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P coda evidence for a layer of anomalous velocity in the upper crust beneath Leduc, Alberta. Somerville, Paul Graham 1971

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P CODA EVIDENCE FOR A LAYER OF ANOMALOUS VELOCITY IN THE UPPER CRUST BENEATH LEDUC, ALBERTA by Paul Graham Somerville B.Sc. (Hons.) University of New England, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of GEOPHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1971 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 r e q u i r e m e n t s f o r an advanced d e g r e e a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t 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 f o r r e f e r e n c e and S tudy . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s thes. is f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada i ABSTRACT Previous seismic studies of crustal structure using short-period P coda recorded in the vicinity of Leduc, central Alberta have indicated that serious discrepancies between the. experimental observations and those based on a horizontally layered model of,the crust exist in both the time and frequency domains. This 'Standard' crustal model is based on well log data for the sedimentary section, and on seismic refraction work in southern Alberta for the lower layers. The principal discrepancy lies in the large amplitude radial motion on the experimental seismograms which lags, vertical arrivals by approxi-mately two seconds. It is concluded that this radial motion, which is absent from synthetic seismograms generated using the 'Standard' crustal model, represents the generation of large-amplitude shear waves within the crust. This; large radial motion.is manifested in the frequency do-main by experimental vertical-radial spectral ratios which are consider-ably, lower than .those, computed using,the 'Standard' crustal model. Using vertical-radial spectral.ratios-and.synthetic seismograms, a modified crustal model has been derived which gives much better agreement between experimental and theoretical results. This model involves the insertion of a layer several kilometers thick having large velocity con-trast with respect to. the surrounding media at the base of the Precam-brian basement (12 km deep). The new crustal, model is discussed in the light of widespread evidence for a low velocity zone in the upper crust in continental regions. Several important.discrepancies, between, experimental and synthetic seismograms remain unresolved: among-these, are the small onset amplitude and the .character of the delayed motion of the radial component. i i CONTENTS Abstract (i) Contents ( i i ) List of Tables ( i i i ) List of Figures (iv) Acknowledgments (vi) Chapter 1: Introduction 1.1 Previous studies. 1 1.2 Data collection and preparation. 2 Chapter 2: Analysis 2.1 Deficiency of 'Standard' crustal model in explaining the character of the earthquake seismograms. 10 2.2 Some explanations of the experimental results. 17 2.3 Effect of the sedimentary section of the crust. 18 2.4 The influence of wavelet shape on the synthetic seismograms. 25 2.5 Modes of generation of shear waves at depth. 26 2.6 Investigations into layers of large velocity contrast. 31 2.7 Comparison of experimental and model seismograms. 49 Chapter 3: Discussion 3.1 Evidence for a low velocity zone in continental regions. 54 3.2 Discussion and interpretation of results. 58 Chapter 4: Concluding Remarks 4.1 Summary. 61 4.2 Conclusions. 63 4.3 Suggestions for further work. 64 References 65 Appendix: Homomorphic deconvolution 68 i i i LIST OF TABLES Table I Parameters of the earthquake events. A Table II Layer parameters of the 'Standard' crustal sections beneath the three stations. 8 Table III Layer parameters of the detailed sedimentary section. 23 Table IV Layer parameters of the low velocity layer and high velocity layer crustal models. 35 Table V Radial-vertical amplitude ratios of the experimental and synthetic Kurile seismograms. 50 i v LIST OF FIGURES Fig. 1. A map of the seismic stations and oilwell locations by which the crustal section was determined. 3 Fig. 2(a) The sedimentary crustal section obtained by interpolation between oilwell geophysical logs. 6 (b) Crustal model for southern Alberta (after Cumming and Kanasewich, 1966). 7 Fig. 3. Vertical and radial components of the Leduc seismograms for the Kurile and Honshu events. 11 Fig. 4. Radial-vertical cross correlations of experimental, standard crust and low velocity layer crust seismograms. 12 Fig. 5. Comparison of the 'Standard' crust seismogram with the experimental Kurile seismogram. 13 Fig. 6. Spectral vertical-radial ratios for the 'Standard' crustal model compared with experimental spectral ratios for i at (a) 20° and (b) 35°. m 15,16 Fig. 7. Impulse response of the 'Standard' crustal model with the major arrivals identified. 21 Fig. 8(a) Synthetic seismogram for a crust having a detailed sedimentary section. 24 (b) Synthetic seismogram for the 'Standard' crustal model using a long wavelet. 24 Fig. 9. Large velocity contrast layer models. 27 Fig.10. Transfer function of a large velocity contrast layer. 28 Fig.11. Generation of shear waves at the boundaries of a large velocity contrast layer. 30 Fig.12(a) Profiles of 'Standard', low velocity layer, and high velocity layer crustal models. 32 (b) Sections of 'Standard', low velocity layer, and high velocity layer crustal models. 33 Fig.13. Radial components of impulse responses for the 'Standard', low velocity layer, and high velocity layer crustal models. 34 Fig.14. Vertical-radial spectral ratios of thick (2.75 km) and thin (1.7 km) 5.3 km/sec low velocity layer crustal models, compared with experimental and 'Standard' results for i at (a) 20° and (b) 35°. m 37,38 Fig.15 Synthetic seismograms for thick (2.75 km) and thin (.1.7 km) 5.2 km/sec low velocity layer crustal models. 39 V Fig.16. Synthetic seismograms for the 5.2 km/sec and 5.5 km/sec low velocity layer crustal models compared with the experimental seismogram. 40 Fig.17. Vertical-radial sepctral ratios for the 5.2 km/sec and 5.5 km/sec low velocity layer crustal models, compared with experimental results for i at (a) 20° and (b) 35°. 41,42 Fig.18. Vertical-radial spectral ratios for perturbed thickness of 5.2 km/sec low velocity layer model for i , at (a) 20° and (b) 35°. m 4 3 ) 4 4 Fig.19. Synthetic seismogram for 7.0 km/see high velocity layer crustal model. 46 Fig.20. Vertical-radial spectral ratios for the 7.0 km/sec high velocity layer model, compared with experimental results for i at (a) 20° and (b) 35°. 47,48 m Fig.21. Azimuth deviations plotted against azimuth for the 15 earthquake events, and..36 of the events. of E l l i s and Basham (1968). 52 Fig.22. Crustal section through, the Rhinegraben. r i f t system (after Mueller et a l , 1969). 55 Al(a) Canonic representation of the homomorphic f i l t e r i n g process. 70 (b) Characteristic system for a homomorphic deconvolution f i l t e r . 70 A2 Synthesis of.a signal using a wavelet and an impulse pair. 72 A3 Complex cepstrum and recovered wavelet from the signal of Fig. A2. 74 A4 The Kurile seismogram, its . complex cepstrum, and the deconvolved impulse-series and wavelet. 75. v i ACKNOWLEDGEMENTS I wish to thank Dr R. M. E l l i s f o r h i s valuable guidance and assistance with t h i s work, and for his c r i t i c a l review of i t s presentation as a the s i s . I am much indebted to Dr 0. G. Jensen for h i s continual discussions and c r i t i c i s m s of the work, and for h i s guidance i n computer analysis techniques. His computer program, together with wavelets deconvolved from earthquake events by Dr T. J . Ulrych, made.possible the computation of synthetic s e i s -mograms. I am also g r a t e f u l to Dr R. M. Clowes for h i s discussions concern-ing c r u s t a l structure i n Alberta, and to Dr H. J . Greenwood for h i s guidance i n p e t r o l o g i c a l i n t e r p r e t a t i o n s . I would also l i k e to acknowledge the contribution of the Department of Geophysics sta f f , i n the preparation of the.seismic f i e l d equipment and i n the recording and. d i g i t i z a t i o n of the data. F i e l d equipment was made a v a i l -able by the Earth Physics. Branch,,Department of Energy, Mines and Resources and both equipment and. . f a c i l i t i e s , were provided by the University of Alberta. This research.was supported by the Defence Research Board of Canada (Grant 9511 - 7 6 ) . I would l i k e to thank.Miss. Wei Hsien Ho for her typing of the manuscript. 1 Chapter One INTRODUCTION 1.1 Previous studies. In an earlier P coda study using seismograms recorded in central Alberta, E l l i s and Basham (1968) compared vertical-radial spectral ratios (V/H) with theoretical ratios calculated using the Thompson-Haskell ma-trix formulation. Although moderate success was achieved in matching curves at several stations, particularly large deviations between the experimental and theoretical curves were observed for seismograms re-corded on the portable seismograph designated LED located in the Univer-sity of Alberta Edmonton Seismological. Observatory. Anomalously large horizontal amplitudes were noted and attributed to shear conversion and scattering within the crust. . For their, theoretical, model,, velocity, logs, from deep wells were used to determine sedimentary P, wave velocities. S.wave velocities were obtained using the relationship v g = v while the formation densities were obtained from the P wave density curves of Nafe and Drake (Grant and West, 1965). Beneath the sedimentsthe crustal layering and velocities used were those of Cumming and Kanasewich (1966) for southern Alberta extrapolated along regional geological strike. 2 1.2 Data c o l l e c t i o n and preparation. During October and e a r l y November 1968, a 15 km-sided t r i p a r t i t e ar-ray of seismographs was operated i n c e n t r a l Alberta* One system (LED) was located i n the University of Alberta Edmonton Seismological Observatory while the other two, CHA and LAR, were i n s t a l l e d i n the basements of abandoned farm houses ( F i g . 1). During t h i s period, 15 teleseismic events of high signal-to-noise r a t i o were recorded and have been used i n time domain studies.. The parameters of these events are presented i n Table I. For frequency domain studies,, the experimental V/H curves of E l l i s and Basham (1968) computed f o r seismograms recorded at LED were used. In t h e i r studies of 41 events, many f e l l i n the same range of incidence angles, i ^ , at the Mohorovicic d i s c o n t i n u i t y . In p a r t i c u l a r , 11 were i n the range 1 8 ° < i 22° and 22 were i n the range 31° < i ^ 45°. They m m were therefore able to average over these events to obtain smoother and presumably more representative experimental V/H curves than those which could be obtained from the more l i m i t e d s e l e c t i o n of data used i n t h i s work. In both studies, the same seismograph system was used at LED, and the CHA s t a t i o n i s of i d e n t i c a l design. The three components of ground motion were detected by seismometers with a natural frequency of 0.5 hz, amplified with phototube-galvanometer amplifiers with a n a t u r a l frequency of 5 hz, and recorded on an FM tape recorder. For a d e t a i l e d d e s c r i p t i o n of the system and the response, the reader i s re f e r r e d to Bancroft and Basham (1967). The LAR system had s i m i l a r c h a r a c t e r i s t i c s . The seismograms were d i g i t i z e d at. a sampling i n t e r v a l of 0.05 second, which corresponds to a Nyquist frequency of 10 hz. Since there i s l i t t l e t e leseismic energy at t h i s frequency, and since i t i s w e l l beyond the cut o f f frequency of the seismograph (5 hz), a l i a s i n g e f f e c t s due to sam-p l i n g are minimal. The ground motion amplitude frequency response for each seismograph was established from transient c a l i b r a t i o n pulses i n the manner described by Deas (1969). The s e n s i t i v i t y of the seismographs varied from one c a l i -b r a t i o n to the next: i n p a r t i c u l a r , the value of the transducer constant K exhibited random f l u c t u a t i o n s of the order of 5% superimposed upon a de-crease i n value of the order of 5% over the f i f t y day period of recording. 7 26 25 24 23 2 2 21 20 RGE ' W4 Fig. 1 A map of the seismic stations and oilwell locations by which the crustal section was determined (after Jensen, 1971) TABLE I Parameters of the earthquake Event Region Lat. (deg 1 Bonin Is. +26.3 2 Hokkaido +42.0 3 Venezuela +10o7 4 F i j i Is, -20.9 5 C h i l e - B o l i v i a -19.6 6 Kuri l e Is. +49*7 7 Honshu +-31*2 8 Alaska +65o4 9 Alaska +65.4 10 Peru -16.3 11 Novaya Zemlya +73.4 12 I l l i n o i s +38.0 13 Alaska Pen. +57.3 14 Honshu +40.1 15 Ryukyu Is. +-27.5 events Long. Depth Date (deg.) (km) Y M D +140.6 516 68 10 7 +140.6 32 68 10 7 - 62.6 80 68 10 12 -178.8 607 68 10 12 - 68.9 107 68 10 24 +155.8 35 68 10 24 +141.6 17 68 10 29 -150.1 7 68 10 29 -150.1 16 68 10 31 - 73.3 67 68 10 31 + 54.9 0 68 11 7 + 88.5 19 68 11 9 -155.3 59 68 11 11 +143.0 35 68 11 11 +128.4 48 68 11 12 H M S 19 20 20.3 6.1 20 49 01.3 5.7 19 19 29.3 4.8 19 17 39.9 5.7 01 29 42.6 5.3 22 35 50.9 5.5 04 06 04.1 5.7 22 16 15.6 6.0 00 25 45.1 4.5 09 15 46.9 5.7 10 02 05.3 6.0 17 01 41.1 5.3 08 53 52.0 5.3 14 41 15.9 5.5 00 44 12.8 5.8 -A i m Azimuth (deg.)(deg.) (LED-event) 78 23 298.2 65 27 307.2 59 30 116.9 93 19 238.4 82 21 138.2 53 31 305.6 74 24 300.5 22 45 318.0 22 45 318.0 77 23 140.7 53 31 004.2 23 44 121.6 24 43 296.7 66 26 305.4 83 21 308.0 5 A s i m i l a r c h a r a c t e r i s t i c i n the v a r i a t i o n of K with time was noted, with-out explanation, by Deas (1969), A decreasing magnetization of the mass may contribute to the decline, while the fluctuations are probably errors i n measurement of parameters of the K. test records (although a f l u c t u a t i n g damping constant could also be contributing to t h i s ) . Accordingly, a least-squares s t r a i g h t l i n e was f i t t e d to the K vs time curves and used i n the c a l c u l a t i o n of the s e n s i v i t y f or each c a l i b r a t i o n . The s e n s i v i t y curves were standardized to. the same gain at 1 hz, the center of the frequency band of .interest, ,and the appropriate corrections were applied to the seismograms., .The response of the three components at each s t a t i o n were thus rendered almost i d e n t i c a l , since for a given s t a t i o n , the s e n s i t i v i t y curves for the three components have very s i m i l a r form. The records were d i g i t a l l y f i l t e r e d using a Lanczos bandpass f i l t e r (Basham, 1967) at 0.5 and 3.0 hz.. .With .the responses of the three compo-nents standardized as described above, the r a d i a l and transverse components of h o r i z o n t a l motion were computed from the North and East components using the great c i r c l e station-.to-event azimuth, . Since e x c e l l e n t frequency and time, domain data are both a v a i l a b l e at LED, the emphasis i n this, study has. been, .concentrated on the c r u s t a l section beneath this station., In the ensuing discussion, the 'Standard' crust i s that portrayed i n Figs. 2(a) and 2(b).. and Table I I , The s e d i -mentary P wave v e l o c i t i e s and layer thicknesses of Jensen and E l l i s (1971) based on w e l l logs have been used and the S wave v e l o c i t i e s and densities obtained i n the same manner as E l l i s and Basham (1968). For. the sub-sedimentary c r u s t a l structure, the model of Cumming and Kanasewich (1966) f o r southern Alberta, extrapolated i n a d i r e c t i o n p a r a l -l e l , to the Rocky Mountains, has been used [Fig. 2(b)], I t i s i n general conformity with the more recent work of Clowes et a l (1968) and of Chandra and Cumming (1971) i n southern Alberta, Both Cumming and Kanasewich (1966) and Chandra and Cumming (1971) have, noted that the base of the Precambrian basement varies i n depth and lacks continuity. The l a t t e r workers also discern a considerable v a r i a b i l i t y in.the thickness of the sub-sediment layer s , and a c r u s t a l thickening which trends eastward from 7 P velocity (km/sec) * Thickness (km) surface }/. j t f r / * > * > * s * * s ' f SEDIMENTS PRECAMBRIAN 6.1 10 SUB-LAYER I ' 6.5 24 SUB-LAYER II 7.2 11 mohorovicic _ _ _ _ _ _ _ _ discontinuity . _ o. Fig. 2(b) Crustal model for southern Alberta (after Cumming and Kanasewich, 1966). TABLE I I Layer parameters of the 'Standard' crustal sections beneath the three stations Layer Formation (top) Thickness (km) P Velocity i (km/sec) Density (gm/cc) LED CHA LAR LED CHA LAR LED CHA LAR 1 Post Alberta 0.62 0.58 0.71 2.7 2.7 2.7 2.16 2.16 2.16 2 Alberta 0.46 0.49 0.49 3.0 3.0 3.1 2.20 2.20 2.22 3 Blairmore 0.31 0.28 0.35 3.7 4.1 4.1 2.32 2.38 2.38 4 Wabumum 0.16 0.23 0.21 5.5 5.6 5.6 2.60 2.62 2.62 5 Ireton 0.17 0.22 0.19 4.0 4.2 4.5 2.36 2.40 . 2.44 6 Cooking Lake 0.26 0.27 0.30 5.3 5.4 5.5 2.58 2.59 2.60 7 Elk Point 0.19 0.19 0.18 4.6 4.7 4.7 2.46 2.48 2.48 8 Cambrian 0.42 0.40 0.37 4.1 4.2 4.2 2.38 2.40 2.40 9 Precambrian 10.0 10.0 10.0 6.1 6.1 6.1 2.70 2.70 2.70 10 Sub-layer I 24.0 24.0 24.0 6.5 6.5 6.5 2.77 2.77 2.77 11 Sub-layer I I 11.0 11.0 11.0 7.2 7.2 7.2 2.88 2.88 2.88 Mantle boundary 8.2 8.2 8.2 3.04 3.04 3.04 9 the Rocky Mountains. In the frequency band of i n t e r e s t , such variations i n thickness have very l i t t l e effect on the f i r s t few seconds of the s e i s -mogram, and the effect on the v e r t i c a l - r a d i a l spectral ratios i s also small. 10 Chapter Two ANALYSIS 2.1 Deficiency of the 'Standard' cr u s t a l model i n explaining the character of the earthquake seismograms„ A s t r i k i n g feature of a l l the seismograms i s the apparent delay of approximately two seconds i n the major a r r i v a l s i n the r a d i a l component compared with the v e r t i c a l . For events with simple waveforms, a detailed correlation of character with this delay i s evident for at least the f i r s t minute of the P coda. For i l l u s t r a t i o n purposes, two earthquakes: a K u r i l e Island event of October 24, 1968 and a Honshu event of October 29, 1968 w i l l be used (Fig. 3). For the Ku r i l e Island event, the delay of the major a r r i v a l s i n the r a d i a l component i s clear. Even detailed character i s replicated with delay for the Honshu event. The r a d i a l -v e r t i c a l r a t i o of the peak amplitudes is. approximately one-half when averaged over a l l seismograms. To quantitatively confirm this v i s u a l observation, cross-correlations between the v e r t i c a l and r a d i a l components were, computed for 6.5-second lengths of data. The. delay of the peak cross-correlation, value generally f e l l between 1.8 and 2.0 seconds for a l l events at a l l stations [Fig. 4(a)]. For the v e r t i c a l and r a d i a l time series normalized to the same power, the value of the cross-correlation i s close to 0,5, indicating the large s i m i l a r i t y i n character. A large discrepancy between the experimental results and theoretical computations based on the 'Standard' crustal model i s evident i n both the, frequency and time domains. A synthetic seismogram for the K u r i l e Island event shown i n Fig. 5, has been generated by the l i n e a r systems technique (Jensen and E l l i s , 19 7.1.) using an input wavelet provided by T„J„ Ulrych. This wavelet was obtained using, the non-linear homomorphic f i l t e r i n g technique (Ulrych, 1971) outlined i n the Appendix. An angle of incidence i of 31.4° upon the Mohorovicic discontinuity i s assumed for this and m a l l subsequent synthetic seismograms, as i t corresponds to the value ap-propriate to the K u r i l e earthquake. The synthetic seismogram does not show the large delayed r a d i a l ar-r i v a l which i s a prominent feature of the K u r i l e seismogram. At a delay KURILE IS Fig. 3 Vertical and radial components of the Leduc seismograms for the Kurile Is. and Honshu events. 12 1.0 0.5 0.0 -3.0 (a) Experimental -7.0 0.5 -6.0 -5.0 -4.0 -3.0 'TIME SHIFT (SEC) 2.0 1.0 0.5 (b) 'Standard' crustal model 0.0 - 3 . 0 -7.0 -6.0 0.5 -5.0 T -4.0 -3.0 TIME SHIFT (SEC) 1.0 0.5 (c) Low velocity layer model 0.0 - 3 . 0 -7.0 -6.0 0.5 n i v — > ' — T — -5.0 -4.0 -3.0 TIME SHIFT (SEC) Fig. 4 Radial-vertical cross-correlations of (a) experimental, (b) 'Standard' crustal model, and (c) low velocity layer model seismograms. KURILE IS - LE D SYNTHETIC - STANDARD CRUST V J I ,5 SEC t WAVELET Fig. 5 Comparison of the 'Standard' crust seismogram with the experimental Kurile seismogram. 14 of about two seconds, i t does, however, e x h i b i t an a r r i v a l that has ap-proximately 1.5 times the amplitude of the r a d i a l onset motion. S i m i l a r l y , the c r o s s - c o r r e l a t i o n of the s y n t h e t i c seismogram [ F i g . 4 ( b ) ] , w h i l e not having a l a r g e maximum at. a 2.0 second l a g , does have i t s maximum near there, although i t i s not very much grea t e r than values at other l a g s . I t was found that these p r o p e r t i e s i n the s y n t h e t i c seismogram and i t s v e r t i c a l - r a d i a l c r o s s - c o r r e l a t i o n are b a s i c a l l y unchanged i f one considers only the sedimentary s e c t i o n of the c r u s t . This suggested that the sedimentary s e c t i o n might be producing the observed e f f e c t s . However, a comprehensive examination of t h i s , p o s s i b i l i t y , w i t h i n the c o n s t r a i n t s imposed by w e l l l o g i n f o r m a t i o n , f a i l e d to produce s i g n i f i -c a n t l y b e t t e r agreement. This i n v e s t i g a t i o n i s the s u b j e c t of S e c t i o n 2.3. The d e f i c i e n c y of r a d i a l motion on the s y n t h e t i c seismograms c a l c u l a t e d f o r the 'Standard' c r u s t a l model i s a l s o evident i n the v e r t i c a l - r a d i a l s p e c t r a l r a t i o s f o r angles of i n c i d e n c e i c l o s e to 20° and 35°. In F i g . 6, m the t h e o r e t i c a l curves f o r the'Standard' c r u s t are compared w i t h the e x p e r i -mental r e s u l t s of E l l i s and Basham (1968) f o r two d i f f e r e n t angles of i n c i -dence. A s l i g h t l y d i f f e r e n t c r u s t a l s e c t i o n and smoothing operator accounts f o r the d i f f e r e n c e between the 'Standard' c r u s t curves presented here and the t h e o r e t i c a l curves of t h e i r work. The c r u s t a l s e c t i o n of Jensen (1971) was p r e f e r r e d s i n c e i t was based on a more d e t a i l e d net of w e l l s around Leduc. Although the peak p o s i t i o n s f o r the two s e t s of curves of F i g . 6 correspond, the amplitudes of the t h e o r e t i c a l curves are too l a r g e , i n d i -c a t i n g that more r a d i a l motion i s present i n the earthquake seismograms than i s p r e d i c t e d by the .'Standard' c r u s t a l model. E l l i s and Basham (1968) used the presence of l a r g e transverse motion to s u b s t a n t i a t e t h e i r i n t e r -p r e t a t i o n of t h i s r a d i a l motion as r e p r e s e n t i n g the conversion of P to S by c r u s t a l inhomogeneities and topographic s c a t t e r i n g . 15 < X CO A -o • m o —i 3D ID 13 CZ a 70° Q experimental 'Standard' crust Q.Q 0 . 5 l.Q n 1 . 5 2 . 0 1 2 . 5 FREQUENCY (HZ) Fig. 6(a) Spectral vertical-radial ratios for the 'Standard' crustal compared with experimental ratios for i «= 20°. m experimental 'Standard' crust 17 2.2 Some explanations of the experimental results. As there are no large amplitude vertical arrivals coincident with the horizontal motions on the simple waveform events (Fig. 3), these displace-ments cannot be attributed to the horizontal component of P motion. The two seconds delay of the prominent horizontal arrivals at a l l stations for a l l azimuths and epicentral distances rules put the possibility that the radial motion consists of scattered P waves. The small delayed arrival on the radial component of the synthetic seismograms for the 'Standard' crust is produced by the reverberations of waves between the surface and the top of the Wabumum (layer 4), and between the surface and the Precambrian basement. These reverberations are sep-arated in time in such a way as to mutually reinforce a wave whose fre-quency content is dominantly 1 hz. The reverberations are shear waves in the upward going leg, thus producing large horizontal motion. A de-tailed analysis of the impulse response of the crust, and its correlation with the crustal section and the synthetic seismograms, is given in Section 2.3. As will be seen in that section, well log information pro-hibits the adoption of a sedimentary model which could enhance this re-verberation effect to a level adequate to explain the appearance of the seismograms. Another possible explanation depends upon shear wave generation, not within the sedimentary section but at a greater depth. In this model, the delay arises not from reverberation times between layers, but in the travel-time difference between S and P waves from the source of shear wave generation. The two-second delay then represents shear wave generation at a depth of approximately 12 km in the 'Standard' crustal ; model, corresponding to the lower boundary of the crystalline basement. This model provides an immediate explanation of the similarity in character and the delay between the vertical and radial motions. Modes of shear wave generation at this depth are examined in Section 2.5, and models obtained which best f i t the experimental results are presented in Section 2.6. 18 2.3 Effect of the sedimentary section. The construction from well log information of a sedimentary section appropriate to studies of short period P coda involves a considerable amount of interpretation. F i r s t l y , the representation of a continuously varying velocity profile as a small number of discrete velocity intervals requires subjective judgement. Secondly, the determination of P wave velocities at approximately 1 hz must be made on the basis of high frequency sonic log measurements. Information upon which to base S velocities and den-s i t i e s for the particular section is absent. The sedimentary section of Jensen (1971) was based on a closer net of wells than that of E l l i s and Basham (1968) and was preferred for this reason. For the purposes of this Investigation, a detailed, accurate sedimentary section at LED was required. For this reason, a slightly d i f -ferent set of c r i t e r i a were used in compiling the section from those used by Jensen (1971). F i r s t l y , whereas Jensen's section was based predomin-antly on geological formations, the new model was based on velocity con-trasts alone. In particular, the actual magnitude of each velocity con-trast was preserved, adjusting the thickness to take account of any fluc-tuations in velocity within a layer. Secondly, rather than compiling an average section from many wells, a single well, G (Fig. 1), the CNS As-sociation Petroleum Corporation Roily View Well, has been used, as i t is within two miles of LED. Correlations between i t and wells H (Big Hay) and I (Imperial Dinant) have been used to complete the section at the upper and lower ends. An empirical study of the relationship between sonic log velocities and those obtained by the conventional 'check shot 1 technique was made by Gretener (1961). Sonic logging tools obtain the average velocity of the rock in a vertical direction over an interval of between three and five feet and at a frequency in the v i c i n i t y of 12 khz. The study of Gretener pertains to measurements made with the Shell sonic logging tool in 50 wells scattered throughout the Alberta plains. The travel time for each check shot interval was computed for both the sonic log and check shot measurements. The times computed conventionally were system-at i c a l l y greater than those from the sonic log by an average of 1.7 ms/1000 f t . with a standard deviation of 1,8 ms/1000 f t . For a material of P wave velocity 3.5 km/sec, this corresponds to a difference of 2% in velocity. Gretener concluded that there existed two p r i n c i p a l causes of this systematic delay. F i r s t l y , he found that at shallow depth a 10% apparent anisotropy was present, consisting of r e a l anisotropy and the effect of layering (ignored i n correcting the check shot times for o f f s e t ) , producing errors i n the conventional survey. Secondly, he assumed that dispersion was contributing to the deviation between conventional and sonic log mea-surements, whose frequencies are separated by a factor of several hundred. Dispersion was studied experimentally by McDonal et a l (1958) and by Wuenschel (1965). McDonal et a l found that i n the range 25 to 400 hz, the phase ve l o c i t y varies by 2.6% i n the Pierre Shale. Wuenschel, using the theory of Futterman (1962) r e l a t i n g the r e a l and imaginary parts of the propagation constant, obtained a t h e o r e t i c a l dispersion r e l a t i o n i n which the phase velo c i t y depends logarithmically on frequency. Using this and the observed dispersion for Pierre Shale, one obtains a difference of the order of 10% for phase v e l o c i t i e s at 1 hz and 10 khz, corresponding to the frequencies of short-period teleseismic signals and the sonic logging instrument respectively. I t i s clear from this that the discrepancies between the phase ve-l o c i t i e s at the two frequencies are much greater i n the Pierre Shale study than i n the findings of Gretener (1961). This may be due to the difference i n Q values of the two sedimentary sections: a higher Q would produce less dispersion. Wuenschel estimates the Q 'of Pierre Shale to be 30, which i s lower than estimates of the Q of the Alberta sediments. Jensen (1971) chose a meanQ of 45 from reports i n the l i t e r a t u r e on lab-oratory studies. This i s i n contrast with a Q of 300 for the southern Alberta sediments as determined by Clowes and Kanasewich (1970) from a comparison of observational and theoretical r e f l e c t i o n seismograms. St r i c k (1971) has found that Q values between 50 and 150, which he considers appropriate to the Alberta sandstones and shales, provide travel-time discrepancies which are compatible with Gretener 1s observations when used i n his PL attenuation function"*" describing the absorption of Strick's power law(PL) attenuation model describes the r e a l and imaginary parts of the propagation constant as having a frequency dependence s l i g h t l y d i fferent from l i n e a r , thus complying with the Paley-Weiner causality condi-tion requiring them to increase slower than the f i r s t power of the frequency as the frequency becomes i n f i n i t e . The demand of constant Q (independent of frequency), with i t s requirement that the r e a l and imaginary parts of the propagation function be l i n e a r functions of frequency, i s s l i g h t l y relaxed. 20 seismic energy in the earth. With this substantiation of the small dis-persion values found by Gretener, i t can be assumed that these values are the most appropriate ones available for the Alberta sediments. This dis-persion i s so small as to be negligible, and velocities obtained from sonic logs can be used in the study of short-period P coda. The appearance of a synthetic seismogram can be explained in terms of i t s impulse response: the impulse response of the 'Standard' crustal model, with i t s major arrivals identified, i s shown in Fig. 7. The large delayed arrivals on the vertical represent reverberations of P-waves between the sedimentary interfaces and the surface. On the radial compo-nent there are two groups of strong arrivals. The f i r s t group, arriving at delays between one quarter and one-half second, are shear waves gen-erated at the sedimentary interfaces, and the second group, arriving between one and two and a half seconds, are reverberations between the surface and sedimentary interfaces, with the upward path as shear motion, and the downward path either as compressional or as shear motion as i n -dicated. It i s the association of this second group of arrivals that produces the large delayed amplitude in the radial component. Dominant among them are the PS reverberation with the base on the sedimentary section, and the PS and SS reverberations with the top of the Wabamum (interface 3). These constructively interfere when the input wavelet has i t s fre-quency content centered at about 1 hz. The impulse response was used as a diagnostic aid in the designing of sedimentary Sections that would enhance the delayed radial motion. As has already been seen, the expected effect of dispersion upon the velocities of a sedimentary section determined from sonic logs can be assumed to be negligible in Alberta. Nevertheless, as a check against gross systematic error in the sonic logs, sedimentary sections were con-structed whose velocities deviated 10% on either side of the sonic log velocities. It was found that delayed radial amplitudes were not en-hanced, and spectral ratios remained unchanged in level and shape, and were only shifted slightly along the frequency axis. The most accurate model derived from the well log data is presented in Table III. It has layers as thin as 12 meters: as was anticipated, 21 P-P reverberations between the surface and the interface numbered. 1 sec Shear conversion from interface numbered. P-S and S-S reverberations between the surface and the interface numbered. Fig. 7 Impulse response of the 'Standard' crustal model with the major arrivals identified. 22 the effect of these in both time and frequency domains is small, because of the high frequency nature of their effects. The synthetic seismogram for a crust with this sedimentary section [Fig. 8(a)] is seen to differ only slightly from that of the 'Standard* crustal model (Fig. 5). Several other models were tried, in which the boundaries of layers were slightly altered, and the number of layers varied. On the suggestion of E. R. Kanasewich (personal communication), a value of Poisson's ratio of 0.3 was used in the sedimentary section. However, none of these models enhanced the delayed radial motion notably, now was any improvement made in the f i t of the spectral ratio curves. The reason is presumably that the delayed ar r i v a l i s indeed due largely to the impulses already described, and the reflection coefficients of the two interfaces that generate them cannot be increased beyond the restraints of well log data. TABLE III Layer parameters of the detailed sedimentary section. Layer P-Velocity S-Velocity Density Thioknet • 1 2.40 1.39 2.10 0.242 V.;. 2.80 .; 1.62 2.17 0.558 . 3 3.20 1.85 2.24 0.107 . " 4 2.80 ;! i.62 2.17 0.149 3.72 2.15 2.31 , 0.034 6 2.82 • 1.63 2.17 0.021 7 • ' 3.82 2.21 2.34 0.207 ' 8 .;. 5.65 ;. ''^'.y- 3.27 2.62 0.168 9 ... ; 4.93 2.95 2.50 0.040 10 . 4.02 2.32 2.36 0.034 11 v 3.77 2.18 2.33 0.052 12 V 4.02 2.32 2.36 0.030 13 4.77 2.76 2.48 0.015 14 4.02 2.32 2.36 . 0.022 15 5.26 3.04 2.58 0.015 16 4.02 • 2.32 2.36 0.012 17 :. 5.26 3.04 2.58 0.030 18 5.87 3.39 2.67 ' 0.241 19 4.60 2.66 2.46 0.192 20 4.10 2.37 2.38 0.421 Basement 6.10 3.52 2.70 24 R -t-INPUT: PULSE ... j • Fig. 8(a) Synthetic seismogram for a crust having a detailed sedimentary section. 5 sec R INPUT PULSE Fig, 8(b) Synthetic seismogram for the 'Standard' crustal model using a long wavelet. 25 2.4 The influence of wavelet shape on the synthetic seismograms. The possiblity that a different wavelet might produce the observed effect in the sedimentary section was also examined. The choice of a wavelet is restricted both in dominant frequency and in duration. A wavelet whose dominant frequency is inappropriate gives a very much diminished resemblance with the Kurile seismogram in the vertical component. Also, the appearance of the vertical component of the Ku-r i l e seismogram (Fig. 3) indicates that a long, 'ringing' wavelet of several cycles duration i s not an appropriate representation of the motion at the base of the crust. If constructive interference phenomena within the sediments are dominant in producing the delayed radial motion, then a longer wavelet would enhance this effect. Accordingly, a synthetic seismogram was generated using Ulrych's wavelet with an additional cycle added onto the t a i l . The five seconds length of this improbably long wavelet ex-ceeds the duration of the dominant radial arrivals of the impulse re-sponse (Fig. 7). The result [Fig. 8(b)] does not display any signif-icant enhancement of the delayed radial motion. It can be concluded that the Ulrych wavelet i s the best available representation of the motion at the base of the crust. In the light of these experiments with the sedimentary section of the crust and with the wavelet, i t is d i f f i c u l t to believe that the sedimentary section i s entirely responsible for the observed de-layed radial motion on the earthquake seismograms. 26 2 . 5 Modes of generation of shear waves at depth. The amplitude of radial motion is far greater than that which could be generated at a single velocity discontinuity of reasonable velocity contrast. For example, i f a P wave is incident upon a velocity decrease of 2 5 % at an angle of 3 0 ° , the ratio of shear to compressional refracted energy is approximately 0 . 1 5 . McCamy et a l ( 1 9 6 2 ) , from whose curves this value is obtained, have shown that variations in Poisson's ratio and density produce very small changes in the Zoeppritz coefficient curves. Thus, in order to explain the experimental data, the effects of inter-ference phenomena within a layer of considerable velocity contrast with the bounding media were examined. F i r s t l y , reverberation effects within a layer were studied. The layer was chosen to have velocity greater than or less than the veloc-i t i e s of the surrounding media, as indicated i n Fig. 9 . In this case, the reflection coefficients at the upper and lower boundaries for waves within the layer have the same sign. For normal incidence of a wave within a layer whose one way travel time i s T , the transmission function varies with frequency as illustrated in Fig. 1 0 , with peaks every J T T 7 corresponding to frequencies at which an integral number of two-way re-verberations represent a travel path of one wavelength. Models were tested which had layers whose thicknesses were chosen so as to optimize the transmission of shear waves at ~ j p p hz. At the same time, the transmission of P waves is near a minimum, as can be seen from Fig. 1 0 . This effect, however, cannot adequately explain the observed shear amplitudes: synthetic seismograms obtained even for extreme Velocity contrast models showed peak radial-vertical ratios of about one-quarter, in contrast with the experimental ratios of one half. A second interference effect has provided good quantitative agree-ment: i t involves the constructive interference between shear waves gen-erated at the upper and lower boundaries of a layer of suitable thickness. For non-critical angles, body waves are transmitted through velocity discontinuities without change of phase, except for the case of shear waves generated by a compressional wave entering a medium of higher veloc-i t y , which suffer a 1 8 0 ° change of phase with respect to the parent com-pressional wave. 27 high velocity low velocity layer high velocity low velocity high velocity layer low velocity Fig. 9 Large velocity contrast layer models. Fig. 10 Transfer function of a large velocity contrast layer. 29 For interleaved layers of alternately high and low velocity (Fig. 9), the phase difference between P and S travel paths within a layer must be (n + 0.5) cycles, n integer, for maximum constructive interfer-ence of generated shear waves for a given harmonic set of frequencies f (see Fig. 11). For a given angle of incidence of upgoing P waves i at the upper boundary of a layer, the travel path must be a fraction r of a P wave-length for constructive interference of S waves to occur, where r is given by r cos i p = (r + n + 0.5)X^ cos i s (1) where and are P and S wavelengths respectively for the nth fre-harmonic, respectively at the upper boundary. The thickness of the layer h i s then given by quency and i and i g are the incident angles of P and S motion , \ (n + 0.5) cos in cos i s h = rAjcos l = — '—— - * b P A^ cos ip - A^ cos i s C2) For near-normal incidence, ( n + o.5) (3) tn oC - P where f is the frequency of the waves, and JL and ft are P and S wave velocities in the layer. Fig, 11 Generation of shear waves at the boundaries of a large velocity contrast layer. 31 2.6 Investigations into layers of large velocity contrast. A modeling investigation was undertaken with the object of exploiting the interference effect that has just been described. Basic crustal mod-els were developed using synthetic seismograms, and these were adjusted to give the best f i t to the experimental data by the use of vert i c a l -radial spectral ratios, which are more sensitive to variations in layer parameters than are synthetic seismograms. Using the delay time of 1.9 seconds obtained from the experimental vertical-radial cross-correlations, the upper surface of the. large ve-locity contrast layer was set. at a depth of approximately 12 km below the surface of the 'Standard1 crustal model. Equation 3 was used to calcu-late the thickness of the layer which would optimize the deepening of one or other of the two troughs, at 1.3 hz and 2.1 hz, in the experimental spectral ratio curves (Fig. 6). The layer which was inserted between the crystalline basement (P velocity 6.1 km/sec) and sub-layer I (P velocity 6.5 km/sec) was allowed to have velocities both lower and higher than those of the surrounding media. The large contrast layer contributes to the large amplitude delayed radial motion by shear wave conversion at i t s upper and lower boundaries. The largest amplitude is obtained i f these arrivals reinforce the rever-berations within the sedimentary column. Thus the optimum depth for a low velocity layer i s 12.2 km, and that for a high velocity layer i s 14.3 km. These depths affect the amplitude of the radial motional c r i t i c a l l y . Profiles and sections of the 'Standard', low and high velocity layer models are shown in Fig. 12, and their radial impulse responses in Fig. 13. The layer parameters of these models, are presented in Table IV. The two impulses representing shear conversion constitute the only significant difference between the high contrast layer and 'Standard' models in the f i r s t few seconds. Models chosen to optimize shear generation at either 1.3 hz or 2.1 hz are comparable in their results. The thin layer gives appreciably better spectral ratio curves for i m = 35° [Fig. 14(b)], especially at the fre-quency of 2.1 hz for which i t is designed. However, E l l i s and Basham (1968) found that for increasing angles of incidence, the coherency 32 10 20 Depth (km) 30 40 50 P velocity (km/sec) 4 • 6 low velocity layer 'Standard' high velocity layer Fig. 12(a) Profiles of 'Standard', low velocity layer, and high velocity layer crustal models. 33 'Standard' model Low velocity layer High velocity layer model model P vel. h P vel. h P vel. h (km/sec) (km) (km/sec) (km) (km/sec) (km) ^3.v•) 3 • S 2• 39 •••« •••• 6.1 10 ' ... 9.6 . . . . 11.7, 5.2 2. 75 ; 7.0 3.7 6.5 24 ... 21.65 ... 18.6 7.2 11 8.2 Fig. 12(b) Sections of 'Standard', low velocity layer, and high velocity layer crustal models. 34 'Standard' model Low velocity layer model 1 sec Fig. 13 Radial components of impulse responses for the 'Standard', low velocity layer, and high velocity layer models. TABLE IV Layer parameters of the low velocity layer (LVL) and high velocity layer (HVL) crustal models Layer Formation (top) P Velocity S V / i l o Density . Thickness (km/sec) (km / sec) (gm/cc) (km) LVL HVL LVL HVL LVL HVL LVL HVL 1 Post Alberta 2.7 2.7 1.56 1.56 2.16 2.16 0.62 0.62 2 Alberta 3.0 3.0 1.73 1.73 2.20 2.20 0.46 0.46 3 Blairmore 3.7 3.7 2.14 2.14 2.32 2.32 0.31 0.31 4 Wabumum 5.5 5.5 3.18 3.18 2.60 2.60 0.16 0.16 5 Ireton 4.0 4.0 2.31 2.31 2.36 2.36 0.17 0.17 6 Cooking Lake 5.3 5.3 3.06 3.06 2.58 2.58 0.26 0.26 7 Elk Point 4.6 4.6 2.66 2.66 2.46 2.46 .0.19 0.19 8 Cambrian 4.1 4.1 2.37 2.37 2.38 2.38 0.42 0 : 4 2 9 Precambrian 6.1 6.1 3.52 3.52 2.70 2.70 9.60 11.70 10 Large vel. contrast 5.2 7.0 3,00 4.05 2.54 2.93 2.75 3.70 11 Sub-layer I 6.5 6.5 3.75 3.75 2.77 2.77 21.65 18.60 12 Sub-layer II 7.2 7.2 4.31 4.31 2.88 2.88 11.00 11.00 Mantle boundary 8.2 8.2 4.73 4.73 3.04 3.04 :i j • '-i j -• ^  ~* 4 1 \ <J1 36 between seismograms for an earthquake event recorded at different stations tended to decrease, implying an increasing amount of scattering. It can be concluded, therefore, that the experimental spectral ratio curves for i = 35° are less reliable than those for i = 20°, and so the better agreement of the thin layer model for i = 35° may not be very significant. Further-more, since the Kurile wavelet has a dominant frequency of approximately 1.5 hz, the synthetic seismogram for the thick layer model (optimizing shear motion at 1.3 hz) shows a larger delayed radial amplitude than that for the thin layer, as is evident-in Fig. 15. The profile of the thick 5.2 km/sec model is shown in Fig. 12. A comparison between the synthetic seismogram for this model and the Kurile earthquake shows that the relative amplitudes of peak radial to vertical motion are equal (Fig. 15). However, while i t is evident that the model produces considerable improvement in the spectral ratio curves, there is s t i l l a large amount of radial energy that remains unexplained: i t may be partially due to scattering caused by inhomogeneities in the crust. Because the peak amplitude motions should be less subjected to scattering than the spectral ratios computed from 20 sec. lengths of record, they are chosen as the criterion for selecting the best model. Increasing the velocity contrast of the layer was found to improve the f i t of model results to experimental ones. For example, in Figs. 16 and 17, the results for thick low velocity layers of velocities 5.2 km/sec and 5.5 km/sec are compared. The improvement of the 5.2 km/sec layer model over the 5.5 km/sec one is clear i n both time and frequency domains. The shape of the spectral ratio curves suffers rapid degradation when the layer thickness is perturbed from i t s optimum value. In Fig. 18, curves are presented for crusts with the thickness of the 5.2 km/sec low velocity layer perturbed by 0.2 km, or less than 10% about the optimum value. For these perturbations, the synthetic seismograms show a cor-responding decrease in delayed radial amplitude. Equally good agreement between model and experimental results is obtained for a high velocity layer modelwith comparable velocity con-trast with the surrounding media. A synthetic"seismogram'and spectral ratio curves for a 3.7 km thick layer of"P wave velocity 7.0 km/sec are presented i n Figs. 19 and 20, and the profile of the model is shown in 37 F i g . H ( a ) Vertical-radial spectral ratios of thick (2.75 km) and thin (1.7 km) 5.2 km/sec low velocity layer crustal models, compared with experimental and 'Standard' results for i m r a 2 0 c . 38 experimental 'Standard' crust — - — • — thin low velocity layer thick low velocity layer o —H ID So r ~ i - — i —i c: a rnN-. X) o o 0 . 5 " T ~ ~ 1 . 0 " ~ 1 2 . 0 0 . 0 " T -1 . 5 I 2 . 5 FREQUENCY (HZ) Fig. 14(b) Vertical-radial spectral ratios of thick (2.75 1cm) and thin (1.7 km) 5.2 km/sec low velocity layer crustal models, compared with experimental and 'Standard* results for in"35* 39 Fig. 15 Synthetic seismograms for thick (2.75 km) and thin (1.7 km) 5.2 km/sec low velocity layer crustal models. V . « Standard' crust 5.2 km/sec low velocity layer R V R 5.5 km/sec low velocity layer Fig. 16 Synthetic seismograms for the 5.2 km/sec and 5o5 km/sec low velocity layer crustal models compared with the espsriscantol seismogram. 41 Fig. 17(a)Vertical-radial spectral ratios for the 5.2 km/sec and 5.5 km/sec low yelocity layer crustal models, compared with experimental results for at 20°. 42 Fig. 17(b) Vertical-radial spectral ratios for the 5.2 km/sec and 5.5 km/sec low velocity layer crustal models, compared with experimental results for i m - 3 5 ° . 43 — experimental — 2.75 km 2.95 km 2.55 km 1 — ~ 1 1 — - i — r~ 0 . 5 1 . 0 1 . 5 2 . 0 2 . S FREQUENCY (HZ) ' Fig. 18(a) Vertical-radial spectral ratios for perturbed thickness of 5.2 km/sec low velocity layer model for. i ^ • 2Q°. 0 . 0 44 experimental 2.75 km ; 2.95 km 2.55 km ° - | , . , , 1 — 1 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 FREQUENCY (HZ) Fig. 18(b) Vertical-radial spectral ratios for perturbed thickness of 5.2 km/sec low velocity layer model for i m • 35°. Fig. 12. The results are comparable with those of the roughly analogous case of a 5.5 km/sec low velocity layer. Notwithstanding this equivalence of results using high and low ve-locity layers, the following chapters w i l l concentrate attention on the low velocity model, as i t appears to conform with other evidence. Using the time-domain amplitude criterion discussed above, the low velocity model that w i l l be examined consists of the standard crust, perturbed so as to contain a layer of velocity 5.2 km/sec and thickness 2.75 km situated below the Precambrian basement at a depth of 12.2 km, as portrayed in Fig. 12 and Table IV. 46 5 sec V R Fig. 19 Synthetic seismogram for the 7.0 km/sec high velocity layer crustal model. Fig. 20(a) Vertical-radial spectral ratios for the 7.0 km/sec high velocity layer model, compared with experimental results for i m - 20°-48 Fig. 20(b) Vertical-radial spectral ratios for the 7.0 km/sec high velocity layer model, compared with experimental results for i m •=> 3 5 ° . 49 2.7 Comparison of experimental and model seismograms. The v e r t i c a l component of the synthetic seismogram for the low v e l o c i t y model gives good q u a l i t a t i v e and quantitative agreement with the K u r i l e earthquake for the f i r s t few seconds. Beyond this point, however, motion of considerable amplitude on the earthquake seismogram i s absent from the synthetic seismogram. This i s understandable since the wavelet used i n the computation of the synthetic seismogram must be much shorter and simpler than the K u r i l e event, which, occurring at a depth of 35 km, must be complicated by motion generated near the source. Furthermore, the synthetic seismogram i s free of scattered energy which i s considered to be p a r t i a l l y responsible f or the large h o r i z o n t a l amplitudes of the seismograms. The r a d i a l motion of the synthetic seismogram does not agree i n character with that of the earthquake. Inaccuracies i n both the c r u s t a l model and the wavelet could be contributing to t h i s . However, the r e l a t i v e amplitudes of peak v e r t i c a l and r a d i a l a r r i v a l s f o r the low v e l o c i t y layer crust are equal to the experimental ones, i n con-t r a s t with those of the 'Standard' crust, as i n d i c a t e d ^ i n Table V. This diff e r e n c e i n delayed r a d i a l amplitude i s evident i f the v e r t i c a l -r a d i a l c r o s s - c o r r e l a t i o n of the K u r i l e seismogram i s compared with those computed from the 'Standard' and low v e l o c i t y layer synthetic seismograms ( F i g . 4). The c o r r e l a t i o n maxima of the experimental and low v e l o c i t y l a y e r model seismograms are equal i n value, being almost twice that f o r the 'Standard', crust case. The delay of the c o r r e l a t i o n peak f o r the synthetic seismogram i s 2.0 s e c , or 0.2 sec. greater than that f or the K u r i l e seismogram. This, delay i s c o n t r o l l e d by the P-S reverberation between the surface and the basement: any attempt to decrease the delay by r a i s i n g the low v e l o c i t y l a y e r results, i n diminishing r a d i a l amplitudes. In order to obtain the observed delay, i t appears that a 10% reduction i n the P t r a v e l time through the sediments i s . required. As was seen i n Section 2.3, such a deviation from known sedimentary p r o f i l e s i s hardly t o l e r -able. The discrepancy i n c o r r e l a t i o n l a g thus remains unresolved. Pos-s i b l y , a more complex large v e l o c i t y contrast structure might resolve TABLE V Radial-vertical amplitude ratios of the experimental and synthetic Kurile seismograms Peak Onset Peak Radial Radial/Vertical Radial/Vertical Onset Radial Ratio Ratio Ratio Kurile Earthquake 0.47 0.13 3.5 'Standard' Crust Synthetic 0.27 0.17 1.6 Low velocity Crust Synthetic 0.46 0.17 2.7 51 both t h i s discrepancy and the lack of s i m i l a r i t y i n character between the synthetic and experimental r a d i a l motions. The main remaining disagreement between a l l the model seismograms and experiment i s i n the v e r t i c a l - r a d i a l onset r a t i o : experimentally, t h i s i s approximately eight, while that calculated (assuming the v e l o c i t y immediately below the Leduc va u l t to be 2.7 km/sec) i s 5.0. Even i f a v e l o c i t y of 2.3 km/sec i s assumed, the r a t i o i s only increased to 5.9. A surface l a y -er of very low v e l o c i t y and thickness at l e a s t one wavelength could cause such an e f f e c t . However, the Leduc vault i s situated on bedrock. The low r a d i a l onset amplitude i s common to a l l events at a l l three s t a t i o n s : the v e r t i c a l - r a d i a l onset r a t i o i s commonly eight to ten, and often cannot be estimated with good accuracy because the noise i s of com-parable amplitude to the r a d i a l onset motion. One explanation of low r a d i a l onset amplitudes i s the p o s s i b i l i t y that seismic waves are t r a v e l i n g upward more st e e p l y than expected. The empirical curve of Ichikawa (Basham, 1967) gives an angle of incidence at the Moho i of 31.4° for the K u r i l e event: this i s confirmed by the m value of 32.75° extrapolated from the angle of incidence at the surface using the tables of Banghar (1970). However, a v e r t i c a l - r a d i a l onset r a t i o of eight,, as observed for the K u r i l e event, requires an i of m 22.5°, for a h o r i z o n t a l Moho, and a dip of 30° of the Moho toward the epicenter for i = 31.4°. The e f f e c t of such a dip on the azimuth m deviations (which: would be. very conspicuous) i s not observed, and furthermore, the v e r t i c a l - r a d i a l onset r a t i o does not show any pro-nounced azimuthal dependence. A systematic east-of-north trend was found i n the azimuths c a l -culated from the h o r i z o n t a l components compared with the great c i r c l e azimuths. The discrepancy was of the order of 10° on the average, which i s i n accord with the measurements by E l l i s and Basham (1968) of an analogous discrepancy averaging 18° for f o r t y events recorded at the same l o c a t i o n . The o r i g i n of such a consistent discrepancy f o r various azimuths i s very d i f f i c u l t to explain. Nevertheless, large discrepancies i n azimuth can be expected i n view of the large dips found by Kanasewich et a l (1969) i n southern Alberta. 0 30 25 20 -I 15 10 Somerville E l l i s and Basham (1969) N A e 6 0 ° ^ dip \ DEVIATION (East of North): \ observed az. - great circle az. ^ A A A A 4 A 4 4 l 0 © / A A' A to 10 o — /-© — A \ 9 \ 4 / \ .ft 30 60 90 120 -5 \ 180 \ AZIMUTH \{deg.) N 210 A 240 274 A 300 330 360 Fig. 21 Azimuth deviations plotted against azimuth for the 15 earthquake events and 36 of the events of E l l i s and Basham (1968). 53 The d i s t r i b u t i o n of azimuth deviation with azimuth i s shown i n F i g . 21, which shows the events of both the present seismograms and those of E l l i s and Basham (1968). Superimposed upon the systematic deviation there appears to be a pseudo-sinusoidal d i s t r i b u t i o n . Interpreted using the tables of N i a z i (1966), t h i s represents a dip of the order of 10° i n a d i r e c t i o n 60° E. of N. Local dips of this magnitude have been determined by Kanasewich et a l (1969) f o r the Mohorovicic and shallower r e f l e c t o r s i n southern Alberta. In the same region, Robertson (1963), using explor-ation seismograms, determined a r e f l e c t o r dipping at 8° to the southeast, from a depth of 8 km to 12 km over a distance of 30 km. 54 Chapter Three DISCUSSION 3.1 Evidence for a low-velocity zone i n continental regions. The assumption that seismic v e l o c i t i e s increase continuously or d i s -continuously with depth i n the crust was f i r s t challenged by Gutenberg (1950). Among other evidence, he observed that P v e l o c i t y measured for shallow earthquakes was 5.5 - 5.6 km/sec, i n c o n f l i c t with the velocity of 6.0 km/sec found i n explosion studies. Mueller and Landisman (1966) summarized evidence from world-wide refraction surveys of large second a r r i v a l s , termed P , which closely c follow the refracted a r r i v a l P i n observations 50 - 150 km from the g shot-point. Further, they presented evidence of strong r e f l e c t i o n ar-r i v a l s at an echo time of 4.0 sec. i n West Germany. They concluded that an explanation of these reflections and refractions required the presence of a we l l developed low velo c i t y channel which overlay a zone of velocity markedly greater than that i n the channel and at least 0.2 km/sec high-er than that above the channel. Landisman and Mueller (1966) showed how the presence of such a layer brought the previously c o n f l i c t i n g e v i -dence of r e f l e c t i o n and refraction determinations of the depth of the discontinuity into accord. More recent seismic r e f l e c t i o n and refraction work i n the Rhine-graben by Mueller et a l (1969) has delineated two velo c i t y inversions i n the upper crust, which e n t a i l velocity contrasts as great as those of the 5.2 km/sec low velocity model proposed i n this thesis. A section through the Rhinegraben r i f t system, taken from the paper of Mueller et a l (1969) i s shown i n Fig. 22. A recent refraction survey i n southern Norway by S e l l e v o l l and Warwick (1971) has indicated the possible existence of a low velo c i t y zone a few km thick at a depth of about 6 km beneath the west coast. The low ve l o c i t y channel (6km/sec) i s overlain by a layer of velocity 6.22 -6.32 km/sec and overlies a layer of velocity 6.51 km/sec. By the inversion of surface wave modes i n southern A f r i c a , Bloch et a l (1969) found that a common cha r a c t e r i s t i c of t h e i r models was the presence of two low v e l o c i t y channels beginning at depths of 12 km 55 Fig. 22 Crustal section through the Rhinegraben r i f t system (after Mueller et a l , 1969). 56 and 24 km. However, models without such channels can be chosen which provide dispersion curves within the error of the experimental data, so t h e i r analysis cannot d e f i n i t e l y e s t a b l i s h the existence of low v e l o c i t y channels. Short-period surface wave, data i n c e n t r a l Europe obtained by Schnei-der et a l (1965) i s consistent with the presence of a low v e l o c i t y chan-n e l . Observed group v e l o c i t y maxima were found to be very close to the resu l t s of t h e o r e t i c a l dispersion computations f o r a model which includes a low v e l o c i t y layer. In North America, comparison between surface wave dispersion data and t h e o r e t i c a l c a l c u l a t i o n s f o r models based on seismic r e f r a c t i o n r e s u l t s have shown discrepancies greater than the uncertainty i n experimental values [Press (1960), O l i v e r et a l (1961), and Dorman and Ewing (1962) J.. At least part of th i s discrepancy could be resolved by the i n c l u s i o n of a low v e l o c i t y zone. The zone should also provide a very e f f i c i e n t channel f o r the transmission of guided waves such as P" and L . g Evidence from other branches of geophysics lends support to the ex-istence of a low v e l o c i t y layer. Steinhart et a l (.1962) have shown that temperature c o e f f i c i e n t s of the seismic wave, v e l o c i t i e s determined i n the laboratory and the known temperature gradients i n the crust make i t prob-able that seismic v e l o c i t i e s decrease with depth. This has led to the suggestion that a low v e l o c i t y channel of thermal o r i g i n might be the s i t e of shallow earthquake a c t i v i t y . A study by Cleary et a l (1964) i n the moderately stable seismic region of south-eastern A u s t r a l i a has shown a very prominent peak at 10 km i n the depth d i s t r i b u t i o n of shallow earthquakes. I t has been found by Mueller and Landisman (1966) that the seismic delay between P^ and P^ measured i n widely separated continental areas i s approximately proportional to the heat, flow through the Earth's sur-face. I t appears that the P -P^ delay may be related to the thermal re-gime i n the low v e l o c i t y channel. The smallest delays are found i n s h i e l d areas, and the l a r g e s t occur predominantly i n t e c t o n i c a l l y active regions. Notwithstanding the above observations,, i t i s d i f f i c u l t to understand how thermal e f f e c t s alone could cause the sharp v e l o c i t y contrasts that most observations require of the low v e l o c i t y channel. 57 Evidence for a low velo c i t y channel has recently been obtained from seismic refraction experiments i n Superior Province, Quebec by Berry (1971). The channel, at a depth of approximately 6 km, has a markedly lower ve-l o c i t y than the surrounding material. Furthermore, deep r e f l e c t i o n results at Yellowknife, N.W.T., by Berry (personal communication, 1971) show re-fle c t i o n s from a depth of approximately 6 km. While evidence for a low velocity layer i n the continental crust accumulates, there appears to be no evidence for a layer of high ve l o c i t y . 58 3.2 Discussion and i n t e r p r e t a t i o n of r e s u l t s . By i n v e s t i g a t i n g synthetic seismograms and v e r t i c a l - r a d i a l s p e c t r a l r a t i o s , a d e f i n i t e preference i s obtained for the large v e l o c i t y contrast model over the 'Standard' c r u s t a l model. The in v e r s i o n i s , by i t s nature, non-unique, and the solutions proposed cannot be considered the best pos-s i b l e ones because parameters such as Poisson's r a t i o and deusity were kept f i x e d and only the gross P wave v e l o c i t y structure of the upper crust was varied. Nevertheless, i n view of the wide geographic occurrence of the shear wave generation phenomenon, i t seems u n l i k e l y that an a l t e r n a -t i v e kind of explanation could prove equally adequate, p a r t i c u l a r l y when the well-defined nature of the sedimentary section i s taken into account. The synthetic seismograms do not show large a r r i v a l s at the t r a v e l time of approximately f i v e seconds f o r P reverberations between the sur-face and the large v e l o c i t y contrast layer. Hence an independent test of the presence of the layer (provided by searching the experimental seismograms f o r these a r r i v a l s ) would not be f r u i t f u l . No c r u s t a l r e f r a c t i o n work has been done i n the Leduc region. The r e f r a c t i o n work of Cumming and Kanasewich (1966) i n southern Alberta does not show any evidence for a large contrast layer. However, Robertson (1963) detected a w e l l defined, continuous r e f l e c t o r over a 30 km p r o f i l e i n southern Al b e r t a , and the work of Kanasewich et a l (1969) i n the same region, while designed for lower c r u s t a l l e v e l s , does i n d i -cate r e f l e c t o r s at a depth of approximately 15 km i n the v i c i n i t y of the base of the Precambrian. I t may be, however, that these r e f l e c t i o n s are due to the v e l o c i t y increase from 6 1 km/sec to 6.5 km/sec at the base of the Precambrian l a y e r . In view of the evidence for a low v e l o c i t y zone i n a number of con-t i n e n t a l regions, and the absence of evidence for a'high v e l o c i t y zone at this depth, i t seems that the large v e l o c i t y contrast l a y e r proposed i n t h i s thesis i s most l i k e l y to be a low v e l o c i t y layer. For thi:, reason, i n the ensuing discussion of the possible composition of the lay e r , a low v e l o c i t y layer w i l l be assumed. This presents more d i f -f i c u l t i e s i n i n t e r p r e t a t i o n than does a high v e l o c i t y l a y e r , to which could be ascribed the composition of some b a s i c material such as gabbro which has a P wave v e l o c i t y of approximately 7;0 km/sec at the appropri-ate depth (Clark, 1966). 59 By virtue of i t s P wave velocity of 6.1 km/sec, i t appears likely that the basement material underneath central Alberta is of granitic composition. The pressure at a depth of 15 km below the earth's surface is of the order of four kilobars: at this pressure, granite has an aver-age P wave velocity of 6,2 km/sec (Clark, 1966). It does not seem possible that the velocity changes entailed i n the low velocity layer model can be explained without resorting to a compo-sitional difference. The possible velocity inversion in granite due to temperature effects proposed by Steinhart et a l (1962) does not seem capable of providing .the sharp velocity decrease that is required by the observations. H.J. Greenwood (personal communication) has pointed out that the only phase change that could occur in the upper crust is the JL~P transition of quartz, which, occurring as temperature is increased, entails a volume increase of 5%. However, for this transition to occur at a depth of 15 km would require a temperature of 700° C (Yoder, 1950) whereas the temperature at this depth is unlikely to be i n excess fof 400° C. Thus the presence of a quartz phase transformation causing a low velocity in the upper crust must be discounted; Turning to a compositional change as an explanation of the low velocity layer, one finds that there are very few materials whose P wave velocities are less than 6.0 km/sec at 4 kb pressure. Greenwood (per-sonal communication) has. suggested the possibility that the material of the low velocity layer might be a greenstone comppsed of chlorite and serpentine derived by the hydration of a rock of gabbroic composition. The work of Fawcett and Yoder (1966) indicates that chlorite i s stable at the pressures and temperatures prevailing at the proposed depth. The gabbro may have been implaced as a s i l l at a depth where the hydrostatic pressure equalled that in the parent magma chamber.. P. wave velocities as low as 5.8 km/sec at 4 kb pressure have been measured for serpentinite (Clark, 1966). This i s , of course, s t i l l considerably larger than the low velocity layer velocities reviewed and proposed in this thesis. Support for the hypothesis of a hydrated layer is lent by recent magnetotelluric work in Alberta. : In a magnetotelluric study at Leduc, Rankin and Reddy (1969) proposed a res i s t i v i t y model in which a downward 60 decrease in resistivity of one or two orders of magnitude occurs at a depth of 5.5 km in a highly anisotropic material. The r e s i s t i v i t y in-' creases again at a depth of 14.5 km. Rankin and Reddy (1969) interpret the results of Vozoff et a l (1963) at the Kavanaugh station (13 km from Leduc, see Fig. 1) as suggesting a low r e s i s t i v i t y zone. The other stations operated by Vozoff et a l (1963), which a l l l i e to the north of the region of the present study, do not indicate the presence of this low res i s t i v i t y zone. Chapter Four CONCLUDING REMARKS 4.1 Summary Using synthetic seismograms and vertical-radial spectral ratios, the 'Standard' crustal model for central Alberta has been shown to be inadequate in explaining prominent features of teleseismic events re-corded in the vi c i n i t y of Leduc. The crustal model, based on the gross crustal structure of Cumming and Kanasewich (1966) for southern Alberta and the sedimentary section of Jensen (1971) does not predict the large delayed radial arrivals that characterize the seismograms. An investigation was made into the possible modes of generation of this radial motion. . Although reverberations within the sediment-ary section cause a considerable amount of shear motion at the observed delay, a detailed examination of the velocity logs in the vi c i n i t y showed i t unlikely that this could be the sole source of the radial motion. The generation of. shear waves at depth was then studied, and the most favourable mechanism was the constructive interference of shear waves generated at the boundaries of a layer of large velocity contrast situated about 13 km below the surface. A layer of low velocity (5.2 km/sec) inserted at the base of the crystalline basement (6.1 km/sec) and overlying sub-layer I (6.5 km/sec) produced delayed radial motion of the required amplitude on the synthetic seismograms and considerably improved the f i t between theoretical and experimental vertical-radial spectral ratios. The choice of a low velocity layer was made in view of the mount-ing evidence in favour of the presence of a low velocity zone of world-wide extent in the upper continental crust. This evidence has recently been augmented by Canadian observations obtained i n Superior Province, Quebec (Berry, 1971) and Yellowknife, N.W.T. (Berry, personal communication). The possible, composition of the proposed low velocity .layer i s extremely problematic. Petrological and magnetotelluric evidence is in favour of a hydrated basic zone as a possible explanation. 62 The anomalously large vertical-radial onset ratios characteristic of the earthquakes recorded in the vi c i n i t y of Leduc remains unexplained. 6 3 4.2 Conclusions. The a p p l i c a b i l i t y of non-normal incidence synthetic seismograms to c r u s t a l studies has been i l l u s t r a t e d . Using a wavelet obtained from the K u r i l e earthquake by the hompmorphic deconvolution technique developed by Ulrych (1971) a synthetic seismogram was obtained whose v e r t i c a l com-ponent agreed very w e l l with the K u r i l e seismogram f o r the f i r s t few sec-onds a f t e r onset. This l e n t confidence to the use of synthetic seismo-grams i n discriminating against unacceptable c r u s t a l models. The v e r t i c a l -r a d i a l s p e c t r a l r a t i o curves for models served as a s e n s i t i v e t e s t of layer parameters. With t h i s integrated approach, a c r u s t a l model was obtained which f i t t e d both synthetic seismograms and s p e c t r a l r a t i o s much more cl o s e l y than the 'Standard' model. The model e n t a i l s a layer of large v e l o c i t y contrast with i t s surrounding layers at a depth of about 12 km. The use of synthetic seismograms lends a considerable amount of confidence to c r u s t a l modelling i n v e s t i g a t i o n s . I t has been demonstrated that at Leduc a h o r i z o n t a l l y layered crust can give f a i r l y good agree-ment with the experimental r e s u l t s . I t should now be possible to de-l i n e a t e more c l e a r l y the l i m i t a t i o n s of a h o r i z o n t a l l y layered model i n c r u s t a l descriptions, and i t can be expected that the quantity and o r i g i n of scattered energy i n the crust may thereby be rendered more susceptible to d e s c r i p t i o n . 64 4.3 Suggestions for further work. The most direct way of confirming the existence of the proposed large velocity contrast layer would be to examine exploration seismograms re-corded near Leduc for deep reflections in the manner of Robinson's work (1963) in southern Alberta. Such records may not be available, or may be of such poor quality that reflections, i f observed at the appropriate depth, may not resolve the high or low velocity nature of--the-layer. A deep reflection survey would probably provide this resolution. Seismic refraction studies could lead to a clearer definition of the velocity of the layer. The areal extent of the proposed layer could be delineated by i n -spection of short period seismograms recorded at other locations, such as those of E l l i s and Basham (1968). However, the characteristic delayed radial motion could be absent in spite of the presence of the layer, since some values of the layer thickness would cause destructive interference of shear waves at about 1 hz. In that case, vertical-radial spectral ratios could be used, although these are always subject to uncertainty as to the quantity of radial motion originating from scattering. In order to find a material having a P wave velocity as low as that suggested for a low velocity zone by many workers (5.5 km/sec), measure-ments of P wave velocity in hydrated ch l o r i t i c greenstones of various compositions should be conducted at about 4 kb pressure and 400° C. 65 REFERENCES Bancroft, A.M. and Basham, P.W. 1967. An FM magnetic tape recording seismograph. Pub. Dom. Obs. 35_, pp. 199 - 217. 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Seismic studies of the earth's crust i n continents, Part I: Evidence for a low-velocity zone i n the upper part of the lithosphere. Geophys. J . R. Astr. S o c , 10, pp. 525 - 548. Mueller, S., Peterschmitt, E.,,, Fuchs, K. and Ansorge, J.. 1969. Cru s t a l structure beneath the Rhinegraben from seismic r e f r a c t i o n and r e f l e c -t i o n measurements. Tectonophysics, J5, pp. 529 - 542. N i a z i , M. 1966. Corrections to apparent azimuths and travel-time gra-dients f o r a dipping Mohorovicic d i s c o n t i n u i t y . B u l l . Seism. Soc. Am., 56, pp. 491 - 509. Oliver,. J . , Kovach, R. and Dorman, J... 1961. Cru s t a l structure of the -New York - Pennsylvania area. J . Geophys. Res. 66, p. 215. Press, F. 1960. Cru s t a l structure, i n the C a l i f o r n i a - Nevada region. J. Geophys. Res., 6\5, p. 1030. Rankin, D. and Reddy, I.K. 1969. A magnetotelluric study of r e s i s t i v -i t y anisotropy. Geophysics, _34_, pp. 438 - 449. 67 Robertson, G. 1963. Intrabasement r e f l e c t i o n s i n southwestern Alberta. Geophysics, 18, pp. 910 - 915. Schneider, G., Mueller, S. and Knopoff, L. 1966. Gruppengeschwindig-keitsmessungen an Kurzperiodischen Oberflachenwellen i n Mitteleuropa, Z. Geophys., 32, pp. 33 - 57. S e l l e v o l l , M. A. and Warwick, R. E. 1971. A r e f r a c t i o n study of the c r u s t a l structure of southern Norway. B u l l . Seism. S o c Am., 61, pp. 457 - 471. Steinhart, J.S., Green, R. , Asada,. T., Rodrigues, A., A l d r i c h , L.T. and Tuve, M.A. 1962. The earth's crust: seismic studies. Carnegie I n s t i t u t i o n of Washington Year Book, 61_, pp. 221 - 231. S t r i c k , E. 1971. An explanation of observed time discrepancies between continuous and conventional w e l l log surveys. Geophysics, _36^  pp. 285 - 295. Ulrych, T.J. 1971. Ap p l i c a t i o n of homomorphic deconvolution to s e i s -mology. Geophysics, 3_6, pp. 650 - 660. Vozoff, K., Hasegawa, H. and E l l i s , R.M. 1963. Results and l i m i t a t i o n s of magnetotelluric surveys i n simple geologic s i t u a t i o n s . Geophysics, 28, pp. 778 - 792. Wuenschel, P.C. 1965. Dispersive body waves - an experimental study. Geophysics, 3i0_, pp. 539 - 551. Y6der, H.S. 1950. High-low quartz in v e r s i o n up to 10,000 bars. Trans. Amer. geophys. Union, _31, pp. 827 - 835. 68 APPENDIX Homomorphic deconvolution. The wavelet used i n the generation of synthetic seismograms was ob-tained by Ulrych (1971) by applying the homomorphic deconvolution tech-nique to the experimental K u r i l e seismogram. This appendix describes the p r i n c i p l e s of this non-linear f i l t e r i n g technique. When approaching the problem of f i l t e r i n g signals that have been ad-d i t i v e l y combined, we usually choose a l i n e a r system because of the ana-l y t i c convenience of the f a c t that l i n e a r systems s a t i s f y the p r i n c i p l e of superposition. This p r i n c i p l e requires that i f <fi i s the system trans-formation, then for any two inputs x^(t) and x^(t) and any s c a l a r c, 0 [ X ; L ( t ) + x 2 ( t ) ] = jrJ [ x r ( t ) ] + <f> [ x 2 ( t ) j and $ [ c.x(t) ] = c . ^ [ x(t) ] This p r i n c i p l e , as i t stands, applies only to signals that have been a d d i t i v e l y combined. For t h i s reason, when a f i l t e r i n g procedure to sep-arate signals that have been combined by other means (such as by m u l t i p l i -cation or convolution) i s to be chosen, i t i s usually more d i f f i c u l t , and often l e s s meaningful, to use a l i n e a r system. In our case, the seismogram i s conceived as being generated by the convolution of a wavelet with an impulse series.. Without making p r i o r assumptions, the separation of these two components i s not possible using the concept of l i n e a r f i l t e r i n g that has been described. Nevertheless, we can expand the concept of l i n e a r f i l t e r i n g so that i t encompasses signals combined i n ways other than by addition. Our f i r s t step i s to generalize the p r i n c i p l e of superposition so that i t applies to operations other than addition of inputs, and m u l t i p l i c a t i o n ' of inputs by s c a l a r s . Such a general p r i n c i p l e of superposition i s 4> [ x±(t) „ x 2 ( t ) ] = <f>[ X ; L ( t ) ] 0 <f> [ x 2 ( t ) ] 69 and ^1 cAx(t) J = c*f[ x(t) J , where ° and * are generalized operations. We now. regard the linear super-position principle as the one in which ° corresponds to + , and * corres-ponds to •. The second step in linearizing our problem is to find a transforma-tion of the inputs into a vector space in which the rule of combination of inputs becomes the rule of vector addition, and the rule of combina-tion of inputs and scalars becomes scalar multiplication. (The equivalent requirements for the set of outputs is guaranteed by the requirement that the inputs constitute a vector space .and that the transformation satisfy the principle of superposition). We are thus selecting from the group of a l l systems which satisfy the principle of superposition the subgroup of homomorphic systems which are those ones which can be represented as linear transformations between vector spaces. A non-linear homomorphic f i l t e r i n g process is one which: i) by A, transforms the space of input vectors into one in which the law of combination of vectors is addition (corresponding to a linear space). i i ) by L, performs the required, f i l t e r i n g operation, linearly, and i i i ) by A \ inverts from the additive space back to the original one. Thus a canonic representation of the non-linear homomorphic f i l t e r i n g process can be depicted as shown in Fig. Al(a). The f i l t e r A i s character-i s t i c of the class in that i t depends only on the operations ° and *, and hot on the details of the required f i l t e r i n g operation. A is such that A[ x 1 » x 2 ] = A[ x 1 ] + A[ x 2 ] and A[ c*x ] = c.A[ x ] A _ 1 is such that A _ 1 [ A[ x(t) ] ] = x(t) 70 o + h + A L A" 1 1 • • 1 F i g . Al(a) Canonic representation of the homomorphic f i l t e r i n g process. Pi 7 tts. + + + + + ,+ +/ + log 7 In L • * Fig. Al(b) Characteristic system for a homomorphic deconvolution f i l t e r . / 71 A common approach to deconvolution i s the method of inverse f i l t e r i n g , which requires a d e t a i l e d d e s c r i p t i o n of the component to be removed. In the case of seismic deconvolution, t h i s d e s c r i p t i o n i s obtained by: i ) assuming a random series of spikes to represent the r e f l e c t i n g l a y e r s , so that the wavelet autocorrelation can be obtained from the trace autocorrelation; and i i ) i n ignorance of the phase spectrum of the wavelet,, assuming i t to be minimum phase, so that i t can be obtained e a s i l y from,its power spec-trum. The homomorphic. method of deconvolution obviates the..need of making such assumptions. The c h a r a c t e r i s t i c system for two convolved signals s and n can be represented as. i n F i g . Al(b) . The t h i r d operation within the c h a r a c t e r i s t i c f i l t e r A produces the 'complex cepstrum' of the input n*s : although i t may appear to. be. operationally superfluous ...to. the f i l t e r -ing process, i t has great a n a l y t i c convenience. Consider the. echoed signal.generated by the convolution of:a. wavelet with a p a i r of impulses as shown i n . F i g . A2, This s i g n a l can be w r i t t e n : x(t) = s ( t ) +. s( t V r ) = s ( t ) * [ J ( t ) +4<J<t-V)] Taking i t s Fourier transform, X ( w ) = S ( W ) [ .1 + - ( e " 1 W r ] The complex logarithm of th i s i s log[ X(w) J = log|s(W)| + log( 1 +o<e" i < 0 r) The second term i n t h i s expression i s p e r i o d i c i n with a r e p e t i t i o n rate p roportional to the delay time . The spectrum of this term would therefore e x h i b i t a peak at this delay time, and would thus prove useful i n i n t e r p r e t a t i o n . Because we are thereby thinking i n s p e c t r a l terms while we are r e a l l y i n a time domain, t h i s i s c a l l e d the complex 'cepstrum'. The cepstrum i s the inverse Fourier transform of the complex logarithm of the Fourier transform of the function x ( t ) , i . e . F" 1 ( log[ F( x(t) •)] j 72 Input wavelet Impulse series Convolution of wavelet with impulse series Fig. A2 Synthesis of a signal using a wavelet and an impulse pair. 73 The complex cepstrum of the s i g n a l of F i g . A2 i s shown i n F i g . A3, In t h i s case, because the spectrum of the wavelet i s of lower frequency content than that of the impulse p a i r , the spike ser i e s i n the complex cepstrum i s w e l l separated from the complex cepstrum of the wavelet, which appears at the c e n t r a l part of the cepstrum. By low-pass f i l t e r i n g the complex cepstrum and applying the inverse characteristic, f i l t e r A: \ the wavelet i s recovered ( F i g . A3). S i m i l a r l y , by applying a high-pass f i l t e r and then the inverse f i l t e r , the impulse p a i r i s obtained. The K u r i l e earthquake seismogram recorded at Leduc,. i t s complex cepstrum, and the deconvolved wavelet and impulse seri e s are shown i n Fi g . A4. 74 Deconvolved wavelet Fig. A3 Complex cepstrum and recovered wavelet from the signal of Fig. A2. 

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