The seismic spectrum, radiated energy, and the Savage and Wood inequality for complex earthquakes

303

Tecronophysics, 188 (1991) 303-320 Elsevier Science Publishers B.V.. Amsterdam

The seismic spectrum, radiated energy, and the Savage and Wood inequality for complex earthquakes Kenneth Seismological

D. Smith, James N. Brune and Keith F. Priestley

Laboratory,

Mackay School of Mines, University of Nevada-Reno,

(Received April 5,198s;

Rena, NV 89557, USA

revised version accepted April 27,199O)

ABSTRACT Smith, K.D., Brune, J.N. and Priestley, K.F., 1991. The seismic spectrum, radiated energy, and the savage and wood inequality for complex earthquakes. Tectonophysics, 188: 303-320. We have integrated velocity squared spectra in order to determine the seismic energy radiated during fault rupture. The high frequency spectral fall-off and the shape of the spectrum at the comer frequency are critical to the energy calculation. High frequency spectral fall-offs of am2 beyond the comer frequency, in a Borne (1970) source model, return radiated energies approximately equal to that of an Orowan (1960) type fault failure, where the final stress level is equal to the dynamic frictional stress. Any spectra with an extended intermediate slope of w-l would therefore result in higher radiated energies. Savage and Wood (1971) proposed a model in which the final stress level was less than the dynamic stress level and that this was the result of “overshoot’. They based their model on the observation that the ratio of twice the apparent stress to the stress drop was typically around 0.3. We show that for such a ratio to exist high frequency spectral fall-offs of = a-3 would be required. Composite spectra have been constructed for several moderate to large earthquakes, these spectra have been compared to that predicted by the Haskell (1966) model and velocity squared spectra have been integrated to determine the radiated energy. In all cases this ratio, twice the apparent stress to the stress drop, is greater than or equal to one, violating the Savage and Wood (1971) inequality, and provides evidence against “overshoot” as a source model.

Introduction As a result of calculations of energy radiation from a deterministic fault model, Haskell (1966) introduced a statistical model of fault rupture to better represent the irregular motions observed on strong motion records (Housner, 1947, 1955; Thompson, 1959) and the observed generation of high frequency energy from earthquakes with large source dimensions. An extension of this model was introduced by Aki (1967). In this model, Haskell (1966) visualized the actual faulting process as a swarm of acceleration and deceleration pulses arising from the variations in the elastic properties along the fault. These pulses propagate along the fault with some mean velocity, but which are highly chaotic in detail. Depending on the spatial and temporal correlation length of these pulses, this model can have a far field displacement amplitude spectra fall-off, beyond the comer 0040-1951/91/%03.50

0 1991 - Elsevier Science Publishers B.V.

frequency, proportional to w-l (spatial correlation length much larger than time correlation wavelength) or wP3 (spatial correlation length comparable

to time correlation

wavelength).

Approaching the problem from a different point of view, Brune (1970) introduced a fractional stress drop model to represent abrupt fault locking or healing, or non-uniform stress drop like a series of multiple events with parts of the fault remaining locked, in either case causing the fault to have less slip than it would have for a uniform static stress drop over the whole fault equal to the dynamic stress drop. Aki (1972) characterized this process as a series of “rapid slips and sudden stops”. In the Brune (1970) model the fractional stress drop introduces an 0-l slope in the displacement spectrum beyond the comer frequency, and thus leads to a considerable increase in high frequency energy over that for an wP2 fall-off model with the same seismic moment and source dimension. This

304

K.D. SMITH

effect is of great importance

in determining

the

volved were so high that uncertainties

ET AL.

in attenua-

level of strong ground motion during large earth-

tion left the results in question (Anderson,

quakes. Some more recent models of earthquakes

Similar weak support for the w-l model was also

have similar features, for example, the asperity models of Hartzell and Brune (1977) and McGarr

found by Anderson

(pers. cormnun.,

1987) in a

study of aftershocks

of the Coalinga

earthquake

(1981)

recorded on the Parkfield strong motion array. It seems that studies of small earthquakes will at

(1983),

the barrier model of Papageorgio and Aki and complex

multiple-event

models

of

Joyner and Boore (1986), and Boatwright (1988). The shape of the spectrum beyond the comer frequency is obviously important of the total radiated

to calculations

energy. The total radiated

energy is given by an integral of the square of the far-field velocity spectrum over frequency. If the displacement amplitude spectrum falls of as a-‘, the velocity spectrum velocity

falls off as w-t

and the

squared

spectrum (proportional to energy) falls off as L2, and thus there is relatively little contribution to the total energy beyond the comer frequency. On the other hand, if the displacement amplitude spectrum falls of as w-l, the velocity spectrum (and velocity squared spectrum) is constant and the contribution to the total radiated energy becomes proportional to the band width of that part of the spectrum. The shape of the spectrum beyond the comer frequency is of crucial importance to the Savage and Wood (1971) hypothesis or inequality, in that the apparent stress is always less than half the

present be limited by similar uncertainties

1986).

in at-

tenuation in most cases, and that we will have to look at larger earthquakes for more definitive evidence. It should be recalled

that many of the

events in the Tucker and Brune (1974) study were large enough (M,_ = 4 to 5) and the comer frequencies low enough (less than 2 Hz) that uncertainties in attenuation were not important, and thus their evidence for partial stress drops is not subject to the same uncertainties.

Unfortunately

Tucker and Brune had only two observing stations and thus the results were not as reliable as, for example, would be the case for similar larger events recorded on the Anza array, with ten high quality digital stations. Vassilou and Kanamori (1982) have published results from a study of energy estimates based primarily on teleseismic body-wave pulse shapes recorded on long period instruments from the WWSSN instruments, which could not give reliable estimates of high frequency radiated energy.

stress drop. Since the apparent stress is proportion to the total radiated energy, it is obviously directly

However, based on strong motion records from four earthquakes they argued that most of the

related to the existence of an w-l band in the displacement amplitude spectrum. In fact, we show

radiated energy in the near field was adequately represented in the far field long period pulse

in the next section that for faults the Savage and Wood (1971) hypothesis is violated directly in

shapes. In this paper we reconsider two of these earthquakes from a different point of view and

proportion to the width of the w-l section of the

conclude that significant energy is radiated energy at frequencies higher than the theoretical Haskell

amplitude spectrum for equi-dimensional faults. The empirical evidence for the existence of an w -i band in actual earthquake displacement spectra remains qualitative, but more data from high dynamic range broad-band digital seismographs may soon provide more objective evidence. In a recent article Brune et al. (1986) gave some preliminary evidence from the Anza, California seismic array (Berger et al., 1984) which suggested that small low stress drop earthquakes had lower spectral fall-offs and thus offered some support for the partial stress drop model for small stress drop events. However the critical frequencies in-

comer frequency for the overall fault dimensions. In a recent study of the 1978 Tabas, Iran earthquake, Shoja-Taheri and Anderson (1988) estimated the radiated energy based on near field strong motion records. They obtained results one to two orders of magnitude higher than corresponding teleseismic energy estimates based on a procedure developed by Boatwright and Choy (1986). Their results dramatically illustrate the importance of reconciling near field and far field energy estimates. Boatwright (pers. commun., 1977) has questioned the Shoja-Taheri and Ander-

SEISMIC

SPECTRUM,

son results,

ENERGY

AND

SAVAGE-WOOD

in part because

INEQUALITY

of the large

FOR

dis-

crepancy. Recently,

Priestley

and

Brune

(1991)

and

305

EARTHQUAKES

the fact that the Brune displacement

spectrum is

rounded at the comer frequency.

This illustrates

the dependence of the calculation

of the seismic

found strong evidence for the existence of a-’ spectral fall-offs for Mammoth Lakes and Round Valley, California earth-

energy on the shape of the spectrum around the

quakes. It was this new evidence from Mammoth

the spectra can be written directly using the defi-

Lakes

nition of apparent stress:

Priestley et al. (1988)

earthquakes

that

motivated

the present

comer frequency. The apparent stress for the asymptotic form of

study. (I

Seismic energy Gutenberg and Richter (1942, 1956) proposed the first dynamic measure of the energy radiated by fault rupture. They related the radiated energy to the earthquake magnitude. Magnitude measures are usually based on information

from a limited

frequency band, and do not adequately represent the contributions of all frequencies to the radiated energy.

However,

integration

of

the

velocity

aPP

=-

4% (3)

43

(Wyss, 1970) eqn. (2) and the definition seismic moment (Keilis-Borok, 1957):

of the

it4, = 47rpp3RQ,

(4)

where p is the shear wave velocity, and in eqn. (3) p is the rigidity. There is some evidence that actual earthquake

spectra have a sharper comer

than for the Brune (1970) 1979).

model (Brune

We will discuss the relationship

et al,

between

of the

spectral shape and radiated energy and the reason

radiated seismic energy, does incorporate the entire frequency band. Wu (1966) derived a simple expression for determining the radiated S-wave energy which incor-

tion but for the following equations, (5) and (6) it is only important to remember that we have assumed a sharp comer. The apparent stress is then

squared seismogram, in the determination

porated the S-wave radiation pattern:

Es=

~~PR~/omlQ(f) I* df

as given by eqn. (12a) of Hanks and Thatcher (1972) (increased by the factor of two), uapp= QpRQ,f:

where p is density, p is the shear wave velocity, R is the hypocentral distance and Q(f) is the spectral amplitude according to Brune (1970). Hanks and Thatcher (1972) obtained an analytic solution to the integration of the velocity squared spectrum in eq. (1) for a simple displacement spectrum in which the asymptotes of the constant long-period level and an a-* (or

for selecting a sharp comer model in a later sec-

amplitude

f-*)

high

frequency fall-off meet at some (sharp) comer frequency fo. The analytic solution of eqn. (1) for an (,IF2 model is:

(5)

Equating (3) with (5), using the definition of the moment in (4) and solving for the energy gives: E = 3-6%=f: S id3

(6)

Note that eqn. (6) was arrived at making no assumptions concerning the relationship of the comer frequency to the source geometry, and the R dependence is now only in the definition of the seismic moment (4). Equation (6) is similar in form to derivations of Randall (1973) and Vassilou and Kanamori (1982).

where P, is the zero frequency spectral amplitude. Hanks and Thatcher (1972) decreased Es in eqn. (2) by a factor of two in order to be consistent with the energy in the Brune (1970) model. The actual difference is a factor of 1.67, resulting from

Radiated energy for the Savage and Wood, Orowan and Brune models Savage and Wood (1971) proposed a faulting model in which the final stress level, S,, (their

306

K.D. SMITH

terminology), is lower than the dynamic frictional stress, St. This results in a static stress drop, S - S, (where S is the initial stress), which is greater than the dynamic stress drop, S - S,. They results from the suggested this “overshoot” momentum of the moving fault block. Savage and Wood (1971) express their model in terms of energy and stress drop, specifically in the ratio of twice the apparent stress to the stress drop. In other words, if:

PES s - s,

-3&----

2

holds, then the final stress, Se, is less than the frictional stress and there is, through their argument, “overshoot” (Savage and Wood (1971) provide a complete derivation). The apparent stress and the static stress drop are measured quantities. Evaluation of eqn. (5) depends on reliable measures of stress drop and radiated energy. Relationship (5) is the Savage and Wood inequality. Savage and Wood determined Es primarily through the Gutenberg-~chter M,-Es relationship (with few exceptions), and measures of static stress drop reported in the literature. They concluded that in most cases the apparent stress was significantly less than half of the stress drop, in support of an “overshoot” model. We believe that recent, more accurate measures of energy and stress drop, as described later, do not support their conclusion. Orowan (1960) proposed a faulting model in which the final stress, Sa, is equal to the frictional stress, S,. In this case, the effective stress is equal to the stress drop and the radiated seismic energy reduces to:

where p is the average slip and A is the fault area. For the Orowan model (5) becomes an equality. In the Brune (1970) model the far field shear wave pulse shape is determined by the effective stress. but the spectrum for the far field pulse accounts for only 44% of the Orowan energy. Most of this difference can be accounted for by the shape of the Brune spectra at the comer

ET AL.

frequency, and this leads to a discussion of energy as a function of spectra1 shapes. Energy and spectral shape The radiated energy is a function of spectral shape. In particular, the shape of the spectrum near the comer frequency and the high frequency spectral fall-off control the measure of the radiated energy, since the displacement amplitude spectrum is multiplied by w and then squared. As discussed earlier, Hanks (1972) integrated the tip2 spectral shape, with a sharp comer frequency, to calculate the radiated energy. If we assume a Bnme (1970, 1971) relationship between corner frequency and source dimension, do not decrease the integral by a factor of two, (that is depart from Hanks and Thatcher, (1972) in this respect), and include P-wave energy (l/18 of that in the S-wave; Wu, 1966), then 83% of the Orowan dislocation energy of eq. (6) is accounted for. Thus the CC spectral shape, with a sharp corner and a Brune (1970, 1971) relations~p between the comer frequency and source dimension, accounts for nearly all of the dislocation energy. It is clear that if the spectral fall-off at high frequency is steeper than w-* there will be less radiated energy. For example, average high frequency spectral fall-offs of ww3, account for only 48% of the Orowan energy, if the comer frequency and source dimension are given by the Brune (1970, 1971) model. For circular fault rupture and a Bnme (1970, 1971) relations~p between comer frequency and source dimension, intermediate spectral slopes of w-’ (or w- 1.5) beyond the initial comer frequency result in higher radiated energies than would be the case for the Orowan model, the amount depending on the band width of this portion of the spectrum. Of course, high frequency spectral falloffs of w-l cannot extend to infinite frequencies, since this would imply infinite energies. In the Brune (1970) model, the bandwidth of the w-’ part of the spectrum is proportional to the fractional stress drop parameter 4, and thus for E = 0.1 the total radiated energy is about ten times as great as for E = 1. Similarly in the Haskell (1966) model if the parameter K,@/K, is 0.007

SEISMIC

SPECTRUM,

(spatial time

ENERGY

correlation

correlation

AND

length

rupture in length.

FOR

INEQUALITY

longer

than

stress drop model, the fault width controls the amount of slip for a given stress drop. Energies

the

see his fig. 2) then

earthquakes,

is usually

is constrained

more

determined

at depth

The spectrum

energies

that

since

sumption.

and only extends

frequency

for a Haskell-type

by integrating

the spectral

shape

re-

sulting from the rectangular source geometry of the Haskell model are consistent with radiated

a rectangular

appropriate,

307

EARTHQUAKES

section of the energy spectrum linearly to the total radiated

energy. For large strike-slip model

much

wavelength,

there is a broad which contributes

source

SAVAGE-WOOD

would

Thus,

result

second

is higher than expected

the fault in the Haskell

rectan-

from

if the

the Orowan (higher)

as-

corner

for the width of

(1966) spectrum,

that

then clearly,

the radia-

intermediate

associated with the length and another with the width of the rupture surface, with the spectrum falling off as CC’ in between. For a constant

again the Savage and Wood inequality is violated. Thus, for rectangular sources we will test whether

1940

ted energy is higher than for the Orowan

the second comer

Imperial

Valley. 3

Momenl(dyne-cm):

26.66

Seismic

Energy(ergs):

22.92

Oronan

Energy(ergs):

22.34

slope longer,

is,

gular rupture theoretically results in two comer frequencies (Haskell, 1966; Savage, 1974a), one

frequency

case, and

is higher than predic-

CA

/

1 /

Stress

Drop(bars):

Apparent

z t;

33

Stress(bars):

& m

62

-1 -2

4 Savage k Wood Ratio: 3.79 Fault ): 760

a

-3 50.0 1

Area(km'

-3

Log

1971

San

Fernando.

-1

-2

0.0

Frequency

1

2

(Hz)

CA

3 log

Moment(dyne-cm):

25 79 s

Log Seismic

Energy(ergs):

22.29

:

Log Oronan

Energy(ergs):

21.23

2 9

Stress

Drop(bars):

2

1

20

Apparent

Stress(bars):

Savage

Wood Ratm:

b k

-31 -3

-2 Log

-1 Frequency

0.0

1

2

(Hz)

Fig. 1. Acceleration spectra. (a) Imperial Valley, California. High frequency level is determined from the corrected acceleration spectra; N-S

component (Mungia and Brune, 1984b). (b) 1971 San Fernando,

El Centro

California. High frequency level is

determined by the corrected spectra of the transverse component of the Pacoima Dam acceleragram (Trifunac, 1972). The dashed line represents the high frequency level that would result from a comer frequency determined from the fault width (Table 1) for a Haskell (1966) model (Savage, 1974a).

308

K.D. SMITH

ted for the Haskell

ET AL.

Although there has been great improvements in

model, and for equi-dimen-

sional sources we will test whether there is any

understanding

w-l section in the spectrum.

frequency near source recordings, large uncertain-

Data

frequency

ties remain.

various factors Recent

advances

weak and strong

which affect in observing motions,

high high

including

down hole recordings, have opened the possibility We have attempted to construct the attenuation corrected far-field radiated energy spectrum for a

of resolving many of these questions.

number of moderate to large earthquakes. At high

example, near site attenuation

frequencies,

amplification,

we have used near source recordings

preliminary

tion. At low frequencies

total radiated energy.

we have used moment

Composite

constraints based on long period seismic waves.

Log Moment(dyne-cm)

Tabas,

Energy(ergs).

23.07

Log

Energy(ergs):

22.43

Fault

Area(km*):

Stress

Savage

to the question

spectra have been constructed

of for

Iran

2400

Drop(bsrs):

Apparent

evidence relating

27.17

Log Sersmic Ororan

and surface layer

we attempt in this study to present

to minimize the effects of uncertainty in attenua-

1978

Although

vigorous debate continues about the effects of, for

13

Stress(bars): & Wood Ratio:

-31 -3

-2

Log

1979

Coyote

Lake,

-1

0.0

1

2

I

I

(Hz)

Frequency

CA

3 Log Moment(dyne-cm)

25.00 z

Log

Sersmic

Lag Ororan

Faull Stress

Energy(ergs).

20.20

Energy(ergs):

20. IS

Area(km’

):

84

Drop(bars).

11


r; t 10

Apparent

Stress(bers):

4 Savage

& Wood Ratm:

2

-l-

-2-

/I

-3

I

I

Log

Frequency

-3

-2

b -1

r

0.0

1

2

(Hz)

Fig. 2. Acceleration spectra. (a) 1978 Tabas, Iran. High frequency level is determined from the corrected Tabas acceleration spectra; transverse component (Shoja-Taheri and Anderson, 1988). (b) 1979 Coyote Lake, California. High frequency level is determined from the corrected spectra of the Gilroy Array No. 1 acceleragram; N40 o W horizontal component (Brady et al., 1980a). The dashed line represents the high frequency level that would result from a corner frequency determined from the fault width (Table 2) for a Haslcell (1966) model (Savage, 1974a).

SEISMIC

SPECTRUM,

ENERGY

AND

SAVAGE-WOOD

the following earthquakes:

INEQUALITY

1940 Imperial

Valley

are acceleration

(Fig. la), 1971 San Fernando, California (Fig. lb),

spectra. This helps to emphasize

the high-frequency

1978 Tabas, Iran (Fig. 2a), 1979 Imperial Valley (Fig. 2b), 1979 Coyote Lake, California

309

FOR EARTHQUAKES

component.

These acceleration

spectra have been corrected

(Fig. 3a),

for free surface effects (a factor of 2) and, for the

1980 Mexicali Valley (Fig. 3b), 1984 Morgan Hill,

Imperial Valley events, an additional correction of

California

Round

Valley,

Cali-

a factor of 3.4 to account for amplification

fornia (Fig. 4b), 1985 Michoacan,

(Fig.

4a),

1984

Mexico

(Fig.

the thick sedimentary

layer (Mungia

5a), and 1987 Edgecombe, New Zealand (Fig. 5b).

1984a). For those acceleration

The

sedimentary

figure

captions

specific acceleration

include

references

to the

records used in constructing

within

and Brune,

records from other

sites, a correction

of a factor 2 has

the high frequency spectra. Although the discus-

been applied along with the free surface correction.

sion of the calculation of radiated energy has been in terms of velocity spectra, plotted in Figs. l-5

For recordings very near to the source the scaling of the energy with distance, eqn. (1) has to

1979

Imperial

Valley,

CA

3 Moment(dyne-cm):

25.77

Seismic

Energy(ergs).

20.69

Orowsn

Energy(ergs):

20.66

Fault

Ares(km’

Stress

).

Drop(bars):

2

;; t \

1

!‘I

540

$

6

2 -6

///,/cc

0.0 -1 r_

/

k WI Apparent

Stress(bsrs):

5

Savage &

Wood Ratio:

u s

1.62

-2

a -3 -3

1980

Mexlcali

-2 Log

-1

0.0

Frequency

1

2

(Hz)

Valley 3-

Log Moment(dyne-cm):

25 69

Energy(ergs):

22.19

s !%

2-

Log Seismic Log Orowan

Energy(ergs):

20.66

‘; v

l-

Fault Stress

Area(km’

)

Drop(bars)-

Apparent

go.0 <

300

t

10

Stress(bars):

P :: 113

-

-1;

.

-2

2 Savage

k Wood Ratio:

21.70

b -3 -3

I

I

I

I

I

-2 -1 0.0 1 2 Log Frequency (Hz) Fig. 3. Acceleration spectra. (a) 1979 Imperial Valley, California. The High frequency level is determined from the corrected spectra of the Keystone Road acceleragram El Centro Array; N140 o E horizontal component (Brady et al., 1980b). (b) 1980 Mexicali Valley, Mexico. The high frequency level is determined from the corrected spectra of the Victoria, Mexico acceleragram; N40 o W horizontal component (Mungia and Brune, 1984b). The dashed line represents the high frequency level that would result from a corner frequency determined from the fault width (Table 3) for a Haskell(l966)

model (Savage, 1974a).

310 TABLE

K.D. SMITH

ET AL.

1

Details of the 1940 Imperial Valley and 1971 San Fernando, California earthquakes (a) 1940 Imperial

Valley, CaIijornia May 19,194O; 04:36:41 UTC

Origin time Epicenter

32.7 o N, 115.5 o W

Magnitude

ML = 6.7

Focal mechanism Moment Field observations

8.4

x

1O26 (Reilinger, 1984)

Surface waves

4.8

x

1O26 (Doser and Kanamori, 1987)

Long-period Body waves Fault dimensions

A fault plane of length 65-km and width 12-km has been estimated from geologic information

Average slip

205 cm

Stress drop

33 bar

(Richter, 1958; Trifunac and Bnme, 1970)

(b) I971 San Fernando, California Origin time

Feb. 9,1971;

Epicenter

34.43ON, 118.23.7OE;

14:00:41.6

UTC

Magnitude

ML = 6.4; nrb = 6.2; Ms = 6.5

Focal mechanism

strikes N67O W, dip: 52O NE

depth: 13.0 km

Moment Field observations

1.5

X

1O26 (Trifunac, 1972)

Surface waves

6.3

x

lO*s (from MS)

Long-period Body-waves From the aftershock locations and the relocation of the main shock (Allen et al., 1971;

Fault dimensions

Hanks, 1974) the fault has an initial dip of 23” and steepens at depth to 52”. The dimension of the aftershock zone is approximately 23 by 14 km*. Average slip:

175 cm

Stress-drop:

25 bar

be modified. The nearest part of the ruptured area may be only several kilometers from the recording site, and the station can be considered to be in the

record by the ratio of twice the rupture area (to account for both sides of the fault) to a sphere of radius 10 km, and then assumed that this was the

near field. For the Michoacan event we are faced with such a source receiver geometry, and have

true amount of energy radiated from a point source. We then applied the Y’ distance scaling in eqn. (1) with respect to the distance between the recording and the fault. The long period level is

attempted to account for it by scaling the high frequency energy contribution appropriately. We have multiplied the integration of the velocity squared spectrum of the near field acceleration

Fig. 4. Acceleration

spectra: (a) 1984 Morgan Hill, California. The high frequency level is determined from the corrected spectra of

the Anderson Dam-Downstream frequency

not affected, since it is determined from the seismic moment, but the high frequency level is increased.

level is controlled

acceleragram, N40 o W component (Brady et al., 1985). (b) 1984 Round Valley, California. The high

by the corrected

spectra of the Paradise Lodge acceleration

record and the intermediate

slope is

determined from long period body waves; transverse component of the acceleration record (Priestley and Smith, 1988). The dashed line represents the high frequency level that would result from a comer frequency determined from the fault width (Table 4) for a Haskell (1966) model (Savage, 1971). (c) The Round Valley composite displacement spectra from Priestley et al. (1988). The long period levels are constrained from 20 s surface waves and 4 s body waves, and the high frequency level is determined is from the near source strong motion record shown above the composite spectra. Also, shown above the time series is the window used for the FFT.

SEISMIC

SPECTRUM.

ENERGY

AND

SAVAGE-WOOD

INEQUALITY

1984

FOR

Morgan

311

EARTHQUAKES

Hill,

CA

3 Log

Moment(dyne-cm).

25.39 C

Log Seismic

Energyjergs).

21.13

Log

Energy(ergs):

20.78

Orowan

Fault

Arcsfkm*

Stress

):

Savage

1

17

Stress(bars): & Wood

2 s

150

Drop(bars):

Appsrent

2

8

Retlo:

20 2.26

-3

-2 Log

1984

Log

Moment(dyne-em):

Log

Seismic

Energy(ergs)

21.54

Log Orowan

Energy(ergs):

20.25

Fault Stress

Aree(km’)

Savage

Wood

1

2

1

2

(Hz)

CA

24.89

.i:‘-

16

Stress(bars): k

Valley,

0.0

110

Dropjbars):

Apparent

Round

-1 Frequency

Ratio:

158 19.32

-3

-2 Log

Log Frequency

(Hz)

-1 Frequency

0.0 (Hz)

312

K.D. SMITH

For all but the 1984 Round earthquake, good

approximation

gested

by aftershock

Valley,

of the acceleration

California,

geometry

is a

proximately

to the fault

geometry

sug-

comer

frequency

The Coyote

Lake

pected

for a Haskell

patterns.

Morgan Hill and Round quakes have particularly

dimensions

for all events. spond tra,

and

Note

other

that

to figure numbers. the intersection

determined

source

Table

corner

In constructing

of the long-period

from the seismic moment

Log Moment(dyne-cm):

spectral

level

Energy(ergs):

23.34

i

;:

20.0 <

8500

2

Apparent

15

fit information

l-5.

Table

and

6 summarizes

subsequent

based on the spectral

energy

shape. The comindi-

Stress(bars):

-

-2-

30

& Wood Ratio:

/----:

-l-

2 R

s” Savage

for each event as referenced

Tables

28.01

Log Orowan

Drop(bars):

model

by the depth extent (width) of line). This corner is fixed by the

Mexico

23.93

Stress

(higher)

Haskell

1985

Michoacan,

ex-

(Savage,

posite spectra of the Coyote Lake earthquake

Energy(ergs):

Area(km*):

length)

model

for the theoretical

ap-

lowest

and the trend

Log Seismic

Fault

frequency

calculations

as

fault

rectangular

depth of fault rupture

the spec-

with

in Figs. 1-5 is the second

in corresponding

corre-

was in all cases

to or consistent (representing

which is fixed rupture (dashed

parameters

numbers

equal

1974a). Plotted

Valley, California earthwell recorded aftershock

sequences which allows a good constraint on the rupture extent. Tables 1-5 include references for source

spectra

source

a rectangular

ET AL.

a

3.90

-3 -3

I -2

I -1

I 00

I 1

I 2

I 1

I

Log Frequency (Hz)

Edgecornbe,

New

Zealand

3Log Moment(dyne-cm).

25 78

Log Seismic

Energy(ergo):

21.90

3 g

Log Orowan

Energy(ergs):

21.50

B v

Fault

Stress

Area(km’):

Apparent

a f;

38

Stress(bars):

l-

$0.0

100

Drop(bars):

2-

:: r? 48

-

-l-

/

-24

Savage& Wood

Ratlo:

b -3

2.51

-3

I

-2 ,A8

I

-1 Frequency

I

0.0

2

(Hz)

Fig. 5. Acceleration spectra. (a) 1985 Michoacan, Mexico. High frequency level controlled by the average spectra (corrected surface amplification) of the Michoacan

acceleration

array;

for

transverse components (Anderson et al., 1986). The dashed line

represents the high frequency level that would result from a comer frequency determined from the fault width (Table 5) for a Haskell (1966) model (Savage, 1974a). (b) 1987 Edgecomb, New Zealand. Fourier acceleration spectra from the horizontal strong-motion records collected near the base of the Matahina dam (Priestley, 1989).

SEISMIC

SPE(JTRUM,

TABLE

2

ENERGY

AND

SAVAGE-WOOD

INEQUALITY

FOR EARTHQUAKES

313

Details of the 1978 Tabas, Iran, and 1979 Coyote Lake, California, earthquakes (a) 1978 Tubas, Iran Origin time

Sept.16,

Epicenter

33.342ON, 57.400°E;

magnitude

M, = 7.7; rnb = 6.5; Ms = 7.4

Focal mechanism

strike: N28O W; dip: 31” NE

1978; 15:35:56.6

UTC

depth: 5 km

Moment

_

Field observations Surface waves

1.5

x

1O27(N&i

Long-period Body-waves

8.2

x

1O26 (Niazi and Kanamori, 1981)

and Kanamori, 1981)

This event was associated with 85 km of discontinuous surface faulting.

Fault dimensions

Extensive zones of bedding-plane slip with thrust mechanism developed in the hanging-wall block, indicating an extensive hanging-wall deformation. The width of the slip surface based on early aftershock locations was about 30-km (Berberian, 1979; 1982) Average slip

196 cm

Stress-drop

14 bar

(b) 1979 Coyore hke,

California

Origin time

Aug. 6,1979;

Epicenter

37.102O N, 121.503O E; depth = 6.3 km

1705 22.7 UTC

magnitude

M, = 5.9; rnb = 5.4; MS = 5.7

Focal mechanism

strike: N30 o W; dip: 80 o NE

Moment Field observations

1.6

Surface waves

1.0 X 10” (from MS)

X

lo*’ (King et al., 1981)

Long-period Body-waves Fault dimensions

The fault surface outlined by aftershock locations principally consist of two right stepping enechelon, NW trending, partially overlapping, nearly vertical sheets. The overlap occurs near a prominent bend in the surface trace of the Calaveras fault. Reasenberg and Ellsworth (1982) infer from the distribution of early aftershocks, that slip during the main shock was confined to a 14-km long part of the northwest sheet between 4- and IO-km depth. Focal mechanisms and hypocentral distributions of aftershocks suggest that the main rupture surface itself is geometrically complex, with left stepping imbricate structures

Average slip

4Ocm

stress-drop

13 bar

cates a second comer frequency very nearly equal to that expected for a Haskell type rupture and is the only event in our study where this is true. The 1984, M, = 5.8, Round Valley, California earthquake is one event for which we have determined a composite spectra that shows a circular or equi-dimensional rupture area. The Round Valley spectrum has the additional constraint, at intermediate frequencies, of teleseismic body wave amplitudes recorded at GDSN (Global Digital Seismic Network) stations, as well as long period surface wave information (20 s) and a near source

acceleration recording (< 5 km epicentral distance) (Priestley et al., 1988). Figure 4c, from Priestley et al. (1988), shows the composite displacement spectra that was used to construct the Round Valley acceleration spectra (Fig. 4b). The displacement spectrum has an initial comer frequency at 0.2 Hz, and a second comer at 4 Hz with an 0-l mtermediate slope. The initial comer frequency slightly overestimates the source dimension, as determined from the aftershock pattern (Priestley et al., 1988), for a Brune (1970) type source model. If the spectral shape was much

K.D. SMITH

314

TABLE

ET AL.

3

Details

of the 1979 Imperial

Valley, California,

and 1980 Mexicalli

Valley, Mexico,

earthquakes

(a) 1979 Imperial Valley, California Oct. 15, 1979; 23:16:54.3 UTC

Origin time Epicenter

32.644O N, 115.309O W; depth:

Magnitude

M, = 6.6; m,, = 5.7; MS = 6.9

Focal mechanism

strike: N28” W; dip: 31° NE

10 km

Moment Field observations Surface

6.0 X 10z5 (Kanamoti

waves

Long-period

and Regan,

1982)

Body-waves Surface

Fault dimensions

breaks

were limited to a zone from approximately

10 to 40-km northwest

of the epicenter. Aftershock

epicenters

geothermal During

within an area 110&m long, from the Cerro Prieto

the first 8 h of the aftershock

approximately Aftershocks Average

occurred

area to the Salton Sea. 45km

extended

northwest

sequence,

activity

of the epicenter

to approximately

was concentrated

(Johnson

and Hutton,

in a zone 1982).

12-km depth

37 cm

slip

6 bar

Stress-drop: (b) 1980 Mexicali Valley, Mexico Origin

June 9,198O;

time

3:28:18.9

UTC

Epicenter

32.220 o N, 114.985 o E; depth:

Magnitude

ML = 6.1; mb = 5.6; Ms = 6.4

5 km

Focal mechanism Moment Field observations Surface

5.0 X 10”

waves

Long-period Fault dimensions

(from

MS)

Body-waves The earthquake The epicenter

did not cause surface was IO-km southeast

and the most intense

aftershock

of Victoria

a rupture

implying

slip:

Stress-drop:

activity

was centered

20&n

northwest

length of 30-km.

The width of the fault is assumed Average

rupture.

of the city of Victoria,

to be lo-km

(Anderson

and Bnme,

1991).

56 cm 11 bar

different and there was an wp2 high frequency fall-off beginning at this initial corner the radiated energy would be significantly lower, but the extended intermediate slope guarantees a high radiated energy and high apparent stress. Provided in Figs. l-5 are the calculations of the radiated energy from eqn. (l), the energy from eqn. (6) that would result for an Orowan type event, the stress drop as determined from the

spectra, seismic moment, the apparent stress, and fault area. The “Savage and Wood Ratio”-that is, ratio of twice the apparent stress to the stress drop-is also included: 2(&/&l) s - s,

(9)

If this number is less than 1, then by the Savage and Wood argument there would be “overshoot”,

SEISMIC

SPECTRUM.

TABLE

4

ENERGY

AND

SAVAGE-WOOD

INEQUALITY

FOR EARTHQUAKES

315

Details of the 1984 Morgan Hill and Round Valley, California, earthquakes 1984 Morgan Hill, California Origin time

April 24, 1984; 21:15:18.8

Epicenter

37.309ON, 121.768OE;

UTC

magnitude

M, = 6.2; mb = 5.7; MS = 6.1

Focal mechanism

strike: N34O W; dip: 84OSE

depth = 8.4 km

Moment Field observations 10” (from MS)

Surface waves

2.5

Long-period Body-waves

2.0 x 10” (Ekstrom, 1985)

Fault dimensions

x

From detailed aftershock locations during the early hours of the From detailed aftershock locations during the early hours of the with dimensions approximately 24&m by 5-km which was surrounded by aftershocks. They inferred this to represent the slip surface of the main shock

Average slip;

70 cm

Stress-drop

22 bar

I984 Round Valley, California Origin time

Nov. 23, 1984; 18:08:25.5

Epicenter

37.455ON, 118.603” W; depth = 13.4 km

UTC

Magnitude

M, = 5.8; nrb = 5.4; MS = 5.7

Focal mechanism

strike: N35 o E, dip: 80 o SE

Moment Field observations

3.8

x

10” (Gross and Savage, 1985)

Surface waves

7.9

x

10z4 (Priestley et al., 1988)

Long-period Body-waves

6.6

x

10” (Priestley et al., 1988)

Fault dimensions

During the 6-h following the main shock, aftershocks outlined approximately a 7

x

7 km* vertical plane, the center section of which was nearly free of activity.

This region which was approximately 36-km3 was inferred to be the slip surface during the main shock by Priestley et al. (1988). Average slip

73 cm

Stress-drop

23 bar

the final stress level being less than the frictional stress. This number is greater than or equal to 1 for all events studied.

drops, would contribute more to the high frequency energy (Boatwright, 1982). With respect to the acceleration spectra of the Michoacan

Large earthquakes as composites and energy implications

earthquake, Anderson et al. (1986) termed this the “roughness” part of the spectra after Gusev (1983) and clearly, this “roughness”, or high frequency detail, can bee seen on their near field displace-

of smaller events

A Savage and Wood ratio greater than 1 (violating the Savage and Wood inequality), implies that the static stress drop S - S, is relatively low, or that a significant amount of extra energy is being radiated at intermediate and high frequencies. Individual sub-events, small with respect to the total fault dimensions but high dynamic stress

ments pulses (fig. 6 of Anderson et al., 1986). The high-velocity pulse observed on the Pacoima Dam record for the 1971 San Fernando, California was interpreted by Hanks (1974), as being due to an initial high stress drop sub-event with a much higher stress drop than determined for the entire faulting event. This in part contributed to the

K.D. SMITH

316

TABLE

ET AL.

5

Details

of the 1985 Michoacan,

Mexico,

and 1987 Edgecombe,

New Zealand,

earthquakes

1985 Michoacan, Mexico Origin

Sept. 19,1985;

time

13 17 49.1 UTC

Epicenter

18.14ON,

102.71° W; depth:

Magnitude

m,, = 6.8; MS = 8.1

16 cm

Focal mechanism: Moment Field observations Surface

waves

Long-period

Body-waves

10.3

x

3.

10” (Priestley

x

lo*’ (Priestley

Early aftershocks

Fault dimensions

approximately Average

and Masters,

and Masters, located 170

1986)

1986)

by Singh (written 50 km* and dipping

x

commun.) about

outlined

a surface

15O beneath

area

the coast

230 cm

slip

19 bar

Stress-drop 1987 Edgecombe, New Zealand Origin

March

time

2, 1987; 01:42:34.0

UTC

Epicenter

37.92OS, 176.79OE;

depth:

Magnitude

M, = 6.3; mb = 6.1; Ms = 6.6

12.0 km

Focal mechanism Moment Field observations Surface

waves

Long-period

Body-waves

7.9

x

1O25 (Priestley,

1989)

6.1

x

1O24 (Priestley,

1989)

Early aftershocks

Fault dimensions:

approximately The teleseismic Average

located 20

x

by Robinson

fault plane solution

extended Pacoima

intermediate slope in the spectrum of the Dam acceleragram, but most of the en-

area

a fault dip of 45 o

simulated

the ground

motion

of large earthquakes

6

Summary

of spectral

fits and energy

Event

calculations

f (i-k,

Year

Valley, Calif.

San Fernando, Coyote

indicated

a surface

12 km depth.

Hartzell (1982), Papageorgiou and Aki (1983), Mungia and Brune (1984b), Joyner and Boore (1986), and Boatwright (1988), among others, have

ergy in this range comes from later high frequency complexities in the record.

Tabas,

outlined

50 bar

Stress-drop

Imperial

commun.)

to approximately

195 cm

slip

TABLE

(written

10 km* and extending

Cahf.

Iran Lake, Calif.

LOis4,

Log Es

s-w

(dyne-cm)

(ergs)

ratio

1940

0.033

1.00

26.68

22.86

3.34

1971

0.100

1.60

25.79

22.29

11.52

1978

0.023

0.30

27.87

23.07

4.26

1979

0.134

0.30

25.00

20.20

1.04

Imperial

Valley, Calif.

1979

0.044

0.35

25.77

20.89

1.62

Mexicah

Valley, Mexico

1980

0.067

5.00

25.69

22.19

21.70

Morgan

Hill, Calif.

1984

0.083

1.00

25.39

21.13

2.26

Round

Valley, Calif.

1984

0.220

4.00

24.89

21.41

10.68

0.20 _

28.01

24.24

7.82

25.78

21.90

2.51

Michoacan,

Mexico

1985

0.011

Edgeeombe,

N. Zealand

1987

0.220

SEISMIC

SPECTRUM,

ENERGY

AND

SAVAGE-WOOD

INEQUALITY

using a summation of small events. Spectral shapes

317

FOR EARTHQUAKES

Discussion

generated by these models relate directly to the calculation

of radiated energy and apparent stress

We have constructed estimates of the spectra of

and therefore evaluation of the Savage and Wood

several large to moderate

inequality (5). For instance, the asperity model of

integrated

Boatwright

termine the radiated seismic energy. Seismic mo-

(1988),

the velocity

size earthquakes

squared

spectra

and

to de-

incorporates an intermediate slope of w-r to w-r5 and is similar in principle,

ments have been used to constrain the long period

as stated by Boatwright, to the “partial stress drop

level (flat part the far field displacement

model”

and near source acceleration

of Brune (1970).

In this model a high

spectra),

spectra to constrain

stress drop event occurs within a larger source

the high frequency amplitudes. Approximate

area of lower stress drop and gives rise to a

rections for free surface and sediment amplificamade.

Significant

cor-

relative increase in high frequency energy. For this

tion have been

case the Savage and Wood inequality is violated.

slopes of 0-l apparently exist in the spectra, with consequent increases in the calculation of the radiated energy. These intermediate

Frequency

band limitations

and the calculation of

radiated energy

slopes extend to

higher frequencies than those predicted for the Haskell (1966) model (Savage, 1974a). Interpreted in terms of apparent stress and static stress drop, these earthquakes

Vassilou

intermediate

and Kanamori

(1982)

calculated

the

violate the Savage and Wood

inequality (5) and provide evidence against “over-

seismic energy radiated by large earthquakes using

shoot”

teleseismic body waves. They determined that most of the energy radiated by large earthquakes is below 1 to 2 Hz and therefore within the band width of GDSN stations, and that this frequency band was sufficient for energy calculations. This method can be applied if the displacement spec-

ML = 5.8, 1984 Round Valley, California eventthe only event considered here that has a more or less equi-dimensional rupture surface (Priestley et

trum falls off as w-’ at frequencies higher than 1 to 2 Hz. However, for several of the events we

as a source model in these cases.

The

al., 1988)-has a composite spectral shape with an extended intermediate slope of w-l. The initial comer frequency of the Round Valley spectrum represents a good approximation to the source dimension, as determined from the aftershock pat-

have studied there are important energy contribu-

tern, for a Brune (1970) type source model. This

tions at frequencies

event is a good example of the calculated

greater than 1 Hz (spectra

energy

with extended intermediate slopes). This especially

being greater than that predicted for an Orowan

can be seen in the spectra of the 1971 San Fernando (Fig. l), 1980 Mexicali Valley (Fig. 3)

type rupture (6).

and 1984 Round Valley (Fig. 4) earthquakes where the second (higher) corner frequencies are greater than 1 Hz. Band limitations can also be a problem at long periods. The integration of velocity squared time series from the acceleration record would return less reliable estimates of radiated energy if the bandwidth of the instrument is at a higher frequency than the lowest comer frequency of the far field spectrum. Therefore, composite spectra or wide band instrumentation would be required to incorporate all the details of the spectral shape, particularly for larger events showing significant intermediate spectral slopes.

On the basis of their data, Savage and Wood (1971) suggested a ratio of twice the apparent stress to the static stress drop of 0.3 as a typical value (that is accounting for only 30% of the Orowan energy), and they used this result as evidence for the “overshoot” model of eqn. (7). Assuming a Brune (1970, 1971) model and a sharp corner frequency, we have shown that such a ratio would require steeper (3 w-‘) high frequency spectra fall-offs, beginning at the comer frequency, than are generally accepted. Hanks (1979) argued against w -3 high frequency spectral fall-offs. In terms of rupture models, focusing due to rupture velocity and the angle with respect to the fault orientation affect the shape of the spectra over the

318

K.D. SMITH

focal sphere. However,

average high frequency

spectral fall-offs of we3 do not exist in the spectra of far-field S-waves for these models (Sato and Hirasawa,

1973;

Madariaga,

1973;

Boatwright,

1980; Joyner, 1984; Joyner and Boore, 1986). Our spectra have typically been constructed from one acceleration record, and we have necessarily made some assumptions about the location of the station with respect to the fault, radiation pattern effects, average spectral shape, and site effects;

thus, there remains considerable

uncer-

tainty in our results. The ideal situation would be to have many stations surrounding the source and be able to account for focusing, site effects, and radiation pattern to a much greater degree. Our purpose in this study has been to show how spectral shape relates to estimates of apparent stress, to further pursue the idea of integration of the entire spectral shape as a method of determining radiated energy, to document cases of intermediate slopes in the spectra of moderate to large earthquakes

and to use energy considera-

tions to show that for many earthquakes the Savage and Wood (1971) inequality is violated. There is no strong evidence that the overshoot mechanism is ever operative.

Quaas, R. and Onate, form the Michoacan

ET AL.

M., 1986, Strong ground motion Mexico,

earthquake.

Science,

233:

1043-1049. Anderson, J.G. and Brune, J.N., 1991. The Victoria

accelera-

gram for the 1980 Mexicali Valley earthquake. Earthquake Spectra, in press. Berberian, M., 1982. Aftershock tectonics of the 1978 Tabas-eGo&an

(Iran) earthquake sequence: A documented active

thin and thick skinned tectonic case. Geophys. J.R. Astron. Sot., 68: 499-530. Berberian,

M., Asudeh, I., Bilham,

R.G.,

Scholl,

C.H. and

Soufleris, C., 1979. Mechanism of the main shock and the aftershock study of the Tabas-aGolsham

(Iran) earthquake

of September 16, 1978: a preliminary report. Bull. Seismol. Sot. Am., 69: 1851-1859. Berger, J., Baker, L.M., Brune, J.N., Fletcher, J.B., Hanks, T.C. and Vernon, F.L., 1984. The Anza array: a high-dynamicrange, broadband, digitally radiotelemetered

seismic array.

Bull. Seismol. Sot. Am., 74: 1469-1481. Boatwright,

J., 1980. A spectral

sources:

simple estimates

theory for circular

of source dimension,

seismic dynamic

stress drop, and radiated energy. Bull. Seismol. Sot. Am., 70: l-26. Boatwright,

J., 1982. A spectral

sources:

simple estimates

theory for circular

of source dimension,

seismic dynamic

stress drop, and radiated energy. Bull. Seismol. Sot. Am., 70: l-26. Boatwright,

J., 1988. The seismic radiation

from composite

models of faulting. Bull. Seismol. Sot. Am., 78: 489-508. Boatwright,

J. and Choy, 1986. The acceleration

subduction

zone earthquakes.

spectra of

Eos, Trans. Am, Geophys.

Union, 76: 310 (abstr.). Brady, A.G., Mork, P.N., Perez, V. and Porter, L.D.,

Acknowledgment

Processed data from the Gilroy

1980a.

array and Coyote creek

records Coyote Lake, California earthquake 6 August 1979.

This research was funded by U.S. Geological Survey research grant No. 140007-G1326.

Calif. Div. Mines Geol., Spec. Pub]., 24. Brady, A.G., Perez, V. and Mork, P.N. 1980b. The Imperial Valley

earthquake,

October

15,

1979.

Digitization

and

processing of acceleragraph records. U.S. Geol. Surv. Open File Rep., 80-703: 197-209.

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