Bulletin of the Seismological Society of America. Vol. 61, No. 2, pp. 343-356. April, 1971

RESPONSE ENVELOPE SPECTRUM AND INTERPRETATION OF STRONG

EARTHQUAKE Gt~OUND ~/[OTION

BY M. D. TRIFUNAC*

A B S T R A C T

A multiple filter technique is developed using a single-degree-of-freedom lightly-damped oscillator as a narrow band-pass filter. It provides for a physi- cally simple approach to the analysis of accelerograph records from the point of view of structural engineering.

The analysis of several typical accelerograph records indicates that a signifi- cant portion of strong earthquake ground motion consists of surface waves. It is concluded that the duration of intense shaking will be determined predominantly by the dispersion properties of the ground.

INTRODUCTION

An understanding of the phenomena associated with strong earthquake motion is important for many areas of research and application. This motion of destructive in- tensity endangers the safety of all man-made structures and has, on several occasions, led to complete or partial devastation of whole cities. The recording and interpreting of strong-motion data is also essential for various studies of earthquake mechanism, for high resolution of the patterns of earthquake energy release and, in general, for better understanding of the close field of earthquake ground motion.

The basic information related to this phenomenon comes from strong-motion ac- celerographs. The first instruments of this kind were installed in the field some 40 years ago, and the first strong-motion record was obtained during the 1933 Long Beach, California, earthquake. Since that time, several hundred earthquakes have been re- corded, predominantly in California and Japan.

Although the history of modern recording of teleseismic earthquake waves is not much older than the measurements of strong, near-field, ground motion, the nature of teleseismic waves and the great number of recorded shocks resulted in early interpre- tations of the different wave forms displayed in those records. This was not the case with the strong-motion records.

The inhomogeneities and discontinuities in the Earth's crust lead to a wide variety of wave velocities. For distant recordings different frequency-wave components are dispersed and cause natural separation of various wave types. This separation, which helps in the record interpretation, is further enhanced by the great distances and hence the long travel times which allow dispersion effects to take place.

Strong earthquake ground motion close to the source of energy release is less ob- vious to interpret. In principle, it is composed of essentially the same types of waves, but often they are not clearly separated in time because of the source proximity. In addition, high frequency motions, whose amplitudes decay rapidly with distance and so cannot be observed on most teleseismic records, are also present. Also source size, relative to the distance to the recording station, and the spatial and temporal distribu- tion of energy release become important. All of these complexities are probably the

* Present address: Lamont-Doherty Geological Observatory, of Columbia University, Palisades, New York.

343

3Axz~ BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA

reason why, so far, no notable attempts have been made to determine the detailed character of strong earthquake ground motion.

Because the collection of recorded strong-motion accelerograms for engineering ap- plications is still far from complete, many research workers have made valuable at- tempts to generate artificial accelerograms. Some of these are based on white noise (Housner, 1947; Rosenblueth, 1956; Rosenbleuth et al., 1962; Bycroft, 1960), Sta- tionary Gaussian process (Tajimi, 1960; Housner and Jennings, 1964; Barstein, 1960) and various nonstationary processes (Bolotin, 1960; Bogdanoff et al., 1961; Cornell, 1964; Amin and Ang, 1966; Shinozuka and Sato, 1967; Jennings et al., 1968). Although these approaches represent a significant step forward in the engineering use of ac- celerograms, they are often based on simplified interpretations of the properties of strong motion. Usually only the simplest features of the recorded accelerograms are employed to model the "rapid build-up" of the motion, the "central portion of strong shaking" and the "decaying tail." The nonstationary time dependence of the intensity and frequency content of strong motion are usually neglected in such models. The reasons for these simplifications come, partly, from the mathematical complexity in generating these processes but are mainly the result of the lack of our knowledge of the actual character of strong-motion aceelerograms.

The area of earthquake engineering research concerned with the effects of local geology on the amplitudes of strong-motion waves is also directly affected by the lack of detailed interpretation of strong-motion accelerograms. Most of these studies are based on models in which earthquake waves are assumed to propagate vertically toward the set of horizontal surface layers, irrespective of the question of whether such motions are predominant or not. However, the simple qualitative analysis of the param- eters which govern the properties of strong-motion waves leads to the conclusion that a predominant part of the near-field ground shaking may be composed of surface waves associated with energy propagating horizontally through soft surface layers as a wave guide (Trifunae, 1969). The following analysis of some typical strong-motion records supports such an interpretation.

RESPONSE ENVELOPE SPECTRUM (RES)

The need for improved methods of assessment of information from seismograms stimulated the development of the moving window Fourier analysis and multiple filter techniques. Alexander (1963) used frequency and velocity windows to extract informa- tion on wave modes. Archambeau, Flinn and Lambert (1966) studied the dispersion of body-wave arrivals by using digital filters. Landisman, Dziewonski, and Sato (1969) developed a process called "moving window analysis" to display the amplitudes and phases of transient seismic signals as functions of period and group velocity. Dzie- wonski, Bloch and Landisman (1969) reported on the "multiple filter technique" as an efficient method for analyzing multiple dispersed signals. These approaches based on different numerical methods are aimed at the same objective, which is to extract a maximum of information present in the seismograms. This information typically con- sists of the dispersion as indicated by arrival times of different frequency components in the seismic surface waves. It further consists of the phase data, ellipticity of the surface particle motion, and other features from which the character of energy release at the source can be determined. Trifunac and Brune (1970), for example, used a simple form of moving window Fourier analysis to study the extended time behavior of seis- mic energy release.

In engineering seismology and in particular in the analysis of strong-motion ac-

INTERPRETATION OF STRONG EARTHQUAKE GROUND MOTION 3 4 5

celerograms, similar information is needed. The knowledge of the time dependent amplitudes and the frequency content of strong-motion records is required for pre- dicting the vibration behavior of engineering structures.

Small motion of a single-degree-of-freedom, viscously-damped, linear oscillator is described by the well-known differential equation

2 2 + 2con~2 ÷ co, x = - a ( t ) (1)

where x = relative motion of the mass,

= fraction of critical viscous damping,

co~ = natural frequency of vibration,

a(t) = absolute acceleration of the ground.

The vibration of several types of structures such as elevated water tanks and small buildings can be described by the single-degree-of-freedom oscillator. For tall build- ings, chimneys, and towers, when contribution of the higher modes of vibration can- not be neglected, each mode of vibration may be represented by the equation of the same form as (1). Then, by combining the response from all higher modes, the result- ing response of the multi-degree-of-freedom system can be determined (Merchant and Hudson, 1962). Hence, from the engineering point of view, the most valuable in- formation about the structural vibrations during earthquake motion will be ob- tained by studying the response of the system described by equation (1). Similar ideas led to the introduction of the concept of the response spectrum by Benioff (1934) and Blot (1941) which was further developed and applied by Housner, Martel and Alford (1953), Hudson (1956), and others.

In order to find a physically simple method for analyzing aeeelerograph records from the point ;of view of structural engineering, we consider again equation (1). It is well known that if for a(t) is substituted -Aco 2 sin cot and if it is assumed that the vibration will go on for a long time, so that transient vibrations generated by the ini- tim conditions have decayed, the steady state part of the relative response x(t) is determined by the following expression

(co)2 x(t) = ~ A-sin (cot - ,1~) (2)

\ c o n / _ 1 COn

where

(2co_. r_ -- tan-1 k, co2 _ co2/" (3)

The factor 1/%/ - in equation (2) represents a dynamic amplieation factor and gives the ratio between the dynamic and static deflections. It is easily seen that for small /', this factor is maximum when w = con and equals ½f. For most engineering structures,

346 BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA

¢ the equivalent viscous damping is in the range between 1 and 5 per cent of critical. For structures oscillating in the nonlinear and plastic range, the equivalent /" in- creases to about 10 per cent and more. Also when co = co~ from equation (3), it follows that the phase shift • is equal to rr/2. Thus we see that a typical lightly-damped single-degree-of-freedom system acts like a narrow band filter which amplifies the input frequencies, centered around co., by 1/2f times and follows the input wave form with 7r/2 delay. When A = A(t) is a slowly changing function of time, the re- sponse x(t) will also have a slowly-varying amplitude. The envelope of the response of the single-degree-of-freedom oscillator is thus approximately A (t)/2~-.

The physical appropriateness of the single-degree-of-freedom system as a narrow band filter suggests tha t the response of a series of such systems can be used to infer the character of the input acceleration. This is, of course, the basic idea behind the

{D ~r--

LJ ©

~ O ~ _m C3

> - -

J

S E C O N D S L I I I I I I I I I 210 I

0 tO

FIG. I. The relative displacement and its envelope for an oscillator with natural frequency o~ = i0 rad/sec and fraction of critical damping ~" = 0.10, for the NS component of the El Centro, 1940, strong motion accelerogram (Figure 5).

Response Spectrum, and the following analysis is simply a means of introducing more detail into the use of such spectra. Computing the response envelopes for many filters for equally-spaced frequencies will enable one to construct a three-dimensional dia- gram which indicates the time and frequency dependence of the instantaneous en- velope A(t, co) of the harmonic component of ground motion with frequency co at time t. We will call A(t, co) the Response Envelope Spectrum (RES) of the ground displacement. Likewise we will call coA (t, co) the RES of the ground velocity and c02A (t, co) the RES of the ground acceleration.

The relationship between the RES and the response spectrum is considered next. A point on the relative displacement spectrum for the frequency co corresponds to the maximum of A (t, co) during the time of excitation. Thus RES contains all of the information required to construct the relative response spectrum (i.e., relative dis- placement, and approximately velocity or acceleration spectrum), and, in addition, retains information about the time at which various responses occur.

The RES technique is closely related to the "multiple filter technique" or the "mov- ing window- analysis." I t is, in fact, simply a multiple filter technique with filter prop- erties specified by the single-degree-of-freedom, viscously-damped oscillator. There- fore the RES can be used as a tool to study dispersion properties of seismic waves and complicated multiple arrivals. Like multiple filter techniques or moving window

INTERPRETATION OF STRONG EARTHQUAKE GROUND MOTION 347

analysis, it enables one to determine group-velocity curves for those cases for which the signal-to-noise ratio is so small tha t the peak and the trough method cannot be applied. Another important merit of the RES is tha t it conveys the information about the length of large amplitude response of structures.

The RES in Figures 3 to 6 were computed in the following way. Recorded ground acceleration, shown on the side of each figure, is substituted in equation (1) as a(t). The a(t) for computation was in the form of equally-spaced data points with At = 0.02 see, interpolated from the original unequally-spaced data. Equation (1) was then integrated by the Adams-Moulton method, for each frequency co and damping f = 0.10 of critical. The envelope of the relative response x(t) was approximated by con- neeting the successive peaks in x(t) by a straight line. This is illustrated in Figure 1, where the response and the envelope are computed for the NS component of the E1 Centro 1940 aeeelerogram for an oscillator with co~ = 10 rad/sec and f = 0.10. The computations were repeated for 100 oscillators with frequencies equally spaced at aco~ = 0.166 rad/see and ranging from 0.166 rad/sec to 16.66 rad/see. All of these

TABLE 1

INFORMATION RELATED TO FIGURES 3, 4, 5 AND 6

Figure No. Event* Component Peak Amplitude w2A EpicentraI Distances (crn/sec 2) A (kin)

3 9 NS 13.58 19 4 9 EW 19.80 19

1A 10.0 5 1B NS 143.29 12.5

1C 15.0 1A 10.0

6 1B EW 89.78 12.5 1C 15.0

* These events were recorded during the Imperial Valley California earthquake, 1940. The names: Event IA, IB, IC and Event 9, are derived from our previous study (Trifunac and Brune, 1970).

envelopes together define J A (t, co). The RES for co2A (t, co) was chosen for this work because, as will be seen later, our main interest in this paper is concerned with studying the properties of accelerograms in which higher frequencies are emphasized. To form a rectangular mesh for Figures 3 to 6, it is necessary to interpolate equally-spaced envelope amplitudes along the time coordinates. This was done by interpolating amplitudes every 0.25 sec. For simplicity of visual presentation, all of the amplitudes were normalized so that the peak equals 5. Intermediate amplitudes were determined in the following way. When the normalized amplitude An was k < A,, < k + ½, for k = 0, 1, 2, 3, 4, 5, A~ was rounded off to k and that value was printed in the cor- responding slot. When lc ÷ ½ < A n ~ k ~- 1, An was substituted by a blank and nothing was printed on the output paper. In this way elevation contours of J A (t, co) became clear and easy to interpret. Because of the photographic reduction of the original scale, all of the contours of JA (t, co) in Figures 3 to 6 are also traced by hand. The actual amplitudes of the RES may be obtained by multiplying normalized am- plitude values by the "peak value" given in Table 1 for each figure. Arrivals of the surface-wave groups were computed using the theoretical dispersion curves (Figure 2) and the epicentral distances shown in Table 1.

348 BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA

ANALYSIS OF ACCELEROGRAMS FOR THE IMPERIAL VALLEY IV~AY 18 1940 CALIFORNIA, EARTHQUAKE

Several typical accelerograms from the E1 Centro accelerograph record of May 18 1940 were chosen as the optimal samples for this analysis for the following reasons. The dispersion of the surface waves can be theoretically estimated, provided the layer thicknesses and velocities of wave propagation are well known in the area between the station and the source. Detailed studies of the local travel times and layer thick- ness in the Imperial Valley were made by Biehler (1964). Trifunac and Brune (1970) used the data from the Westmoreland seismic profile (Biehler, 1964) to compute the group-velocity curves for several Love and Rayleigh modes (Figure 2). These dis-

4.0 I

5 2

4/ j 9.0 /

>:

> j~4 ~ 2

o I

LO ~ ~

LOVE WAVES ----- RAYLEIGH WAVES

I 0 I 2 B 4 5

PERIOD, seconds

FIG. 2. Love- and Rayleigh-wave dispersion curves for the structure corresponding to the West- moreland profile (Biehler, 1964) in the Imperial Valley.

persion curves are based on the five-layer model. Layer properties were taken as shown in Table 2 (Trifunac and Brune, 1970).

The flat character of the fundamental-mode group velocity indicates that most of the energy in the short-period range (shorter than about 5 see) will travel with a velocity of about 1 km/sec. In the same paper, approximate distances and tentative locations for all shocks recorded on the E1 Centro strong-motion accelerograph were determined. Trifunac and Brune (1970) also interpreted all obvious (P and S) phases for all recorded events. Therefore, these accelerograms permit a comparison of an interpretation based on RES with that carried out in the previous study.

We first analyze an aftershock referred to as Event 9 (Trifunac and Brune, 1970). Figure 3 shows the trace and RES of the NS acceleration component and Figure 4 shows the same for the EW acceleration component. Epicentral distance for this event was estimated (Trifunac and Brune, 1970) to be about 13 to 19 km from El Centro. If it is assumed that the actual distance was about 19 km, it is possible to

I N T E R P R E T A T I O N O F S T R O N G E A R T H Q U A K E G R O U N D M O T I O N 349

calculate the expected arrival times for all surface phases for which theoretical dis- persion curves were calculated (Figure 2). I t should be pointed out tha t our interpre- tat ion here is only qualitative since we have estimated only the theoretical group velocities in the absence of actual measurements. Nevertheless, when the expected arrivals for the surface phases are superimposed on RES, one can easily associate various bursts of arriving energy in the aecelerogram with the estimated arrivals. In this way, we interpret the RES high amplitudes between 17 and 22 sec (Figure 3) to be associated with the fundamental Love and Rayleigh waves. Likewise, waves arriving during the time interval from about 6 to 16 see are interpreted to be the higher surface wave modes, as may be seen in the same Figures 3 and 4.

I t may be difficult and perhaps useless to draw a clear line in the RES separating surface waves from body waves. This is clear from the Figures 3 and 4 where it may be seen that some surface-wave phases (in RES from 6 to 10 sec and for frequency co < 8 rad/see, Figures 3 and 4) do arrive simultaneously with the higher frequency energy (in RES from 6 to about 10 see and for co > 8 tad/see, Figures 3 and 4) which

T A B L E 2

LAYER PROPERTIES

Layer Thickness (kin) a (km/sec) fl (km/sec) ~" (gr/cm~)

1 0.18 1.70 0.98 1.28 2 0.55 1.96 1.13 1.36 3 0.98 2.71 1.57 1.59 4 1.19 3.76 2.17 1.91 5 2.68 4.69 2.71 2.19 6 ~ 6.40 3.70 2.71

we may call S wave. The arrivals after 22 see (Figures 3 and 4) cannot be interpreted in a simple way and probably represent reflections which traveled along the indirect paths and hence arrived later at the recording station.

In the light of these interpretations, based on the calculated arrivals of surface waves, we can conclude that , except for the high frequency arrivals from about 6 to 8 see (Figures 3 and 4), all other motions recorded during Event 9 are represented by surface waves. The first burst of higher frequencies which we may call the S-wave arrival is the prominent feature of an accelerogram in almost every case. However, from the RES, one can find that , from the point of view of the vibration amplitudes of the wide class of simple oscillators, surface waves may be more significant. During surface-wave accelerations, structures may experience the longest and biggest oscilla- tions. In Figure 3, for example, the response spectrum peak would occur for a fre- quency 15.3 rad/sec at a time t = 13.25 sec which is definitely in the region of surface- wave excitation.

The duration of strong earthquake ground motion is one of the most important parameters governing destruction of man-made structures. In terms of duration, the body, P and S, waves are recorded on most accelerograms in the form of relatively higher frequency bursts of energy, arriving at the station for several seconds. The surface waves, on the other hand, usually composed of somewhat longer period waves, arrive during considerably longer time intervals. This extended arrival of the energy in the form of the surface waves is the consequence of the dispersion through the hori- zontal waves guides which are usually characterized by the low velocity of wave

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~o ococooccocccccccoc¢cccoccooooooooo ooooocoooccooooo oco~ooooooooooooo~gg~ooo occcoococcccccccocccccccoocGooco~cooo ocoooo¢oooooooo ooococccocoooooooooocc¢c~o~oooccc cccccoccoccoocccocooocccccoooocooooco o¢oooococcooooooccoooccccccccoGooooooooococooccooooo OCCCCOCOGCCOCC¢CCCCOOC¢CGGO000CO000CO OOCO00CCOOCO00COOCCOCOOCCCCCCCCCOOCC¢O00C¢OCOCOCO00¢O

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0 I 2 3 4 5

FIG. 4. The co2A (t, ~o) Response Envelope Spectrum (RES) for Even t 9, EW component.

351

352 BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA

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FIG. 5. The co2A (t, ~) Response Envelope Spectrum (RES) for Event~ 1A, 1B, and 1C, NS component.

INTERPRETATION OF STRONG EARTHQUAKE GROUND ~[OTION 353

~L CENTRD MAIN EVENT EW pEAK VALUE IS 89.78

RELATIVE RESPONSE aCCELER. ENVELOPE SPECTRUM, R~$PDNSE ~*#4PLITUDE IN CM. IS OBTAINED By MULTIPLYING 5PECTR~t VALUE BY THE PEAK VALUE, D~MPING IS 0,).00 OF THE CRITICC~L

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354 BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA

propagation. Dispersion curves in Figure 2 can serve as one example of this. The sur- face-wave motions characterized by such a dispersion will thus last for

(1 1) A V rain V max

time units, where A is the epicentral distance and V is the group velocity. The main part of the energy release during the Imperial Valley, 1940 earthquake

was recorded on the accelerograph at E1 Centro for about 25 sec (Figures 5 and 6). The approximate analysis of this record (Trifunac and Brune, 1970) indicates that at least four distinct shocks (Events 1A, 1B, 1C, and 2) may be associated with indi- vidual bursts of S-wave arrivals on the NS and EW accelerograph records at about 2, 8, 13 and 23.5 sec. Corresponding P-wave arrivals for each of these events cannot be unequivocally identified because of the high level of short-period vibrations produced by the previous events. For this reason, the estimated distances based on the S-P times may be in error. In this study, we interpret RES for these events in an attempt to find the best fit of the observed and expected arrivals of surface-wave energy. The results of this fit are shown in Figures 5 and 6 which indicate that our previous esti- mates (Trifunac and Brune, 1970) for the Events 1A and 1C were probably within the correct range. The tentative interpretatiou based on the surface-wave arrivals indicates that Event 1B could have occurred within a smaller epicentral distance than estimated previously. Surface-wave arrivals for Event 2 are not interpreted here separately. The epicentral distance for this event is probably 35 to 44 km which places it at the southernmost end of the observed surface fault. For this distance, fundamental Love and Rayleigh modes would arrive more than 50 sec after the strong motion was triggered and would be mixed with the arrivals from Event 3. If we as- sume the epicentral distances for Events 1A, 1B and 1C to be about 10, 12.5 and 15 kin, then the expected surface-wave arrivals are as indicated in Figures 5 and 6.

Although the accuracy in the above interpretations depends on a knowledge of the correct epieentral distances, the applicability of our theoretical dispersion curves, and on the properties of the RES, the most important tentative conclusion of this work seems well established. This conclusion is, as already mentioned, that the pre- dominant part of strong earthquake ground motion consists of surface waves. The form of these waves is considerably more complicated than for teleseismic surface waves because of the proximity and size of the source of the seismic energy release and the frequency domain considered. In our interpretation these waves are surface waves in the sense that their arrivals are governed by the travel-time-distance rela- tionships based on a simplified model for surface-wave dispersion.

CONCLUSIONS

The main results of this study can be summarized as follows: The Response Envelope Spectrum (RES), a multiple filter technique, represents a

physically simple and informative tool for the analysis of strong earthquake ground motion accelerograph records. I t contains all of the information necessary to deter- mine the relative response spectra and is a logical extension of response spectrum tech- niques. The most important property of this method is that it displays the time de- pendence of the amplitude and frequency changes in the accelerograph record. It is particularly useful in detecting arrivals of different wave forms and may be used as a

INTERPRETATION OF STRONG EARTHQUAKE GROUND MOTION 355

tool in dispersion analysis. Another important property of RES is that it conveys the information about the length of large amplitude response of structures.

The analysis of several strong-motion accelerograph records using the RES indi- cates that the predominant part of the strong earthquake ground motion, generated by a near-shallow earthquake energy release is represented by surface waves. This is important from the point of view of destructiveness and also from the point of view of the establishment of the correct simplified wave-propagation models for studying the effects of local geology. The analysis indicates that the duration of the intense shaking will be mainly governed by the dispersion properties of the ground.

ACKNOWLED G3/IENTS

I am indebted to Professor D. E. Hudson and Mr. F. E. Udwadia for a critical reading of the manuscript.

This study was supported in part by the National Science Foundation grant (Engineering Mechanics Program) and the Earthquake Research Affiliates Program at the California Institute of Technology.

REFERENCES

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356 BULLETIN OF THE SEIS~[OLOGICAL SOCIETY OF A~IERICA

Merchant, H. C. and D. E. Hudson (1962). Mode superposition in multi-degree of freedom systems using earthquake Response Spectrum data, Bull. Seism. Soc. Am. 52,405-416.

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CALIFORNIA INSTITUTE OF TECHNOLOGY I~ASADENA, CALIFORNIA

Manuscript received August 31 1970