1. Automation, Vol. 26, No. 4, pp. 721-737, 1990
Print¢d in Great Britain.
0005-1098/90 $3.00 + 0.00
Pergamon Press pie
(~) 1990 International Federation of Automatic Control
Analysis and Simulation of Strong Earthquake
Ground Motions Using ARMA Models*
TH. D. POPESCUt and S. DEMETRIUS:
Segmentation of the nonstationary time series and representation of the
quasi-stationary data blocks through ARMA models provides an efficient
and flexible procedure for characterization of earthquake ground motions.
Key Words--Time-series analysis; signal processing; parameter estimation; simulation; ARMA
models; nonstationarity analysis;geophysics;earthquake ground motion.
Abstract--The acceleration record of an earthquake ground
motion is a nonstationary process with both amplitude and
frequency content varying in time. The paper presents a
general procedure for the analysis and simulation of strong
earthquake ground motions based on parametric ARMA
models to be used in computing structural response. Some
computational results obtained in the analysisand simulation
of ground acceleration, recorded during the Romanian
Earthquake of 4 March 1977,are also included.
1. INTRODUCTION
IT HAS aECOME relatively common practice in
recent years to use stochastic processes as
models of dynamic loadings that are subject to
considerable uncertainty, such as earthquake
ground motion, or forces caused by wind or
waves. The particular problem considered in this
paper is the analysis and simulation of
earthquake ground motions, although the results
may be also applicable to other similar
problems.
The analysis and simulation of strong
earthquake ground motions may prove very
useful, especially for the design of complex
structures that are expected to have nonlinear
behaviour during seismic motion, allowing a
statistical treatment of the response characteris-
*Received 25 October 1988; revised 29 March 1989;
receivedin finalform4 October 1989.The originalversionof
this paper was presented at the 8th IFAC/IFORS
Symposiumon Identificationand System Parameter Estima-
tion which was held in Beijing, People's Republic of China
during August 1988.The PublishedProceedingsof this IFAC
Meeting may be ordered from: Pergamon Press pie,
Headington Hill Hall, Oxford OX3 0BW, U.K. This paper
was recommended for publication in revised form by
Associate Editor Y. Sunahara under the direction of Editor
P. C. Parks.
"I"Institute for Computers and lnformatics 8-10 Miciurin
Bird, 71316Bucharest, Romania.
¢ Faculty of Civil Engineering 124 Lacul Tei Blvd, 72302
Bucharest, Romania.
721
tics. The analysis and simulation problems of
seismic signals can be stated as follows:
(a) Analysis problem: to develop a method of
nonstationary characterization of earthquake
accelerograms that can describe the time-varying
nature of the mean square amplitude and
spectral content.
(b) Simulation problem: to recover the
original accelerogram or a class of compatible
time functions, given the solution for (a).
Historically, the modelling and application of
nonstationary covariance time series in en-
gineering applications have been approached via
the fitting of locally stationary models, via
orthogonal polynomial expansion of AR
coefficient models, and other analyses of random
coefficient AR models. Locally, stationary AR
modelling was shown by Ozaki and Tong (1975)
and Kitagawa and Akaike (1978),
The analysis and simulation of nonstationary
processes have been studied mostly in connec-
tion with earthquake ground motions. The
common feature of these studies is that a
nonstationary process can be simulated by
multiplying by an envelope function of a
stationary process generated either by filtering a
white noise (Jennings et al., 1968), or by a series
of waves with random frequency and random
phase (Kitada et al., 1983). Some relatively new
methods employ the class of autoregressive
models (AR) (Kozin, 1979; Jurkevics and
Ulrich, 1978; Gersh and Kitagawa, 1985) and the
general class of autoregressive moving average
(ARMA) models for the analysis and simulation
of strong motion accelerations (Nau et al., 1980).
Kozin (1979) shows an orthogonal polynomial
expansion of the AR coefficients of a time-
varying AR model. The method is extended by
Gersh and Kitagawa (1983) to a multivariate
time-varying AR model in the context of

2. 722 TH. D. PoPESCU and S. DEMETRIU
econometric data analysis. Jurkevics and Ulrich
(1978) use an "adaptive filtering" method for
estimating time-varying parameters in an AR
model. The similarities and differences between
adaptive filtering methods and Kalman filtering
methods for estimating time-varying AR para-
meters have been discussed in a paper by Nau
and Oliver (1979). In Gersh and Kitagawa
(1985), the nonstationary time series is modelled
by a time-varying autoregressive (AR) model
with smoothness constraints on the time-evolving
AR coefficients. The analysis yields the sequence
of instantaneous time-varying AR coefficients
and the process variance and may have numerous
engineering applications. The major contribu-
tions of the report elaborated by Nau et al.
(1980) are the use of Kalman filters for
estimating time-varying ARMA model para-
meters, and the development of an effective
non-parametric method for estimating the
variance envelopes of the accelerogram records.
The AR and ARMA models represent filters
used for the generation of the synthetic
accelerograms by passing an approximation of a
white noise through them. Also, the evaluation
of ARMA and related parameters, on their own,
has physical significance for certain model
structures and is related to parameters such as
earthquake intensity and duration, distance to
the fault and local geology (Cakmak and Sheriff,
1984).
In particular, the application of ARMA
models to strong motion accelerograms can be
performed after processing the seismic signals by
a variance stabilizing transformation (Polhemus
and Cakmak, 1981).
In the present paper the problem of signal
nonstationarity is solved by segmenting the
original data, using the evaluation of Akaike
Information Criterion (AIC) for different data
blocks (Kitagawa and Akaike, 1978), so that
each data block could be considered quasi-
stationary. For each quasi-stationary data block
an ARMA model is fitted by canonical
correlation analysis and exact maximum like-
lihood method. These models are used for
simulation purposes, to recover the original
seismic signal. Finally, the original and synthetic
data, obtained from ARMA models in a case
study, are compared by evaluating a number of
statistical characteristics and parameters, com-
monly used to characterize strong motion
accelerograms. An acceptance match is found
in all cases. The application of the time domain
technique for the analysis and simulation of
digitized earthquake accelerograms, presented in
the paper, seems to be a potentially useful
method of characterizing earthquake ground
motions by constant linear models with a small
number of parameters.
2. MODELLING OF NONSTATIONARY
TIME SERIES
Nonstationarity analysis
In certain situations the statistical characteris-
tics of a natural time series are a function of
time. To model an observed time series that
possesses nonstationarity, a common procedure
is to first remove the nonstationarity by invoking
a suitable transformation and then to fit a
stationary stochastic model to a transformed
sequence. When modelling certain types of
geophysical time series, it is often reasonable to
assume that the time series are approximately
stationary over a specified time interval. The
problem of signal nonstationarity is solved in this
paper by segmenting the original data, so that
each data block can be considered quasi-
stationary.
The nonstationarity analysis procedure used
(Kitagawa and Akaike, 1978), originally pro-
posed by Ozaki and Tong (1975), is suitable for
application to a nonstationary situation where
the analysis of a short span of data is necessary.
The procedure is discussed in detail in Kitagawa
and Akaike (1978). Here we will give only a
conceptual description of the procedure.
Suppose we have a set of initial data
Y~,Y2 ..... Yr., and an additional set of S
observations YL+I..... YL+S is newly obtained,
where S is a prescribed number. The procedure
is as follows:
(1) Fit an autoregressive model ARo, y, =
A,,,y~_,,, + e,, with tr2 the innovation vari-
m=l
ance, where Mo is chosen as the one which gives
the minimum of the criterion AIC, for the set of
data y~, Y2.... , yL.
AICo = L- log ~o + 2(Mo + 2). (1)
(2) Fit an autoregressive model AR1, Yn=
M1
E A,,,yn-,~ + en, with 02 the innovation vari-
m=l
ance, and Mt chosen as giving the minimum of
the criterion AIC, for the set of data
YL+I, • • • , YL+S.
AlCt = S. log o2+ 2(M1 + 2). (2)
(3) Define the first competing model by
connecting the autoregressive models ARo and
AR1. The AIC of this jointed model is given by:
AICo., = L. log O~o+ S. log o2+ 2(Mo + M, + 4).
(3)

3. Analysis of ground motions using ARMA models 723
(4) Fit an autoregressive model AR2, y, =
A,yn-, + en, with 0~2 the innovation vari-
ance, where M2 is chosen in the same way as
above, for the set of data Yt..... YL,
YL+I, • • • , YL+S.
(5) Define the model ARz as the second
competing model with the AIC given by:
AIC.2 = (L + S). log o~2+ 2(M2+ 2). (4)
(6) If AIC2 is less than AICo,t, the model AR2
is accepted for the initial and additional sets of
observations and the two sets of data are
considered to be homogeneous. Otherwise, we
switch to the new model ARz. The procedure
repeats these steps whenever a set of S new
observations is given. S is called the basic span.
The procedure is so designed as to follow the
change of the structure of the signal, while if the
structure remains unchanged it will improve the
model by using the additional observations.
The numerical procedure used for AR model
computation is the method of least squares
realized through Householder transformation
(Golub, 1969). This approach provides a very
simple procedure of handling additional new
observations which is useful for the on-line
nonstationarity analysis by autoregressive model
fitting. This approach also provides a very
flexible computational procedure for the selec-
tion of regressors.
Some Bayesian-type model fitting procedures,
developed by Akaike (1978b, c), can be also
used for AR modelling. It is expected that these
procedures would reduce the risk of adopting
models of too high an order by the simple
minimum AIC procedure.
Concerning the procedure of choosing a model
with the minimum value of AIC, it does not
have a clearly defined optimal property.
Especially when it is applied to the determina-
tion of the order of an AR model, the order
chosen by the minimum AIC procedure does not
produce a consistent estimate. Results of some
Monte Carlo experiments are reported by Jones
(1975). In spite of this inconsistency, the
corresponding estimate of the spectrum is
consistent (Shibata, 1976).
A modification of AIC for the fitting of AR or
ARMA models to univariate time series, BIC
criterion, is suggested by Akaike (1977).
The extension of the nonstationarity analysis
procedure, used in the paper, to the multi-
variante case is direct.
ARMA modelling
The ARMA model's suitability for describing
strong earthquake ground motions has been
studied in the reports of Change et al. (1979) and
Nau et al. (1980). There are two important
justifications for the ARMA model approach in
modelling of quasi-stationary data blocks,
obtained by the procedure previously described.
First, considered simply as an empirical method
of time series analysis and simulation, the
ARMA model framework makes it possible to
proceed directly and systematically from the
analysis of historical discretized accelerograms to
the synthesis of artificial discretized accelero-
grams with similar specific statistical properties.
Secondly, there is an important theoretical
basis for using ARMA models to simulate
sampled continuous time random processes. It is
well known [Bartlett (1946) and Gersh and Luo
(1970)] that an exact ARMA (n, n- 1) process
results from the equispaced sampling of con-
tinuous random processes generated by passing
stationary white noise through a linear time-
invariant filter with a rational transfer function,
whose denominator is n. Since continuous
processes of this type include those representing
the response of noise-driven multi-degree-of-
freedom linear oscillators, and integrals thereof,
this correspondence is of great theoretical and
practical importance for the modelling of ground
acceleration, velocity and structural response.
For example, maximum-likelihood estimates for
the parameters of an ARMA model fitted to
sampled data may be converted directly into
maximum-likelihood estimates of the natural
frequency and damping parameters of a continu-
ous-time linear model for the underlying physical
system (Gersh et al., 1973; Gersh, 1974).
Some authors, in the analysis and simulation
of earthquake accelerograms, use purely auto-
regressive (AR) models, which may be con-
sidered as ARMA (p, q) models in which q = 0.
AR models are sometimes favoured over more
general ARMA models, in certain applications,
because they allow more efficient parameter
estimation. Technically, for a fixed number of
parameters, a pure AR model gives a "maxi-
mum entropy" representation for a random
process. However, a model with both AR and
MA parameters will often provide as good a
representation with fewer total parameters. It
should be noted that, in general, both
autoregressive and moving average terms arise,
in a complex and interconnected way, when an
ARMA model is sought to describe a time series
resulting from equispaced sampling of a
continuous-time random process.
The use of the ARMA models given by:
P q
Y,,- Z A,,,y,,_,,, =e,,- E Bke,,-k (5)
m=l k=l
AUTO26:4-F

4. 724 TH. D. POPESCU and S. DEMETRIU
as ultimate parametrically parsimonious models
for the quasi-stationary data blocks, obtained by
the previously described procedure, is hindered
by both the numerical difficulty of the necessary
maximum likelihood computation and the
difficulty in choosing the order of AR and MA
model parts. From the theoretical analysis
(Akaike, 1974) it resulted that a canonical
correlation analysis procedure would produce a
useful initial guess for the model. Also, a
criterion developed for the evaluation of the
goodness of fitting a statistical model obtained
by the method of maximum likelihood provided
a solution to the order determination problem.
There is a technical difficulty in fitting an
ARMA model. This is the problem of
identifiability or of uniqueness of the model. In
the univariate case, by choosing the lowest
possible values of the Ar order p and the MA
order q, the parameters are uniquely specified.
In the multivariate case, where the coefficients
A,, and Bk are matrices, the difficulty is avoided
by using a canonical Markovian representation
of the analysed time series. The assumption of
finitenesss of the dimension for the state vector
is fundamental for producing a finite parameter
model. This subject is discussed in detail by
Akaike (1974; 1976). The conceptual descrip-
tion of the organization of the procedure for
ARMA model fitting, in univariate case, is as
follows (Akaike, 1978a):
M
(1) Fit an AR model Yn= ~ Amyn-m+en,
m=l
where M is chosen as the one which gives the
minimum of AIC.
(2) Do the canonical correlation analysis
between yn, Yn-1..... Yn-M and yn, Y~+t.....
By this analysis, determine z. as the vector of
predictors of these variables within the future set
Yn, Yn+~..... that are associated with positive
canonical correlation coefficients. For the multi-
variate case yn is replaced by Y'n, the transpose
of Y., the multivariate time series under
consideration. By this analysis the estimates of
the matrices F and G of a Markovian
representation are obtained;
z~+~ = Fz~ + Ge~
(6)
y~ = Hzn
where zn is the state vector of the system in time
and H takes a prescribed form.
(3) Do the maximum likelihood computation
for the Markovian representation.
(4) Try several possible alternative structures
of the state vector zn and choose a final estimate
using AIC criterion.
(5) If necessary, transform the Markovian
representation into an ARMA representation.
For a univariate time series y,, direct ARMA
parameter estimates computation is possible.
ARMA models are attractive because they are
characterized by a small number of parameters,
lend themselves to digital simulation in the time
domain and can be easily adapted to include
changes in frequency contents of correlated
random processes that characterize the ground
motion. The coefficients of the ARMA model
can be set to account for filtering a ground
motion due to transmission path and local site
condition effects. Hence, the ARMA formula-
tion has a rational basis for the modelling of
ground motions with considerable flexibility.
3. SIMULATIONOF NONSTATIONARY
TIME SERIES
Simulated earthquake motions can be con-
sidered as samples of a random process with a
prescribed power spectral density, multiplied by
envelope functions chosen to model the changing
intensity of real accelerograms. In our approach
the simulated accelerograms are obtained by
shifting, with the mean values of quasi-stationary
data blocks, the time series resulting after the
passing of a white noise through the filters
represented by ARMA models determined for
each stationary data block.
The general procedure for the generation of
an artificial accelerogram is as follows:
(1) Select the parameters and the innovation
variance for the ARMA model associated with a
quasi-stationary data block of the original
accelerogram.
(2) Generate a Gaussian white noise series
having zero mean and variance given by (1)
above.
For the multivariance case a multivariate
normal noise series with zero mean and
covariance matrix obtained in the ARMA model
fitting procedure is generated.
(3) Pass the white noise series through the
appropriate ARMA model to obtain a simulated
accelerogram; the values for ARMA model
parameters may be estimated from a particular
real accelerogram or else chosen to correspond
to a particular continuous time model.
(4) Shift the simulated accelerogram (3) by
corresponding mean value of quasi-stationary
data block, removed during the phase of ARMA
model parameter estimation (canonical correla-
tion analysis and maximum likelihood method).
The ARMA model simulation scheme can be
diagrammed, for a quasi-stationary data block,
as shown in Fig. 1. Here a discrete stationary
white noise sequence {e,,} is served as input to
an ARMA filter. The output from this filter is

5. Analysis ot ground motions using ARMA models 725
Sfafion,~ry
While Nois~
Poromefers Pc~ramefers
' I Average regressive
L. Fifief Filler
ARMA Filler
FK~. ]. ARMA simulation model for discrete acceleration.
Yn
Discrefe
Accelerctfion
considered to represent a discretized (sampled)
acceleration record which is shifted by the
corresponding mean value of the quasi-
stationary data block to obtain the simulated
accelerogram.
Thus, for a real accelerogram with B
stationary data blocks, all the information used
in simulation is contained in B mean values of
quasi-stationary data blocks and B models,
ARMA (p,q) (p-autoregressive parameters;
q-moving average parameters and innovation
variance). ARMA modelling has a distinct
advantage in the fact that the model parameters
can be used for the simulation of the original
acceleration series in a recursive manner. The
phase characteristics are also conserved in this
approach. The simulated accelerograms are
consistent with the time variation of the spectral
content of the original signal.
4. CASE STUDY
This section contains some results obtained in
the analysis and simulation of 40 s of N-S, E-W
horizontal and vertical component records of
strong ground motions of the Romanian
Earthquake of 4 March 1977. The data are based
on corrected accelerograms digitized at 0.01 s.
The evolution of seismic signals is represented in
Fig. 2 and reflects the time-varying nature of the
amplitude and frequency content of the com-
ponents. Because the statistics characteristics of
the time series are slowly varying, we can admit
the quasi-stationarity of these time series on
short data blocks, referred to the entire records.
400
300
E zooL3
1O0
_g o
-100
-200
<C -300
-400
400
' 300u~
E 200
1O0
g o
8 -1 oo
~ - 200
~ -300
-400
(a)
I i I
2.5 5.0 Z5
(I0)
I I I I I I I I I I I I
10.0 1z.5 15.0 17.5 20.0 z2.s z5.0 27.5 30.0 32.5 35.0 37.5 40.0
Time (s)
0.0 2.5 5.0 z.5 10.0 12.5 15.0 17.5 200 22.5 250 27.5 300 32.5 35.0 3r.5 40.0
Time (s)
(c)
400
N
'~ 3(30
E 200
1oo
.~ o
O
-100
(o
-ZOO
(,3
<~ - 300
-- 400 i i I I I | I I I I I I I I |
0 '0 2. 5 ~ .0 7.5 10.0 12.5 15.0 17.s 20.0 22 .S 25.0 ~? 5 30.0 32 ~ 35.0 ~ .5 40. 0
Time (s)
FIG. 2. Evolution of recorded seismic signal. (a) N-S component; (b) E-W component; (c) vertical
component.

6. 726 TH. D. POPESCUand S. DEMETRIU
11/I
o)
10
8
~ 4 "--
~ 2
~ o
0 - 2
> - 4C ~--J
-10 ~ I I
0.0 Z.5 5.0
(b)
150
? 135 -
u,i
E 120 -
105 -
r-
.2 90 -
75 -
"o 60 -
45 -
"o
c 30 -
9
15 2.~-'-'-"
....... ...... _J .... .........
I I I I I I I I I I I I
r5 100 12.5 150 17s zo.o zz.5 zs.o zr.5 300 3~.s 350 ~7.5 40.0
Time (s)
....... i ............. --- 1. _ . .L.__
t I J ~...... i ...... ~ ...... r ...... J...... i:'-':'~'-"::~-'-'-i .... :~-"-'-"
0 2 .S ~ .0 7 .s 10.0 12 l~ ~~10 1~.S 20.0 2 ~.~ Z~ 10 Zr, ~ 30.0 ] ~ .~ ] ~10 ~71~ 40. 0
Time (s)
FIG. 3. Mean values and standard deviations. (a) Mean values; (b) standard deviations.
component; - - - E-W component; .... vertical component.
N-S
On each data block statistical structure of the
series is considered constant, and this is modified
only by passing from one block to another.
After a preliminary stationarity analysis of the
seismic components, for different basic spans
between 1 and 5 s, we found the basic span of
2.5 s as the best for this practical application.
Using the nonstationarity analysis procedure
presented in the first part of the paper, for basic
span of 2.5 s, the original records were divided
into data blocks, correponding to stationarity
intervals. The quasi-stationary data blocks are
separated by vertical lines, as shown in Fig. 2.
For these quasi-stationary data blocks, the mean
values and evolution of the standard deviations
of seismic data for all components are
represented in Fig. 3. These representations
reflect the nonstationarity character of the
analyzed time series. The locally stationary AR
models obtained from Canonical correlation
analysis are presented in Table 1, Table 2 and
Table 3; locally stationary ARMA models
resulting are given in Table 4, Table 5 and Table
6. The necessary computer programs for
nonstationarity analysis and canonical correla-
tion analysis are contained in the TIMES
program package (Tertisco et al., 1985; Popescu
1981) and EARTS (Demetriu, 1986).
It should be noted that, in general, the AR
models obtained in stationarity analysis for each
TABLE1. LOCALLY STATIONARY AR MODELS RESULTING IN CANNONICALCORRELATION ANALYSIS (N-S COMPONENT)
Model Innovation
Interval order AR1 AR2 AR3 AR4 AR5 AR6 AR7 AR8 AR9 AR10 variance
1-250 8 1,064 0.146 -0.038 -0.099 -0.181 -0.062 -0,015 0.120 1.732
251-500 5 1.469 -0.251 -0.135 -0.240 0.116 2.012
501-750 4 1,416 -0.117 -0.176 -0.130 26.485
751-1000 4 1.373 -0.162 -0.121 -0.122 26.629
1001-1500 9 1.410 -0.110 -0.204 -0.103 -0.086 0.086 -0.028 -0.072 0.092 7.208
1501-1750 5 1.116 0.032 0.046 -0.112 -0.099 9.963
1751-2000 6 1.346 -0.139 -0.101 -0.014 -0.220 0.095 3.813
2001-2250 4 1.115 0.155 -0.125 -0.216 4.434
2251-2500 4 1.201 0.064 -0.126 -0.186 2.569
2501-3000 7 1.322 -0.015 -0.196 -0.058 -0.148 0.016 0.066 0.960
3001-3500 6 1.107 0.097 -0.025 -0.131 0.033 -0.093 0.837
3501-3750 4 1.095 0.668 0.006 -0.188 1.550
3751-4000 5 1.093 0.077 -0.096 0.079 -0.176 0.673

7. Analysis of ground motions using ARMA models 727
TABLE 2. LOCALLY STATIONARYAR MODELSRESULTINGIN CANONICALCORRELATIONANALYSIS(E-W COMPONENT)
Model Innovation
Interval order ARI AR2 AR3 AR4 AR5 AR6 AR7 AR8 AR9 ARI0 variance
1-250 3 1.307 --0.064 -0.351 1.I00
251-500 3 1.131 --0.030 -0.151 12.227
501-1000 8 1.649 -0.500 -0.096 -0.095 0.004 -0.008 -0.050 0.079 20.819
1001-1250 7 1.714 -0.598 -0.076 -0.210 0.209 -0.198 0.130 9.162
1251-1500 3 1.612 -0.410 -0.236 6.856
1501-1750 5 1.617 -0.383 -0.238 -0.155 0.126 4.103
1751-2000 10 1.461 --0.174 -0.171 -0.102 -0.095 0.069 -0.121 -0.102 0.410 -0.197 1.913
2001-2750 6 1.498 -0.219 -0.256 -0.036 -0.133 0.122 1.295
2751-4000 10 1.239 0.047 -0.174 -0.069 -0.045 --0.059 0.018 0.008 -0.035 0.054 0.885
TABLE 3. LOCALLY STATIONARYAR MODELSRESULTINGIN CANONICALCORRELATIONANALYSIS(VERTICALCOMPONENT)
Model Innovation
Interval order AR1 AR2 AR3 AR4 AR5 AR6 AR7 AR8 AR9 AR10 variance
1-250 4 1.636 -0.802 -0.067 0.100 3.435
251-500 7 1.241 -0.263 -0.079 -0.057 -0.054 -0.006 5.952
501-750 6 1.307 -0.212 -0.225 -0.046 -0.062 0.140 8.216
751-1000 9 1.647 --0.616 -0.225 0.081 0.048 -0.150 0.228 0.028 -0.123 14.121
1001-1250 5 1.644 -0.596 -0.147 --0.103 0.128 6.471
1251-1500 10 1.521 -0.385 -0.253 0.090 -0.144 0.094 -0.134 0.257 0.038 -0.157 5.472
1501-1750 3 1.350 --0.188 -0.266 2.321
1751-2000 4 1.354 --0.195 -0.026 -0.194 1.589
2001-2500 8 1.194 0.020 -0.195 -0.107 -0.009 -0.010 -0.086 0.140 1.373
2501-2750 4 1.188 0.005 -0.116 -0.136 0.787
2751-3000 5 1.373 --0.266 0.007 -0.320 0.152 0.820
3001-3250 3 1.200 0.144 --0.425 1.037
3251-3500 9 1.057 0.206 -0.096 -0.073 -0.025 -0.021 -0.004 -0.054 0.153 1.175
3501-3750 4 1.027 0.091 --0.018 --0.189 0.594
3751-4000 3 1.308 -0.242 --0.161 1.137
TABLE 4. LOCALLY STATIONARY ARMA MODELS RESULTING FROM CANONICAL
CORRF-LATIONANALYSIS(N-S COMPONENT)
Model
Interval order AR1 AR.2 AR3 MA1 MA2
1-250 (2, 1) 1.929 -0.956 0.865
251-500 (2, 1) 1.857 -0.888 0.388
501-750 (2, 1) 1.940 -0.944 0.523
751-1000 (2, 1) 1.872 -0.890 0.499
1001-1500 (2, 1) 1.901 -0.916 0.491
1501-1750 (2, 1) 1.930 -0.951 0.765
1751-2000 (2, 1) 1.896 -0.913 0.549
2001-2250 (3, 2) 1.892 -0.994 0.083 0.743
2251-2500 (2, 1) 1.926 -0.945 0.724
2501-3000 (2, 1) 1.907 -0.916 0.585
3001-3500 (2, 1) 1.972 -0.976 0.866
3501-3750 (2, 1) 1.961 -0.965 0.866
3751-4000 (2, 1) 1.953 -0.959 0.861
-0.295
TABLE 5. LOCALLY STATIONARY ARMA MODELS RESULTING FROM CANONICAL
CORRELATIONANALYSIS(E-W COMPONENT)
Model
Interval order AR1 AR2 AR3 MA1 MA2
1-250 (3, 2) 2.110 -1.509 0.344 0.803 -0.396
251-500 (2, 1) 1.791 -0.813 0.661
501-1000 (2, 1) 1.858 -0.874 0.209
1001-1250 (3, 2) 2.782 -2.637 0.852 1.068 -0.209
1251-1500 (3, 2) 2.697 -2.459 0.757 1.084 -0.300
1501-1750 (2, 1) 1.866 -0.898 0.249
1751-2000 (3, 2) 2.655 -2.398 0.738 1.194 -0.480
2001-2750 (3, 2) 2.155 -1.480 0.309 0.657 -0.277
2751-4000 (3, 2) 1.982 -1.155 0.162 0.744 -0.280

8. 728 TH. D. POPESCU and S. DEMETRIU
TABLE 6. LOCALLYSTATIONARY ARMA MODELS RESULTINGFROM CANONICALCORRELATION ANALYSIS
(VERTICALCOMPONENT)
Model
Interval order AR1 AR2 AR3 AR4 MA1 MA2 MA3
1-250 (2, 1) 1.527 -0.696 -0.109
251-500 (2, 1) 1.602 -0.713 0.361
501-750 (2, 1) 1.675 -0.774 0.368
751-1000 (3, 2) 2.403 -2.123 0.692 0.756
1001-1250 (2, 1) 1.719 -0.810 0.076
1251-1500 (3, 2) 2.531 -2.289 0.742 1.010
1501-1750 (3, 2) 0.475 1.322 -0.963 -0.876
1751-2000 (2, 1) 1.868 -0.901 0.514
2001-2500 (3, 2) 2.120 - 1.615 0.455 0.926
2501-2750 (2, 1) 1.879 -0.903 0.691
2751-3000 (4, 3) 1.258 -0.318 0.086 -0.108 -0.114
3001-3250 (3, 2) 1.166 0.293 -0.537 -0.035
3251-3500 (2, 1) 1.922 -0.941 0.865
3501-3750 (2, 1) 1.920 -0.939 0.894
3751-4000 (2, 1) 1.619 -0.688 0.311
-0.261
-0.368
0.328
-0.530
-0.209
0.107
-0.178
quasi-stationary data block, are different from
the AR models obtained in canonical correlation
analysis. This is because in the stationarity
analysis the data obtained after removing the
global mean value from all original data
available are used, while in the canonical
correlation analysis the data obtained after
removing the local mean value from data
belonging to a quasi-stationary data block are
used; thus the time series used practically in
stationarity analysis and in the canonical
correlation analysis are different.
The parameters of ARMA models, obtained
by canonical correlation analysis, have been used
as preliminary estimates in the maximum
likelihood method, to obtain the final estimates
of ARMA parameters for simulation. The final
ARMA models used in simulation are given in
Table 7, Table 8 and Table 9. The software
support for exact maximum likelihood method
was assured by the AUTOB & J program
package (Popescu, 1985).
The simulation results, obtained for the
ARMA models of the quasi-stationary data
blocks using the previously discussed procedure
are given in Fig. 4.
By the visual analysis of the original and
simulated data we can point out that the time
evolution of the signals is similar. Also, the
parametric representations of quasi-stationary
data blocks conserve stochastic properties of
original data. While these models are admittedly
not perfect, they reflect the main statistical
features of real ground motions. To validate the
experimental results it is important to show that:
(a) the model captures the relevant dynamic
structure of the original series;
(b) simulated series from the model have
some characteristics which are similar to the
original series;
(c) the modelling procedure if applied to the
simulated series will recover reasonable esti-
mates of the relevant parameters.
In order to compare original record and
simulation results, different characteristics of
these time series, in the time and frequency
domain, were computed. The results are
presented in the following subsections.
TABLE 7. LOCALLY STATIONARY ARMA MODELS RESULTING FROM EXACT MAXIMUM
LIKELIHOODPROCEDURE(N-E COMPONENT)
Model Innovation
Interval order AR1 AR2 AR3 MA1 MA2 variance
1-250 (2, 1) 1.718 -0.759 0.283 1.160
251-500 (2, 1) 1.867 -0.897 0.398 1.757
501-750 (2, 1) 1.941 -0.945 0.519 9.374
751-1000 (2, l) 1.886 -0.901 0.472 9.136
1001-1500 (2, 1) 1.908 -0.921 0.470 4.043
1501-1750 (2, 1) 1.933 -0.952 0.749 6.330
1751-2000 (2, 1) 1.904 -0.920 0.531 1.826
2001-2250 (3, 2) 1.911 -0.993 0.066 0.734 -0.306 2.300
2251-2500 (2, 1) 1.930 -0.947 0.710 1.454
2501-3000 (2, 1) 1.908 -0.916 0.578 0.826
3001-3500 (2, 1) 1.972 -0.975 0.864 0.732
3501-3750 (2, 1) 1.962 -0.965 0.864 0.853
3751-4000 (2, 1) 1.954 -0.959 0.858 0.437

9. Analysis of ground motions using ARMA models
TABLE 8, LOCALLY STATIONARY ARMA MODELS RESULTING FROM EXACT MAXIMUM
LIKELIHOOD METHOD (E-W COMPONENT)
Model Innovation
Interval order AR1 AR2 AR3 MAI MA2 variance
1-250 (3, 2) 2.104 -0.504 0.345 0.805 -0.430 1.014
251-500 (2, 1) 1.820 -0.826 0.617 2.147
501-1000 (2, 1) 1.869 -0.883 0.184 12.693
1001-1250 (3, 2) 2.781 -2.637 0.853 1.067 -0.209 8.567
1251-1500 (3, 2) 2.697 -2.459 0.756 1.084 -0.300 5.844
1501-1750 (2, 1) 1.869 -0.900 0.227 3.578
1751-2000 (3, 2) 2.656 -2.398 0.737 1.193 -0.482 1.923
2001-2750 (3, 2) 2.155 - 1.480 0.307 0.658 -0.284 1.179
2751-4000 (3, 2) 1.990 -1.156 0.157 0.737 -0.284 0.812
729
TABLE 9. LOCALLYSTATIONARYARMA MODELSRESULTINGFROMEXACTMAXIMUMLIKELIHOODMETHOD
(VERTICALCOMPONENT)
Model Innovation
Interval order AR1 AR2 AR3 AR4 MA1 MA2 MA3 variance
1-250 (2, 1) 1.535 -0.703 -0.134 3.249
251-500 (2, 1) 1.735 -0.820 0.379 5.452
501-750 (2, 1) 1.716 -0.811 0.392 7.983
751-1000 (3, 2) 2.399 -2.122 0.694 0.736 -0.287 13.636
1001-1250 (2, 1) 1.746 -0.834 -0.044 4.692
1251-1500 (3, 2) 2.532 -2.292 0.742 0.993 -0.377 4.350
1501-1750 (3, 2) 0.803 0.729 -0.683 -0.676 0.145 1.476
1751-2000 (2, 1) 1.875 -0.907 0.500 1.259
2001-2500 (3, 2) 2.139 - 1.621 0.444 0.919 -0.511 1.183
2501-2750 (2, 1) 1.875 -0.899 0.676 0.749
2751-3000 (4, 3) 1.258 -0.315 0.091 -0.116 -0.108 -0.215 -0.177 0.808
3001-3250 (3, 2) 1.158 0.292 -0.530 -0.019 0.118 0.984
3251-3500 (2, 1) 1.915 -0.923 0.830 1.418
3501-3750 (2, 1) 1.916 -0.932 0.879 0.622
3751-4000 (2, I) 1.611 -0.680 0.292 1.131
Cumulative energy
The energy contained in the real discrete
waveform y, during interval 1 ~<n ~<N is defined,
in general, by:
~(n) = ~ y~ (7)
rPl=l
and the normalized cumulative energy function
is given by:
e(n) = E(n)/E(N) (8)
in which N designates the entire duration of
motion in sampling intervals. This function is
related to the amplitude variation of seismic
motion. The evolution of this function, for
original and simulated signals, is represented in
Fig. 5 and reflects the amplitude properties of
original and simulated seismic motions. The
general tendency of normalized cumulative
energy is the same for both time series.
Root Mean Square acceleration
The r.m.s, acceleration is defined as follows;
r 1 n -11/2
Y(n)=In ~__y~j (9)
for 1 ~<n ~<N.
This measure represents an index of the strong
ground motion severity. The evolution of this
measure for original and simulated accelero-
grams is presented in Fig. 6. This representation
points out the fact that the model captures the
essential transient character of the seismic
motion.
Short-time energy
For nonstationary signals such as accelero-
grams, it is often more appropriate to consider a
time-varying energy calculation such as the
following:
M--1
E(n) = ~_~ [w,~y,_m] 2 (10)
rrl = O
where w,, is a weighting sequence of window
which selects a segment of Yn and M is the
number of samples in the window (M = 250).
This function shows the time-varying ampli-
tude properties of the seismic signal. Figure 7
shows the energy function for original and
simulated seismic signals for a Hamming
window. It is easy to see the similarity between
the analyzed signals.

10. 730 TH. D. POPESCU and S. DEMETRIU
400
'¢~ 300
E 200
100
g o
P -loo
-200
-300
-400
0
(a)
V
__ iI I I I I I I I I i l I 1 I I
7.5 50 75 10.0 125 15.0 175 :~0.0 225 25.0 L:r?',5 30.0 325 35.0 37.5 40.0
Time ( s )
4O0
300
100
.f, o
-200
-300
-4 O0
0
{b)
i I 1
2.5 5.0 75
I I I f I I I | I l l i
10.0 12.5 ls.o 17.5 20.0 22.5 250 27.5 30.0 325 35.0 37.5 40.0
Time (s)
(c)
400
v~ 3OO
E 200u
100
.g o
~ -~0o
-2oo
-300<
-400
0 2~.s s'.o 71s ~10.0 125
I L , i i I i I I i i
15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5 35.0 37.5 40.0
Time is)
FIG. 4. Evolution of simulated seismic signal. (a) N-S component; (b) E-W component; (c) vertical
component.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
(a)
*--J I I
2,5 50 7.5 10.0
I t ! I I I I I I I I
12.5 15.o 17.s 20.0 22.5 2s.o 27.s 30.0 32.5 30.0 37.s 40.0
Time (s)
FIG. 5. Normalized cumulative energy function. (a) N-S component; (b) E-W component; (c) vertical
component. . real signal; .... simulated signal.

11. Analysis of ground motions using ARMA models 731
(b)
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
100
gO
80
60
E
u 50
~ 4o
2O
10
70
~...:°
I I I I I I I I I I I I I I I
2.5 5.0 r.5 I00 12.~ I~.0 I?.5 ZO.O ~.5 2~.0 :~r.5 3O.O 3Z~ 3~,.0 ~.~ 40.0
Time (S)
(b)
0.4
0.3
0.2
0.1
1.0
0.9
0.8
0.7
0.6
0.5
I I I I I I I I , I,
2.5 5.0
"(C )
__ i ~ .J*';"~ I I I I I
2.5 5.0 7.5 10.0
7.5 10.0 Iz,5 1'~.0 17.5 20.0 22.5 zs.0 27,5 30.0 325 35.0 37,5 40o
Time (s)
r
I I I I I I I I
12.5 150 I"r.5 20.0 225 25.0 27.5 30.0 32.5 35.0 37.5 40.0
Time (s)
FIG. 5. (continued)
(a)
63
56
49e,
42
5
~ 2a
(1:
14
| I I I I I I I I I I I I I I
2,5 5,0 7,5 10,0 lZ-~ 15.0 11",5 20.0 22,5 25,0 2?,5 30,0 32.5 35,0 37,5 40O
Time ( s )
Fro. 6. Cumulative r.m.s, acceleration. (a) N-S component; (b) E-W component; (c) vertical component;
real signal; .... simulated signal.

12. 732 TH. D. POPESCUand S. DEMETRIU
40
(c)
36
32
28
7
ZO "''"'"
U 12
8
4
I I I I I I l I I I I
0 2,5 5,0 7`.5 10.0 12.5 15.0 17`.5 20.0 22.5 25.0 L:~.3 30.0 32,5 35.0 37`.5 40.0
Time (s)
FIG. 6. (continued)
(al
1.0
0.8
0.7`
0.6
O.m, i
O, 4 ~"~
03
0.2
0.1
--- J I I I '- ---~i-'~
O 2.5 5.0 7.5 10.0 12,5 15.0 17`,5 20.0 22.5 25.0 z7.5 3o.0 32.5 35.0
Time (s)
(b)
1.o
0.9
o.8
0.7`
0.6
0,5
o,4
0.3
02
0.1
G
:~: - ..,, : ",
j"j ~ ....2.5 5.0 7`.5
(CI
lo.o 12.5 15.0 17`.~ 2o.o 22.5 2s.o 27`.5 30.o 32.s 35.0
Time (s)
1.'0" ..;
o, i0.8
0.6 ::,.
: 5
0,0.34 .,..,,..,,
0.2 ~j .,..,
O.I ~,,....... . .,,".........
I I I I I ..... " -~"'; -'-.~F'"'~. "~" .~"T~' -"t"- "''
0 2,5 S.O 7".5 10.0 12.5 15.0 17,5 200 225 25.0 27.5 30.0 32.5 35,0
Time {s)
FIG. 7. Normalized short-time energy. (a) N-$ component; (b) E-W component; (c) vertical component;
real signal; .... simulated signal.

13. Analysis of ground motions using ARMA models 733
(a)
--~ "--_____--~-
1.0 [-x
o.ol -_
oo - o.'--~ 0.5
Lag (s)
..,..
--_,___"-,...
"------__------
(b) -, ~ ~-V/"'--'--
o Oo.o, _i; c o ,
Log (s)
-'%,
-., --,---.--._
( C ) " "~--'---~"
~020[I" ...." ~ "
O0 ~ ,~
O0 0.2 0.5
Log (s)
FIG. 8. Short-time autocorrelation function for recorded seismic signal. (a) N-S component; (b) E-W
component; (c) vertical component.
Short-time autocorrelation analysis
From the viewpoint of a stochastic repre-
sentation of the ground motion data, the
autocorrelation function and the power spectral
density provide statistical characterizations of
the time series to be analyzed. The two functions
are closely related to the second-order moments
of a random process and are sufficient to provide
a complete statistical description for a local
Gaussian process like a segment of a motion
earthquake accelerogram.
The conservation of stochastic properties of
original and simulated seismic signal resulted,
also, from the representation of the evolutionary
normalized correlation functions in Figs 8 and 9,
respectively.
This function is computed for each stationary
data block by:
where
ryy(k) = Ryy(k)/Ryy(O)
1 N-k
Ry,(k) = ~[ ~ [y, - )T][y,÷k -- )71
1 N
k=0,1 ..... K and )7 -- T, ~'~ y~.
IV i=l
(11)

14. 734 TH. D. POPESCUand S. DEMETRIU
oo~ " - - - " ~
0.0 0.2 0.*~
Loci ( s )
(b)
2.o[ ~
1.o ~'K ~"--'~-""
0.5
o.o =._.~--~'~--'-
o.o o.1 o.z 0.3 0.4 0.5
Lag (s)
-.-._.
°
--~ __-----_.~.
co,
2.0
1.0 k
0.0 --' ~ - - ,
0.0 0.2 0.5
Laq (s)
F=G. 9. Short-time autocorrclation function for simulated seismic signal. (a) N-S component; (b) E-W
component; (c) vertical component.
N is the number of samples for a stationary data
block.
Short-time spectrum analysis
Short-time spectrum analysis has traditionally
been one of the most important seismic signals
processing techniques. The fundamental as-
sumption underlying any short-time analysis
method is that over a long-time interval, the
signal is nonstationary, but that over a
sufficiently short-time interval it can be con-
sidered stationary.
The evaluation of power spectral density
function for the original and simulated signals
was performed by the relation:
I (-i2~'fk) 2 21- ~ B k exp
k=l - - ; 5 cr (12)
s(f) = 1 k=l~Akexp(-i2~fk)
(-1/2 ~<f ~<1/2)
where Ak, B k represent ARMA parameters and
o2 is innovation variance.
The results are represented in Figs 10 and 11.
Frequency peaks of spectra appear in a similar
mode for both spectral representations, accord-
ing to the evolution of the original accelerogram.
Starting from the results presented in this case

15. Analysis of ground motions using ARMA models 735
8
7
-.,6
"-'5
n 4
/ 2
8
7
6
~4
~3
o
/ 2
1
0
(a)
L--f--f
5 I0 15 20
Frequency (Hz)
(b)
A "~'--~---'~
O 5 10 15 20
Frequency (Hz)
(c)
0 5 10 15 20
Frequency (Hz)
FIG. 10. Short-time spectrum for recorded components of seismic signal. (a) N-S component; (b) E-W
component; (c) vertical component.
o.
(a)
~ 10 15 20
Frequency (Hz)
(b)
, , ~-__~ , , ~...~------
O 5 10 15 20
Frequency CHz]
FIO. 11. Short-time spectrum for simulated components of seismic signal. (a) N-S component; (b) E-W
component; (c) vertical component.

16. 736 TH. D. POPESCUand S. DEMETRIU
O
7
C6
5
O_ 4
o3
~2
_J
1
(C) ~
... ~. -I",, "-,... x... --~--_.._~----_...__ ~--~.-:.---~-
-~.. J . ~--...~_ .... ~ ~'~..,..~ --'--,--~
-...
5 10 15 20
Frequency (Hz
FIG. 11. (continued)
study we can consider that the original and
simulated signals are similar as concerns
evolution in time and frequency, and are
characterized by the same statistical properties
of stationarity data blocks; the nonstationarity
character of the original seismic signal is also
conserved.
It may be emphasized that a complete
agreement between the real and artificial
accelerograms can never be expected since some
approximations have been introduced while
building the model and evaluating the
parameters.
5. CONCLUSIONS
The analysis and simulation method presented
in the paper seems to be a successful approach.
Nonstationarity analysis technique and charac-
terization of quasi-stationary data blocks of
strong ground motions through parametric
ARMA models provide an efficient and flexible
description of the observed motion by a small
number of parameters. The results presented in
the case study, as well as other results obtained
for several strong earthquake ground motions
analyzed, are promising and could have sig-
nificant utility in the design of engineering
structures.
Potentially there are also other applications of
the described approach to problems in aerody-
namics, meteorological, oceanographic, wind,
vibration and econometric data for nonstationary
processes with both amplitude and frequency
contents varying in time.
REFERENCES
Akaike, H. (1974). Markovian representation of stochastic
processes and its application to the analysis of autoregres-
sive moving average processes. Ann. Inst. Statist. Math.
26, 363-387.
Akaike, H. (1976). Canonical correlation analysis of time
series and the use of an information criterion. In Lainiotis,
D. G. and R. K. Mehra (Eds), System Identification:
Advances and Case Studies. Academic Press, New York.
Akaike, H. (1977). On entropy maximization principle.
Proc. Syrup. Applic. Statist., Wright State University,
Dayton, OH. June 1976.
Akaike, H. (1978a). Time series analysis and control through
parametric models. In Findley, D. F. (Ed.), Applied time
Series Analysis. Academic Press, New York.
Akaike, H. (1978b). A Bayesian extension of the minimum
AIC procedure of autoregressive model fitting. Research
Memo. No. 126, The Institute of Statistical Mathematics,
Tokyo.
Akaike, H. (1978c). Likelihood of a model. Research Memo.
No. 127, The Institute of Statistical Mathematics, Tokyo.
Bartlett, M. S. (1946). On the theoretical specification and
sampling properties of autocorrelated time series. J. R.
Statist. Soc., Series B, Vol. 27.
Cakmak, A. S. and R. I. Sheriff (1984). Parametric time
series models for earthquake strong ground motions and
their relationship to site parameters. Proc. 8th World
Conf. on Earthquake Engng, 21-28 July, San Francisco,
CA.
Chang, M. K., J. W. Kwiatkowski, R. F. Nau, R. M. Oliver
and K. S. Pister (1979). ARMA models for earthquake
ground motions. Report ORC 79-1, Operations Research
Center, University of California, Berkeley.
Demetriu, S. (1986). EARTS---Program package for analysis
and modelling of earthquake ground motions. Internal
Report, Faculty of Civil Engineering, Bucharest,
Romania.
Gersh, W. (1974). On the achievable accuracy of structural
system parameter estimates. J. Sound Vibrat., 34, 63-79.
Gersh, W. and G. Kitagawa (1983). A time varying
multivariate autoregressive modelling of econometric time
series. Technical Paper 9, Statistical Research Division,
Bureau of the Census, U.S. Department of Commerce.
Gersh, W. and G. A. Kitagawa (1985). Time varying AR
coefficient model for modelling and simulating earthquake
ground motion. Earthquake Engng Structural Dyn., 13,
243-254.
Gersh, W. and S. Luo (1970). Discrete time synthesis of
randomly excited structural response. J. Acoust. Soc. Am.,
51, 402-408.
Gersh, W., N. N. Nielsen and H. Akaike (1973). Maximum
likelihood estimation of structural parameters from
random vibration data. J. Sound Vibrat., 31, 295-308.
Golub, G. E. (1969). Matric decompositions and statistical
calculations. In Milton, R. C. and J. A. Nelder Statistical
Computation. Academic Press. New York.
Jennings, P. C., G. W. Housner and N. C. Tsai (1968).
Simulated earthquake motions. Report of the Earthquake
Engineering Research Laboratory, California Institute of
Technology, Pasadena, CA.
Jones, R. H. (1975). Fitting autoregression. J. Am. Statist.
Assoc. 70, 590-592.
Jurkevics, A. and T. J. Ulrich (1978). Representing and
simulating of strong ground motions. Bull. SeL~mol. Soc.
Am., 68, 781-801.
Kitada, Y, et al. (1983). Simulated earthquake ground
motions by Gauss wave superposition method. Proc. 7th
SMiRT, Vol. K(a), 53-61, Chicago.
Kitagawa, G. and H. Akaike (1978). A procedure for the
modelling of nonstationary time series. Ann. Inst. Statist.
Math., 30, B, pp. 351-363.
Kozin, F. (1979). Estimation and modelling of non-stationary
time series. Proc. Symp. Comput. Meth. Engng,
University of Southern California, pp. 603-612.

17. Analysis of ground motions using ARMA models 737
Nau, R. F. and R. M. Oliver (1979). Adaptive filtering
revisited. I. Operat. Res. Soc., 30, 825-831.
Nau, R. F., R. M. Oliver and K. S. Pister (1980). Simulating
in analysing artificial non-stationary earthquake ground
motions. Report UCB/EERC-80/36, University of Cali-
fornia, Berkeley.
Ozaki, T. and H. Tong (1975). On the fitting of
non-stationary autoregressive models in time series
analysis. Proc. 8th Hawaii Int. Conf. Syst. Sci., Western
Periodical Company, Hollywood, CA.
Polhemus, N. W. and A. S. Cakmak (1981). Simulation of
earthquake ground motions using autoregressive moving
average (ARMA) models. Earthquake Engng Structural
Dyn. 9, 343-354.
Popcscu, Th. D. (1981) TIMES--Program package for time
series analysis and forecasting. Internal Report, Central
Institute for Management and Informatics, Bucharest,
Romania.
Popescu, Th. D. (1985). AUTOB & J--A computer-aided
procedure for time series analysis and forecasting.
Preprints 3rd IFAC Syrup. Computer Aided Design in
Control and Engng Syst., 31 July-2 August, Lyngby,
Denmark.
Shibata, R. (1976). Selection of the order of an
autoregressive model by Akaike's information criterion.
Biometrika, 63, 117-126.
Tertisco, M., P. Stoica and Th. D. Popescu (1985). Time
Series Modelling and Forecasting (in Romanian). Acade-
mic Publishing House, Bucharest.
*** Digitized Data of Strong Motion Earthquake Accelero-
grams in Romania, 4 March 1977. Building Research
Institute, Tokyo, 1977.