ARMA Models for Earthquake Ground Motions Seismic Safety

104 Pages·2012·2.9 MB·English

XA04NO351
NUREG/CR-1751
UCRL-15084
NIS-XA-N--052
ARMA Models for
Earthquake Ground Motions
Seismic Safety Margins
Research Program
Mark K. Chang, Jan W. Kwiatkowski, Robert F. Nau, Robert M. Oliver, Karl S. ister
University of California, Berkeley, California
Prepared for
U.S. Nuclear Regulatory Commission
LAMIRENCE
UVERMORE
LABORATORY
NOTICE
This report was prepared an account of work sponsored by
an agency of the United States Government. Neither the
United States Government nor any agency thereof, or any of
their employees, makes any warranty, expressed or implied, or
assumes any legal liability or responsibility for any third
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apparatus product or process disclosed in this report. or
represents that its use by 3uch third party would not infringe
privately owned rights.
Available from
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U. S. Nuclear Regulatory Commission
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Printed copy price: 400
and
National Technical Information Service
Springfield, Virginia 22161
NUREG/CR-1751
UCRL-15084
RA, RD, RM, R6
ARMA Models for
Earthquake Ground Motions
Seismic Safety Margins
Research Program
Manuscript Completed: July 1979
Date Published: February 1981
Prepared by
Mark K. Chang, Jan W. Kwiatkowski, Robert F. Nau, Robert M. Oliver, Karl S. Pister
University of California, Berkeley, California
Lawrence Livermore Laboratory
7000 East Avenue
Livermore, CA 94550
Prepared for
Office of Nuclear Regulatory Research
U.S. Nuclear Regulatory Commission
Washington, D.C. 20555
NRC FIN No. A0139

iii
ABSTRACT
This report contains an analysis of four major California earthquake
records using a class of discrete linear time-domain processes comm nly
referred to as ARMA (Autoregressive/Moving-Average) models. It has
been possible to analyze these different earthquakes, identify the
order of the appropriate ARMA model(s), estimate parameters and test
the residuals generated by these models. It has also been possible
to show the connections, similarities and differences between the
traditional continuous models (with parameter estimates based on spec-
tral analyses) and the discrete models with parameters estimated by
various maximum likelihood techniques applied to digitized acceleration
data in the time domain. The methodology proposed in this report is
suitable for simulating earthquake ground motions in the time domain
and appears to be easily adapted to serve as inputs for nonlinear dis-
crete time models of structural motions.

v
ACKNOWLEDGEMENT
ORC 79-1
This research effort is part of the Seismic Safety Margins Research Program
conducted by the Lawrence Livermore National Laboratory for the U. S. Nuclear
Regulatory Commission under Purchase Order No. 1980709.

vii
TABLE OF CONTENTS
Page
CHAPTER 1: INTRODUCTION AND SUMMARY . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Highlights and Summary of Findings 4
1.3 Contents of the Report . . . . . . . . . . . . . . . . . . . 6
CHAPTER 2 ARMA MODELS FOR EARTHQUAKE GROUND MOTIONS . . . . . . . . 8
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 The ARMA (2,1) Model . . . . . . . . . . . . . . . . . . . . 11
2.3 Relations Between Discrete and Continuous
2nd-Order Models . . . . . . . . . . . . . . . . . . . . . . 16
2.4 The ARMA 41) Process and Its Interpretation . . . . . . . 24
CHAPTER 3 ANALYSIS OF EL CENTRO AND SAN FERNANDO EARTHQUAKE
RECORDS . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1 Data . . . . . . . . - - . . . . . . . . . . . . 27
3.2 Identification and Es;im'a(cid:1)i;n Tchniques . . . . . . . . . . 28
3.3 General Results . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Discussion of ARMA (2,l) Models . . . . . . . . . . . . . . 32
3.5 Discussion of the ARMA 41) Models . . . . . . . . . . . . 34
3.6 Time-Variation of Parameter Estimates Within Components 38
3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 41
CHAPTER 4 DIRECTIONS FOR FUTURE RESEARCH . . . . . . . . . . . . . . 59
APPENDIX A: EXISTING MODELS FOR EARTHQUAKE GROUND MOTIONS . . . . . . 63
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 63
A.2 Analysis and Characterization of Earthquakes . . . . . . . . 63
A.3 Existing Statistical Models for Earthquake Ground Motions 66
A.4 Structural Response . . . . . . . . . . . . . . . . . . . . 72
APPENDIX B: BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . 73
viii
INDEX OF FIGURES AND TABLES
Page
CHAPTER 2
FIGURE 21: A GENERAL GROUND MOTION MODEL . . . . . . . . . . . 16
TABLE 21: COMPARISON OF DISCRETE AND CONTINUOUS
2ND-ORDER RANDOM PROCESSES . . . . . . . . . . . . 22
TABLE 22: EXAMPLE OF CONVERSION BETWEEN CONTINUOUS
2ND-ORDER MODEL AND DISCRETE ARMA (2,I) MODEL
(At = .02 SEC.) . . . . . . . . . . . . . . . . . . 23
CHAPTER 3
FIGURE 3.1a: ACCELEROGRAMS OF EARTHQUAKES STUDIED IN THIS
REPORT . . . . . . . . . . . . . . . . . . . . . . 42
FIGURE 3.1b: ACCELEROGRAMS OF EARTHQUAKES STUDIED IN Tills
REPORT . . . . . . . . . . . . . . 43
FIGURE 3.2a: IIA001 S90W ACCELERATION DATA 5_ lo iCO'ND'S . . . . .
AUTOCORRELATION FUNCTION . . . . . . . . . . . . . 44
FIGURE 3.2b: IIA001 S90W ACCELERATION DATA 5_10 SECONDS
LOG 10 SPECTRUM 95 P.C. CONFIDENCE LIMITS,
SMOOTHING BANDWIDTH = 0.1) . . . . . . . . . . . . 45
FIGURE 3.2c: THEORETICAL SPECTRUM OF ARMA 21) MODEL FOR
IIA001 S90W 510 SECONDS . . . . . . . . . . . . . 46
FIGURE 3.2d: IIA001 S90W ARMA 21) MODEL RESIDUALS 10 SECONDS
LOG 10 SPECTRUM 95 P.C. CONFIDENCE LIMITS,
SMOOTHING BANDWIDTH = 0.1) . . . . . . . . . . . . 47
FIGURE 3.3a: IIC041 S16E ACCELERATION DATA -10 SECONDS
AUTOCORRELATIONS FUNCTION . . . . . . . . . . . . . 48
FIGURE 3.3b: IIC041 S16E ACCELERATION DATA 5_10 SECONDS
LOG 10 SPECTRUM 95 P.C. CONFIDENCE LIMITS,
SMOOTHING BANDWIDTH = 0.1) . . . . . . . . . . . . 49
FIGURE 3.3c: THEORETICAL SPECTRUM OF ARKA 41) MODEL FOR
IIC041 S16E 5-10 SECONDS . . . . . . . . . . . . . 50
FIGURE 3.3d: IIC041 S16E ARMA 41) MODEL RESIDUALS 10 SECONDS
LOG 10 SPECTRUM 95 P.C. CONFIDENCE LIMITS,
SMOOTHING BANDWIDTH = 0.1) . . . . . . . . . . . . 51
TABLE 31: ARMA 21) MODELS FOR IIA001 S90W COMPONENT
(IMPERIAL VALLEY 1940 EARTHQUAKE,
EL CENTRO STATION) . . . . . . . . . . . . . . . . 52
TABLE 32: ARMA 41) MODELS FOR IIA011 S90W COMPONENT
(EL ALAMO B.C. 1956 EARTHQUAKE,
EL CENTRO STATION) . . . . . . . . . . . . . . . . 53
TABLE 33: ARMA (4,I) MODELS FOR IlCO41 S16E COMPONENT
(SAN FERNANDO 1971 EARTHQUAKE,
PACOIMA DAM STATION) . . . . . . . . . . . . . . . 54