The rotationally and elliptically split normal modes of the earth are observed for the 1960 Chilean earthquake by analysis in the time domain. One hundred and fifty hours of the Isabella, California, strain record are narrow band filtered about the central frequency of each split multiplet to isolate the complex wave form resulting from the interference of the different singlets. We compute synthetic seismograms using our previous theoretical results, which show the dependence of the amplitude and phase of the singlets on source location, depth, mechanism, and the position of the receiver. By comparing these synthetics to the filtered record, we conclusively demonstrate the splitting of modes whose splitting had not been definitely resolved: torsional modes (_0T_3, _0T_4) and spheroidal modes (_0S_4, _0S_5). The splitting of _0S_2 and _0S_3 is reconfirmed. We obtain good agreement between the synthetics and the filtered data for a source mechanism (previously determined from long-period surface waves) of thrust motion on a shallow dipping fault

Geller, Robert J.

Stein, Seth

Caltech Authors - Main

Bulletin of the Seismological Society of America, Vol. 68, No. 2, pp. 325-332, April, 1978 
TIME-DOMAIN OBSERVATION AND SYNTHESIS OF SPLIT 
SPHEROIDAL AND TORSIONAL FREE OSCILLATIONS OF THE 1960 
CHILEAN EARTHQUAKE: PRELIMINARY RESULTS 
BY SETH STEIN AND ROBERT J. GELLER* 
ABSTRACT 
The rotationally and elliptically split normal modes of the earth are observed 
for the 1960 Chilean earthquake by analysis in the time domain. One hundred 
and fifty hours of the Isabella, California, strain record are narrow band filtered 
about the central frequency of each split multiplet to isolate the complex wave 
form resulting from the interference of the different singlets. We compute 
synthetic seismograms using our previous theoretical results, which show the 
dependence of the amplitude and phase of the singlets on source location, 
depth, mechanism, and the position of the receiver. By comparing these syn- 
thetics to the filtered record, we conclusively demonstrate the splitting of modes 
whose splitting had not been definitely resolved: torsional modes (0T3, oT4) and 
spheroidal modes (0S4, 0S5). The splitting of 0S2 and 0S3 is reconfirmed. We 
obtain good agreement between the synthetics and the filtered data for a source 
mechanism (previously determined from long-period surface waves) of thrust 
motion on a shallow dipping fault. 
INTRODUCTION 
Split peaks in the earth's normal mode spectrum were first observed for the 
1960 Chilean earthquake. The splitting of 0S2 and 0S3 was reported by Benioff, Press, 
and Smith (1961) and Ness, Harrison, and Slichter (1961). Slichter (1967) observed 
the splitting of oS3 for the 1964 Alaskan earthquake. Splitting has never been 
conclusively demonstrated for other spheroidal modes or for any torsional modes. 
Extensive theoretical efforts (summarized in our earlier papers) have been devoted 
to the computation of the eigenfrequencies of the 2l ÷ 1 individual singlets into 
which the multiplet of angular order l (e.g., nSz) is split. This frequency splitting of 
the very long-period modes is primarily due to the earth's rotation (Pekeris et al., 
1961 and Backus and Gilbert, 1961) and ellipticity (Dahlen, 1968). In an earlier 
paper (Stein and Geller, 1977) we derived theoretical results that allow us to 
calculate the amplitudes and phases of the split normal modes (or equivalently, 
their time history) excited by a double couple of arbitrary orientation resulting 
from slip on a fault plane. These results were applied successfully (Geller and 
Stein, 1977) to split spectra of the Chilean and Alaskan earthquakes. 
In this paper we use source models derived from long-period surface wave studies 
to generate synthetic seismograms for each of six multiplets (oS2 - 0S5, 0T3, 0T4), and 
compare them to a time-domain record of the Chilean earthquake which has been 
narrow band filtered to isolate that multiplet. This procedure yields several results 
which were not obtained by earlier studies in the frequency domain. We show that 
splitting is present for modes for which it has not been previously resolved, including 
torsional modes. 
Our results are derived entirely from wave-form shapes, and make no use of 
absolute amplitude information, due to calibration complexities. The absolute 
amplitudes are used in a forthcoming study (Stein and Geller, 1978) to determine 
the moment at very long periods. We also determine accurate Q's for the low-order 
modes. 
* Also Department of Geophysics, Stanford University, Stanford, California 94305. 
325 
326 SETH STEIN AND ROBERT J. GELLER 
DATA AND ANALYSIS 
Figure 1 shows 150 hr of the strain record (filtered electronically to reduce the 
tidal amplitudes) from Isabella, California (Benioff et al., 1961) for the great 1960 
Chilean earthquake. The record (top) was processed to remove tides by twice 
subtracting 3-h running averages. The resulting record (bottom) was tapered at both 
ends and then filtered to isolate and resolve individual low angular order modes. 
The resulting narrow band filtered data are shown in the upper traces of Figures 2 
through 7, for four spheroidal and two torsional modes. All these wave forms show 
the general (e -wt/2q) attenuation ofthe entire mode superimposed on a complicated 
pattern which results from the interference of the split singlets. 
The middle trace of each figure shows a synthetic seismogram for each mode 
which includes the effects of splitting. The results of Stein and Geller {1977) for 
CHILEAN EARTHQUAKE Mey 22, 1960 
STRAINMETER AT ISABELLA, CALIFORNIA 
E l  t/': 'i/ ' 
~t I ' :~l,t 
b J 
'" 't/ ~, 
i . ,  
j:, '~' 4 
NETWORK STRAIN 
'[ll!!ir• I 
TIDES REMOVED 
L L__~ ~_ . . . . . . .  ~ ~ ~j~ 
0 50 hr I00 150 
FIG. 1. Isabella strain record of the Chilean earthquake (top trace) and the high-passed record with 
tides removed (bottom trace). The origin time of this figure and all others, is 1911 hours, 22 May 1960, 
the origin time of the main shock. The digitized record (top) begins 289 min later, at 0000 hr, 23 May. 
The high-passed record (bottom) starts another 3 hr later. 
the excitation of split modes are used to calculate the synthetics. The effect of 
splitting can be seen by examining the lower traces, which are calculated without 
splitting. In this case each mode appears as a pure damped harmonic oscillator, 
since all its singlets have the same frequency. The data are fit far better by the 
synthetics with splitting than by those without it, and thus splitting is demonstrated. 
The eigenfrequency of each singlet is computed using Anderson and Hart's (1978) 
values for the unperturbed eigenfrequencies and Dahlen's {1968) rotational splitting 
parameters. The elliptical splitting parameters are not used, due to the difficulties 
involved in previous calculations of these parameters (Woodhouse, 1976). Dahlen 
{personal communication) states that more accurate recent computations show 
that the spheroidal mode elliptical splitting parameters may be neglected for our 
purposes. Accurate values have not yet been published for the elliptical splitting of 
torsional modes, or for splitting due to lateral heterogeneities. 
Synthetic seismograms are computed for a range of source parameters; good 
agreement is obtained for the source mechanism determined by Kanamori and 
TIME DOMAIN OBSERVATIONS OF SPLIT FREE OSCILLATIONS 327 
Cipar {1974) from long-period surface waves. The rupture was initiated at 38°S, 
286.5°E and propagated at 3.5 km/sec to 46°S, 286.5°E, on a fault plane dipping 
10 ° east and striking N10°E. (We approximate the finite source by five point 
sources at a depth of 55 km.) The slip angle is 90 °, a pure thrust motion. We are 
also including a precursory slip (Kanamori and Cipar, 1974; Kanamori and Anderson, 
1975) at 41.5°S, 285.7°E with a rise time of 5 min starting 15 min before the main 
shock, and with a moment equal to that of the main shock. 
Excellent agreement between the synthetics and data is obtained for the longest 
period spheroidal modes, oS2 and oS3 (Figures 2 and 3). The absolute amplitude of 
oS2 T=55.8 rain 
VVF I I  f~ JU l  ~ I  L t  l J ] l~ ' , J  
L ~  ~ .L~L_  ~ r ~ ~ ,  .,~ ~ _~ 
0 50 hr I00 150 
FIG. 2. Data and synthetics for 0S2. The top trace is filtered data from the high-passed Isabella strain 
record of the Chilean earthquake. The middle trace is the synthetic seismogram, including the effects of 
splitting, and the bottom trace is the synthetic without splitting. Both synthetics are computed using the 
same Q. The synthetics were tapered and filtered in the same way as the data. The filtering and tapering 
are responsible for reducing the initial amplitude (especially visible in the bottom trace). 
these modes on the Isabella record is about three times greater than any of the 
other modes (Geller, 1977). Thus the general envelope shape of these modes match 
the data quite well. Note the large peak at 35 hr for oS2 (Figure 2) and the peak at 
80 hr for oS3 (Figure 3). The nodal times of the data nd synthetics also agree quite 
well. 
We obtain acceptable fits for the other modes. Their absolute trace amplitudes 
are much lower, and in the case of oT3, oT4 and oS4 and oS5 are barely resolvable above 
the noise (Benioff et al., 1961). Nonetheless, the beat pattern is clearly evident 
(Figures 4 to 7), demonstrating splitting. The discrepancies between the data and 
synthetics are probably due to the noise, and, in the case of torsional modes, 
328 SETH STE IN  AND ROBERT J. GELLER 
ellipticity splitting. Given these difficulties, the agreement ofdata and synthetics i
quite acceptable. 
We tested a wide variety of source geometries and obtained good results for the 
source parameters given previously. We thus conclude that the fault mechanism 
derived at periods of several hundred seconds is consistent with the data at far 
longer periods. This does not, of course, demonstrate that this is the only acceptable 
source mechanism. A more accurate mechanism ight result from use of better 
splitting parameters, but will still be limited primarily by the noise in the data. 
A prior study of split spectra (Smith, 1961) suggested the possible splitting of o&, 
oS5 T=35.6 m~n 
,,;:~ ........ gl(@2il,:~ al,~ . . . . .  
DATA 
, v , .  ,, / ii~ 
~7.g~*k~:~ ,  ~%, ~7{~ • 
SYNTHETIC 
WITHOUT SPLITTING 
,bo h~ ~00 150 
FIG. 3. Data and synthetics for 0Sa. Details are as in Figure 2. Note that the maximum trace amplitude 
occurs 80 hr into the r cord. 
oT2, and oT3, but Smith noted that the data for these modes were marginal. The high 
resolution spectra presented by Wiggins and Miller (1972) for the UCLA gravity 
record of the 1964 Alaskan earthquake also suggest the splitting of oS4. Our time 
domain analysis hows the beat pattern, and thus the splitting, of o&, o&, oT3, 
and oT4. 
APPARENT Q OF SPLIT MODES 
We study the attenuation ofsplit modes by comparing the data to the synthetics. 
Splitting is no longer visible when the beat time is longer than the Q decay time. [In 
the frequency domain, this occurs when the broadening of spectral peaks due to 
T IME DOMAIN OBSERVATIONS OF SPL IT  FREE OSCILLATIONS 329 
attenuation is much greater than the frequency separation of singlets resulting 
from rotation and ellipticity (Gilbert and Backus, 1965).] Figures 5 and 7, respec- 
tively, show that splitting is barely detectable for 0S~ and o2'4, because the synthetics 
with and without splitting are much more similar than those for the longer period 
modes. 
The synthetics are computed by using a Q of 400 in Figures 2, 4, and 5; a Q of 300 
for Figure 3; and a Q of 450 for Figures 6 and 7. These Q values are nominal and 
cannot be well constrained by using only 150 hr of data. 
It is important to note that because the amplitude of the record does not decrease 
monotonically with time, it is difficult to determine Q by examining the amplitude 
of spectral peaks from successive time windows. This is demonstrated by 0,93 (Figure 
oS4 T :25 .8  rain 
~ ~ I ~ ~ I , ~ t  ~ I 
0 50  hr  I00  150 
FIG. 4. Data and synthetics for 0S4. Details are as in Figure 2. Substantial  noise appears to be present in 
the data. 
3), in which the maximum trace amphtude occurs 80 hr into the record. Thus the 
peak spectral amplitude will not even be a monotonically decaying function f time. 
A more subtle difficulty in amplitude decay analysis occurs even when beats are 
not apparent on the filtered seismogram. For these long-period modes, the amplitude 
as a function of time is controlled at least as much by the interference of the 
singlets as by attenuation. This can be seen clearly by comparing the decay of the 
two synthetics for 0S5 (Figure 5), computed with the same Q. Destructive interference 
due to splitting makes the synthetic with splitting (middle trace) decay much 
oS5 T=i99  mm 
.... DAT,~ 
(J,,~, 
i ~" ~;,:~ ~:,~' ~ ~, 
I~ ~',~,',  
SYNTHETIC 
..... WITHOUT SP[ ITT',NG 
O 50 hr Ioo 15o 
FIG. 5. Data and synthetics for oS5. Details are as in Figure 2. The two synthetics computed with the 
same Q decay at different rates as a result of the destructive interference caused by splitting. 
oT3 T=28,4 min 
, , I 
J 
0 50 hr IOO 150 
FIG. 6. Data and synthetics for oT3. Details are as in Figure 2. 
T IME DOMAIN OBSERVATIONS OF SPL IT  FREE OSCILLATIONS 331 
faster than the unsplit synthetic ( bottom trace). Thus if splitting is neglected, the Q 
measured from the amplitude decay would be far too low. 
It is thus clear that splitting must be correctly included for accurate Q determina- 
tions. Such determinations are difficult and beyond the scope of this paper. Our 
intent here is to discuss the importance of splitting for the measurement of the Q's 
of long-period modes. 
Alsop et al. (1961) discuss the measurement of Q for split modes. They show that 
Q can be correctly measured under several conditions: either if the record length is 
adequate to completely resolve individual singlets, or the record length is much 
greater than the beat time, or if the same portion of a cycle is being averaged over. 
Our method is applicable in general even if these special circumstances are not 
fulfilled. 
oT4 T=2:7 m~n, 
... "J.4 4..:.-~ 
,~ ,~I  . . . . . . . . . . . . . . .  = , . . , ,  t=.+,;,, 
s:,i?i:~i ~ , :  !4.'~ .... 
DAT#~ 
I 
. ,¢ . . F I .  1..." , 
i; t. "/! . 
SYNTHETIC 
u . i - -~  + = * ' : .  
=~#)*C£, . "  *,,i . "  + 
W:THOUT SPLITTING 
& so --hi :oo ~so 
FIG, 7. Data  and synthet ics for 0T4. Detai ls are as in Figure 2. 
CONCLUSION 
The comparison of narrow band filtered records to computed synthetic seismo- 
grams provides a new method for the analysis of normal modes when splitting is 
present. This technique allows the resolution of splitting in cases when spectral 
amplitude analysis (which does not incorporate phase information) is insufficient. 
We confirm the splitting of 0S2 and oSm and demonstrate he splitting of oS4, oS5, oT3, 
and oT4. Time domain analysis of split normal modes represents a powerful new 
approach to ultra-long period source and attenuation studies. 
332 SETH STEIN AND ROBERT J. GELLER 
ACKNOWLEDGMENTS 
We thank Stewart Smith and Hiroo Kanamori  for providing us with data from and 
information about the Isabella strain record. Yoshio Fukao, Hiroo Kanamori ,  and Emile 
0kal  critically read the manuscript. We have also benefited from the comments of an 
anonymous reviewer. Seth Stein was supported by a fellowship from the Fannie and John 
Hertz Foundation. This research was supported by the Division of Earth  Sciences, National 
Science Foundation, NSF  Grants EAR76-14262, EAR74-22489, and EAR77-14675. 
REFERENCES 
Alsop, L. E., G. H. Sutton, and M. Ewing (1961). Measurement ofQ for very long period free oscillations, 
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Backus, G. and F. Gilbert (1961). The rotational splitting of the free oscillations of the Earth, Proc. Natl. 
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Benioff, H., F. Press, and S. Smith (1961). Excitation of the free oscillations of the Earth by earthquakes, 
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Geller, R. J. (1977). Amplitudes of rotationally split normal modes for the 1960 Chilean and 1964 Alaskan 
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excited by a double couple, J. Phys. Earth, 25, 117-142. 
Stein, S. and R. J. Geller (1978). Attenuation measurements of split normal modes for the 1960 Chilean 
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SEISMOLOGICAL LABORATORY 
CALIFORNIA INSTITUTE OF TECHNOLOGY 
PASADENA, CALIFORNIA 91125 
CONTRIBUTION NO. 2892 
Manuscript received May 24, 1977 


Time-domain observation and synthesis of split spheroidal and torsional free oscillations of the 1960 chilean earthquake: Preliminary results

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Time-domain observation and synthesis of split spheroidal and torsional free oscillations of the 1960 chilean earthquake: Preliminary results

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