Bulletin of the Seismological Society of America, Vol. 69, No. 4, pp. 1055-1079, August 1979

FREQUENCY DEPENDENCE OF Qscs

BY STUART A. SIPKIN AND THOMAS H. JORDAN

ABSTRACT

Data from HGLP instruments at KIP and MAT and W W S S LP and SP instru- ments at KIP and GUA have been used to study the amplitude characteristics of Sc$ and multiple ScS waves from deep-focus earthquakes. The data at low frequencies (0.006 to 0.06 Hz) are consistent with our previously published estimate, Qscs = 156 _+ 13 (Jordan and Sipkin, 1977). However, at high frequen- cies (>0.1 Hz), Qscs appears to increase rapidly with frequency. Lower bounds on Qscs are obtained by assuming a flat source spectrum and ignoring any energy losses due to scattering; we find that Qscs must be greater than 400 at frequencies between 1 and 2.5 Hz. Correcting for a source spectrum with a corner at 0.16 Hz and an asymptotic roll-off of ~ - 2 considered appropriate for these events, raises this estimate to about 750. The increase in Qscs at frequen- cies above 0.1 Hz is consistent with a spectrum of strain retardation times which has a high-frequency cutoff in the range 0.2 to 1.0 sec. At very low frequencies Qscs can be estimated from normal mode data; the best available models yield values of about 230. Comparison of these estimates with our data suggests that Qs~s decreases with frequency in the vic ini ty of 0.01 Hz. Because the scattering coeff ic ient increases rapidly with frequency, the fact that signif icant $cS ampli- tudes are observed at high frequencies implies that any bias in Qs~s measure- ments due to scattering at low frequencies is probably small. We show that, although our data provide only integral constraints on the variation of Q~ with depth, the regions in which Q~ is frequency dependent occupy a substantial portion of the mantle, probably including at least part of the mantle below 600 km depth.

INTRODUCTION

In the study of seismic-wave propagation the intrinsic quality factor Q is usually assumed to be independent of frequency. A constant-Q model of seismic attenuation is both theoretically convenient and empirically consistent with most available data on wave amplitude decay (Knopoff, 1964; Anderson and Archambeau, 1964; Kana- mori and Anderson, 1977; Jordan and Sipkin, 1977). Nevertheless, the possibility that Q varies with frequency has intrigued seismologists for many years. Gutenberg (1958) was evidently the first to advocate that the apparent Q of teleseismic P waves (Qp) increases with frequency, and his hypothesis has received additional support from later studies (Kurita, 1968; Archambeau et al., 1969; Solomon and ToksSz, 1970; Solomon, 1972; Der and McElfresh, 1977; Lundquist, 1977). The variation of Qp is difficult to measure, however, because Qp is generally large (-1,000 at 1 Hz) and even substantial variations in its value have only small effects on P-wave amplitudes and wave forms. These are easily obscured by the uncertainties in source excitation and propagation effects other than anelastic attenuation. Consequently, the dependence of Qp on frequency has not been precisely quantified.

SH-polarized shear waves are more severely attenuated than compressional waves, and their structural interactions are simpler; hence, they are often more suitable for the study of anelastic structure. In a previous report (Jordan and Sipkin, 1977; hereafter referred to as Paper I), we recovered the attenuation operator for multiple ScS waves propagating in the western Pacific by applying a spectral stacking technique to digitally recorded data from High-Gain Long-Period (HGLP) stations

1055

1056 STUART A. S I P K I N AND THOMAS H. J O R D A N

in Hawaii and Japan. The spectral modulus of this attenuation operator yields the apparent Q of SH-polarized ScS waves (Qscs) as a function of frequency. We observed no significant frequency dependence of Qscs; the data were consistent with the estimate Qscs = 156 _+ 13 throughout the band 0.006 to 0.06 Hz.

The purpose of this paper is to extend our earlier work on Qscs to higher frequencies. As we shall demonstrate, the low value of Qscs obtained in Paper I, if extrapolated to frequencies greater than about 0.1 Hz, is inconsistent with ScS amplitudes observed on instruments of the World Wide Standardized Seismographic Network (WWSSN). We infer from this discrepancy that Qscs increases with frequency in the band 0.1 to 2.5 Hz.

ScS S P E C T R A

The basic data set for this study comprises the ScS wave forms from three deep- focus Tonga-Fiji earthquakes recorded on both the long-period (LP) and the short- period (SP) WWSS instruments at Kipapa, Hawaii (KIP) and/or Guam (GUA) (Table 1), as well as the ScS2 wave forms from one of these events (event 3, Table 1) recorded on the WWSS LP instruments at KIP. The geometry (Figure 1) was chosen so that the ray paths are similar to those used in Paper I. Deep-focus events

T A B L E 1

EARTHQUAKES USED IN THIS STUDY*

Event Date Origin Time (UT) Lat. (°S) Long. Depth rot, No. (°W) (km)

1 9 Oct. 1967 17:21:46.2 21.10 179.18 605 6.2

2 24 Jan. 1969 02:33:03.4 21.87 179.54 587 5.9 3 28 June 1970 11:09:51.3 21.66 179.42 587 5.8

* Source parameters from Bulletins of the International Seismological Centre.

of intermediate magnitude were selected for two reasons: (1) their source mecha- nisms are relatively simple and have short rise times, producing impulsive wave forms with moderate trace amplitudes which are easily digitized and are nearly ideal for studying high-frequency attenuation; and (2) their use eliminates an upper mantle path through a Benioff zone whose attenuative properties may not be representative of the western Pacific and whose velocity structure may be complex. The LP ScS wave forms from KIP for event 1 and from GUA for event 3 were not digitized, because the former is not completely recorded (the trace left the paper) and the latter has a poor signal-to-noise ratio; these event-station pairs are thus excluded from the analysis. The epicentral distances for the remaining four event- station pairs range between 47.6 ° and 49.5 ° .

The horizontal-component WWSSN seismograms were digitized on a Bendix tablet digitizer in the scan mode with continuous recording of points at ½-sec intervals in real time. These series were interpolated and resampled at uniform trace-time intervals of 0.75 sec for the LP records and 0.2 sec for the SP records. Three independent digitizations were made for each record, and the results were averaged. The averaged seismograms were then rotated to yield the transverse (S/-/) component of motion. The time series and their spectra, uncorrected for instrument response, are shown in Figures 2 and 3. Time windows were selected to maximize the ScS energy without including too much extraneous signal-generated noise; the window lengths are 41.2 and 16.8 sec for the LP and SP records, respectively. For the LP records this window length excludes energy from the sS and the SS phases,

FREQUENCY DEPENDENCE OF Qscs 1 0 5 7

which follow ScS by 30 to 50 and 60 to 80 sec, respectively. To compute the spectra, the mean was removed from each signal and a 10 per cent cosine-squared taper was applied. The spectra were interpolated by padding the time series with zeroes to ten times their original lengths prior to transformation, and the logarithms of the spectra were smoothed by running-mean filters with lengths of 0.066 and 0.25 Hz for the LP and SP signals, respectively. The noise spectra shown in Figures 2 and 3 were obtained by applying the same procedures to time windows either preceding or following the ScS signals.

The uncorrected ScS spectra for the various event-station pairs show similar features. The LP spectra reach peak values between 0.06 and 0.10 Hz and roll off

" 7

0 : IP

/ /

¥

FIG. 1. Map of western Pacific showing epicenters, stations, and surface projections of ray paths used in this study. Numbers correspond to event number in Table 1. Shaded bands indicate region sampled in Paper I.

approximately exponentially with frequencies greater than the peak frequencies; the amplitudes at the Nyquist frequency (0.66 Hz) are down from the peak amplitudes by about 50 dB. The SP spectra have peak amplitudes in the range 0.25 to 0.60 Hz, beyond which there are exponential roll-offs similar to the LP spectra; again, their -50 dB points occur near their Nyquist frequency (2.5 Hz). The signal-to-noise ratio (SNR) is variable, but for both the LP and the SP ScS spectra its average value is about 10 dB between the peak and Nyquist frequencies.

The SNR for the LP ScS2 spectrum from event 3 (Figure 2) is 20 dB at the peak frequency (0.05 Hz) and decreases to 0 dB at about 0.13 Hz. This observation is consistent with the fact that no ScS2 signals are observed on the SP records from this event.

1 0 5 8 STUART A. SIPKIN A N D THOMAS H. J O R D A N

0

-2-

- 4 -

- 6 -

O-

- 2 -

- 4 -

__J Q_ -

_ 2 ¸

~- -4 <~ ...J hA

v

t 2 3rain

Event 0 2 "',.

0 0.2 0.4 0.6

FREQUENCY (Hz)

FIo. 2. Transversely polarized long-period WWSSN ScS seisomograms and spectra, uncorrected for in s t rument response. Sol id curves are the ScS spectra from windows indicated on the se i smogram by the solid overbar; dashed curves are the noise spectra from windows indicated by the dashed overbar.

FREQUENCY DEPENDENCE OF Qsc$ 1059

O-

- g

-6

,-' ' , , . , , 1 m i n

Event #i \ . . _ ~ _ .

LLI CZ:}

I--

.._I (3_ :E <

LLJ >

I-- <

LLJ cr" v

(.--

0-

- 2 -

- 4 "

-6

O-

-2-

-4-

F---H

GUA "-'J -',_,_ ~ Event ~ 2 ",-, . , . . ~

-61 KIP'""" ,,_.....,_~~ Event #2 - ' ,_.

0- -2- ' , , ~ -4

Event # 3

f t T

' '.o . . . . 2'.5 0 0.5 1 1.5 2.0

F R E Q U E N C Y ( H z )

FI6. 3. Transversely polarized short-period WWSSN ScS seismograms and spectra, uncorrected for instrument response.

1060 STUART A. SIPKIN AND THOMAS H. JORDAN

The observations of significant S c S energy at high frequencies are critical to our arguments for the frequency dependence of ScS. As an independent check on the spectra in Figure 3, we applied a series of high-pass, convolution filters to the SP record from KIP for event 2 (Figure 4). Both S c S and sS are observed on the unfiltered trace, but only the S c S phase has significant amplitude on the filtered

ScS sS

I I

- - O . , , , , .

(D

Q .

E

0 . t 4 ~

-0.22 . . . . . .

0 . 0 4 ~ 0

-0.04 . . . . .

I

0 20 40 60 80 100

Time (sec) Fro. 4. (a) Trace is the transversely polarized SP WWSS record from KIP for event 3 showing ScS

and sS phases. (b), (c), and (d) Traces were obtained from (a) by applying high-pass, zero-phase convolution filters with corner frequencies at 0.5, 1.0, and 2.0 Hz, respectively. All filter responses are down 7 dB at the corner and down 27 dB one octave below the corner.

traces. In fact, the SNR for the S c S phase is approximately constant on all traces, which is consistent with the spectral estimates shown in Figure 3.

Qscs AT LOW FREQUENCIES (0.006 to 0.06 Hz)

Let Sn(w) and Sn+1(¢o) be the spectra of two successive multiple S c S waves recorded on the same instrument. To a good approximation these spectra can be related by the linear equation

bnAo(cO)Sn(OO) = bn+lSn+l(a~) (1)

w h e r e bn and b~+l are geometrical spreading factors, A0(~) is the attenuation operator for zero-distance, surface-focus S c S waves, and ¢o is angular frequency. The

FREQUENCY DEPENDENCE OF qscs 1061

modulus of this attenuation operator can be expressed in terms of the vertical ScS travel time o Q ScS Tscs and a quality factor o

= Tscs/Qs~s(o~)] IAo(w)[ exp[ - ½ [wl o o

o ~b ~ Ts~s = 2 dr/v~(r)

2 f; [Q°cs(O~)]-I =- ~ Q.-l(r, ~o) dr/v~(r).

T ScS

(2)

(3)

(4)

The range of integration in equations (3) and (4) extends from the core-mantle boundary, at radius b, to the surface, at radius a. Generally, both the shear velocity v, and the intrinsic quality factor Q, depend on geographical position as well as radius, but we shall assume that their lateral variations within the western Pacific are negligible. The possible dePendence of Q°scs and Q, on frequency is explicitly noted in equation (4). A weak (at most logarithmic) dependence of T°~s and v, on frequency can also be expected (e.g., Liu et al., 1976), but this enters the expression for IAol as terms of order Q-2. We consistently ignore such terms; that is, we assume Q>>I .

In Paper I we estimated the modulus and phase of Ao for the western Pacific in the frequency band 0.006 to 0.06 Hz from a spectral stack consisting of 17 pairs of multiple ScS phases recorded at HGLP stations in Hawaii and Japan. We demon- strated that our results were consistent with a linear, causal, constant-Q (LCCQ) model of seismic attenuation. A least-squares regression fit to the modulus data yielded Q°scs = 156 + 13 (Figure 5). We have confirmed this value of Qscs by further HGLP experiments utilizing sScS, phases as well as ScSn phases (Sipkin and Jordan, 1979). For example, an augmented spectral stack comprising 21 ScSn phase pairs and 21 sScSn phase pairs gives 0 Qscs = 155 ___ 11 in the band 0.006 to 0.06 Hz. Again, no significant frequency dependence is required by the data in this band.

As a further check on the results of Paper I, we show in Figure 5 the logarithm of IAol estimated from the ScS2/ScS spectral ratio for event 3-KIP using equation (1). A least-squares regression fit to these amplitude data yields Q°scs = 170 _ 15, which is slightly higher than, but clearly compatible with, the HGLP results.

Qscs AT HIGH FREQUENCIES (0.06 to 2.5 Hz)

At frequencies greater than 0.1 Hz, the attenuation of ScS2 phases from earth- quakes of intermediate magnitude (mb ~ 6) typically reduces their spectral ampli- tudes to noise levels (Figure 2), and the multiple ScS spectral ratio techniques discussed in the previous section cannot be applied. Therefore, to estimate Qscs at frequencies greater than 0.1 Hz, we must examine the spectral decay characteristics of the primary ScS phase.

The spectrum of an ScS phase, S~(w), can be written in the form

81(6o) = Rl(~o)Pl(cO)Al(~O)I(w) (5)

where R1(¢o) is the far-field source spectrum, PI@) is the ScS transfer function for a perfectly elastic Earth, A1(¢o) is the ScS attenuation operator, and I(¢o) is the

1062 STUART A. SIPKIN AND THOMAS H. JORDAN

instrument response spectrum. ]A11 can be estimated from observations of ISI[ if the moduli of each of the functions R1, P1, and I are known.

We assume that the frequency dependence of the source spectrum is specified by the simple expression (Aki, 1967; Brune, 1970)

]R~(ag]=Ro[ Wc2 ] ~2 + j - (6)

where O~c is the corner frequency and R0 is the zero-frequency modulus. At frequen- cies much less than the corner frequency the source spectrum given by equation (6)

. 5 i i i ~ i

O9

_1 -0.5

E3 0

-1.0

c Q=156

-1.5

-2.0 i i i i , I 10 20 50 40 50 60 70

FREQUENCY (mHz)

FIG. 5. Modulus of WWSSN ScS2/ScS spectral ratio for 3-KIP, corrected for geometrical spreading (heavy solid line). Discrete points with lo error bars are estimates of the modulus of the multiple ScS attenuation operator from Paper I.

is nearly flat, and at frequencies much greater than o~c it rolls off as w -2. This source parameterization adequately describes the spectral data for deep-focus Tonga-Fiji earthquakes obtained by Wyss and Molnar (1972); their data for shear waves from intermediate magnitude events yield estimates of fc = ~c/27r which fall in the range 0.1 to 0.2 Hz. As a typical value for our discussions we shall adopt f~ = 0.16 Hz.

The elastic transfer function P1 accounts for frequency independent geometrical spreading, as well as other, possibly frequency dependent, propagation effects such as reflection and transmission through discontinuities and scattering from other velocity heterogeneities. For the moment we ignore any frequency dependence of P1, although we return to this question in a later section.

To calculate the response spectrum of the WWSS LP instruments we make the usual low-frequency approximations (Hagiwara, 1958; Mitchell and Landisman, 1969): their response is parameterized by the nominal peak magnification and the free periods of the seismometer and galvanometer assuming critical damping,

FREQUENCY DEPENDENCE OF Qscs 1063

negligible coil inductances and a coupling factor of 0.05 (Mitchell and Landisman, 1969). However, in parameterizing the WWSS SP response, we include the effects of transducer inductance, which is significant at high frequencies (Chakrabarty and Choudhury, 1964); in this case the response is calculated from the parameters determined by L. J. Burdick and G. Mellman (unpublished manuscript), using their computer algorithm. For both the LP and SP instruments, the relative values of the magnifications and free periods were checked by measuring the amplitudes and widths of the calibration pulses; the discrepancies were found to be less than 5 per cent, which is sufficiently small for this experiment. As a further check, we compared the instrument-corrected LP and SP spectral amplitudes in their band of overlap (see, for example, Figure 6). Between 0.40 and 0.66 Hz (the Nyquist frequency for the LP spectra), the agreement is very good--the spectra generally differ by less than a few decibels. At frequencies below 0.40 Hz, the SP spectral amplitudes tend to be somewhat larger than the LP amplitudes; this discrepancy averages about 10 dB at 0.20 Hz. We attribute the differences between the instrument-corrected SP and LP spectral amplitudes at these low frequencies to inaccuracies in the computed SP response.

The factor A1 in equation (5) is, for the data shown in Figures 2 and 3, the spectrum of the ScS attenuation operator corresponding to a deep-focus source at an epicentral distance of about 50 ° and thus differs from the surface-focus, zero- distance attenuation spectrum A0 discussed earlier. If we write the modulus of A1 in a form similar to equation (2),

Tscs/Qscs(~z)] iAl(w) l = e x p [ _ ½ 1 ~ l 1 1 (7)

then 1 Tics is the travel time and 1 Qs~s is the quality factor obtained by integration along the actual ScS ray path.

To permit the comparison o of Qs~s estimates at low frequency with Q~s~s estimates at high frequency we make the approximation

1 -~- Qs~s(w) yQ°~s(~) (8)

where y is a parameter which depends on source depth and distance but is independent of frequency. Crude bounds on y are easily established. Models of Q, obtained from body-wave, surface-wave and free-oscillation data (Anderson and Archambeau, 1964; Julian and Anderson, 1968; Anderson and Hart, 1978a, b; Sailor and Dziewonski, 1978) generally have an average upper-mantle Q which is less than the average lower-mantle Q. If this is true beneath the western Pacific, then y > 1. An upper bound on y is obtained by assuming that all of the attenuation occurs in the upper mantle above the source depth (~600 km). From this fact we find that ~/ ~ 2 .

Much more precise estimates of y can be computed from published shear velocity and attenuation models. Two quite different Q models representative of recent work are QKB (Sailor and Dziewonski, 1978) and SL1 (Anderson and Hart, 1978a). From these models and the shear velocities of model 1066A (Gilbert and Dziewonski, 1975), we obtain the following results

Q Model Ql~c~, Q,~'~.s y

QKB 230 286 1.24 SL1 287 336 1.17

(In this table, Qlcs was calculated for a source depth of 600 km and an epicentral distance of 50°.) Despite the fact that QKB and SL1 have values of o Qscs which

1064 S T U A R T A. S IP KIN AND THOMAS H. J O R D A N

differ by 25 per cent, their y values are nearly identical, differing by less than 6 per cent. This difference is small compared with the uncertainties in estimates of Qscs, and hence the error made in applying equation (8) can probably be neglected. In our discussion we quote only the estimates of Q°scs; when appropriate, these have been computed from estimates of Q~scs assuming y = 1.17, the value from model SL1.

Two analytical procedures are used to estimate Qs~s at high frequencies. The first is based on the decay of the instrument-corrected ScS spectra (spectral decay method), whereas the second uses the energy ratio computed from the LP and SP time series by direct integration of the squared ScS wave forms (energy ratio method).

A O I B ~ ~ - - l l l ,

~ ~ 7000

- 8

-10

-1

\

0.5

,, %

" - " f c = c o

I I L "

1.0 t .5 2.0 2.5

Frequency (Hz) FIG. 6. Averaged long- and shor t -per iod W W S S N ScS spec t ra corrected for i n s t r u m e n t response.

D a s h e d curves are spec t ra uncor rec ted for source exci tat ion (ft = oo); solid curves have been source corrected a s s u m i n g fc = 0.16 Hz. Po in t s A, B, and C are explained in the text.

Spectral decay method. The raw ScS amplitude spectra shown in Figures 2 and 3 were corrected for instrument response and normalized. The normalization con- stants, one for each event-station pair, were computed by integrating the amplitude spectra across two frequency bands (0.06 to 0.25 Hz for the LP and 0.3 to 1.0 Hz for the SP) and summing the results. The logarithms of the LP normalized, instrument- corrected spectra and those of the SP spectra were then independently averaged over the four event-station pairs, with each event-station pair weighted by its mean signal-to-noise ratio. The averaged spectra were corrected for source roll-off using equation (6), assuming an arbitrary value of the scaling constant, Ro. The results of this averaging and correction procedure are displayed in Figure 6. By equation (5) these curves are, to within a constant factor, estimates of J A1 I, the amplitude spectrum of the ScS attenuation operator.

FREQUENCY DEPENDENCE OF Qscs 1065

In Figure 6 two sets of curves are shown: one set corresponding to f~ = ~ (flat source; constant source correction) and one set corresponding to fc = 0.16 Hz (the best es t imate for deep-focus Tonga-Fij i events of this magnitude). At these frequen- cies, the former value of fc gives est imates of Qscs which are lower, and presumably less realistic, than those obtained from the latter. Est imates of Qscs obtained assuming tha t fc = ~ are, therefore, to be regarded as lower bounds on the actual values.

In any f requency band where Qscs is constant ln [Al[ decreases linearly with f requency and its slope is inversely proport ional to Qscs

d 1 1 Ts~s Tscs d--~ In [ A~ (~) 2Q~cs 2yQscs°, (Qscs : constant, ~0 > 0). (9)

1 0 0 0

og, 0

1 0 0

1C

I , I r I ] i f I I I f i , i , , I I

I I I I I J I t J , , J L , J

. 0 t .1 1

Frequency (Hz)

FIG. 7. Qscs° versus frequency. Points with lo error bars are the estimates from Paper I; solid curve corresponds to the ScS2/ScS ratio shown in Figure 5. Estimates bounding the shaded fields were derived by regression fits to the spectra in Figure 6 assuming Qscs is independent of frequency [equation (9)].

I t is obvious from Figure 6 tha t a constant-Q model of ScS at tenuat ion cannot satisfy the ampli tudes across the entire f requency of the W W S S N data; in particular, the ampli tude curves show a change of slope near 1 Hz, corresponding to an increase in Qscs with frequency. This curvature is equally evident in the spectra from individual event-s ta t ion pairs and is not an artifact of the averaging procedure. Moreover , at all frequencies shown, even those below 1 Hz, the magni tude of the slope is less than tha t predicted by the low-frequency (multiple ScS) est imate of Q%s = 156 ± 13, again suggesting tha t Qscs increases with frequency.

1066 S T U A R T A. S I P K I N A N D T H O M A S H. J O R D A N

To illustrate these facts, straight-line segments were fit by least-squares regression to the amplitude curves in Figure 6 over three frequency bands, and the values of Qscs were computed assuming a constant-Q model [equation (9)]. We obtained the following estimates of Q°cs

B a n d (Hz) re, -- oo f~. = 0.16 Hz

LP

0.1-0.5 383 1402 sP

o.4-1.o 329 517 1.0-2.5 1037 2270

These results are compared with the low-frequency estimates in Figure 7. As Figure 7 clearly shows, a substantial increase in Qzcs with frequency is required

by the WWSSN data, and the application of a constant-Q model to these data is inappropriate. Instead we appeal to the direct solution of equation (7). Let IA1 ] be our estimate of I All from Figure 6; then,

w T~cs Q°cs(~) ~- - - [ln A - In I A~(~)I]-'. (10)

2T

The term involving the constant A is included to emphasize that ]A~] generally differs from ] A~ ] by a scaling factor A -1. This factor arises because, in correcting for the source function [equation {6)], the zero-frequency source amplitude is unknown and is thus set to some arbitrary value. A can be thought of as the value of IA~ ] extrapolated to zero frequency. In Figure 8 Q°scs has been calculated from equation (10) for three values of In A equally spaced at 7-dB intervals and labeled as the points A, B, and C on the zero-frequency axis of Figure 6. Point B (ln A = - 7 dB) is presumably the best of these, since it provides the most continuous interpolation of Qscs values between very low and very high frequencies (Figure 8). Above 1 Hz, however, the values of Qscs calculated from equation (10) are fairly insensitive to the choice of A: for f¢ = 0.16 Hz, moving the 0 dB point from A to C in Figure 6 induces a variation in Qscs which exceeds 350 per cent at 0.1 Hz but is less than 40 per cent at 1 Hz and less than 25 per cent at 2.5 Hz. Therefore, the estimates of Qscs at high frequencies are only weakly coupled to the estimates of Qs~s at low frequencies.

of Qscs in Using the curves labeled B in Figure 8, we find that the average value 0 the band 0.4 to 1.0 Hz is about 300 for f~ = ~ and about 600 for fc = 0.16 Hz. In the band 1.0 to 2.5 Hz, the corresponding values are 400 and 750, respectively.

Energy ratio method. Parseval's theorem states that the squared modulus of an ScS time function Sl(t) is related to the squared modulus of its spectrum Sl(oz) by the equation

F 1; ] S,(t)l 2 dt = ~ ] $1(¢o)} 2 d~.

o o

(11)

Because the source function Rl(t) is time-limited, the ScS signal is pulse-like, and its significant amplitudes are confined to some time interval [tl, t2]. Similarly, because the instrument transfer function I(60) is band-limited, the significant spectral

FREQUENCY DEPENDENCE OF Vscs 1067

amplitudes of the ScS wave are confined to the frequency intervals [-o~2, -o~1] and [o~, ~2], where ¢o2 > wl > 0. We thereby define two quadratic forms

U -= I S , ( t ) l 2 dt (12) 1

~ 2 (13)

and write Parseval's formula approximately as U ~ V. From equation (5) we have

f ) 2 v = 1 iRl(,O)l~lpl(,,)l~lA~(,,)121I(,,)12 d, o. (14)

V is a nonlinear functional of Qscs(O~) through its dependence on the squared modulus of Al(o~); for a given Qscs(o~), V is specified to within a constant factor by the expressions and assumptions discussed previously. As before, the constant scaling factor arises because R0 is chosen arbitrarily. For any particular model of Qscs(W) and any choice of Ro, let VLP be the value of V calculated from equation (14) using the WWSS/LP instrument response, and let Vse be the corresponding SP value. The ratio of these intergrals is a (nonlinear) functional of Qscs which is independent of Ro

F[ Qscs(~o) ] =- Vsp/VLp. (15)

Let ULp and Usp be the time-domain integrals [equation (12)] computed from the LP and SP time series, respectively, and define the ratio

f = Usp/ULp. (16)

Then, as a constraint on Qscs(O~) we have

F[Qscs(¢)] = f. (17)

A similar equation has been applied by Burdick (1978) to the problem of determining t* (the ratio of travel time to Q) for S waves propagating beneath North America. To obtain a unique solution he assumed that t*, and hence Q, is independent of frequency.

A constant-Q model is not an essential, nor in our case a desirable, assumption. Instead, we use the constraints on Qscs in the LP band to estimate Qscs in the SP band.

For either the LP or SP spectrum of an ScS wave, we can define the pth moment of IS11 about the fiducial frequency ~o by the integral

f ~2

/~p(eoo) = I 81(~o)I2(~o - ~oo) p deo. i

(18)

1068 STUART A. S IPKIN AND THOMAS H. JORDAN

Let us choose our fiducial f requency such tha t the first momen t is zero, i.e., ttl(~O) = 0, and let us define the paramete r o z -- # J ~ . Then, in the positive f requency interval [~,, w2], [$1 [2 can be approximately described as a spectral peak centered at a f requency ~0 ~ (~1, ~2), with an area #o = ~rV and a spectral half-width o. If Qs~s is a slowly varying funct ion of frequency, then [ $1 [2 is approximately parameter ized by an average of Qscs over the interval [~o - o, w0 + a], and equat ion (17) can be rewri t ten in the form

LP SP F[ Qs~s, (19) Qs~s] = f

where LP SP qscs are Qscs and the appropria te LP and SP averages. F rom the LP spectra (Figure 2) we find tha t the mean value (± s tandard deviation) of the center

O3

o~ 0

1000

t00

' ' ' ' ' 1 i i , i , , i I i i i i i i r i ] i

/ ~ " ~ L fc~,~6 HZ

I~ WWSS SP ~1 I~ WWSS LP :',

1 0 i I i L i l l I i i i l l l l I I i i i l l [ I i r I .01 .t 1

Frequency (Hz)

FIG. 8. Q%s versus frequency. Light solid curves (fc = 0.16 Hz) and dashed curves (f~ = oo) derived from the averaged spectra using equat ion (10). A, B, and C correspond to values of A indicated in Figure 6. Heavy solid line and points with error bars same as in Figure 7.

f requency ~o is 0.123 ± 0.021 Hz and the mean value of the half-width a is 0.046 ± 0.005 Hz; f rom the SP spectra (Figure 3) these statistics are 0.438 +_ 0.043 Hz and

Qscs to be an 0.149 ± 0.006 Hz, respectively. Therefore , in equat ion (19) we in terpre t LP Qscs to be an average of average of Qscs(~) over the interval 0.08 to 0.17 Hz and sp

Qscs(~) over the interval 0.29 to 0.59 Hz. For each event-s ta t ion pair, the energy ratio f was computed by direct quadra ture

of the squared ScS t ime series over the t ime windows shown in Figures 2 and 3. Because the ins t ruments are standardized, the square-roots of these ratios are direct ly comparable if each is first multiplied by its appropria te L P / S P ins t rument gain ratio gLP/SP. This yields

FREQUENCY D E P E N D E N C E OF Qscs 1069

Event-Stat ion gLP/SP "~

1-GUA 17.42 x 10 -3 2-GUA 9.40 x 10 -3 2-KIP 8.67 x 10 -3 3-KIP 7.76 × 10 .3

T h e observed values show dispersion which m ay be at tr ibuted, at least in part, to differences in the source spectra for the three events. For example the est imates of gLP/spJf for 1-GUA and 3-KIP differ by a factor of 2.2, about the variat ion expected iffc = 0.20 Hz for event 1 and fc = 0.10 Hz for event 3. These comer frequencies fall within the range observed by Wyss and Molnar (1972). Moreover, the two inde- pendent est imates of gLP/SP'Jf for event 2 differ by only 8 per cent, a fact consistent with this explanation.

0

n

E 0 t-

O3

100

300

200

3OO

Long Period QO $eS

fc=CO

(D

o~ 0

400 200

Fro. 9. Contour plot of @ as a function Le " of Qscs and Q'~Ps for f,. = o¢. Dashed curves are values of q~ corrected for noise according to the procedure described in the test. Hatchured region indicates subjective error bounds on the locus q) = 1. Est imate of QscsSe derived assuming 150 _<- QLPs <_ 200.

For fur ther calculations we adopt the v a l u e g L P / S P ~ - = 9.04 × 10 -3 which is the median of all four observations and the mean of the two observations for event 2. For this es t imate we can compute the ratio

q) = f / F (20)

as a function of LP SP Qscs and Qscs (Figures 9 and 10). To satisfy equation (19) exactly, the pair LP SP (Qscs, Qscs) must lie on the locus ~ = 1. Most of the uncer ta in ty in est imating this locus is induced by the uncer ta in ty in the assumed source spec-

1070 S T U A R T A. S I P K I N A N D T H O M A S H. J O R D A N

t rum IR1 I; 0 has therefore been contoured for both fc = = (Figure 9) and fc = 0.16 Hz (Figure 10).

The locus 0 = 1 is also biased by any systematic differences between the LP and SP signal-to-noise ratios, but we can argue that this bias is probably small. A time series containing an ScS pulse can generally be written

91(t) = Si(t) + N(t) (21)

where $1(t) is the ScS wave form and N(t) is the noise (in our case, primarily signal- generated noise). We assume that $1 and N are uncorrelated; then, the expected value of the observed energy O - f~l Sl(t)l z dt is approximately U + N, where N is

m 6oo o~ 0

" 0

0 . _

~_ 400

0 ¢-

(/3 2OO

fc=O.16Hz

200 400 600

Q ScS Long P e r i o d o

Qscs a n d sv 0.16 C o n v e n t i o n s s a m e FIG. 10. C o n t o u r p l o t of ~ as a f u n c t i o n of LP (~]cS for fc = Hz. as in Figure 9.

the expected value of f~lIN(t)l ~ dt. Using the noise samples indicated in Figures 2 and 3 to estimate N/we computed the noise-corrected energy ratio

f _- _-- Osp - Nsp (22) ULp -- NLp"

This ratio reduces to the uncorrected ratio OSP/OLP if the signal-to-noise ratio U /N is the same for both the LP and SP records. In fact, the LP and SP signal-to-noise ratios nearly are equal: for all four event-station pairs the estimate calculated from equation (22) differs from the uncorrected ratio by less than 25 per cent. The median value of gLP/SP'v/f obtained from the corrected ratios is 8.63 x 10 -3, only 4 per cent

FREQUENCY DEPENDENCE OF Qscs 1071

less than the median value of 9.04 x 10 -3 computed from the raw ratios. Using the former to calculate ~ yields the dotted contours in Figures 9 and 10.

We have argued that reasonable variations in source spectra can account for the observed variations in the energy ratios. However, to illustrate the possible effects of other errors, we also show, as cross-hatchured bands in Figures 9 and 10, the loci of (~ = 1 generated by allowing the instrument-corrected ratio gLP/Sp~g~ - to vary within the interval 6.64 - 11.38 x 10s; i.e., a 26 per cent variation about the median value of 9.04 x 10 -3. This range spans three of the four observations and provides a conservative estimate of the uncertainties in the locus ~ = 1 induced by non- systematic errors.

1 0 0 0

U3

o~ 0

100

= r = i ~ I r I I I I L I I I I r l I [

f c = .16 Hz

f c = co

' [ 0 ' , , , , , r l I I I I I I I I [ I I I t r I I I I I i I

.01 .I I

Frequency (Hz)

FIo. 11. Q°scs versus frequency. Bounds on Q%s in the frequency interval 0.29 to 0.59 Hz are derived by the energy ratio method. Heavy solid line and points with error bars same as in Figure 7.

The assumption that Qscs is independent of frequency LP sP = Qscs) yields ( Qscs esti- mates of 270 to 320 (fc = ~ ) and 590 to 800 (fc = 0.16 Hz). These values of Qscs° are significantly greater than those found at low frequencies from multiple ScS data but are generally consistent with those derived from spectral decay method (Figure 7). As in the case of the spectral decay method, if Qscs increases with frequency, the assumption that Qscs is frequency-independent biases the results to high values.

On the other hand, an increase of Qscs with frequency also implies that the Qscs at low frequencies (0.006 to 0.06 Hz) is less than QLPs, which is an average over the band 0.08 to 0.17 Hz. Underestimating LP Qs~s results in low estimates of sp QScS. Therefore, let us make the conservative assumption that QLPs lies in the range 150 to 200, which includes the low-frequency results. From the confidence bands on the locus ~ = 1 in Figures 9 and 10 we obtain the bounds 236 < QSPs < 293 ( f~ = ~) and

1072 S T U A R T A. S I P K I N AND T H O M A S H. J O R D A N

8P 385 < Qscs < 535 (fc = 0.16 Hz), which are compared with the low-frequency data in Figure 11. The estimate LP Qscs = 236 is considered to be an unconditional lower bound; to derive this estimate no source correction was applied, a low value of

sP Qscs was used, and a large uncertainty in the locus (~ = I was allowed. Even so, this estimate is significantly greater than either of the low-frequency estimates, 156 _+. 13 (HGLP data from Paper I) and 170 +_. 15 (WWSS LP ScS jScS ratio), which confirms our conclusion that Qses must increase with frequency.

F R E Q U E N C Y D E P E N D E N C E OF Qscs The available constraints on Q°¢s(~) for the western Pacific region are summarized

in Table 2 and Figure 12. In the discussion of these data we shall comment on the estimation errors, particularly on the possible sources of bias.

At frequencies greater than about 0.01 Hz, Qs~s can be obtained directly from ScS and multiple ScS data using the methods outlined above. However, at frequencies

T A B L E 2

CONSTRAINTS ON THE FREQUENCY DEPENDENCE OF Q,~.~ IN THE WESTERN PACIFIC

Data Set Method Frequency Band f, (Hz)* Q!~,,~ (Hz}

Norma l modes Gross ea r th invers ion (Sailor & Dzie- 0.0003-0.01 - - 190-240 wonski, 1978; Ander son & Hart , 1978b)

H G L P Spectra l s tacking (Paper I) 0.006-0.06 - - 156 +_ 13 t W W S S LP ScS2/ScS ratio 0.03-0.07 - - 170 + 15t W W S S S P / L P Ene rgy ratio 0.29-0.59 ¢o 236-293 W W S S S P / L P Energy ratio 0.29-0.59 0.16 385-535 W W S S SP Spectral decay 0.4-1.0 ~¢ 300 W W S S SP Spectra l decay 0.4-1.0 0.16 600 W W S S SP Spectral decay 1.0-2.5 ~ 400 W W S S SP Spectral decay 1.0-2.5 0.16 750

* Corner f requency a s s u m e d in source spec t rum. t Uncer ta in t i e s quoted are s t anda rd errors.

below 0.01 Hz, the only good constraints on Qscs come from the inversion of normal Qscs in the mode data. According to Sailor and Dziewonski (1978), the value of 0

normal mode band lies between 200 and 230; the estimates computed for their various models span the interval 195 to 232. These are somewhat lower than those found by Anderson and Hart (1978b), who have used a different normal mode data set; their most recent models, SL7 and SL8, yield 243 and 235, respectively. We therefore adopt the estimate 190 < Q°,.s < 240 in the normal mode band 0.0003 to 0.01 Hz (Figure 12) for the purposes of comparison with the ScS data at higher frequencies.

The validity of such a comparison may be questioned on the grounds that the normal mode data sample the entire Eart h , not just the western Pacific. However, we contend that the difference between Qscs for the average Earth and that for the western Pacific is small. In a study of lateral attenuation variation (Sipkin and Jordan, 1979) we have estimated the multiple ScS attenuation operator in the HGLP band (0.006 to 0.06 Hz) for a number of geographic and tectonic regions. Generally speaking, young oceanic regions such as the eastern Pacific are charac-

of Qscs (<150), whereas old continental terrains are character- terized by low values 0 ized by high values (>180). The data imply that, in the HGLP band, Q°scs for the

FREQUENCY DEPENDENCE OF Qscs 1073

average Earth is intermediate to these extremes, with a value similar to that observed for old ocean basins such as the western Pacific (~160). This inference is consistent with our studies of lateral variations in ScS travel times (Sipkin and Jordan, 1975, 1976), which demonstrated that the times for the western Pacific and the average Earth are nearly identical.

If we accept as valid the inference that the western Pacific and the average earth are characterized by the same Qs~s, then we must conclude that Qscs decreases with

of Qscs derived from frequency somewhere in the vicinity of 0.01 Hz. The estimates 0 the best available normal mode models (characterized by nonzero bulk dissipation)

o~ 0

1 0 0 0 0

1 0 0 0

100

1C

. . . . ' ' " 1 . . . . . . . . . I . . . . . . . . I

/ ' / / / / / / / / / / / ' / / / / / ' / " / ' ~ / ~ W W S S S P ---J

I _ _ H G L P i i P 8~ WWSS LP WWSS

SP/LP

I t i ~ i l f l I I t I I l l l l I I I r r l l l l I I i I I I J

])01 .01 .1 1 0

Frequency (Hz)

FIG. 12. Summary of constraints on Q°cs. Lower bounds on WWSS estimates derived by assuming fc = oo; best estimates are indicated by heavy solid lines, derived by assuming f%-- 0.16 Hz. Solid curves computed from the absorption band model, equation (28), for a = 0, "/'M = 10 sec, and the T~ values shown.

are 230 (model QKB of Sailor and Dziewonski, 1978) and 235 (model SL8 of Anderson and Hart, 1978b), both of which lie near the high end of the normal mode range shown in Figure 12. On the other hand, the multiple ScS data require that Q°scs = 160 at frequencies greater than 0.01 Hz, considerably below the range permitted by the mode data. The attenuation data for the modes may be corrupted by multiplet splitting due to lateral heterogeneity and rotation (Alsop et al., 1961; Dahlen, 1968), but such splitting should bias mode-derived estimates of Qscs to low values, which only reinforces our conclusion.

In Figure 12 the bounds placed on Q°¢s in the range 0.006 to 0.07 Hz allow considerable room for error (130 to 200); they include, for example, the 95 per cent confidence intervals for both the HGLP estimate from Paper I (156 _+ 13) and the WWSS LP estimate from the S c S j S c S spectral ratio for event 3 (170 _ 15). Based

1074 STUART A. SIPKIN AND THOMAS H. JORDAN

on our further work with western Pacific HGLP data (quoted previously), we believe that the actual average value for this band probably lies in the more restricted interval 140 to 180.

We have examined in detail the possibility that the values of Qses retrieved in our multiple ScS experiments are biased by frequency dependent propagation effects and/or radiation-pattern variability ignored by the estimation procedures. We tested these procedures by applying them to synthetic data. Synthetic seismograms were computed by normal-mode summation and propagator-matrix methods using a double-couple source parameterization and realistic earth models (including models with oceanic crustal structures); the geometry was chosen to closely duplicate the geometry of the actual experiments. Generally, Qscs estimated from the synthetic data differed from its model value by less than 10 per cent. In particular, no significant systematic errors were caused by interference of other arrivals, diffraction effects associated with curved interfaces, or resonances within the crust and upper mantle. A fuller account of these experiments is given elsewhere (Sipkin and Jordan, 1979).

Analysis in a preceding section has demonstrated that the low values of Qses required by the multiple ScS data are inconsistent with the large ScS amplitudes observed on WWSS SP records, implying that Qscs increases with frequency above about 0.1 Hz. In fact, no sophisticated analysis is required to argue this point: if Qs¢s at high frequencies were as low as 160, or even 200, the short-period ScS signals which are so prominent in Figures 3 and 4 would be well below noise levels!

However, much uncertainty accompanies any quantitative estimate of Qscs at these high frequencies. The largest uncertainty arises through the crude and naive parameterization of the source spectrum in terms of a single corner frequency whose value is poorly constrained [equation (6)]. Two values have therefore been used in our analysis of the high-frequency data: fc = oo and fe = 0.16 Hz. If the far-field source amplitude spectrum is indeed a monotonically decreasing function of fre- quency, which asymptotically it must be (e.g., Hanks and Wyss, 1972), then the former provides a lower bound on Qs~s, but the latter is probably more realistic.

Assuming that f~ = 0.16 Hz gives the best estimate, we find the Q°scs increases, apparently monotonically, from values less than 200 at frequencies below 0.1 Hz to values in excess of 700 at frequencies greater than 1 Hz. Moreover, for any finite value of the corner frequency, our estimates of Q°~s above 1 Hz exceed 400.

In evaluating these high-frequency estimates we have considered other potential sources of bias. The algorithm used in deriving Qscs by spectral decay was tested by applying the windowing, transformation, and smoothing procedures to synthetic data, with good results--no significant bias was introduced. The effects of crustal layering beneath the receivers were also investigated with synthetics. We computed the transmission response of simple three-layer crustal models for Oahu (Furumoto et al., 1968) using propagator-matrix methods. The response spectra have amplitudes which vary by factors of 2 to 2.5 over the frequency interval 0.1 to 2.5 Hz, but these variations introduce no systematic trends to the ScS spectra, and they are effectively eliminated by our smoothing and averaging procedures.

More difficult to evaluate is the possibility that the SH-reflection coefficient for the core-mantle boundary is not unity for all frequencies, as we have implicitly assumed in our calculations. Most seismic observations are consistent with the notion of a simple core-mantle interface (e.g., Kanamori, 1967; Choy, 1977), perhaps the most spectacular being the high-frequency observations of internal multiple reflections such as P5KP (Engdahi, 1968). Furthermore, theoretical calculations

F R E Q U E N C Y D E P E N D E N C E OF Qs~s 1 0 7 5

show that any frequency dependence of the reflection coefficient due to nonzero core viscosity is small (Sato and Espinoza, 1966, 1967). But we cannot eliminate the possibility that significant high-frequency ScS energy is scattered by unresolved fine structure in the vicinity of the core-mantle boundary. In fact, some sort of small- scale, lateral heterogeneity near this boundary evidently is needed to explain the high-frequency precursors to PKIKP at distances less than 140 ° (Haddon and Cleary, 1974).

Scattering by velocity heterogeneities near the core-mantle discontinuity, as well as elsewhere along the ScS path, is strongly dependent on frequency and thus may bias our estimates of Qscs. The importance of this bias can be roughly assessed from Chernov's (1960) formulas for the scattering of scalar waves by weak, random heterogeneities. For a plane wave with wave number k, we write the fractional energy loss per cycle due to scattering as 2 ~ra/k, where a is a scattering coefficient. We suppose that the heterogeneity is described by a Gaussian correlation function, parameterized by a root-mean-square fluctuation in the refractive index, r, and a correlation distance, d. Using Chernov's (1960, p. 55) formula for a, we can relate the apparent quality factor of the plane wave, Qapp, to the intrinsic quality factor of the medium, Q, by the equation

-1 _- Q-1 e-h2d~). Qapp + ~ ~2.kd(1 - (23)

For this formula to be applicable, the heterogeneity must be weak in the sense that the scattering term on the right-hand side of equation (23) must be small compared with unity. At low frequencies (kd << 1, Rayleigh scattering) the correction term increases as the third power of frequency, whereas at high frequencies (kd >> 1) it is proportional to frequency.

For heuristic purposes we consider a wave traveling at 6 km/sec through a heterogeneous region characterized by 1 per cent velocity variations (~ -- 0.01) which have a scale length d = 10 km. Then equation (23) yields

Qa~o = Q-1 + 2.0 × 10 -7

--1 ~ V-1 Qaoo + 1.9 × 10 -3

(0.01 Hz)

(1.0 Hz).

In this case, the scattering term is clearly negligible for frequencies as low as 0.01 Hz but it is important at 1 Hz. For example, if the intrinsic Q of the medium is 1000, scattering reduces the apparent Q of a 1-Hz wave to about 350.

We use this hypothetical calculation for scalar waves to motivate two important conclusions

1. Because the scattering coefficient in a random medium increases rapidly with frequency, the fact that significant ScS amplitudes are observed at high frequencies (>1 Hz) implies that the bias in Qscs measurements due to random scattering at low frequencies (<0.1 Hz) is probably small.

2. Although scattering may indeed be important at high frequencies, its net effect is to lower wave amplitudes and thus bias the estimates of Qscs to low values. Consequently, the high-frequency values given in Table 2 and Figure 12 must be strictly regarded as lower bounds, and the increase in Qscs with frequency is more rapid than our estimates imply.

We note, however, that if a linear viscoelastic rheology adequately explains anelastic dissipation, then the anelastic quality factor in equation (23) cannot

1076 STUART A. SIPKIN AND THOMAS H. JORDAN

increase with frequency more rapidly than a function proportional to frequency (see inequality 25). In this case, the fact that Qscs increases at high frequencies places a severe constraint on the amount of elastic scattering which occurs at these high frequencies.

As a final caveat, we should admit that bias can also be introduced into our measurements of Qscs by the lateral variations in the crust and upper mantle which must certainly exist beneath the recording sites (both are perched on volcanoes). Such variations may contribute to the complexities observed in the short-period ScS wave forms and could amplify high frequencies. For example, some P-SV energy could be preferentially scattered into SH by the heterogeneity. Fortunately, in all cases but one (2-GUA) the ScS amplitudes on the transversely polarized components exceed those on the radial components by a factor greater than two, so we do not expect this kind of bias to be large, but other frequency dependent propagation effects associated with nonplanar layering are more difficult to evaluate.

IMPLICATIONS

The data on Qscs in the western Pacific now span more than three decades in frequency and should be helpful in evaluating the various physical mechanisms proposed for seismic attenuation (e.g., Knopoff, 1964; Jackson_and Anderson, 1970; O'Connell and Budiansky, 1977; Goetze, 1977; Anderson and Minster, 1979}. Of course, the problem of the dissipation mechanism is difficult, and its solution obviously requires more data than those presented here; but some tentative state- ments may be justified.

Although our data provide only integral constraints on the variation of Q, with depth, these constraints imply that the regions in which Q, is frequency dependent occupy a substantial portion of the mantle. For example, let us suppose that above a radius ro the dependence of Q, on frequency is arbitrary, but that below ro, Q, is independent of frequency. Then, at any frequency, an upper bound on Qs~s is obtained from the assumption that Q~ = ~ for r > r0

Q°~s < [ Ts~s Jb Q"-l(r) dr/v~(r) (24)

We have computed this bound assuming that the Q~ structure below ro is given by the average earth model QKB (Sailor and Dziewonski, 1978). If ro = 6100 km, approximately the base of the Gutenberg low-velocity zone, then Qscs must be less than 300 for all frequencies, which clearly violates the constraints displayed in Figure 12. At frequencies of 1 to 2.5 Hz, our lower bound on Q°cs is 400 and our best estimate is 750, which require that ro < 5950 km and r0 < 5210 km, respectively. Thus, the region of frequency dependence must extend below the depth of the olivine-spinel phase transition and probably includes at least part of the mantle below a depth of 600 km.

The relationship between depth and frequency dependence is undoubtably com- plex. Since Qscs in the western Pacific is not a monotonic function of frequency, Q~ obviously cannot be a monotonic function of frequency everywhere in the mantle. The variation of Q~,, implied by the data may be describable in terms of the absorption band models investigated by Anderson and his co-workers (Anderson et al., 1977; Anderson and Hart, 1978a; Anderson and Minster, 1979). The strain retardation spectrum of a general linear viscoelastic solid can be specified as a

FREQUENCY DEPENDENCE OF Qs~s 1077

distribution of relaxation times, D(r), from which the solid's response to transient disturbances can be deduced (e.g., Liu et al., 1976). Following Liu et al. {1976) and Anderson et al. (1977), we consider a model of weak shear dissipation (Q, >> 1) in which the nonzero values of D(z) are confined to the interval ~ _-< ~ _-< VM, called the absorption band. Such a model has the following important features

} d l n Q , / d l n ~ } < l , if w > 0 (25)

Q , - 1/~0, i f 0 < 0) <<~ TM -1 (26)

Q~-w, if w >> ~-1. (27)

Anderson and Minster (1979) discuss a distribution function of the form

OLT a - I

D(r) ~'i" -- ~',n" H(r ~',~)H(I"M r) (28)

where H is the Heaviside distribution and a is a constant. This model implies that

Q~ - w ~, if VM-' << ~o << rm -1 (29)

which yields a constant Q if a -- 0. Anderson and Minster (1979) suggest that the value of a is approximately 1, which they argue is consistent with the available Chandler wobble, tidal, and free oscillation data, as well as laboratory measurements of transient creep and internal friction at high temperatures.

If such a model is applicable to the Earth, we can infer that, at least in some regions of the mantle, both of the cutoff frequencies Tm -1 and ~M -~ must lie within our band of observation. This follows from the fact that Qscs does not vary as ~/3 across the seismic band, nor is it even a monotonic function of frequency. The rapid increase of Qscs begins somewhere between 0.1 and 1 Hz. In Figure 12 we model this increase by a single cutoff time rm using equation (28) with a = 0. The data are best described by cutoff times in the interval 0.2 to 1.0 sec. Values in this range are also consistent with other observations on the frequency dependence of Q (Kurita, 1968; Solomon and ToksSz, 1970; Minster, 1978).

Similarly, our observation that Qscs apparently decreases with frequency in the vicinity of 0.01 Hz is inconsistent with the Anderson-Minster model unless TM -~ lies within the seismic band for some portion of the mantle. Since dissipation is probably governed by thermally activated processes, the cutoff times should depend exponen- tially on inverse temperature and hence should decrease as temperature increases (Anderson et al., 1977; Anderson and Hart, 1978a). Therefore, the observed decrease in Qs~s with frequency may arise from parts of the mantle where ZM is small; i.e., regions near their melting point, such as the Gutenberg low-velocity zone.

Our results have implications for seismology which are independent of these speculative remarks. Because Q, evidently varies with frequency throughout large portions of the mantle, attenuation models based on the constant-Q assumption must be used with caution. In particular, as emphasized by Solomon (1972), such models may not be adequate for correcting body-wave spectra for dissipation in seismic source studies. The frequency dependence of Q must also be accounted for

1078 STUART A. SIPKIN AND THOMAS H. JORDAN

in correcting eigenfrequencies and travel times for the effects of attenuative disper- sion (Kanamori and Anderson, 1977; Hart et al., 1977).

ACKNOWLEDGMENTS

The authors thank D. L. Anderson and J. B. Minster for their preprint and for useful discussions and L. J. Burdick and G. Mellman for the use of their WWSS SP instrument response algorithm. Critical comments by G. MeUman are also gratefully acknowledged. This research was supported by the National Science Foundation under Grants EAR77-00897 and EAR78-12962.

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GEOLOGICAL RESEARCH DIVISION SCRIPPS INSTITUTION OF OCEANOGRAPHY LA JOLLA, CALIFORNIA 92093

Manuscript received December 6, 1978