the a v a i l a b l e i n f o r m a t i o n contained i n the seismogram. The d i s l o c a t i o n theory o f earthquakes mathematically d e s c r i b e s the source as a ' d i s l o c a t i o n ' or d i s c o n t i n u i t y i n displacement a c r o s s a f a u l t s u r f a c e . There i s abundant evidence t h a t t h i s c o n s t i t u t e s a good d e s c r i p t i o n of shallow earthquakes. I t i s u s u a l l y assumed t h a t the s l i p i s p a r a l l e l to a plane f a u l t s u r f a c e (a p h y s i c a l l y reasonable assumption f o r f a u l t i n g i n s i d e the e a r t h ) , i n which case i t i s a pure shear d i s l o c a t i o n . The shear d i s l o c a t i o n t h e o r y i s confirmed by t h e c o m p a t a b i l i t y of an overwhelming p r o p o r t i o n of o b s e r v a t i o n s with a double-couple 2 source mechanism. Within the framework of shear d i s l o c a t i o n theory, t h e r e are two d i s t i n c t approaches to the study of earthquake r u p t u r e . The f i r . s t i s concerned with the i n v e r s e problem of d e s c r i b i n g the d i s l o c a t i o n f u n c t i o n on the f a u l t plane from a n a l y s i s of the seismograms. Since the s l i p motion i s a f u n c t i o n of time and two space c o o r d i n a t e s , a complete i n v e r s i o n i s i m p r a c t i c a l . A t r a c t a b l e compromise i s to model the rupture k i n e m a t i c a l l y by a s m a l l number of parameters, which may then be determined from the seismograms. Kinematic models have been proposed f o r s t r i k e - s l i p s u r f a c e f a u l t s (Knopoff and G i l b e r t , 1959), u n i l a t e r a l l y propagating b u r i e d f a u l t s ( H a s k e l l , 1964), and rupture growing c i r c u l a r l y from a p o i n t and t e r m i n a t i n g on an e l l i p t i c a l ^ b o u n d a r y (Savage, 1966) among o t h e r s . The weakness of these models l i e s i n t h e i r a r b i t r a r y p r e s c r i p t i o n of the d i s l o c a t i o n f u n c t i o n . The second approach to the study of earthquakes as shear d i s l o c a t i o n s a v o i d s t h i s d e f i c i e n c y by s o l v i n g the p h y s i c a l problem of rupture i n a s o l i d . C o n t r i b u t i o n s toward the s o l u t i o n have been made by Kostrov (1964), Archambeau (1968), Weertman (1969), B u r r i d g e and H a l l i d a y (1971), Hanson e t a l (1971), and Ida and A k i (1972) among o t h e r s . The s o l u t i o n e n t a i l s the .treatment of n o n - l i n e a r and a n e l a s t i c p r o c e s s e s s a s s o c i a t e d with f a i l u r e . Although a complete s o l u t i o n has not been achieved (and remains as a major c h a l l e n g e of t h e o r e t i c a l s e i s m o l o g y ) , p a r t i a l s o l u t i o n s can provide r e a l i s t i c d e s c r i p t i o n s of the d i s l o c a t i o n f u n c t i o n f o r kinematic models. There have been many s u c c e s s f u l d e s c r i p t i o n s of the s u r f a c e 3 wave and l o n g - p e r i o d body wave r a d i a t i o n from l a r g e earthquakes i n terms of propaqating shear d i s l o c a t i o n models. A p i o n e e r i n g and comprehensive example i s the study of G waves from the N i i g a t a earthquake by Aki (1966). Berckhemer and Jacob (1968) and B o l l i n g e r (1968) have performed time-domain a n a l y s e s of the l o n g - p e r i o d P wave r a d i a t i o n from earthquakes having dimensions of some t e n s o f k i l o m e t r e s . The present s t u d y - i s a s h o r t - p e r i o d analoque of these two i n v e s t i g a t i o n s , usinq earthquakes havinq dimensions of a few k i l o m e t r e s . There are s e v e r a l p a r t i c u l a r l y v a l u a b l e a s p e c t s of the approach o u t l i n e d i n t h i s t h e s i s . F i r s t l y , i t i s a p p l i c a b l e to many earthquakes s i n c e i t i s best s u i t e d to r u p t u r e s having s h o r t d u r a t i o n . Secondly, by using t e l e s e i s m i c r e c o r d i n g s , i t provi d e s access t o the study of earthquakes i n remote or i n a c c e s s i b l e p l a c e s . T h i s enables earthquakes of d i f f e r e n t t e c t o n i c o r i g i n t o be compared. T h i r d l y , i t employs r e a d i l y a v a i l a b l e r e c o r d i n g s at permanent s t a t i o n s . The rupture processes of two earthquakes of moderate maqnitude are s t u d i e d i n t h i s t h e s i s . F i r s t l y we show how shear d i s l o c a t i o n theory a l l o w s us to c a l c u l a t e the dynamic displacement f i e l d and s e v e r a l important parameters of the s t a t i c displacement f i e l d from a p r e s c r i b e d d i s l o c a t i o n f u n c t i o n . We next d e s c r i b e the v a r i o u s analyses of the seismoqrams t h a t are r e q u i r e d b e f o r e they can be a p p l i e d to source s t u d i e s . The most c r u c i a l of these i s the d e c o n v o l u t i o n of the e f f e c t of the t r a n s m i s s i o n path from the seismograms which would i d e a l l y y i e l d the t r u e f a r - f i e l d displacement. 4 Having reduced the seismograms to the kinds of ' o b s e r v a t i o n s ' r e l e v a n t to the source model, we next i n v e s t i g a t e the g e n e r a l l i n e a r i n v e r s e as a means of s t u d y i n g how well and i n what manner the parameters of the source model are determined by the o b s e r v a t i o n s . Besides p r o v i d i n g us with best f i t t i n g models, the i n v e r s i o n scheme permits us to f i n d the range of a c c e p t a b l e source models, and to see how p a r t i c u l a r o b s e r v a t i o n s c o n t r i b u t e to p a r t i c u l a r model parameters. D i s c r e p a n c i e s between the o b s e r v a t i o n s and the model c a l c u l a t i o n s are used t o i d e n t i f y d e f i c i e n c i e s i n the model. One p o s s i b l e source of d e f i c i e n c y i s suggested, together with the means whereby i t can be c o r r e c t e d . F i n a l l y , the source parameters of the two earthquakes are d i s c u s s e d i n the c o n t e x t of t h e i r t e c t o n i c environments. 5 CHAPTER 2. SOURCE THEORY 2 . 1 . The Knopoff-de Hoop r e p r e s e n t a t i o n theorem The wave motions of s m a l l amplitude i n a homogeneous e l a s t i c s o l i d s a t i s f y Newton's equation of motion f o r an element of volume where U- = displacement v e c t o r Tjj = s t r e s s tensor = d e n s i t y of body f o r c e s per u n i t volume = r e c t a n g u l a r C a r t e s i a n c o o r d i n a t e s "t = time <5" = d e n s i t y The s t r e s s t e n s o r may be r e p r e s e n t e d i n terms of s t r a i n s through the e l a s t i c c o n s t a n t s ^ij»pv IX T . - C.. d u , 9x. i where = A £ • d" + so. ( £• S- -t- S. J. ) ^,^0. = Lame cons t a n t s of the medium cV. = u n i t tensor E l i m i n a t i n g T- • from Equations 2.1 and 2.2 l e a d s to the elastodynamic wave equ a t i o n 6 •> V The s o l u t i o n i s given by the Knopoff-de Hoop r e p r e s e n t a t i o n theorem (de Hoop, 1958) which pro v i d e s the displacement a t any po i n t U - ( x , t/) due to a f o r c e §(%,T} i n a volume V bounded by a s u r f a c e S U ; ( X , t ) - | Gi} [ f ] ] du V. ( Loclij forces j '5 (displacement di.scon+l'Mti.tCzs } where i s the u n i t outward normal on S • T n e Green's op e r a t o r transforms a f u n c t i o n \js {% y t) i n t o J 1 ^ '] . oCr . • - ( f a r f i e l d P) 2-5" 4 - - a ^ ( * , t - f ) + 3X1% - hi > ( 5 , t - ? ) 7 < / 7 -where ^ = | 3C (Y- = *; - "*b» are the d i r e c t i o n c o s i n e s of r. 7 2.2. Representing a f a u l t by a Somigliana d i s l o c a t i o n He now seek a p h y s i c a l i n t e r p r e t a t i o n of t h i s mathematical model. F i r s t l y , l e t us c o n s i d e r the body f o r c e term i n Equation 2.4. The only body f o r c e a c t i n g i n a t e c t o n i c s e t t i n g i s g r a v i t y , which does not c o n t r i b u t e d i r e c t l y t o the t r a n s i e n t s o l u t i o n , and may be neglected f o r our purposes. We are now l e f t with two s u r f a c e i n t e g r a l s . Let us now I t i s convenient to d i v i d e the boundary i n t o an outer s u r f a c e which may be allowed to recede to i n f i n i t y (where i t c o n t r i b u t e s nothing at any f i n i t e time) , and an i n n e r surface" w i t h i n which the n o n - l i n e a r deformation accompanying f a u l t i n g takes place. I f t h i s r e g i o n i s c o n s i d e r e d to be a t h i n , p l a n a r s l a b (Figure 2.1) whose width i s l e s s than the s h o r t e s t wavelength of i n t e r e s t , then we need only c o n s i d e r the i n t e g r a l s over the s u i t a b l y d e f i n e the boundary of the e l a s t i c volume l a r g e f l a t s u r f a c e s Tn7 S " K- D -I F i g u r e 2 . 1 . Cross s e c t i o n of the source r e g i o n 8 Using the s t r e s s - s t r a i n r e l a t i o n s of Equation 2.2 we may w r i t e the second i n t e g r a l as MTjn,ds - [^.riJds where the t r a c t i o n F ~ J l ^ i s the f o r c e per u n i t area t h a t the m a t e r i a l i n the n o n - l i n e a r r e g i o n i n s i d e 5 e x e r t s on the e l a s t i c m a t e r i a l e x t e r n a l to 5 • We may d i v i d e t h i s i n t e g r a l i n t o two components, one i n v o l v i n g the f o r c e R a c t i n g a c r o s s the S s u r f a c e and the other the f o r c e Fj a c r o s s the 3 s u r f a c e . I t has been p o i n t e d out by H a s k e l l (1964) t h a t any p h y s i c a l l y reasonable process of f a i l u r e w i t h i n the narrow zone between the two s u r f a c e s i s unable to produce f o r c e s Fj and Fj t h a t are not equal i n magnitude and o p p o s i t e i n d i r e c t i o n , at l e a s t over i n t e r v a l s of time no s m a l l e r than the s e i s m i c t r a v e l times between the s u r f a c e s . We thus assume t h a t the d i s l o c a t i o n i s a Somigliana d i s l o c a t i o n (across which f o r c e s and thus s t r e s s e s are continuous) and so the two i n t e g r a l s c a n c e l . We are then l e f t with the f i n a l term i n Eguation 2.4 which may be w r i t t e n where we have r e p l a c e d U.- by the d i s l o c a t i o n D- = w.*"- u. which i s the r e l a t i v e displacement of the S s u r f a c e with r e s p e c t to the S s u r f a c e , and n£ i s the u n i t normal d i r e c t e d outward from the S s u r f a c e . I t does not matter whether we regard the displacement D: as being produced by an a c t u a l d i s c o n t i n u i t y a t a f r a c t u r e s u r f a c e w i t h i n 5 or by 9 a continuous p l a s t i c deformation w i t h i n -3 2.3. Displacement s o l u t i o n s from Savage's f a u l t model We now proceed to s e l e c t a model of r u p t u r e , t h a t i s , we p r e s c r i b e the d i s l o c a t i o n v e c t o r D ( * , t ) which i s a f u n c t i o n both of p o s i t i o n and time. Savage (1966) proposed an elegant and v e r s a t i l e model i n which rupture begins at a focus and spreads r a d i a l l y at a uniform speed "V" u n t i l i t covers a plane e l l i p t i c a l s u r f a c e . The e p i c e n t r e ' s l o c a t i o n at the focus of the e l l i p s e , b e s i d e s p r o v i d i n g mathematical convenience, a l l o w s the modelling of a wide v a r i e t y of rupture c o n f i g u r a t i o n s , r a n g i n g from an e g u i d i m e n s i o n a l r a d i a l l y propagating one ( e l l i p t i c i t y near zero) to a u n i l a t e r a l l y propagating one ( e l l i p t i c i t y near u n i t y ) . F i g u r e 2.2. Rupture Model F i g u r e 2.3. F a u l t C o o r d i n a t e s The rupture model i s i l l u s t r a t e d i n F i g u r e 2.2. At a p a r t i c u l a r i n s t a n t , the r u p t u r e d area l i e s w i t h i n both the 1 0 c i r c l e p = v t and the e l l i p t i c a l boundary p - /b(^) • a e wish t o compute the motion t h a t i s r a d i a t e d t o the f a r - f i e l d p o i n t P (Figure 2.3) whose s p h e r i c a l p o l a r c o o r d i n a t e s are (r0 j (9O )$>) • The d i s t a n c e P from an a r b i t r a r y p o i n t on the f a u l t s u r f a c e to P i s expressed through the p o l a r c o o r d i n a t e s [py(f)) i n the f a u l t plane 2.7 = iY + ~ J\nf>rsLn0o c o s where pr - V- ( t - X ) whichever i s ph[(j>) j l e a s t = compressional wave v e l o c i t y and the s u b s c r i p t r denotes the i n s t a n t a n e o u s rupture boundary, and b the f i n a l boundary. The e x p r e s s i o n p " 'v~(t~Z<;) d e s c r i b e s an e l l i p s e whose major a x i s l i e s along the p r o j e c t i o n of P0 on the f a u l t plane, and r e p r e s e n t s the l o c u s of p o i n t s on the f a u l t s u r f a c e beginning to c o n t r i b u t e to the s i g n a l seen at P at a given i n s t a n t . Let us assume f o r the present t h a t the d i s l o c a t i o n i s r e a l i s e d i n s t a n t a n e o u s l y as the rupture f r o n t passes each p o i n t . The s u p e r p o s i t i o n theorem w i l l permit us to subsequently f i n d the s o l u t i o n f o r an a r b i t r a r y time f u n c t i o n of d i s l o c a t i o n , which must however be the same at a l l p o i n t s on the f a u l t . For the case of ,a ste p time f u n c t i o n , the d i s l o c a t i o n v e c t o r D , t) may be w r i t t e n i n p o l a r c o o r d i n a t e s as 2 . 8 D{P,4,t) - H (t-£)[.- H (/>-/*)] where the vector \J ( P, (f>) i s a f u n c t i o n of p o s i t i o n on the f a u l t s u r f a c e . 11 He now e v a l u a t e t h e G r e e n ' s o p e r a t o r i - H ( / > - / ! ) } 2.9 ) w h e r e g . P w U iC/^ H ( S u b s t i t u t i n g t h i s o p e r a t o r i n t o t h e e x p r e s s i o n f o r t h e d i s p l a c e m e n t , we o b t a i n s e p a r a t e e x p r e s s i o n s f o r t h e P a n d 3 m o t i o n s . W r i t i n g ds = pdj>d(J) , we o b t a i n f o r P waves (?) 1 '-v F r o m t h e d e f i n i t i o n o f p , we may i n c o r p o r a t e t h e s t e p f u n c t i o n ' s d e p e n d e n c e on r u p t u r e v e l o c i t y a n d on t h e f a u l t b o u n d a r y i n t o t h e l i m i t o f t h e i n t e g r a l o v e r p -\ '-ir D i f f e r e n t i a t i n g w i t h r e s p e c t t o Dc^ . we o b t a i n 12 bird U; (P) 2.13 W ] ^1 <a.„^ ^ K / " ^ | H ^ J I n t h e f a r - f i e l d a p p r o x i m a t i o n we n e g l e c t a l l t e r m s w h i c h - i f a l l o f f more r a p i d l y w i t h d i s t a n c e t h a n u . T h i s a p p r o x i m a t i o n i s a p p r o p r i a t e i f t h e o b s e r v e r i s a t a d i s t a n c e f r o m t h e f a u l t t h a t i s much g r e a t e r t h a n t h e f a u l t d i m e n s i o n s . W i t h t h i s a p p r o x i m a t i o n we o b t a i n w h e r e e)/>P AW w h i c h e v e r i s l e a s t a c c o r d i n g t o w h i c h o f t h e v a l u e s o f pr(tf>) a p p l i e s f - i - J £ S t n 0 O c o s [4'<}>*) A f u r t h e r s i m p l i f i c a t i o n r e s u l t s b e c a u s e we may r e g a r d r a n d i t s d i r e c t i o n c o s i n e s ^ L a s b e i n g v i r t u a l l y c o n s t a n t o v e r t h e f a u l t s u r f a c e , t h a t i s , t h e ^ may be e v a l u a t e d a t p - O a n d t h u s b e c o m e t h e d i r e c t i o n c o s i n e s o f P 0 . W e a r e t h u s e n a b l e d t o r e m o v e t h e d i r e c t i o n c o s i n e s f r o n t h e i n t e g r a l . F u r t h e r , t h e s e c o n d i n t e g r a l may be n e g l e c t e d 13 » because ^<^if} ^c^X^ f a l l s o f f as F X , and so the f a r -f i e l d approximation f o r P waves becomes The i n t e g r a l i s over a l l values of f o r which the rupture f r o n t i s a c t i v e , t h a t i s , f o r ^3, < . We now assume t h a t the d i r e c t i o n of d i s l o c a t i o n has the same d i r e c t i o n everywhere on the f a u l t s u r f a c e . We may then express the d i s l o c a t i o n vector as U(f>rj) = t> . UUJ) and the displacement may then be w r i t t e n as The P motion i n the r a d i a l d i r e c t i o n (with u n i t v e c t o r ) i s given by 2..7 Pit) = r j ^ i ± c . w u2(i-z){UiMLtt 2.1* . - r _ ^ L _ GM Rfat) T M For tel'eseisms we may use an approximate e x p r e s s i o n f o r the v e r t i c a l f r e e s u r f a c e displacement which i s exact only f o r normal i n c i d e n c e upon a h a l f - s p a c e boundary 2.19 S(t) Z P ( t ) The expression f o r the displacement i s the product of three 14 independent terras. The f i r s t , ~ra , r e p r e s e n t s the g e o m e t r i c a l s p r e a d i n g i n an i n f i n i t e medium and w i l l be r e p l a c e d by the term a f u n c t i o n of f o c a l depth and e p i c e n t r a l d i s t a n c e , f o r the r e a l e a r t h . The term yi ¥f> ^ C.^ • b- f)^ r e p r e s e n t s the r a d i a t i o n p a t t e r n , which depends on the o r i e n t a t i o n of the f a u l t s u r f a c e and the d i r e c t i o n of s l i p on i t . The t h i r d term 2.20 T(t./.) • - y ( t - l ) ^  UiPr.t) c o n t a i n s the time dependence, and expresses the shape of the s i g n a l as a f u n c t i o n of the o r i e n t a t i o n of the observer i n space. The s i g n a l shape does not depend on the d i r e c t i o n of s l i p , nor even whether the rupture i s s l i p or t e n s i l e . Me have yet to p r e s c r i b e the displacement ^J(p,<fi) as a f u n c t i o n of p o s i t i o n on the f a u l t plane, and the time f u n c t i o n d e s c r i b i n g how i t i s r e a l i s e d at any p o i n t . We s h a l l r e t u r n to these s u b j e c t s i n S e c t i o n 2.6. The i n t e g r a l of the pulse T(t,«(.j i s the same f o r a l l o b s e r v e r s , that i s j T"(t^) dt = const. The d i f f e r e n c e i n s i g n a l shape at d i f f e r e n t p o s i t i o n s merely r e p r e s e n t s d i f f e r e n t p e r c e p t i o n s of the e v o l u t i o n of rupture over the f a u l t s u r f a c e . In the i n t e g r a l [ i - ir / o C sin Qa cos ( ^ - f ^ J J 2 -the denominator r e p r e s e n t s the apparent source d e n s i t y on the f a u l t s u r f a c e as seen by a p a r t i c u l a r observer. T h i s i s 15 d e p i c t e d i n F i g u r e 2 . 6 . when t h e e x p r e s s i o n f o r T i s i n t e g r a t e d o v e r t i m e we w i l l h a v e o b t a i n e d , f o r a n y o b s e r v e r , a q u a n t i t y e q u i v a l e n t t o t h e p r o d u c t o f t h e mean d i s l o c a t i o n U a n d t h e s o u r c e a r e a A 2X\ T ( t < ) At - 0 A We now d e m o n s t r a t e t h a t t h i s q u a n t i t y i s p r o p o r t i o n a l t o t h e s e i s m i c moment M c . 2 . 4 . S e i s m i c moment We now i n v e s t i g a t e t h e s e i s m i c moment a s a m e a s u r e o f t h e s i z e o f t h e r u p t u r e p r o c e s s . I t h a s b e e n shown b y Maruyama (1963) a n d B u r r i d g e & K n o p o f f (1964) t h a t a d i s c o n t i n u i t y i n p u r e s l i p d i s p l a c e m e n t on a f a u l t s u r f a c e c a n be r e p r e s e n t e d by a d o u b l e c o u p l e i n t h e a b s e n c e o f a d i s p l a c e m e n t d i s c o n t i n u i t y . T h e c o u p l e s , h a v i n g e q u a l m a g n i t u d e a n d o p p o s i t e s e n s e , p r o d u c e no n e t t o r q u e . We now c a l c u l a t e t h e moment o f t h e two c o m p o n e n t c o u p l e s . A s s u m e t h a t i n a r e c t a n g u l a r c o o r d i n a t e s y s t e m ( s c , x , X 5 ) , a l o c a l t a n g e n t i a l s t e p w i s e d i s p l a c e m e n t o f t h e m a t e r i a l i n t h e h a l f - s p a c e > O o c c u r s r e l a t i v e t o t h a t i n X3 < O » a s s h o w n i n F i g u r e 2 . 4 ( a ) . 16 ( x , t ) a o F i g u r e 2.4a. A point d i s l o c a t i o n a t 0 Then the set of body f o r c e s e q u i v a l e n t to t h i s d i s l o c a t i o n i s shown i n F i g u r e 2.4(b) fat) - ^-WwfeHfo F i g u r e 2.4b. E q u i v a l e n t double couple The e x p r e s s i o n f o r -fy r e p r e s e n t s a point f o r c e d i s t r i b u t e d on the plane 3C3 - O and an o p p o s i t e p o i n t f o r c e d i s t r i b u t e d on the plana 3c 3 = -O . The f o r c e d i r e c t i o n i s Jc ( and the arm d i r e c t i o n i s 3C-3 . The t o t a l f o r c e vanishes 17 The t o t a l moment of t h i s couple about the X z a x i s i s 22Z dx, dx, ^ £ (x , t j * 3 d x 3 ) cbc3 s i n c e dx, ) I f we d e f i n e the average d i s l o c a t i o n L ^ , as where /\ i s the f a u l t area, we then have 2.23 M . • ^ 0 , A The opposing c o u p l e £ } with f o r c e i n the OCi d i r e c t i o n and arm i n the Xx d i r e c t i o n c o n t r i b u t e s an egual and opp o s i t e moment. He now r e c a l l our previous r e s u l t to o b t a i n 18 E x p r e s s i n g T (t »c) i n terms of the v e r t i c a l s u r f a c e displacement u s i n g Eguations 2.18 and 2.19 we o b t a i n an e x p r e s s i o n f o r the moment t h a t can be d e r i v e d from a seisroogram 2.5. E s t i m a t i n g source parameters He have j u s t o b t a i n e d an e x p r e s s i o n f o r the moment i n terms of the time i n t e g r a l of the seismogram. T h i s i n t e g r a l i s e q u i v a l e n t to the magnitude of the zero freguency component of the spectrum of the seismogram. P r a c t i c a l l y , t h i s may be estimated from the low-freguency asymptote of the spectrum, which i s independent of freguency f o r a d i s l o c a t i o n model, as shown i n F i g u r e 2.9. At f r e q u e n c i e s g r e a t e r than some ' c o r n e r 1 freguency, the spectrum begins t o decrease. S e v e r a l r a t e s of decay may be apparent, to each of which a cor r e s p o n d i n g c o r n e r freguency can be a t t a c h e d . The lower corner i s o f t e n used as an estimate of the f a u l t dimension, s i n c e i n a broad sense i t r e f l e c t s the d u r a t i o n of the s e i s m i c p u l s e . The d e t a i l s of the i n t e r p r e t a t i o n of the lower c o r n e r vary among d i f f e r e n t models, as was po i n t e d out by Savage (19,72). The r e l a t i o n s h i p between P and S c o r n e r f r e q u e n c i e s has a l s o been d i s c u s s e d by Savage (1974). 2.2.6 o 19 In t h i s work, f a u l t dimension has been estimated by time domain a n a l y s i s , with a rough check being provided by the s p e c t r a . I f the moment and f a u l t area are known, we are then able to f i n d s e v e r a l a d d i t i o n a l source parameters which are of c o n s i d e r a b l e s i g n i f i c a n c e . From Equation 2.23 we can o b t a i n an estimate of the average s l i p U , having assumed some value f o r the r i g i d i t y modulus As was shown by Fukao (1972), Eshelby's a n a l y s i s (1957) of the e l a s t i c f i e l d of an e l l i p s o i d a l c a v i t y enables us to o b t a i n an e x p r e s s i o n f o r the s t r e s s drop a s s o c i a t e d with f a u l t i n g . For a disc-shaped f a u l t , the s t r e s s drop i s (Eshelby (1957), Equation 5.7) r disc ra.(ii\A.S c M. c The corresponding s t r a i n drop Y is Y -S i.u He can write the s t r e s s drop as the d i f f e r e n c e between the i n i t i a l s t r e s s and the f i n a l s t r e s s 20 2.2.9 S = S. - S, The average a c t i n g shear s t r e s s i s then S, t 2..30 s - ^ The e l a s t i c energy El a s s o c i a t e d with f a u l t i n g i s the work done by the average a c t i n g shear s t r e s s d u r i n g r u p t u r e 2.31 S O A S u b s t i t u t i n g f o r Li i n t h i s e x p r e s s i o n we o b t a i n the energy r e l e a s e d d u r i n g the formation of a disc-shaped f a u l t E = I Q A A (So - S, / 2.3* L~ ZC/x I f the s t r e s s f a l l s t o zero d u r i n g f a u l t i n g , t h i s y i e l d s Eshelby's e x p r e s s i o n (Eshelby (1957), Equation 5.6) i n which the enerqy r e l e a s e i s p r o p o r t i o n a l t o the sguare of the i n i t i a l s t r e s s . The s e i s m i c enerqy r e l e a s e d E.$ i s a f r a c t i o n A? of the t o t a l energy El , where y i s termed, the s e i s m i c e f f i c i e n c y S u b s t i t u t i n g e x p r e s s i o n s f o r E and (JA from Equations 2.34 and 2.23 i n t o Equation 2.31 we have 2 1 2.35 »jS (S. - 5.) 2.36 s yu M. The e x p r e s s i o n S i s termed the apparent s t r e s s and denotes the s t r e s s a v a i l a b l e f o r s e i s m i c r a d i a t i o n . We may regard the remainder of the s t r e s s as being opposed d u r i n g f a u l t i n g by a dynamic f r i c t i o n a l s t r e s s SF such t h a t 2.37 *7 S = S ~ S F Savage and Wood ( 1 9 7 1 ) have p o i n t e d out t h a t t h i s model of f a u l t i n g r e s u l t s i n the i n e q u a l i t y 2.38 7S < i f i t i s assumed t h a t the dynamic f r i c t i o n 5F i s g r e a t e r than the f i n a l s t r e s s S f » T h i s would be the case i f rupture t e r m i n a t e s as a r e s u l t of the a c t i n g shear s t r e s s dropping below the dynamic f r i c t i o n a l shear s t r e s s . The i n e q u a l i t y of Equation 2 . 3 8 expresses the i d e a t h a t g e n e r a l l y , the i n e r t i a of the moving f a u l t b l o c k s w i l l cause an overshoot i n s l i p beyond the p o i n t at which the dynamic f r i c t i o n b alances the a c t i n g s t r e s s . The s e i s m i c energy E- s may be e s timated from the e m p i r i c a l magnitude-energy r e l a t i o n of Gutenberg and H i c h t e r ( E i c h t e r , 1 9 5 8 ) 22 Assuming t h a t the s t r e s s drop and apparent s t r e s s have been c a l c u l a t e d using Equations 2.27 and 2.36, then we have two equations 2.29 and 2.35 r e l a t i n g the three unknown q u a n t i t i e s i n i t i a l s t r e s s So » f i n a l s t r e s s S, , and s e i s m i c e f f i c i e n c y . These three important q u a n t i t i e s remain indeterminate. 2.6. P r e s c r i b i n g the d i s l o c a t i o n f u n c t i o n The Savage model t r e a t s rupture k i n e m a t i c a l l y by a r b i t r a r i l y p r e s c r i b i n g the e v o l u t i o n of d i s l o c a t i o n . The model l a c k s p h y s i c a l l e g i t i m a c y i n a fundamental sense. T h i s e s s e n t i a l d e f i c i e n c y can i n some measure be compensated f o r by a s s i g n i n g a s p a t i a l and temporal d e s c r i p t i o n of the d i s l o c a t i o n f u n c t i o n LJ(/^t) which i s i n accord with the p h y s i c a l c o n d i t i o n s t h a t are l i k e l y t o accompany ru p t u r e . The s o l u t i o n t h a t was obtained i n S e c t i o n 2.3 i s f o r a step time f u n c t i o n of d i s l o c a t i o n everywhere on the f a u l t s u r f a c e . As was i n d i c a t e d b e f o r e , we may use the s u p e r p o s i t i o n theorem to o b t a i n the s o l u t i o n f o r an a r b i t r a r y time dependence R.(t) °f the d i s l o c a t i o n . I f we r e p l a c e H(t - t r ) b y i n Equation 2.8, the time dependent p a r t of the displacement s o l u t i o n i s modified such that 2.39 X(t,4 • 0-TH ( t - T , * ) dPdH Ar That i s , the s t e p response i s convolved with the time d e r i v a t i v e of the d e s i r e d time f u n c t i o n Rw. The time f u n c t i o n i s r e q u i r e d t o be the same at every point on the f a u l t s u r f a c e . Kostrov (1964) s t u d i e d the p h y s i c a l problem of the 23 propagation of a c i r c u l a r shear c r a c k . His work c o n s t i t u t e s a p h y s i c a l l y reasonable d e s c r i p t i o n of the dynamics of rupture growth, but i s unable to address the problem of the t e r m i n a t i o n of r u p t u r e . Savage's model p r e s c r i b e s the s t o p p i n g of rupture k i n e m a t i c a l l y . F u r t h e r , i t p r e s c r i b e s that the rupture speed be c o n s t a n t with time and the same i n a l l d i r e c t i o n s , thereby m a i n t a i n i n g a u n i f o r m l y expanding c i r c u l a r rupture f r o n t . However, Kostrov found t h a t f o r h i s model, the r u p t u r e f r o n t remains c i r c u l a r f o r only one value of the rupture speed, v i z . V~ = 85P f o r * Z5 . Under t h i s c o n d i t i o n , the time f u n c t i o n of displacement along the d i r e c t i o n of s l i p has the form 2 . 4 0 UU) X . ' ( t * - f / t f which i n time a s y m p t o t i c a l l y reaches a ramp f u n c t i o n a t a l l p o i n t s on the f a u l t s u r f a c e . However i t has an i n i t i a l l y p a r a b o l i c r i s e which steepens away from the c e n t r e . At the c e n t r e i t s e l f , the r i s e i s ramp-like from the i n i t i a t i o n of r u p t u r e . These r e s u l t s i n d i c a t e t h a t i t i s i n c o r r e c t to r e s t r i c t the time f u n c t i o n to be the same at a l l p o i n t s on the f a u l t . But given t h a t r e s t r i c t i o n , i t seems reasonable to use a ramp time f u n c t i o n of d i s l o c a t i o n . The r u p t u r e i s assumed to cease as a b r u p t l y as i t s t a r t e d , with a d i s c o n t i n u i t y i n v e l o c i t y . The only parameter r e q u i r e d to s p e c i f y t h i s time f u n c t i o n i s the d u r a t i o n of the ramp. The maqnitude of the d i s l o c a t i o n L / ( P , © J as a f u n c t i o n 24 of p o s i t i o n remains to be s p e c i f i e d . I f we assume i t to be uniform everywhere, then the term f e t y - [ [i " I s t n 0 o COS ((j) - <j)0)] d<j> 11 - b c o s x j d x has an a n a l y t i c i n t e g r a l (Gradshteyn S Ryzshik (1965), p. 148, equations 2.553.3 and 2.554.3) 2.4f ( d x • a -1 f b s i n x - £ arcfo ( l - b T W f ) ( i ^ b c o s x ) 1 ( F F j ( u b c o s x ( i - b 2 ) * J i + b f o r b* < 1 , which i s a p p r o p r i a t e f o r subsonic r u p t u r e . In p a r t i c u l a r , f o r the i n t e r v a l p r i o r to the commencement of s t o p p i n g , 2 r r d x » 2rr _ ( 1 + b c o s x ) * ( < ~ b 1 ) * " o which i s a c o n s t a n t . C l e a r l y , t h i s assumption r e q u i r e s a d i s c o n t i n u i t y i n d i s l o c a t i o n a t the boundary of the f a u l t . A more r e a l i s t i c e x p r e s s i o n f o r the magnitude of the s l i p , suqgested by Savage (1966), i s provided by the formula *3 U U0 1 - [P cos <f) - e a j - (psi£$] where Q. and b are the major and minor semi-axes of the e l l i p s e and Q, i s i t s e l l i p t i c i t y . T h i s i s d e r i v e d from Eshelby's s t a t i c s o l u t i o n f o r the d i s c o n t i n u i t y i n displacement a c r o s s an e l l i p s o i d a l c a v i t y i n an e l a s t i c medium s u b j e c t to shear s t r e s s (Eshelby (1957), Equation 5.7). For our purposes, 25 one a x i s of the e l l i p s o i d has been s e t to zero, thereby p r o v i d i n g an e x p r e s s i o n f o r the s l i p on a shear f a u l t between whose f a c e s there i s no f r i c t i o n . S t r i c t l y , then, t h i s e x p r e s s i o n i s a p p r o p r i a t e only f o r complete s t r e s s drop. F i g u r e 2.5 shows the manner i n which the d i s l o c a t i o n d i e s away r a p i d l y near the f a u l t boundary. Although the use of t h i s d i s l o c a t i o n d i s t r i b u t i o n e n t a i l s numerical i n t e g r a t i o n s , i t s use can be j u s t i f i e d by the l a r g e c o n t r i b u t i o n i t makes toward r e n d e r i n g the model r e a l i s t i c . X F i g u r e 2.5. Contours of equal d i s l o c a t i o n on the f a u l t 2.7. The form of the r a d i a t i o n from Savage's model By s t u d y i n g the rupture p a t t e r n we can r e a d i l y e x p l a i n the form o f the p u l s e s r a d i a t e d by the model. As we have a l r e a d y seen, the apparent l o c u s of r u p t u r e onset a t any i n s t a n t i s an e l l i p s e whose major a x i s i s the p r o j e c t i o n onto the f a u l t plane of the d i r e c t i o n t o the observer. S u c c e s s i v e apparent f r o n t s are i l l u s t r a t e d i n F i g u r e 2.6 f o r a ray d e p a r t i n g at t h i r t y 26 0 s e c 1 I I F i g u r e 2 . 6 . S u c c e s s i v e a p p a r e n t r u p t u r e f r o n t s 27 degrees from the f a u l t normal and towards the nearest p a r t of the f a u l t boundary. I n i t i a l l y , the a c t u a l rupture f r o n t p" , as w e l l as the apparent rupture f r o n t p = ^ ( t ~ ^ . ) , i n c r e a s e i n l e n g t h l i n e a r l y with time. Hence i n the case where ^ ( p , ^ ) i s uniform over the f a u l t s u r f a c e , the r a d i a t e d displacement a l s o i n c r e a s e s l i n e a r l y with time. E v e n t u a l l y the rupture : f r o n t reaches the boundary of f a u l t i n g beyond which rupture ceases, and from then on an i n c r e a s i n g l y l a r g e s e c t i o n of the e f f e c t i v e l o c u s i s e l i m i n a t e d by the boundary. There i s a p a r t i c u l a r l y abrupt decrease i n the l e n g t h of the e f f e c t i v e l o c u s of r u p t u r e at f i r s t and l a s t c o n t a c t between the e f f e c t i v e l o c u s and the f a u l t boundary because at those i n s t a n t s the two curves o s c u l a t e r a t h e r than i n t e r s e c t . In summary, then, the r a d i a t e d s i g n a l f i r s t i n c r e a s e s s t e a d i l y but changes a b r u p t l y when the boundary i s f i r s t encountered and e v e n t u a l l y decreases to zero a b r u p t l y when rupture over the whole f a u l t i s complete. Savage has termed the l a t t e r two events 'stopping phases', which may be i d e n t i f i e d as d i s c o n t i n u i t i e s i n s l o p e of. the displacement p u l s e . The f u l l range of p o s s i b l e c o n g r e s s i o n a l pulse shapes f o r a f a u l t s u r f a c e having an e l l i p t i c i t y of 0.6 and a r a t i o of r u p t u r e to compressional wave speed of one h a l f i s shown i n F i g u r e 2.7. The v a r i a b l e 0 occurs only i n the e x p r e s s i o n V" S i * l G t so these p u l s e s c o u l d be a p p r o p r i a t e l y s c a l e d f o r d i f f e r e n t v a l u e s of the speed r a t i o . The p u l s e s show i n c r e a s i n g azimuthal v a r i a t i o n as t h e i r departure d i r e c t i o n s become more p a r a l l e l to the f a u l t plane or, a l t e r n a t i v e l y , as the rupture 28 uniform s t a t i c d i s l o c a t i o n 0 sec 1 1 I E s h e l b y ' s s t a t i c d i s l o c a t i o n F i g u r e 2.7. Displacement p u l s e s from a f a u l t model 29 speed approaches the wave speed. U n l e s s depth phases are used, t e l e s e i s m i c r e c o r d i n g s of shallow events sample a r e l a t i v e l y narrow cone of t a k e o f f angles (roughly one r a d i a n i n diameter) and so we would not expect to f i n d as much v a r i a t i o n as i s p o r t r a y e d i n F i g u r e 2 . 7 . 2.8. The spectrum of the r a d i a t i o n from Savage's model The s p e c t r a of the pu l s e s computed from Savage's model e x h i b i t two components of b a n d - l i m i t a t i o n which may be a s s o c i a t e d with two i n t e r v a l s i n the time domain. The f i r s t i s a s s o c i a t e d with the time i n t e r v a l preceding the f i r s t s t opping phase, and the i n t e r v a l s over which the f i r s t and f i n a l s t o p p i n g phases occur. The second i s a s s o c i a t e d with the time i n t e r v a l between the st o p p i n g phases. A pu l s e may be c r u d e l y approximated by the c o n v o l u t i o n of two boxcar f u n c t i o n s having these i n t e r v a l s as shown i n F i g u r e 2.8. The amplitude spectrum F i g u r e 2.8. C o n v o l u t i o n a l model of a displacement p u l s e of such a pulse w i l l be the product of two s i n e f u n c t i o n s whose f i r s t zeros occur at f r e q u e n c i e s t h a t are i n v e r s e l y p r o p o r t i o n a l to the time i n t e r v a l s . Thus the spectrum beqins a CO ' r o l l o f f 30 F i g u r e 2.9. Spectrum of a d i s p l a c e m e n t p u l s e 31 at a frequency of l/7j hz , and a r o l l o f f of u; at a frequency of l / l ^ hz. The two m u l t i p l i e d s i n e lobe p a t t e r n s are e v i d e n t i n the spectrum of a model pulse i n -Figure-2.9. The p u l s e , whose e v o l u t i o n i s d e s c r i b e d i n F i g u r e 2.6, i s d i s p l a y e d i n F i g u r e 2.7 having Q =-.30° and (j) = 180°. 2.9. Choosing model parameters f o r i n v e r s i o n The parameters of the Savage model l e n d themselves d i r e c t l y to a convenient p a r a m e t e r i s a t i o n f o r the purposes of i n v e r s i o n . The seven model parameters t h a t were chosen are l i s t e d i n Table 5.3. Rupture v e l o c i t y V and dimension, expressed as the major sem i - a x i s A, are o b v i o u s l y a p p r o p r i a t e parameters. The ramp time f u n c t i o n of d i s l o c a t i o n i s c h a r a c t e r i s e d by RT.IH, the d u r a t i o n of the ramp. The shape of the e l l i p s e i s most s u i t a b l y r e p r e s e n t e d by the e l l i p t i c i t y E, s i n c e i t i s l i n e a r l y r e l a t e d to the d i s t a n c e between the c e n t r e and a focus of the e l l i p s e . The o r i e n t a t i o n of the f a u l t plane i s expressed through the d i p angle DIP and the azimuthal down-dip d i r e c t i o n DIPAZ. The o r i e n t a t i o n of the e l l i p t i c a l f a u l t s u r f a c e i s expressed through the d i r e c t i o n along the major a x i s from the e p i c e n t r a l f ocus to the other focus of the e l l i p s e . T h i s i s the d i r e c t i o n to the f u r t h e s t p o i n t on the f a u l t boundary. The o r i e n t a t i o n ELAZ of the major a x i s i s measured c l o c k w i s e from the downdip d i r e c t i o n on the f a u l t plane. The three f a u l t s u r f a c e o r i e n t a t i o n angles are i l l u s t r a t e d i n F i g u r e 2.10. 32 F i g u r e 2.10. O r i e n t a t i o n angles of the f a u l t s u r f a c e 33 2.10. A s t a t i o n ' s p o s i t i o n i n f a u l t s u r f a c e c o o r d i n a t e s For our model computations, we need to express the p o s i t i o n of a s t a t i o n i n s p h e r i c a l c o o r d i n a t e s (Q, 0] with r e s p e c t to the f a u l t c o o r d i n a t e frame of F i g u r e 2.11. Here, Z i s the downward normal to the f a u l t plane, <7j i s measured c l o c k w i s e from the major a x i s , and 0 i s measured from the Z a x i s . T h i s i s an i n v e r t e d v e r s i o n of F i g u r e 2.3. W >- N Z verticql F i g . 2.11. F a u l t C o o r d i n a t e s F i g . 2.12. S t a t i o n C o o r d i n a t e s The f a u l t frame c o o r d i n a t e s Q and (f> are c a l c u l a t e d from two a n g l e s d e s c r i b i n g the d i r e c t i o n of the d e p a r t i n g ray (Figure 2.12) and the t h r e e a n g l e s d e s c r i b i n g the o r i e n t a t i o n of the f a u l t s u r f a c e (Figure 2.10). These f i v e angles are summarised i n Table 2.1. The task of f i n d i n g Q and (f> i s posed as a problem i n s p h e r i c a l trigonometry i n Table 2.1 and F i g u r e 2.13. From a knowledge of B, a, and c we wish to determine A and b, where we may t h i n k of A and B as angles between planes and a, b, and c as 34 angles between d i r e c t i o n s . The s o l u t i o n of the problem i s given by the Cosine formula (Dupuis and Hatheson, 1911) Cos b - Cos a cos c * Sen a s in c cos B c o s A = Cos a - cos b cos c Sin b sin c from which we may o b t a i n 0 and (j> as shown i n Table 2.1. OQUS y e r i i c a io sfofcon downward •fault normal F i g u r e 2.13. S p h e r i c a l trigonometry at the e p i c e n t r e i '• : •—; 1 I GIVEN ANGLES (Fig u r e s 2.10 and 2.12) I | XI = T a k e o f f angle of d e p a r t i n g ray I | AZ = Azimuth of d e p a r t i n g tay I | DIP = Dip angle of f a u l t plane I | DIPAZ = Azimuth of down-dip d i r e c t i o n of f a u l t plane | | ELAZ = Angle on f a u l t s u r f a c e from down-dip d i r e c t i o n | I to major a x i s d i r e c t i o n ( e p i c e n t r a l focus | | to other focus) I | SPHERICAL TRIGONOMETRY PROBLEM (Figure 2.13) | | Known: I | a = XI I | c = jj- - DIP Plunge angle of f a u l t plane | | B = DIPAZ - AZ Angle between the v e r t i c a l planes | J c o n t a i n i n g the down-dip d i r e c t i o n and I | the d e p a r t i n g ray j | C a l c u l a t e d : I | b = Angle between the normal to the f a u l t s u r f a c e [ | and the d e p a r t i n g ray I | A = Angle between the v e r t i c a l plane c o n t a i n i n g I | the d i p d i r e c t i o n and the plane c o n t a i n i n g j | the d e p a r t i n g ray and the f a u l t normal I I DESIRED ANGLES (Figure 2.11) I | 9 - b Takeoff angle measured from the f a u l t normal | | (f> - A - ELAZ Azimuthal angle i n the f a u l t plane | L Table 2.1. Computing f a u l t s u r f a c e c o o r d i n a t e s of a ray 36 CHAPTER 3. SEISMOGRAM ANALYSIS 3.1. Choosing earthquake events The short- p e r i o d seismograph i s most a p p r o p r i a t e l y a p p l i e d t o the study of earthquakes having a r u p t u r e d u r a t i o n w i t h i n the s h o r t - p e r i o d band, whose c e n t r e i s roughly one second (Figure 3.1). In order that reasonably l a r g e amplitudes be recorded t e l e s e i s m i c a l l y , the earthquakes s e l e c t e d by t h i s c r i t e r i o n must a l s o have a l a r g e enerqy r e l e a s e r e l a t i v e t o t h e i r dimensions. In summary, s h o r t p e r i o d instruments are best s u i t e d to the study of earthquakes w i t h , l a r q e s t r e s s drops and s m a l l f a u l t a r eas. Such earthquakes are r e c o g n i s e d by t h e i r s h o r t coda (on at l e a s t some r e c o r d s ) , the r e l a t i v e l y high frequency content of the waves and t h e i r l a r g e amplitudes a t t e l e s e i s m i c d i s t a n c e s . While s e l e c t i n g events i t i s o f t e n found t h a t a s i n g l e r e c o r d i n g of an event has great s i m p l i c i t y , having j u s t a few c y c l e s of l a r g e amplitude motion. However, other r e c o r d i n g s of the same event o f t e n have g r e a t complexity, with l a r g e i n t e r f e r i n g phases a r r i v i n g soon a f t e r onset, or a l o n g , pseudo-s i n u s o i d a l coda. In these cases, the complexity of the seismogram must r e s u l t from the t r a n s m i s s i o n path of the waves, r a t h e r than from the source (Thirlaway (1966), Douglas et a l (1973) ) . I t i s d e s i r a b l e t o avoid such r e c o r d i n g s s i n c e the recovery of source i n f o r m a t i o n from them i s d i f f i c u l t . The p a r t i c u l a r d i f f i c u l t y of source s t u d i e s using s h o r t -p e r i o d seismograms l i e s i n the i n t e r f e r i n g e f f e c t of the c r u s t a l s t r u c t u r e underneath the r e c e i v e r and of inhomogeneities near 37 o i 1 1 1 1 1 1 1 -1.5 -].0 -0.5 0.0 0.5 1.0 1.5 LOG FREQUENCY (HZJ F i g u r e 3.1. Displacement s e n s i t i v i t y of the seismographs (set e g u a l at 1hz) the source. C r u s t a l inhomogeneities have dimensions of some k i l o m e t e r s , and so cause l a t e r a r r i v a l s through multipath t r a n s m i s s i o n , r e f l e c t i o n from o b l i q u e i n t e r f a c e s , or r e v e r b e r a t i o n s between sedimentary l a y e r s which have d e l a y s on the order of a second. An array of seismometers which have sp a c i n g on the order of a k i l o m e t e r can be used t o average over the e f f e c t s of such inhomogeneities and thereby o b t a i n a b e t t e r estimate of the source. An a r r a y can d e t e c t the presence of secondary a r r i v a l s which have an unexpected phase v e l o c i t y . We are then a b l e to apply v e l o c i t y f i l t e r i n g to the r e c o r d i n g s , or to exclude them from our a n a l y s i s . Earthquakes i n e a s t and c e n t r a l A s i a are a t t e l e s e i s m i c d i s t a n c e s from a l l f o u r UKAEA a r r a y s , which are l o c a t e d a t Eskdalemuir, S c o t l a n d ; Y e l l o w k n i f e , Canada; Gauri b i d a n u r , I n d i a and Warramunga, A u s t r a l i a . The two events s t u d i e d were chosen because of t h e i r simple a r r a y seismograms and because reasonably good f a u l t plane s o l u t i o n s .were a v a i l a b l e . The a r r a y data were augmented by a s e l e c t i o n of good q u a l i t y WWSS r e c o r d i n g s chosen to g i v e as broad an azimuthal d i s t r i b u t i o n as p o s s i b l e . The h y p o c e n t r a l and nodal plane c o o r d i n a t e s of the two earthquakes are shown i n T a b l e 3.1. The Eat I s l a n d earthguake of June 2nd 1966 was grouped by Stauder (1968a,b) among events having a s i m i l a r mechanism which i s c h a r a c t e r i s e d by h o r i z o n t a l t e n s i o n a l s t r e s s p e r p e n d i c u l a r to the s t r i k e of the A l e u t i a n Trench. These events, whose hypocentres l i e underneath the t r e n c h , are a t t r i b u t e d to the 39 I — - - - •• j Parameter | Rat I s l a n d | Broach , i , 1 i J J 1 I | L a t i t u d e (N) I 51.1 I 21.75 | Longitude (E) | 176.0 | 73.06 j Depth (km) | 41 | 21 | O r i g i n Date | 1966 June 2 | 1 9 7 0 March 23 1 O r i g i n Time | 03:27:53 | 01:53:05 | Magnitude (m(,, USCGS) I 6.0 I 5.4 j Nodal Plane 1 : I I | Dip D i r e c t i o n | 5 0 0 | 0 0 I Dip Angle | 4 0 0 | 4 0 0 j Nodal Plane 2 : I I | Dip D i r e c t i o n | 2 1 1 0 | 1 8 0 0 | Dip Angle | 5 1 0 | 5 0 O • i T a b l e 3.1. Hypocentral and nodal plane data f o r the events e x t e n s i o n a l f r a c t u r e of the P a c i f i c P l a t e as i t i s t h r u s t under the American P l a t e . T h e i r mechanism i s q u i t e d i s t i n c t from the tin d e r t h r u s t events whose hypocentres l i e under the A l e u t i a n I s l a n d s . T h i s a c t i v e t e c t o n i c environment i s l i k e l y t o possess s t r o n g l y heterogeneous p h y s i c a l p r o p e r t i e s . The complex r e c o r d i n g of the Rat I s l a n d event at the Y e l l o w k n i f e a r r a y may be the r e s u l t of anomalous propagation through the downthrust s l a b . The hypocentre was at a depth of 41km, where the compressional wave v e l o c i t y was assumed to be 8.05km/sec. The c r u s t a l s t r u c t u r e i n the v i c i n i t y of the Broach no thickness ( Km) 15? 6 0 6-5-1 6-75" 25-G a u r i b i d a n u r H e r r i n F i g u r e 3.2. The Gauribidanur and H e r r i n c r u s t a l models e p i c e n t r e may be estimated from the s t r u c t u r e underneath the G a u r i b i d a n u r a r r a y (Arora 1971) which i s approximately one thousand k i l o m e t r e s away. The s t r u c t u r e underneath G a u r i b i d a n u r i s shown a l o n g s i d e the H e r r i n c r u s t a l model {Herrin et a l , 1968) i n F i g u r e 3.2. The two s t r u c t u r e s are s u f f i c i e n t l y s i m i l a r t h a t the H e r r i n model can be used. The depth of the hypocentre was determined to be 21km by Arora e t a l (1971) using a c e p s t r a l t echnique. T h i s depth, which was confirmed by r e a d i n g s of the phase pP, p l a c e s the hypocentre i n the lower c r u s t a l l a y e r . 3.2. D i g i t i s i n g the seismograms The analog magnetic tape a r r a y data were f i l t e r e d with a lowpass f i l t e r whose ga i n was -3db at 9hz and -3Udb at 12.5hz. The data were then sampled a t a r a t e of 25/second y i e l d i n g a Nyquist frequency of 12.5hz. The WHSS p h o t o g r a p h i c a l l y c o p i e d paper r e c o r d s have a time 41 s c a l e of roughly 1mm/sec. These r e c o r d s were d i g i t i s e d using the U n i v e r s i t y of B r i t i s h Columbia's D i g i c o n d i g i t i s e r which has a r e s o l u t i o n of one thousandth of an i n c h . The i n t e r p o l a t i o n scheme of R. A. Wiggins was used, i n which only peaks, troughs and i n f l e x i o n p o i n t s are sampled. A piecewise continuous c u b i c p o l y nomial f u n c t i o n i s then f i t t e d through these p o i n t s i n such a way th a t the f i r s t d e r i v a t i v e i s continuous a t the knots of the s p l i n e . The f u n c t i o n i s then i n t e r p o l a t e d at the d e s i r e d r a t e of 25 samples per second. I t i s worthwhile t o note some of the l i m i t a t i o n s on accuracy of the d i g i t i s a t i o n process. F i r s t l y , i t i s noteworthy t h a t the .001 inch r e s o l u t i o n of the instrument p r o v i d e s the p o s s i b i l i t y f o r very a c c u r a t e r e p r o d u c t i o n , s i n c e i t i s on the order of the s m a l l e s t i n t e r v a l p e r c e p t i b l e t o the naked eye. The major f a c t o r m i l i t a t i n g a g a i n s t the achievmerit of optimal accuracy i s the t h i c k n e s s of the l i n e on the seismograms, which o f t e n exceeds t e n thousandths of an i n c h , w h i l e the s c a l e i s roughly f o r t y thousandths per second. A second source of e r r o r l i e s i n the f a i l u r e of the seismograph galvanometer's displacement to be e x a c t l y p e r p e n d i c u l a r to the time t r a c e on the drum. A u s e f u l r e f e r e n c e f o r the displacement d i r e c t i o n i s provided by the c a l i b r a t i o n p u l s e s . In -this s i t u a t i o n , i t i s best to a l i g n the d i g i t i s e r ' s o r d i n a t e with the displacement d i r e c t i o n , and l a t e r to remove the mean trend from the time a x i s . I d e a l l y , the skewness of the time a x i s with r e s p e c t to the a b s c i s s a should be c o r r e c t e d by a r o t a t i o n . A t h i r d and u s u a l l y minor d i f f i c u l t y i s presented by the 42 warping of the photographic paper on which the seismograms are reproduced. T h i s i s u n l i k e l y t o be a s e r i o u s problem l o c a l l y where only a few tens of seconds of the P coda are of i n t e r e s t . Time marks were removed by measuring the displacement of a number of neighbouring time marks. The averaged displacement was then s u b t r a c t e d from the d i g i t i s e d o r d i n a t e values which l a y w i t h i n the time mark. 3.3. Array beamforming In order to employ the redundancy a v a i l a b l e i n the array data we must c a l c u l a t e the a p p r o p r i a t e time c o r r e c t i o n s to be a p p l i e d to the i n d i v i d u a l channels before they are summed. T h i s may be done with a knowledge of the geographic c o o r d i n a t e s of the seismometers, and of the azimuth and phase v e l o c i t y of the incoming waves. I f we wish to c o r r e c t f o r v a r i a t i o n s i n e l e v a t i o n of the seismometers, we a l s o need to know these e l e v a t i o n s and the wave v e l o c i t y of the s u r f a c e m a t e r i a l . Assume that plane P waves are impinging upon a f l a t s u r f a c e with azimuth ^ and phase v e l o c i t y V- , as shown i n F i g u r e 3.3a. We wish to c a l c u l a t e the time d i f f e r e n c e between the The equation of the i n t e r s e c t i o n of the wavefront with the t s u r f a c e i s given by a r r i v a l a t an a r b i t r a r y p o i n t and the o r i g i n 0 b The d i s t a n c e S between the wavefront and the o r i g i n i s 43 S - b | Cos^| = (y * X t a n ^ j \cos<f>\ The time advance a t P with r e s p e c t to the o r i g i n i s thus ( lj «• X f a n (/ ) |6t93^ / A n \ base level (a) H o r i z o n t a l plane (b) V e r t i c a l plane F i g u r e 3 . 3 . Geometry f o r a r r a y beamforming Suppose that the seismometer a t P i s a t an e l e v a t i o n K above the base l e v e l P0 . Then a f t e r the wavefront has a r r i v e d at - R i t s t i l l must t r a v e l the d i s t a n c e 0^ = h cos i as shown i n F i g u r e 3 . 3 b . The angle C i s g i v e n by s<-n i - — where «< i s the P wave v e l o c i t y . The a r r i v a l at P i s thus delayed by 44 h The t o t a l time delay to be a p p l i e d as a c o r r e c t i o n i s thus t = Oy + x f<^  ^) J CoS <f> | h IT IT t&s\L 3.4. Homomorphic d e c o n v o l u t i o n The e f f e c t of the earth's s t r u c t u r e near the earthquake source and the r e c o r d i n q s t a t i o n s p r e s e n t s a major o b s t a c l e t o the i n v e s t i g a t i o n of earthquake source p r o p e r t i e s usinq s h o r t p e r i o d seismograms. I t i s probable t h a t the h o r i z o n t a l s t r u c t u r e near shallow sources has an i n f l u e n c e that decreases with depth. However the e f f e c t of the inhomoqeneous B e n i o f f zone i n which most events occur w i l l s t i l l be present, and w i l l probably be the dominant d i s t o r t i n q i n f l u e n c e upon waves r a d i a t e d from such zones. The s t r u c t u r e of the c r u s t near the r e c e i v e r c o n t r i b u t e s s t r o n q l y t o the d i s p e r s a l of incoming waves. In h o r i z o n t a l l y l a y e r e d s t r u c t u r e s such as sedimentary sequences, energy a r r i v e s at r e v e r b e r a t i o n d e l a y s which are t y p i c a l l y on the order of a second (Jensen and E l l i s , 1971). These delayed a r r i v a l s i n t e r f e r e with our attempts to determine a source pulse which, f o r earthquakes havinq sma l l source dimensions, has a s i m i l a r d u r a t i o n . I t appears that r e c o r d i n g s at s t a t i o n s s i t u a t e d on basement 45 rock, while l a c k i n g r e v e r b e r a t i o n s , are n e v e r t h e l e s s degraded by l a t e r a r r i v a l s . These may be generated by the complex f a u l t i n g i n such s t r u c t u r e s . Let us assume t h a t a seismogram i s c o n s t i t u t e d by the c o n v o l u t i o n of a source wavelet with an impulse s e r i e s r e p r e s e n t i n g the impulse response of the s t r u c t u r e s through which the wavelet t r a v e l s . We are faced with the task of d econvolving the e f f e c t of these s t r u c t u r e s which i n most i n s t a n c e s are p o o r l y determined. C l e a r l y we need a method t h a t does not r e q u i r e a d e t e r m i n i s t i c d e s c r i p t i o n of the s t r u c t u r e . In e x p l o r a t i o n seismology, the source i s most commonly estimated v i a the seismogram a u t o c o r r e l a t i o n f u n c t i o n . T h i s method assumes t h a t the impulse response of the s t r u c t u r e behaves l i k e white n o i s e , and t h a t the source i s minimum phase. N e i t h e r of these assumptions appears to have v a l i d i t y f o r t e l e s e i s m i c r e c o r d i n g s . The homomorphic de c o n v o l u t i o n method proposed by Oppenheim (1967) and a p p l i e d to t e l e s e i s m i c r e c o r d i n g s by O l r y c h e t . a l . (1.972) avoids both of these assumptions. I t i s used i n t h i s work although i t e n t a i l s s e v e r a l p r a c t i c a l d i f f i c u l t i e s when a p p l i e d to t e l e s e i s m i c data. These d i f f i c u l t i e s w i l l be d i s c u s s e d l a t e r . We may inodel a n o i s e - f r e e seismogram x (t) by the c o n v o l u t i o n of a source f u n c t i o n s ( t ) with an impulse response i ( t ) 3 . 1 , X W = S ( t ) * L ( t ) By the homomorphic d e c o n v o l u t i o n method we are a b l e t o t r a n s f o r m the seismogram i n t o a • c e p s t r a l ' time s e r i e s i n which 46 the m o d i f i e d components are i n an a d d i t i v e r e l a t i o n s h i p 31 X H = J (r) - h r ) The time v a r i a b l e f of the cepstrum r e p r e s e n t s the •frequency' content of the l o q s p e c t r a and i s termed 'quefrency'. The c e p s t r um O C ( r ) of x ( t ) i s obtained by the f o l l o w i n g sequence of o p e r a t i o n s : 3-3 X W = J { ex. log [ J (x(t)j where denotes F o u r i e r t r a n s f o r m a t i o n and ex. l o q denotes the complex l o g a r i t h m i n which both amplitude and phase i n f o r m a t i o n are r e t a i n e d . S e p a r a t i o n of the two c e p s t r a d and $ i s achieved using l i n e a r f i l t e r i n g , f o r which some c r i t e r i o n i s r e q u i r e d . The usu a l assumption i s t h a t the source cepstrum has a lower quefrency content than the impulse s e r i e s cepstrum , as d e s c r i b e d by Schafer (1969) and Ul r y c h (1971). I n v e r s i o n t o the o r i g i n a l time domain i s performed u s i n q . the same sequence of o p e r a t i o n s as i n Equation 3.3. S e v e r a l problems are encountered when the homomorphic method i s a p p l i e d to t e l e s e i s m i c r e c o r d i n g s . The f i r s t i s due t o the f a c t t h a t the complex l o g a r i t h m i s not unique with r e s p e c t to the phase, and y i e l d s a d i s c o n t i n u o u s 'raw' phase curve composed of seqments t h a t are continuous over the i n t e r v a l 0 to 2Tr, as i l l u s t r a t e d i n Fiq u r e 3.4. The phase curve must be made continuous by the a d d i t i o n of some m u l t i p l e of 2Tr at each p o i n t , thereby 'unwrapping' the curve. The phase i s s e t to zero at z e r o frequency and the l i n e a r t r e n d , c o r r e s p o n d i n q t o a time-s h i f t i n the s i g n a l , i s removed. 47 The presence of a d d i t i v e noise comparable to the s i g n a l amplitude a t some freguency causes a l a r g e d e v i a t i o n i n the phase of the s i g n a l at t h a t freguency, as i l l u s t r a t e d by C l a y t o n (1975). T h i s may l e a d t o e r r o r s i n the unwrapping of the phase curve. T h i s c o n d i t i o n e x i s t s at holes i n the spectrum, whose presence i n earthguake s p e c t r a i s p r e d i c t e d by d i s l o c a t i o n models such as Savage's (1966). C l a y t o n (1975) has examined t h i s s i t u a t i o n i n the Z transform domain. The hole i n d i c a t e s the presence of a z e r o near the u n i t c i r c l e , and w i l l cause the phase to change r a p i d l y by T T i n the v i c i n i t y of the c o r r e s p o n d i n g frequency. Small d e v i a t i o n s i n the p o s i t i o n of the zero due t o a d d i t i v e n o i s e may f o r c e i t to c r o s s the u n i t c i r c l e , c a u s i n g the d i r e c t i o n of the phase jump to change. Such behaviour i s evident i n the s p e c t r a of the AKU r e c o r d i n g of the Rat I s l a n d event, shown i n F i g u r e 3.4. The seismogram used i n the computation of spectrum (a) c o n t a i n e d one l e s s data p o i n t at i t s onset than f o r spectrum (b), g i v i n g i t a h i g h e r s p e c t r a l l e v e l at higher f r e q u e n c i e s . The phase jumps of T T a t 3hz are i n the same sense f o r both s p e c t r a . However the jumps at 6hz are i n o p p o s i t e senses. There i s a l s o a f u r t h e r decrease of 2T T i n the phase curve of (b). Large d i f f e r e n c e s i n the phase curves and c e p s t r a r e s u l t , but the recovered wavelets are not r a d i c a l l y d i f f e r e n t . T h i s i s probably due to the f a c t t h a t the holes are at f r e q u e n c i e s beyond those c o n t a i n i n g g e n e r a l l y l a r g e s i g n a l amplitudes, and so the e r r o r s that they might i n t r o d u c e i n phase e s t i m a t i o n are of l i t t l e consequence. Wavelets are shown f o r cepstrum c u t o f f s between 10 and 20 u n i t s . 48 LOG FREQUENCY (HZ) F i g u r e 3.4a. D e c o n v o l u t i o n o f t h e H a t I s . s e i s m o g r a m a t AKO ( o n s e t c h o s e n t o be m o r e a b r u p t ) 49 SE1SMQGRRM 0 sec 1 1 I CEPSTRUM I — i 1 -1.0 0.0 1.0 LOG FREQUENCY (HZ) WAVELET F i g u r e 3.4b. D e c o n v o l u t i o n of the Rat I s . seismogram at AKU (onset chosen to be more gradual) 50 The e f f e c t of a d d i t i v e n o i s e on the unwrapping of the phase curve g e n e r a l l y caused few d i f f i c u l t i e s i n the d e c o n v o l u t i o n of the other seismograms. A second, and f o r t h i s work, more s e r i o u s problem r e s u l t e d from the use of a simple lowpass c e p s t r a l f i l t e r i n g c r i t e r i o n . The presence of sharp f e a t u r e s such as h o l e s i n the s i g n a l spectrum makes c o n t r i b u t i o n s to the high guefrency part of the cepstrum. T h i s contravenes the low guefrency assumption f o r the recovery of the s i g n a l . I t i s a l s o p o s s i b l e t h a t a d d i t i v e noise and the impulse response w i l l have low guefrency c o n t r i b u t i o n s . During the a n a l y s i s of the seismograms, two predominant phase curve shapes were encountered. The f i r s t , i l l u s t r a t e d i n F i g u r e 3.4, decreases from zero at zero frequency. The recovered wavelets computed f o r d i f f e r e n t c e p s t r a l c u t o f f s vary mainly i n the second peak. The second, i l l u s t r a t e d i n F i g u r e 3.5, i n i t i a l l y i n c r e a s e s from zero and then f a l l s . The r e c o v e r e d wavelets vary i n both peaks. The l a t t e r phase curve type u s u a l l y o c c u r r e d when the s i g n a l to n o i s e r a t i o was low at low f r e q u e n c i e s , s u g g e s t i n g t h a t i n these cases i t may be d i s t o r t e d s e r i o u s l y by n o i s e . The seismograms and the wavelets deconvolved from them are shown i n F i g u r e s 3.6 and 3.7. For each seismogram, the wavelet chosen was the one which occupied an approximately median p o s i t i o n within the envelope of wavelets computed f o r a wide range of c e p s t r a l c u t o f f s . We now i n v e s t i g a t e the nature of the u n c e r t a i n t i e s i n the ' o b s e r v a t i o n s ' ( i . e . the deconvolved seismograms) t h a t are r e g u i r e d f o r our study. Besides p o s s i b l e e r r o r s i n measurement 51 SE1SM0GRRM o L sec CEPSTRUM I 1 1 -].o o.o 1.0 LOG FREQUENCY (HZ) WAVELET F i g u r e 3.5. D e c o n v o l u t i o n of the Broach seismogram a t SNG 54 and d i g i t i s a t i o n , we are faced with an i n t r i n s i c u n c e r t a i n t y because n e i t h e r of the components t h a t are separated are known beforehand. T h i s u n c e r t a i n t y may be p a r t l y d i s p e l l e d by the study of s y n t h e t i c examples, such as those of U l r y c h (1971), or of a c t u a l r e c o r d i n g s upon well determined c r u s t a l s t r u c t u r e s . In one such study by Dlrych e t a l (1972), seismograms recorded near Leduc, A l b e r t a were r e s y n t h e s i s e d using the deconvolved seismogram and an impulse response f o r the c r u s t a l model. Good agreement between the o r i g i n a l seismograms and the r e s y n t h e s i s e d ones was o b t a i n e d . Thus d e s p i t e the d i f f i c u l t i e s e n t a i l e d i n the use of the homomorphic d e c o n v o l u t i o n method, a u s e f u l estimate of the source can be o b t a i n e d . I t i s most l i k e l y that the major u n c e r t a i n t y i n source e s t i m a t e s o b t a i n e d using the method i s i n c u r r e d i n the c h o i c e of the c u t o f f of the c e p s t r a l lowpass f i l t e r . 3.5. A t t e n u a t i o n The a t t e n u a t i o n of e l a s t i c waves may be d e s c r i b e d i n terms of the Q f a c t o r of the absorbing m a t e r i a l . From measurements on a wide v a r i e t y of m a t e r i a l s i t appears t h a t Q i s independent of frequency over a wide band of f r e q u e n c i e s (Knopoff and HacDonald, 1958). I f Q i s independent of frequency f o r a l l f r e q u e n c i e s , then the a b s o r p t i o n process i s l i n e a r , and the r e a l s p a t i a l a b s o r p t i o n c o e f f i c i e n t i s a l i n e a r l y i n c r e a s i n q f u n c t i o n of frequency <X.(CJ) = Wy/ 2 C & where c i s c o n s t a n t . In t h i s s i t u a t i o n , however, we have a b s o r p t i o n without d i s p e r s i o n , i m p l y i n q a n o n - l i n e a r wave eq u a t i o n . 55 Futterman (1962) showed how t h i s d e s c r i p t i o n of a t t e n u a t i o n can be extended t o i n c l u d e d i s p e r s i o n i n such a way t h a t a l i n e a r wave equation i s s t r i c t l y obeyed while the a b s o r p t i o n i s l i n e a r over an a r b i t r a r i l y broad frequency,range of i n t e r e s t . The d i s p e r s i v e component of the complex index of r e f r a c t i o n i s d e r i v e d from the a b s o r p t i v e component using the Kramers-Kronig r e l a t i o n s , which are d e r i v e d s o l e l y from the p r i n c i p l e of c a u s a l i t y . T h i s q u a s i - l i n e a r f o r m u l a t i o n permits us t o use the s u p e r p o s i t i o n p r i n c i p l e t o o b t a i n the time domain response. Wuenschel (1965) subsequently showed t h a t Futterinan's f o r m u l a t i o n provided an adequate d e s c r i p t i o n of the d i s p e r s i o n i n a p l e x i g l a s p l a t e , and of the d i s p e r s i o n measured i n the P i e r r e s h a l e by HcDonal et a l (1958). Lomnitz (1957) proposed a law of i n t e r n a l f r i c t i o n based on creep o b s e r v a t i o n s which a l s o e x p l a i n s the a t t e n u a t i o n of s e i s m i c waves. Savage and O ' N e i l l (1975) have shown t h i s law to g i v e r e s u l t s t h a t are so s i m i l a r to those of the Futterman model t h a t they are, at l e a s t at present, i n d i s t i n g u i s h a b l e . They suggested t h a t the Lomnitz law should be p r e f e r r e d because of i t s e m p i r i c a l b a s i s . Carpenter (1967) showed t h a t Futterman's a b s o r p t i o n model obeys a p r i n c i p l e of s i m i l a r i t y such t h a t the pulse shape depends only on the r a t i o t * = T/Q, where T i s t r a v e l time and Q the q u a l i t y f a c t o r of the medium. I f we model the earthquake source by a pulse having a d u r a t i o n of roughly one second, then we can model the observed p u l s e s of t h i s study s a t i s f a c t o r i l y without a l l o w i n g f o r 56 F i g u r e 3.8. Hatching an a t t e n u a t o r to the Rat I s . AKU wavelet 57 a t t e n u a t i o n . The best f i t i s ob t a i n e d , however, i f t * i s .1 or .2 which e n t a i l s an average Q of s e v e r a l thousand f o r t e l e s e i s m s . T h i s i s i n accord with s e v e r a l measurements such as t h a t of F r a s i e r and F i l s o n (1972) . A s e r i e s of t e s t s using the deconvolved AKU r e c o r d i n g of the Rat I s l a n d earthquake (Figure 3.8) i n d i c a t e s that a value of 1 f o r t * (implying an average Q of s e v e r a l hundred) i s unacceptably h i g h . ' In t h i s study, t * was assigned the value 0.1. The e f f e c t of t h i s a t t e n u a t o r on a spectrum i s i l l u s t r a t e d i n F i g u r e 2.9. I d e a l l y , t * c o u l d be i n c l u d e d i n the i n v e r s i o n , and t h i s might be p r a c t i c a l i f a r a d i a l l y symmetric Q s t r u c t u r e could be assumed. Such an assumption, however, would c o n f l i c t with the known h e t e r o g e n e i t y of the Q s t r u c t u r e , e s p e c i a l l y near i s l a n d a r c s (Barazangi and I s a a c s , 1971). The u n c e r t a i n t y i n Q p e r s i s t s as a s i g n i f i c a n t l i m i t a t i o n upon the study of s e i s m i c sources using t e l e s e i s m s , and can l e a d t o gross e r r o r s i n near f i e l d s i t u a t i o n s . 3.6. Ray geometry c a l c u l a t i o n s We now b r i e f l y o u t l i n e the c a l c u l a t i o n of s e v e r a l q u a n t i t i e s t h a t depend on the g e o m e t r i c a l p o s i t i o n of the seismograph s t a t i o n with r e s p e c t t o the earthquake e p i c e n t r e . The v a l u e s of these q u a n t i t i e s are l i s t e d i n Table 3.2. Angles of i n c i d e n c e at the focus were obtained using the t a b l e s of Banghar (1970). These are based on the H e r r i n P t a b l e s (Herrin et a l , 1968). For the Rat I s l a n d event, the values f o r 40km depth were adopted. Values f o r the Broach event 58 I 1 1 ~ T 1 1 T 1 1 | S t a t i o n | L a t . | Long. |Azimuth| Delta | Ray | G (A,h) | R (Q,<t>) | I I (N) | (E) | | | angle |»10-«/cm| I i i i i i i i i i Rat I s l a n d earthquake I AKU T " -| 65.68 T T -18.10| 6.52| 63. 15 T 28.93 T .70 | .79 | | DEC | 32.38 ! -64.691 48.06| 81. 35 I 22.71 ! .61 | .79 | | FLO 1 38.30 ! -90.37| 61.081 63. 25 ! 28. 93 I .70 | .64 | 1 DUG | 40.20 I -112.821 73.38| 49. 26 ! 33.72 ! .92 | . 60 | | CT.A 1-20.08 ! 146-25| 208.97| 75. 69 i 24. 49 l .64 | .46 | I SNG | 7.17 ! 100.62| 263. 40 | 75. 48 ! 24.81 I .64 | .67 | I SHI | 29.64 ! 52.52| 313.391 35. 65 I 20.82 I . 53 | .95 | | STU 1 48.77 ! 9.20| 351.20| 80. 00 ! 23.03 ! .61 | .93 | ! EKA 1 55.33 ! -3.17| 359.501 73. 90 I 25. 13 ! . 63 | . 88 | | WRA 1-19.95 i 134.33| 219.50| i 79. 60 23.17 .61 | . 51 | Broach earthquake I COL | 64.90 -147.80| 16.27| 88. 44 16..80 .46 | ... . . . . , _ . 1 .73 | | HAT I 36.53 ! 138.20| 5 9.90| 57. 81 ! 25. 08 ! .8 1 | . 70 | | ADE |-34.97 ! 13 3.721 130.88| 83. 64 1 17.90 ! . 64 | . 90 | I WIN 1-22.57. I 17.10| 234.85| 70. 00 i 21. 97 ! .72 | . 87 | 1 MAI • 1 -1.27 ! 36.80| 242.26| 42. 33 29. 73 * .98 | .76 | I EIL | 21.70 ! 73.00 | 290.81 | 35. 07 i 31.53 1 1.12 | . 62 | I TRI | 45.72 I 13.77| 311.14| 53. 66 i 26. 23 I . 84 | .62 | | GDH j 69.25 i 53.371 343.09| 31. 55 ! 18.52 I .64 | .79 J | WRA 1-19.95 I 134.33| 120.20| 72. 80 1 21.03 I . 66 | . 38 | | EKA i . . . . . . | 55.33 -3.17| j . , 322.10| 64. 60 .i-23.33 .71 | .63 | j T a b l e 3.2. Geom e t r i c a l values at the s t a t i o n s at 20km depth were computed using the ray parameter e q u a t i o n . For the Broach event, some s t a t i o n s w i t h i n 20° of the 59 e p i c e n t r e were used i n the nodal plane s o l u t i o n . In these cases the f i r s t a r r i v a l was assumed t o be Pn which has a t a k e o f f angle of 57° w i t h i n the lower l a y e r of H e r r i n ' s c r u s t a l model. The g e o m e t r i c a l spreading c o r r e c t i o n G - ( A , h ) was c a l c u l a t e d using Banghar's t a b l e s . From Ben Menahem et a l (1965) G " i s given by &(A,h) -C \ cos i 0 sin A d A / and f o r shallow events, G ( A ) • \ f fan L r . [ Sin A" dA The s u b s c r i p t h r e f e r s to the f o c a l depth while the s u b s c r i p t o r e f e r s to the base of the mantle low v e l o c i t y l a y e r , which was taken to be 125km deep. The r a d i a t i o n p a t t e r n F^  ^9, y4J of Equation 2.18 i s given i n e x p l i c i t formulas by Ben Menahem et a l (1965). I t should be noted t h a t while the formulas i n t h a t paper are c o r r e c t , there are e r r o r s i n the examples given. 3.7. Nodal plane s o l u t i o n s The nodal plane s o l u t i o n of Stauder (1968), i l l u s t r a t e d i n F i g u r e 3.9, was used f o r the Rat I s l a n d earthquake. There are two poorly determined p u b l i s h e d s o l u t i o n s f o r the Broach earthguake (Arora et a l (1971), Gupta e t a l (1972)). The HWSS r e c o r d i n g s of t h i s event were read, and supplemented by the b u l l e t i n s of the USCGS, the ISC, and the USSR network. A somewhat b e t t e r determined s o l u t i o n (Figure 3.10) was o b t a i n e d , but a l a r g e degree of u n c e r t a i n t y remains. The p r e f e r r e d 6 0 •+• c o m p r e s s i o n p r e f e r r e d s o l u t i o n ta d i l a t a t i o n - a n o t h e r a d m i s s i b l e s o l u t i o n F i g u r e 3 . 1 0 . N o d a l p l a n e s o l u t i o n o f t h e B r o a c h e a r t h q u a k e 61 P f i r s t motion S p o l a r i s a t i o n A = d i l . o =comp. F i g u r e 3.9. Nodal plane s o l u t i o n of the Rat I s . earthquake ( a f t e r Stauder) s o l u t i o n was a t h r u s t f a u l t with a s t r i k e of 90°E, but a t h r u s t f a u l t s t r i k i n g a t any angle between 40°E and 90°E i s a c c e p t a b l e . 3.8. Computing s p e c t r a An i n t e r v a l of f o u r seconds of the seismogram, beginning at the onset of the event, was used i n the computation of the spectrum. The l e a s t squares l i n e a r trend was removed, and the data i n t e r v a l o f 100 p o i n t s was smoothly tapered to zero using a c o s i n e window. The time s e r i e s was then padded with zeros to a l e n g t h of 250 p o i n t s . The F o u r i e r t ransform was computed using the FOURT r o u t i n e of Norman Bremmer, fl.I.T. Department of Geophysics, which i s i n the p u b l i c l i b r a r y of the U n i v e r s i t y o f B r i t i s h Columbia Computing Centre. No smoothing was used i n the computation of the spectrum. The n o i s e spectrum was estimated from the fou r seconds of data immediately preceding the onset of the event. 62 CHAPTER 4 . INVERSION THEORY 4 . 1 . Formulating the problem Suppose that we wish to determine a model c o n s i s t i n g o f a set of IT) unknown parameters pj t j - 1 .... m from a s e t of H o b s e r v a t i o n s , i - 1 . . . . D , where, a c c o r d i n g to the model, each of the o b s e r v a t i o n s i s f u n c t i o n a l l y r e l a t e d t o the parameters i n a known way. That i s , the model p r o v i d e s c a l c u l a t e d values C; , <- * 1 .... n corres p o n d i n g to the observed values Ot- : c- = f- (p.) i = i . . . . n , j - t . . . . m I f the f u n c t i o n a l s are l i n e a r f u n c t i o n s i n pj , we may wr i t e the problem i n matrix form AA C = F P I f the f u n c t i o n s "f- ( p j ) are not s t r i c t l y l i n e a r but vary smoothly enough, they may be expanded i n a T a y l o r s e r i e s o about some s e t of i n i t i a l v alues of the parameters pj , so that 4 - 2 c:. = f t( PJ) * K Z\Pj .+ r I f we d e f i n e C ; s ' i ( Pj ) + a n < 3 i 9 " n o r e terms above the f i r s t order, we o b t a i n 63 4-3 A C - • Ap u - where Q ; j = X <^ Pi A c = A Ap * ^ Pj o We are i n t e r e s t e d i n f i n d i n g t h a t set of parameter c o r r e c t i o n s Apj which w i l l minimise the d i f f e r e n c e s between the observed values and those c a l c u l a t e d from our s t a r t i n g model, i . e . we wish to minimise A^C;., - 0K - . To do t h i s we have a set of n simultaneous e q u a t i o n s to s o l v e f o r m unknown parameter c o r r e c t i o n s Apj , using the known c a l c u l a t e d values f o r the s t a r t i n g model p° and t h e i r p a r t i a l d e r i v a t i v e s — — „ . d?i PJ We may f o r m a l l y write the s o l u t i o n t o our problem as 4-4 Ap 5 5 H A A p = HAC where the operator H w i l l be a good i n v e r s e i f i t s a t i s f i e s the f o l l o w i n g c r i t e r i a , as summarised by Jackson (1972): (a) 3 ~ A H ^ I N t h i s i s a measure of how w e l l the model f i t s the data, s i n c e i f A H - L . (b) t h i s i s a measure of the uniqueness of the s o l u t i o n , s i n c e there e x i s t s only one s o l u t i o n ^ i f H A I ^ (c) The u n c e r t a i n t i e s i n Ap are not too l a r g e . For s t a t i s t i c a l l y independent d a t a . 64 4-5 var(zfPj) = Z h.. va r (AC t ) In the equation «Ap = H A A p , Backus and G i l b e r t (1968) showed that we may reqard the matrix R - HA as a matrix whose rows are 'windows' or r e s o l v i n g k e r n e l s throuqh which the q e n e r a l s o l u t i o n i s viewed. The degree to which ^ approximates the i d e n t i t y matrix i s a measure of the r e s o l u t i o n of the parameters o b t a i n a b l e from the data. S i m i l a r l y i n the r e l a t i o n A c A H A C , Wiggins (1972) showed t h a t the matrix i s a matrix whose rows are windows through which the o b s e r v a t i o n s (expressed throuqh the c a l c u l a t e d value r e s i d u a l s &C ) are viewed. The degree to which 5 approximates the i d e n t i t y matrix i s a measure of the independence of the o b s e r v a t i o n s f o r the model chosen. 4.2. G e n e r a l i s e d e i g e n v e c t o r a n a l y s i s For our s o l u t i o n , we now seek the i n v e r s e matrix H of A . F i r s t we s t a t e how a g e n e r a l r e a l n « rn matrix can be f a c t o r e d i n t o two s e t s of e i g e n v e c t o r s which share a common s e t of e i g e n v a l u e s . T h i s i s the ' s h i f t e d ' e igenvalue problem of Lanczos (1961). I f t here are p independent equations among the s e t of simultaneous equations A A p - A C (which i s to say t h a t t h e r e are p deqrees of freedom i n the d a t a ) , then the matrix A h a s rank p and can be f a c t o r e d as: 65 A <te. dc, dp, dPj, dp, <?C. dPr U t l K p 1 "1 1 1 1 1 I I I _ 1 1 ) U , U f - U p 1 1 1 1 1 1 I 1 1 I I 1 A pxp A, o vT p* - — V, T r L A - • U L V U c o n t a i n s p e i g e n v e c t o r s U; o f l e n g t h n a s s o c i a t e d w i t h t h e c o l u m n s ( o b s e r v a t i o n s ) o f A . A i s a d i a g o n a l m a t r i x o f p e i g e n v a l u e s ( d e c r e a s i n g i n s i z e down t h e d i a g o n a l b y c o n v e n t i o n ) , a n d V c o n t a i n s p e i g e n v e c t o r s Vj o f l e n g t h rn a s s o c i a t e d w i t h t h e r o w s ( p a r a m e t e r s ) o f A A f t e r U a n d V a r e f o r m e d f r o m t h e e i g e n v e c t o r s c o r r e s p o n d i n g t o t h e p n o n - z e r o e i g e n v a l u e s , t h e r e r e m a i n n-p e i g e n v e c t o r s U i and m~p e i g e n v e c t o r s Vj c o r r e s p o n d i n g t o z e r o e i g e n v a l u e s . We may a s s e m b l e t h e s e i n t o t h e c o l u m n s o f m a t r i c e s U e (an x ( H - p ) m a t r i x ) and V0 (an r r v x ( rn - p ) m a t r i x ) . T h e m o d e l i s n o t s e n s i t i v e t o t h e s e two s e t s o f e i g e n v e c t o r s . L e t us e x a m i n e t h e f a c t o r i s a t i o n o f /A b y e i g e n v a l u e s T. a n d e i g e n v e c t o r s . I n t h e m a t r i x e q u a t i o n A - U A V e a c h row o f A i s a w e i g h t e d sum o f t h e e i g e n v e c t o r s V- , J ' *....p t h e w e i g h t s b e i n g g i v e n by U A . T h e s e e i g e n v e c t o r s r e p r e s e n t a s e t o f p s p e c i f i c l i n e a r c o m b i n a t i o n s o f t h e o r i g i n a l m p a r a m e t e r s t h a t a r e f i x e d by t h e 66 o b s e r v a t i o n s , and may be regarded as a new p a r a m e t e r i s a t i o n of the model. The e i g e n v e c t o r c o r r e s p o n d i n g to the l a r g e s t e i g e n v a l u e i s the v e c t o r that i s most n e a r l y p a r a l l e l to the rows (parameters) of A The next e i g e n v e c t o r i s most n e a r l y p a r a l l e l to the remainder, and so on. He may s i m i l a r l y regard the p e i g e n v e c t o r s Uj, as a set of p l i n e a r combinations of the n o b s e r v a t i o n s t h a t are r e l e v a n t to the i n v e r s e problem. 4.3. O v e r c o n s t r a i n e d and underdetermined systems F o l l o w i n g Jackson (1972) we s h a l l study the r e l a t i o n s h i p of the U 0 and V0 v e c t o r s to the c o n d i t i o n of the i n v e r s e system. Si n c e the rn v e c t o r s V ; , j * i....rn form a complete s e t , we may express the parameter c o r r e c t i o n v e c t o r A p of l e n g t h m as a sum of the Vj , „ P w v— V"* 4.7 <Ap - I«jVj - 2>jYi- * L+jVj = V<*<-V«(e where the v e c t o r s <A and ©< 0 have p and r i - p components r e s p e c t i v e l y . By the o r t h o n o r m a l i t y of the V- e i g e n v e c t o r s we have V V = Ip V V0 - 0 p x (m-p) V. V. * L-p V„ V ' O (m-p) x p Using these r e l a t i o n s h i p s we can r e p r e s e n t oC and d0 by 67 4-9 so we may express A p i n the form 4.10 A p - V V A p * V„ V 0 A p S i m i l a r l y , 411 AC = U UAC - U 0 U.Ac S u b s t i t u t i n g these e x p r e s s i o n s i n t o the d e f i n i t i o n of the r e s i d u a l v e c t o r £ s A A p - A C , we may write the square e r r o r as e e A VAp - U A C * U 0 A C The l e a s t squares s o l u t i o n i s t h a t which minimises \ l £ | and r e q u i r e s that 4..3 V A P - A ' U A C T Z l e a v i n g the l e a s t square e r r o r (J EAC T e x i s t an exact s o l u t i o n only i f U„AC " O t h i s c o n d i t i o n must h o l d , f o r then U 6 = 0 p < n , there may be an exact s o l u t i o n only i f the data c o n t a i n no c o n t r i b u t i o n from the e i g e n v e c t o r s i n U 0 ; t h i s l e a d s to the n-p c o n s t r a i n t s A C = O , i - p + 1 n . In t h i s s i t u a t i o n , the number of o b s e r v a t i o n s exceeds the number of degrees of freedom t h a t the o b s e r v a t i o n s possess, and the system i s o v e r c o n s t r a i n e d . There w i l l For p = H , For 68 The V Q v e c t o r s do not appear i n Equation 4 . 1 2 and thus may c o n t r i b u t e a r b i t r a r i l y to the s o l u t i o n . ' The l e a s t squares s o l u t i o n w i l l be unique only i f p e r n . When p < rr\ f there are more model parameters than degrees of freedom i n the data, and the system i s underdetermined. I f p<t*i and p<n the system i s both o v e r c o n s t r a i n e d and underdetermined, and an exact s o l u t i o n may not e x i s t . However there w i l l e x i s t an i n f i n i t e number of s o l u t i o n s s a t i s f y i n g the l e a s t squares c r i t e r i o n . 4 . 4 . The g e n e r a l i n v e r s e Lancos ( 1 9 6 1 ) i n t r o d u c e d the ' n a t u r a l ' i n v e r s e T h i s i n v e r s e , which always e x i s t s , i s the unique g e n e r a l i s e d i n v e r s e which s a t i s f i e s the Moore-Penrose c r i t e r i a . L e t us examine the p r o p e r t i e s of t h i s i n v e r s e , bearing i n mind the c r i t e r i a f o r a good i n v e r s e which we e s t a b l i s h e d i n S e c t i o n 4 . 1 . These p r o p e r t i e s have been e l u c i d a t e d by Jackson ( 1 9 7 2 ) . The model o b t a i n e d by using t h i s i n v e r s e w i l l be - i T 4-/5' Apu - H L A C - V A U A C we o b t a i n T By comparing t h i s r e s u l t with Equation 4 . 1 3 i t i s ev i d e n t that the n a t u r a l i n v e r s e i s a ' l e a s t squares' i n v e r s e , and i s 69 thus an exact s o l u t i o n , i f any e x i s t s . Since i t i s a l e a s t squares i n v e r s e , the n a t u r a l i n v e r s e f o r a p u r e l y o v e r c o n s t r a i n e d system w i l l be i d e n t i c a l to the i n v e r s e provided by the standard l e a s t squares procedure. P r e m u l t i p l y i n q by V we o b t a i n 4.17 T h i s i n d i c a t e s t h a t our l e a s t squares s o l u t i o n i s the one |2 which minimises A p , s i n c e from Equation 4 . 1 0 , a l e a s t squares s o l u t i o n occurs when 4 - . I8 A P 2 - A 'Ac * YAP i n which the f i r s t term i s f i x e d , and the second term i s r minimised when V A p * 0. The property of minimising JAp| i s a u s e f u l one i f A p r e p r e s e n t s a p e r t u r b a t i o n to some s t a r t i n g model i n a problem that i s n o n - l i n e a r to some extent. The r e s o l u t i o n matrix f o r the n a t u r a l i n v e r s e i s g i v e n by 4.« R - HA - VAU-UAV • VV T h i s i s the optimum r e s o l v i n g matrix i n the sense t h a t i t minimises 4-20 r \ » T. ( Q £ J - SKi ) f o r eacn value of k , which i s to say that each row of IR i s the best f i t t o the corres p o n d i n g row of the i d e n t i t y matrix, i n the l e a s t squares sense, which may be formed from the rows of A-70 S i m i l a r l y the n a t u r a l i n v e r s e provides the best i n f o r m a t i o n d e n s i t y matrix w S * AH L - Ulf i n the sense t h a t i t minimises These r e s u l t s show that the n a t u r a l i n v e r s e i s endowed with many d e s i r a b l e p r o p e r t i e s . Besides p r o v i d i n g optimum r e s o l v i n g and i n f o r m a t i o n d e n s i t y matrices as p r e s c r i b e d i n S e c t i o n 4 . 1 , i t i s a l s o a l e a s t squares, i n v e r s e which minimises the change to the model. He s h a l l examine the f i n a l c r i t e r i o n of model v a r i a n c e i n S e c t i o n 4.7. 4.5. The p a r t i c u l a r s o l u t i o n We now f o r m a l l y present the p a r t i c u l a r s o l u t i o n to the problem. S t a r t i n g with the l i n e a r equation 423 A A p - A C we s u b s t i t u t e the decomposition of to o b t a i n UAVz\P - Ac From t h i s we may o b t a i n the s o l u t i o n f o r A p i n the form 71 4-25- A p VAUAC P - I vtUulAc} T h i s i s p r e c i s e l y the s o l u t i o n that we encountered before when studyinq the p r o p e r t i e s of the n a t u r a l i n v e r s e of a matrix. The s o l u t i o n f o r the parameter c o r r e c t i o n s i s c o n s t i t u t e d i n the f o l l o w i n g way. The s c a l a r q u a n t i t y i n s i d e the b r a c k e t s i s a p a r t i c u l a r combination of the o b s e r v a t i o n s as s e l e c t e d by the e i g e n v e c t o r U T , and serves as a weightinq of the i th V e i g e n v e c t o r ' s c o n t r i b u t i o n to the parameter c o r r e c t i o n s . That v e c t o r ' s c o n t r i b u t i o n i s f u r t h e r s c a l e d i n i n v e r s e p r o p o r t i o n to the magnitude of i t s corresponding e i g e n v a l u e . I t w i l l be r e c a l l e d from S e c t i o n 4.2 that the V e i g e n v e c t o r s are a s e t of p l i n e a r combinations of the model parameters t h a t are determined by the o b s e r v a t i o n s . A g e n e r a l s o l u t i o n to the problem c o n s i s t s of t h i s p a r t i c u l a r s o l u t i o n augmented by a r b i t r a r y c o n t r i b u t i o n s from 4.6. Weighting the parameters So f a r , we have assumed that the data are d i m e n s i o n l e s s and s t a t i s t i c a l l y independent. I f the g e n e r a l i s e d i n v e r s e i s found d i r e c t l y without weighting the equations, the r e s u l t w i l l be s t r o n g l y dependent on the d i m e n s i o n a l i t y of the components of the v e c t o r s /\p and A C . We now examine m o d i f i c a t i o n s to the i n v e r s e t h a t encompass t h i s s i t u a t i o n , f o l l o w i n g JacJcson (1972) and Wiggins (1 972) . the v e c t o r s . 72 F i r s t c o n s i d e r an o v e r c o n s t r a i n e d system f o r which the data are s t a t i s t i c a l l y independent but have d i f f e r e n t u n i t s . We might then g e n e r a l i s e the l e a s t squares e r r o r c r i t e r i o n so t h a t i t minimises 4.26 £T D e where D i s a d i a g o n a l matrix whose elements are 4-27 cf- = 1 var ( A C i ) T h i s enables us to compare the r e s i d u a l f o r each data p o i n t with i t s expected e r r o r . When the data are not s t a t i s t i c a l l y independent, i t i s a p p r o p r i a t e to use the i n v e r s e of the c o v a r i a n c e matrix of the data f o r D as e x p l a i n e d by Kaula (1966) . The symmetric, p o s i t i v e d e f i n i t e matrix D may be decomposed i n t o i t s e i g e n v e c t o r s and e i g e n v a l u e s (which w i l l be p o s i t i v e ) : D - VAV 0 D D 4.-28 so we can always f i n d E - AIV 1 such t h a t 4.30 ETE • D D O I f we now m u l t i p l y our system of e q u a t i o n s ~ A C by E , we have 73 431 E £ = E A A p - E A c E ' - A'zfp - Ac which i s a new weighted system of equations in.which £ * E £ A • EA A c * E Ac We can now proceed to find A p which minimises 4.33 £'T£' - sT D e The weighting matrix E by which we multiply our system of equations puts the data in dimensionless, s t a t i s t i c a l l y independent form. We may s i m i l a r l y analyse the underdetermined problem, where we may wish to find that solution which miminises 4.34 A p T F A p The symmetric p o s i t i v e d e f i n i t e matrix F* w i l l be the inverse covariance matrix of the model parameters: 74 4 3 S - ATAl" IS For example the k th diagonal element of A A given by (~jp^) ' w n i c h i s an average of the squares of the changes in a l l the computed values produced by a v a r i a t i o n in the k th parameter. By assigning weights in t h i s way we are seeking parameter changes that w i l l change the calculated values by the same amount for each of the d i f f e r e n t parameters. The d i f f e r e n t s e n s i t i v i t i e s of the model's calculated values to i t s parameters are thus counteracted. As before, we modify the system of equations so that A f > = &Ap and A = A G , where (j i s chosen so that G G ~ F . He then proceed to find A p which minimises |Ap | and obtain from i t the desired solution -I A p = GAP We may combine these two transformations to find that Ap which simultaneously minimises and A p F A P . Let A • E A G and AC = EAC so that our system of equations become 4.36 AC - A A p Finding the solution 4-37 A p » HLAC where H L i s the natural inverse of A , we f i n a l l y obtain 75 4-38 « G'A^P • & 1H L A c Thesa t r a n s f o r m a t i o n s a f f e c t the r e s o l u t i o n and i n f o r m a t i o n d e n s i t y m atrices as f o l l o w s : R - G' R G s • E'S'E 4.7. Choosing between v a r i a n c e and r e s o l u t i o n We w i l l now examine the s e l e c t i o n of the number of model parameters t h a t i t i s a p p r o p r i a t e to use i n the i n v e r s i o n when t h e i r v a r i a n c e i s taken i n t o account. I t has j u s t been shown th a t we may apply t r a n s f o r m a t i o n s to render the data s t a t i s t i c a l l y independent and u n i v a r i a n t . The v a r i a n c e from Eguation 4.5. i s then 4.4.0 var A p k = V /Vfc,-| C l e a r l y , s m a l l e i g e n v a l u e s may make the v a r i a n c e unacceptably l a r g e , so we may wish to d i s c a r d those e i g e n v a l u e s beyond the <^th one which r a i s e the v a r i a n c e above some 76 t h r e s h o l d of a c c e p t a b l e values , as suggested by Wiggins (1972), 4.4. t ("T)2 < h for «H k j ' l 1 The e f f e c t of r e d u c i n g 0^  i s to reduce the number of e i g e n v e c t o r s belonging to U and V and c o r r e s p o n d i n g l y i n c r e a s e those i n l_J0 and V . T h i s l i m i t a t i o n upon the variance of the parameter c o r r e c t i o n s degrades the r e s o l u t i o n and i n f o r m a t i o n d e n s i t y . The i n t e g e r <^  i s the e f f e c t i v e number of degrees of freedom i n the data, and depends on the u n c e r t a i n t i e s i n the data as w e l l as on our need f o r c e r t a i n t y i n determining the model. The c h o i c e of the value of 0^  has an important e f f e c t on how u s e f u l our i n t e r p r e t a t i o n from the model w i l l be. For a b a s i c a l l y underdetermined system, we may be l i m i t e d by a l a c k of r e s o l u t i o n or by a l a r g e v a r i a n c e i n A p i f c» i s p o o r l y chosen. In a b a s i c a l l y o v e r c o n s t r a i n e d system, the model may be s u b j e c t to u n n e c e s s a r i l y l a r g e variance i n an e f f o r t to s a t i s f y poorly determined f e a t u r e s of the data unless cj i s a p p r o p r i a t e l y chosen. 4.8. Extremal i n v e r s i o n Once a model i s found that f i t s the data r e a s o n a b l y w e l l , then our i n t e r e s t t u r n s to f i n d i n g the l i m i t s t h a t the values of the v a r i o u s model parameters can assume while s t i l l remaining c o n s i s t e n t with the o b s e r v a t i o n s . T h i s procedure has been a p p r o p r i a t e l y termed 'extremal i n v e r s i o n ' by Wiggins, McMechan 77 and Toksoz (1973). The f o l l o w i n g e x p o s i t i o n of the method i s based on Jackson (1975). Let us assume t h a t the experimental e r r o r s i n the ob s e r v a t i o n s are Gaussian i n d i s t r i b u t i o n and have zero mean, and that a s u i t a b l e t r a n s f o r m a t i o n as p r e v i o u s l y d e s c r i b e d has been a p p l i e d to render the o b s e r v a t i o n s s t a t i s t i c a l l y independent and u n i v a r i a n t . Then, f o r a given set of parameter c o r r e c t i o n values A p , the l i k e l i h o o d that the r e s i d u a l s are due e n t i r e l y t o random e r r o r s i n the o b s e r v a t i o n s i s r e l a t e d to the sum of the squares of the r e s i d u a l s Q - £t which may be expressed i n the q u a d r a t i c form 443 Q = ApVAVAp - MpVAUAC - AC AC By minimising t h i s q u a d r a t i c form, we o b t a i n the l e a s t squares s o l u t i o n 4-44-f o r which the sum of squares of r e s i d u a l s i s denoted by Q.uS • The u n c e r t a i n t y i n the parameter c o r r e c t i o n e s t i m a t e s A p L S i s d e r i v e d from u n c e r t a i n t i e s i n the o b s e r v a t i o n s , and has mean 4.45- <Ap t s> = V A l / <AC> and c o v a r i a n c e 78 -2 T 4 t 6 C - V A V The square r o o t s of the d i a g o n a l elements of t h i s matrix are r e f e r r e d to as standar d d e v i a t i o n s of the l e a s t square estimates and are sometimes used as es t i m a t e s o f the ranqe of values which the parameter c o r r e c t i o n s might assume. An a l t e r n a t i v e approach i s to f i n d the extremal values which a p a r t i c u l a r parameter or l i n e a r combination of parameters may take on s u b j e c t t o the c o n s t r a i n t t h a t the r e s i d u a l s are l e s s than or equal t o some t h r e s h o l d Q Q . More g e n e r a l l y , we could extremise a l i n e a r combination of the parameters Z T Ap ' b . As long as V A V i s p o s i t i v e d e f i n i t e the d e s i r e d extremum w i l l always occur when the c o n s t r a i n t i s e x a c t l y s a t i s f i e d , t h a t i s when Q • Q. . S i n c e the r e s u l t i n g s o l u t i o n thus has the maximum t o l e r a b l e sum of r e s i d u a l s , Jackson (1975) has suggested the t i t l e 'most squares'. Usinq the v a r i a t i o n a l method, we minimise U 7 Ap"b * ^ [ApVAVAp " 2Ap T VAUAC •ACAC-Q, with r e s p e c t t o A p , where 1 /' Z/^ i s a Laqrangian m u l t i p l i e r . The r e s u l t i s 79 -2 T 4 . 4 . * w h e r e A p x = V A U A C - ^ V A V b = 7 V.U u T A C - f t v b 4.49 ^ = -/Q. - Q.. = 1 / Q 0 - Q u b T VAVb >/ v a r ( A p b ) I f t h e d e s i r e d i s s m a l l e r t h a n t h e l e a s t s q u a r e s v a l u e Q LS t h e r e w i l l be no r e a l s o l u t i o n t o E q u a t i o n 4 . 4 8 . When Q 0 > Q L S , t h e r e w i l l be two s o l u t i o n s f o r y^, c o r r e s p o n d i n g t o t h e maximum a n d minimum v a l u e s o f A p ' b T h e l e a s t s q u a r e s s o l u t i o n o c c u r s when LS § t h a t i s , when / U ' O , l e a v i n g o n l y t h e f i r s t t e r m i n E q u a t i o n 4 . 4 8 , w h i c h t h e n b e c o m e s i d e n t i c a l t o E q u a t i o n 4 . 2 5 . T h e s e c o n d t e r m o f E q u a t i o n 4 . 4 8 c o n t a i n s a n a d d i t i o n a l c o n t r i b u t i o n o f t h e V* v e c t o r whose s i z e i s p r o p o r t i o n a l t o f/A.^, w h i l e t h e l e a s t s q u a r e s c o n t r i b u t i o n i s p r o p o r t i o n a l t o 1/Aj . L a r g e c h a n g e s w i l l t h u s o c c u r i n p a r a m e t e r s a s s o c i a t e d w i t h s m a l l e i g e n v a l u e s when e x t r e m a l s o l u t i o n s a r e s o u g h t . T h e s i z e o f t h e s e c o n d t e r m i s a l s o d e t e r m i n e d by yu. , w h i c h e x p r e s s e s t h e d e g r e e o f d e v i a t i o n f r o m t h e l e a s t s q u a r e s s o l u t i o n t h a t w i l l be t o l e r a t e d . The t h i r d i n f l u e n c e o n t h e s i z e o f t h i s t e r m c o n s i s t s o f w e i g h t i n g s , p r e s c r i b e d by t h e Vt e i g e n v e c t o r s , o f t h e p a r a m e t e r c o m b i n a t i o n b t h a t i s c h o s e n t o be e x t r e m i s e d . S u p p o s e t h a t we c h o o s e Q „ = n , t h a t i s , we w i s h t o t o l e r a t e 80 one standard d e v i a t i o n of e r r o r i n each of the measurements (whose e r r o r s are assumed u n i v a r i a n t ) . Then Jackson has shown t h a t , i f the data are c o r r u p t e d only by measurement e r r o r (and i f the f i t i s unhindered by inadequacies i n the model) then Q u s i s a c h i - s q u a r e d v a r i a b l e whose expected value i s n - r n , and the most squares l i m i t of A p b w i l l be g r e a t e r than the s t a n d a r d d e v i a t i o n by the f a c t o r J rn . We may g e n e r a l l y expect to f i n d t h a t GILS i s s i g n i f i c a n t l y d i f f e r e n t from i t s expected value, because of l i m i t a t i o n s i n the f o r m u l a t i o n of the model, or erroneous e s t i m a t e s of measurement e r r o r s . Where measurement e r r o r s are known a c c u r a t e l y , we may use Q u s as a t e s t of how w e l l the model has been formulated. I f Q u s <K o-m , the l e a s t squares model f i t s b e t t e r than i t should, i n d i c a t i n g t h a t the data do not c o n s t r a i n the model parameters very s t r o n g l y . I f a LS » n-n , the model i s i n c o n s i s t e n t with the data, i n d i c a t i n g t h a t t h e data are g r o s s l y i n e r r o r or t h a t the f o r m u l a t i o n of the model does not provide an adeguate d e s c r i p t i o n of the o b s e r v a t i o n s . 4.9. Handling a n o n - l i n e a r i n v e r s i o n problem We have a l r e a d y i n d i c a t e d how a moderately n o n - l i n e a r problem can be l i n e a r i s e d using a T a y l o r s e r i e s expansion. But i because the system i s s t i l l e s s e n t i a l l y n o n - l i n e a r , we cannot r e l y on our i n v e r s i o n scheme f i n d i n g us a s o l u t i o n modal from an a r b i t r a r y s t a r t i n g model. Instead, we must seek a s t a r t i n g model t h a t i s i n reasonable agreement with the o b s e r v a t i o n s . In i t e r a t i n g from the s t a r t i n g model, we must be watchful of any s i g n s of g r o s s l y n o n - l i n e a r behaviour . We are aided i n 81 t h i s r e s p e c t by the property that the l e a s t squares s o l u t i o n possesses of minimisinq parameter chanqes. N e v e r t h e l e s s , l i m i t a t i o n s upon the t o l e r a b l e s i z e of the parameter changes may have to be imposed. Wiggins (1972) has suggested that these l i m i t a t i o n s may be imposed by a c c e p t i n g c o n t r i b u t i o n s to the parameter c o r r e c t i o n v e c t o r A p from the V e i g e n v e c t o r s i n d e c r e a s i n g order of t h e i r a s s o c i a t e d e i g e n v a l u e s u n t i l the maximum permitted parameter c o r r e c t i o n f o r a parameter (the j t h one) has been reached. A f r a c t i o n a l c o n t r i b u t i o n CX from the l a s t e i g e n v e c t o r accepted ( the kth one) a l l o w s the exact attainment of the d e s i r e d parameter c o r r e c t i o n , as shown i n Equation 4.50 fe-i 4-so L%i. u*Ac * a-V(J. xk u;Ac - A p j ) ( T h i s technique a l l o w s the parameter c o r r e c t i o n to reach i t s d e s i r e d l i m i t with the l e a s t r e l i a n c e on the \/ e i g e n v e c t o r s a s s o c i a t e d with the most poorly determined parameters. P a r t i c u l a r care must be taken when extremal s o l u t i o n s are being sought, because under these c o n d i t i o n s very l a r g e parameter changes may occur. Such changes must be checked by comparing t h e i r r e s u l t i n g c a l c u l a t e d values with the o b s e r v a t i o n s . 82 CHAPTER 5. INVERSION PRACTICE 5.1. Representing the seismogram by measurements The seismogram was modelled by f i v e measurements as shown i n F i g u r e 5.1 and Table 5.1. The f i r s t t h r e e measurements are F i g u r e 5.1. Seismogram measurements Cpt. No. D e s c r i p t i o n of Measurement' Value i n F i g . 5.1 1 2 3 4 5 Time between onset and f i r s t peak: Time between f i r s t peak and f i r s t trough Time between f i r s t trough and second peak Ratio of h e i g h t s of 1st peak and 1st trough Ratio of h e i g h t s of 2nd peak and 1st trough a b c d/e f/e Table 5.1. Seismogram measurements 83 time i n t e r v a l s i n seconds from onset to f i r s t peak, from f i r s t peak to f i r s t trough, and from f i r s t trough to second peak. The f i n a l two are the r e l a t i v e h e i g h t s of the two peaks compared to the i n t e r v e n i n g trough. 5.2. A s s i g n i n g standard d e v i a t i o n s to the o b s e r v a t i o n s The l a r g e s t source of u n c e r t a i n t y i n the o b s e r v a t i o n s l i e s i n the d e c o n v o l u t i o n of the response of the r e c e i v e r c r u s t from the seismograms. The i n t r i n s i c u n c e r t a i n t y i n the a p p l i c a t i o n of t h i s method has already been d i s c u s s e d . The dominant source of u n c e r t a i n t y , however, i s most l i k e l y i n c u r r e d i n the p r a c t i c a l implementation of the method, e s p e c i a l l y i n the c h o i c e of the c u t o f f of the c e p s t r a l f i l t e r . U n c e r t a i n t i e s were estimated from the envelope of the decon v o l u t i o n s obtained using a broad range of c u t o f f v a l u e s . T h i s envelope was a r b i t r a r i l y a s s i g n e d a 95% c o n f i d e n c e l e v e l , which i s 1.96 times the standard d e v i a t i o n . Estimated minimum e r r o r s based upon the u n c e r t a i n t y i n v o l v e d i n d i g i t i s a t i o n and upon the v a r i a t i o n of component channels of the a r r a y data were s u b s t i t u t e d i f these exceeded the envelope width. The standard d e v i a t i o n s , shown together with the observed v a l u e s i n Table 5.2, were then assigned by h a l v i n g the d e v i a t i o n of the envelope from i t s mean. The values f o r the array measurements average between one-half and o n e - f i f t h those f o r the WWSS s t a t i o n s . For some WWSS s t a t i o n s , measurements such as th a t of the second peak height were assigned such l a r g e standard d e v i a t i o n s as to impose almost no c o n s t r a i n t whatever. 84 RAT ISLAND EARTHQUAKE BROACH EARTHQUAKE T T T 1 I T T T — |Station|Measure Value |St.Dev.| |Station|Measure] Value I St.Dev. i f— -+ -j i _ i t 1 l + ~j 1 1 1 .270 | .025 | | | 1 | . 344 I . 025 ( 1 2 I . 475 | .025 | j | 2 ! .591 | . 025 I AKU | 3 .553 I .025 | I COL | 3 I . 596 | . 050 1 1 4 .520 | .050 | | | 4 I . 783 | . 100 1 I 5 .531 I -075 | 5 I .647 I . 100 1 | 1 i .296 I .025 | | | 1 I . 283 I . 025 1 1 2 I .554 | .025 | | | 2 I .589 I . 025 I BEC | 3 I . 564 I -025 | | MAT | 3 I . 660 | . 050 1 1 4 I .785 I .025 | j j 4 I . 497 I .050 1 I 5 I .659 | .100 | | | 5 I .498 | . 100 1 1 1 | .34 4 | .025 | | | 1 | . 282 j . 025 1 I 2 I .435 I -025 | | | 2 1 .514 | . 025 1 FLO | 3 I .502 | .050 | | ADE | 3 1 .530 J . 050 I I I .353 | .050 | | | 4 1 .705 I . 025 1 1 5 I .549 I .150 | | | 5 1 .751 I . 050 1 I 1 | . 362 | .025 | 1 1 1 | . 295 I . 025 1 I 2 1 .4 56 I .025 | | | 2 I .570 | .025 I DUG | 3 1 .522 | .050 | | WIN | 3 I . 571 | . 050 1 1 4 I .273 I .025 | | | 4 I .722 | . 050 1 I 5 1 .400 I .200 | 5 I .722 | . 075 1 I 1 | .498 | .025 | | | 1 I . 295 I . 025 1 I 2 I .415 | .025 | | | 2 I .510 I . 025 I CT A | 3 I .425 | .025 | | NAI | 3 I . 567 | . 025 1 | 4 I .525 | .050 | | | 4 I .414 | . 025 1 | 5 I .767 | .050 | 5 I .616 | . 100 1 | 1 | .363 | .025 | | | 1 | .375 | . 025 1 I 2 I .466 | .025 | | | 2 I . 500 | . 025 | SNG | 3 I .471 I .025 | I EIL | 3 I .521 | . 050 1 | <* I .658 | .050 | | | 4 I .736 | . 050 1 | 5 I .770 | .050 | | | 5 I .676 | . 075 1 I 1 I .265 I .025 | j | 1 | . 267 I . 025 1 I 2 I .500 | .025 | | | 2 I . 599 | . 025 I SHI | 3 I .503 | .050 | I TRI | 3 I .706 I . 025 1 | 4 I .635 | .050 | | | 4 I .419 | .025 1 I 5 I .830 | .100 | | j 5 I . 300 I .050 1 | 1 | .381 | .050 | | | 1 | .316 | . 025 1 I 2 I .490 | .050 | ) | 2 I .524 I .025 | STU | 3 I .625 | .100 | I GDH | 3 I .582 | . 025 1 | I .772 | .050 | | | 1 1 . 332 | . 025 1 | 5 1 .76 1 | .100 | 1 1 5 1 . 394 | . 025 1 | •1 I .193 I .010 | | | 1 I . 443 | .025 ! I 2 I .398 | .010 | | | 2 I .413 | . 025 | EKA | 3 I .312 I .050 | | WRA | 3 I . 431 I . 050 1 | 4 I .517 | .025 | | | 1 I .477 | .025 I I 5 I .38 9 | .050 | | | 5 i .858 | . 100 I | 1 | .259 | .010 | f | 1 | . 162 | . 025 1 I 2 I .340 I .025 | { | 2 I .468 J . 025 | WRA | 3 I .433 | .025 | | EKA | 3 I . 520 | . 025 1 I 4 I .666 | .025 | | | 4 I .367 | .050 I 1 5 I .906 | .050 | 5 I .318 | . 050 J L J L 1 1 1 j Table 5.2. Observed values and t h e i r standard d e v i a t i o n s 85 5.3. F i n d i n g a s t a r t i n g model As we have seen, the parameters of Savage's model l e n d themselves d i r e c t l y t o a convenient p a r a m e t e r i s a t i o n f o r the purposes of i n v e r s i o n . The seven parameters of the model are l i s t e d i n Table 5.3. 1 - T — — 1 | Parameter j D e s c r i p t i o n j 1 E 1 E l l i p t i c i t y of the f a u l t s u r f a c e j 1 V | Rupture v e l o c i t y (km/sec) j 1 A 1 Major semi a x i s of e l l i p s e (km) j f RTIM | Duration of ramp source time f u n c t i o n (sec) | | ELAZ | O r i e n t a t i o n of the major a x i s of the e l l i p s e | on the f a u l t plane (radians) | 1 DIP | Dip angle of f a u l t plane (radians) j | DIPAZ | L ,. ,JL.. Azimuthal d i r e c t i o n of dip (radians) j Table 5.3. F a u l t model parameters A f t e r some ex p e r i e n c e at c a l c u l a t i n g seismograms, i t was p o s s i b l e to a n t i c i p a t e the appearance o f seismograms t h a t would r e s u l t from a proposed model, by c o n s i d e r i n g the z e n i t h and azimuth angles of the d e p a r t i n g rays measured with r e s p e c t to the e l l i p t i c a l f a u l t s u r f a c e . For example, displacement s t o p s g r a d u a l l y on a ray t h a t t r a v e l s away from the f u r t h e s t boundary of the f a u l t , but stop s r a p i d l y on a ray t h a t t r a v e l s toward the f u r t h e s t boundary. The r a t e of s t o p p i n g has a s t r o n g i n f l u e n c e on the p o s i t i o n and h e i g h t of the seismogram's second peak. By 86 t h i s method, s t a r t i n g models f o r both earthquakes were found i n which rupture propagated f u r t h e s t i n the updip d i r e c t i o n on the n o r t h e r l y d i p p i n g nodal plane. For both events, seven f u r t h e r s t a r t i n g models were t e s t e d , such t h a t f u r t h e s t r u p t u r e i n the updip, downdip, and the two ( h o r i z o n t a l ) s t r i k e d i r e c t i o n s on each of the nodal planes was i n v e s t i g a t e d . I t was found t h a t the p r e v i o u s l y chosen s t a r t i n g models were the only ones which provided a reasonably good f i t . An exception o c c u r r e d with the Rat I s . event, f o r which a rupture propagating f u r t h e s t h o r i z o n t a l l y along the 140° s t r i k e of the no r t h e a s t d i p p i n g plane was a moderately good f i t . During extremal i n v e r s i o n , t h i s model was found to be c l o s e to the a d m i s s i b l e s o l u t i o n s . I t e r a t i o n s from the s t a r t i n g models were f i r s t made using a l l seven model parameters, but i n c l u s i o n of the nodal plane s o l u t i o n provided too much freedom f o r the d e t e r m i n a t i o n of a s o l u t i o n . S p e c i f i c a l l y , the d i p c o u l d take on v e r t i c a l v a l u e s i n both cases. The nodal plane parameters were t h e r e f o r e f i x e d s i n c e they-had been estimated from independent o b s e r v a t i o n s . I d e a l l y , s t e p s i z e c o n t r o l c o u l d be used t o keep these parameters wit h i n the boundaries determined by the maximum l i k e l i h o o d f o c a l plane s o l u t i o n method of D i l l i n g e r , Harding and Pope (1972). New s t a r t i n g models employing the remaining f i v e parameters were chosen using e x p e r i e n c e gained from the seven parameter model i n v e s t i g a t i o n . 87 5.4. Computing the p a r t i a l d e r i v a t i v e s P a r t i a l d e r i v a t i v e s were computed by v a r y i n g each parameter i n turn by one tenth of i t s value, o r , i n the case of a n g l e s , by one tenth of i t s average value. F u l l computation of the seismogram from the perturbed model was performed, and the d e v i a t i o n of i t s f i v e component measurements from t h e i r i n i t i a l values was d i v i d e d by the parameter increment. A method was sought whereby seismogram components could be c a l c u l a t e d from a few parameters such as the two s t o p p i n g times together with the time during s t o p p i n g at which the displacement has decreased to h a l f i t s peak value. However, no simple l i n e a r r e l a t i o n s h i p between these three parameters and the f i v e seismogram measurements could be found. The t i n e taken f o r the computation of a s e t of p a r t i a l d e r i v a t i v e s using the f u l l c a l c u l a t i o n s was about one hundred seconds on the U n i v e r s i t y of B r i t i s h Columbia's IBM 370/168 computer. 5.5. Least squares i n v e r s i o n L e a s t squares i t e r a t i o n s from the s t a r t i n g model ware performed i n order to seek b e t t e r f i t t i n g models. However, the s t a r t i n g models f o r both events were w i t n i n the a c c e p t a b l e range of s o l u t i o n s , and consequently no l a r g e changes o c c u r r e d . The i t e r a t i o n s are t a b u l a t e d i n Table 5.4, and the seismograms computed from the models are shown i n F i g u r e 5.2. V i s u a l e v a l u a t i o n of the improvement produced i n an i t e r a t i o n was u s u a l l y d i f f i c u l t , because the changes were 88 |It.|Frora| D e s c r i p t i o n | I I I , I I I I E i J j 1 I S t a r t i n g Model l e a s t Squares I n v e r s i o n I T T 1 I o I I I I 1 I 0 | 1st i t e r a t i o n I I I | 2 | 1 | 2nd i t e r a t i o n L J I Parameter Values -j • 1  I -+-, n I RTIM | ELAZ | 4 j 1 .3 | 3.0 I .319 | 3.287 .250 | 3.351 1.8 | .45 | 3.141 I I 1.841 | .445 | 2.014 I I 1.965 | .447 | 2.400 Fin d i n g extremal values of E and ELAZ (EZ = ELAZ) I 2a | 1 max E & min EZ .323 3.463 I 2b | 1 min E & max EZ . 178 3. 239 I 3a| 2a max E . 325 ! 3.637 I 3b | 2a min EZ .317 3. 597 I 3c| 2b min E S max EZ .215 | 3.292 I 4a| 3c max EZ . 285 | 2. 837 2.013 1.917 2.121 2. 116 1.889 1.593 . 446 .449 . 440 .439 .440 . 436 1.962 2.839 1.939 1.898 3.880 2.736 Finding extremal values of V and A T I 2c | 1 1 min V & A | .216 1 3.078 | 1. 873 | . 449 | 2.600 | I 2d | 1 1 max V 6 A | . 285 ! 3.624 | 2. 057 | .446 I 2.200 | I 3d | 2c 1 min V | .265 I 3. 200 | 1. 907 | . 449 | 2.649 | 1 3e| 2c I min A I .278 I 3.321 | 1. 854 | .442 | 2.549 | 1 3 f | 2d 1 max V & A | .299 I 3. 904 | 2. 222 | . 439 | 2.038 | 1 4b| 3f i max V | .264 I 3. 950 | 2. 148 | .434 | 2.192 | 1 i i . 3f max A | . 293 f 3.801 | 2. 219 | f . 441 | 2.006 | 1 i F i n d i n g extremal values of RTIM i i 1 2e| 1 • min RTIM | . 256 i 3.428 | 1. 960 | ;432 "I T | 2.364 | 1 2f| 1 ! max RTIM I .245 1 3. 274 | 1. 970 | .462 | 2.436 | 1 3g| 2e i min RTIM | .230 i 3.595 | 2. 031 | .429 | 2.666 | 1 3h| 21 ! max RTIM | . 182 ! 3. 330 | 1. 997 | .468 | 3.267 | 1 4d| t i 3h max RTIM | .289 i 3. 105 | 1. 795 | .457 | 2.323 | .i i Table 5.4a. D e s c r i p t i o n of i t e r a t i o n s f o r the Rat I s . event 89 |It.|From| D e s c r i p t i o n ( E t T l o I I I Least Squares I n v e r s i o n 1 . I -+-Parameter Values j I RTIM | ELAZ I S t a r t i n g Model | .3 j 2.5 | 1.5 T i I I 3 . 1 4 1 j J 1 j 0 j 1 s t i t . E=.33 | .330 | 2.458 | 1.523 | .464 | 3.113 I 2 | 1 j 2nd i t . E=.36 | .360 j 2.485 | 1.491 | .487 | 3.121 I 3 | 2 | 3rd i t . V=2.4 \ .376 | 2.400 | 1.563 | .508 | 3.046 I I 4 I 3 i 4 t h i t e r a t i o n | .372 | 2.431 | 1.565 | .504 | 3.217 Maximising E I T T | 4a| 3 | max E I I I | 5a| 4a | max E I I I | 6a| 5a | max E t i i - j Maximising ELAZ I .391 | 2.460 | 1.600 | .511 | 3.207 I .401 | 2.567 | 1.642 | .514 | 3.254 . I -397 | 2. 686 | 1.672 | .516 | 3.243 r ^ n  I 4b| 3 | max ELAZ I I I | 5b| 4b | max ELAZ I 1 L I .371 | 2.313 | 1.557 | .507 | 3.527 I I I I I I .375 | 2.305 | 1.580 | .528 | 3.464 - X 1 1 j x  Minimising E and ELAZ I T T ' T T T T T I 1a I 0 | min S & ELAZ | .319 | 2.074 | 1.597 | . 524 | 2.642 I I I I I I I I I 2a| 1a | min ELAZ | .323 | 2.292 | 1.518 | .523 | 2.8o8 i l l I I I ' l l I 2b| 1a | min E I I I I 3b| 2b | min E i 1 1 Minimising V and A • T T -, P I 1c| 0 | 1st i t . 7=2.1 | .355 | 2.100 I I I ' I I | .290 | 2.166 | 1.448 | .525 | 3.394 I I I I I I .305 | 2.027 | 1.539 | .534 | 2.796 I 2c| 1c | min V I I I | 2d| 1c | min A I I I I 3d| 2d | min A i i i I .337 | 2.141 | .342 | 2.199 I .314 | 2.014 -i j 1 ) 1.581 | .520 | 3.033 | 1.488 | .532 | 3.680 | 1.443 | .521 | 3.660 | 1.467 | .525 | 3.033 | 1 i i Table 5.4b. D e s c r i p t i o n of i t e r a t i o n s f o r the Broach event 90 JIt.|From| D e s c r i p t i o n j I I I r-I I I I Parameter Values RTIM Maximising V and A -+ 1 1e| I max V s A | . 367 I 2. 312 | 1. 639 | .518 3.084 I 2e| 1e I max V & A | .349 | 2. 556 | 1. 683 I .513 3.090 I 3e| 2e I max V s A | .361 I 2.743 | 1.716 | .510 2.991 I 4e| 3e I max V & A | .379 | 2. 896 | 1.769 | .506 2.989 I 5e| 4e 1 max V & A | .375 I 3. 042 | 1.794 I .498 2.947 I 6e| 5e I max V & A | .385 I 3.142 | 1. 339 | .506 2.932 I 7e| 6e I max V & A | .384 | 3. 306 | 1. 382 I .497 2.851 I 8e| 7e I max V s A | . 385 | 3. 434 j 1.924 | .495 2.849 I 9e| 8e I max V I .345 | 3. 244 | 1. 924 1 .512 2.931 I 9f| 8e I max A | .350 j 3.119 j 1. 943 I -519 | 2.944 I 10 f | 9f I max A | .333 | 2.994 | 1. 372 1 .530 j 3.084 ELAZ | .j Minimising RTIM I 3g| 2 | min RTIM I I I I 4gl 3g | min RTIM i i i I .364 J 2.466 ] 1.555 T~.487 T 3.157 I I I I I | .375 j 2.464 | 1.578 | .505 | 3.170 -1 J 1 L 1 Maximising RTIM 3h | 2 max RTIM | .379 2. 396 | 1. 575 | .521 3. 277 1 4h| 1 3h max RTIM I .374 2. 378 | 1. 573 j .547 3. 176 5h| 1 4h max RTIM I .392 2. 404 | 1. 6T6 .56 1 3. 133 6h| 1 5h max RTIM, f i t unacceptable | .396 2. 455 | 1. 638 i .574 3. 191 Table 5.4b.(cont.) I t e r a t i o n s f o r the Broach event 91 AKU BEC FLO DUG CTA 5 N G SHI STU EKA WRA S t a t i o n S t a r t i n g Model 2 sec 1 sec I I F i r s t I t . Second I t , model seismogram o b s e r v a t i o n F i g u r e 5.2a. Model seismograms f o r s u c c e s s i v e i t e r a t i o n s - Rat I s . (the d i s p l a c e m e n t p u l s e i s f o r a st e p d i s l o c a t i o n f u n c t i o n ) F i g u r e 5.2b. Model seismograms f o r s u c c e s s i v e i t e r a t i o n s - Broach (the d i s p l a c e m e n t pulse i s f o r a s t e p d i s l o c a t i o n f u n c t i o n ) 93 u s u a l l y g u i t e s m a l l . Furthermore, i t was d i f f i c u l t to mentally weight the f i t of the model a p p r o p r i a t e l y a c c o r d i n g to the estimated standard d e v i a t i o n s of the o b s e r v a t i o n s . I t i s apparent, however, that s e n s i b l e improvements d i d occur. For example, the t o t a l sum of squares (SST) decreased from 565 to 547 during the f i r s t i t e r a t i o n of the Eat I s . event. During the course of the f i r s t two i t e r a t i o n s , the parameters E and V, whose s t a r t i n g values were c l o s e to t h e i r extremal l i m i t s , took on more moderate v a l u e s . S i m i l a r t r e n d s w^re observed f o r the Broach event. During subsequent i t e r a t i o n s , the SST f o r t h i s event was 621, 550, 523 and 498, showing a mor.otonic d e c l i n e . The improvement i n t h i s case can be seen i n the seismograms. 5.6. Extremal i n v e r s i o n The extremal i n v e r s i o n i t e r a t i o n s f o r the two earthquakes are shown i n Table 5.4, and the extremal parameter v a l u e s o b t a i n e d are l i s t e d i n Table 5.5. For the Rat I s . event, extremal s o l u t i o n s were sought s t a r t i n g from the f i r s t i t e r a t i o n model. The s t a r t i n g model was used as a base f o r extremal i n v e r s i o n f o r the Broach event because l e a s t squares i t e r a t i o n s from t h i s model r e q u i r e d s t e p s i z e c o n t r o l f o r s e v e r a l s t e p s before a s t a b l e domain was encountered. I t e r a t i o n s were pursued, checking the model o b t a i n e d a g a i n s t the o b s e r v a t i o n s at each step and l i m i t i n g s t e p s i z e i f necessary, u n t i l each parameter e i t h e r passed an extremum or d i d not proceed s i g n i f i c a n t l y f u r t h e r from i t s p r e v i o u s v a l u e . S t e p s i z e c o n t r o l , when necessary, was a p p l i e d to any parameter f o r which a l a r g e e x c u r s i o n caused a poor f i t to the 94 I Event V— Rat I s . Broach minimum maximum minimum maximum I 18 33 29 40 -+-3. 0 4.0 2.0 3.5 -+-1.8 2.2 1.4 2.0 RTIM .43 .47 46 56 ELAZ | 1 Table 5.5. Extremal parameter values o b s e r v a t i o n s . I t was sometimes found, to a good approximation, t h a t a p a i r of parameters took on extreme v a l u e s t o g e t h e r . For example, the rupture v e l o c i t y and major se m i - a x i s were f r e q u e n t l y extremised together f o r both events, and f o r the Rat I s . event, the e l l i p t i c i t y and o r i e n t a t i o n of the major a x i s took on extreme val u e s of the o p p o s i t e sense t o g e t h e r . For the l a t t e r case, Table 5.6 shows that the parameter values o b t a i n e d by j o i n t e x t r e m i s a t i o n are w i t h i n a few percent of the i n d i v i d u a l l y determined extremal v a l u e s . T h i s s i m p l i f y i n g and c o s t - s a v i n g approach was used wherever such c l o s e agreement was obta i n e d . U s u a l l y only a few i t e r a t i o n s were r e q u i r e d before an extremal value was determined. However, the maximization of rupt u r e v e l o c i t y and semi-major a x i s f o r the Broach event proceeded i n s m a l l steps from low i n i t i a l v a l u e s , and r e q u i r e d ten i t e r a t i o n s . O c c a s i o n a l l y an extremal value was unexpectedly found i n 95 I t , From D e s c r i p t i o n I •+-Parameter Values I -+-2u | 1 | max E .325 3. 476 2.009 . 446 I 1 -995 2v| 1 I min EZ ( = 3 L A Z ) .317 3. 446 2.015 .446 I 1- 951 2a | 1 I max E & min EZ .32 3 3. 463 2.013 , .446 I 1. 962 2x| 1 I min E . 176 | 3. 227 1.922 .449 I 2. 805 2yl 1 I max EZ . 184 | 3. 256 i 1.915 i .449 I 2. 849 2b 1 1 I rain E & max EZ . 178 | 3. 239 1.917 | .449 I 2. 839 RTIM | ELAZ j T a b l e 5.6. J o i n t e x t r e m i s a t i o n of two Rat I s . parameters one parameter while the extremal value of a d i f f e r e n t parameter was being pursued. T h i s occurred d u r i n g the search f o r minimum values of rup t u r e v e l o c i t y and major semi-axis f o r the Broach event. A f t e r a minimum i n rupture v e l o c i t y was found, the sea r c h f o r the minimum value of the major semi-axis produced a s l i g h t l y lower value f o r rupture v e l o c i t y than the s t e p s i z e -l i m i t e d one that had been p r e v i o u s l y determined. A l s o , d u r i n g the second i t e r a t i o n m i n i m i s a t i o n s of both of these parameters, v a l u e s s l i g h t l y higher than the e x t r e m a l l y determined one were obtained f o r the major a x i s o r i e n t a t i o n . However, proceeding i n the minimisation of the major sem i - a x i s , the value of major a x i s o r i e n t a t i o n returned to i t s pr e v i o u s v a l u e , and i t s maximum value in i t e r a t i n g from that model d e c l i n e d s l i g h t l y . S e v e r a l such p u r s u i t s of unexpected extrema were conducted, and i n a l l cases i t was found that extreme values were not extended f u r t h e r . T h i s i n d i c a t e d t h a t no gross n o n - l i n e a r i t i e s e x i s t e d i n the v i c i n i t y of the pool of extremal 96 s o l u t i o n s found. The i r r e g u l a r behaviour that has j u s t been d i s c u s s e d can be understood to a r i s e i n the f o l l o w i n g manner. Because of the n o n - l i n e a r i t y of the problem, the p a r t i a l d e r i v a t i v e s computed from the a r b i t r a r y 1 0 % increments to the model parameters do not a c c u r a t e l y d e s c r i b e the changes i n c a l c u l a t e d values t h a t w i l l r e s u l t from the a c t u a l parameter increments determined by the i n v e r s i o n . I t i s thus always p o s s i b l e that a parameter c o r r e c t i o n w i l l be o b t a i n e d which w i l l l e a d to unacceptable c a l c u l a t e d v a l u e s . T h i s n e c e s s i t a t e s a c h e c k i n g of the c a l c u l a t e d values f o r each model found. 5.7. V e i g e n v e c t o r a n a l y s i s of the model parameters The V e i g e n v e c t o r s from the f a c t o r i s a t i o n of the A matrix p r o v i d e a means of a s s e s s i n g the a p p r o p r i a t e n e s s of the chosen p a r a m e t e r i s a t i o n of the model. I f the V v e c t o r s are s p i k e l i k e , then t h i s i n d i c a t e s t h a t the model parameters are l a r g e l y independent of one another. The V v e c t o r s f o r both e v e n t s , shown i n Table 5.7, c l e a r l y d i s p l a y t h i s c o n d i t i o n . The secondary terms are g e n e r a l l y no g r e a t e r than o n e - t h i r d the p r i n c i p a l ones. T h i s enables us t o a s s o c i a t e the model parameters with the e i g e n v a l u e s of t h e i r v a r i o u s l i n e a r combinations s p e c i f i e d by the V e i g e n v e c t o r s . V a r i o u s measures of the v a r i a n c e of the model parameters f o r both events are shown i n Table 5.8. The d i s t r i b u t i o n o f eigenvalues and t h e i r a s s o c i a t e d model parameters are v i r t u a l l y the same f o r both earthquakes. The e i g e n v a l u e s i n d i c a t e the r e l a t i v e importance of that parameter i n determining the 97 | Vector | Assoc. | L a r g e s t | I number j model | element r I I param. | j Model parameter Rat I s l a n d Earthquake 1 2 3 4 5 RTIM E A V ELAZ L. Broach Earthquake . 946 . 874 .914 .910 . 885 1 2 3 4 5 758 775 951 966 977 I 239 -123 182 -999 999 -132 195 331 289 40 302 -999 -150 -999 -294 43 | 379 297 210 I 692 r | 999 -118 I I 233 24 52 782 •999 -225 I 212 999 100 -113 184 -103 50 -7 Table 5.7. V e i g e n v e c t o r s (normalised to 999) RTIM ELAZ | u 105 | H 337 | H -9 | -344 | -13 -999 | + -287 379 -999 -184 | j -192 | H 7 I 58 999 s o l u t i o n . As would be expected, the weights a s s i g n e d to the parameters as d e s c r i b e d i n S e c t i o n 4.5 are very c l o s e i n value to the e i g e n v a l u e s . H i t h i n c r e a s i n g l y s m a l l e i g e n v a l u e s , the parameter c o r r e c t i o n standard d e v i a t i o n (normalised by d i v i s i o n by the parameter value) shows a c l e a r t rend t o i n c r e a s e , i n d i c a t i n g the i n c r e a s i n g u n c e r t a i n t y i n the d e t e r m i n a t i o n of t h a t parameter. The parameter d e v i a t i o n s determined by the extremal i n v e r s i o n method d e s c r i b e d i n Chapter 4 are g e n e r a l l y twice as l a r g e as the parameter c o r r e c t i o n standard d e v i a t i o n s . The f a i l u r e of these two measures to be e q u i v a l e n t may i n d i c a t e the 98 r T T T " |Para-| Assoc.) Weight) |meter| eigen | | | | value | | i l l i I 1 I L_ Rat I s l a n d Earthquake Val u e | E x t r . |Extremal (Parameter |Norm. | I range | i n v e r s i o n | c o r r e c t i o n | p a r . ) I I s t a n d a r d l s t a n d a r d | c o r r . | I I d e v i a t i o n | d e v i a t i o n |st.Dev| I RTIM| 13.00 | 12. 65 | 1 E 1 9.26 | 8. 74 j 1 A | 5.21 | 5. 54 | 1 V | 2.36 j 3. 21 | I ELAZ| 1.25 | 3. 58 | Broach Earthquake I T 7 RTIM| 14.50 E | 8.53 A | 6.30 V | 2.33 ELAZ| 1.60 i 445| .04 | ! I 1.341| .40 | .082 . 008 I .007 I .015 .034 | .033 J . 132 . 082 j .046 I . 024 . 223 I .138 I . 04 1 .447 I .201 I . 087 I 12. 23 .5 | | 10. 19 . 3 | I 7. 31 1.5 | I 3. 97 , 2.5 | I 2. 86 3.141| T T . 10 | .11 | .60 | 1.5 | u . 223 I . 009 | .017 .025 | .017 J . 049 .134 j . 022 | .014 . 335 | .106 I . 050 .245 I .214 1 .070 Table 5.8. Measures of the d e v i a t i o n of the parameters e x i s t e n c e of c o n s i d e r a b l e e r r o r i n the assumption t h a t the data are c o r r u p t e d only by measurement e r r o r s having a Gaussian d i s t r i b u t i o n . A l t e r n a t i v e l y , i t may be t h a t the n o n l i n e a r i t i e s have a la r g e e f f e c t on the a c t u a l standard d e v i a t i o n s . 5.8. U e i g e n v e c t o r a n a l y s i s of i n f o r m a t i o n d i s t r i b u t i o n The d i a g o n a l of the U.U matrix, which p r o v i d e s i n f o r m a t i o n about the d i s t r i b u t i o n of i n f o r m a t i o n among the o b s e r v a t i o n s , i s shown i n Table 5.9 f o r both earthquakes. The d i a g o n a l value c o r r e s p o n d i n g to a p a r t i c u l a r o b s e r v a t i o n i s termed i t s 1 1 S t a t i o n 1 r 1 Se | " " ' 1 1 1 .Ml. ismogram Mea surement 1 • 1 1 2 3 4 5 J Rat I s . | — — * Earthquake I AKU I 40 17 365 8 89 | 1 BEC I 41 18 80 97 63 j I FLO | 40 16 21 32 30 | 1 DUG | 47 14 21 173 —I 16 | 1 CTA | 44 54 78 8 20 | 1 SNG j 46 48 195 7 75 | I SHI I 34 52 90 6 39 | I STU I 8 12 20 7 55 | I EKA | 330 463 858 137 302 | I WRA L I 81 70 136 135 87 | Broach E arthquake 1 I COL I 71 52 27 2 16 | I MAT I 71 52 57 18 72 | I ADE I 71 44 37 87 123 | I WIN I 77 I— -48 8 21 12 | I NAI | 86 60 54 147 22 | 1 I EIL | 86 67 18 29 1 99 | I TRI I 75 68 83 75 j 160 | 1 I GDH I 72 65 109 47 4 14 | I WRA I 25 91 86 304 - '— 1 112 | l I EKA | 70 » 100 923 65 1 229 | • Table 5.9 Diagonal of the U.U T matrix I n il m i J 100 •importance 1 by Minster et a l (1974). I t i s apparent i n each case t h a t the a r r a y measurements, with t h e i r s m a l l e r d e v i a t i o n s , provide the most s i g n i f i c a n t i n f o r m a t i o n . The WWSS s t a t i o n s are ordered i n i n c r e a s i n g azimuth eastward from the dip d i r e c t i o n of the f a u l t plane. There i s no obvious importance a s s o c i a t e d with p a r t i c u l a r o r i e n t a t i o n s of the s t a t i o n with r e s p e c t t o the source. I t can be concluded t h a t the best way to o b t a i n more c e r t a i n t y i n the d e t e r m i n a t i o n of the model i s to employ more arr a y measurements, making use of t h e i r g r e a t e r accuracy. I t seems u n l i k e l y t h a t much b e n e f i t would be d e r i v e d from attempting to o b t a i n a b e t t e r d i s t r i b u t i o n of measurements using the WWSS network. We may study the s e n s i t i v i t y of the model to the d i f f e r e n t T components of the wavelet p a r a m e t e r i s a t i o n by summing the U.U di a g o n a l a c r o s s a l l s t a t i o n s . The r e s u l t s , shown i n Table 5.10, i r Earthguake | Seismogram Measurement r 1 2 3 4 5 | .+ 4 Kat I s . | 383 410 999 329 417 | I I Broach | 504 462 999 568 897 | I I I L J r T a b l e 5.10. U.U d i a g o n a l summed acr o s s s t a t i o n s i n d i c a t e t h a t the most important component i s the t h i r d one, which measures the i n t e r v a l between the f i r s t trough and the second peak. The othe r components have approximately h a l f the importance of t h i s one, with the e x c e p t i o n of component 5 (the second peak h e i g h t ) , which f o r the Broach event i s almost as 101 important as component 3. I t may be concluded that a l l of the wavelet component measurements play a s i g n i f i c a n t r o l e i n determining the model. We can i n v e s t i g a t e the i n f l u e n c e of the v a r i o u s component measurements upon each model parameter by summing i n d i v i d u a l U v e c t o r s a c r o s s s t a t i o n s as shown i n Tabl e 5.11. The value s f o r I Vector | Assoc. T Largest ] ~ I number | model | element r  I I param. I I i « , 1 ' l Seismogram measurement j Rat I s l a n d Earthquake 1— —,— — 1 — 1 1 i RTIM | 2. 173 I 2 | E | 1. 966 I 3 | A I 1.374 I 4 | V | 1.883 I 5 | l— —L. ELAZ | L 1.757 17 999 474 I 130 616 I 959 342 | _ 999 128 622 I 999 ! 279 207 999 621 221 392 272 999 680 Broach Earthquake 1 1 1 - 1 — RTIM | 2. 123 I 2 1 E | 2. 107 I 3 1 A | 1.326 I 4 « V | 2.453 I 5 — i ELAZ | L 1.900 4-12 635 999 463 554 561 999 106 778 207 999 I— 241 314 207 35 565 806 Table 5.11. u e i g e n v e c t o r s summed ac r o s s s t a t i o n s j 587 —+ 532 ] 877 ] -j 324 | 1 717 | 999 1 244 j H 647 263 ] 999 I the two events are broadly s i m i l a r . Our a b i l i t y to a s s o c i a t e model parameters with the U e i g e n v e c t o r s i s based upon the s p i k e - l i k e c h a r a c t e r of the V e i g e n v e c t o r s . G e n e r a l l y , the summed U e i g e n v e c t o r s are not s p i k e - l i k e , and so no c l e a r correspondence between a p a r t i c u l a r parameter and a p a r t i c u l a r 102 wavelet component i s apparent. An e x c e p t i o n i s the parameter V which f o r both events i s q u i t e c l o s e l y l i n k e d with component 1, the time between onset and the f i r s t peak. The dominance of component 3 has a l r e a d y been noted. For the Rat. Is. event, i t i s the most important determinant of 3 , A and ELAZ, while f o r the Broach event, i t i s the most important determinant of A. 5.9. C o n c l u s i o n s Models were found which gave a moderately good f i t to the o b s e r v a t i o n s f o r both earthquakes. Extremal i n v e r s i o n i n d i c a t e d t h a t a f a i r l y wide range of models was a c c e p t a b l e . N e v e r t h e l e s s , i t does seem that v a r i a t i o n s from s p h e r i c a l l y symmetric r a d i a t i o n are measurable and can c o n t r i b u t e to the d e t e r m i n a t i o n of the shape and o r i e n t a t i o n of the source. I t i s important to i n v e s t i g a t e the o r i g i n of the f a i l u r e of the model to f i t the o b s e r v a t i o n s , as expressed i n the sura of squares of d e v i a t i o n s from the r e g r e s s i o n . The sum of squares f o r both events i s about ten times the number o f degrees of freedom (45) a s s o c i a t e d with i t , i n d i c a t i n g t h a t t h e r e are s u b s t a n t i a l e r r o r s i n the data, or that the model i s i n a d e q u a t e l y formulated to c o r r e c t l y d e s c r i b e the o b s e r v a t i o n s , or both. I t may be t h a t the standard d e v i a t i o n s o f the o b s e r v a t i o n s were somewhat underestimated.. However the most obvious d i s c r e p a n c y i s a c o n s i s t e n t e r r o r i n the h e i g h t and time p o s i t i o n of the f i r s t peak. These measurements are the two that are l e a s t a f f e c t e d by homomorphic d e c o n v o l u t i o n , and t h e i r 103 c o n s i s t e n c y w o u l d s u g g e s t an i n a d e q u a c y i n t h e m o d e l . A d i s c r e p a n c y i n t h e f i r s t p e a k o f t h e s e i s m o g r a m s c o r r e s p o n d s t o a n i m p r o p e r d e s c r i p t i o n o f t h e i n i t i a t i o n a n d e a r l y e v o l u t i o n o f r u p t u r e , p e r h a p s i n b o t h i t s s p a t i a l a n d t e m p o r a l a s p e c t s . We h a v e a l r e a d y s e e n i n S e c t i o n 2 . 6 t h a t o u r a s s u m p t i o n o f a d i s l o c a t i o n f u n c t i o n t h a t i s t h e same e v e r y w h e r e o n t h e f a u l t s u r f a c e i s i n c o r r e c t . T h e a d o p t i o n o f K o s t r o v ' s (1964) d i s l o c a t i o n f u n c t i o n w o u l d c o n s t i t u t e a f i r s t s t e p t o w a r d a c h i e v i n g a more c o r r e c t d e s c r i p t i o n . I t s a l t e r a t i o n o f t h e p u l s e s h a p e w o u l d t e n d t o c o r r e c t t h e d i s c r e p a n c y b e c a u s e i t w o u l d c a u s e t h e r i s e s l o p e o f t h e f a r - f i e l d d i s p l a c e m e n t t o r e a c h a h i g h e r maximum v a l u e . H o w e v e r , t h e u s e o f a s p a c e -d e p e n d e n t d i s l o c a t i o n f u n c t i o n w o u l d e n t a i l a p r o h i b i t i v e l y h i g h c o s t i n c o m p u t i n g t i m e i f i t were d e s i r e d t o e m p l o y a n i n v e r s i o n s c h e m e a s i n t h i s w o r k . 104 CHAPTER 6. SOURCE PARAMETERS Using the g e n e r a l l i n e a r i n v e r s e scheme we have ev a l u a t e d model parameters f o r two earthquakes by f i t t i n g s i g n a l shapes. The parameters, whose value ranges are l i s t e d i n Table 5.5, can be grouped i n the f o l l o w i n g way. There are t h r e e o r i e n t a t i o n a l parameters: the f o c a l plane parameters (which were kept f i x e d ) and the o r i e n t a t i o n of the e l l i p t i c a l f a u l t s u r f a c e . The parameters r u p t u r e v e l o c i t y and d i s l o c a t i o n r i s e t i m e p e r t a i n e x c l u s i v e l y t o the dynamic aspects of f a u l t i n g , and f o r an earthquake of f i x e d dimension, have no i n f l u e n c e on the s t a t i c p r o p e r t i e s of the displacement f i e l d . The remaining two parameters, major semi-axis and e l l i p t i c i t y , determine the shape of the f a u l t and i t s area. From the f a u l t area and from measurements of the a b s o l u t e amplitudes of the recorded s i g n a l s we may use the theory of S e c t i o n s 2.4 and 2.5 to determine s e v e r a l p r o p e r t i e s of the s t a t i c displacement f i e l d . In S e c t i o n 2.4 we saw how the moment of an earthquake can be estimated from the low-frequency zero s l o p e asymptote of i t s amplitude spectrum. However, the b a n d l i m i t e d nature of the instrument response h i n d e r s s p e c t r a l a n a l y s i s using s h o r t p e r i o d records alone. I d e a l l y , long p e r i o d r e c o r d s should a l s o be used, but f o r shallow events we are thwarted by the presence of depth phases which i n t e r f e r e with the d i r e c t a r r i v a l . Short p e r i o d instruments are most u s e f u l i n p r o v i d i n g i n f o r m a t i o n about the dynamical a s p e c t s of r u p t u r e , s i n c e these a f f e c t the r a t e at which the amplitude f a l l s o f f with frequency. S p e c i f i c a l l y , the order of a d i s c o n t i n u i t y ( f o r example, rupture onset or stopping) i n the displacement pulse i s equal to the 105 s l o p e of the s p e c t r a l f a l l o f f which i t generates ( B r a c e w e l l , 1965). Thus the u l t i m a t e r a t e of d e c l i n e of the spectrum i s c o n t r o l l e d by the lowest order d i s c o n t i n u i t y (that i s , the sharpest f e a t u r e ) of the s i g n a l . For moderate earthquakes, the corner f r e q u e n c i e s markinq the onsets of these s p e c t r a l trends are a l s o present i n the s h o r t - p e r i o d band. The corner f r e q u e n c i e s provide measures of the d u r a t i o n of coherent rupture. I f i t i s assumed t h a t r u p t u r e occurs c o h e r e n t l y over the f a u l t s u r f a c e a t some s p e c i f i e d r u p t u r e v e l o c i t y , the dimension of the f a u l t may then be c a l c u l a t e d . In t h i s study, we have estimated f a u l t dimensions throuqh a time domain a n a l y s i s of seismoqraras. A rough check t h a t the e s t i m a t e s are i n o rder can be made by comparing the amplitude s p e c t r a of the model displacement p u l s e s with those of the recorded seismoqrams. For the purposes of comparison, the e f f e c t of an a t t e n u a t i o n o p e r a t o r havinq t * = 0.1, and of the instrument response, has been removed from the seismogram s p e c t r a . The e f f e c t of the t r a n s m i s s i o n path, however, i s s t i l l p r esent i n the spectrum. I d e a l l y , to avoid t h i s problem, the spectrum of the deconvolved pulse should be used. However, homomorphic de c o n v o l u t i o n e n t a i l s an indeterminate d e g r a d a t i o n of the spectrum which renders i t s i n t e r p r e t a t i o n somewhat u n c e r t a i n . The s p e c t r a are compared i n F i g u r e 6.1 f o r the SHI r e c o r d i n g of the Rat I s l a n d event, and f o r the WRA r e c o r d i n g of the Broach event. I t i s apparent i n each case t h a t the c o r n e r f r e q u e n c i e s of the model spectrum and the seismoqram spectrum 106 F i g u r e 6.1a. Observed and model s p e c t r a of Rat I s . event a t SHI 107 6.1b. Observed and model s p e c t r a of Broach event at WRA 1 0 8 LOG FREQUENCY (HZ) Figure 6.2a. Observed spectrum of the K u r i l I s . event at YKA 109 F i g u r e 6.2b. Observed spectrum of the San Fernando event a t EKA 110 are comparable. T h i s confirms that the source d u r a t i o n (and l e s s c e r t a i n l y , the source dimension) estimated from the time domain a n a l y s i s i s not g r o s s l y i n e r r o r . Such a check i s by no means redundant i n the context of t h i s study s i n c e s h o r t - p e r i o d seismograms, e s p e c i a l l y those of the WWSS network, are so b a n d - l i m i t e d that the d u r a t i o n of the pulse i s e a s i l y d i s g u i s e d . The UKAEA i n s t r u m e n t s , with t h e i r g r e a t e r bandwidth, provide a c o n s i d e r a b l y b e t t e r d i s c r i m i n a t i o n between events of d i f f e r e n t d u r a t i o n . T h i s i s i l l u s t r a t e d i n a comparison of the magnitude 5.7 K u r i l I s l a n d s earthquake of 10 June 1970, recorded a t YKA, with the magnitude 6.2 San Fernando earthquake of 9 February 1971, recorded at EKA (Figure 6.2). The former event i s b r i e f i n comparison with the l a t t e r , and i t s s p e c t r a l 'corner' i s at a high frequency, compared with the i n d e t e r m i n a t e l y low corner frequency of the l a t t e r event. The low-frequency asymptote of the spectrum was estimated by matching the model spectrum with the seismogram spectrum. In many c a s e s , the seismoqram spectrum was not s u f f i c i e n t l y f l a t at long p e r i o d s to permit a rea s o n a b l y a c c u r a t e f i t . At these long p e r i o d s , the gain of the instrument i s s e v e r a l decades below that i n the c e n t r e of the band, and l a r g e e r r o r s may occur when the spectrum i s c o r r e c t e d f o r instrument response. Estimates of the moment c a l c u l a t e d u s i n g Equation 2.26 are shown i n Table 6.1. The moment of the Rat I s l a n d event appears to be w e l l determined, while t h a t of the Broach event i s r a t h e r poorly determined. From these moment v a l u e s , toqether with estimates of s e i s m i c enerqy, we may e v a l u a t e the average 111 S t a t i o n n , I — , J . X 1 0 - 9 | cm - 1 | X 1 0 ~ 3 | cm.sec| _. J . . x10&cm 2.s9c. | x102*dyne.cm | Rat I s l a n d r- ~ •— Earthquake — ~ i I AKU I .79 | . 70 i . 10 j . 18 | 1.9 | I FLO I .64 | .70 1 . 10 | .22 | 2.4 | I DUG I . 60 i .92 1 .13 | .24 | 2.6 | I SHI I .95 | .53 1 . 10 | .20 | 2.2 | I EKA I . 88 | .63 ! .12 | .22 | 2.4 | I WRA I . 51 j .61 ! .07 | .22 | 2.4 | I mean i _ — i . J . j .21 | i 2.3 | Broach Earthquake — • — . , i I COL • .73 | .46 1 .023 | T ' . 07 | . ....^  .38 | I WRA I .88 | .66 1 . 10 | . 17 | .92 | I EKA I .63 | .7 1 ! .075 | .17 | .92 | I mean • . i _ j J . . 14 | .75 | i Table 6.1. Moment c a l c u l a t i o n s d i s l o c a t i o n LJ , s t r e s s drop S , apparent s t r e s s 7 5 » and s t r a i n drop Y using the equations of S e c t i o n 2.5. The source parameters of the two events are summarised i n Table 6.2. The r e s u l t s c o n f i r m our e x p e c t a t i o n , expressed i n S e c t i o n 2.1, t h a t events of a p p a r e n t l y short d u r a t i o n t h a t are well recorded on s h o r t - p e r i o d instruments should have r e l a t i v e l y s mall source dimensions and r e l a t i v e l y l a r g e s t r e s s drops. In each case, the s i z e of the s t r e s s drop i s a few percent of the value of the l i t h o s t a t i c s t r e s s . 112 r I I I t, T 1 P h y s i c a l Q u a n t i t y | 1 Rat I s . ~ T -I I I | Broach Assumed values [ Body wave mag. ™ b 1 6.0 T " 5.4 P wave v e l o c i t y 1 8.05 km/sec ! 6.75 km/sec Density ! 3.3 gm/cm3 i 2.8 gm/cm3 J Shear modulus A 1 .7x10^2 dyne/cm 2 | . 4 x 1 0 1 2 dyne/cm 2 i S t r e s s drop c o e f f c 1 2.4 ! 2.4 4 L i t h o s t a t i c s t r e s s sL j 1 12 kfaar 5 kbar Measured values i | Major semi-axis , ~—"T -a 1 2.0 km 1.6 km E l l i p t i c i t y e 1 .25 1 . 375 E l o n g a t i o n d i r n . elcu | updip j updip Rupture speed V | 3.5 km/sec 1 2.5 km/sec Risetirae rtirn j .45 sec 1 .50 sec u. Moment < 2. 3r.10 2*dyne. cm _|__ . 7 5 x 10 2 * d y n e . c m C a l c u l a t e d values i F a u l t area A * irab | 12 km2 _ T_ 7 km2 Average s l i p U'Mo/^A | 27 cm J 27 cm S t r e s s drop* S * 130 bars ! 100 bars S t r a i n drop* V' S/A ! 1.9x10-* j 2.5x10-* Seism i c energy Es ' • Z-if "U) 1.6x10zOergs ! .063x102o ergs Apparent s t r e s s yS " ic/» A 1 S i — _ . i , . 50 bars I 3 bars i T o t a l energy* E w 2x10 2°ergs i 1x1Q2o ergs * s t r i c t l y a p p r o p r i a t e only f o r complete s t r e s s drop Table 6.2. Source Parameters 1 1 3 The two events have s i m i l a r f a u l t areas and moments, and thus have s i m i l a r average d i s l o c a t i o n and s t r e s s drop. The l a r g e d i f f e r e n c e i n magnitude (and energy) between the events i s r e f l e c t e d mainly i n the apparent s t r e s s , which was very low f o r the Broach event. In f e r e n c e s about the t e c t o n i c environment can be made from the s t r e s s e s t i m a t e s . In p a r t i c u l a r , i t i s of i n t e r e s t to i n t e r p r e t the r e l a t i v e l y l a r g e s t r e s s drops of the two events. In the case of the Rat I s l a n d earthquake, i t may be a t t r i b u t e d to the e x t e n s i o n a l f r a c t u r i n g of the downthrust s l a b . T h i s e n t a i l s the c r e a t i o n of f r e s h f r a c t u r e s i n the rock, which i s l i k e l y to r e l e a s e more s t r e s s than would the p e r i o d i c rupture on a p r e - e x i s t i n g t h r u s t f a u l t . For the Broach earthquake, the l a r g e s t r e s s drop i s a s s o c i a t e d with a low apparent s t r e s s . T h i s i m p l i e s t h a t a l a r g e r e l e a s e o f s t r e s s o c c u r r e d because the m a t e r i a l became q u i t e weak durin g f a u l t i n g . However, most of the energy a s s o c i a t e d with the drop i n s t r e s s was not r a d i a t e d from the sudden c r e a t i o n of the f a u l t zone, but was expended a n e l a s t i c a l l y through the p l a s t i c deformation of an extended r e g i o n of the medium. The Savage i n e g u a l i t y (Equation 2.38) i s s a t i s f i e d f o r both earthquakes. The e f f e c t i v e s t r e s s i s n e a r l y h a l f the s t r e s s drop f o r the Rat I s l a n d event. According t o the f r i c t i o n a l model of S e c t i o n 2.5, only about one-tenth of the s l i p occurred a f t e r the a c t i n g shear s t r e s s dropped below the dynamic f r i c t i o n a l s t r e s s . The low value of the apparent s t r e s s f o r the Broach event r e l a t i v e to i t s s t r e s s drop, on the other hand, i n d i c a t e s that almost h a l f ( i . e . almost the maximum p o s s i b l e 114 f r a c t i o n ) of the s l i p occurred as overshoot. A noteworthy f e a t u r e of the two earthquakes i s the e l o n g a t i o n of rupture i n the updip d i r e c t i o n . T h i s might be expected f o r moderate shallow earthquakes s i n c e the l i t h o s t a t i c s t r e s s has a negative g r a d i e n t upwards. On t h i s h y p o t h e s i s , great shallow earthquakes would propagate updip to the s u r f a c e i n i t i a l l y , and would then continue t h e i r p ropagation h o r i z o n t a l l y , g i v i n g a net e l o n g a t i o n along the s t r i k e . The study of K e l l e h e r e t a l (1973) l e n d s s t r o n g support to t h i s h y p o t h e s i s . They used a f t e r s h o c k zones to d e l i n e a t e the f a u l t boundaries of shallow t h r u s t earthquakes i n t r e n c h environments. The e p i c e n t r e was c h a r a c t e r i s t i c a l l y at a depth of roughly f o r t y k i l o m e t r e s , with rupture extending updip f o r moderate earthquakes, and both updip and u n i l a t e r a l l y f o r l a r g e earthquakes. 115 CHAPTER 7. CONCLUSIONS I t has been shown that the form of the s h o r t - p e r i o d P wave r a d i a t i o n from- two earthquakes i s l a r g e l y i n accord with a kinematic propagating d i s l o c a t i o n model of earthguake r u p t u r e . The seismograms are shown to possess the k i n d of departure from s p h e r i c a l l y symmetric r a d i a t i o n t h a t i s e n t a i l e d i n the model. S t a t i c source parameters such as moment, s t r e s s drop, f a u l t area and average d i s l o c a t i o n , which are u s u a l l y estimated from the amplitude spectrum, have been determined. The a n a l y s i s of the form of the s i g n a l s , however, has provided a d d i t i o n a l i n f o r m a t i o n which p e r t a i n s t o the dynamics of r u p t u r e . Thus i t has been p o s s i b l e t o estimate rupture v e l o c i t y , the d u r a t i o n of steady s l i p at a p o i n t on the f a u l t , and the d i r e c t i o n i n which the rupture propagated f u r t h e s t . In demonstrating that t h i s kind of i n f o r m a t i o n i s a v a i l a b l e i n s h o r t - p e r i o d seismograms, we have pr o v i d e d a means of st u d y i n g the r u p t u r e processes of a l a r g e c a t e g o r y of moderate earthquakes o c c u r r i n g i n a l l kinds of t e c t o n i c environments. A g e n e r a l l i n e a r i n v e r s e scheme has enabled us to study the r e l a t i o n s h i p s between the o b s e r v a t i o n s and the parameters of the model. Best f i t t i n g models have been e s t i m a t e d , t o g e t h e r with the range of v a l u e s t h a t the model parameters can assume. By s t u d y i n g the d i s t r i b u t i o n of i n f o r m a t i o n among the o b s e r v a t i o n s , the value of s e i s m i c array measurements f o r source s t u d i e s has been demonstrated. Those d i s c r e p a n c i e s between observed motions and model computations which p e r s i s t e d through the i n v e r s i o n i t e r a t i o n s 1 16 have been used t o i d e n t i f y d e f i c i e n c i e s i n the model. The major d e f i c i e n c y has been a t t r i b u t e d to a known i n a c c u r a c y i n the d e s c r i p t i o n of the d i s l o c a t i o n time f u n c t i o n . The manner i n which Kostrov's p h y s i c a l l y proper (but c o m p u t a t i o n a l l y inconvenient) d e s c r i p t i o n would tend to remove the d i s c r e p a n c y has been i n d i c a t e d . In the context of the kinematic d e s c r i p t i o n of shear f a u l t i n g , i t has been demonstrated t h a t Savage's model can be f r u i t f u l l y a p p l i e d t o the d e t e r m i n a t i o n of parameters r e l a t e d to the dynamics of r u p t u r e . The use of t h i s model t h e r e f o r e seems well j u s t i f i e d , i n the absence of a comprehensive p h y s i c a l theory of rupture i n a s o l i d . 117 REFERENCES Aki K. 1966. Generation and propagation of G-waves from the N i i g a t a earthquake of June 16, 1964. B u l l . E a r t h q . Res. I n s t . , Tokyo Dniv., 44, 23-68. Archambeau, C. B. 1968. General theory of elastodynamic source f i e l d s . Rev. Geophys. Space Phys., 6, 241-288. Arora, S. K. 1971. A study of the e a r t h ' s c r u s t near Ga u r i b i d a n u r i n Southern I n d i a . B u l l . Seism. Soc. Amer., 61, 671-683. Arora, S. K., G. J . N a i r and T. G. Varqhese. 1971. Broach earthquake of March 23, 1970. E a r t h q . 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