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UBC Theses and Dissertations

Evaluation of a rectilinear motion detector Fuchs, Jens Peter 1969

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EVALUATION OF A RECTILINEAR MOTION DETECTOR t>y JENS PETER FUCHS Vordiplom, Teehn-ische Un-iversitSt Munich, 1964 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF . MASTER OF SCIENCE i n the Department of Geophysics We accept th i s thesis as conforming to the required standard University of B r i t i s h Columbia A p r i l , 1969 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d 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 t u d y . 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 t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t 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 . 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 V a n c o u v e r 8, Canada D e p a r t m e n t ABSTRACT One c r i t e r i o n f or distinguishing underground nuclear explosions from earthquakes i s that most earth-quakes occur at considerably larger depths. Focal depth can he determined with seismograms i f both the P and the pP a r r i v a l are c l e a r l y distinguishable. "REMODE 5A" (REctilinear Motion DEtector), a time-varying p o l a r i z a t i o n f i l t e r f or the detection of P-type motion, i s applied to a r t i f i c i a l inputs and earth-quakes. It i s investigated how va r i a t i o n of the f i l t e r parameters affects the output. The length of the time window within which the cross-correlation f i l t e r operator i s calculated must not be small compared to the signal period, especially i f the onset of a quake i s to be enhanced. If the window i s very short, the f i l t e r outputs w i l l be e r r a t i c . It makes l i t t l e difference whether the truncated f i l t e r operator i s tapered at the ends or not. REMODE 5A removes background noise and much of the signal-generated noise, but i t s e f f i c i e n c y i n picking pP from complicated P coda i s not s i g n i f i c a n t l y d i f f e r e n t from more elementary versions of REMODE f i l t e r s . i i T h r o u g h c h o i c e o f p a r a m e t e r s , REMODE 5A c a n be made a r b i t r a r i l y s e l e c t i v e t o t h e d e g r e e o f r e c t i l i n e a r i t y and t h e i n c i d e n t a n g l e o f s i g n a l and n o i s e . However, s i n c e t h e s o u g h t - f o r pP and P w i l l o f t e n be c o n t a m i n a t e d by s i g n a l -g e n e r a t e d n o i s e , t o o h i g h l y s e l e c t i v e a f i l t e r may e a s i l y r e j e c t s i g n a l . i i i TABLE OF .'CONTENTS CHAPTER I . INTRODUCTION 1-1 Problem and Background : 1 1-2 -REMODE F i l t e r s A g a i n s t Frequency Band F i l t e r s 4 1-3 Reference to P r e v i o u s P u b l i c a t i o n s 6 1- 4 Aim, O u t l i n e , and Scope of Th i s Study 8 CHAPTER I I REMODE FILTERING 2 - 1 F i l t e r Inpu t from Rotated." Seismograms 9 2-2 C r o s s - C o r r e l a t i o n , the B a s i s o f REMODE 11 2-3 N o r m a l i z e d REMODE F i l t e r s 13 2-4 " A " M o d i f i c a t i o n to Suppress SV 15 2-5 "REMODE 5A" , a H ighe r Order F i l t e r 17 2 - 6 D i r e c t i o n a l P r o p e r t i e s 19 CHAPTER I I I TESTING TECHNIQUES 3- 1 REMODE 5A Parameters . . 2 2 3-2 Purpose and Procedure of T e s t i n g 25 3- 3 F i l t e r Inpu ts as Employed f o r T e s t i n g 26 CHAPTER IV RESULTS OF TESTS WITH REMODE PARAMETERS 4 - 1 D i r e c t i o n a l P r o p e r t i e s : INTANG 29 4-2 D i r e c t i o n a l P r o p e r t i e s : M 37 4-3 T a p e r i n g the Edges of the F i l t e r Opera tor 45 i v 4-4 Normalization : Summation Limits 5 3 4-5 Time'Window and Number of Lags 56 4-6 Number of Convolutions 6 2 CHAPTER V . SUMMARY AND CONCLUSIONS ( 5-1 Summary N 6 8 -5-2 Conclusions 7 1 REFERENCES 7 2 V LIST OP FIGURES FIGURE 1 Schematic i l l u s t r a t i o n of P and pP trav-e l paths. FIGURE 2 Orientation of the P ray towards the seismogram coordinate system a f t e r r o -tat i o n s . 10 FIGURE 3 Detection of P and SV motion by product of R and Z component of motion ( a f t e r Basham, 1967). 16 FIGURE 4 Ha l f - c y c l e cosine function i l l u s t r a t i n g approximate response of normalized RE-MODE processors ( a f t e r Basham, 1967). 18 FIGURE 5 An example of an a r t i f i c i a l f i l t e r i n -put c o n s i s t i n g of three s i n u s o i d a l seg-ments. 27 FIGURES 6 THROUGH 11 An example of the e f f e c t of varying the s p e c i f i e d angle of incidence INTANG-FIGURE 6 F i l t e r input, f o r the outputs shown i n Figures 7 through 11. FIGURE 7 FIGURE 8 FIGURE 9 FIGURE 10 FIGURE 11 Output f o r INTANG=15 Output, f o r INTANG=40C Output f o r INTANG=45C Output f o r INTANG=50C Output f o r INTANG=75C 31 32 33 34 35 36 FIGURES 12 THROUGH 16 An example of the e f f e c t of varying the cosine power. M governing d i r e c t i o n -a l d i s c r i m i n a t i o n . FIGURE 12 FIGURE 13 FIGURE 14 F i l t e r input f o r the outputs shown i n Figures 13 through 16. Output f o r M=0, and' no convolution. Output f o r M=3, and no convolution. 38 39. 40 v i FIGURE 15 FIGURE 16 FIGURE 17 FIGURES 18 FIGURE 18 FIGURE 19 FIGURE 20 FIGURE 21 FIGURE 22 FIGURE 23 FIGURE 24 Output f o r M=0, and. 3 convolutions. Output f o r M=3, and 3 convolutions. Schematic representation of cos ( I I N T A N G ' -B E T A I ) ( a f t e r G r i f f i n , 1966a). THROUGH 24 Examples of the e f f e c t of dif f e r e n t , types of tapering. F i l t e r input f o r the outputs shown i n Figures 19 through 24. Output f o r NO TAPERING, and no convolution Output f o r HAMMING window, and no convo-l u t i o n . Output f o r DANIELL -window, and no convo-l u t i o n . Output f o r NO TAPERING, and 2 convolutions Output f o r HAMMING window, and 2 convo-l u t i o n s . Output, f o r DANIELL window, and 2 convo-l u t i o n s . FIGURE 25 Schematic representation of the ef f e c t s of d i f f e r e n t types of normalization app-l i e d to an input with extremely low noise l e v e l . FIGURES 26 THROUGH 29 Examples of the e f f e c t of varying window v/idth LVY and maximum l a g LAGS. FIGURE 26 FIGURE 27 FIGURE 28 FIGURE 29 F i l t e r input f o r the outputs shown i n Figures 27 through 29. Output f o r 1W=40, and LAGS=24. Output f o r LW=20, and LAGS=10. Output f o r LW=4, and LAGS=4. FIGURES 30 THROUGH 32 Examples of the e f f e c t of varying the number NO of convolutions. FIGURE 30 FIGURE 31 F i l t e r input f o r the outputs shown i n Figures 31 and 32. Output.. FZ f o r NC=0 Output FZ f o r NC=I (31a). (31b). v i i FIGURE 32 Output FZ f o r NC=2 ( 3 2 a ) Output FZ f o r NC=3 ( 3 2 b ) v i i i ACKNOWLEDGEMENTS . The author wishes to thank Dr. R.M. E l l i s f o r his assistance i n defining the project f o r t h i s thesis, fo r his valuable advice during the research, and for his c r i t i c a l appraisal of the manuscript. Gratitude i s also expressed to Dr. R.D. Russell, Head of the Department of Geophysics, and to his predeces-sor Dr. J.A. Jacobs, for the use of departmental f a c i l i t i e s . The author was supported f i r s t by a joint schol-arship of World University Service, UBC, and German Academic Exchange Service, and l a t e r on by a National Research Council graduate studentship. 1 CHAPTER I ... INTRODUCTION 1-1 Problem and Background Seismic records are always noise-contaminated. Sometimes the noise l e v e l i s so high that i t i s impossible to recognize any s i g n a l on the record. Fortunately, through f i l t e r i n g one can produce a modified record on which the s i g n a l i s easier to recognize by in c r e a s i n g the s i g n a l - t o -noise r a t i o . To detect the pP phase and the onset of the P phase on such noise-contaminated record^ c a r e f u l f i l t e r i n g i s often needed. I f both these phases are d i s c e r n i b l e one can determine the f o c a l depth of the source from the time elapsed between the two. Accurate knowledge of f o c a l depth i s an important, clue f o r deciding, on the basis of seismic records, whether a given event was an earthquake or a nuclear explosion. The p r i n c i p l e of how to determine f o c a l depth from the time elapsed between the a r r i v a l s of P and pP i s i l l u s t r a t e d i n Figure 1. The pP i s an echo of the E = Centre of the explosion or earthquake (Focus). R = Point of r e f l e c t i o n at the surface. S = Seismograph s t a t i o n . This f i g u r e i s not drawn to true s c a l e . Figure 1. Schematic i l l u s t r a t i o n of P and pP trav-e l paths. 3 P, by r e f l e c t i o n at the surface above the source. Imagine the same f i g u r e drawn to true scale. Obviously then the path ES i s almost i d e n t i c a l to AS, and so i s ER to AR. Thus, the f o l l o w i n g approximation w i l l apply, and i t w i l l be the more accurate the shallower the source w i l l be : The r e f l e c t i o n point: i s v e r t i c a l l y above E, and the time d i f f e r e n c e between the P and the pP a r r i v a l s i s the time i t takes a compressional wave to t r a v e l twice the epicen-t r a l depth. I.e., the f o c a l depth i s h ^ t ( p P - P ) V/2 where t(pP-P) i s the time difference between the a r r i v a l s of P and pP, and V i s the average source-to-surface ve-l o c i t y . More accurate f o r determining the f o c a l depth h i s the equation t ( p P - P ) = V cose " where $A = 2h'tan 9 , h' i s a f i r s t , approximation to h , B i s the i n c l i n a t i o n to the normal of the ray r e f l e c t i n g from the surface, and: dt/dA i s the inverse phase veloc-i t y of P as a fun c t i o n of A . The i n c l i n a t i o n angle can be determined from the equation s i n B = V (dt/d^) (Basham and E l l i s , 1 9 6 9 ) . V/e must, understand that the P and the pP can be 4 buried: under noise of two d i s t i n c t l y d i f f e r e n t , kinds, background noise and: signal-generated noise. Background, noise i s a l l those v i b r a t i o n s which are picked up and recorded by the seismograph and do not originate-from, the.-earthquake or. the explosion, but from wind or water waves, or i s c u l t u r a l noise l i k e t r a f f i c and construction. Signal-generated noise i s the product of m u l t i -ple r e f l e c t i o n s and r e f r a c t i o n s that take place at the i n t e r f a c e s and inhomogeneities of c r u s t a l , l a y e r s . Most " of t h i s o r i g i n a t e s underneath the s t a t i o n . The primary P, being the f i r s t of a l l pulses to a r r i v e at; the s t a t i o n , w i l l only be affected by back-ground noise and not by the l a t e r a r r i v i n g signal-gener-ated noise. The problem i s to pick the l a t e r a r r i v i n g pP from the P coda. The. P coda i s formed by the reverber-ations near, the source and the signal-generated noise from the v i c i n i t y of the r e c e i v e r . 1-2 REMODE F i l t e r s Against. Frequency Band F i l t e r s D i g i t a l f i l t e r s f o r the computer processing of seismic records can be designed to pass any s p e c i f i e d frequency band: and to r e j e c t , to a greater or l e s s degree, 5 other frequencies. The spectrum of background noise i s often concentrated around frequencies d i f f e r i n g from the dominant-period of P and pP signals (Basham, 1967, Chapter I I I ) . Consequently, frequency band f i l t e r i n g can serve to reduce background'noise that, contaminates pP and P. Unfortunately, frequency f i l t e r i n g i s not e f f e c -t i v e i n separating the pP from that, part of the P coda which i s signal-generated noise. This i s so because,in p r i n c i p l e , t h e r e i s no frequency d i f f e r e n c e between P, pP, and the signal-generated noise of the P coda : we remem-ber that: the pP as well as the signal-generated noise (SGN) stem from r e f l e c t i o n s or r e f r a c t i o n s of the P. Neither r e f l e c t i o n s nor r e f r a c t i o n s cause any changes i n frequen-c i e s . However, there-' i s one strong point of d i s t i n c t i o n between pP and SGN that can be exploited f o r f i l t e r i n g purposes : the motion of pP i s r e c t i l i n e a r l y p olarized, whereas the motion of SGN i s l i k e l y to be e l l i p t i c a l l y or. randomly p o l a r i z e d . The random or e l l i p t i c a l p o l a r i z a -t i o n of the SGN i s caused by superposition of many r e c t i -l i n e a r components which stem from the multiple r e f l e c t i o n s and r e f r a c t i o n s i n the c r u s t a l l a yers. REMODE f i l t e r s ("REctilinear MOtion DEtector") discriminate against s i g n a l f o r which the input motion i s not. r e c t i l i n e a r along the ray path, but random and e l l i p t i c a l . 6 When the input motion i s predominantly r e c t i l i n e a r large spikes are produced f o r output. These spikes do not. ne c e s s a r i l y have the true shape of the input, pulses. Shear waves and compressional waves, i . e . S and P, are both r e e t i l i n e a r l y p o l a r i z e d and thus v / i l l both be .amplified by a.REMODE f i l t e r . But i n S waves p a r t i c l e s move along s t r a i g h t l i n e s perpendicular to the d i r e c t i o n of propagation, whereas i n P waves the p a r t i c l e s move c o l i n e a r l y v/ith the propagation vector. This d i f f e r e n c e permits one to suppress S and; pass P (includi n g pP) by adding to the REMODE f i l t e r a r e l a t i v e l y simple modifica-t i o n c a l l e d a "motion product operator". The motion product operator- i s explained i n Chapter I I . REMODE f i l t e r s are time-varying and thus need f a r more computer time than the simpler frequency f i l t e r s which remain unchanged along the record. 1-3 Reference to Previous Publications The p r i n c i p l e of REMODE f i l t e r s was developed by researchers at Teledyne Industries : Mims and Sax (1965), G r i f f i n (1966a, 1966b), Sax (1966). An account i n phys i c a l and mathematical terms of how the REMODE process works i s given i n Chapter II of the present study. At t h i s point, i t s h a l l only be stated that the mathematical formulation of a REMODE processor contains several parameters. Their 7 magnitudes can be i n d i v i d u a l l y varied to a considerable extent; t h i s allows a wide choice of REMODE types d i f f e r -i n g i n t h e i r f i l t e r i n g q u a l i t i e s . G r i f f i n (1966a) discusses a group of such f i l t e r s (REMODEs 2,2A, 3,3A) of intermediate s e n s i t i v i t y . Based on t h e o r e t i c a l considerations he recommends a set of para-meter values l i k e l y to produce good r e s u l t s i n seismic data processing. Basham (1967) applied one of these f i l t e r s (REMODE 3A) , v/ith parameters as suggested, by G r i f f i n , to the records of 4-0 teleseismic events; he detected the phase pP on the records of 25 events, 16 of which had reported depths of l e s s than 40 km. G r i f f i n (1966b, and 1966a) also suggests a f i l t e r d i s c r i m i n a t i n g much more strongly than those previous types against, frequency components which are not purely r e c t i l i n e -ar i n t h e i r p o l a r i z a t i o n , or which a r r i v e from d i r e c t i o n s d i f f e r i n g from a s p e c i f i e d angle of incidence. In order to evaluate the p o t e n t i a l of REMODE 5 , he performed t e s t -runs with a r t i f i c i a l records. The l a t t e r were steady state sinusoids under r e c t i l i n e a r l y p o l a r i z e d noise. But because t h i s a r t i f i c i a l noise was r e c t i l i n e a r , G r i f f i n ' s r e s u l t s are not an evaluation of how powerful REMODE 5 might be f o r p i c k i n g pP from n o n - r e c t i l i n e a r signal-generated noise. 8 1-4 Aim, Outline, and Scope of This Study The purpose was to determine how much better than the l e s s s e n s i t i v e REMODE 3A might REMODE 5A be f o r detecting pP. Spe c i a l a t t e n t i o n was given to two questions : (I) -Is i t . worthwhile to have the f i l t e r very s e n s i -t i v e ? (Higher s e n s i t i v i t y requires more com-puter time). ( i i ) Which parameter Values are a good compromise between the power of dis c r i m i n a t i o n against, non-r e c t i l i n e a r motion and computational expense ? Chapter II presents i n a condensed form the mathematical p r i n c i p l e s underlying REMODE f i l t e r i n g i n general, and i n p a r t i c u l a r the highly s p e c i a l i z e d REMODE 5A. • In order to answer questions ( i ) and ( i i ) , vary-in g modifications of REMODE 5 A were applied to test, inputs. Chapter I I I outlines t e s t i n g techniques. The r e s u l t i n g f i l t e r outputs are plotted and discussed i n Chapter IV. Summary and. conclusions c o n s t i t u t e Chapter V. 9 CHAPTER II REMODE FILTERING Since the:: mathematics of REMODE have been d i s -cussed i n previous p u b l i c a t i o n s , ( G r i f f i n 1966a and 1966b, Basham 1967), only a b r i e f account w i l l be given here. 2-1 F i l t e r Input from Rotated.Seismograms REMODE f i l t e r s are two-dimensional : two input traces and two output traces. The input i s the represen-t a t i o n by two perpendicular components, R and Z, of the wave motion i n c i d e n t at. the s t a t i o n . R and Z are i n the v e r t i c a l plane through the incident, ray, and the ray i s the b i s e c t o r between the two (Figure 2). For a seismic ray a r r i v i n g at the s t a t i o n from any d i r e c t i o n t h i s R,Z representation can be obtained from a three-component s e i s -mogram by two successive rotations of the coordinate system. The R,Z b i s e c t o r i s rotated into the apparent d i r e c t i o n of incidence which i s determined by f i t t i n g a s t r a i g h t l i n e to the v e r t i c a l - r a d i a l p a r t i c l e motion. (For d e t a i l s see/ Basham, 1967). 10 Figure 2. Orientat ion of the P ray towards the seismogram coordinate system after r o t a t i o n s . 11 Assuming that, the l a y e r i n g of the crust, under-neath the s t a t i o n i s h o r i z o n t a l , the boundary conditions at. the in t e r f a c e s require that, i n theory, only P-type and- SV-type motion w i l l be generated as a result, of mul-t i p l e r e f l e c t i o n s and r e f r a c t i o n s of. an incident, compres-s i o n a l wave. A l l such P and SV motion w i l l be i n the ver-t i c a l plane through the ray and. thus w i l l be represented i n t o t a l by the two components R and Z. Thus, with com-pr e s s i o n a l phases, a l l a v a i l a b l e s i g n a l energy i s u t i l i z e d . In p r a c t i c e , some scattered SH energy w i l l also be present and record on the three-component seismogram. But i t . w i l l not. show on e i t h e r of the two REMODE input traces because the R-Z plane i s perpendicular to such SH motion. A c t u a l l y a side effect, of the seismogram r o t a t i o n s , t h i s suppression of SH i s desirable when one wishes to apply REMODE to the detection of pP and. P. 2-2 Cross-Correlation, the Basis of REMODE : The even part of the c r o s s - c o r r e l a t i o n f u n c t i o n c(T) between R(t) and Z(t) i s large compared to the odd part when the p o l a r i z a t i o n of motion i n the R-Z plane i s predominantly r e c t i l i n e a r ; and vic e versa, when motion of e l l i p t i c a l and random p o l a r i z a t i o n p r e v a i l s , the even part of c(T) i s small r e l a t i v e to the odd part ( G r i f f i n , 1966a, Basham, 1967) . Consequently, i f we convolve ( i . e . , use as a f i l t e r operator) t h i s even part of c(T) with the 12 input R ( t ) , Z ( t ) , the output w i l l be large only where the p a r t i c l e motion i s predominantly r e c t i l i n e a r (e.g. P,pP, SV). E l l i p t i c a l motion w i l l be suppressed. With various seismic phases a r r i v i n g at d i f f e r -ent times t , the type of p o l a r i z a t i o n and so the r e l a t i v e magnitude of the even part of c(T) w i l l vary with time. In order to have large output only from the r e c t i l i n e a r phases, a time-varying f i l t e r i s required. The f i l t e r operator c(T) i s calculated anew at every point t wit h i n a segment (or "time window") LW• around t, : t+LW/2 (1) c(+T) = X ± Z(t)R(t+T) t-Lw72 Instead of e x t r a c t i n g the even part from c(T) we use a simpler technique which was demonstrated ( G r i f f i n 1966a, Basham 1967) to r e s u l t , i n a s a t i s f a c t o r y f i l t e r f u nction : c(T) i s computed f o r p o s i t i v e lags +T , and then an even function i s generated by r e f l e c t i n g c(+T) at, the o r i g i n according to (2) c(-T) = c(+T) Because of t h i s symmetry, REMODE f i l t e r s are phase d i s t o r -t i o n l e s s . • The l a r g e s t l a g T f o r which c(T) i s calcu-13 l a t e d must, be smaller than the window length LW since f o r larger' lags e i t h e r R or Z would contribute to the c r o s s - c o r r e l a t i o n v/ith terms which are outside the i n t e r -v a l LW around the point, t, f o r which the p a r t i c u l a r f i l t e r operator- c(T) a p p l i e s . G r i f f i n (1966a) recommends to keep T = LW/2 i n order to r e t a i n within the sum (1) max ' v ' s i g n i f i c a n t portions of the record segment centered on t f o r both R(t) and Z(t) . The f i l t e r output FR(t),FZ(t) i s computed accord-ing to T=+LAGS F R ; T=-LAGS (t) = 3* R(t-T) c(T) T=+LAGS (3) FZ(t) = ^> Z(t-T) c(T) T=-LTGS The f i l t e r operator c(T) i s a d i f f e r e n t function f o r every data point, t. For t h i s reason, R E M O D E processing requires much more computer time than frequency band f i l t e r s which apply the. same operator along the entire record. 2 - 3 Normalized R E M O D E F i l t e r s The output, of the f i l t e r described i n Section 2-2 14 w i l l depend on the amplitudes of R and Z as well as on the degree of r e c t i l i n e . a r i ty of the motion. In prac-t i c a l a p p l i c a t i o n s (e.g. picking pP) one may wish to discriminate only "by r e c t i l i n e a r i t y and not by amplitudes. Then, large-amplitude n o n - r e c t i l i n e a r noise camouflaging a small-amplitude pP could be suppressed and the ( r e c t i -l i n e a r ) pP detected. This feature i s r e a l i z e d through m u l t i p l y i n g the unnormalized output by normalization f a c t o r s XNORM(t). PR(t) = PR( t)-XNORM(t) (4) PZ(t.) = PZ(t)-XNORM(t) (XNORM(t)) are approximately proportional to the r e c i p -r o c a l s of s i g n a l energy averaged over the time windows LW within which the c r o s s - c o r r e l a t i o n f i l t e r functions are computed;. : ' t+LW/2 t+LW/2 (5) (XNORM(t)O2 = 1 / l / ^ > R 2 ( t ) Z 2 ( t ) t-LW/2 t-LW/2 Disadvantages of normalization are that scattered pulses (SGN) of weak amplitude but high degree of r e c t i -l i n e a r i t y may be unduly enhanced, and the o r i g i n a l shapes of pulses are not, l i k e l y to be preserved. 15 2-4 "A" M o d i f i c a t i o n to Suppress SV SV, being r e c t i l i n e a r , w i l l pass through an unmodified REMODE equally w e l l as P and pP. The modi-f i c a t i o n "A" (so designated by G r i f f i n ) exploits the fact, that. SV motion i s perpendicular to the ray, where-as compressional P p a r t i c l e motion i s along the ray. The product, R(t)Z(t) i s > 0 when the motion at. t i s compressional, and < 0 f o r SV ; Figure 3 (adopted from Basham, 1967)• In order to suppress• SV a simple "motion product operator" might, thus be used : FR A(t) = FR(t) f o r R(t)Z(t) £ 0 F Z A ( t ) = FZ(t) FR A ( t ) = 0 f o r R(t ) z ( t ) £ 0 F Z A ( t ) = 0 Instead of the motion product R(t)Z(t.) of a si n g l e point t , i n p r a c t i c e we use the zero-lag cross-c o r r e l a t i o n c a l c u l a t e d within the usual time window 1Y/ : F R A ( t ) = FR(t) t+LY//2 f o r c (0) =• ^ > R(t)Z(t) F Z A ( t ) = FZ(t) i ^ l W 7 2 " (6) F R A ( t ) = 0 F Z A ( t ) = 0 f o r c (0) < 0 16 SURFACE SIGNS OF COMPONENTS PHASE z R ZXR +P + + + -P - - + +SV - + --SV + _ — Figure 3. Detection of P and. SV motion by product of R and Z component of motion ( a f t e r Basham, 1 9 6 7 ) . 1 7 The f i l t e r could he made to r e j e c t P,pP and pass SV by simply s e t t i n g the output zero f o r c(0)s£o , and v i c e versa. 2-5 "REMODE 5A" , a Higher Order F i l t e r G r i f f i n (1966a) shows that the response of a normalized A-type REMODE f i l t e r can be approximated by a quarter-cycle cosine f u n c t i o n (Figure 4). Frequency components R(fe>),Z(oa) of the input pass i n proportion to the cosine of the phase difference &y~( R(<w) - ^ z(eo)) between them. This of course means passing r e c t i l i n e a r r ather than e l l i p t i c a l motion : &^f=0 , cosA^=l when the frequency component's motion i s purely r e c t i l i n e a r , and ^f=90°, c o s ^ f = 0 f o r c i r c u l a r p o l a r i z a t i o n . E l l i p -t i c a l l y p o l a r i z e d motion may s t i l l come through at appre-c i a b l e amplitilde, e.g. 1/2 f o r A j f=60°. Higher order REMODE f i l t e r s , l i k e REMODE 5A, discriminate more r i g i d l y against frequency components f o r which (&)) i s between 0° and 90° : t h e i r response 2 v a r i e s as c o s ^ C + 1 ^ | instead of cos( ^ • - £ f z ). This narrowing of the "phase window" i s achieved by convolving the f i l t e r f u n c t i o n ( i . e . the symmetric cro s s - c o r r e l a t i o n ) c(T) with i t s e l f i n NO stages ( G r i f f i n , 1966b) : ( 7 ) ( c ( T ) ) n + 1 = ( c ( T ) ) n ^ f ( c ( T ) ) n where n = 0 NC 18 0 40 80 120 160 PHASE DIFFERENCE BETWEEN Z AND R INPUT SIGNALS (DEGREES) Figure 4. Half-cycle cosine function i l l u s t r a t i n g approximate response of normalized REMODE processors ( a f t e r Basham, 1967). 1 9 Truncation to o r i g i n a l length 2 I n v a f t e r max each s e l f - c o n v o l u t i o n i s needed because each such convo-l u t i o n doubles the length of the f i l t e r operator. Taper-ing the truncated ends i s optional (Section 4-3). The l a r g e r NC the more s e n s i t i v e the f i l t e r w i l l be against n o n - r e c t i l i n e a r motion. But every increase i n NC i s expensive i n computer time since the e n t i r e process i s c a r r i e d out f o r every s i n g l e point t anew. 2-6 D i r e c t i o n a l Properties REMODE 5A discriminates, by m u l t i p l i c a t i o n with a f a c t o r l i k e (8) , against input a r r i v i n g at an apparent angle of incidence BETA whenever BETA d i f f e r s from a s p e c i -f i e d d i r e c t i o n ANGINC : (8) c o s M ( | BETA - ANGINC | ) Any value f o r M can. be chosen, thus allowing the pass range of t h i s d i r e c t i o n a l f i l t e r to be as narrow as desired. BETA, the apparent angle of incidence at each point t , i s found from the input R,Z by t+LW/2 X Z R 2 ( t ) ( 9 ) BETA = BETA(t) =. t a n " 1 -z 2 ( t ) t-LW/2 20 Instead, of (8) we use (10,11) ( G r i f f i n , 19661)), thus i n c l u d i n g a d i s c r i m i n a t i o n against S phases, while the d i r e c t i o n a l c h a r a c t e r i s t i c s of (8) remain un-changed. (10) . c o s M ( | BETA + ANGINC (2 REJECR - 1) | ) where 0 f o r P-type input (11) REJECR = -1 f o r S-type input The d i s t i n c t i o n (11) i s made according to the motion product operator (pp 15-17) being p o s i t i v e (P) or negative (S). The two possible cases f o r (10) are (10a) c o s M ( | BETA - ANGINC | ) f o r P or (10b) c o s M ( | BETA + ANGINC | ) f o r S Assuming that the seismogram rotations (p9) placed the b i s e c t o r between R and. Z ( i . e . BETA=45°) exactly into the i n f l u x d i r e c t i o n of seismic energy, we s h a l l s p e c i f y ANGINC=45°. Then, any phase approaching along the b i s e c t o r i a l ray w i l l be passed v/ith f u l l ampli-tude i f i t i s of the P type (10a) , or t o t a l l y suppress-ed i f i t i s S (10b). Pulses a r r i v i n g at BETAS not equal to ANGINC, i . e . noise, w i l l be suppressed the more strongly the l a r -2 1 ger M i s . However, M should not he chosen too large i n order to r e t a i n P motion of which the BETAs d i f f e r only l i t t l e from ANGINC; f o r due to scattered energy the BETAs from ( 9 ) may o s c i l l a t e about the average value ANGINC even i f the P i t s e l f i s exactly within'the bisec-t o r i a l ray. 22 CHAPTER I I I TESTING TECHNIQUES 3-1 REMODE 5A Parameters In the author's REMODE 5A program, the s i x i n t e -ger parameters LW, LAGS, NC, M, IPTAP, INTANG (=ANGINC) could he i n d i v i d u a l l y adjusted. Each of them would i n f l u -ence the f i l t e r c h a r a c t e r i s t i c s . Their magnitudes were printed out together v/ith the plotted outputs of each test run (Chapter 4). LW i s the length ( i n data i n t e r v a l s ) of the time window within which the c r o s s - c o r r e l a t i o n c(T) i s taken (pl2) . G r i f f i n (1966a) points out. that, the r e l a t i o n of LW to s i g n a l duration and period of noise w i l l deter-mine how e f f e c t i v e l y a REMODE f i l t e r w i l l i s o l a t e a weak s i g n a l . He argues that the window should he short enough to he just f i l l e d by the s i g n a l being sought because a long window over short and weak s i g n a l can sample polar-i z a t i o n more representative of noise than of s i g n a l and as a consequence the f i l t e r might well r e j e c t both s i g n a l and noise; too short a window, however, might miss part of. 22 a s i g n a l . LAGS i s the l a r g e s t l a g T m a x f o r which the c r o s s - c o r r e l a t i o n c(T) i s calcula t e d ( p l 2 f ) . The length of the f i l t e r operator i s 2 LAGS. NC , being e i t h e r a one or a two d i g i t number, indicates the number NC of convolutions of the f i l t e r operator with i t s e l f (pp 17,19) as well as the kind of normalization applied ( p l 4 f ) . Three types of normaliza-t i o n were t r i e d ; they d i f f e r i n the l i m i t s of the sums i n formula (5) . These l i m i t s as determined by the f i r s t d i g i t (or by NC being one d i g i t instead of two) are i l l u s t r a t e d i n the f o l l o w i n g table. FIRST DIGIT = 1 , or NC ONE DIGIT ONLY t t-LW/2 FIRST DIGIT = 2 t+LW/2 > n t-LW/2 FIRST DIGIT = 3 t+LW/2 > t •The second d i g i t , or NC i t s e l f where only one d i g i t , i s the actual number of convolutions. Where f u r t h e r on we r e f e r to NC we mean the l a t t e r . 24 LW , LAGS , and NC strongly influence f i l t e r response, whereas the remaining three parameters M , IFTAP , INTANG are of l e s s e r significance'. This, w i l l he I l l u s t r a t e d by test r e s u l t s i n Chapter 4-_M , we remember, determines the d i r e c t i o n a l properties of the f i l t e r (Section 2-6). Possible unde-s i r a b l e consequences of having too large an M were pointed out on p21. INTANG (=ANGINC) designates i n degrees the " s p e c i f i e d d i r e c t i o n " as explained i n Section 2-6. IFTAP determines the form of tapering which may be applied to the truncated f i l t e r operator before i t i s convolved v/ith the time s e r i e s . Tapering i s c a r r i e d out by superimposing on the f i l t e r operator a m u l t i p l i c a t i v e window of equal length. IFTAP WINDOW 0 NO TAPERING 1 D(T) = 1 - T / T m a x LENGTH=2T m a x '2 "HAMMING" D(T) = 0.54 + 0.46 cos(jrT/T ) 3 "HANNING" D(T) = 0.50 ( l + cos ( T T T / T m a x ) 4 DANIEL! D(T) = sin( T/T f f i a x) / ( T / T m a x ) 25 The reason f o r tapering i s that the sharp edges of a truncated and untapered time domain f i l t e r operator may introduce undesirable GIBBS1 o s c i l l a t i o n s into i t s frequency response. Tapering those edges strongly reduces GIBBS' o s c i l l a t i o n s . ( A-more de t a i l e d dissuss.ion of tapering Is given by Blackman and: Tukey ( 1 9 5 9 ) ) . 3 - 2 Purpose and Procedure of Testing The o v e r a l l goal of this study was to f i n d as t e f f i c i e n t as possible a f i l t e r which would detect pP and P onsets of earthquakes and explosions. An attempt v/as made to determine how the f i l t e r parameters (Section 3 - 1 ) influence i t s c h a r a c t e r i s t i c s . The l a t t e r v/ere evaluated by applying REMODE 5A to selected input, and then from the output determining the e f f e c t of the p a r t i c u l a r set of parameters employed. Performing several test, runs with i d e n t i c a l input, but varying parameters would demonstrate trends. A r t i f i c i a l records (Section 3 - 3 ) v/ere employed f o r the majority of te s t s , and earthquake records only f o r a few. The reasons f o r this are the fol l o w i n g . Detec-t i o n of pP and. P means to i d e n t i f y presence and exact a r r i v a l time. I t i s important to v e r i f y that the f i l t e r output w i l l not r e g i s t e r P type pulses which are not . 26 part of the input, or suppress them when they are present, or record t h e i r a r r i v a l times i n c o r r e c t l y . These q u a l i t i e s of the f i l t e r can only be v e r i f i e d i f every aspect of the input Is known ( A r t i f i c i a l Inputs) ; earthquake records, since they are d i s t o r t e d by noise, do not f u l f i l l the l a t t e r requirement. 3-3 F i l t e r Inputs as Employed f o r Testing Three types of inputs v/ere used : ( i ) EARTHQUAKE RECORDS Three component seismograms d i g i t a l l y stored on magnetic tape were bandpass p r e f i l t e r e d against back-ground noise, and properly rotated as to convert them into the two component REMODE inputs. The procedure f o r p r e f i l t e r i n g and r o t a t i o n was the same as described by Basham (1967). ( i i ) ARTIFICIAL EVENTS WITHOUT NOISE Up. to three s i n u s o i d a l segments of dif f e r e n t , p e r i o d i c i t i e s , s t a r t i n g points, and lengths, would be added to generate eit h e r input trace R,Z (Figure 5 ) . Varying degrees i n r e c t i l i n e a r i t y of p o l a r i z a t i o n could be chosen i n d i v i d u a l l y f o r each s i n u s o i d a l segment by ad-j u s t i n g the phase diff e r e n c e between i t s R and Z com-ponents . Figure 5. An example of an a r t i f i c i a l f i l t e r input c o n s i s t i n g of three s i n u s o i d a l segments. 28 The symbols printed on the l a b e l s that go with the computer-plotted inputs are explained i n the f o l l o w i n g table. The l a b e l ' s f i r s t , second, and t h i r d l i n e s of numerical values each describe one of the three s i n u s o i d a l segments (Figure 5). Total length of each R,Z i s 1,000 data points. ( i i i ) ARTIFICIAL EVENTS WITH BACKGROUND NOISE Seismic noise as plo t t e d on traces labeled NOISR and NOISZ was added to the R and Z sums of the sinus-o i d a l segments. In order to be as r e a l i s t i c as possible, background noise from quakeless periods on s-ei sinograms was chosen instead of random noise. S Y M B O L MEANING NUMAQK IDENTIFICATION NUMBER OF RECORD NT PERIOD OF SINUSOID (DATA POINTS) AR (AZ) AMPLITUDE OF R (Z) SEGMENT JSTTR (JSTTZ) START POINT OF R (Z) SGMT JSTPR (JSTPZ) END POINT OF R (Z) SGMT FILEG PHASE DIFFERENCE (DEGREES) BE-TWEEN THE SEGMENT'S R , Z 2 9 CHAPTER IV RESULTS OP TESTS WITH REMODE PARAMETERS 4-1 D i r e c t i o n a l Properties : INTANG INTANG (=ANGINC) designates the " s p e c i f i e d " d i r e c t i o n as explained i n Section 2-6. Any P-type. pulse a r r i v i n g at an apparent angle of incidence BETA i s atten-uated by a f a c t o r (see p20) (10a) cos M( | BETA - INTANG | ) V a r i a t i o n of INTANG v/hen f i l t e r i n g an earth-quake (Figure 6) affec t e d the outputs as recorded i n Figures 7 through 11. The d r a s t i c reduction of the input to e s s e n t i a l l y three spikes A,B,C i s due to the number of convolutions NC=3 which narrows the "phase window" ( P i 7 ) to cos16(yR - y z ) . FIGURE INTANG M NC IFTAP 7 15° 8 40° 9 45° 6 3 0 10 50° 11 75° 30 The absolute magnitudes of the spikes f o r d i f -f e r e n t INTANGs s h a l l not be compared to each other since the p l o t t i n g required i n d i v i d u a l s c a l i n g (to 38mm maximum amplitude on each trace) . The maxima of each A,B,C r e l a t i v e to the other two spikes occur at d i f f e r e n t values of INTANG , as i s l i s t e d i n the fol l o w i n g table. Based on the i m p l i c a t i o n of (10a), the apparent angles of incidence f o r A,B,C can be estimated. , • SPIKE INTANG f o r which the r e l a t i v e maximum occurs Estimated BETA A 15° Below 40° B 50° Between 45° and 75° C 75° Above 50° This p l a c i n g of the BETAs belonging to the spikes A,B,C had to be crude because of wide spacing of the INTANG values t r i e d . BETA could be determined more accurately i f we would cover the range from 0° to 90° more c l o s e l y with d i s c r e t e values of INTANG , e.g. 0 , ... , n AOC, ... , 90° I f then the r e l a t i v e maximum (compared to the other spikes) 31 Figure 6. F i l t e r input f o r the outputs shown i n Figures 7 through 11. 32 1 eg A < Figure 7. S p e c i f i e d angle of incidence: output f o r INTANG=15°. 33 A m Figure 8 . S p e c i f i e d angle of incidence: output f o r INTANG=40°. 34 Figure 9. S p e c i f i e d angle of incidence: output f o r INTANG=45°. 35 Figure 10. Spe c i f i e d angle of incidence: output f o r INTANG=50°. 3 6 Figure 11. S p e c i f i e d angle of incidence: output f o r INTANG=75°. 3 7 of a p a r t i c u l a r pulse v/ere found on the output f o r INTANG = n <AoC, then i t s apparent angle of incidence could be assigned within (n-l ) A O C 6^. BETA ^ (n+1) £>><X This technique might also be applied to determine the azimuths of such i n d i v i d u a l spikes. The two components used f o r input would then have to be i n that plane through the ray which i s perpendicular to the one employed so f a r . If. a BETA thus determined d i f f e r s very much from 45° th i s i s strong evidence that such pulse c o n s t i -tutes scattered energy (SGN). Further c l a r i f i c a t i o n of whether or not such a spike would represent one of the basio seismic phases can be obtained from t r a v e l time tables i f the f o c a l depth of tlie event; i s known. 4-2 D i r e c t i o n a l Properties ; M V a r i a t i o n of the cosine power M (Section 2-6) when f i l t e r i n g an earthquake (Figure 12) affected the outputs as recorded i n Figures 13 through 16. FIGURE M INTANG NC LW LAGS IFTAP 13 0 0 14 3 45 0 20 10 0 15 0 3 16 3 3 38 Figure 12. F i l t e r input f o r the outputs shown i n Figures 13 through 16. 2,9 Figure 13. D i r e c t i o n a l d i s c r i m i n a t i o n : output M=0, and no convolution. 40 Figure 14. D i r e c t i o n a l d i s c r i m i n a t i o n : output f o r M=3, and no convolution. 41 o It) Figure 15. D i r e c t i o n a l d i s c r i m i n a t i o n : output f o r M=0, and 3 convolutions. Figure 16. Di M= r e c t i o n a l d i s c r i m i n a t i o n : 3, and 3 convolutions. output f o r 43 For e i t h e r NC=0 or NC=3 , the outputs pro-duced with M=0 or- M=3 are p r a c t i c a l l y i d e n t i c a l . This may he explained, as follow s . Inspection of those pulses that are passed by the f i l t e r reveals that f o r a majority the R amplitudes approximately equal the Z amplitudes; i . e . these pulses • apparent incident angles BETA £ ^ 4 5 ° . Together with the s p e c i f i e d INTANG=45° , the d i r e c t i o n a l f a c t o r (10a) then assumes the fo l l o w i n g values : M DIRECTIONAL FACTOR 0 3 cos°( | INTANG - BETA J ) = 1 cos 3( J INTANG - BETAj ) & 1 Such minor differences f o r M=0 or M=3 explain the qu a s i - i d e n t i t y of records with eith e r value M . The r e l a t i v e lack of d i r e c t i o n a l discrimination, even f o r M=3 , against pulses a r r i v i n g at an apparent angle of very close to 4 5 ° , can be q u a n t i t a t i v e l y v e r i f i e d i n Figure 17 (Adapted from G r i f f i n , 1966b). Greater d i r e c -t i o n a l s e n s i t i v i t y can be achieved through large M . How-ever, this may have disadvantages which were pointed out on ,p21. 44 Figure 17. Schematic representation of cos (jINTANG BETA|) ( a f t e r G r i f f i n , 1966a). 45 4-3 Tapering the Edges of the F i l t e r Operator ; IFTAP Reasons f o r tapering the truncated f i l t e r oper-ator, as well as the fu n c t i o n of the parameter IFTAP , were explained i n Section 3-1. D i f f e r e n t types of tapering (also no tapering), with a non-convolved as well as a twice-convolved f i l t e r , were t r i e d on an a r t i f i c i a l record that consisted of three s i n u s o i d a l segments of d i f f e r e n t p e r i o d i c i t i e s superim-posed; on seismic noise (Figure 18). : Some r e s u l t s are presented i n Figures 19 through 24. FIGURE IFTAP TYPE OF TAPERING CONVOLUTIONS 19 0 NO TAPERING 20 2 HAMMING 0 21 . 4 DANIELL 22 0 NO TAPERING 23 2 HAMMING 2 24 4 DANIELL Within each of the two s e r i e s (NC=10, NC=12) the three output records processed with d i f f e r e n t types of tapering are, i f not exactly then f o r a l l p r a c t i c a l purposes, i d e n t i c a l . In f u r t h e r tests i t v/as established that t h i s holds true also f o r two other types of tapering 46 Figure 18. F i l t e r input f o r the outputs shown i n Figures 19 through 24. 47 Figure 19. E f f e c t of tapering: output f o r NO TAPER-I N G , and no convolution. Figure 2 0 . E f f e c t of tapering: output f o r HAMMING window, and no convolution. 49 Figure 21. E f f e c t of tapering: output f o r DANIELL window, and no convolution. 50 Figure 22. E f f e c t of tapering: output f o r NO TAPER-ING, and; 2 convolutions. 51 Figure 23. E f f e c t of tapering: output f o r HAMMING window, and 2 convolutions. 52 Figure 24. E f f e c t of tapering: output f o r DANIELL window, and 2 convolutions. 53 (IFTAP=1 and IFTAP=3 ; see p24) , and f o r higher order f i l t e r s ( i . e . a larg e r number of convolutions). 4-4 Normalization : Summation Limits (XNORM(t)) are proportional to the r e c i p r o c a l s of averaged: s i g n a l energy (Formula 5, pl4) : (5) (XNORM(t)) 2 = 1 / I J ~ R 2 ( t ) » y ~ Z 2 ( t ) For-simple harmonic (sinusoidal) o s c i l l a t i o n s , the XNORM(t) are p roportional to the r e c i p r o c a l s of averaged s i g n a l amplitude; f o r non-harmonic seismic motion this w i l l s t i l l be approximately true. Such averaged amplitude should represent as close-l y as possible the s i g n a l amplitude at point t; f o r thi s reason, the sums are taken over a time window i n c l u d i n g t . Tried: v/ere three window types to which v/e assign the f o l -lowing symbols : WINDOW SYMBOL t; t+LW/2 t+LW/2 I ~z 5 4 Through t r i a l runs we found that f o r sev e r a l earthquakes tested i t did not matter whether <£^-4~y , or — 7 — ^ was employed. I f , however, earthquakes with sharp onsets on a very low-noise l e v e l (per se, or a f t e r p r e f i l t e r i n g ) are to he processed, the 4^-^> type i s to he preferred. This i s 'suggested by the fo l l o w i n g considerations (Figure 25). T h e o r e t i c a l l y , we can predict the e f f e c t of < ^ , —^ > , or ^ on such input; t h i s i s i l l u s t r a t e d i n Figure 25, and was confirmed by experiments v/ith a r t i -f i c i a l inputs. ( i ) < ^ ( i ± ) < ^ : XNORM v / i l l be extremely large v/hen the f i l t e r operator reaches A , because the l e f t oriented window samples only very small amplitudes into the denominator's sum (5) while to the l e f t of A ; t h i s causes the spike A' . Once to the r i g h t of A , la r g e r values are sampled, thus preventing f u r t h e r spikes l i k e A' . : The r i g h t - o r i e n t e d h a l f of the centered window prevents case ( i ) at A , and the l e f t - o r i e n t e d h a l f prevents case ( i i i ) at B . ( i i i ) ^ : The counterpart to ( i ) , with spike B' 55 A INPUT J U l f B OUTPUTS FOR DH ^ I ( i ) For < ^ A _ : ^ J CERENT TYPES OF NORMALIZATION ( i i ) For f ( i i i ) For • B' Figure 25. Schematic representation of the e f f e c t s of different, types of normalization app-l i e d to an input v/ith extremely low noise l e v e l . 56 from B "because of the r i g h t - o r i e n t e d window then covering the very-small amplitude segment to the r i g h t of B . f o r an onset detector. However, as we understand from the previous explanations, the l e f t - o r i e n t e d spike A' w i l l only occur when f i r s t l y the background noise i s of extreme-l y small amplitudes, and secondly the onset A i s sharp. When-these two conditions are f u l f i l l e d , an onset detector i s unnecessary. In order then to avoid seemingly e r r a t i c f i l t e r responses of types ( i ) or ( i i i ) , we recommend to always 4 - 5 Time Window and Number of lags : LW and LAGS (=T ) ZI—IL . 2 . i—max— From G r i f f i n ' s comment, (p22) on the most e f f e c -t i v e length LW of the time window we. understand that t h i s strongly depends on the p a r t i c u l a r signal-noise character-i s t i c s of a record. Consequently, we f e e l that t r i a l and error, rather than a t h e o r e t i c a l c a l c u l a t i o n , may be the best v/ay to optimize LW f o r any p a r t i c u l a r a p p l i c a t i o n . should not be much la r g e r than LW/2 i n order to r e t a i n within the c r o s s - c o r r e l a t i o n s i g n i f i c a n t portions of both the R and Z segments centered on t . On a f i r s t look, i t i s tempting to t r y use For LAGS , G r i f f i n (1966a) recommends that i t 57 An earthquake record (Figure 26) v/as subjected to f i l t e r i n g with d i f f e r i n g LW and LAGS as l i s t e d i n the f o l l o w i n g table. FIGURE LW LAGS NO IFTAP 27 40 24 28 20 10 1 0 29 4 4 LW=40 (Figure 27) confirms G r i f f i n ' s point that the length of the time window should be such that i t . i s of the same length as the s i g n a l period. The dom-inant s i g n a l period, i s 1cm , and 1W=40 corresponds to s l i g h t l y l e s s than 1cm (1cm = 46.3 po i n t s ) . This ' output shows the onset of the quake very conspicuously and! also enhances other r e c t i l i n e a r motion that i s less well d i s c e r n i b l e on the input. LW=20 . This record, even though i t r e t a i n s most of the pulses which v/ere conspicuous on the preceding one, i t does not enhance but rather minimize the onset. An explanation may be that the shorter window could not sample enough r e c t i l i n e a r P motion to the r i g h t of the onset point; consequently, at this point the even part of the c r o s s - c o r r e l a t i o n c(T) f a i l e d to be large, r e s u l t i n g i n the output's small onset. \ Figure 26. F i l t e r input f o r the outputs shown i n Figures 27 through 29. Figure 27. E f f e c t of varying window width LW and maxi-mum l a g : output f o r LW=40, and LAGS=24. 60 Figure 28. E f f e c t of varying window width LW and maxi-mum l a g : output f o r LW=20, and LAGS=10. 61 62 LW=4 , LAGS=4 (Figure 29) makes the outputs FR,FZ look rather uncorrelated, even though some of the la r g e r peaks show on "both traces simultaneously. This suggests that the f i l t e r response i s random and dominated by noise, thus again supporting G r i f f i n ' s argument that LW should be long enough to cover a f u l l s i g n a l period. 4-6 Number of Convolutions : NC The s e n s i t i v i t y of the "phase window", i . e . the pov/er to discriminate- against motion which i s not s t r i c t l y r e c t i l i n e a r (Section 2-5) , i s determined by the number of convolutions of the f i l t e r operator v/ith i t s e l f : An increase of NC by only one u n i t may make the f i l t e r very much sharper, as i s i l l u s t r a t e d i n th i s table : FILTER RESPONSE oC cos NUMBER OF CONVOLUTIONS PHASE WINDOW NC 0 1 2 3 NC=3 seems to be the maximum number of convol-utions to produce outputs that are r e a l i s t i c i n terms of 63 pulses, passed or r e j e c t e d v/hen the f i l t e r i s used as a P wave detector. This v/as established i n t e s t s . G r i f f i n (1966a) states : "As the number of convolutions i s i n -creased to two and three, ... , and s i g n a l waveform (be-comes) more e r r a t i c . The o v e r a l l e f f e c t i n t h i s case i s ... a l o s s i n s i g n a l coherence between s t a t i o n s . " One reason i s that many P pulses which v/e wish to r e g i s t e r on the o u t p u t . w i l l , as they reach the s t a t i o n , be s l i g h t l y n o n - r e c t i l i n e a r due to contamination by S-wave-converted phases. Such P motion would e a s i l y , and unwantedly, be r e j e c t e d i f we kept the phase window too narrow by using too large an NC . An example of how the power of d i s c r i m i n a t i o n against n o n - r e c t i l i n e a r motion increases with NC i s shown i n Figures 31 and 3 2 . The input (Figure 30) con-' s i s t e d of three s i n u s o i d a l segments of d i f f e r e n t p e r i o -d i c i t i e s , with 45 degrees phase diffe r e n c e between each segment's R and Z components. The FZ output traces f o r d i f f e r e n t NCs are presented' as follows : FIGURE NC LW LAGS INTANG 31a 0 20 10 45 31b 1 3 2 a 2 32b 3 Figure 30. F i l t e r input f o r the outputs shown i n Figures 31 and 32. 65 Output FZ f o r NC=0 (31a). Output FZ f o r NC=1. (31b). Figure 31. E f f e c t of varying the nuraber of convolutions. 6 6 Output FZ f o r NC=2 ( 3 2 a ) . Output FZ f o r NC=3 ( 3 2 b ) . Figure 3 2 . E f f e c t of varying the number of convolutions. 1 67 The leftmost s i n u s o i d a l segment with a period of NT = 200 points i s not affected by the' f i l t e r , no matter how large NC . The reason i s that here the time window i s too short compared: to the period' (LW/NT = 1/10). Such short a window- cannot sample enough of the o s c i l l a t i o n as to e s t a b l i s h the odd or even function c h a r a c t e r i s t i c s of the corresponding c r o s s - c o r r e l a t i o n operator which i s the basis of REMODE f i l t e r i n g . . For the two segments with periods comparable to or shorter than LW (NT=50 , NT=5) , the f i l t e r works. The expected e f f e c t of sticcessively stronger suppression v/ith successively l a r g e r NCs i s v e r i f i e d . 6 8 CHAPTER V SUMMARY AND CONCLUSIONS 5-1 Summary In testruns, two-component d i g i t a l records were f i l t e r e d with the "REMODE 5A" p o l a r i z a t i o n f i l t e r ( G r i f f i n , 1966a and 1966b). The inputs were e i t h e r earthquakes (three-component seismograms reduced to two by r o t a t i n g the coordinate system) , or a r t i f i c i a l s i n u s o i d a l wave-t r a i n s with equal amplitudes on both traces. A r t i f i c i a l records w i t h and v/ithout background noise were used. The time domain REMODE f i l t e r s pass or enhance seismic motion only i f the p o l a r i z a t i o n i s more r e c t i l i n e a r than random and. e l l i p t i c a l . They are time varying which makes them expensive i n computer time. REMODE 5A i s the most sharp-l y d i s c r i m i n a t i n g , most complicated, and computationally most expensive type within the REMODE c l a s s . Previously, other workers had s u c c e s s f u l l y applied simpler REMODE' f i l t e r s to seismograms (e.g. G r i f f i n , 1966a and 1966b ; Basham, 1967 and 1969) • The aim of th i s study v/as to investigate whether REMODE 5A would have much advantage over the sim-6 9 p i e r types, and to determine optimal values f o r the f i l t e r parameters. These questions v/ere seen against the background of nuclear t e s t detection. One point of d i s t i n c t i o n be- . tvveen earthquakes and explosions i s that the l a t t e r take place at much shallower depths than most quakes. Focal depth can be determined from a seismogram i f both the P and pP phases are c l e a r l y distinguishable. On r e a l seismograms these phases ( e s p e c i a l l y pP) are often cover-ed by background noise and signal-generated noise (SGN) . REMODE 5A can suppress much noise provided that the po-l a r i z a t i o n of the la t t e r - i s d i f f e r e n t from that of the ( r e c t i l i n e a r ) P phases. SGN , generated by multiple r e f r a c t i o n s and r e f l e c t i o n s i n the crust's layers and inhomogeneities, i s not suppressed with such c e r t a i n t y since i t may be rather r e c t i l i n e a r . REMODE 5 A , through i t s self-convolution provid-i n g f o r stronger d i s c r i m i n a t i o n against motion which i s not. purely r e c t i l i n e a r , v/as found to be better at enhanc-ing r e c t i l i n e a r phases from strongly noise-contaminated records than the non-convolved types. However, i f the s e n s i t i v i t y was strongly increased by r a i s i n g the number of convolutions above NC=2 , much r e c t i l i n e a r motion would be deleted as w e l l , and those output spikes which i n d i c a t e r e c t i l i n e a r phases would become quite e r r a t i c . 70 S i m i l a r l y , s i g n a l may be l o s t i f the power M governing d i r e c t i o n a l d i s c r i m i n a t i o n i s chosen too large. The reason i s that because of the complexity of the earth we cannot exactly s p e c i f y the incident angle of the P,pP ray(s) . " . The apparent, i n c i d e n t angles of i n d i v i d u a l spikes can be roughly determined i f the range of 0 to 90 degrees i s scanned with the s p e c i f i e d angle of incidence (INTANG) by processing the record s e v e r a l times. The length LW of the time -window within which the c r o s s - c o r r e l a t i o n between the. two input traces i s taken must not be small compared to the period of a sought-for r e c t i l i n e a r phase i n order that the l a t t e r be r e g i s t e r e d . E s p e c i a l l y , .LW has to be large enough i n order to enhance onsets. The optimum LY/ i s best determined by t r i a l and e r r o r . The character of the output v/as not s i g n i f i c a n t l y a ffected by tapering the truncated f i l t e r operator. We therefore recommend that tapering be deleted. Even though REMODE 5A proves successful i n remov-ing background noise and much of the SGN , i t does not solve the problem of p i c k i n g pP from an extended P coda. Por very shallow events, pP follows so s h o r t l y a f t e r the P onset that pP may be hidden i n the extended P v/avetrain, 71 and', s i n c e pP i s an echo of P , the two phases are of the same ( r e c t i l i n e a r ) p o l a r i z a t i o n . 5-2 Conclusions -REMODE 5A has p o t e n t i a l l y u n l i m i t e d s e n s i t i v i t y i n terms of d i s c r i m i n a t i o n a g a i n s t motion which i s not of e x a c t l y l i n e a r and l o n g i t u d i n a l p o l a r i z a t i o n , and which does not. a r r i v e from a s p e c i f i e d d i r e c t i o n . However, a p p l y i n g the f u l l power of REMODE 5A leads to disappointment. Too h i g h a s e n s i t i v i t y can cause the l o s s of pP or P . Thus,, i n p r a c t i c a l a p p l i c a t i o n s one should: use the f i l t e r v/ith parameters that, render i t l e s s s e n s i t i v e (e.g. NC=1 , M=6). Comparing the out-puts produced by such REMODE 5A v/ith those from the c o n s i d -e r a b l y s i m p l e r REMODE 3A (Basham, 1967) , the 5A outputs look c l e a r e r because of a more r i g i d s u ppression of s m a l l -amplitude o s c i l l a t i o n s which on the 3A outputs are s t i l l p r e s e n t between the l a r g e (and only s i g n i f i c a n t ) s p i k e s . However, 5A would not d e t e c t r e c t i l i n e a r motion ignored by 3A . Taking i n t o account a l s o t h a t 5A r e q u i r e s more computer time than 3A we would not, f o r the r o u t i n e p r o c e s s i n g of s e i s m i c r e c o r d s , p r e f e r REMODE 5A over 3A . I t . then appears to the author t h a t REMODE f i l t e r s of the here i n v e s t i g a t e d form w i l l not a t t a c k the problem of n u c l e a r t e s t d e t e c t i o n more s u c c e s s f u l l y than shown by G r i f f i n and Basham. 72 REFERENCES Basham, P. V/., Time domain studies of short period t e l e -seismic P phases, M.Sc. thesis, Department of Geo-physics, University of B r i t i s h Columbia, September, Basham, P. W., and R. M. E l l i s , The composition of P codas using magnetic tape seismograms, B u l l . Seism. Soc. Am., 5_2_, ( i n press). Blackman, R.B., and J.W. Tukey, The measurement of power spectra, Dover Publications, New York, 1959. G r i f f i n , J. N., Application and development of p o l a r i z a -No. 141, Earth Sciences Division, Teledyne, Inc., A p r i l , 1966a. G r i f f i n , J. NY, REMODE signal/noise tests i n polarized noise, Seismic Data Laboratory Report No. 162, Earth Sciences Division, Teledyne, Inc., September, 1966b. 1967. ti o n Laboratory Report 

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