Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

A feasibility study for a portable long period seismograph Bolduc, Pierre-Michel 1971

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1972_A6_7 B64_3.pdf [ 3.06MB ]
Metadata
JSON: 831-1.0053438.json
JSON-LD: 831-1.0053438-ld.json
RDF/XML (Pretty): 831-1.0053438-rdf.xml
RDF/JSON: 831-1.0053438-rdf.json
Turtle: 831-1.0053438-turtle.txt
N-Triples: 831-1.0053438-rdf-ntriples.txt
Original Record: 831-1.0053438-source.json
Full Text
831-1.0053438-fulltext.txt
Citation
831-1.0053438.ris

Full Text

A FEASIBILITY STUDY FOR A PORTABLE LONG PERIOD SEISMOGRAPH by PIERRE-MICHEL BOLDUC B. Sc., Un i v e r s i t e Laval, 1965 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of GEOPHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1971 In presenting t h i s thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The University of B r i t i s h Columbia Vancouver 8, Canada Date I ABSTRACT As a p r e l i m i n a r y s t u d y t o t h e o p e r a t i o n o f a p o r t a b l e l o n g p e r i o d s e i s m i c s y s t e m , t h e e f f e c t s o f e n v i r o n m e n t on t h e p h a s e r e s p o n s e o f t h e S p r e n g n e t h e r 201 a n d G e o t e c h S L -210 l o n g p e r i o d v e r t i c a l s e i s m o m e t e r s a r e i n v e s t i g a t e d . I t i s f o u n d t h a t c h a n g e s i n t h e p h a s e r e s p o n s e o f t h e s e i s m o m e t e r s a r e r e l a t e d t o t h e z e r o i n s t a b i l i t y o f t h e m o v i n g m a s s . P r e s s u r e c h a n g e s a r e n o t i m p o r t a n t i n t h i s r e g a r d ; h o w e v e r , t e m p e r a t u r e v a r i a t i o n s s h o u l d b e h e l d t o l e s s t h a n 1°C. C a l c u l a t i o n o f t h e p h a s e r e s p o n s e d i r e c t l y f r o m t h e a m p l i t u d e r e s p o n s e i s i n v e s t i g a t e d . I t i s c o n c l u d e d t h a t i t i s more p r a c t i c a l t o c a r r y o u t a p h a s e c a l i b r a t i o n d i r e c t l y . The t e c h n i q u e may h o w e v e r b e u s e f u l when o n l y a n a m p l i t u d e c a l i b r a t i o n i s known. A l s o i n v e s t i g a t e d a r e t h e n o i s e p r o p e r t i e s o f two a m p l i f i e r s : t h e s o l i d s t a t e G e o t e c h AS-330 a n d t h e p h o t o t u b e G e o t e c h 12613 a m p l i f i e r s . The f u n d a m e n t a l n o i s e o f t h e s e i s m o m e t e r - a m p l i f i e r c o m b i n a t i o n i s n e g l i g i b l e . The e x c e s s n o i s e o f t h e AS-330 a m p l i f i e r i s o f t h e same o r d e r as s e i s m i c n o i s e a t l o n g p e r i o d s a t a r a t h e r n o i s y s i t e . II TABLE OF CONTENTS Page I INTRODUCTION 1 1.1 Background 1 1.2 General Problem 2 1.3 Investigations of this Thesis 2 II THE LONG PERIOD SEISMOMETER 2 2.1 The La Coste suspension 4 2.2 Adjustment Procedures 8 2.3 Temperature E f f e c t s 10 2.4 Pressure E f f e c t s 13 2.5 The Seismometer Constants 15 II I THE AMPLITUDE RESPONSE 24 3.1 The Response of the Seismometer 24 3.2 The Response of the Seismograph 26 IV THE PHASE RESPONSE 4.1 Introduction 33 4.2 Amplitude-Phase Relations 33 4.3 Minimum-Phase 34 4.4 Method of Integration 35 4.5 The T a i l E r r o r 36 4.6 Truncation and Round-Off Errors 39 4.7 E f f e c t s of Errors i n the Amplitude Data 43 4.8 The Phase Response 45 V THE NOISE 52 5.1 The Thermal Noise 52 5.2 The Excess Noise 53 5.3 The Noise Spectrum of the Amplifier 54 5.4 Comparison with Seismic Noise 55 VI SUMMARY AND CONCLUSIONS 62 6.1 R e l i a b i l i t y of Phase V e l o c i t y 62 6.2 Environmental Problems 63 6.3 Noise 63 6.4 Determination of Phase Response from the Amplitude Response 64 APPENDIX 65 REFERENCES 68 I l l LIST OF FIGURES Page F i g . 1 Geometry of the La Coste suspension 4 F i g . 2 Diagrammatic representation of the p o s i t i v e , 7 negative and zero length springs and t h e i r force-extension r e l a t i o n s F i g . 3 Period vs. mass p o s i t i o n f o r the Sprengnether and 11 Geotech seismometers. F i g . 4 Mass p o s i t i o n vs. temperature f o r the Sprengnether 12 and Geotech seismometers. F i g . 5 The equivalent e l e c t r i c a l c i r c u i t f o r the 16 seismograph. F i g . 6 The Maxwell bridge c a l i b r a t i o n system with the 17 AS-330 a m p l i f i e r and chart recorder as the detection device. F i g . 7 Equivalent c i r c u i t of the c o i l of seismometer. 18 F i g . 8 Resistance and conductance of the c o i l of the 20 Sprengnether and the Geotech at center p o s i t i o n . F i g . 9 Equivalent impedance of the Sprengnether seismometer. 21 F i g . 10 Equivalent impedance of the Geotech seismometer. 22 F i g . 11 C i r c u i t of the balanced seismograph. 24 F i g . 12 Gain of the Geotech phototube a m p l i f i e r (PTA). 27 F i g . 13 Gain of the Geotech s o l i d state a m p l i f i e r (SSA). 28 F i g . 14 Asymptotic behaviour of the v e l o c i t y s e n s i t i v i t y 30 curve of the e l e c t r o n i c seismograph. F i g . 15 Acceleration s e n s i t i t y (AS), v e l o c i t y s e n s i t i v i t y (VS)31 and magnification (MS) of the Sprengnether s e i s -mometer-solid state a m p l i f i e r (SSA) combination at center p o s i t t i o n . F i g . 16 Acceleration s e n s i t i v i t y (AS), v e l o c i t y s e n s i t i v i t y 32 (VS) and magnification (MS) of the Geotech seismometer-s o l i d state a m p l i f i e r (SSA) combination at center p o s i t i o n . F i g . 17 The two components of the integrand of minimum- 37 phase formula (4.9) at lnw = -2. Fig. 18 Phase response of the normalized second order system 41 computed using minimum-phase program. Fig. 19 Error in the phase response of the normalized second 42 order system for a seven significant digits logarithmic amplitude. Fig. 20 Error in the phase response of the normalized second 44 order system for a two significant digits logarithmic amplitude. Fig. 21 Phase response of the Sprengnether seismometer- 49 solid state amplifier (SSA) combination at center position. Fig. 22 Phase response of the Geotech seismometer-solid state amplifier (SSA) combination at center position. 50 Fig. 23 Noise spectrum of the Geotech phototube amplifier (PTA) 56 with 40ft input resistance at any gain. Fig. 24 Noise spectrum of the Geotech solid state amplifier 57 (SSA) with 300ft input resistance at 100000 gain. Fig. 25 Comparison of seismic noise with the fundamental noise (1) of the Sprengnether seismometer and the excess noise (2) of the SSA. 2a is determined from 5.10; 2b i s obtained from the SSA spectrum. 58 Fig. 26 Comparison of seismic noise with the fundamental noise (1) of the Geotech seismometer and the excess noise (2) of the solid state amplifier (SSA). 2a i s determined from 5.10; 2b i s obtained from the SSA spectrum. V L I S T OF TABLES Page T a b l e I . S e i s m o m e t e r c o n s t a n t s . 23 T a b l e I I . The p h a s e r e s p o n s e o f t h e S p r e n g n e t h e r - S S A 47 c o m b i n a t i o n : $ - t h e p h a s e s h i f t , c e n t e r p o s i t i o n ; A c o l u m n s r e p r e s e n t t h e d i f f e r e n c e f r o m $ f o r t h e mass a t t6mm. T a b l e I I I . The p h a s e r e s p o n s e o f t h e G e o t e c h - S S A 48 c o m b i n a t i o n : $ - t h e p h a s e s h i f t , c e n t e r p o s i t i o n ; A c o l u m n s r e p r e s e n t t h e d i f f e r e n c e f r o m $ f o r t h e mass a t +6mm. VI ACKNOWLEDGEMENT The author wishes to acknowledge the advice and encouragement of Dr. R. D. R u s s e l l and Dr. R. M. E l l i s during the course of t h i s research. S p e c i a l thanks are given to Dr. F. K o l l a r of the Earth Physics Branch, Department of Energy, Mines and Resources and Mr. Robert Meldrum for f r u i t f u l discussions on instrumentation problems. The author would l i k e to acknowledge also the a i d of Mr. John Blenkinsop who was of great assistance i n programming, Mr. Jim Misener who provided the temperature c o n t r o l l e r s and Mrs. Corinne McAdam who typed the manuscript. 1. Chapter I Introduction 1.1 Background The mechanism involved i n the present models of g l o b a l tectonics (Isacks et a l , 1968) are not w e l l understood. To gain i n s i g h t i n t o these mechanisms,the determination of the p h y s i c a l properties of the mobile regions beneath the c r u s t a l plates i s of p a r t i c u l a r importance. Present data i n d i c a t e that marked v a r i a t i o n s i n seismic properties between regions do occur at these depths (Wickens and Pec, 1968). Regional studies are therefore required. The optimum seismic technique presently a v a i l a b l e to i n v e s t i g a t e upper mantle properties i s surface wave studies; further the costs are modest. Most previous studies have used records from permanent seismic s t a t i o n s . However for re g i o n a l studies, portable arrays are required as the locatio n s and spacing of permanent stat i o n s are not generally s u i t a b l e . The a n a l y t i c a l techniques of surface wave studies are quite w e l l developed. Using the Haskell (1953) formulation f o r e l a s t i c wave propagation i n layered media, surface wave dispersion curves may be r e a d i l y determined. For more complex g e o l o g i c a l s i t u a t i o n s , s i g n i f i -cant contributions to the theory of Rayleigh wave propagation across s t r u c t u r a l boundaries have been made by Mai and Knopoff (1965) and McGarr and Alsop (1967). Data analysis by Fourier techniques (Sato, 1955) and time v a r i a b l e f i l t e r s ( P i l a n t and Knopoff, 1965) are generally used with the data being f i t t e d to models through l e a s t square i n v e r s i o n procedures (Dorman and Ewing, 1962). 1.2 General Problem Studies i n B r i t i s h Columbia would be of p a r t i c u l a r i n t e r e s t since we are adjacent to the ridge system of the northeast P a c i f i c . Compared to eastern North America, i t i s known that the upper mantle e l e c t r i c a l c o n d u c t i v i t i e s and heat flow are high and Pn amplitudes are low (Caner, 1970). On Vancouver Island, the Pn phase was not even observed i n e a r l i e r explosion studies (White and Savage, 1965). This may be i n t e r p r e t e d as a th i n or missing high v e l o c i t y upper mantle cap or a very thick c r u s t a l s e c t i o n . For the reasons given above, the Department of Geophysics of the U n i v e r s i t y of B r i t i s h Columbia proposes Rayleigh wave studies using data recorded by a portable t r i p a r t i t e array for the Vancouver Island region. 1.3 Investigations of this Thesis For i n t e r p r e t a t i o n to depths much i n excess of 100 km, the period of Rayleigh waves must approach 100 sec. I t i s w e l l known that at these periods, temperature and pressure f l u c t u a t i o n s and instrument noise are important f a c t o r s . As portable stations must operate i n adverse environments compared to that of seismic v a u l t s , a study of these e f f e c t s was considered necessary. In t h i s t h e s i s , a_,preliminary instrumentation study i s c a r r i e d out with the Sprengnether 201 and Geotech SL-210 long period v e r t i c a l seismometers. The above considerations have lead to i n v e s t i g a t i o n of the following p o i n t s : (1) the e f f e c t of temperature and pressure v a r i a t i o n s i n the zero s t a b i l i t y and fundamental constants of the seismometers. (Note: Since the p r e c i s i o n attainable i n phase v e l o c i t y measurements i s ulti m a t e l y dependent on the exact 3. knowledge of the phase response of the instrument p a r t i c u l a r care i s taken to relate those effects to variations in the phase response.) (2) the noise characteristics of the solid state Geotech AS-330 (SSA) and photoelectric Geotech 12613 (PTA) amplifiers. (3) . calibration of the seismometer-amplifier-recorder system. (4) determination of the phase response from the amplitude response. . 4. Chapter II The Long Period Seismometer 2.1 The La.Coste Suspension To understand the detailed behaviour of the Geotech and Sprengnether seismometers, one must understand in detail the functioning of the La Coste suspension system. This suspension consists of a boom and a mass supported by a spring and hinged so that i t i s free to oscillate in a vertical plane. Consider such a seismometer t i l t e d at an angle £ and with an angle a between the mast and the boom (Fig. 1). These angles are measured positive clockwise. 4 Fig. 1 Geometry of the La Coste suspension From the e q u i l i b r i u m equation f o r the torques, i t i s found (Melton, 1970) T = S a 2 s i n ot-Sal cos ^  - Mg d sin(ct+6) + h(a- a ) (2.1) O 2. O where: S = Spring rate (spring constant) M = I n e r t i a l mass of the moving part h = Hinge rate 1 = I n i t i a l length of the spring o d = Distance from hinge to center of mass of the moving part X = Distance from hinge to seismometer scale a = Distance from hinge to point of suspension of the spring The La Coste seismometers use a spring with 1 = 0 i . e . , the p h y s i c a l o contraction of the spring to "zero length" corresponding to zero tension i s prevented only by the interference of the c o i l s with one another. I f i t i s furt h e r assumed that the hinge rate i s n e g l i g i b l e , equation (1.2) becomes, at eq u i l i b r i u m : 2 T = Sa s i n a - Mgd s i n ( a +<5) = 0 (2.2) o o Although t h i s equation can be s a t i s f i e d i n an i n f i n i t e number of ways by varying the parameters, a Q , M and 6, the only way i t can be s a t i s f i e d at <5 = 0 i s by s e t t i n g the mass equal to: «! 2 Mj. = m = — - • (2.3) 6=0 gd The advantage of th i s design i n the La.Coste seismometer is-that, at zero t i l t , there i s equ i l i b r i u m at a l l p o s i t i o n s , i . e . , the n a t u r a l period i s i n f i n i t e . With small departures i n m or 6 from that e q u i l i b r i u m state, equation (2.2) w i l l be s a t i s f i e d at only one a . For both the Sprengnether and Geotech seismometers a Q = TT/2. In t h i s case, Equation (2.2) reduces to: M - — ( 2 . 4 ) cos 0 Thus, when t i l t i n g the seismometer by 6, the mass has to be increased from m to M to keep ct Q at 90°. The change i n mass, or more p r e c i s e l y the change i n the e f f e c t i v e md product, i s accomplished by s l i d i n g a small weight along the boom of the seismometer. In angular harmonic motion, the period P i s given by P = 2ir /F (2.5) where: K = Moment of i n e r t i a of the moving part U = Change i n torque per radian ( t o r s i o n a l constant) For small displacements from the eq u i l i b r i u m p o s i t i o n : 9a a=a , (2.6) o From (2.2) : U = Mgd s i n 6 (2.7) The moment of i n e r t i a can be written as: K » Mp2 (2.8) where p= Radius of gyration, whence the equation for the period reduces to: P = 2 v P \ s \ 1 n x (2-9) s i n 0 The n a t u r a l period decreases for an increase i n <5 . With 1 =0, o only p o s i t i v e 6 are considered as negative 6 would give a r e s u l t a n t 7. torque i n the same d i r e c t i o n as the displacement i n (2.2) and hence i s unstable. The geometry described here i s that of the Sprengnether but i n the case of the Geotech, the same p r i n c i p l e s apply. Although the geometry i s changed, the r e l a t i o n M = m/cos 6 again r e s u l t s by s e t t i n g the c o e f f i c i e n t s of s i n a and s i n (a +6) equal at zero t i l t . o o In the r e a l instruments the springs have negative i n i t i a l length, which means that the tension i s so great that the two terminals, when released, would pass each other i f p o s s i b l e . The zero length i s obtained by use of a s t i f f l i n k whose v a r i a b l e length i s set equivalent to the negative length of the spring ( F i g . 2). P o s i t i v e length  h 1 >o Negative length m m T T r r i r Y i n r d — • F Zero length Jtfff - > ' n h ^ T r r n w — > F —x-F i g . 2 Diagrammatic representations of the p o s i t i v e ,negative and zero length springs and t h e i r force-extension r e l a t i o n s . 8. 2.2 Adjustment Procedures Although manuals are provided with the seismometers, a more detailed description i s given here in terms of the equations developed in the previous section in order that future workers may benefit from the author's experience. It i s suggested that the instruments be assembled according to the manufacturer's instructions with one exception. For the Sprengnether, i t i s preferable to postpone the mounting of the magnet to the end of the adjustment procedures since a small drag, probably of magnetic origin, hampers the free motion of the c o i l inside the magnet. This makes the measurement of the natural period d i f f i c u l t at small amplitudes since a sufficient number of free oscillations i s necessary to obtain an average value for i t . After the seismometers have been leveled in both horizontal directions and the length of the s t i f f link adjusted so that the mass seems to be "about" to float at any position, they should in theory be close to the state represented by equation (2.2) with <S=0. In practice, the i n f i n i t e period condition i s not attainable and after a certain time, the mass w i l l start to d r i f t to one of the stop positions. This is because 1 ^ 0 and therefore the term - Sal cos -r-r o o 2 in (2.1) must be retained and/or (2.3) does not hold. One effect i s indistinguishable from the other at this stage. In the next operation, the seismometer is given a small positive t i l t (about two turns of the single levelling screw) and 9. the n a t u r a l period measured by gi v i n g a small impulse to the mass and feeding the s i g n a l to an o s c i l l o s c o p e . I f the na t u r a l period i s too f a r from the desired period (say more than two seconds), the t i l t i s changed i n the d i r e c t i o n given by equation (2.9). I f the mass cannot be centered by s l i d i n g the small weight, the length of the s t i f f l i n k i s corrected accordingly. At t h i s point, e q u i l i b r i u m i s reached as i n equation (2.1) with T = 0 rather than as i n (2.2). The zero length condition i s not exactly s a t i s f i e d however and t h i s i s very important because the period versus p o s i t i o n curve w i l l be symmetrical and f l a t r e l a t i v e to ct Q = 90° only under the zero length condition (Melton, 1970). Therefore, the next step i s to s l i d e the small weight i n order that the mass f l o a t s i n both high and low posit i o n s r e l a t i v e to the scale center and to measure the period. I f the period i s longer when the mass i s f l o a t i n g high, 1 i s smaller than zero and vice-versa. The s t i f f l i n k i s o lengthened i f 1 Q <0 and shortened i f 1 Q >0 and the mass readjusted. This step i s very c r i t i c a l and a f t e r some t r i a l s , only extremely small changes are required to get a f l a t response. The center period might change s l i g h t l y because of the v a r i a t i o n i n M; and the t i l t has to be corrected once again. This should f i n a l l y reduce the equ i l i b r i u m equation to (2.2) with the value of M (or <5) that gives the proper period. I t i s not an easy task to optimize the adjustment of the instruments because when one parameter i s changed, i t a l t e r s a l l the others. I t i s emphasized that one should consider each step i n terms 10. of the e f f e c t s on equations (2.1) and (2.2). More d e t a i l e d consideration of these equations and worked out examples w i l l be found i n Melton (1970). F i g . 3 shows the actual period vs p o s i t i o n curves obtained from the Sprengnether and the Geotech before proceeding to the other measurements. I t i s l i k e l y that a d d i t i o n a l adjustment on the Sprengnether would y i e l d smaller deviations from the zero p o s i t i o n value of period. However, v a r i a t i o n s i n phase response as a function of mass p o s i t i o n was found to be smaller than that of the Geotech f o r which the period changes l e s s with p o s i t i o n . The phase i s also a f f e c t e d by the change i n seismometer constants (Table I ) . I t should be noted that the Geotech curve has a l a r g e r deviation than that shown i n t h e i r manual. F. K o l l a r (private communication) also reports that the instruments of the Earth Physics Branch do not meet the manufacturer's s p e c i f i c a t i o n s . 2.3 Temperature E f f e c t s An important cause of v a r i a t i o n s i n a i s thermal i n s t a b i l i t y . To f i n d the e f f e c t s of temperature f l u c t u a t i o n s on the mass p o s i t i o n , the seismometers were covered by a styrofoam case containing e l e c t r i c l i g h t bulbs as a heat source. The temperature was regulated by a thermostat and measured with a d i f f e r e n t i a l thermocouple. F i g . A shows the mass p o s i t i o n vs temperature curves f o r the Sprengnether and the Geotech. The Sprengnether spring has a p o s i t i v e c o e f f i c i e n t of thermal expansion while that of the Geotech i s negative. I t i s noted that the mass p o s i t i o n as a function of temperature i s approximately l i n e a r with deviations from the s t r a i g h t l i n e normally being less than 1 mm. S p r e n g n e t h e r G e o t e c h MASS POSITION (MM) -10 -bu - 10 MASS POSITION (MM) Fig. 3 Period vs. mass position for the Sprengnether and Geotech seismometers. Fig. 4 Mass position vs. temperature for the Sprengnether and Geotech seismometers. In an operational system an a d d i t i o n a l complication a r i s e s due to d r i f t i n the temperature c o n t r o l l e r s themselves. Over a short period (1 week), t h i s d r i f t may lead to a change i n mass p o s i t i o n of ±3 mm. Over an i n t e r v a l of a month d r i f t s as large as ±6 mm were observed. As t h i s corresponds to s i g n i f i c a n t changes i n phase response, the system should be checked at frequent i n t e r v a l s . Further e f f o r t might also be expended i n c o n t r o l l e r design. 2.4 Pressure E f f e c t s In the eq u i l i b r i u m equation f o r the torques (2.1), i t was assumed that the density of the a i r p was a constant. Since a pressure o change dP Q m a y change p Q , i t was necessary to f i n d the e f f e c t of such a change on the zero s t a b i l i t y of the seismometer. With p , the density of the mass, equation (2.1) i s now w r i t t e n as: T = Sa 2 s i n a - Sal cos £ - Mgd (1 ) s i n (a + 6) + h(a - a ) (2.10) o 2 p o With p and a as v a r i a b l e s : o dT = | ^ dp +'|^ da 3p o 3a (2.11) At e q u i l i b r i u m (dT=0) -3T da = From (2.6) 3p. a=a 3T 3a a=a dp. (2.12) da = - 1 U 3T 3P, a=a dp. (2.13) From (2.10): d a = - sin (a + 6) dp (2.14) Up o o Assuming isothermal pressure changes: O O P P o o o P o o (2.15) dp = =2- dP (2.16) Furthermore: da = - (2.17) From (2.14), (2.16) and (2.17) at aQ = 90°: J x = MgdXp cos<5 d x ^ dP (2.18) UpP o For example, for the Sprengnether: M = 11.2 kg 2 g = 9.8 m/s d = .308 m X = .454 m 6 = 3° U = .121 kg m2/s2 P q= 1.293 kg/m3 p, = 11300 kg/m3 lead For a large pressure change of 2 cm at a normal pressure of 76 cm: dx = .4 mm (2.19) With a d r i f t of the same order f o r the Geotech, i t i s concluded that pressure changes of this magnitude cannot a f f e c t the zero s t a b i l i t y s i g n i f i c a n t l y . However, i t i s s t i l l very important to have the seismometer t i g h t l y sealed since a rapid change i n pressure be equivalent to a strong s i g n a l from the ground. 2.5 The Seismometer Constants As the general theory for seismometer c a l i b r a t i o n can be found i n R u s s e l l and K o l l a r (1966) , a d e t a i l e d development w i l l not be given here. Their equations for a "mass on a s p r i n g " seismometer have been modified to s u i t the pendulum seismometer. Consider F i g . 1 with an i n i t i a l frame of reference y ( p o s i t i v e downward) r e l a t i v e to the surface of the Earth and the pendulum hinge at the o r i g i n 0 of an x, <f> system moving with the seismometer case. A c o i l of impedance f i x e d to the boom swings through a magnet and produces a current I through a load Z . The following symbols are adopted: <t>: Displacement from e q u i l i b r i u m p o s i t i o n : p = a-a^ 2 D: Damping moment constant (kg m /s) 2 2 U: Spring moment constant (kg m /s ): 2 2 G: C o i l motor constant (kg m /As ) l c : Distance from hinge to center of gravity of the moving part 1 : Distance from hinge to center of o s c i l l a t i o n (reduced P pendulum length) T^: Natural period For small displacements the equations of motion are: K«f + Dg + U«5 = Ml y" - GI (2.2C G$ = (Z + ZT) I (2.2] C L 16. The l o a d i n c l u d e s the a t t e n u a t i n g r e s i s t o r s , t h e i n p u t impedance o f t h e a m p l i f i e r and the f i l t e r s i f any. These two e q u a t i o n s a r e c o m p l e t e l y e q u i v a l e n t t o the n o d a l and l o o p e q u a t i o n s o f the c i r c u i t o f F i g . 5. ® )Ml cy I L = U Z s F i g . 5 The e q u i v a l e n t e l e c t r i c a l c i r c u i t f o r t h e s e i s m o g r a p h . S i n c e t h e e l e c t r o m e c h a n i c a l s e i s m o m e t e r i s i n d i s t i n g u i s h a b l e f r o m i t s p u r e l y e l e c t r i c a l e q u i v a l e n t , t h e above c i r c u i t w i l l be used i n the a n a l y s i s . F o r c a l i b r a t i o n p u r p o s e s t h e seismometer i s p l a c e d i n t h e M a x w e l l b r i d g e (Fig..6). To f i n d Z , t h e s e i s m o m e t e r i s clamped and t h e b r i d g e i s b a l a n c e d . Then Z ^ i s g i v e n by: Z C = V R / Z B ' (2.22) The l o c k i n g mechanism o f t h e S p r e n g n e t h e r i s n o t t i g h t enough t o a v o i d the mass moving s l i g h t l y under b a l a n c e c o n d i t i o n and an a d d i t i o n a l heavy w e i g h t must be p l a c e d on t h e mass t o make i t c o m p l e t e l y immovable and p r e v e n t a f a l s e b a l a n c e c o n d i t i o n . F o r b o t h s e i s m o m e t e r s , t h e b a l a n c e c o n d i t i o n i s f o u n d t o be f r e q u e n c y dependent. The e f f e c t s o f Fig. 6 The Maxwell bridge calibration system with the AS-330 amplifier and chart recorder as the detection device. 18. hysteresis and eddy current losses must be taken into consideration. Those effects are observed at frequencies much higher than that for which the seismometers are designed to operate. However, we shall see that the amplitude response must be determined over a wide frequency range i f the phase response i s to be determined directly from i t . represented by a series connection of a resistance r and an inductance At higher frequencies, Shima (1959) uses a loss resistance q in parallel with the inductance as in Fig. 7(a). This i s easily transformed in the form of Fig. 7(b) by: At low frequencies the c o i l of the seismometer i s adequately q (2.23) V I (2.24) Z r + r' + V (2.25) I r XJLAJL/ A M / W A A A / V V q (a) A A A / V V W W V r r'(u)) i'(co) (b) Fig. 7 Equivalent circ u i t of the c o i l of seismometer Formulas (2.23) and (2.24) show that the resistance r + r' seen by the bridge should increase with frequency and the inductance lx should decrease. This i s observed experimentally ( F i g . 8) although no s i n g l e l o s s resistance that would f i t both curves simultaneously could be found. C l e a r l y the model of Shima i s only an approximation. The Willmore method i s used to f i n d the other constants. The seismometer i s undamped and a se r i e s of output voltages from the main (V^) and s u b s t i t u t i o n (V ) inputs are recorded. Ru s s e l l and K o l l a r (1966) point out that V M/V i s completely independent of Z and i s given by: Z (2.26) From F i g . 5, i t i s seen that i n terms of the complex frequency s =a + jto: G 2s Z ^ (2.27) Ks + Ds + U The |Z g| curve f o r the center p o s i t i o n of the Sprengnether and the Geotech are shown i n F i g . 9 and F i g . 10. A l l the constants were ca l c u l a t e d and c o l l e c t e d i n Table I. The moment of i n e r t i a was found by the r e l a t i o n : K = M l c l ^ (2.28) The other parameters were determined from the asymptotes and the maximum of the Z curve and from the n a t u r a l period of the 1 s' seismometer. Those parameters are those that w i l l be used i n the next two chapters to f i n d the amplitude and phase response of the seismometers at the +6mm, 0mm, -6mm equi l i b r i u m p o s i t i o n s . Fig. 8 Resistance and conductance of the c o i l of the Sprengnether and the Geotech at central position. 2 2 . Seismometer Position(mm) Temperature (°C) G(kgm2/As2) 2 D(kgm /s) U(kgm2/s2) T N(sec) Sprengnether: +6 -23.5 27.7 .0640 .131 19.2 1 : .3564m P 0 -25. 27.7 .0318 .121 20.0 K = 1.23 kgm2 -6 -26.5 27.7 .0248 .098 22.2 Geotech -6 -23.5 21.5 .00220 .0114 20.4 1 = .30m P 0 -25. 22.4 .00142 .0121 19.8 K = .12 kgm2 +6 -26.5 20.9 .00406 .0128 19.2 Table I. Seismometer constants. Chapter III The Amplitude Response 3.1 The Response of the Seismometer As w e l l as being used to determine the seismometer constants, the Maxwell bridge can be used to f i n d the amplitude response of the complete seismograph. With the bridge balanced at a l l frequencies and an appropriate choice of impedances i n the c i r c u i t , the currents through R^, CQ, and R^ and R^ may be neglected and the equivalent of the seismometer i n the bridge may be reduced to a ladder c i r c u i t (Russell and K o l l a r , 1966). For a n a l y t i c a l purposes, i t i s convenient to divide the seismograph i n t o i t s three components ( F i g . 11) and f i n d the trans f e r functions of the seismometer F c , of the a m p l i f i e r F. and the recorder F . o A K I Fs = F i g . 11 C i r c u i t of the balanced seismograph For convenience we define an e f f e c t i v e input impedance of the a m p l i f i e r by Z. = R i R i l (3.1) R7 + R. 1 l 25. The damping r e s i s t a n c e i s then given by: = r + r' + ^ + R 2 + Z ± (3.2) From the c i r c u i t of F i g . 11 the i n p u t v o l t a g e to the a m p l i f i e r i s r e a d i l y found as: VZ.Z ' i - R ^ . V - * ' + V < 3- 3 > and from F i g . 5, the source c u r r e n t i s e q u i v a l e n t t o : V_ = (3.4) h G By i n t e g r a t i n g (3.4) and using (2.28), the e q u i v a l e n t ground v e l o c i t y i s found to be: V i G * - v T < 3 - 5 ) By (2.27), (3.3) and (3.5), the t r a n s f e r f u n c t i o n ( y ^ / y ) o f the seismometer i s given by: „ GKZ.s F„ -S l p [ K Z ' s 3 + (D^.,+KRD) s 2 + ( G 2 + i ' U + DR D)s + R ^ ] ( 3 * 6 ) Since the V terms are s m a l l and decrease r e l a t i v e l y w i t h i n c r e a s i n g frequency, they can be dropped and equation (3.6) reduces t o : GKZ.s 2 i FS l ^ K R ^ 2 + (G 2 + DR D)s + R^U] ( 3 - 7 ) At low fr e q u e n c i e s : GKZ.s 2 At high frequencies: F, (3.9) On a Bode diagram, F_ w i l l be a s t r a i g h t l i n e of slope +2 and F a s t r a i g h t l i n e of slope 0 as long as r' does not grow i n d e f i n i t e l y . In Shima's model, i t s l i m i t i n g value i s that of the loss r e s i s t o r . The corner i s at the n a t u r a l frequency of the seismomter: 3.2 The Response of the Seismograph In neither case was the complete t r a n s f e r function (V Q/V )of the PTA and SSA ampl i f i e r s given by the manufacturer. The t r a n s f e r function of the PTA at maximum gain i s given i n F i g . 12; the slopes of the low and high frequency asymptotes are +3 and -4 r e s p e c t i v e l y with corner frequencies f = .0093 and f = 6.00. L tl The t r a n s f e r function of the SSA i s given i n F i g . 13 at the maximum gain a v a i l a b l e from the low l e v e l output; i n t h i s case the low and high frequency asymptotes have slopes of +2 and -2 with corners at f ^ = .0094 and f = 4.77. F i n a l l y , the t r a n s f e r function of the recorder (v/v ) H O w i l l a f f e c t the amplitude response of the seismograph. The t o t a l v e l o c i t y response becomes: S (3.10) F = v (3.11) o v y V. 1 V o (3.12) --4 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 LN to (co in rad/sec) Fig. 12 Gain of the Geotech phototube amplifier (PTA) 9 oo L N oj ( to in rad/sec) Fig 13: Gain of the Geotech solid state amplifier (SSA). Or f r o m ( 3 . 5 ) : v R_.Ks F - v r h - ( 3 - 1 3 ) P The c o u p l i n g o f the s e i s m o m e t e r t o t h e a m p l i f i e r i s t a k e n i n t o a c c o u n t t h r o u g h Z ; i t i s assumed t h a t the i n p u t impedance o f t h e r e c o r d e r i s h i g h and p u r e l y r e s i s t i v e . I f n o t , o r i f t h e r e i s a f i l t e r i n between, t h e t r a n s f e r f u n c t i o n o f the a m p l i f i e r must be t a k e n t h r o u g h t h e a c t u a l impedance o f t h e r e c o r d i n g s t a g e . By (3.12) the a s y m p t o t i c b e h a v i o u r o f t h e whole s y s t e m i s e a s i l y f o u n d on a Bode p l o t ( F i g . 1 4 ) . The s l o p e s o f each component a r e s i m p l y added between each c o r n e r . The a c c e l e r a t i o n , v e l o c i t y and a m p l i t u d e r e s p o n s e s o f t h e s e i s m o g r a p h s w i t h t h e S p r e n g n e t h e r and t h e G e o t e c h a t c e n t e r p o s i t i o n a r e shown i n F i g . 15 and F i g . 16. The a m p l i f i e r used i s the AS-330 and t h e r e c o r d e r i s t h e B r u s h 480 model 15-6687-00. The o s c i l l a t o r y b e h a v i o u r o f the S p r e n g n e t h e r c u r v e a t h i g h e r f r e q u e n c i e s i s a t t r i b u t e d t o r e s o n a n c e s a t t h e n a t u r a l f r e q u e n c i e s o f i t s s u s p e n s i o n s p r i n g . T h i s was deduced f r o m t h e f a c t t h a t when th e s l i d i n g w e i g h t was s h i f t e d t o t h e r i g h t and t h e s p r i n g became t i g h t e r , the r e s o n a n c e peaks were s h i f t e d t o h i g h e r f r e q u e n c i e s and v i c e v e r s a f o r a l e s s t i g h t s p r i n g . I t was a l s o o b s e r v e d t h a t when the s p r i n g was t a p p e d , b e a t s were p r e s e n t i n t h i s f r e q u e n c y r a n g e . Slope of seismometer (+2) Slope of seismometer (+2) + low frequency/ slope of amp l i f i e r Into i—i > M H I H Z w CO o o I-l w > a). : lower corner frequency L of the amplifier to : corner frequency of S the seismometer to 'A^rhigher corner frequency of the am p l i f i e r ' UR:corner frequency of the recorder lnco In co High frequency slope of amplifier LN to (co i n rad/sec) 1/4 o High frequency slope of am p l i f i e r + slope \of recorder. In w R F i g . 14 Asymptotic behaviour of the v e l o c i t y s e n s i t i v i t y curve of the e l e c t r o n i c seismograph. Fig. 15 Acceleration sensitivity (AS), velocity sensitivity CVS) and magnification (MS) of the Geotech seismometer-solid state amplifier(SSA) combination at center position. Fig. 16 Acceleration sensitivity (AS), velocity sensitivity(VS'> and magnification (MS) of the Geotech seismometer-solid state amplifier (SSA) combination at center position. 3 3 . Chapter IV The Phase Response 4.1 Introduction The d i r e c t procedure f o r obtaining the phase response of the seismograph i s to measure the phase di f f e r e n c e between the voltage applied to the bridge and the output voltage from the seismograph. However, the phase response may also be obtained from the amplitude response without a d d i t i o n a l measurements. In the following sections t h i s procedure w i l l be i n v e s t i g a t e d i n some d e t a i l . 4.2 Amplitude-Phase Relations In t h i s s ection several basic ideas of l i n e a r systems theory are o u t l i n e d . For a more d e t a i l e d exposition the reader i s r e f e r r e d to Papoulis (1962). In general, f o r a causal system a unique r e l a t i o n between the amplitude and phase of the t r a n s f e r function F(s) does not e x i s t . For a stable system, F(s) contains no poles i n the r i g h t h a l f plane or on the u — a x i s . In t h i s case s = JOJ and the response i n terms of the amplitude H(aj)and phase <3>(u)) can be w r i t t e n F ( j U ) = A(u) e j < K u ) ) (4.1) Taking the logarithm of both sides y i e l d s : lnF(jco) = inA(uj) + j$(ou) (A.2) For a complex function of s, a unique r e l a t i o n e x i s t s between the r e a l and imaginary parts provided i t has no s i n g u l a r i t i e s i n the r i g h t half-plane. That i s , §(oj) may be found d i r e c t l y from A(u) (and vice versa) provided that lnF(s) has no s i n g u l a r i t i e s i n the r i g h t half-plane. For this to hold an a d d i t i o n a l condition on F(s) i s required; namely ,F(s) has no zeros f o r Re s>0.The r e l a t i o n s h i p s between A(OJ) and $(co) known as H i l b e r t transforms are: In A(co) = - - f du (4.3) i j U - ID 4(w) 1+f lnA(u)_ d u v _& u - co Henceforth, the frequency for which the phase i s c a l c u l a t e d i s designated co and the frequency of the recorded amplitude i s designated u. Further, we define L(u) = lnA(u). 4.3 Minimum-Phase Functions of the type described above are c a l l e d "minimum-phase". The reason f o r t h i s w i l l become cl e a r with the following argument. Consider a t r a n s f e r function which has a zero i n the r i g h t h a l f plane: T(s) = N ( s ) p j s 1 - a s ) (a>0) (4.5) This can be rewritten as: T(s) = N(s) (1 + as) D(s) (1 - as) (4.6) (1 + as) T(s) i s the product of two complex functions, but the second term i s a pure phase s h i f t e r , that i s i t s modulus i s always equal to unity and i t s phase varies from 0 to -TT. This means that given a function with a zero i n the r i g h t h a l f plane, one can always f i n d another function with the same gain but with a smaller l a g . In the i l l u s t r a t i o n N(s) (1 + as)/D(s) i s the minimum phase function with the same amplitude as T ( s ) . Since there are no zeros i n the r i g h t h a l f plane, a pure phase s h i f t e r cannot be extracted which decreases the phase. The trans f e r function of the seismograph i s minimum phase since the a m p l i f i e r and the recorder can be represented by ladder c i r c u i t s whose tr a n s f e r functions contain no zeros i n the r i g h t h a l f plane, and the seismometer i n the bridge i s also a ladder c i r c u i t when and only when the bridge i s p e r f e c t l y balanced. The t r a n s f e r function of the seismometer has two zeros at s = 0 and s = 0 0 which are s i n g u l a r i t i e s of In F (s) , s but i t can be shown (Papoulis, 1962) that f o r zeros of F(s) on the CD axis i t s e l f , the H i l b e r t transforms are s t i l l a p p l i c a b l e . 4.4 Method of Integration Due to the s i n g u l a r i t i e s (u= 0 ,u>°°) > formula (4.4) i s d i f f i c u l t to i n t e g r a t e . The numerical i n t e g r a t i o n procedure used i s based on an a l t e r n a t i v e formula proposed by Bode (1945) which reduces the number of s i n g u l a r i t i e s to one. A new v a r i a b l e i s introduced: x = ln(u/u) (4.7) A f t e r i n t e g r a t i o n by parts (Solodovnikov, 1960), the H i l b e r t transform takes the form: ^ »( U) = - / In coth|f| dx (4.8) ir I dx Z Since the i n t e g r a t i o n includes the complete frequency spectrum, the phase c h a r a c t e r i s t i c at any point depends upon the slope of the gain c h a r a c t e r i s t i c i n a l l parts of the spectrum. The r e l a t i v e importance of the slope i s given by the weighting f a c t o r In coth which becomes l o g a r i t h m i c a l l y ( F i g . 17) i n f i n i t e at x = 0 and goes to zero as x becomes large. The i n t e g r a l of 2 the weighting f a c t o r i s already known (Bode, 1945)and i s 7 1 /2. Therefore the asymptotic behaviour of the phase i s simply the slope of the corresponding asymptote m u l t i p l i e d by ir/2. 36. For numerical integration i t i s possible to obtain a smooth integrand by subtracting the singularity at x = 0. Equation (4.8) i s now written as: } TT 7 Ldx d x X==0J In coth 2 dx x=0 (4.9) weighting factor Fig. 17 shows the form of the logarithmic derivative and the in the "integrand of (4.9) at In co = -2. It should be noted that under the change of scale from u to x (4.7), the value of dL(x) dx dL(u) _ i s the same as — x=0 dlnu u=co The logarithmic derivative used i s that of the velocity sensitivity curve of Fig. 16. The FORTRAN program for the computation of (4.9) i s given i n the Appendix. Values of the amplitude in the interval between the l e f t and right asymptotes are used to compute the derivative by the 3-point formula: D. ( L i + l " Li-1> i 2h (4.10) where h i s the interval between two consecutive pivotal points. On the asymptotes, the derivatives are generated by the program. After tabulation of the weighting function, the integrations are carried out for each co by Simpson's composite formula: I = h/3 ( f n + 4f, + 2f. +...+2f , + 4f . + f ) (4.11) (J JL 2. n—/ n—i. n 4.5 The Tail Error The integral (4.9) i s evaluated inside f i n i t e limits a and b with the error introduced by the neglected part given exactly by: r r.rt - - 1 j f f d L ( x ) d L ( x ) t a K } ~ i W dx • dx x=0 In coth CO 2 j J I dx dx In coth — dx (4.12) Fig.17 The two components of the integrand of minimum-phase formula (4.9) at lnu) = -2. 38. I n p r a c t i c e t h e i n t e r v a l a,b i s chosen l a r g e enough so t h a t f o r a n y as-: d L ( x ) dx d L ( x ) dx x < a " a i a l l co x>b = a r (4.13) (4.14) a l l oo c o t h C h o o s i n g a and b w i t h e q u a l a b s o l u t e v a l u e s and r e p l a c i n g by i t s d e f i n i t i o n , t h e e x p r e s s i o n f o r e ( to) r e d u c e s t o : L 3L t a TT 2 d L ( x ) dx 00 x = o - a r a 2 ] / l n 7 7 7 d x < 4 - 1 5 > b The s e r i e s e x p r e s s i o n f o r t h e i n t e g r a n d i s g i v e n by: (x>0) x , i » ( l - 2 k ) x — e e - 1 (2k - 1) k = l I n t e g r a t i n g ( 4 . 1 5 ) term by t e r m y i e l d s : £ O) = -t a K 2 d L ( x ) dx x=0 " V a 2 " ( l - 2 k ) b e (4.16) (4.17) "J £i (2k - 1) The s e r i e s i s r a p i d l y c o n v e r g i n g and when a p p l i e d t o the s e i s m o g r a p h c u r v e s , t h e c o n t r i b u t i o n o f e (w) i s f o u n d t o be v e r y s m a l l (See S e c t . 4 . 8 ) . ca I f t h e c o n d i t i o n s (4.13) and (4.14) do n o t a p p l y , i n s p e c t i o n o f (4.12) shows t h a t f o r any c o n f i g u r a t i o n an upper bound on the t a i l e r r o r i s g i v e n by: t a max • K h l + N) ( l - 2 k ) b ( 2 k - I K k = l (4.18) 39. 4.6 Truncation and Round-off Errors An estimate of the truncation e r r o r i n the d i f f e r e n t i a t i o n i s given by McCormick and Salvadori (1964) : 2 £ t r = J~ ( ° ( 4 - 1 9 ) For each double i n t e r v a l of i n t e g r a t i o n , a s i m i l a r estimate i s given by McCormick and Salvadori 0.964): e t r =to" f C 4 ) ( 5 ) ( A ' 2 0 ) where f ^ ( £ ) are the higher order d e r i v a t i v e s evaluated within the i n t e r v a l . These errors are determined by the choice of i n t e r v a l at which one reads the amplitude values from the Bode p l o t . The program was tested with systems described by a n a l y t i c a l functions and the computed phase was compared with the true phase f o r d i f f e r e n t values of h. I t appears that the choice of h i s not p a r t i c u l a r l y c r i t i c a l provided i t i s not extreme. The value of 0.2 was used. A worked out example w i l l show how the errors propagate. The system chosen i s the normalized second order system with damping r a t i o equal to one. In the frequency domain the gain and phase are given r e s p e c t i v e l y by: A(u) =/^ — (4.21) 4(u) = (4.22) On the Bode p l o t the v a r i a b les are changed from l i n e a r to logarithmic and the gain and i t s d e r i v a t i v e become: 4 0 . L C u ) " l n ' 21nu ( 4 - 2 3 ) 1 + e £ i H l . _Z2 ( 4 . 2 4 ) dlnu .. , -21nu 1 + e The logarithmic gain L(u) has a corner at u = 1 and the slopes of the l e f t and right asymptotes are respectively 0 and -2. Further inspection of (4.24) shows that the slope i s exactly -1 at ln u = 0 (u=l) and completely symmetrical relative to that point. With the limits of integration fixed at a = -10 and b = +10, the phase response w i l l be computed from x . = -7 (x . =lnco . ) to x = +7 (x = lnco ) which corresponds to mxn min min max max max a lower frequency of 1.45 x 10 \ z and an upper frequency of 1.75 x 10 2 hz. The logarithmic derivative at each point i s given exactly by (4.24) and w i l l be kept at seven significant digits to eliminate the input errors which w i l l be considered i n the next section. *- . r- d]l ( U ) . . . , Because of the symmetry of - — — one can readily see that the dlnu . phase should be exactly 0 at co=l as should the t a i l and truncation errors. Moreover, these errors should also show symmetry relative to co=l. The actual results are presented in Fig. 2.8 and Fig. 19. The results for both the phase and the error are those expected with their values at co=l slightly different from zero due to roundoff error. The li m i t on the t a i l error given by (4.18) amounts to: e ? - i n = - (0 + 2) (e ) = .0000578rad =.00331deg. (4.25) max tr ta The error in Fig. 19 tends to that value at the extremities of the spectrum. This was also expected since for small or large values of to (relative to co=l) the contribution of the integral in (4.9) becomes zero leaving only the LN to ( to in rad/sec) -7 -6 -5 - 4 - 3 - 2 - 1 0 1 2 3 -200 I Fig. 18 Phase response of the normalized second order system computed using minimum-phase p r o g r a m . .006 -.005 w Di g .004 o o T a i l error limit 10 < w w 003 5 .002 g .001 -9 -8 -7 -6 -5 -4 -3 -2 -1 o 0 1 o 2 -.oor -.0002 -.003 -.004 -.005 -.006 5 6 7 8 9 10 LN to ( co in rad/sec) Tail error limit o o o o o o Fig.19 Error in the phase response of the normalized second order system for a seven significant digits logarithmic amplitude. 4 3. t a i l e r r o r . Furthermore, the maximum absolute value of e i s .0056; i t shows th a t the t r u n c a t i o n and roundoff e r r o r s are very s m a l l . S i m i l a r r e s u l t s were obtained using other a n a l y t i c a l systems. I t i s concluded that these e r r o r s are completely n e g l i g i b l e compared to e r r o r s i n t r o d u c e d by u n c e r t a i n t i e s i n the data. 4.7 E f f e c t s of E r r o r s i n the Amplitude Data Formula (4.9) i s composed of two terms and s i n c e the a l g o -r i t h m f o r numerical d i f f e r e n t i a t i o n magnifies input e r r o r s more than the a l g o r i t h m f o r numerical i n t e g r a t i o n , i t might appear at f i r s t s i g h t term. that the main source of e r r o r would come from the TV dL(x) 2 dx x=0 However, i f an u n c e r t a i n t y $ i s assumed f o r the d e r i v a t i v e term at x=0, the e r r o r eJ i s zero due to c a n c e l l a t i o n by the i n t e g r a l term: + » 1 f dL(x) TfJ [ dx x=0 + 6 o In coth dx + -dL(x) dx x=0 o = 0 (4.26) The e r r o r a r i s e s from the i n t e g r a l term only and i s expected to be s m a l l due to the smoothing e f f e c t of i n t e g r a t i o n . More e x p l i c i t c o n s i d e r a t i o n w i l l be presented now. In p r a c t i c e , a smooth curve i s t r a c e d on the Bode p l o t through the unequally spaced amplitude p o i n t s and the i n p u t data are obtained by reading t h i s curve at the chosen i n t e r v a l of 0.2 u n i t . I f a new curve i s drawn from a d i f f e r e n t s e r i e s of measurements performed under the same c o n d i t i o n s , i t i s found to f i t the o r i g i n a l one to at l e a s t two s i g n i f i c a n t d i g i t s accuracy. The method was t e s t e d once again w i t h a system whose answer i s known but t h i s time by reading the amplitude w i t h only two s i g n i f i c a n t d i g i t s . F i g . 20 shows the e r r o r s obtained f o r the second order system s t u d i e d i n s e c t i o n 4.6. Even at t h i s low degree of accuracy o w p 2.Or 1.5 pi Q W o O o rt 12—Q_i— -5 -4 1.0| o 0.5f -2 -1 b 1 LN oj (OJ in rad/sec) _Q ^ -l.CT -1.5 -2.Cr 3 o o 4° "^V7 "6 - 7 ° o ° Fig. 20 Error in the phase response of the normalized second order system for a two significant digits logarithmic amplitude. 4 5 . on the amplitude, the maximum error that can be found i s 1.71 degree and that only f o r two of the frequencies, the average er r o r being much smaller. The method was also tested with f i r s t and t h i r d order systems i n the same manner and an errpr as large as two degrees never appeared. The seismograph curves are quite s i m i l a r to these t h e o r e t i c a l curves; they are also smooth and monotonically decreasing and they behave as fourth and second order systems at low frequencies, and as second order systems at higher frequencies ( F i g . 17). They are known to at l e a s t two s i g n i f i c a n t d i g i t s and on a s u f f i c i e n t l y wide band to determine the p o s i t i o n of t h e i r asymptotes. I t i s b e l i e v e d that the e r r o r on the phase c a l c u l a t e d by t h i s method i s at most ±2°, with ±1° l i k e l y to be a more r e a l i s t i c f i g u r e . 4.8 The Phase Response The phase response was found by the method described above for the Sprengnether and the Geotech at the temperatures that brought t h e i r e q u i l i b r i u m p o s i t i o n to -6mm, Omm and +6mm. The l i m i t s of i n t e g r a t i o n were reduced to a = -7 and b = +7 which gives a maximum t a i l e r r o r of about .05 degree f o r a seismograph curve. The resonances of the Sprengnether (Fig. 15) are not taken i n t o account since they appear outside the range of the frequencies of i n t e r e s t , that i s , f a r from the large values of the weighting f a c t o r and further t h e i r contribu-tions to the remaining part of the i n t e g r a l l a r g e l y cancel because of the r a p i d l y o s c i l l a t i n g character of the d e r i v a t i v e i n that region. The transfer function of the a m p l i f i e r i s therefore used beyond In u = 2.6. 46. The corner frequency of the recorder i s not v i s i b l e on the v e l o c i t y s e n s i t i v i t y curves because of the wide band response of the Brush 480 ( f u = 125 cps). N e g l i g i b l e e r r o r i s introduced by n e g l e c t i n g H. i t s e f f e c t . I t should be noted t h a t an e r r o r which m u l t i p l i e s the whole set of p o i n t s by a c e r t a i n value e.g. an e r r o r i n the determination of the constants of the seismometer (see 3.13), does not a f f e c t the l o g a r i t h m i c d e r i v a t i v e . The r e s u l t s f o r periods between 100 sec. and 1 sec. are c o l l e c t e d i n Table I I and Table I I I . The phase responses at the center p o s i t i o n of the seismometer s c a l e s are p l o t t e d on F i g . 21 and F i g . 22. The c i r c u l a r p o i n t s on these f i g u r e s represent experimental values obtained by measuring the phase d i f f e r e n c e by z e r o - c r o s s i n g between the v o l t a g e a p p l i e d to the bridge ( a c c e l e r a t i o n ) and the output v o l t a g e from the seismograph. The phase s h i f t of the v e l o c i t y i s obtained by adding 90°. The A columns of Table I I and Table I I I represent the d i f f e r e n c e i n the phase response at ±6mm r e l a t i v e to the phase response at 0mm. I t can be seen t h a t the re g i o n most a f f e c t e d by the zero i n s t a b i l i t y i s the band between 5 sec. and 100 sec. This represents two octaves on each s i d e of the n a t u r a l frequency of the seismomter. I t i s the re g i o n where the slope of the t r a n s f e r f u n c t i o n of the seismometer, i . e . the d e r i v a t i v e of (3.7) i s most a f f e c t e d by a change i n i t s constants. The other p a r t s of the curve l a r g e l y depend on the a m p l i f i e r - r e c o r d e r slopes and should not be much d i s t u r b e d by a change i n the mass p o s i t i o n of the seismometer For the Sprengnether the change i n phase response i s about 3° and f o r the Geotech, i t i s about 2° on one s i d e and 4° on the other. This i s a s i g n i f i c a n t change and p o i n t s out the n e c e s s i t y of keeping 47 . In co F (hz) T (sec) (degreef (degree) . ,A+6mmN (degree) -2.8 .010 103.3 0 240 3 -2.6 .011 84.6 -2 222 2 -2.4 .014 69.3 -2 204 3 -2.2 .018 56.7 -3 188 3 -2.0 .022 46.4 -4 172 3 -1.8 .026 38.0 -3 155 4 -1.6 .032 31.1 -5 140 3 -1.4 .039 25.5 -4 123 3 -1.2 .048 20.9 -3 106 3 -1.0 .058 17.1 -4 91 2 -0.8 .072 14.0 -3 76 2 -0.6 .087 11.4 -2 62 3 -0.4 .107 9.4 -1 50 2 -0.2 .130 7.7 -1 40 1 0.0 .159 6.3 0 31 1 0.2 .194 5.1 0 24 1 0.4 .237 4.2 0 18 0 0.6 .290 3.4 0 13 0 0.8 .354 2.8 0 8 1 1.0 .433 2.3 0 4 1 1.2 .528 1.9 0 0 1 1.4 .645 1.6 0 -3 0 1.6 .788 1.3 1 -8 1 1.8 .963 1.0 0 -12 0 2.0 1.176 0.8 0 -17 1 Table I I . The phase response of the Sprengnether-SSA combination: $-the phase shift center position; A columns represent the difference from $ for the mass at ±6mm. 48. ln F (hz) T (sec) , ,—6mm , (degree) (degree) (degree^1 -2.8 .010 103.3 -2 246 -8 -2.6 .011 84.6 1 225 -7 -2.4 .014 69.3 1 206 -4 -2.2 ,018 56.7 1 189 -2 -2.0 .022 46.4 1 173 1 -1.8 .026 38.0 2 157 2 -1.6 .032 31.1 4 140 4 -1.4 .039 25.5 4 124 4 -1.2 .048 20.9 2 109 2 -1.0 .058 17.1 1 94 -1 -0.8 .072 14.0 -2 81 -4 -0.6 .087 11.4 -4 69 -7 -0.4 .107 9.4 -4 57 -8 -0.2 .130 7.7 -4 46 -7 0.0 .159 6.3 -3 36 -6 0.2 .194 5.1 -2 27 -4 0.4 .237 4.2 -1 20 -3 0.6 .290 3.4 -1 15 -3 0.8 .354 2.8 -1 10 -2 1.0 .433 2.3 -1 6 -2 1.2 .528 1.9 -1 2 -2 1.4 .645 .1.6 0 -3 -1 1.6 .788 1.3 0 -7 -1 1.8 .963 1.0 0 -11 -1 2.0 1.176 0.8 0 -16 -2 Table I I I . The phase response of the Geotech-SSA combination: $-the phase shift,center position; A columns represent the difference from $ for the mass at ±6mm. Fig. 21 Phase response of the Sprengnether seismometer-solid state amplifier(SSA) combination at center position. -30 Fig. 22 Phase response of the Geotech seismometer-solid state amplifier (SSA) combination at center position. the seismometer in a stable temperature environment. It has been observed that the amplitude response does not vary linearly with the deviation of the mass from the zero position, but that the rate of change increases. For a ±3 mm d r i f t , which corresponds to a temperature fluctuation of less than 1°C, the phase change can be held to less than 1.5° provided the unsymmetrical response of the Geotech can be corrected. It i s expected that this standard can be achieved with an improved temperature controller. The geophysical implications of these errors w i l l be discussed in Chapter VI. Chapter V The Noise 5 . 1 The Thermal Noise A fundamental theorem due to Nyquist ( 1 9 2 8 ) states that a noise source r e s u l t s from thermal a g i t a t i o n associated with any d i s s i p a t i v e element of an e l e c t r i c a l network. This source can be represented by an rms voltage V i n se r i e s with the output impedance Z q (admittance Y q ) of the equivalent e l e c t r i c a l c i r c u i t or by an rms current I across the same impedance. The corresponding s p e c t r a l d e n s i t i e s are given by: G = 4kTRe(Z ) ( 5 . 1 ) nV o G _ = AkTRe(Y ) ( 5 . 2 ) n i o where: k = Boltzman's constant T = absolute temperature As pointed out by Russ e l l and K o l l a r ( 1 9 6 6 ) i n t h e i r analysis of a "mass on a s p r i n g " seismometer, the source current analysis can be made with ease. From the c i r c u i t of F i g . 5 , Re(Y Q) i s simply the damping r e s i s t o r (R i s p a r a l l e l with R^) i n the p a r a l l e l resonant c i r c u i t and i s independent of frequency. At long periods, the terms 1 , 1 ' and r' are completely n e g l i g i b l e . Referring to the c i r c u i t of F i g . 5 and using ( 2 . 2 8 ) , the mean square current across the damping r e s i s t o r i s equivalent to: i 2 . 4 4 <=-3> 1 G P Moreover: 1 - / G n l d f ( 5 ' 4 ) f l 2 Combining (5.2), (5.3) and (5.4) and s u b s t i t u t i n g s y f o r y*: f2 72 = ( V V kT(R+Rp) /-df ( } y l ^2T?K/ RR D J f4 f l For a bandwidth of one octave about a c e n t r a l frequency f, the lower l i m i t f ^ i s equal to f/-^*2 and the upper l i m i t i s equal to yjT f. With these l i m i t s , equation (5.5) becomes: T 2 - A/I l p 2 k T ( G 2 + D V { 5 6 ) 1 4 2 3 48TT K R ^ f J For comparison w i t h s e i s m i c n o i s e , i t i s more convenient to express the e q u i v a l e n t ground displacement as a peak-to-peak amplitude. The peak i s taken as that value whose magnitude i s exceeded 5% of the time. For t h i s d e f i n i t i o n , the r a t i o of peak-to-peak amplitude to rms value i s 3.28 (Byrne, 1961). In microns: 6 1 r7\£" k T ( G 2 + D R r ) l l / 2 3.28x10 Ip Y P p i 4 A 5.2 The Excess Noise 3 V 3 (5.7) U s u a l l y , the noise measured at the output of an a m p l i f i e r i s much higher than the thermal noise of the i n p u t r e s i s t a n c e . Many sources are r e s p o n s i b l e f o r t h i s excess noise and i t i s common i n e l e c t r o n i c s to represent i t by an equivalent, rms v o l t a g e source i n s e r i e s w i t h the i n p u t r e s i s t a n c e of the a m p l i f i e r . A corresponding s p e c t r a l d e n s i t y of the ground displacement, G > may be obtained by d i v i d i n g the s p e c t r a l d e n s i t y of the excess noise G g at the input of the a m p l i f i e r by the square of the m a g n i f i c a t i o n of the seismometer: G » \ (5.8) ne G e 2 S 5 4 . From (3.7) G 1 ' e P ne 6_2 2 2 64TT G K Zj 1 6 I T 4 K 2 I ^ 2 4 T T 2 ( G 4 + 2 D R D G 2 - 2 K R D 2 U + D 2 R D 2 ) R p V For one octave bandwidth around f, the corresponding peak-to-peak ground amplitude, i n microns, i s given by: PP-3.28x10 1 E 87T 3GKZi 8 ^ 2 i r 4 K Z I ^ Z 7> j2Tr 2(G 4+2DR DG^-2KR R ) ZU+D^R D /) 2 „ 2, 3 f 2 „ 2 lhJ2 R ^ U 4 0 f U/2 ( 5 . 9 ) ( 5 . 1 0 ) 5.3 The Noise Spectrum of the Amplifier To determine the noise introduced by the a m p l i f i c a t i o n stage exclusive of the long-period seismometer, the spectra of the PTA and SSA am p l i f i e r s were measured. The spectra G were obtained by taking the Fourier transform of the time au t o c o r r e l a t i o n function of the noise N ( t ) : G A C £ ) = l 7 f N ( 2 t ) N ( t + x ) e - J 2 i r f T dtd, - - - r / 2 ( 5 . 1 1 ) The s i g n a l from the a m p l i f i e r was recorded d i g i t a l l y and standard techniques used to evaluate ( 5 . 1 1 ) . The band of the s i g n a l was l i m i t e d by the low c u t o f f frequency of the a m p l i f i e r (-.01 hz) and the high cutoff frequency of the d i g i t a l voltmeter ( ~1.5 hz) which was sampling at an i n t e r v a l of 0.14 sec. A one hour record length was used to obtain s u f f i c i e n t p r e c i s i o n f o r the long-period components. The spectrum of the PTA at any gain (the attenuator i s at the input of the a m p l i f i e r ) w i t h an input of 4 0 n i s shown i n Fig.23. The 5 5 . spectrum of the SSA (Gain: 100000) with an input resistance of 300p, i s shown i n F i g . 24. In both cases, the spectrum was smoothed by tapering the a u t o c o r r e l a t i o n function with a Hanning window. 5.4 Comparison with Seismic Noise S t a t i s t i c a l data on microseismic noise has been c o l l e c t e d by Brune and O l i v e r (1959). Their r e s u l t s are summarized by three curves representing maximum, average, and minimum values of surface displacement i n a one octave bandwidth ( s o l i d p o r t i o n of upper 3 curves i n F i g . 25 and 26). Savino and Hade (1970) and Sayino et a l (1971) have observed a pronounced minimum i n the spectrum of earth noise between 30 and 40 sec and an increase of 14 db/octave noise between 50 to 100 sec at Ogdensburg, N.J. S i m i l a r r e s u l t s are found elsewhere i n the world. This increase i n noise i s a t t r i b u t e d to loading of the earth's surface by atmospheric pressure v a r i a t i o n s . On t h i s basis the curves of Brune and O l i v e r are continued to longer periods i n F i g . 25 and 26 (dotted sections of curves). To determine whether microseismic or instrument noise predominates at a given frequency, the curves f o r earth noise are compared with the apparent noise at the surface of the earth due to the fundamental" thermal noise of the seismometer and the excess noise of the SSA a m p l i f i e r (Gain=3000) f o r both the.Sprengnether ( F i g . 25) and Geotech ( F i g . 26). Curve 1 i s obtained d i r e c t l y from equation (5.7) using the values of Table I for the center p o s i t i o n . Curve 2a i s obtained from equation (5.10) where G^ i s the average s p e c t r a l density given by the mean square noise divided by the bandwidth of the noise and r e f e r r e d to the input of the a m p l i f i e r : G = " (5.12) 6 Gain Af 5 6 . 58. 2.0 u > u 0.5 1.0 1.5 2.0 2.5 L0G 1 Q PERIOD ( p e r i o d i n s e c ) F i g . 25 Comparison o f s e i s m i c n o i s e w i t h the f u n d a m e n t a l noise(1) o f t h e S p r e n g n e t h e r s e i s m o m e t e r and the e x c e s s n o i s e (2) o f t h e s o l i d s t a t e a m p l i f i e r ( S S A ) . 2a i s d e t e r m i n e d f r o m 5.10; 2b i s o b t a i n e d f r o m the SSA s p e c t r u m . 59. 2.0 r -6.0' ' ' ' ' • 0.0 0.5 1.0 1 > 5 2 Q 2 5 L0G 1 Q PERIOD (period in sec) Fig. 26 Comparison of seismic noise with the fundamental noise(1) of the Geotech seismometer and the excess noise(2) of the solid state amplifier (SSA). 2a i s determined from 5.10; 2b is obtained from the SSA spectrum. 60. The mean square noise at the output of the SSA corresponds to 2 the zero lag autocorrelation function and i s equal to 34.3 mV . For a gain of 100000 and a bandwidth of 1.5 cps, the average spectral density -15 2 is 2.28 x 10 V s . From this, the spectral density of the funda-mental noise at the input of the amplifier must be subtracted, i t i s —18 2 equal to 4kTR^ or 5.28 x 10 V s . Since i t i s three orders lower than Ge, i t can be considered negligible and a l l the output noise i s excess noise. The spectra of Fig. 23 and Fig. 24 are not exactly white, i n particular at low frequencies, where the 1/f component dominates. Curve 2b i s a better estimate of the corresponding ground displacement. I t i s obtained by a numerical integration of the SSA spectrum divided by the square of the magnification curves of the Sprengnether and the Geotech. Because of the changing limits of integration, i t is easier to do this integration on a l o g ^ f scale. On that scale: 1/2 y = 3.28 x 10 6  P P2 , ^ ^ ( x H O ^ i x 9 x 10 In 10 MS 00 X-.15 (5.13) -4 The 9x10 factor i s to scale the power spectrum to the amplification level of the SSA (3000) used in the measurement of the magnification MS. 61. From the curves of Fig. 25 and Fig. 26, one can conclude that the fundamental noise of the seismometer-amplifier combination (without f i l t e r ) is negligible in comparison with the excess noise of the SSA amplifier. The excess noise should be of the same order as the seismic noise at longer periods and less important at periods below the 30-40 sec minimum. For any period the sum of a l l noises i s equal to: In the above discussion i t i s assumed that the seismometer cases are airtight and perfectly r i g i d i.e.pressure effects have not been considered. (5.14) 62. Chapter VI Summary and Conclusions 6.1 Re l i a b i l i t y of Phase Velocity In phase velocity measurements, the phase shift A $ ( < D ) of a component of the wavetrain spectrum over a known distance Ax i s related to the travel time by: A$ = coAt (6.1) The phase velocity c(co) is given by: c - ff (6 .2 , 'if- ( 6 ' 3 ) When measuring A $ , the phase shift introduced by the seismograph must be subtracted at both stations. As the phase response of the instruments are known only within certain li m i t s , i t i s important to determine the error in phase velocity due to errors i n the phase response. By (6.3) dc = ^ dA* (6.4) A<JT Assuming that the instrumental phase at one station can be determined to ±2°, and allowing an additional degree for phase inaccuracies due to temperature fluctuations, the most probable error for 2-station measurements i s given by: M • ± 2W2(236V l2) (6.6) = ± 0.055 (6.7) For a two significant digit velocity determination, we have by (6.5) c 2 T < 1 2TTAX At 20 sec period t h i s requires f o r t y p i c a l v e l o c i t i e s a s t a t i o n separation of only 40 km but at 80 sec a separation of 200 km i s required to achieve t h i s accuracy. Some errors may also be introduced by geologic e f f e c t s (e.g. r e f r a c t i o n ) and analysis procedures. 6.2 Environmental Problems In t h i s f e a s a b i l i t y study of a portable long period seismic array, i t i s found that achievement of temperature s t a b i l i t y to w i t h i n 1°C i s necessary. Fluctuations l a r g e r than this lead to changes i n the instrumental phase response i n excess of 1°. Large pressure f l u c t u a t i o n s (2 cm Hg) w i l l not influence the zero s t a b i l i t y s i g n i f i c a n t l y . Rapid f l u c t u a t i o n s may however be a noise source i f the seismometer case i s not a i r t i g h t or i f i t i s not s u f f i c i e n t l y r i g i d . The seismometer should be located i n an environment where short term f l u c t u a t i o n s are minimized. 6.3 Noise At the periods of i n t e r e s t f o r Rayleigh wave studies, the instrumental and seismic noiseare of the same order. The fundamental noise i s not s i g n i f i c a n t although i t w i l l increase when f i l t e r s are introduced to shape the seismograph responses to give maximum s e n s i t i v i t y i n the 50 to 100 sec band. 64. 6.4 Determination of Phase Response from the Amplitude Response For a minimum-phase system, determination of the phase response from the amplitude response has been shorn to be f e a s i b l e . However, the amplitude response i s required over a much wider frequency band than the region of i n t e r e s t of < i > ( c o ) . In the p a r t i c u l a r systems under consider-ation, t h i s has lead to several problems • c a l i b r a t i o n due to eddy currents, noise l e v e l s , and resonances i n the system. For the portable system , i t i s recommended that the phase response of the system be determined d i r e c t l y . However, i t i s expected that the techniques given here w i l l be u s e f u l when only the amplitude curves are a v a i l a b l e as i n the case f o r many permanent s t a t i o n s . Further these procedures serve as an a i d to r e l a t e the form of the phase response to the basic p h y s i c a l c h a r a c t e r i s t i c s of the system. 65. References Bode, H.W., Network analysis and feedback a m p l i f i e r design, D. van Nostrand Co., Inc., New York, N.Y,, 1945. Brune, J.N. and J . O l i v e r , The seismic noise of the earth's surface, B u l l . Seism. Soc. Am., 49, 349-353, 1959. Byrne, C.J., Instrument noise i n seismometers, B u l l . Seism. Soc. Am., 51, 69-84, 1961. Caner, B., E l e c t r i c a l conductivity structure i n Western Canada and 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 , J . Geomagn. Geoelectr., 22, 113-129, 1970. Dorman, J . and M.Ewing, Numerical i n v e r s i o n of seismic surface wave dispe r s i o n data and crust-mantle structure i n the New York-Pennsylvania area, J . Geophys. Res., 6_7, 5227-5241, 1962. Haskell, N.A., The dispersion of surface waves i n multilayered media, B u l l . Seism. Soc. Am., 43, 17-34, 1953. Isacks, B. J . , J . O l i v e r , and L.R. Sykes, Seismology and the new g l o b a l t e c t o n i c s , J . Geophys. Res., 73, 5855-5899, 1968. K o l l a r , F. and R. D. R u s s e l l , Seismometer analysis using an e l e c t r i c current analog, B u l l . Seism. Soc. Am., 5_6, 1193-1205, 1966. McCormick^J. M. and M.G. Salvadori, Numerical methods i n FORTRAN, Pre n t i c e -H a l l , Inc., Englewood C l i f f s , N.J., 1964. McGarr, A. and L. Alsop, Transmission and r e f l e c t i o n of Rayleigh waves at v e r t i c a l boundaries, J . Geophys. Res., 72_> 2169-2180, 1967. Mai, A. K. and L. Knopoff, Transmission of Rayleigh waves past a step change i n e l e v a t i o n , B u l l . Seism. Soc. Am., 55, 319-334, 1965. Melton, B. S., Technical note 2/70, The La Coste suspension-Principles and p r a c t i c e , Teledyne Geotech, 3401 Shiloh Road, Garland, Texas, 1970. Nyquist, H., Thermal a g i t a t i o n of e l e c t r i c charge i n conductors, Phys. Rev., 32, 110-113, 1928. Papoulis, A., The Fourier i n t e g r a l and i t s a p p l i c a t i o n s , McGraw-Hill Book Co., Inc., New York, N.Y., 1962. 66. P i l a n t , W. L. and L. Knopoff, Observations of multiple seismic events, B u l l . Seism. Soc. Am., 54, 19-39, 1965. Sato, Y., Analysis of dispersed surface waves by means of Fourier Transform I, B u l l . Earthq. Res. Inst., 33, 33-50, 1955. Savino, J . and G. Hade, Long-period (15-150 sec) seismic noise observations at the Ogdensburg Mine Observatory, Trans. Amer. Geophys. Union, 574, 1970. Savino, J . , K. McCamy and G. Hade, An improved high-gain, long-period, seismograph system, I I I . A pronounced minimum i n the spectrum of long-period earth noise between 30 and 40 s e c , Trans. Amer. Geophys. Union, 558, 1971. Shima, E., T h e o r e t i c a l and experimental approach to the designs and c a l i b r a t i o n s of electro-magnetic seismograph: I. Voltage s e n s i t i v i t y of the moving-coil type seismometer, B u l l . Earthq. Res. Inst., 38, 29-39, 1960. Solodovnikov, V.V., Introduction to the s t a t i s t i c a l dynamics of automatic c o n t r o l systems, Dover P u b l i c a t i o n Inc., New York, N.Y., 1960. White, W. R. H. and J . C. Savage, A seismic r e f r a c t i o n and gravity study of the earth's crust i n B r i t i s h Columbia, B u l l . Seism. Soc. Am., 55, 463-486, 1965. Wickens, A. J . and K. Pec, A crust-mantle p r o f i l e from Mould Bay, Canada, to Tucson, Arizona, B u l l . Seism. Soc. Am., 58, 1821-1831, 1968. Appendix A l = SLOPE OF LEFT ASYMPTOTE . A2 = SLOPE OF RIGHT ASYMPTOTE H = INTERVAL AT WHICH DATA ARE READ k t B = LIMITS OF I NT RAT I ON ( MUST BE SYMMETRICAL RELATIVE TO 0 ) Ct D = LIMITS 3F AVAILA3LE 3 A TA DM BODE PLOT In' = ANGULAR FREQUENCY FOR WHICH WE SEEK PHASE RESPONSE kw =,LOG OF W XMIN = L3G OF MINIMUM W XMAX = LOG OF MAXIMUM W NW = TOTAL NUMBER OF POINTS FROM A TO B N = TOTAL NUMBER OF PCI NTS FROM C TO D NT = T3TAL NUMBE* OF POINTS F RDM A+XMIN TQ B + XMAX L = TOTAL NUMBER OF POINTS FOR WHICH HE SEEK PHASE RESPONSE A.B.CiDtXW MUST BE INTEGRAL MULTIPLES OF H v OTHERWISE LIMITS WILL NOT BE AT DATA POINTS OR DERIVATIVE AND WEIGHTING FUNCTIONS WILL NOT LINE-UP WHEN SHIF TED RELATIVE TO EACH OTHER IF ANY CONSTANT IS CHANGED IN THE PROGRAM ONLY THE FIRST PART MUST BE REVIS5 DIMENSIONS ARE AS FOLLOWS: AMP(N)» DER(N)• DE(NT), WE(NW), XW(L)t PHA SE(L ) DETERMINATION OF THE CONSTANTS A l = +4. j . A2= - 2 . H = 3 .2 B = +7. C = - 4 . D = +5. XMIN = -2 .8 XMAX = +2. N = ( D - O / H + l .1 NT = { [B + XMAX)-(A+XMIN))/H+L.1 MW = . ( B - A ) / H + l . 1 L=25 DIMENSION AMP(46)t DER{46), DE(95), W E {71 )» XW(25), P HASE(25) COMPUTATION OF THE D E R I V A T I V E R E A D(5 , 9 1 ) AMP 91 F O R M A K 1 3 F 6 . 2 ) V\ = N - l D E R ( i ) = A l DO 15 I = 2 ? M 15 D E R ( I ) = ( A M P ( I + 1 ) - A M P ( I - l ) ) / C 2 . * H ) DER(N) = A2 W R I T E ( 6 , 8) DER - 7 . C O M P I L E * MA JN 0 6 - 1 6 - 7 1 0 2 : 2 6 : 1 9 P A G E 0 0 0 2 C R M A T I / / 2 6 X t ' D E R ' / ( 2 0 X t F 1 0 . 2 ) ) EXTENSION OF THE RANGE OF THE D E R I V A T I V E \i 1 = ( C - (A+XMIN) ) / H + . 1 DO 25 J = 1,N1 25 O E ( J ) = A l N2 = M l+1 N3 = N U N DO 30 J = N2.N3 I = J - N l 30 D E I J ) = D E * ( I ) m = N3* i DO 40 J = N 4 , N T 40 D E ( J ) = A2 TABULATION OF THE WEIGHTING FUNCTION PI = 3 . 1 4 1 5 9 3 X = A DO 5 0 K = 1,NW WE(K) = l . / P U A L O G U . / T A N H ( A B S I X / 2 . ) ) ) 50 X = X+H d R I T E ( 6 , 7 7 ) 77 F QRMAT{* 1 • 1 O X , ' X W 1 1 3 X , • F • 1 5 < , « T • 1 3 X t ' P H A S E • ) S H I F T I N G AND COMPUTATION OF THE INTEGRAL * E A Q ( 5 , 9 2 ) XW 92 F O R M A T C 2 0 F 4 . 1 ) MW = NW-1 KW = NW-3 DO 5 I = 1 »L JA = (XW ( I ) - X M I N ) / H + l . l JO = JA+KW/2 FA = ( D E I J A ) - D E ( J O ) ) * W E ( 1 ) E\/EM = 0 . ODO = 0 . DO 65 K = 2»KW»2 EVEM = EVE^+t DE( J A - U K ) - D E ( JO) ) *WE( K) 65 ODD = O D D + ( D E ( J A + K ) - D E ( J O ) ) * W E ( K + l ) F MW = ( D E ( J A + N W - 2 ) - D E ( J D ) )*WE(MW ) FB = ( D E ( J A + M W ) - D E ( J O ) ) * W E ( N W ) PHASEJ I) = (FA + 4 . * ( E v E N + F M W U 2 . * O D D * - F 3 ) * H * 6 0 . / P I + DE( J 0 U 9 0 . F=EXP(XW ( I ) ) / ( 2 . * P I ) T = 1 . / F 5 W « I T E ( 6 t 9 7 ) X W ( I ) , F , T , P H A S E ( I ) 97 F ORMAT ( / F U . 11 F l 6 .4 f 2F16 • 2 ) PLOTT ING DF THE PHASE RESPONSE .0MPILER MAIN 0 6 - 1 6 - 7 1 0 2 : 2 6 : 1 9 PAGE 0003 CALL P L O T S CALL AX I S O . f l . ? ' L O G OF ANGULAR FREQUENCY* , - 2 4» 8. » 0 . » - 4 . , 1 . ) C A L L A X I S O . t O . , » PHASE RESPONSE ( SPRENGNETHER : OMM ) • , 3 3, 10 . , 9 0 . , - 3 I 1 0 . , 3 0 . ) 03 18 I = 1 ,L XW ( I ) = X W ( I ) +4 . 18 P H A S E ( I ) = ( P H A S E d ) * 3 0 . ) / 3 D . CALL PLQT(XW( 1) tPHASEt 1) t + 3) DO 20 I = 2 , L 20 CALL P L Q T ( X H ( I ) t P H A S E t I ) f + 2 ) CALL PL0TM3 STOP END REQUIREMENTS 000E02 BYTES 3 .2 SECONDS 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0053438/manifest

Comment

Related Items