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A method for simulating and representing strong ground motion Jurkevics, Andrejs 1978

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A METHOD FOE SIMULATING STRONG GROUND AND REPRESENTING MOTION by ANDREJS JUBKEVICS B. Sc., University of Toronto, 1976 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department of Geophysics and Astronomy) He accept t h i s t h e s i s as conforming to the reguired standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1978 Andrejs Jurkevics, 1978 In present ing t h i s thes is in p a r t i a l fu l f i lment of the requirements f o r an advanced degree at the Univers i ty of B r i t i s h Columbia, I a g r e e that the L ibrary sha l l make it f ree l y ava i lab le for r e f e r e n c e and s t u d y . I fur ther agree that permission for extensive copying o f t h i s t h e s i s for scho la r l y purposes may be granted by the Head o f my Department o r by h is representat ives . It is understood that c o p y i n g o r p u b l i c a t i o n o f th i s thes is for f i n a n c i a l gain sha l l not be allowed without my wr i t ten permiss ion. Department of ^^^e^7^A -^ X2^cc- -o "4- dr- S (y^n^ery^^^ The Univers i ty of B r i t i s h Columbia 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 P a t , ^ - t t . 1 l/vtf ABSTRACT A method for representing and synthesizing strong motion accelerograms i s proposed i n t h i s thesis. The procedure models an acceleration time-history as a non-stationary second order autoregressive (AR) process. Three AR parameters are determined from the data i n a time-adaptive manner. They provide a guantitative description of the time-varying spectral content of the recording. The AB parameters may also be u t i l i z e d as prediction f i l t e r c o e f f i c i e n t s , enabling one to generate a suite of a r t i f i c i a l accelerograms, each having the same time-dependent spectral content as the target record. The simulated time-h i s t o r i e s may be used for computing s t r u c t u r a l response i n earthquake-prone areas. This analysis has been extended to include a number of recordings obtained during earthguakes of various magnitudes (M) at a variety of epicentral distances (D).,As a r e s u l t , 'type curves' representing the empirical behaviour of the three AB parameters have been determined. Although incomplete, t h i s information may be used to generate a r t i f i c i a l accelerograms for ar b i t r a r y combinations of M and D. i i i TABLE OF CONTENTS Page ABSTRACT . , , . i i TABLE CF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i i i LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . V • LIST OF FIGURES , v i ACKNOWLEDGEMENTS .... .................. X I. INTRODUCTION 1 II . THEORETICAL BACKGSOUND 7 A. Random Processes and Entropy ........................ 7 B. The AR Model 11 C. Determining the AR Parameters .......................,14 D. AR Order ............................................ 19 III. APPLICATIONS TO STRONG MOTION ........................ ,21 A. AR Order for Accelerograms .......................... 24 B. Physical Interpretation .........,................... 28 C. Segmented Analysis of Accelerograms ................. 31 IV. TIME-ADAPTIVE APPROACH ................................ 36 A. Time-Adaptive Maximum Entropy Analysis ..............36 B. The LMS Adaptive Technigue 41 V. SIMULATING GROUND MOTION 55 VI. EMPIRICAL AR ANALYSIS OF STRONG MOTION ACCLELEROGRAMS . 71 A. The Approach ............. r.......................... 71 B. The Analysis .......................................• 73 C. The Results ...... ..... , , .... 86 VII. CONCLUSIONS APPENDIX REFERENCES V LIST OF TABLES Table Page I. Earthguakes used i n empirical AR analysis ............ 75 II . Accelerograms used i n empirical AR analysis .77 I I I . Summary of graphs containing r e s u l t s of empirical AR analysis ...... ...................... .... 87 v i LIST OF FIGURES Figure Page 1. ;K-S component of acceleration recorded at E l Centre, C a l i f o r n i a during the 1940 Imperial Valley earthquake. .................................... 2 2. Acceleration Response Envelope Spectrum for the El Centro accelerogram. ............................... 4 3., K-S component of ground acceleration recorded at 8244 Orion Blvd., Los Angeles, during the 1971 San Fernando earthquake. ......................... 22 4. Periodcgram power spectrum estimated from the data between t = 6 and t = 12 seconds of the Orion Blvd. recording, ................................ 23 5. *R spectral estimates for orders 2, 4 and 8, evaluated between t = 6 and t = 12 seconds of the Orion Blvd. accelerogram. ......................... 25 6. AB spectral estimates for orders 2, 4 and 8, evaluated between t = 15 and t = 20 seconds of the Orion Blvd. accelerogram. ......................... 26 7. AR spectral estimates for orders 2, 4 and 8, evaluated between t = 8 and t = 10 seconds of the Orion Blvd. accelerogram. .......................... 27 8. Second order f i l t e r poles z, and z L i n the imaginary z plane. ................ .................... 29 9. AR spectral parameters obtained by a segmented analysis of the Orion Blvd. accelerogram. ............. 33 10. Integrated power, P(t) as a function of time for the Orion Blvd. accelerogram 35 11. AR parameters of the Orion Blvd. recording as calculated by the Riley-Burg adaptive technique. ....................... 38 12. AR parameters as determined by the Gutowski adaptive algorithm using a gate length of 11 points. ............ 40 v i i Figure Page 13. Sixth order AR power spectra of a synthetic s i g n a l estimated by the one-sided and two-sided LMS processors. .... .................................... 47 14. Eighth order AR power spectra of a synthetic sig n a l estimated by the one-sided and two-sided LMS processors. . . 49 15. Tenth order AR power spectra of a synthetic s i g n a l estimated by the one-sided and two-sided LMS processors. ....................................... 50 16. AR spectral parameters of the Orion Blvd. accelerogram as determined by the two-sided LMS algorithm with Y = 0.1 ..52 17. AR spectral parameters of the Orion Elvd. accelerogram evaluated by means of the two-sided LMS algorithm with X = 0.01. 53 18. A r t i f i c i a l accelerogram fashioned a f t e r the Orion Blvd. recording. .......................... ...... 57 19. A r t i f i c i a l accelerogram fashioned a f t e r the Orion Blvd. recording. ................................ 58 20. A r t i f i c i a l accelerogram fashioned a f t e r the Orion Blvd. recording. ....•-......... .................. 59 21. AR spectral parameters evaluated from six a r t i f i c i a l accelerograms fashioned af t e r the Orion Blvd. recording. ................ ................ 60 22. Acceleration Response Envelope Spectrum of the Orion Blvd., accelerogram. ............................ 61 23. ARES of six a r t i f i c i a l accelerograms fashioned a f t e r the Orion Blvd. recording. 63 24. Tangential component of acceleration recorded at Cal. Tech, , Pasadena, during the 1971 San Fernando earthguake. .................... .............. 64 25, AR spectral parameters as determined by an adaptive analysis of the Cal. Tech. accelerogram. •. ...... ........ 65 26. A r t i f i c i a l accelerogram fashioned a f t e r the Cal. Tech. recording. .<«... 66 V X l i Figure Page 27. A r t i f i c i a l accelerogram fashioned after the Cal. Tech. recording. ..... ............ ................ 67 28. ARES for the Cal. Tech. accelerogram 68 29. ARES of six a r t i f i c i a l accelerograms fashioned after the Cal. Tech.,recording. ....................... 69 30. Resolving the two horizontal components of acceleration into r a d i a l and tangential directions. ................ ....... .... .. ...... ... . ..... ;.. 76 31. Radial component of ground acceleration recorded at the Pacoima Dam, C a l i f o r n i a , during the 1971 San Fernando earthguake. ..................... 80 32. AR spectral parameters corresponding to the Pacoima Dam accelerogram. 81 33. Radial component of ground acceleration recorded at Temblor, C a l i f o r n i a , during the 1966 Par k f i e l d earthquake. 83 34. AR spectral parameters corresponding to the Temlor recording. ................. .................... 84 35. Radial component of ground acceleration recorded at s i t e no. 4, Lake Hughes, during the 1971 San Fernando earthguake. 85 36. AR spectral parameters corresponding to the Lake Hughes recording. ..•',......................... •... 86 37. AB spectral parameters for M = 6.4 and D = 9 km. ........................................... ........ 88 38. AB spectral parameters for M = 6.4 and D = 25 -35 km. ............ . . ...... . 89 39. AR spe c t r a l parameters f o r M = 6.4 and D = 28 -38 km. 90 40. AR spectral parameters for M = 6.4 and D = 34 -50 km. ., ................ 91 41. AR spectral parameters for H = 6.4 and D - 6 0 -76 km, 92 42. AR spectral parameters for M = 6.4 and D = 64 -86 km. ,93 ix F i g u r e Page 4 3 . AR s p e c t r a l p a r a m e t e r s f o r M = 6 . 4 a n d D = 135 km, . . . . . . . . , . . . . . . . . . . . . . . , , . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4 4 . AR s p e c t r a l p a r a m e t e r s f o r M = 6 . 0 a n d D = 8 km. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 45 . AR s p e c t r a l p a r a m e t e r s f o r M = 5 . 5 and D = 11 -21 km 96 46 . AR s p e c t r a l p a r a m e t e r s f o r M = 5 . 5 and D = 15 -25 k m . 97 4 7 . AR s p e c t r a l p a r a m e t e r s f o r M = 5 . 3 a n d D = 12 km. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4 8 . Maximum v a l u e s o f ^ ^ ( t ) a s a f u n c t i o n o f M a n d D. ...... , . 1 0 2 4 9 . D u r a t i o n o f s h a k i n g as a f u n c t i o n o f H a n d D . . . . . . . . . . 1 0 6 X ACKNOWLEDGEMENTS F i r s t and foremost, I would l i k e to express my gratitude to my supervisor. Professor Tad Olrych, who turned me on to time serie s analysis and who was my guiding l i g h t during the course of t h i s research., I would l i k e to acknowledge some enlightening discussions held with the fo l k s at the Dept. of C i v i l Engineering, OBC, concerning earthquake engineering. In p a r t i c u l a r . Professor Sheldon Cherry provided some valuable information and ideas during the early stages of t h i s study. I thank Professor Bob E l l i s for reading t h i s thesis and providing some constructive suggestions. My gratitude also goes to Garry Rogers, Al Thorleifson, Jean^Gerard Napoleoni, Mathew Yedlin, Barry Narod, and a l l the other graduate students at the Geophysics Department f o r t h e i r inspiration and for creating such a f i n e working atmosphere. F i n a l l y , I would l i k e to express my appreciation to the National Research Council of Canada for supporting me during the course of t h i s study with a post-graduate scholarship and for bearing the computing costs. 1 Ii. INTRODDCTION The object of t h i s thesis i s to apply some well-known time-series analysis technigues to strong motion acceleration data i n order to simulate r e a l i s t i c a r t i f i c i a l accelerograms for use i n earthguake design of c i v i l structures. The accelerograms are modelled by time-varying autoregressive processes which, as an addit i o n a l benefit, simultaneously y i e l d a quantitative estimate of the time-varying spectral content of the time s e r i e s . Figure 1 shows one of the f i r s t strong motion accelerograms ever to be recorded. I t i s the N-3 component of ground motion recorded at E l Centro, C a l i f o r n i a during the 1940 Imperial Valley earthguake (M = 6.7). (In th i s thesis, M denotes the l o c a l magnitude.) Good accelerograms were hard to come by i n those days, so f o r many years the El Centro recording was used as a design earthguake f o r engineering structures a l l over the world. However, the station at E l Centro was located over some 6 km. of unconsolidated sediments ( F a c c i o l i and Rasendiz (1976)), so t h i s recording was in fact quite a t y p i c a l of ground motion which could be expected during medium-sized earthquakes elsewhere. It soon became apparent that the strong motion catalogue was expanding extremely slowly, and methods had to be developed to specify the important c h a r a c t e r i s t i c s of ground motion which could occur under other recording conditions. Housner (1941) made the f i r s t s i g n i f i c a n t contributions to the engineering interpretation of strong motion accelerograms. He introduced the concept of a Response Envelope Spectrum (RES) ACCELERATION.CM/SEC/SEC (XlO 1 -45.0 -30.D -15.0 0.0 15.0 -I '—i J L. 30.0 •L-45 J H cn G w (0 n as OJ ro i n » w c* rt-tr t-1 O iQ O O c •» B p w o o (D (» a H (0 »-» H - O Mi r+ O II H O S3 !™h H -• CD PJ O — pu O • C (0 H 1-H - (D O r-( rt tr o ro y - » n •P O o o n M P< B (0 TJ P ' 3 i n order to d i f f e r e n t i a t e the s i g n i f i c a n t features of strong motion records and to provide a simple method for evaluating the response of engineering structures to ground shaking. An Acceleration Response Envelope Spectrum (ARES), for example, i s evaluated as follows (Cherry (1970)): an accelerogram i s used to excite a suite of single degree-of-freedom harmonic o s c i l l a t o r s , each having a d i f f e r e n t natural freguency. Some measure of damping i s usually introduced as well, in order to approximate the response of real structures. The maximum acceleration experienced by each o s c i l l a t o r i s then evaluated and plotted as a function of the corresponding natural period. As the damping i s increased, the maximum response decreases, and the envelope becomes f l a t t e r . To i l l u s t r a t e , the ARES for 2 per cent damping has been evaluated f o r the E l Centro accelerogram, and i s presented in Figure 2. Due to the fact that the maximum response of any pa r t i c u l a r o s c i l l a t o r i s somewhat phase dependent, response envelopes tend to be 'peaky* i n nature. Hence, dif f e r e n t accelerograms with s i m i l a r freguency and amplitude contents may exhibit d i f f e r e n t response envelopes. A further problem arises i n conjunction with the information content of an RES. Since the response envelope i s , s t r i c t l y speaking, a freguency spectrum, there are an i n f i n i t y of time-histories with dif f e r e n t amplitude configurations for which the response spectra may be i d e n t i c a l . For rigorous solutions to the st r u c t u r a l dynamics problem, actual time-histories of motion must be used as input to excite the system. Hence as earthguake design developed into a f i e l d of c i v i l engineering, the stage was set for the introduction of r e a l i s t i c a l l y simulated 4 FIGURE 2. Acceleration Response Envelope Spectrum f o r the E l Centro accelerogram. 5 accelerograms. Housner (1941) was the f i r s t to recognize the extremely i r r e g u l a r nature of earthguake ground motion and proposed using stationary Gaussian white noise to represent i t . Housner (1955) l a t e r suggested a simulation technigue which involved the addition of a large number of one-cycle sine waves occurring randomly i n time. With a few variations, t h i s approach was used for a number of years. As more recordings became available, i t became clear that t h i s method was inadeguate, since the white noise pulses exhibited a freguency content unlike that of recorded accelerograms. With the advent of the computer, more sophisticated simulation technigues became possible. Housner and Jennings (1964) introduced a r t i f i c i a l accelerograms formed by f i l t e r i n g Gaussian white noise in such a way that the spectral content of the records matched the undamped v e l o c i t y spectra of recorded ground motions. However, the freguency content was s t i l l stationary in time and the amplitude envelope was a simple box-car function which approximated the large amplitude portions of actual recordings. At t h i s point i t was recognized that stationary processes ignore many of the important features exhibited by r e a l earthguake time-histories. Amin and Ang (1966) were among the f i r s t to generate a r t i f i c i a l recordings which contained time-varying amplitudes as well as response spectra compatible with those of actual accelerograms. They used a f i l t e r e d Poisson process which was modified by an envelope function consisting of: a) an i n i t i a l guadratic amplitude b u i l d -up; b) a stationary 'strongest motion* central section; and c) an exponentially decaying t a i l . This envelope could be modified 6 at w i l l to represent recordings corresponding to d i f f e r e n t earthguake magnitudes and e p i c e n t r a l distances. A number of simulation technigues were subseguently proposed which were l i t t l e more than minor variations of the above approach. Non-stationary freguency content was f i r s t introduced by Beaudet (1970). He simulated r e a l i s t i c seismograms of nuclear explosions by passing random noise through a f i l t e r which had a freguency and amplitude response which could be varied in time. The f i l t e r parameters were determined by examining actual recorded events. Saragoni and Hart (1974) i n a recent paper proposed a s i m i l a r approach to generating a r t i f i c i a l earthquake records. F i n a l l y , Trifunac (1971) introduced a method for incorporating r e a l i s t i c frequency and amplitude by adding together a series of sine waves, each of which had a time-dependent amplitude. The amplitude functions were determined on the basis of surface wave dispersion c h a r a c t e r i s t i c s for the p a r t i c u l a r region under consideration. The accelerogram simulation approach developed i n this study i s more e f f i c i e n t and at the same time more powerful than any of the previous methods. It i s based on a prediction f i l t e r whose c o e f f i c i e n t s are related to the time-varying power spectrum of a target recording. Since t h i s power spectrum must be determined f i r s t , the technique also results i n an economic quantitative description of the target acclerogram. The earthquake data used i n t h i s study were obtained from tapes of Volume I I , Corrected Acclerograms. V e l o c i t i e s and Displacements, supplied by the Earthquake Engineering Besearch Laboratory, C a l i f o r n i a I n s t i t u t e of Technology, Pasadena. 7 H i THIOMIICiL BACKGROUND In t h i s chapter, an overview of the t h e o r e t i c a l aspects of autoregressive <AR) data modelling and AR power spectral estimation i s presented. For completeness, some s t a t i s t i c a l f i r s t . II.A. RANDOM PROCESSES AND ENTROPY An i n f i n i t e , discrete time series y^  i s said to be random i f the l i k e l i h o o d of occurrence of any p a r t i c u l a r value y^  from the sample space of a l l possible values (y^ , ... y^  , ... y^) i s described by an associated p r o b a b i l i t y P (y t c). The function P(y t) i s a probability d i s t r i b u t i o n with Following Lathi (1968), the mean/< of the process at time t i s given by the expectation E of yt : p'roperties of the data which are to be modelled are discussed II-1 M II-2 and w i l l be assumed to be zero. The variance cr" i s given by 8 i-1 X - V- < I.. 1 5. - - \ I . .L 1 r I ..<• I II-3 which together with Eguation II-2 defines the f i r s t order s t a t i s t i c s of the process y t. The j o i n t p r o b a b i l i t y density function P lYtl *Y-tt) provides a measure of the c o r r e l a t i o n between the random variable y t at times t, and t i . The l i k e l i h o o d of occurence of y£ at time t/ and at time tL i s given by P (yt* ). The j o i n t p r o b a b i l i t y function i s used i n the evaluation of the autocorrelation of y± ; /»U,-g = E [ y t . , , . J = Y,L yi, yi, P(ii., , j= i <.=/ which i s a second order s t a t i s t i c s parameter. If the f i r s t and second order s t a t i s t i c s / * , a~x and p (t, ,t z) are independent of time, the process i s termed wide-sense or weakly stationary. In other words, the relationships P(u.J =• P(u t l) U-5 and must hold for a l l t, and t z , where f •= t t - t, . , Hence for a weakly stationary process, the autocorrelation function depends only on the lag t : II-7 9 A s t r i c t l y stationary process i s one for which s t a t i s t i c s of a l l orders are independent of time., Por an ensemble of weakly stationary random processes, the ergodic theory states that the s t a t i s t i c s II-2, II-3 and which are given i n terms of the probability d i s t r i b u t i o n , may be evaluated using summations in time. Then, for instance, an estimate of the mean of the process becomes N II-8 M It and the autocorrelation i s estimated by N- f . \ I V 1 1 - 9 -t= I A random time seri e s f o r which the autocorrelation function i s zero f o r a l l non-zero lags i s c a l l e d a purely random process, t = i Such a time series i s also termed white noise. Jenkins and Watts (1969) have pointed out that another important class of random processes may 4>e obtained by passing white noise e^ through a l i n e a r operator ¥V z .11-11 10 The resu l t i n g time series xt i s c a l l e d a linear random process. In t h i s research, acceleration time h i s t o r i e s w i l l be represented by x t and w i l l be assumed to be l i n e a r random processes. The concept of entropy as related to data analysis has been discussed in d e t a i l by Ulrych and Bishop (1975). In t h i s context, entropy i s rel a t e d to the information content of a message. The amount of available information, I, contained i n a process i s linked to the probability of occurrence, P; , of a part i c u l a r state of the system at any time by I - - log, (PL) . H-12 If the process i s observed for a time i n t e r v a l T, the information gained i s m t— I II-13 where m i s the number of possible states. The guantity •information per unit time* i s termed entropy, H, where Entropy, therefore, i s a measure of the uncertainty described by a probability d i s t r i b u t i o n , or a measure of our ignorance about the actual structure of a system. In the r e a l world where f i n i t e length processes are dealt with, the term •maximum entropy' i s given to a probability assignment which describes the available 11 information but i s maximally non-commital with regard to the unavailable information. Obviously, an approach incorporating maximum entropy i s desireable when r e a l data sets are being dealt with. II.B. THE AR MODEL In z-transform notation, the random process x t defined by Equation 11-11 becomes where T(z) i s a l i n e a r f i l t e r , T{z) = 1 • ^z • ^ z 2 + ... , and z i s the unit delay operator. If the l i n e a r process V(z) i s assumed to be i n v e r t i b l e then Equation 11-15 may be written as A c i ) X c i ) = Ect) , II-16 where A(z) = ¥ (z). In the time domain the l i n e a r random process described by Eguation 11-16 becomes x t =. y-t-, +• a"xt-z t- • • . + e.± . 11-17 Eguation 11-17 i s an i n f i n i t e - o r d e r autoregressive representation of the data, i n which the present value x t i s expressed as a line a r combination of the preceding values xt_^ plus an error term e t. The a~ are c a l l e d autoregressive 12 c o e f f i c i e n t s and the e t are a white noise series c a l l e d the innovation. The innovation has zero mean and a well-defined variance a^1. However, most l i n e a r random processes, including strong motion accelerograms, may be adeguately described by f i n i t e AR models as depicted by Eguation 11-18; L * t = Z °-\  xt-i + e t , 1 1 - 1 8 where L i s the order of the process. A necessary assumption made in modelling x t by the AR method i s that the time series i s weakly stationary. In continuous time, a general Lth order AR process i s defined as the output from a l i n e a r system whose input i s white noise. The r e l a t i o n s h i p between input and output i s given by the Lth order d i f f e r e n t i a l eguation 4 L +. <*•_, Q/ L xti) 4- . . . + *,?c(t) - e ( t ) , n - 2 0 where e(t) changes -^-£^discontinuously. (Jenkins and Watts, d t (1969)). An autoregressive process may also be understood i n terms of a whitening or prediction error operator, where the predicted value of the time series at time t , ai xt_, + &\ xt_r + ... + a^x^, i s subtracted from the actual term x t to y i e l d the prediction error e t : •t-i 11-20 13 The c o e f f i c i e n t s 1, -a}", -a L z, ... -a"[ comprise the prediction error f i l t e r and the f i l t e r output e t i s white noise. In z-transform notation, Equation 11-17 becomes Ecu = Xc*> - a , L 2 : - 11-21 or E ( 2 ) = Xii) A (2) , " - 2 : where A(z) i s the transfer function of the prediction error f i l t e r . The roots of the polynomial A(z) must l i e outside the unit c i r c l e f or s t a t i o n a r i t y . The power spectrum of the process X(z) i s e a s i l y obtained from Equation 11-22: 11-23 Writing | E ( z ) j 2 as , the variance of the innovation, and z as exp(-2Jrif), the power spectrum becomes S(-f) = 2. (Te i-i 11-24 The power S(f) evaluated by means of Equation 11-24 i s actually a spectral density estimate. The actual power i s given by the area under t h i s curve. Van den Bos (1971) has pointed out that f i t t i n g an AR model to a random process i s equivalent to maximizing the entropy of that process. The concept of entropy 14 i s also entailed in the method by which the autoregressive c o e f f i c i e n t s are evaluated. II.C. DETERHINING THE AR PARAMETEBS Optimum estimates a£ of the autoregressive f i l t e r c o e f f i c i e n t s a"[ may be obtained by minimizing the prediction error f i l t e r output power, D (a^). Following Olrych and Bishop (1975), D(a-;) i s calculated as the mean sguared error between the desired data values x t and the predicted values: N — a t X v c - J ' 11-25 or N 11-26 where N i s the number of data points. D i f f e r e n t i a t i n g D(a-) with A respect t o a^ and setting t h i s to zero results i n the well-known set of L eguations c a l l e d the normal or Yule-Halker equations M — a, x, - L) = O ; r =1, L, H-27 K = L which may be rewritten i n matrix form as 15 p (o) p (i) p(o) P(L-\) p(L-l) p(L-i) p(L-i) • • • p(o) A A at — pU) 11-28 A 0-u PCL) The terms /0(r) are the autocorrelations of the input data x t at lag r ; pC r) 11-2 9 k = o If the Lth order AR process A 11-30 i s multiplied throughout by x t and expected values are taken, the relationship fi(o) — al~ pi \) +• . . . -H p C L) -v- o-i7" 11-31 i s obtained, since E{x ie t} = E{e|} = &e .Combining eguations 11-28 and 11-31 gives pCi) pCo) p(L) p(L~l) p(Q p(o) 11-32 16 Levinson (1947) has proposed an e f f i c i e n t algorithm f o r solving these simultaneous eguations which exploits the symmetrical Toeplitz properties of the LxL autocorrelation matrix. Claerbout (1976) i l l u s t r a t e s that, due to t h i s symmetry, the system of eguations 11-32 may be decomposed into a •forward and reversed' system which incorporates the c o e f f i c i e n t s of the L-1st order f i l t e r ; /o(o) /o(\) pi\) /oio) p(L-l) p(l-t) p(D P(L-I) PCL) p(L-\) pCO '/ \ I / N O A -a\-' A -at A A ~< o J o o o 11-33 The reversed component i n the system 11-33 i s , i n e f f e c t , the result of running the prediction f i l t e r over the data i n a reverse d i r e c t i o n . Hence the Lth order autoregressive c o e f f i c i e n t s are obtained by f i r s t solving a l l successive orders m = 1, L through the Levinson recursion re l a t i o n A A m - i 11-34 which does not d i r e c t l y involve the autocorrelation values. A However, the c o e f f i c i e n t s a"£ must be determined independently through A 11-35 a H e r e ro i s t h e p r e d i c t i o n e r r o r p o w e r c o r r e s p o n d i n g t o o r d e r A, m. T h i s t e r m i s o b t a i n e d b y c o m b i n i n g t h e c o e f f i c i e n t s a^ w i t h t h e a u t o c o r r e l a t i o n v a l u e s P[r) t h r o u g h T h e a u t o c o r r e l a t i o n s a r e g e n e r a l l y e s t i m a t e d b y s e t t i n g t h e d a t a v a l u e s o u t s i d e t h e f i n i t e d a t a w i n d o w e q u a l t o z e r o , i . e . , xt = 0 f o r t < 0 a n d t > N . C l e a r l y t h i s i s a n u n r e a l i s t i c a s s u m p t i o n r e g a r d i n g t h e u n a v a i l a b l e d a t a . F o r t h i s r e a s o n , t h e Y u l e - W a l k e r e s t i m a t e o f t h e a u t o r e g r e s s i v e c o e f f i c i e n t s i s n o t c o n s i s t e n t w i t h t h e p r i n c i p l e o f m a x i m u m e n t r o p y . B u r g ( 1 9 6 7 , 1 9 6 8 ) h a s p r o p o s e d a m e t h o d f o r d e t e r m i n i n g t h e c o e f f i c i e n t s a £ w h i c h d o e s n o t r e q u i r e p r i o r e s t i m a t i o n o f t h e a u t o c o r r e l a t i o n v a l u e s . H e s u q q e s t e d e v a l u a t i n g t h e o u t p u t p o w e r D ( a ^ ) by a p p l y i n q t h e p r e d i c t i o n e r r o r f i l t e r t o t h e d a t a i n b o t h t h e f o r w a r d a n d b a c k w a r d d i r e c t i o n s , b u t n o t o f f t h e d a t a . I n t h i s c a s e , t h e p r e d i c t i o n o u t p u t p o w e r b e c o m e s 11-36 A - a 11-37 o r N II-3 8 K=nrv 18 The backward error terra e b imposes an additional constraint on the system so that the /<?(r) are not reguired 'a p r i o r i ' . The A J T)(r/r) c o e f f i c i e n t a£ i s obtained by s e t t i n g , > ' - 0 and solving for a£, which yie l d s N4 A N T~ 11-39 K = rr\ where the terras e^ k and e^ K are the prediction error values A corresponding to order m-1. The terms af, i=1, m-1 are obtained by the Levinson recursion, Eguation 11-34. As indicated previously, the Levinson formula i s consistent with the pri n c i p l e of maximum entropy. In the Burg scheme, the zero lag autocorrelation /o(0) i s estimated i n the usual way: N The remaining autocorrelation values are obtained recursively according to r L-/ The power spectral density evaluated by Eguation 11-24 with the AS parameters obtained by the Burg scheme i s the estimate incorporating maximum entropy. In applications to extremely short data lengths, i t has been shown that the Burg spectral 19 estimator p r o v i d e s a b e t t e r r e s o l u t i o n than the c o n v e n t i o n a l AE method (Ulrych and Bishop (1975)). II.D. AS ORDER So f a r , no mention has been made r e g a r d i n g the s e l e c t i o n o f an optimum order f o r the AR process. In many cases, i n d i v i d u a l judgement must be used. Often, a p r i o r knowledge of the shape of the power spectrum i s u s e f u l . A lower bound on the order i s determined by the r e s o l u t i o n d e s i r e d ; t o o b t a i n a spectrum with H peaks, an AR order of at l e a s t 2M i s r e g u i r e d . However, using too high an order r e s u l t s i n wasted computing e f f o r t and p o s s i b l y i n i n s t a b i l i t i e s (Davies (1976)). A q u a n t i t a t i v e estimate of optimum AR order i s p o s s i b l e due to the work of Akaike (1969,1970). His F i n a l P r e d i c t i o n E r r o r (FPE) c r i t e r i o n has been d i s c u s s e d i n d e t a i l by Ulrych and Bishop (1975). The FPE i s de f i n e d as the mean squared e r r o r between the a c t u a l data term x t and the value p r e d i c t e d by a f i l t e r of l e n g t h L : 11-42 T h i s can be shown to g i v e F P E N 4 - L - I 11-43 L N - L -HI z. where o~e i s the i n n o v a t i o n v a r i a n c e and N i s the number of 20 data points. The optimum order i s the one for which the FPE i s a minimum, although the FPE c r i t e r i o n has proved to be useful in many applications, experience has shown that a more subjective estimate of as order i s often preferable. 21 III.. APPLICATIONS TG STRONG MOTION Figure 3 displays the N-S component of ground acceleration recorded at 8244 Orion Blvd., Los Angeles during the 1971 San Fernando earthquake (M = 6.4). This recording w i l l be used as input data for the analyses described i n the following chapter. The recording s i t e was under-lain by a deep s o i l deposit and was located at an epicentral distance of about 20 km. This same accelerogram was used by Saragoni and Hart (1974) to i l l u s t r a t e their simulation approach. Although strong motion accelerograms exhibit a well-defined frequency-amplitude content as a function of time, the actual time history of motion i s highly random i n nature. For t h i s reason, acceleration power spectra obtained by means of response envelopes or periodograms are extremely 'peaky* and must be smoothed by an averaging process. For example, the periodogram power evaluated from the data between t = 6 and t = 12 seconds of the Orion Blvd. accelerogram i s presented in Figure 4. I t i s obvious that t h i s type of spectrum i s suited for l i t t l e more than a q u a l i t a t i v e estimate of the power content. However, an autoregressive representation requires the input data to be random in nature. The technigue i s therefore well suited for application to strong motion data. The corresponding AR power spectral estimate for the accelerograms i s the one which i s optimally smooth. In the following chapter, acceleration time h i s t o r i e s w i l l be modelled by the autoregressive technigue. It w i l l be shown FIGURE 3. N-S component of ground a c c e l e r a t i o n recorded at 8244 Orion Blvd., Los Angeles, d u r i n g the 1971 San Fernando earthguake. 23 FIGURE 4. Periodogram power spectrum estimated from the data between t = 6 and t = 12 seconds of the Orion Blvd. recording. 24 that an e f f i c i e n t quantitative description of the non-stationary spectral content of the records can be obtained by tracking the AR parameters with time. III.A. AS ORDER FOR ACCELEROGRAMS Since an AR representation of a time series assumes weak st a t i o n a r i t y of the data, the target accelerogram was subdivided into a series of short time segments before the processor was applied. The Burg algorithm f o r c a l c u l a t i n g the AR c o e f f i c i e n t s , l i s t e d as SUBROUTINE MEMPR i n Ulrych and Bishop (1975), was used to calculate the power spectrum for a number of orders in each i n t e r v a l . Figures 5, 6, and 7 i l l u s t r a t e a few spectral estimates t y p i c a l of those found i n other segments of the accelerogram. The curves corresponding to a l l orders are of the same general form; a broad, s i n g l e peak decreasing assymptotically to zero at higher freguencies. These uni-modal shapes can be recognized as being c h a r a c t e r i s t i c of second order AS processes. I f was found that the eight point f i l t e r s resolved no spectral components which were absent from the second order power estimates, other authors, f o r example Saragoni and Hart (1974), and Vanmarcke (1976) have also found the power spectral shape for most accelerograms to be uni-modal. However, there are some exceptions to t h i s rule. For instance. Seed and I d r i s s (1969) and Seed et. a l . (1976b) have pointed out that spectral curves for accelerogams recorded over deep unconsolidated s o i l deposits may deviate from t h i s form. This i s due to the f a c t 25 F IGURE 5. AR spectral estimates for orders 2 ( s o l i d ) , H (dashed) and 8 (dotted), evaluated between t = 6 and t = 12 seconds of the Orion Blvd. accelerogram. GORE 6. AR spectral estimates for orders 2 ( s o l i d ) , 4 (dashed) and 8 (dotted), evaluated between t = 15 and t = 20 seconds of the Orion Blvd. accelerogram 27 FIGURE 7. fiR spectral estimates for orders 2 ( s o l i d ) , 4 (dashed) and 8 (dotted), evaluated between t = 8 and t = 10 seconds of the Orion Blvd. accelerogram. 28 that the s o i l layers may amplify some freguencies not close to the dominant spectral components of the motion i n the underlying bedrock. Accelerograms of t h i s nature need to be given s p e c i a l considerations. Since a uni-modal power spectrum can be resolved egually well by a l l AR orders, a second order AR process i s selected to model the acceleration. E s s e n t i a l l y , the motion i s allowed to be less deterministic i n nature, since the f i l t e r length i s shorter. A v i s u a l examination of an accelerogram confirms the high degree of randomness in the motion. The advantage i n choosing a second order model l i e s i n i t s being the most parsimonious representation of the data. This i s an important factor, since a knowledge of the AR c o e f f i c i e n t s i s reguired in order to simulate a time-history of motion. The FPE c r i t e r i o n was also used to estimate the optimum AR order. In most cases i t indicated that any order above t h i r d i s egually acceptable. However, the convenience and s i m p l i c i t y of the second order model out-weigh any possible increased resolution afforded by higher order AR processes. III.B. PHYSICAL INTERPRETATION Once the prediction error f i l t e r c o e f f i c i e n t s a, and a z have been determined, the f i l t e r transfer function A (i) = I ~ a, z - a2. 2 . 2 n i - 1 29 i s known. The polynomial A(z) =0 may then be solved f o r i t s two roots z, and z 2 , which represent the two f i l t e r poles. These are complex conjugates of each other, located i n the complex z plane just outside the unit c i r c l e , as shown in Figure 8. A lm(z) FIGURE 8. Second order f i l t e r poles z, and z 2 in the imaginary z plane.,R and 9 are the polar coordinates of the poles. An alternative way of characterizing the AR model i s through the parameters R and 8, which are the polar coordinates of the f i l t e r poles z, and z 2 . I t can be e a s i l y shown that the relationship between the AR c o e f f i c i e n t s and S and 9 i s 30 R = III-2 and & — cos -i - a, III-3 z (a^r The parameters R and 0 provide an i n t u i t i v e physical description of the power spectrum. The term 0 represents the location of the peak of the power spectral curve along the frequency a x i s . The units of 0 may be deduced as follows. The Nyquist frequency of the data i s obtained from where At i s the d i g i t i z i n g i n t e r v a l . Since f N corresponds to 0 -77, the freguency coordinate of the spectral peak i s f N0/ff Hz or, i n t h i s case, 250/1? Hz, since At = 1/50 sec. The dimensionless parameter R i s related to the sharpness or half-width of the curve. As R approaches the value one, the curve becomes a delta function at angular frequency 0. To ensure that the prediction error f i l t e r i s minimum phase, R must remain greater than 1.0. The innovation variance Oe7- , i n units of (ci/sec/sec) 2 , i s a scaling factor representing the amplitude of the spectrum, as can be seen from Equation 11-24, It was pointed out i n Chapter I I that the physical model corresponding to an Lth order AR process i s an Lth order d i f f e r e n t i a l eguation with a white noise input. For a second + N - Y^t III-4 31 order process, the analagous continuous time model i s a damped harmonic o s c i l l a t o r excited by random impulses, f ( t ) : . r ^ l U t ) r c u 0 occt) ~ Ht) n i - 5 ~ J t r ~ A harmonic o s c i l l a t o r has a unimodal freguency response with peak amplitude located at the resonance freguency The ha l f -width of the amplitude spectrum i s governed by the damping term r and the displacement amplitude i s proportional to <Xf.z , the variance of f ( t ) . E g u a t i o n III-5 may be approximated by the method of f i n i t e differences, in which case r e l a t i o n s h i p s between the AR parameters B, 0 and and the harmonic o s c i l l a t o r c o e f f i c i e n t s r 0 u>0 and <yf~ may be deduced. , This has been carried out in Appendix I, where a correspondence to second order in a Taylor expansion series i s shown. III.C. SEGMENTED ANALYSIS OF ACCELEROGRAMS In order to accomodate the extreme non-stationarities found i n strong motion accelerograms, the i n t e r v a l over which the AR c o e f f i c i e n t s are calculated should be made quite short. In that case, the averaging of the s t a t i s t i c s over the segment length becomes a plausible assumption. A lower bound for the gate length i s determined by the number of points available for evaluating the f i l t e r c o e f f i c i e n t s . Since the d i g i t i z i n g i n t e r v a l i s 0.02 seconds, the segment length could be made as short as 0.1 seconds, say, and enough data would s t i l l be 32 a v a i l a b l e f o r the c a l c u l a t i o n s . However, i f the i n t e r v a l i s made too s h o r t , low freguency s p e c t r a l components corresponding to the n a t u r a l p e r i o d s of some l a r g e r s t r u c t u r e s may be l o s t . Some h i g h - r i s e b u i l d i n g s , f o r example, may e x h i b i t modes of v i b r a t i o n with p e r i o d s as l a r g e as a few seconds (Cherry (1970)). In a d d i t i o n , extremely s h o r t - p e r i o d n o n - s t a t i o n a r i t i e s c r e a t e a • s c a t t e r * i n the time-dependent s p e c t r a l parameters and these w i l l be ignored. The i n t e r e s t i n g n o n - s t a t i o n a r i t i e s a r e the l a r g e r s c a l e t r e n d s such as the b u i l d - u p of amplitude, the d u r a t i o n of s t r o n g e s t shaking and the decaying of motion toward the end of the r e c o r d . Taking a l l of these c o n s i d e r a t i o n s i n t o account, a reasonable v a l u e f o r the segment l e n g t h i s one second (50 data p o i n t s ) . In the segmented a u t o r e g r e s s i v e a n a l y s i s of the Orion Blvd. r e c o r d i n g , the mean was f i r s t removed from t h e data i n each i n t e r v a l . F i g u r e 9a shows the time dependence of E and 8 f o r the 30 seconds of r e c o r d l e n g t h . The freguency c o o r d i n a t e 8 i s i n i t i a l l y about 0.8 x 25/fl =6.37 Hz at the beginning of the r e c o r d and decreases to about 1.10 Hz toward the end. The parameter B remains roughly constant but i s always g r e a t e r than 1,0, as r e g u i r e d . F i g u r e 9b shows the time-dependent i n n o v a t i o n variance a-g-, p l o t t e d on a l o g s c a l e to accomodate the extreme range i n values. The i n n o v a t i o n v a r i a n c e i s seen to behave i n a manner s i m i l a r t o the amplitude o f the i n p u t accelerogram; i t s t a r t s t o i n c r e a s e q u i c k l y a t about t = 3 seconds and e x h i b i t s the l a r g e s t values between 3.5 and 13 seconds. I t then decays away again toward the end of the r e c o r d . The i n t e g r a t e d power P(t) of the accelerogram, e v a l u a t e d as 13.0 IB 0 T I M E . S E C x.o 24.0 28.0 33.0 0.0 4.0 17.0 IB.O T I M E , S E C a. R (above) and 8 (below) b. Innovation variance FIGURE 9.; Time dependence of the AR spectral parameters as determined by segmenting the Orion Blvd. recording into one second i n t e r v a l s and f i t t i n g 2nd order AR processes to the data in each segment. The units of 9 are 1/50th radians/sec. 34 r e f l e c t s the 'energy* of the surface motion which has occurred prior to time t.,This parameter i s shown as a function of time for the Orion Blvd. recording i n Figure 10. The slope of t h i s curve r e f l e c t s the severity of ground motion occurring at any part i c u l a r instant. The time i n t e r v a l necessary for P(t) to assume, say, 80 per cent of i t s f i n a l value i s a measure of the duration of shaking. It can be seen from Figure 10 that the most severe shaking occurs between times t = 3.5 and t = 13 seconds of the record. This measure of duration compares well with the information obtained from the cr£ curve in Figure 9b. It i s seen that such a segmented autoregressive analysis of accelerograms i s superior to conventional methods for estimating the power spectra. I t yiel d s a much smoother spectral curve than the periodogram method and i s more informative than the response envelope technigue since i t doesn't integrate the properties of the data over the e n t i r e time s e r i e s . w m r t H fx CJ fD r t (0 OXi ri n o r t t3 (D CO HtJ < O o< * • fD n o o *o (D — f - ' r t n o CO l - h c o o ft u » o Hi rt H-B (D M> O H M CD -O L O iv) 36 IVj. TIME-ADAPTIVE APPROACH The time-varying power spectrum o f a n o n - s t a t i o n a r y time s e r i e s may be estimated through e i t h e r of two b a s i c approaches. One has alre a d y been d i s c u s s e d . The data s e t i s d i v i d e d i n t o a s e r i e s of sh o r t , * roughly s t a t i o n a r y • time i n t e r v a l s and the power spectrum i s estimated from the data i n each i n t e r v a l . A maximum entropy approach i s d e s i r a b l e i n t h i s case because i t e l i m i n a t e s the •end e f f e c t s * problems encountered i n e v a l u a t i n g the a u t o c o r r e l a t i o n f u n c t i o n f o r a s h o r t data sample. However, the i d e a l way to deal with n o n - s t a t i o n a r y data i s t o update the processor at each time step o f the time s e r i e s . In t h i s way, a continuous d e s c r i p t i o n of the time-varying p r o p e r t i e s o f the i n p u t data i s obtained. IV.A. TIME-ADAPTIVE MAXIMUH ENTROPY ANALYSIS R i l e y and Burg (1971) have proposed a technigue f o r c o n t i n u o u s l y updating the a u t o r e g r e s s i v e f i l t e r c o e f f i c i e n t s i n order to e v a l u a t e the t i m e - v a r y i n g maximum entropy power spectrum of a time s e r i e s . The method i s s i m i l a r to the one used f o r a s t a t i o n a r y time s e r i e s , except t h a t the AR parameters are evaluated from the data w i t h i n a short time window centered about the time p o s i t i o n o f the adaptive processor. The length of t h i s gate i s egual to L * 1 , where L i s the order of the AR 37 process used. The autoregressive c o e f f i c i e n t s are calculated i n the same manner as for the stationary analysis. The Levinson recursion r e l a t i o n 11-34 i s used to obtain the mth order c o e f f i c i e n t s from the m-1st order ones, which must be evaluated f i r s t . The term a£(t) i s obtained by minimizing the forward and backward errors of the prediction error f i l t e r . Since no u n r e a l i s t i c assumptions are made regarding the data outside the rectangular data window, the scheme i s consistent with the pri n c i p l e of maximum entropy. Biley and Burg (1971) suggest weighting the forward and backward error terras exponentially i n order to emphasize the s t a t i s t i c s close to the time location of the f i l t e r . However, due to the extremely short f i l t e r length used i n t h i s application, no such weighting factors were applied. When the Biley-Burg processor was i n i t i a l l y applied to the Orion Blvd. recording, the B and 9 values were found to exhibit an extremely 'scattered' time dependence. For thi s reason, a s l i d i n g one second (50 point) averaging window was used to smooth the values. The same one second gate length was used to evaluate the innovation variance from the error terms. The parameters fi, 9 and cr^ for the Orion Blvd. accelerogram evaluated i n t h i s manner are shown i n Figures 11a and 11b. When these r e s u l t s are compared to those obtained by the segmented method, a substantial deviation i s found. In spite of the assumptions regarding s t a t i o n a r i t y of the data made by the segmented method, i t probably provides a superior estimate of the spectral content. The adaptive processor lacks s t a b i l i t y i n t h i s application and cannot be expected to resolve spectral to a. R (above) and 9 (below) b. Innovation v a r i a n c e FIGURE 1 1 . Time-dependent s p e c t r a l parameters of the Orion Blvd. r e c o r d i n g as c a l c u l a t e d by the R i l e y - B u r g adaptive t e c h n i g u e . 39 components which have a period much longer than the gate length {3/50 s e c ) , Lines (1976) has discussed a modification of the Riley-Burg adaptive approach which was o r i g i n a l l y suggested by P. Gutowski. 1 This algorithm eliminates some of the i n s t a b i l i t y problems associated with the extremely short data window by evaluating the f i l t e r c o e f f i c i e n t s in a s l i d i n g time gate which i s somewhat larger than the f i l t e r length L. Since the AR parameters are calculated from the data i n the same way as i n the Riley-Burg scheme, the technigue i s consistent with the p r i n c i p l e of maximum entropy. However, for economic reasons, the c o e f f i c i e n t s are generally evaluated at every mth time step, where 1 < m < L. , The Gutowski adaptive algorithm was implemented on the Orion Blvd. recording for a number of window lengths ranging from 3 to 50 points. The re s u l t s were as expected. For a gate length of 3 points, the parameters were i d e n t i c a l to those found by the Riley-Burg algorithm. As the window length increased, the res u l t s became analogous to those found by a segmented analysis. The parameters corresponding to a data i n t e r v a l of 11 points are shown in Figures 12a and 12b. As i n the previous analysis, a one second smoothing operator was applied to R and 8. I t i s seen that the i n s t a b i l i t y associated with the Riley-Burg processor has decreased and the values are guite si m i l a r to those obtained by the segmented method. 1 P. Gutowski i s p r e s e n t l y with AMOCO O i l i n T u l s a , Okla. a. B (above) and 6 (below) b. I n n o v a t i o n v a r i a n c e FIGURE 12. AB parameters as determined by the Gutowski adaptive a l g o r i t h m u s i n g a gate l e n g t h of 11 p o i n t s . 41 The Gutowski algorithm i s e s s e n t i a l l y a compromise between the two previous approaches. Although i t tracks the time-dependent power spectrum more c l o s e l y than does the segmented method, i t has the same short-commings associated with the Riley-Burg algorithm since i t i s unable to resolve low-freguency spectral components. IV. B. THE IMS ADAPTIVE TECHNIQUE If an adaptive processor was to simply 're-tune* the f i l t e r c o e f f i c i e n t s i n response to the l o c a l properties of the data instead of completely recomputing them, some of the problems associated with the previous adaptive technigues may be eliminated. ,The previous approaches take no account of the properties of the data prior to the time location of the processor. However, i f the c o e f f i c i e n t s were to 'remember' the nature of the foregone data, a measure of s t a b i l i t y or damping would be introduced. An adaptive scheme of t h i s nature, c a l l e d the LMS algorithm, has been proposed by Widrow and Hoff (19 60) and i s discussed i n d e t a i l by Widrow, et. a l . (1967) and Widrow (1970). In t h i s approach, the the f i l t e r c o e f f i c i e n t s at time step t are obtained from the c o e f f i c i e n t s at time t-1 by the addition of a vector guantity which represents the change i n the properties of the data. The vector increment i s chosen in such a way that the mean squared error between the desired and the predicted f i l t e r output i s minimized. This i s accomplished by a gradient search 42 technigue based on the method of steepest descent. The LMS algorithm has been successfully applied to the autoregressive modelling of non-stationary data by G r i f f i t h s (1975) and to AS analysis of natural seismic events by G r i f f i t h s and Prieto-Diaz (1977) . A brief description of the LMS algorithm follows. Let the vector of autoregressive c o e f f i c i e n t s at time k be represented by A(k), where AM = Ct,(k) a t(k) oL(i<) IV-1 In the LMS algorithm, changes i n the f i l t e r c o e f f i c i e n t s at each time step are made along the dire c t i o n of the estimated gradient vector. Following Widrow, et. a l . (1967), the updated values at time k+1 are given by / U k + i) = A (K) +• * V [ E{ ej (k)J I V - 2 where VC E {e^2 <k) 3 1 1 S t h e gradient d i r e c t i o n . This term i s calculated as the derivative of the expectation of e^ >2 (k) with respect to the f i l t e r c o e f f i c i e n t s A(k), where e^ (k) i s the forward error. e f ( k ) = ^ c k ) a i U ) • x ( K-<I) IV-3 In vector notation, Eguation IV-3 becomes 43 e.p (K) - X ( K ) - A T ( K ) X ( k - i ) IV-4 where X(k - 1 ) i s the vector composed of 1 data samples; X ( K - I ) X ( K - Z ) I V - 5 X ( K - L ) and T denotes transpose. The term d\ , where <*< 0, i s a scalar constant c o n t r o l l i n g the rate of convergence and s t a b i l i t y . Through proper choice of o{, the mean squared error i s minimized and the mean of the f i l t e r parameters converges to the solution of the normal eguations by the Yule-Walker method, for a stationary time series. During t h i s application of the LMS algorithm, i t was noted that the gradient term incorporates only the forward prediction error (see Eguation IV-2). The approach of Burg i n formulating a maximum entropy solution inspired a modification to t h i s algorithm: evaluation of the gradient vector term through both the forward and backward errors. This scheme i s termed the two-sided LMS algorithm i n contrast with the o r i g i n a l formulation incorporating the one-sided error. The i t e r a t i o n eguation now becomes IV- 6 44 where the backward error term i s given by l_ e b (k) = %CK-L) — ^ a; CiO X ( K - L + L ) t = i II-7 In vector notation t h i s becomes e b C k ) = x(k-L) - A T ( K ) X CK-u + l) IV-8 As indicated by Widrow (1970), the gradient of the mean sguared error functions may be obtained by taking the gradients of single time samples of the sguared errors: V e f (k) + 2 . e b \7 IV-9 which may be obtained from Eguations IV-4 and IV-8; V e n ( k ) = - XCK-O IV-10 and V IV- 11 Thus the i t e r a t i o n formula IV -2 becomes A ( K + / ) = A ( K ) + # e ^ C k ) X ( k - i ) -r-/? e b ( K ) X ( K - L - H ) . I V " 1 2 The corresponding equation for each f i l t e r c o e f f i c i e n t i s a ^ k + i ) = a t - ( K ? +/c?e^..ck) x ( k - 0 e.b (k) x ( K - L + O / I v " 1 3 which together with Eguations IV-3 and IV-7 completes the two-sided LMS algorithm. For convergence of the one-sided processor, G r i f f i t h s (1975) has indicated that cA - )(' IV-14 . L p(o) , where 0 < 0 < 2. Here L i s the f i l t e r length and /?(0) i s the zero lag autocorrelation, or power l e v e l , h modification of this r e s u l t for the two-sided case implies that a = y IV-15 A L p i o ) , where 0 < ^ < 1. The choice of an appropriate step-size parameter within these bounds depends upon the nature of the input time s e r i e s . Applications to high freguency data which are very non-stationary require a fas t adaptation rate. However, analysis of noisy or highly random data with low freguency spectral components c a l l s for a small step-size parameter to dampen the noisy 'overshoots' of the adapting c o e f f i c i e n t s . For the applications i n t h i s study, one further modification to the o r i g i n a l LMS algorithm was made. The 46 instantaneous power l e v e l , /D{0), of an accelerogram may be highly non-stationary. For t h i s reason, /O(0) was evaluated from the 100 data points (2 sec.) surrounding the time location of the adaptive f i l t e r . Thus the processor was allowed to vary i n response to the l o c a l amplitude of the input. In order to compare the two LMS schemes, the algorithms were f i r s t applied to synthetic data sets. In most instances, there was no difference between the output of the two methods. However, when the l i m i t s of resolution for processors were approached, the two-sided scheme performed s l i g h t l y better. This i s i l l u s t r a t e d by the analysis of the synthetic signal y t composed of two sinusoids with additive noise; tj t - sin ( i T i - f . t ) -r- s i n ( z T T ^ t ) +- e t . IV-16 Here f, = 0.2, f ^ = 0.22, the variance of the noise i s 0.1 and the sampling rate i s 1 second. The step-size parameters were obtained through Eguations IV-14 and IV-15, with # = 0.2 f o r the one-sided case and # = 0.1 for the two-sided algorithm. Figure 13 i l l u s t r a t e s the power spectra estimated with a si x t h order AR f i l t e r . The spectra were evaluated well i n t o the time series in order to eliminate any effects due to i n i t i a l i z a t i o n of the autoregressive c o e f f i c i e n t s . In t h i s particular example, the AR order i s too low to resolve the two component frequencies. Both methods give i d e n t i c a l peaks at 0.21 Hz. However, when the order i s increased to eight, the two-sided algorithm gives a better resolution than does the one-sided, as i l l u s t r a t e d in Figure 14. The 10th order spectral estimates, shown i n Figure 15, have 47 FIGURE 13. Sixth order AS power spectra estimated by the one sided (dotted) and two-sided (solid) LSS processors. The input signal i s composed of two sinusoids of freguencies 0.02 and 0.22 Hz. 48 sharper peaks and narrower bases than the 8th order ones. Again, the two-sided method yields a s l i g h t l y better resolution. When the adaptive algorithms were applied to strong motion data, the two-sided approach gave R values which were somewhat closer to those obtained by the segmented maximum entropy method. For these reasons, the two-sided algorithm i s considered marginally superior, and w i l l be used in a l l subseguent analyses of strong motion data., In the application of the adaptive processor to accelerograms, i t was discovered that the preliminary output i s affected by the choice of i n i t i a l f i l t e r c o e f f i c i e n t s . , I n order to minimize th i s transient decay, the s i z e of the adaptation rate parameter for the f i r s t 100 time steps was made 10 times larger than the optimum value used over the remainder of the record.,A further complication arises due to the nature of many strong motion records, where large amplitudes are encountered at the beginning of the accelerogram. This i s due to the fact that the ground acceleration must bu i l d up to some f i n i t e amplitude to trigger the recording instruments. Because the AB parameters should be determined accurately i n t h i s region, the adaptive f i l t e r was run across the data in the reverse d i r e c t i o n , s t a r t i n g at the end of the time series. D i r e c t i o n a l i t y does not a f f e c t t h i s algorithm because i t incorporates both forward and backward prediction e rrors symmetrically. As i n the previous cases, a 50 point s l i d i n g window, centered about the time position of the f i l t e r , was used to evaluate the innovation variances from the error terms e t. This window was also used to smooth the parameters a,, and a2# since any non-49 FIGURE 14. Eighth order spectral estimates obtained by analysing the same synthetic data as for Figure 13 by the one-sided (dotted) and t»o-sided (solid) LMS algorithms. 50 0.1 0.15 0.2 0.25 0.3 0.35 0.4 FREQUENCY (HZ) FIGURE 15. Tenth order spectral estimates obtained by analysing the same synthetic data as for Figures 13 and 14 by the one-sided (dotted) and two-sided (solid) LMS algorithms. 51 s t a t i o n a r i t i e s with period shorter than one second are not interesting. The optimum step-size parameter was determined by t r i a l and error. A number of values i n the range 0.8 to 0.001 were t r i e d . Values larger than about 0.5 caused the processor to be unstable, y i e l d i n g r e s u l t s d i s s i m i l a r to those obtained by the segmented method. However, f o r values l e s s than about 0.005, the adaptation rate was too slow to respond to the important non-stationarities i n the input data. The AR parameters obtained with ^ i n the range 0.1 to 0.01 compared favourably with the r e s u l t s of the segmented method. The R, 9 and <y& values obtained with ^=0.1 are shown i n Figures 16a and 16b, and the corresponding curves for = 0.01 are presented i n Figures 17a and 17b. Due to the broad power sp e c t r a l content of earthguake accelerograms, some of the non-stationarities in the data may have the same period as the low freguency s p e c t r a l components of record. In fact, the difference between these two phenomena i s not always clear. A processor with a high adaptation rate t r e a t s both these g u a l i t i e s as non-stationarities. This i s i l l u s t r a t e d by the behavior of the parameter 9 in Figure 16a. In spite of the 50-point smoothing operator, 9 r e f l e c t s a short-period •wobble* which could be c l a s s i f i e d as a spectral component i n the 0.5 to 1.0 Hz range. Clearly, t h i s analysis overlooks valuable information regarding the low freguency spectral content of the data. However, a smaller step-size parameter smooths the 0 curve, since the adapting f i l t e r cannot track the short-period non-stationarities. This can be seen by comparing TIME.SEC a, S (above) and 6 (below) "1 I 1 1 1 1 1 1 1 0.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 T I M E . S E C b. Innovation variance FIGURE 16. Time dependence of the AR spectral parameters of the Orion Blvd.,accelerogram, as dstermined by the M two-sided LMS algorithm with step-size parameter 0.1 to a. fi (above) and 9 (below) b. Innovation variance FIGURE 17. Time dependence of the A R spectral parameters of the Orion Blvd. accelerogram, evaluated by means of the two-sided LMS algorithm with ^ = 0.01. 54 the curves i n Figures 16a and 17a. The slower adaptation rate also causes the 0 values to decrease somewhat (R has increased), in d i c a t i n g that the power spectrum now incorporates some lower freguencies. However, the AR parameters obtained with ^ = .01 s t i l l r e f l e c t the gross non-stationarities in the acceleration data. In summary, then, the two-sided LMS algorithm with a step-size parameter of 0.01 i s the optimum method for determining the time-varying second order power spectrum of an accelerogram. Once t h i s has been accomplished, the corresponding autoregressive c o e f f i c i e n t s may be used to simulate a time history of motion with s i m i l a r c h a r a c t e r i s t i c s as the target record. This i s carried out i n the following chapter. 55 V. SIMULATING GROUND MOTION-Once the time-dependent autoregressive parameters of an accelerogram have been determined, they may be used to re-construct a time-history of motion which has the same non-stationary spectral content as the o r i g i n a l record. This approach i s used i n t h i s chapter to simulate accelerograms modelled after actual recordings obtained during the 1971 San Fernando earthguake. The simulated and target time-^histories are compared by evaluating a number of diagnostic parameters commonly used to characterize ground motion. The spectral parameters R(t), 9{t) and Oe"(t) f or the Orion Blvd. accelerogram are shown i n Figure 17. These curves were d i g i t i z e d at one second in t e r v a l s and then l i n e a r l y interpolated to 0.02 second time steps. The time-dependent AR c o e f f i c i e n t s were obtained through the rel a t i o n s a , (t) - Z cos ( © C t ) ) V-1 RU) and azC-t) = V-2 A time history was then generated according to •jct-t) - at(t) ic(t-i) -h QZL±)-)L(±-Z) +• e(±j t V - 3 56 where the additive noise has a variance <^(t). The random term e(t) enables one to generate an entire suite of a r t i f i c i a l accelerograms, each exhibiting d i f f e r e n t phase r e l a t i o n s h i p s . Figures 18, 19 and 20 show three time-histories generated through the AR parameters obtained from the Orion Blvd. recording, A v i s u a l comparison with the target shows that they a l l possess similar amplitude-freguency c h a r a c t e r i s t i c s as a function of time. A more rigorous comparison i s obtained by determining the time-variant AR parameters of the simulated records, as was o r i g i n a l l y done for the target accelerogram. Figures 21a and 21b show the average ± 1 standard deviation of the R (t) , 9 (t) and 0e~it) values for s i x a r t i f i c i a l time-histories. They compare favourably with the values obtained from the Orion Blvd. recording., The integrated power P( t ) , defined by Eguation III-6, i s an indication of the destructive capacity of the ground motion. The value of P(t) integrated over the entire 30 second duration of the Orion Blvd. record i s 3.89x10s (cm/sec/sec) 2 . An ensemble of 15 simulated records shows a mean power value of 3.51x10* (cm/sec/sec) 2 and a standard deviation of 10 per cent. As was pointed out i n Chapter I, the Acceleration Response Envelope Spectrum of an accelerogram i s an important parameter r e l a t i n g ground motion to expected st r u c t u r a l response. An algorithm supplied by the U.B.C. Department of C i v i l Engineering, based on work by Nigam and Jennings (1968), was used to evaluate the ARES of the target and a number of simulated accelerograms. Figure 22 shows the ARES of the Orion FIGURE 18. A r t i f i c i a l accelerogram generated with the AR parameters shown i n Figure 17, corresponding to the Orion Blvd. recording. Ln FIGURE 19, A r t i f i c i a l accelerogram generated i n the same way as the one shown i n Figure 18. L n OO FIGURE 2 0 . A r t i f i c i a l accelerogram generated i n the same way as the ones shown in Figures 1 8 and 1 9 . a. R (above) and 8 (below) b. Innovation variance FIGURE 2 1 . Average (solid) ± 1 standard deviation (dotted) of the spectral parameters evaluated from six a r t i f i c i a l accelerograms fashioned after the Orion Blvd. recording. 61 FIGURE 2 2 . Acceleration Response Envelope Spectrum of the Orion Blvd. accelerogram. Blvd. record and Figure 23 shows the average ± 1 standard deviation of the response envelopes for six a r t i f i c i a l time-h i s t o r i e s . The damping value i s 2 per cent i n a l l cases. As was mentioned previously, the detailed structure of any p a r t i c u l a r response envelope spectrum i s dependent upon the phase c h a r a c t e r i s t i c s of the input accelerogram. If the ARES curves i n Figures 22 and 23 are compared on the basis of t h e i r o v e r - a l l forms, an acceptable match i s found. In order to further evaluate this simulation technigue, another accelerogram was used as a target time history. This recording was obtained during the 1971 San Fernando earthguake at the Seismic Laboratory at the C a l i f o r n i a I n s t i t u t e of Technology, Pasadena. The epice n t r a l distance was about 35 km and the station was located on rock. This accelerogram has a smaller amplitude and a higher freguency content than the Orion Blvd. record, as can be seen from Figure 24. The parameters R (t) , 9 (t) and cr^ (t) calculated by the two-sided LMS algorithm with ^ = 0.01 are shown i n Figures 25a and 25b. The corresponding AR c o e f f i c i e n t s were used to generate a number of a r t i f i c i a l time-histories, two of which are shown i n Figures 26 and 27. A l l of the records have the same o v e r - a l l features. The ARES for the Cal. Tech. accelerogram i s shown i n Figure 28. For comparison, the response envelopes for six simulations are summarized i n Figure 29. The correspondence i s excellent. The value of P (t) evaluated over the entire 20 seconds of the Cal. Tech. record i s 8.63x10s (cm/sec/sec) 2, compared with a mean of 7,34x105 (cm/sec/sec) 2 and a standard deviation of 15 per cent for 15 a r t i f i c i a l time-histories. As i n 63 FIGURE 23. Average ± 1 standard deviation of the ARES corresponding to six a r t i f i c i a l accelerograms. *79 a. fi (above) and 9 (below) b. Innovation variance FIGURE 25. AR spectral parameters as determined by an adapti analysis of the Cal. Tech. accelerogram with # 0.01. o CD . FIGURE 26. A r t i f i c i a l accelerogram generated with the AR parameters shown i n Figure 25, corresponding to the Cal. Tech. recording. O N FIGURE 27. A r t i f i c i a l accelerogram generated i n the same way as the one shown i n Figure 26. 68 F I G U R E 2 8 . ARES for the Cal. Tech. accelerogram. 69 FIGURE 29., Average ± 1 standard deviation for the ARES of six a r t i f i c i a l accelerograms generated by the AR parameters obtained from the Cal. Tech. recording. 70 the previous example, the t o t a l integrated power i s les s for the simulated ensemble than for the target record. I t i s not clear at t h i s point just why t h i s i s so. It i s seen from the analyses presented above that the AR spectrum-matching technique can successfully generate a r t i f i c i a l time h i s t o r i e s which have the same important c h a r a c t e r i s t i c s as the actual acceleroqrams. 71 ILs. I I H I I C A I . AR ANALYSIS OF STRONG MOTION AG CEL EROGSA MS -VI.A. THE APPROACH One method for prescribing s i t e - s p e c i f i c strong ground motion i s by solving the forward problem through a deterministic approach. The f a u l t i n g process and the radiating energy f i e l d may be mathematically modelled to generate complete time-h i s t o r i e s of seismic wave a r r i v a l s at a distant design s i t e . This i s a promising way to approach the problem, since i t incorporates the tectonic and geologic c h a r a c t e r i s t i c s of the design s i t e directly.„However, i t i s also quite d i f f i c u l t . Solutions to many of the inherent problems are barely state-of-the-art at the present time. In addition, wide uncertainty margins are associated with quantitative descriptions of geology. Due to the extremely complex nature of earthquakes and seismic waves propogating within the earth, numerical methods must be used. Hence computation e f f o r t i s large and simplifying assumptions must invariably be made. The resultant ground motions may be correspondingly s i m p l i s t i c and may reguire substantial modification before they are suited f o r earthguake design. An alternative way of obtaining strong ground motion records for design purposes i s by selecting and inputing appropriate pre-recorded earthquake records. In order to ensure that the h i s t o r i c a l recordings r e a l i s t i c a l l y portray the 72 potential ground motion at the design s i t e , the earthquake magnitudes, nature of f a u l t i n g , epicentral distances and geologic enviroments must be comparable for a l l cases. The success of t h i s type of approach c l e a r l y depends upon the selection of available earthquake records. Although the strong motion catalogue has greatly expanded i n recent years, the range of magnitudes, epicentral distances, geologic environments, etc. which the recordings cover i s far from complete. In order to compensate for t h i s lack of data, new accelerograms may be formed by simply modifying e x i s t i n g ones. For instance, the amplitudes may be scaled by a factor representing the empirical r e l a t i o n s h i p between peak accelerations and earthguake magnitudes, epicentral distances and l o c a l geology. However, t h i s i s only a f i r s t approximation to a solution. To ensure that the complete time-dependent freguency and amplitude content of the input record incorporates conditions s p e c i f i c to the design s i t e , more elaborate modifications are needed. Empirically scaled Fourier amplitude and response envelope spectral curves are a step i n t h i s d i r e c t i o n . McGuire (1974), Trifunac (1976) and Seed, et. a l . (1976b) have derived attenuation laws for these spectral estimators. However, t h e i r spectral evaluation methods neglect such time-domain information as non-stationary freguency content and duration of shaking..Trifunac and Hestermo (1976) have recently completed a separate study of the empirical behaviour of shaking duration, but their r e s u l t s are d i f f i c u l t to combine with the spectral information. In any case, these approaches do not readily lend themselves to simple modification of ex i s t i n g 73 time h i s t o r i e s . In order for any probabilistic method of generating accelerograms to be successful, the parameters which are empirically modelled must be chosen with care. They must be simple to evaluate, yet easy to apply. Also, they should contain a l l the important time-domain information about the records and at the same time they should have some physical meaning. The AR spectral parameters R (t) , 6{t) and it) s a t i s f y a l l of these conditions. As was pointed out in previous chapters, these c o e f f i c i e n t s quantify the time-dependent spectral content of an accelerogram and at the same time may be used to re-create complete time-histories of motion. The empirical behaviour of these parameters may be rea d i l y deduced through the analysis of a large number of earthguake accelerograms by the time-adaptive AR modelling technigue. The r e s u l t s of an analysis of t h i s nature are presented i n the following two sections. VI.B. THE ANALYSIS In order to determine empirical r e l a t i o n s h i p s for the spectral parameters R (t) , 9{t) and <5£*{t) , a number of strong motion accelerograms from the Cal. Tech. Report series have been analysed by the two-sided LMS adaptive technigue. Since most of the recordings were obtained during shallow earthquakes located in southwestern U.S.A., the r e s u l t s are not d i r e c t l y applicable to areas whose tectonic settings are d i s s i m i l a r to those found i n t h i s region. The accelerograms have been c l a s s i f i e d according 7a to the following three parameters: earthguake magnitude {&), epicentral distance (D) and the nature of the l o c a l geology. A rigorous analysis would also include such important factors as source mechanisms and energy propogation paths. However, due to the paucity of the data at t h i s stage, t h i s preliminary analysis considers only the variables M, D, and l o c a l geology, A si m p l i f i e d approach such as t h i s w i l l , of course, y i e l d generalized r e s u l t s , with a correspondingly large scatter i n the time-dependent spectral parameters. One of the more s i g n i f i c a n t factors which a f f e c t s the over-a l l nature of the ground surface motion i s the type and extent of the unconsolidated material beneath the recording s i t e . Due to the extreme range in different s o i l types, depths, water contents, degrees of consolidation and grain sizes which are commonly encountered, no attempt was made to c l a s s i f y l o c a l geology into s i m p l i s t i c , discrete d i v i s i o n s for t h i s study. Moreover, the information regarding the nature of the under-ly i n g s o i l for the records was q u a l i t a t i v e and extremely sketchy. For these reasons, only those acceleroqrams recorded on rock or over very s t i f f s o i l were analysed. The r e s u l t s are thus v a l i d f o r firm ground or bedrock motions only. Surface accelerations for i n d i v i d u a l s i t e s situated over loose s o i l deposits may be obtained by an appropriate modification of the bedrock time h i s t o r i e s . F a c c i o l i and Hesendiz (1976) review various techniques for incorporating e f f e c t s of l o c a l geology, for example, by propogating S-H waves non-linearly through horizontal s o i l layers., The * rock-site* time-histories for t h i s study were selected 75 on the basis of s i t e descriptions provided by Trifunac and Brady (1975). A l l of the accelerograms were recorded during one of the six earthquakes l i s t e d i n Table I . Hith the exception of the Table I., Earthquakes used in empirical AR analysis. EARTHQUAKE « I i YEAR -j~ I i DATE T 1 T~ MAGNITUDE | NO. i I i Helena Montana i 1935 i Oct. 31 J 6. 0 | 1 a San Francisco C a l i f o r n i a i 1957 Mar. 22 | 5.3 | 2 Park f i e l d C a l i f o r n i a r 1966 i Jun. 27 | 5.6 | 3 Borrego Mtn. C a l i f o r n i a | 1968 ! Apr. 8 i 6.4 | 4 Lytle Creek C a l i f o r n i a ! 1970 ! Sep. , 12 | 5.4 | 5 San Fernando C a l i f o r n i a 1971 Feb. 9 t _ j. 6. 4 | - . . i . 6 _j 1935 Helena, Montana earthquake, they were a l l located i n C a l i f o r n i a . The earthquakes range i n s i z e from H = 5 , 3 to 8 = 6.4. The records are unevenly distributed i n t h i s range; a majority are from the 1971 San Fernando earthguake (M = 6.4). The range of epicentral distances spanned by the accelerograms i s also uneven; most of the San Fernando earthguake accelerograms f a l l i n the range 25 to 45 km. However, there i s a good coverage of source-receiver azimuths, so a systematic bias due to d i r e c t i o n of f a u l t s l i p i s not expected. A l i s t i n g of the accelerograms which were analysed, along with the corresponding 76 earthquakes and epicentral distances i s provided in Table I I . Since i t i s the horizontal ground motion which i s of most concern to earthguake engineers, only the two horizontal components of acceleration were dealt with i n t h i s study. In order to discriminate any systematic trends inherent i n the dif f e r e n t directions of surface motion, the horizontal accelerations were resolved i n t o r a d i a l and tangential components before the analysis was performed. This was done as follows; once the the azimuth from the station back to the epicenter and the orientation of the two recorded components are known, the r a d i a l and tangential components may be deduced by simple vector addition, as i l l u s t r a t e d in Figure 30. i center N , E : m e a s u r e d c o m p o n e n t s T N 4 R,T: d e s i r e d components R T — Es in f l - t -Ncosf? FIGURE 30. Resolving the two horizontal components of acceleration into r a d i a l and tangential directions 77 Table I I . accelerograms used i n empirical AB analysis. Both r a d i a l and tangential components were u t i l i z e d i n a l l cases. 1 | EABTH-| QOAKE | NO. "T-BECORD NO. -T- — - — - r-| STATION LOCATION \ EPICENTRAL DISTANCE (KM) • S-WAVE | ABBIVAL | (SEC) | 1 2 j A015 I Golden Gate Park | \ San Francisco | 12 ! - ] | 1 ! B025 J C a r r o l l College 1 | Helena, Montana | 8 ! 1. 5 | I 3 I B037 J Sta.2, Temblor | I C a l i f o r n i a | 17 ! 2.0 | I 3 1 B038 ] Bee. Bldg., San j | Luis Obispo, Cal.. | 60 i - I | 4 ! BO 40 | S.C. E. Power Plant | i San Onefre, Cal. | 135 ! 15. 4 | I 6 i C041 I Pacoima Dam | | C a l i f o r n i a | 10 ! 1.9 I I 6 j C054 I 445 Figueroa St. | I Los Angeles ) 44 I 1. 8 | I 6 ! D056 I Castaic Old Bidge | | Route, C a l . | 30 I 0.8 | I 6 j . E078 | Water & Power Bldg. J | Los Angeles | 44 I •1. 2 I I 6 ; E081 | Santa F e l i c i a Dam | | C a l i f o r n i a | 33 J - | i 6 i F092 \ 2011 Zonal Ave., | I Los Angeles ] 45 ! - | I 6 j F102 | Fort Tejon j I C a l i f o r n i a I 72 ! - | i 6 ! G106 | Caltech Seismic Lab. J J Pasadena, C a l . | 36 ! 2. 4 I I 6 ! J141 | Lake Hughes Sta . 1 | J C a l i f o r n i a J 30 ! 2. 1 1 ! 6 ! J142 | Lake Hughes Sta.4 | J C a l i f o r n i a | 29 i 2.0 | I 6 i J143 J Lake Hughes Sta.9 | | C a l i f o r n i a \ i _ , . J. 28 i i 78 Table II. Continued 1 \ 1 1 EARTH-QUiKE NO. 1 RECOBD NO. •— — - T~ | STATICS LOCATION | EPICENTBAL DISTANCE (KM) T ^ I S-WAVE | | ARRIVAL J I (SEC) J r" 1 1 l 6 1 J144 1 Lake Hughes Sta.12 J I C a l i f o r n i a | 25 I 0.8 | 1 j 1 4 6 ! H179 1 Tehachapi Plant | J Grapevine, Cal. | 83 I " ! 1 1 i 1 6 1 M183 J 6074 Park Drive | I Irightwood, Cal. | 70 I 6.6 | 1 J I 1 6 j 0198 I G r i f f i t h Park Obs. | j Los Angeles | 33 | 3.7 | 1 1 1 | 6 0207 | F a i r IB ont Reservoir J I C a l i f o r n i a | 35 i " I T 1 1 1 6 ! P221 | Santa Anita Res. | | Arcadia, Cal. | 45 i " ! 1 i 1 1 6 i P223 I Puddingstone Res. | i San Dimas, Cal. | 63 ! " ! J— 1 i 1 6 ! Q241 | 800 H. F i r s t St. | 1 Los Angeles j 44 J 6. 2 | I \ i 1 5 ! W334 J 6074 Park Drive | I Srightaood, C a l . | 13 !• " i 1 I 1 1 5 j S335 | Allen Ranch, Cedar | | Springs, Cal. | 19 ! " ! T 1 I 1 . 5 W336 1 Bight Abut., C.f.R. J J S i t e , Cedar Springs | _JL 23 -t J The three spectral parameters r e s u l t i n g from t h i s time-domain analysis are obtained as functions of time. In order to compare r e s u l t s from d i f f e r e n t accelerograms, the AS parameters must i n some way be 'synchronized* along the time axis. Aligning the o r i g i n times t = 0 of d i f f e r e n t records i s unsatisfactory because i t i s unlikely that a l l the s e l f - t r i g g e r i n g recording instruments started at exactly the same 'instant* r e l a t i v e to 79 the entire envelope of motion. The P-wave a r r i v a l s are of no use since they occur p r i o r to time t = 0. However, the S-wave a r r i v a l i s usually present and has an important r e l a t i o n to the record as a whole. Moreover, i t can be accurately i d e n t i f i e d from the plots of ground velocity vs. time, A double check on the time picks i s possible by estimating t h i s event for each component independently. For these reasons, the S a r r i v a l time was selected as an 'absolute* time reference. For accelerograms st a r t i n g after the S-waves have arr i v e d , i t was assumed that the recording instruments triggered no more than 2 seconds l a t e r . Hence the start t = 0 of these records was shifted by less than 2 seconds r e l a t i v e to the S a r r i v a l of the other accelerograms. These recordings were synchronized with the others by matching times for which &&~{t) i s a maximum. The AR spectral parameters of the accelerograms l i s t e d in Table II have been s t a t i s t i c a l l y summarized i n the following section. The results f o r only a few i n d i v i d u a l records are presented here. Figure 31 shows the r a d i a l component of acceleration recorded at the Pacoima Dam (record C041) during the 1971 San Fernando earthguake. The S a r r i v a l time has been estimated at t = 1.9 sees, from the v e l o c i t y time-histories (not shown). Figure 32 shows the corresponding parameters R ( t ) , 9 (t) and o-^-(t) . The values corresponding to the P-wave data are shown as dotted l i n e s . The maximum value of the innovation variance for t h i s record i s about 5x10*.,,It decreases by three orders of magnitude between times t = 8 and t = 15 sees. Figure 33 displays the r a d i a l component of motion recorded at Temblor, C a l i f o r n i a (record B037) during the 1966 Parkfield earthguake. FIGURE 31. Radial component of ground acceleration recorded at the Pacoima Dam (record C041) during the 1971 San Fernando earthguake. 00 o a. B (above) and 0 (below) b. Innovation variance 00 FIGURE 32. AR spectral parameters corresponding to recording C041, shown in Figure 31. 82 The S a r r i v a l i s estimated at t = 2.0 sees. Figure 34 shows the corresponding spectral parameters. I t i s seen that the area under the innovation variance curves i s substantially d i f f e r e n t for the two accelerograms. Note the difference i n scale for oV(t) i n the two cases. . F i n a l l y , Figure 35 depicts the r a d i a l acceleration for record J142, obtained at s i t e no. 4, Lake Hughes array, C a l i f o r n i a , during the 1971 San Fernando earthguake. The corresponding AR spectral parameters are presented i n Figure 36. By comparing these three examples, i t can be seen that a substantial deviation may be found between the AR spectral parameters corresponding to d i f f e r e n t earthguake records. This indicates that the method i s well suited for d i f f e r e n t i a t i n g between accelerograms obtained under di f f e r e n t recording conditions., VI.C. THE RESULTS The * rock-site* accelerograms l i s t e d i n Table II have been grouped into a number of discrete magnitude (M) and epicentral distance (D) i n t e r v a l s , which are l i s t e d i n Table I I I . The average time-dependent spectral parameters have been evaluated for the recordings i n each group and are presented in Figures 37 - 47. Both the r a d i a l and tangential components for each station have been used to obtain these re s u l t s , since a preliminary examination of the AR parameters revealed no trends inherent i n the d i f f e r e n t d i r e c t i o n s of motion. Thus, for example, both the r a d i a l and tangential components of recording CO41 were used to 83 FIGUEE 33. Badial component of ground acceleration recorded at Temblor, C a l i f o r n i a {record B037) iu r i n g the 1966 Parkfield earthguake. 84 to •-1 •* -I a. S (above) and 9 (below) b. Innovation variance FIGURE 34. AR spectral parameters corresponding to recording B037, shown in Figure 33. -240.D H a w w Ul Ul -* w W <£> H- ») rt-~* CD H-W 3 H Hi O B • O O •XJ s ID h o P tr" E3 fu f» CD a A - S3 Pu CD rt-o D3 O (D C Mi n crxj r»* co H cr w O C C ~ P B> rj Pi ?!• CD CD o P> • o o n o & CD CD h P> rt-i - " O pi S3 C n n CD » n VQ o M «+ Pi tr CD CD Pu p) rt-o CD O m mo-<~>o a a c n i o -c s -fe-ci C O ' o -1B0.0 J RCCELERRTIQN.CM/SEC/SEC -80.D 0.0 80.0 160.0 240.0 i j _ _ I I I ro o o £8 0.0 — I — 4.0 —I 1 e.o 12.0 TIME,SEC - i — 16.0 -1 20.0 0.0 — I — 4.0 —I 1 e.o i2.o TIME,SEC — i — 16.0 20.0 a. B (above) and 0 (below) b. Innovation variance FIGUfiE 36. AB spectral parameters corresponding to recording J142, shown in Figure 35. 87 Table I I I . Summary of graphs containing r e s u l t s of empirical AH analysis. r — r -| FIGURE | 1 NO. | MAGNITUDE i EPICENTRAL DISTANCE (KM) r ••' — I ACCELEROGRAMS | USED i 37 | 6.4 1 9 | C041 1 38 | 6 . 4 | 25 - 35 J D056 | E081 | J141 \ J142 | J143 | J144 I 0198 1 39 J 6.4 j 28 - 38 | D056 | E081 | G106 | J141 | 0198 I 0207 1 40 | 6.4 i 34 - 50 J C054 | E078 { F092 | G106 I 0207 | P221 I Q241 1 41 | 6.4 1 60 - 76 | F102 j M183 I P223 1 42 | 6.4 | 64 - 86 | F102 | M179 f M 1 83 1 ^3 I 6.4 1 135 | B040 1 44 | 6 . 0 8 | B025 1 45 | 5.5 11 - 21 j | B037 | S3 34 | W335 1 46 | 5.5 i 15 - 25 | B037 | H335 J 8336 i 1 47 | * i 5.3 T" 1 _JL-12 | A015 i -1 I _ j 88 FIGURE 37. Average (solid) ± 1 standard deviation (dashed) of the AR sp e c t r a l parameters f o r H = 6.4 and D = 9 km. 89 FIGURE 38. Average ± 1 standard d e v i a t i o n parameters f o r M = 6.4 and D = of the AR s p e c t r a l 25 - 35 km. 90 FIGUBE 39. Average ± 1 standard deviation of parameters for M = 6.4 and D = 28 the AB spectral - 38 km. F I G U R E 40. Average ± 1 standard deviation parameters for M = 6.4 and D = of the AH spectral 34 - 50 km. 92 FIGURE 41. Average ± 1 standard deviation of the AR spectral parameters for K = 6.4 and D = 60 - 76 km. 93 0 . 0 4 . 0 8 . 0 1 2 . 0 J 6 . 0 0 . 0 4 . 0 6 . 0 3 2 . 0 16 .0 FIGURE 42. Average ± 1 standard deviation of the AR spectral parameters for M = 6.4 and D = 64 - 86 km. 94 FIGUBE 43. Average ± 1 standard deviation of the AB spectral parameters for M = 6.4 and D = 135 km. F IGDHE 44. Average ± 1 standard deviation of the AR spectral parameters for M = 6.0 and D = 8 km. 96 Average ± 1 standard deviation of the AB spectral parameters f o r n - 5.5 and D = 11 - 21 km. 97 FIGURE 46. Average ± 1 standard deviation of the AR spectral parameters f o r M = 5.5 and D = 15 - 25 km. , 98 FIGURE 4 7 . A v e r a g e ± 1 s t a n d a r d d e v i a t i o n o f t h e AR s p e c t r a l p a r a m e t e r s f o r M = 5 ,3 a n d D = 12 km. 99 o b t a i n t h e s p e c t r a l c u r v e s c o r r e s p o n d i n g t o M = 6.4 and D = 9 km., shown i n F i g u r e 37. The c a l c u l a t i o n s were c a r r i e d o u t a s f o l l o w s : f i r s t , t h e a c c e l e r o g r a m s i n e a c h g r o u p were a l i g n e d i n th e t i m e domain w i t h r e s p e c t t o one a n o t h e r u s i n g t h e S-wave a r r i v a l s . Then t h e mean and s t a n d a r d d e v i a t i o n o f e a c h p a r a m e t e r R ( t ) , 8{t) and os^(t) were e v a l u a t e d a t e v e r y o n e - s e c o n d t i m e s t e p . The o r i g i n t i m e s , t = 0, f o r t h e s p e c t r a l c u r v e s i n F i g u r e s 37 - 47 have been a r b i t r a r i l y p o s i t i o n e d so t h a t t h e S-wave a r r i v a l t i m e s f a l l a t t = 2 s e c o n d s . F i g u r e 47 c o n t a i n s no i n f o r m a t i o n i n t h e r a n g e 0 < t < 2 s e c o n d s b e c a u s e r e c o r d A015 b e g i n s a f t e r t h e S-wave has a r r i v e d . I n o r d e r t o make t h e most e f f i c i e n t use o f t h i s l i m i t e d d a t a s e t , t h e D i n t e r v a l s were made t o o v e r l a p whenever p o s s i b l e . R e c o r d i n g O207 (D = 35 km.), f o r i n s t a n c e , i s us e d t o e v a l u a t e t h e AS p a r a m e t e r s f o r two e p i c e n t r a l d i s t a n c e i n t e r v a l s ; D = 28 - 38 km. and D = 34 - 40 km. The D v a l u e s c o r r e s p o n d i n g t o e a c h g r o u p of a c c e l e r o g r a m s i n T a b l e 3 a r e a c t u a l l y ± 2 s t a n d a r d d e v i a t i o n s from t h e mean c f t h e e p i c e n t r a l d i s t a n c e s f o r t h e r e c o r d i n g s i n e a c h i n t e r v a l . The s p e c t r a l p a r a m e t e r s i n F i g u r e 38 e x h i b i t a l a r g e r v a r i a t i o n a t any t i m e s t e p t h a n t h e o n e s i n F i g u r e 37. However, t h e f o r m e r r e s u l t s (H = 6.4* D = 26 - 35 km.) a r e l i k e l y t o be more r e l i a b l e . T h i s i s b e c a u s e t h e s p e c t r a l c u r v e s i n F i g u r e 38 have been c o m p i l e d f r o m a t o t a l o f 14 a c c e l e r o g r a m s r e c o r d e d a t 7 d i f f e r e n t s t a t i o n s , whereas t h e r e s u l t s i n F i g u r e 37 were o b t a i n e d f r o m 2 a c c e l e r o g r a m s , b o t h r e c o r d e d a t t h e same l o c a t i o n . S i n c e one s t a t i o n c a n h a r d l y be c o n s i d e r e d r e p r e s e n t a t i v e o f a m a g n i t u d e - e p i c e n t r a l d i s t a n c e r a n g e , t h e r e s u l t s f o r c a s e s s u c h a s t h i s must be i n t e r p r e t e d w i t h c a r e . 100 Any systematic variation i n R{t) with M and D seems to be absent from t h i s data set. I t i s seen from Figures 37 - 47 that R<t) tends to assume a variety of values and changes s i g n i f i c a n t l y with time. For example, the behaviour of R{t) i n Figure 44 i s s i g n i f i c a n t l y d i f f e r e n t from the values of R(t) for the other fl, D combinations. However, i n Figures 38 - 43, R(t) i s approximately 1.2 and remains guite stationary i n time. As a tentative conclusion regarding the empirical behaviour of R(t),• i t may be said that R(t) becomes increasingly stationary with increasing epicentral distance. The freguency coordinate 9 (t) also seems to exhibit l i t t l e or no systematic variation f o r t h i s data set. Any s i n g l e 13, D combination may exhibit wide variations i n 9(t). :, For example, 9 <t) i n Figure 45 has a standard deviation of about 50 per cent of the mean value. Prior to t = 3 seconds of t h i s example, the range in 9 (t) drops, since only 2 of the 6 accelerograms i n t h i s H, D i n t e r v a l contributed to the AR parameter values prior to t = 2. The most r e l i a b l e estimates for 9(t) are found i n Figures 38 - 41, since each of these graphs i s composed of a larger number of recordings.,The freguency content i n these cases i s seen to somewhat decrease with time. This behaviour f o r 9<t) may be related to the attenuation and dispersion of seismic waves within the earth. Although both R {t) and 9 (t) seem to have l i t t l e or no systematic dependence on M and D in these r e s u l t s , i t i s important to note that they do tend to assume a range of values within certain well-defined bounds. Any ensemble of a r t i f i c i a l accelerograms should r e f l e c t these variations in order to r e a l i s t i c a l l y portray the variety of ground motions 101 which may possibly occur. The parameter most affected by changes i n M and D i s the innovation variance, < 3 e v(t). In general, <3el(t) remains r e l a t i v e l y small for times t = 0 to t = 1 seconds, which correspond to the P-wave content of the accelerograms. At t = 2 seconds, which marks the S-wave a r r i v a l s , o a l ( t ) has substantially increased. The largest values for the innovation variance are found i n the few seconds immediately following the S-wave a r r i v a l s , representing the large amplitude surface waves. With a few exceptions, the build-up and decay portions of the <3e"(t) envelopes tend to be guite symmetric about t h e i r peak positions. In order to guantify the empirical behaviour of the innovation variance, the maximum values of oe'it) from Figures 37 - 47 plus record B038 have been plotted as functions of M and D i n Figure 48. The v e r t i c a l bars indicate one standard deviation of the peak values. The data have been plotted at D values corresponding to the mid-points of the epicentral distance i n t e r v a l s f o r each group of accelerograms. Straight l i n e s have been f i t t e d to the data sets corresponding to B = 6,| and M = 5,5, as shown in Figure 48, Eguations of the form may be deduced for these l i n e s , i n which case the relationships a = -3.23, b = 6.19x107, for M = 6.4, and a = -3.34, b = 7.01x10*, for M = 5.5 are obtained. Since only one datum value i s available for each VI-1 102 5 10 5 0 100 5 0 0 J 1 I l 1 1 I I I I l l I I I 1 1 1 L. un 1 — I — I I I | 1 1 1 1 — I I I I | 1 1 1 — r c 5 10 5 0 100 5 0 0 E P I C E N T R A L D I S T A N C E , K M F I G U R E 4 8 . M a x i m u m v a l u e s o f Oe~It) a s a f u n c t i o n o f e p i c e n t r a l d i s t a n c e f o r v a r i o u s M v a l u e s . T h e v e r t i c a l b a r s d e n o t e o n e s t a n d a r d d e v i a t i o n a n d t h e s t r a i g h t l i n e s r e p r e s e n t t h e b e s t f i t t o t h e p e a k v a l u e s i n e a c h m a g n i t u d e r a n g e . 103 of H = 6.0 and H = 5.3, no l i n e s sere f i t t e d to these points. The datum point corresponding to M = 6.0» which i s the Helena, Montana recording, has a maximum Oel(t) value somewhat smaller than the peak value for M = 6.0 as predicted by the other data. This may be related t c an anomalous tectonic s e t t i n g and/or an incorrect magnitude determination for t h i s particular earthguake. The attenuation laws f o r M - 6.4 and M = 5. 5 may be combined to y i e l d an average relationship mot*. ^ between peak innovation variance and D. Since the units of a£*(t) are (cm/sec 2) 2, taking the sguare roots of both sides of Eguation VI-2 e f f e c t i v e l y gives an attenuation r e l a t i o n s h i p for peak acceleration with D; This r e s u l t may be rea d i l y compared to strong-motion amplitude decay laws determined by other authors. For example, Richter (1958) has found that the c o r r e l a t i o n between n e a r - f i e l d wave amplitude and epicentral distance for C a l i f o r n i a earthguakes i s A„ oi D~' FL . Seed, et, a l . (1976a) have derived the relationship 104 max. occe I D - I . 5-q VI-5 for maximum acceleration as a function of D for * rock-site* recordings i n western U.S.A. F i n a l l y , Davenport (1972) has examined the c o r r e l a t i o n between maximum acceleration and epicentral distance for of 46 earthquakes i n western U.S.A. and found that A number of additional studies have given r e s u l t s s i m i l a r to the above. Thus the c o r r e l a t i o n between peak innovation variance and epicentral distance as determined in t h i s study i s consistent with amplitude attenuation laws obtained in previous work. One of the more important parameters of a strong motion time-history i s the duration of shaking. Structural members weaken with time i f subjected to r e p e t i t i v e loading, and the number of cycles through which a structure i s stressed c l e a r l y depends upon the duration of ground motion. This parameter, i n turn, depends on earthguake magnitude, epicentral distance and possibly the l o c a l geology at the recording s i t e . An actual d e f i n i t i o n for shaking duration i s guite ar b i t r a r y . Once the acceleration amplitude has decreased by some factor r e l a t i v e to the peak value, the potential f o r further s t r u c t u r a l damage has also diminished. Trifunac and Westermo (1976) have defined duration of acceleration i n terms of the rate of growth of the power i n t e g r a l P(t) , where m a x . a c c e l . cA D VI-6 105 t P U ) = f X V T W - C . VI-7 Once P{t) i s known f o r an accelerogram, i t s time derivative may be evaluated. This guantity, which i s the slope or rate of growth of P ( t ) , i s a measure of the •energy* of ground motion ocurring at any time. The duration of shaking i s taken as the sum cf the time i n t e r v a l s during which th i s derivative i s greater than some a r b i t r a r y percentage of i t s maximum value., A more elegant estimate of shaking duration may be obtained from the innovation variance envelope of an accelerogram. The period of time over which oe\t) remains larger than some arbitrary value i s an excellent measure of duration and can he e a s i l y measured from a plot of Oe\ty vs. time. For convenience, this arbitrary l e v e l has been chosen as one {logarithmic) decade less than the maximum value of log ( ot?(t)) , since t h i s represents a decrease to 10 per cent of the peak value. The shaking duration has been measured i n t h i s manner from the curves in Figures 37 - 47 and i s plotted as a function as M and D in Figure 49. One standard deviation for each of these datum points i s approximately 20 per cent of i t s duration value. Due to the paucity of these data, no attempt was made to quantitatively describe the c o r r e l a t i o n between shaking duration and H or D. , However, i t can be seen from Figure 49 that the duration tends to increase with both M and D. Becord B025 (M = 6.0) i s again an exception. At short epicentral distances, the duration seems to l e v e l out with further changes i n D. This may be attributed to the f a c t that the shaking duration for small values of D i s determined by the f a u l t rupture time, whereas the 106 5 10 i i I i 1 CN O CN" CJ LU CO CN I CC cn ID O o o H 5 0 J I L _ 100 I I I  M = 6.4 O M = 6.0 • M= 5.5 • M = 5.3 • O I I I I I "I 1 1 1 1 I I I I 5 10 50 100 E P I C E N T R A L D I S T A N C E , K M 5 0 0 j i u CN O CN .CN r-00 T 1 r 5 0 0 FIGURE 4 9 . Duration of shaking as estimated from the envelopes of <?ez{t), plotted as a function of epicentral distance and earthquake magnitude. 107 d u r a t i o n f o r l a r g e r e p i c e n t r a l d i s t a n c e s i s governed by the d i f f e r e n c e s i n v e l o c i t i e s and t r a v e l paths f o r d i f f e r e n t s e i s m i c waves. The ru p t u r e time of the 1971 San Fernando earthquake, f o r example, can be estimated to be about 6 - 8 seconds (Mikumo (1973)). Smaller earthquakes tend t o have c o r r e s p o n d i n g l y s h o r t e r r u p t u r e times. These estimates are c o n s i s t e n t with the trends of the d u r a t i o n data i n F i g u r e ag. I t i s thus seen t h a t the s i n g l e parameter <xe"(t) i s a very powerful t o o l f o r c h a r a c t e r i z i n g the amplitude of s t r o n g motion r e c o r d i n g s . By performing AR a n a l y s e s of strong-motion accelerograms recorded under a v a r i e t y of H and D values, e m p i r i c a l r e l a t i o n s h i p s f o r t h e s p e c t r a l parameters B (t) , 9(t) and o& i±) have been determined. ,.,As a r e s u l t , g u a n t i t a t i v e d e s c r i p t i o n s of p o s s i b l e v a r i a t i o n s i n freguency content, a c c e l e r a t i o n amplitude and shaking d u r a t i o n f o r a l i m i t e d range i n H and D have been obtained. From t h i s i n f o r m a t i o n , r e a l i s t i c a r t i f i c i a l accelerograms f o r a r b i t r a r y combinations of M and D w i t h i n t h i s range may now be c o n s t r u c t e d . As more earthguake r e c o r d i n g s become a v a i l a b l e , these e m p i r i c a l curves may be extended, making i t p o s s i b l e to c o n f i d e n t l y generate a r t i f i c i a l accelerograms f o r a wider v a r i e t y of r e c o r d i n g c o n d i t i o n s . In a d d i t i o n , our understanding of how earthguake energy propogates w i t h i n the e a r t h w i l l then be g r e a t l y enhanced. 108 IIIJL. C O H C J J U S I O N S To. summarize the res u l t s of this study, the following important points are emphasized: 1) The optimum way of determining the spectral content of a strong motion accelerogram i s by f i t t i n g a second order autoregressive process to the data. Only three c o e f f i c i e n t s are then needed to quantitatively describe the power spectrum. This method yields a spectral estimate which i s optimally smooth. However, since a second order AR power spectrum i s necessarily uni-modal in form, t h i s analysis should be extended only to those accelerograms recorded ort rock. This i s because s u r f i c i a l s o i l layers tend to modify the spectrum i n such a way that i t i s no longer uni-modal. 2) In order to account for the non-stationarities i n the accelerograms, the power spectrum should be determined i n a time-varying manner. This i s best achieved by f i t t i n g the second order AB model to the data with a two-sided L M S time-adaptive algorithm, using a step-size parameter of 0.01, 3) Once the optimum AR model for an accelerogram i s known, i t may be used to generate an ensemble of a r t i f i c i a l accelerograms, each d i f f e r i n g i n phase content but exhibiting the same time-dependent power spectrum as the target record. 109 H) Using the AH f i l t e r c o e f f i c i e n t s , the three spectral parameters R, 9 and may be evaluated. These terms characterize the strong motion recording i n a very e f f i c i e n t manner. 9(t) represents the freguency coordinate of the peak of the power spectrum. I t indicates the dominant frequencies present i n the record at any time. R{t) i s related to the h a l f -width of the power spectrum and i s a measure of the range of freguencies present. The innovation variance a*~(t) i s dependent on the instantaneous amplitude of the accelerogram. The envelope of o~e (t) describes the build-up and decay of ground motion with time and i s therefore a convenient parameter by which to characterize the duration of shaking. 5) In order to examine the empirical behaviour of R, 9 and cr&, a suite of 'ro c k - s i t e 1 accelerograms from south-western U.S.A. have been analysed by the time-adaptive AR technigue. From the limited data set a v a i l a b l e , i t was found that R and 9 show l i t t l e systematic v a r i a t i o n with earthguake magnitude or epicentral distance., However, the innovation variance shows a strong dependence on M and D. The peak value of oV" f a l l s o f f with epicentral distance as rr\a\ D and the shaking duration as measured from the envelope of ^ V(t) decreases with both M and D. 6) As a r e s u l t of t h i s empirical analysis, the spectral curves may be interpolated so that a r t i f i c i a l accelerograms may 110 be constructed f o r a rb i t r a r y M and D combinations. However, th i s excerc ise i s severely data-l imited at t h i s time, and can be conf ident ly performed only when the set of empir ica l curves becomes more complete; i . e . , when more strong motion accelerograms have been recorded. 111 APPENDIX The analytic correspondence between a continuous second order li n e a r d i f f e r e n t i a l eguation and a discrete second order AB process i s examined here. The d i f f e r e n t i a l eguation representing the behaviour of a harmonic o s c i l l a t o r with damping P and natural freguency at, driven by continuous white noise f (t) i s Sx-w +. r J * m + coi x(±) = i ( t ) . A"1 The terms may be approximated by the method of f i n i t e differences incorporating the discrete values x t. Since a second order AB process u t i l i z e s only three data points at any instant, a three-point f i n i t e difference representation w i l l be used. According to Abromowitz and Stegun (1965), the f i r s t derivative i s d -x.Ci) _ _ -x-t-, _ J_ d -x-tt) A-elt 2. Z. 6 dt3 ' where the l a s t term on the r i g h t i s an error term. S i m i l a r l y , the 3-point f i n i t e difference representation of the second derivative i s 112 dt1- 2-* dt1* A-4 In addition. A-5 and A-6 Substituting Eguations A-3 to A-6 into A-1 and c o l l e c t i n g l i k e terms gives + (I - "f) Z t - , = -Pt A-7 where the error terms in Eguations A-3 and A-4 have been combined into the parameter € f. Since f t i s white, the input may be shifted by one time step.,Thus t -1 J2 2. 7C *-z j2 z. A-8 I t can be shown that the AB f i l t e r c o e f f i c i e n t s are related to the power spectral parameters B and 9 through 1 1 3 a, — 1 cos e A - 9 and R 1 A - 1 0 Comparing Equations A - 2 and A - 8 , i t can be seen that A- 1 1 R = Z 4- P z - r A - 1 2 and Z COS © =. Cu0 A - 1 3 Some of the terms i n Eguations A - 1 2 and A - 1 3 may be approximated by a Taylor expansion series about zero: 2 4- r z. - P + = \ + 2. + 6, A - 1 4 and Z cos & — + _e2. \Z 114 — Z - e2- + £ & . A - 1 5 Taking only the f i r s t two terms of each se r i e s , the relationships R - I t r +-eP , A-16 and A - 1 8 R - £ P are obtained. Hence f i r s t order c o r r e l a t i o n s between the harmonic o s c i l l a t o r c o e f f i c i e n t s and the A B s p e c t r a l parameters are R = I 4- - £ A - 1 9 e - <^ o A - 2 0 and 115 Any mathematical calculations involving relationships between these parameters should, of course, include higher order terms. 116 REFER ENCES Abromowitz, M., and I. Stegun (1965). Handbook of Mathematical Functions. Dover Publications, New York, p. 914. Akaike, H. (1969). Power spectrum estimation through autoregressive model f i t t i n g . ,^nnA Inst, S t a t i s t , Math.. 21, p. 407. Akaike, H.,(1970). S t a t i s t i c a l predictor i d e n t i f i c a t i o n . Ann. Inst. S t a t i s t . Math.. 22, p. 203. Amin, M. and A. H. Ang (1966). A nonstationary stochastic model fo r strong motion earthquakes. University of I l l i n o i s , Dept. Civ. Eng., Structural Research Series, no. 306. Beaudet, P. R, (1970). Synthesis of non-stationary seismic signals. Bull^ Seis. Soc. Aix# 60, p. 1615.. Burg, J. P. (1967). Maximum entropy spectral analysis, paper presented at the 37th Annual International Meeting, Soc. Explor. Geophys., Oklahoma City, Oklahoma. Burg, J . P . (1968). A new analysis technigue for time series data, paper presented at the Advanced Study I n s t i t u t e on Signal Processing, NATO, Enshede, Netherlands. Burg, J. P. (1972). The re l a t i o n s h i p between maximum entropy and maximum l i k e l i h o o d spectra. Geophysics. 37, p. 375. Cherry, S. (1970). Earthguake and dynamic design of structures including NBC 1970 requirements, paper presented at Canadian Structural Engineering Conference, Toronto, Ontario. Claerbout, J. F. (1976). Fundementals-of Geophysical Data Processing. McGraw-Hill International Series i n the Earth and Planetary Sciences, San Francisco, p. 55. Davenport, A. G. (1972). A s t a t i s t i c a l relationship between shock amplitude, magnitude, and epicentral distance and i t s application to seismic zoning. Univ. Western Ontario, Faculty Eng. S c i . , BLWT-4-72. Davies, J. C. , (1976) . Maximum entropy s p e c t r a l analysis of free o s c i l l a t i o n s of the earth; the 1964 Alaska event. M. Sc. t h e s i s , University of B r i t i s h Columbia, p. 28. F a c c i o l i , E. and D, Resendiz (1976). S o i l dynamics: behaviour including liquefaction, i n Seismic Risk and Engineering Decisions, edited by Lomnitz, C. and E. Rosenblueth, Developments i n Geothechnical Engineering 15, Elsevier S c i e n t i f i c Publishing Co., Amsterdam, p. 71. 117 G i f f i t h s , L. J, (1975). Sapid measurement of d i g i t a l instantaneous freguency. IEEE Trans* on Accoustics. Speech and Signal Processincj, A SSP-23, p. 207. G r i f f i t h s , L. J., and R. Preito-Diaz (1977) , Spectral analysis of natural seismic events using autoregressive techniques..IEEE Trans, on Geoscience E l e c t r o n i c s , GE-15, p. 13. Housner, G. W. (1944). An investigation of the e f f e c t s of earthquakes on buildings. Ph. D. thesis, C a l i f o r n i a I n s t i t u t e of Technology, Pasadena. Housner, G. W, (1947). Charac t e r i s t i c s of strong motion earthguakes. Bullj. S e i s x S G£i , 4 s A , 37, p. 19. Housner, G. . ff. .. (1955) .. Properties of strong motion earthguakes. B u l l . Seig,. Soc. Am. , 4 5, p. ,187. Housner, G. W. and P.C. Jennings (1964). Generation of a r t i f i c i a l earthguakes. Jour. Eng. 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Dependence of the duration of strong earthquake ground motion on magnitude, epicentral distance, qeologic conditions at the recording station and frequency of motion. University of Southern C a l i f o r n i a , Dept. Civ. Eng., report CE76-02. rjlrych, T. J. and T. N. Bishop (1975) . Maximum entropy spectral analysis and autoregressive decomposition. Bev. •• Geophys. and S£ace Phys,, 13, p.,183. Van den Bos, A. (1971). Alternative interpretation of maximum entropy spectral analysis. IEEE Trans. Inform. Theory. IT-17, p. 493. 119 Vanmarcke, E. V. (1976). Structural response to earthquakes, i n Seismic Risk and Engineering Decisions, edited by Lomnitz, C. and E. Rosenblueth, Developments i s Geotechnical Engineering 15, Elsevier S c i e n t f i c Publishing Co., Amsterdam, p. 287., Widrow, B. (1970) . Adaptive f i l t e r s , i n Aspects of Network and System Theory., edited by Kalman, S. ,E. and N. DeClaris, Holt, Rinehart and Winston, New York, p. 563. Widrow, B., P. E. Mantey, L. J. G r i f f i t h s and B. B. Goode (1967). Adaptive antenna systems. Proc^ IEEE, 55, p. 2143. Widrow, B. and M. E. Hoff (1960). Adaptive switching c i r c u i t s . IRE WESCON Con. Record, part 4, p. .96. 

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