Time-dependent power spectral density of earthquake ground motion

Time-dependent power spectral density of earthquake ground motion F. E. Elghadamsi, B. Mohraz and C. T. Lee

School of Engineerin9 and Applied Science, Southern Methodist University, Dallas, TX 75275, USA P. Moayyad

Structural Dynamics Group, Bell Helicopter Textron, Inc., PO Box 482, Fort |lbrth, TX 76101, USA This paper investigates the stationarity of earthquake accelerograms and shows that the strong motion segments constitute a locally stationary random process. For such a process, a timedependent power spectral density may be formulated as the product of: (a) a mean-square acceleration; (b) a time scale factor describing the variation of the local mean square acceleration; and (c) a normalized power spectral density describing the frequency structure of the motion. A segmental time averaging procedure is proposed for obtaining a time scale factor, an envelope function, and a normalized power spectral density from the strong motion segment of accelerograms. The procedure is used to compute smooth normalized power spectral densities from a total of 367 horizontal and vertical components of accelerograms recorded on alluvium, rock, and alluvium underlain by rock.

INTRODUCTION It has long been recognized that earthquake accelerograms have nonstationary characteristics. However, several investigators have assumed the strong motion segment of the record to be stationary. This paper investigates the stationary characteristics of accelerograms and shows that it is more appropriate to consider the strong motion as locally stationary. In such cases, even though the auto-correlation and power spectral density functions are time-dependent, the accelerationtime history may be expressed as a uniformly modulated excitation, i.e., the product of an envelope function and a stationary component. Based on the results of the investigation, a procedure is established for computing the envelope function of the strong motion segment and the normalized power spectral density of an ensemble of acclerograms. BACKGROUND Assuming that the strong motion segment '~g(t) of a record is locally stationary, the auto-correlation function R~ (t, z) and the power spectral density G~,(t,f ) can be expressed (see Bendat and Piersol 1) as

R:~,(t,z) = S(t)R~,(~) (1) G~,(t, f ) = S(t)G~ (f) (2) where R;(z) and G%(f) are the auto-correlation and the power sl~ectral density functions of a stationary process {:~9(t)} and S(t) is a slowly varying time scale factor. Accepted February 1987. Discussion closes March 1988.

9 1988 Computational Mechanics Publications

Equation (2) can be rewritten in terms of a normalized power spectral density function G~:)(f) (area = 1.0) of the stationary process as

G.~,(t,f ) =-~.~S(t)G~ ) (f)

(3)

where ~,~ is the mean square value of the stationary process {zo(t)}. A zero time lag 9 in equation (1) gives the local mean square ~,z (t) in terms of the mean square of the stationary process as

~, (t)=S(t)72.:,

(4)

The above equation implies that the scale factor S(t) can be obtained if the local mean square ~p~,(t) 2 and the mean square of the stationary process if2 are known. The strong motion segment c a n be expressed as a uniformly modulated excitation in terms of an envelope function e(t) and a stationary component ~g(t); thus, _

_

:'r ) = e(t)~O(t)

(5)

In the above equation, e(t) is a slowly varying time function which can be assumed constant over short time intervals. Therefore, squaring both sides of equation (5) and averaging over short time intervals gives 2 O:~, ( t )_- e 2 (t)~%

(6)

Comparing equations (4) and (6), one obtains the envelope function as .,.G = (~:~, (t)/~_)

(7)

It should be noted that the envelope function is needed for

Soil Dynamics and Earthquake Engineering, 1988, Vol. 7, No. 1 15

Time-dependent power spectral density: F. E. Elghadanlsi et al. computing the mean square response of a system and for generating artificial accelerograms. EXTRACTING THE STRONG MOTION Since the duration of an observation affects the computation of both the power spectral density and the mean square value, it is necessary to select a consistent duration of strong motion for all records. Several investigators have proposed procedures for extracting a strong motion segment from an accelerogram. Bolt ~ proposed the 'bracketed duration' which is the time interval between the first and the last acceleration peaks greater than a specified value. Trifunac and Brady ~ defined the duration of the strong motion as the time interval in which a significant contribution to the integral of the square of acceleration ([ a 2 dr) takes place. They selected the time interval between the five percent and the ninety-five percent contributions as the duration of strong m6tion. A third procedure suggested by McCann and Shah 5 is based on the average energy arrival rate. The duration is obtained by examining the cumulative rootmean square acceleration (RMS) of the accelerogram. A search is performed on the rate of change of the cumulative RMS to determine the two cut-off times. The final cnt-offtime T2 is obtained when the rate ofchange of the cumulative RMS acceleration becomes negative and remains so for the remainder of the record. The initial time T~ is obtained in the same manner except that the search is performed starting from the 'tail-end' of the record. An examination of these procedures shows that

they result in different durations. This is to be expected since the procedures are based on different criteria. Therefore, no 'standard' procedure can be defined for extracting a strong motion segment and the selection of a procedure for certain study depends on the purpose for which the study is intended. McCann and Shah's procedure was selected in this study since it is based on the cumulative root-mean square acceleration which is closely related to both the autocorrelation and the power spectral density functions. It was, however, determined that introducing a minor modification to the procedure would result in strong motion segments with improved stationary characteristics 7. The modification consists of selecting the cutoff points at times when the rate of change of the cumulative RMS acceleration approaches 1.0cm/secZ/sec and remains less than this value for the remainder of the record. Fig. 1 shows comparison between the strong motion durations extracted using the original and the modified McCann and Shah procedures for the S00E component of El Centro, 1940, and the $69E component of Taft, 1952. Table 1 gives the cut-off times, durations, and RMS accelerations computed from the two procedures for eight accelerograms. The figure and the table show that while for some accelerograms the two procedures result in close durations, for others the difference is significant. It is noted that the modification results in strong motion durations which are on the average twenty-three percent shorter than those computed by the original McCann and Shah's procedure but with a reduced RMS value of only seven percent.

176

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Modified McCann& Shah

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0

:,

,~

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Time,

;5. ,'o ,,'s 5'0

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(o) El Centro, 1940 - SOOE

Fig. 1.

~, ,~ ,~, 2'o 2'5 3'o 3'5 ,`O

Time.s e c

45

5'0

(b) Toil, 1952 - $69E

Comparison of strong motion durations using tlre original aml modified McCann and Shah's procedures

16 Soil Dynamics aml Earthquake Engineering, 1988, Vol. 7, No. 1

Time-dependent power spectral density: F. E. Elghadamsi et al. Table 1.

Comparison o f initial aml final thnes, duration aml root mean square value for eight records Record

Comp.

El Centro, 1940

S00E

S90W

Taft, 1952

N21E

$69E

El Centro, 1934

S00W

S90W

Olympia; 1949

N04W

N86E

T1 (sec)

T, (sec)

AT (sec)

RMS (cm/sec2)

0.00 0.88 1.38 0.00 0.80 1.32 0.00 2.14 3.46 0.00 2.34 3.18 0.00 1.92 1.96 0.00 1.62 2.00 0.00 0.08 1.06 0.00 0.28 4.34

53.74 26.32 26.30 53.46 26.62 26.42 54.34 36.46 20.66 54.38 35.30 17.34 90.28 23.88 14.98 90.22 20.10 17.78 89.06 23.02 20.18 89.02 21.80 20.46

53.74 25.44 24.92 53.46 25.82 24.92 54.34 34.32 17.20 54.38 32.96 14.16 90.28 21.96 13.02 90.22 18.48 15.78 89.06 22.94 19.12 89.02 21.52 916,12

46.01 65.60 65.88 38.85 54.73 55.14 25.03 30.85 40.19 26.10 32.71 46.20 19.48 38.38 46.83 20.76 44.26 46.80 22.98 43.73 46.59 28.10 55.48 61.50

1 - Entire record 2 - McCann and Shah 3 - Modified McCann and Shah

Table 2.

Results o f the run test on entire accelerogram Test interval I sec Record

El Centro, 1940 Taft, 1952 El Centro, 1934 " Olympia, 1949

S00E S90W N21E $69E S00W S90W N40W N86E

1.5 sec

m

n

r

rI

ru

m

n

r

rI

ru

m

n

r

rI

ru

38 37 41 41 71 73 69 69

14 15 13 13 19 17 19 19

9 I1 5 3 5 5 7 5

11 12 10 10 13 13 13 13

23 24 22 22 26 25 26 26

24 22 27 27 48 47 44 44

10 12 9 9 12 13 14 14

7 8 5 3 5 3 4 4

9 I0 8 8 7 10 11 11

19 21 17 17 18 22 23 23

19 17 19 19 35 35 33 34

7 9 7 7 9 9 11 10

6 8 3 3 5 3 4 4

6 7 6 6 8 8 9 9

15 17 15 15 17 17 20 19

E X A M I N I N G THE S T A T I O N A R I T Y OF ACCELEROGRAMS

3.

A number of statistical tests can be used to examine the stationary characteristics of a record. The run test (see Bendat and Piersol t) which is indepdent of the statistical distribution was used in this study. Although one of several statistical averages may be selected as the testing parameter, the mean square value which is related to the auto-correlation and power spectral density functions was used in this study. The following summarizes the test:

4.

1. 2.

The accelerogem or the extracted strong motion segment is divided into N equal time intervals. One, one-and-a-half, and two second intervals w___ereused. For each interval, the local mean squares ~,2, if2, ~,2, . . . . ~,2 are computed, and used to obtain the overall mean square

"

2 see

v r

,~I N

(8)

Each ~2 is compared to $2. A ' + ' or a ' - ' is assigned depending on whether ~,2 is greater or less than $-T. The number of ' + ' s ' or ' - ' s ' is denoted by m and n, respectively. The number of runs (clusters of pluses or minuses) is denoted by r. For a level ofsignificance ct (considered as 0.05 in this study), the region of acceptance is given as rm.n;(1 -=12) "~ r < rm,n;=l2

(9)

The region of acceptance can be obtained from texts on statistics (for example9). If the number of runs r lies in the acceptance region between the lower limit r~ and the upper limit r,, the record is considered stationary. The above procedure was used to test the stationarity of the entire accelerogram and the strong motion segment for eight records. The results are shown in Tables 2 and 3. The test confirms that while the entire accelerogram is nonstationary, the strong motion segment may be considered stationary. It is noted, however, that for the

Soil Dytlamics and Earthquake Engineering, 1988, Vol. 7, No. 1 17

Time-depemlent power spectral density: F. E. Elghadamsi et al. Table 3.

Results of the

run

test

on

strong motion segment

Test interval 1sec Record El Centro, 1940 Taft, 1952 El Centro, 1934 Olympia, 1949

S00E S90W N21E $69E S00W S90W N40W N86E

1.5 sec

2 sec

m

n

r

r~

ru

m

n

r

r!

r.

m

n

r

rI

r.

6 7 7 7 4 6 7 6

18 17 I0 7 9 9 12 10

6 7 6 2 5 7 5 8

5 6 5 3 3 3 5 4

13 15 13 12 9 10 13 12

5 5 5 4 3 5 4 7

11 11 6 5 5 5 8 3

4 5 6 2 4 6 5 4

4 4 3 2 2 3 2

11 11 9 8 7 9 9 7

3 5 5 3 2 5 4 3

9 7 3 4 4 2 5 5

4 7 4 2 4 5 5 4

2 3 2 -

7 10 7 7 5 5 8 7

majority of cases, the number of runs r for the strong motion segment tends to approach the lower limit of the region rt indicating that the stationary characteristics are in general very weak. Therefore, it was decided to examine whether the strong motion segment can be considered as a uniformly modulated excitation.

$oo.

}

~ STRCHGMOT/ON -500

32|,.

S T R O N G M O T I O N AS A M O D U L A T E D EXCITATION If the strong motion segment is expressed as a product of an envelope function and a stationary component (equation (5)), then the run test may be used to examine the stationarity of s According to equation (7), once the local mean square ~,z (t) is computed (over short time intervals), the envelope fdnction e(t) can be obtained if the 2 of the stationary component s mean square value ~p,is known. However, equation (5) lnd~cates that to compute s one needs e(t). Therefore, an iterative procedure is required to compute e(t) and s In such a procedure, one may use the overall mean square value ~,.~ from equation (8) in the first trial for ~ in equation (7) to compute e(t). The resulting e(t) is used in equation (5) to compute s The procedure is continued until a close agreement between two successive values is reached. The above procedure was applied to the eight records in Tables 2 and 3 and it was found that after one iteration, the difference between ~.z and ~ is approximately five percenL Therefore, it was decided to substitute ~k~.2for ~,2 in equation (7). With this substitution, the following procedure was used to obtain the envelope function and the stationary component of a record: I.

2.

3.

The strong motion segment 5~g(t)is extracted from the accelerogram using the modified McCann and Shah procedure. The ordinates are squared, averaged over consecutive time intervals and smoothed (a spline interpolation was used in this study). The scale factor S(t) is obtained by dividing (normalizing) the ordinates of the smooth curve in step 2 by its mean square value. The envelope function e(t) is then obtained as the square-root of

s(t). 4.

The stationary component ~g(t) is determined by dividing the ordinates of .~g(t) by the corresponding ordinates of e(t). Figs 2 and 3 show the envelope function and the

18

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2.0 s e c a v e r a g i n g .

Fig. 2. Acceleratiott-time history f o r the SOOE component of El Centro, 1940, as a product of an envelope fimetion and a stationary component stationary component for the S00E component of El Centro, 1940and the $69E component of Taft, 1952. Both a 1.0 sec and a 2.0 sec segmental averaging were used to obtain the envelope function. It is apparent from the figures that while the envelope function from a 2.0sec segmental averaging is flatter than that from a 1.0sec, the corresponding stationary components exhibit similar characteristics. To examine the stationary characteristics of the modulated excitation, the run test was applied to 2g(t). The results are shown in Tables 4 and 5. Comparisons of these tables with Table 3 indicate that the value of r for ~o(t) are closer to the upper limit of the acceptance region !", than the corresponding values of r for ~o(t). This implies that ~g(t) exhibits stronger stationary characteristics than Jg(t) and suggests that the locally stationary assumption is more appropriate with the modulated excitations. It should be noted t.hat the results of the run test for Yg(t)

Soil Dynanffcs and Earthquake Engineering, 1988, Vol. 7, No. 1

T i m e - d e p e m l e n t p o w e r s p e c t r a l d e n s i t y : F . E. E l y h a d a m s i et al.

o b t a i n e d from a segmental averaging over intervals greater than 2.0sec (not shown here) indicate that t h e characteristics of s a p p r o a c h those of 2o(t). It was concluded that, in general, a 2.0 sec segmental averaging gives the best results. It should be noted that the envelope

functions such as those in Figs 2 a n d 3 have a m e a n square value of one. NORMALIZED POWER SPECTRAL DENSITY The n o r m a l i z e d power spectral density G~">(f) for an ensemble of accelerograms was o b t a i n e d by estlmatlng the power spectral densities for i n d i v i d u a l accelerograms a n d averaging them at various frequencies. Since the strong m o t i o n segments extracted from different accelerograms do not have the same length, they were padded with zeros to o b t a i n records with equal d u r a t i o n s . The equal d u r a t i o n s are needed so that the spectral ordinates for different accelerograms c a n be estimated at identical frequencies. T h e ensemble power spectral density was then o b t a i n e d using a weighted averaging based o n the d u r a t i o n of the strong m o t i o n segment to a c c o u n t for the u n e q u a l d u r a t i o n s (see M o a y y a d a n d M o h r a z 7 for the details of c o m p u t a t i o n ) . T h e n o r m a l i z e d power spectral densities for the ensemble of the e i g h t records c o m p u t e d from the strong m o t i o n segment directly a n d from the s t a t i o n a r y c o m p o n e n t are s h o w n in Fig. 4. The figure indicates that the shape of the n o r m a l i z e d power spectral density is not affected by the d e c o m p o s i t i o n , a n d that one m a y c o m p u t e the n o r m a l i z e d power spectral density directly from the extracted strong m o t i o n segment. Fig. 5 shows the n o r m a l i z e d power spectral density for the horizontal c o m p o n e n t s of accelerograms recorded on deep alluvium. N o r m a l i z e d power spectral densities were also c o m p u t e d for the vertical c o m p o n e n t s of these accelerograms as well as for the h o r i z o n t a l a n d vertical c o m p o n e n t s of two other ensembles of accelerograms recorded o n rock a n d o n a l l u v i u m layers of a p p r o x i m a t e l y 30-200 feet thick u n d e r l a i n by rock. These three categories which will be referred to, in further 9

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Fig. 3. A c c e l e r a t i o n - t h n e history for t h e $ 6 9 E c o m p o n e n t o f Taft, 1952, a s a p r o d u c t o f a n e n v e l o p e f i m c t i o n attd a s t a t i o n a r y c o m p o n e n t Table 4.

~

.

~

Resuhs of the run test on 2g(t) obtained from 1 second segmental at'eragin 9

Test interval I sec Record El Centro, 1940 Taft, I952 El Centro, 1934 Olympia, 1949

Table 5.

S00E S90W N21E $69E S00W S90W N40W N86E

1.5 sec

2 sec

m

n

r

rI

ru

m

n

r

rt

ru

11 16 7 6 6 6 6 6

13 8 10 8 7 9 14 10

12 10 11 11 10 9 12 10

7 6 5 3 3 4 5 4

18 16 13 I1 11 12 13 12

7 9 6 4 4 5 5 5

9 7 5 5 4 5 7 5

11 12 8 7 6 8 9 7

4 4 3 2 2 3 2

13 13 9 8 8 .9 10 9

m

n

r

rI

r,

6 9 3 4 2 5 3 3 .

6 3 5 3 4 2 6 5

4 4 6 4 3 3 5 6

3 2 2 -

10 7 7 7 5. 5 7 7

Resuhs of the nm test on ~g(t) obtained from 2 second segmental averaging

Test interval I sec Record El Centro, 1940 Taft, 1952 El Centro, 1934 Olympia, 1949

S00E S90W N12E $69E S00W S90W N40W N86E

1.5 sec

2 sec

m

n

r

rt

r.

m

n

r

rl

ru

m

n

r

rl

r.

12 15 9 6 7 7 11 8

12 9 8 8 6 8 8 8

16 15 12 10

7 7 5 3

18 17 13 11

7 7 5 3

9 9 6 6

12 I1 7 7

4 4 3 2

13 13 9 7

6 5 3 3

6 7 5 4

10 8 6 4

3 3 -

10 10 7 7

8

3

11

4

4

4

-

8

3

3

6

-

11 12 13

4 5 4

12 14 13

4 8 5

6 4 5

7 7 -

2 3 9

8 9 2

4 4 9

3 5 4

6 5 4

2 8

-

S o i l D y n a m i c s a n d E a r t h q u a k e E n g i n e e r i n g , 1988, Vol. 7, N o . 1

6

7 8 -

19

73me-dependent power spectral density: F. E. Elghadamsi et al. discussion, as alluviuni, rock, and alluvium/rock are similar to those u s e d in a previous study 6 with the exception that a three- instead of a four-fold classification is used in this study. In addition, this study includes many more accelerograms and as in the previous study, the accelerograms were selected from those recorded in the free-field and on the basement or first floor of buildings. For a detailed description of the recording stations and the selection of accelerograms, one should refer to Moayyad and Mohraz 7. For practical applications, it is desirable to present the power spectral density by an analytical expression which can reflect the characteristics of the ground motion on different sites. Consequently, the computed normalized power spectral densities were smoothed using the

-~

0.3

"6

~

Frm origlnal strong mol,on. --" From stationary c~ponent (t ser averaging).

0.2-

I

~

"

-

r m stat " ionar y comDohent (z sec. av9raging) o

e.

~ 0.1-

frequency. Hz

Fig. 4. Comparison of power spectral density for the ensemble o f eight accelerograms 0.~"

0.3-

analytical expression proposed by Kanai 3 and Tajimi t~ in the form I + 4~2(f/fo) 2 G~> ( f ) = [(1 - (fifo)z] 2 + 4r

2 Go

(10)

where fo and s represent ground frequency and damping, respectively, and G Ois the spectral intensity. A procedure similar to that used by Lai 4 based on Vanmarcke's method of spectral moments 12, was used to compute the parameters. Table 6 gives the values of fg, ~_gand G O for different ensembles. Also given are the three spectral moments 2 o, 2 a, and 2z, the central frequency fc, and the shape factor ,5. The table indicates that the harder the site, the greater the ground frequency fg- For the vertical components, the ground frequencies are greater than the corresponding values for the horizontal. It should be noted that the ground damping ~g remains the same for both the horizontal and for the vertical components and does not change with the firmness of the site. The table also shows that G O for horizontal components is greater than that for vertical and that Go increases as the site becomes less stiff. This indicates that the soil condition affects the response of a system through G Oand fg and not ~g. A comparison of fg, ~g, and Go, computed from this study with the corresponding values reported by Lai 4 is given in Table 7. The table indicates that the values from the two studies are in close agreement. Figs 6 and 7 present the normalized smooth power spectral densities for horizontal and vertical components of accelerograms recorded on alluvium, alluvium/rock, and rock. The figures show that as the site gets stiffer, the predominant frequency increases and the power spectral densities spread over a wider frequency range. The figures also indicate that the normalized power spectral densities for horizontal components have a sharper peak and span over a narrower frequency region than the normalized power spectral densities for vertical components. These observations are consistent with the studies on response spectra (see for example Seed et al. 8 and Mohraz6).

] =; E

Table 7. Comparison of Kanai-Tajimi parameters computed in tiffs study and those reported by Lai4

0.1"

0.0 frequency. HZ

Fig. 5. Raw power spectral density for the ensemble of 161 horizontal components of accelerograms on alhwium

Study

Site

Lai4

Alluvium Rock Alluvium Rock

This study

No. of Rec.

fg

~g

GO

118 22 161 26

3.04 4.25 2.92 4.30

0.32 0.32 0.34 0.34

0.095 0.068 0.102 0.070

Table 6. Stmzmaryof Kanai-Tajhni parameters computed from the normalized power spectral density of the ensemble of accelerograms Site category Horizontal Alluvium Alluvium/rock Rock Vertical Alluvium Alluvium/rock Rock

20

No. of records

20

21

22

f,

6

fg

~.g

Go

161 60 26

1.00 1.00 1.00

3.12 3.69 4.39

16.86 20.99 29.46

4.10 4.58 5.41

0.65 0.59 0.59

2.92 3.64 4.30

0.34 0.30 0.34

0.102 0.078 0.070

78 29 13

1.00 1.00 1.00

4.86 5.23 6.28

39.30 44.62 56.74

6.27 6.68 7.53

0.63 0.62 0.55

4.17 4.63 6.18

0.46 0.46 0.46

0.080 0.072 0.053

Soil Dynamics attd Earthquake Engineering, 1988, Vol. 7, No. 1

Time-depemlent power spectral density: F. E. Elghadamsi et al. c o m p u t e d a n d given for h o r i z o n t a l a n d vertical ensembles of a c c e l e r o g r a m s r e c o r d e d on alluvium, a l l u v i u m / r o c k , a n d rock. C o m p a r i s o n o f these p a r a m e t e r s with those r e p o r t e d b y Lai ~1 indicates close agreement.

O3O

~

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r; ~4 o2o "o

ACKNOWLEDGEMENT T h e s t u d y r e p o r t e d herein was S u p p o r t e d by the N a t i o n a l Science F o u n d a t i o n G r a n t s P F R 8004824 a n d C E E 8300461.

o.o5,

Frequency, HZ

Fig. 6. Normalized smooth power spectral densities horizontal components

REFERENCES 1 2

o

3O

3

Aflu,;um

c~ o=o

Rock

4 5

o.o~

6

Frequency, HZ

Fig. 7. Normalized smooth power spectral densities vertical components

7

8 CONCLUSION U s i n g the run test, the s t a t i o n a r y characteristics of the s t r o n g m o t i o n segment o f a c c e l e r o g r a m s is e x a m i n e d . It is s h o w n that it is m o r e a p p r o p r i a t e to c o n s i d e r the s t r o n g m o t i o n segment as a m o d u l a t e d excitation in the form of an e n v e l o p e function a n d a s t a t i o n a r y c o m p o n e n t . A p r o c e d u r e for c o m p u t i n g the e n v e l o p e function of the s t r o n g m o t i o n segment a n d the n o r m a l i z e d p o w e r spectral density of an ensemble of a c c e l e r o g r a m s is presented. T h e p a r a m e t e r s of n o r m a l i z e d p o w e r spectral densities in the form p r o p o s e d by K a n a i a a n d T a j i m i 1~ are

9 10

I1 12

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