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Observational study of terrestrial eigenvibrations
Physics of the Earth and Planetwy Interiors, 28 (1982) 27—69 Elsevier Scientific Publishing Company, Amsterdam — Printed in The Netherlands
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Observational study of terrestrial eigenvibrations Roger A. Hansen Seismographic Station, University of California at Berkeley, Berkeley, CA Q4720((J.S.A.) (Received August 24, 1981; revision accepted November 24, 1981)
Hansen, R.A., 1982. Observational study of terrestrial eigenvibrations. Phys. Earth Planet. Inter.. 28: 27—69. A comprehensive analysis has been made of analog and digital recordings of eigenvibration ground motion obtained following four great earthquakes; August 1976 (Philippines), August 1977 (Indonesia), September 1979 (West Irian), and December 1979 (Colombia). The time series (ranging in length from —28 to 140 h) are assumed to be linear combinations of damped harmonics in the presence of noise. Tables are calculated from values of the four parameters: 8, used in describing eigenvibrations, period of oscillation, amplitude, damping factor Q, and phase together with their statistical uncertainties (53 spheroidal modes, 054 to 0~48’ and 13 torsional modes. 0T8 to 0T45). The estimation procedures are by the methods of complex demodulation and non-linear regression that specifically incorporate into the basic model the decaying aspect of the oscillations. These methods, extended to simultaneous estimations of groups of modes, help to eliminate measurement error and measurement bias from estimations of 8. The result is that overtone modes veiy near in frequency to fundamental modes can, under certain conditions, be resolved through a non-linear regression technique, although parameter uncertainties are underestimated in general. Of the time series analyzed, 17 were from a northern California regional network of ultra-long period seismographs at Berkeley (three components), Jamestown (vertical component), and Whiskeytown (vertical component) following the four listed earthquakes. The other 7 time series were recorded digitally by the worldwide IDA network following the 1977 Indonesian earthquake. Weighted regional and worldwide averages were made for period and Q of each eigenvibration mode. From the theoretical viewpoint, comparisons of measured period, Q, amplitude, and phase for all modes analyzed led to five conclusions. First, there are no detectable systematic shifts in period. Q, or phase of eigenvibrations within a region whose dimensions are less than a wavelength. Second, though not conclusive, there may be slight systematic shifts in period (<0.65 s) and relative amplitudes within the California regional network due to different source positions and mechanisms. Differences in Q values are not statistically significant. Third, even though differences in period obtained worldwide were as great as 1.33 s (~‘0.33%),differences between Q values (as great as 20%) for the same mode were not significant. The conclusion is that the damping characteristics of singlet eigenfunctions are not observed to be significantly different. Fourth, the assumption that a multiplet ,, 5, behaves as a single oscillation is valid from at least 0~7 through 0S30. Fifth, no systematic pattern emerged for the shift of eigenperiod as a function of order / or position on the Earth.
1. Introduction This study consists of a statistical analysis of 24 time series of ground motion obtained following four large earthquakes. The analysis yields tables of the four parameters (denoted by 8) used in describing eigenvibrations, period of oscillation, *
Present address: NRC Research Associate, NOAA/ CIRES/NGSDC, Boulder, CO 80303, U.S.A.
amplitude, damping factor Q, and starting phase together with their statistical uncertainties. The estimation procedures are by the methods of cornplex demodulation (Tukey, 1961) and non-linear regression that specifically incorporates in the basic model the decaying aspect of the oscillations (penodogram analysis does not). Complex demodulation displays are used not only to locate as precise a value of eigenfrequency as possible, but also to ascertain the nature of an oscillation that corre-
0031-9201 /82/0000—0000/$02.75 © 1982 Elsevier Scientific Publishing Company
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sponds to a particular spectral peak in a periodogram. These displays exhibit, as a function of time, the amplitude and phase of a particular harmonic oscillation isolated in frequency. This is used to determine whether difficulties in resolution arise from such physical causes as multiple energy sources, interference of neighboring fundamental and overtone modes, or splitting of peaks due to Earth inhomogeneities and rotation. The instantaneous phase is especially useful in making decisions on resolution. Final estimates of the above four parameters are found through a non-linear regression scheme in the frequency domain. Small frequency intervals which contain the spectral peak or peaks of interest are fitted to a model of exponentially decaying cosines. The statistical assumptions made in the formulation of this procedure allow estimates of formal uncertainties to be made. In this way, more reliable estimates than previously available have been made for a set of both spheroidal and torsional eigenvibrations. A description of the analysis methods is given in Hansen (1981 b) hereafter referred to as Paper 1. Using the vanances, statistical comparisons are made between each of the estimates of the four parameters. Complex demodulation and non-linear regression are used on two different data sets in order to systematically assess the effects of the Earth’s regional variations on its eigenvibrations. One of the data sets provides information regarding the differences in the eigenvibrations as recorded within a region whose spatial dimension is small relative to the wavelength of the oscillation. This set relates to oscillations from four different great earthquakes recorded by a network of three broad-band seismographic stations (one three-component and two vertical components) located in northern California. The stations are separated by distances of only 200—350 km, yet each station samples motions in different tectonic settings. The observed spectral variations may be related to different tectonic regions which change quickly with distance. In addition, by considering four earthquake sources, variations may be assessed for different source positions, orientations, and time functions as observed within this regional network of receivers. The results aid in aswering the follow-
ing question: given that the eigenfunctions fix the positions of nodal lines, do different source parameters cause shifts in multiplet locations at a particular receiver site due to different relative excitement of singlet amplitudes? A special feature of the discussion is a comparison of parameters made with statistical significance tests which use the calculated formal uncertainties. The second data set comes from a global network of receivers which recorded the large 1977 Indonesian earthquake. This earthquake is the most efficient event since 1964 for exciting free oscillations of the Earth. Records from the three California stations are compared with those obtained from a global network of seven gravimeters. These gravimeters are from the IDA Network (International Deployment of Accelerometers operated by the University of California at San Diego) and have proved most valuable in gathering spheroidal mode data. The IDA gravimeters are not sensitive to torsional oscillations so that no comparisons could be made with the spectra based on measurements by the horizontal pendulums at northern California stations (i.e. at Berkeley). The IDA records proved to be rich in overtone data as well as the fundamental modes of spheroidal oscillations. When interference between closely spaced modes were recognized with the complex demodulation, non-linear regression was successfully used to solve for groups of spectral peaks yielding estimates for previously unresolved modes. Direct comparisons of frequency shifts of multiplets at different receiver sites are made for a single source. Implications on the spatial separations of receivers required to see frequency shifts are explored with these two different scale networks of receivers. As a by-product, this study of free oscillation modes with the above complexities in mind also provides some measurements of values of period, amplitude, and Q for modelling source mechanisms and a laterally varying and attenuating Earth.
2. Use of networks The California regional network consists of a set of three-component ultra-long period seismom-
29
eters in Berkeley (free pendulum period 7~= 100 s), a vertical component seismometer at Jamestown (7~= 40 s), and a vertical component seismometer at Whiskeytown (7~ 40 s). Coordinates of these stations are given in Table I. For all stations the signals are continuously recorded on analog magnetic tape and for the present study were digitized to produce ±2048 counts. Response curves converting counts to microns of ground motions for the frequency range of interest are shown in Fig. 1. The distances between these three sites range from 200—350 km. This distance is much less than half a wavelength of 0S~,for example. At these separations, each site is located on a different geologic structure within the rapidly changing tectonics of northern California. A theory developed by Mendiguren (1972) for treating the free oscillations of a laterally heterogeneous earth provides some selection rules between degree of oscillation 1, and order s of a spherical harmonic expansion of the lateral heterogeneity (see Aki and Richards, 1980, p. 771): (1) s is even; (2) s < 21; (3) m’ t + m. Rule (1) indicates that lateral inhomogeneity expressed by spherical harmonics of odd-order number does not show an observable effect on freeoscillation periods. Rule (2) indicates that free oscillations of order 1 are affected only by the lateral inhomogeneity harmonics of orders lower
than 2!. Rule (3) indicates that the coupling between the unperturbed eigenfunctions ~U,m and ~U,m’occurs only through inhomogeneity with the azimuthal order number t equal to m’ m. From rule (3) it follows that the average of the singlet frequencies comprising a multiplet is equal to the frequency for the unperturbed, spherically symmetric Earth model. This averaging is called the diagonal sum rule, and was first applied to the perturbations of free oscillations by Gilbert (1971). Of particular interest is rule (2) which states that those heterogeneities whose scale length is shorter than half the wavelength of the free oscil—
lation do not affect the eigenfrequency. Specifically, the aim of the regional comparison is to determine if significant differences in observed eigenspectra can be detected within the regional network. Support for no difference in period is found from comparisons of observations of 0 obtained following each earthquake. The question raised concerning shifts in multiplet frequencies due to different source parameters is addressed by the comparison of all observations of 0 following the four earthquakes. In the second part of the study, the global network of accelerometers called IDA is used. Information concerning this network can be found in Agnew et al. (976). A list of the stations used and their coordinates are given in Table I. Response curves converting counts to microns of ground motion are also found in Fig. I. Five of the stations have such similar response curves that
TABLE I Seismographic stations Location
Code
Latitude
Longitude
Elevation (m)
Jamestown, California Whiskeytown, California Berkeley, California Nana, Peru Sutherland, S. Africa Halifax, N.S., Canada Rarotonga, Cook Islands Garm, USSR Brasilia, Brazil College, Alaska
JAS WDC BKS NNA SUR HAL RAR GAR BDF CMO
37°95’N 40°58’N 37°88’N ll099~S 32°38’S 44°64’N 2l°2l’S 39°00’N I 5°66’S 64°86’N
120°44’W 122°54’W l22°24W 76°84’W 20°80’E 63°59’W 159°77’W 70°32’E 47°90’W l47°84’W
457 300 276 575 1770 38 28 1300 1260 183
30 o o.
______________________________________ JAS woC -
_____
—
-
—
—
/ ,/. / /.~1/
55 NNA US
/:
/.‘ /“ ,~I ,,.‘ /“,‘/ /•‘‘~‘/
GA R I 0.
/ // /
i..’ /‘I
/.. /
,7’ / / /
/.;‘)‘ f/I
ft/I ,)~/// f//f
I.
,f/,// ~‘2/ i/ ,‘//.~,~f
/ p.1/ // j
,///.J
.1
/I ~
/ / I/i /~/,/ 1/ f / ./
~/J. // oI 000I
,!/
~
00 FR ~ 15 U EN C V (H Z I
Fig. I. Response curves of seismometers and filters.
only one curve in Fig. I represents them, The separation of stations in the IDA network are on the order of a few wavelengths of normal modes up to the order of interest (0S~).Analysis is made of recordings from seven stations (see Table I) following the 1977 Indonesian earthquake. Comparisons of frequency shifts of multiplets at different receiver sites are made for this single source. Spatial separations of receivers, required to see frequency shifts, are explored with these two different scale networks of receivers.
3. Preparation of seismograms In this study of long-period seismograms, data were collected from two different networks of seismographic stations. Seventeen vertical and horizontal seismograms were recorded on a regional network of broadband seismometers in northern California (operated and maintained by the U.C. Berkeley Seismographic Station) following four large shallow earthquakes: the Philippine earthquake of August 16, 1976 (M5 = 7.9 BRK); the Indonesian earthquake of August 19, 1977 (M~= 8.0 BRK); the West Irian earthquake of September 12, 1979 (M~= 7.9 BRK); and the Columbian earthquake of December 12, 1979 (M5 = 7.7 BRK). Seven seismograms were obtained corresponding to the vertical ground motions on the world-wide IDA network of gravimeters following’ the Indonesian earthquake of August 19, 1977. Spectral preparation of each of these seismograms is necessary to obtain a usable time series of the form of eq. 1 of Paper 1. The data obtained from the Berkeley network were recorded continuously on analog tape. Thus, it was necessary to use an analog-to-digital converter to obtain the digital time series for use on a high-speed computer. A sample interval of 10 s was the largest available with the existing machine. A digital interval of 20s was obtained by low-pass anti-alias filtering (at 100 s) the digital time series with an eight-pole Butterworth filter and taking every other point. The IDA data are recorded in digital form with a time interval of 20 s so no analog-to-digital conversion, filtering, or decimation was necessary. In each case the usable time series was begun just after the instrument was no longer clipping, which varied from 4 to 11 h. For the regional network, all starting times are the same for each earthquake. Since eq. 1 of Paper 1 represents only decaying cosinusoids in the presence of random noise, any irregularities such as glitches, aftershocks, or nondecaying harmonic components (e.g. long-period tides) must be removed. The four largest amplitude tides are M2, S2, 0, and K with periods of 12.42, 12.00, 25.829, and 23.934h, respectively (Melchior, 1966). Cosine and sine waves of these
31
periods were fitted in a least squares sense for the amplitudes. The time series X( t) is then formed by subtracting the fitted tides from the observed data at each point. Since glitches in the data can affect the fit of these tides, any aseismic irregularities must be removed before the least-squares fit is made. This is done by removing the contaminated section of data and replacing it with a straight line of approximately the same slope. Simply adding zeros or a constant in a non-zero or sloping region is not as satisfactory because this would also affect the fit of the tides, It was necessary to make these replacements in seven of the seismograms. Following is a list of what was replaced in those seismograms identified by earthquake and station name. 3.1. Philippine earthquake The starting time for all records is Year 1976, Day 229, Hour 22, Minute 20, Second 00.0. The length of each of five records (JASZ, WDCZ, BKSZ, BKST, BKSR) is 45.51 h. The only contaminated record was WDC. There were four aseismic spikes within the 45.51 h record. The first occurred 21 h into the record for a length of 54 mm. The other three were at 35.7, 39.3, and 42.6 h into the record and had durations of 24, 14, and 10 mm., respectively. 3.2. Indonesian earthquake JAS vertical component—Starting time is 1977, 231, 14, 36, 40. Length of record is 45.51 h. This case calls for special attention. A large section of record, 15.1 h from the start, was contaminated by a large offset, 58% of full scale, which then returned to the mean position in about 11 h with what appears to be an exponential decay. Rather than discard 11 h of data, the offset was removed in the following manner. A function of the form A exp( — bi) was fitted to this 11 h stretch of data in a least-squares sense and then subtracted point by point from the original data. All oscillations present in the data were preserved, except for a short section 26 mm. long at the beginning of the offset. This section was replaced with a straight line, WDC vertical component—Starting time is 1977, 231, 14, 36, 40. Length of record is 45.51 h.
One and three quarters hours into the record a 0.5 h glitch was repaired, and 13.5 h into the rëcord, a 22 mm section was repaired. BKS vertical northwest and northeast—Starting time is 1977, 231, 14, 36, 40. Length of record is 45.51 h. There were no defects requiring treatment in these records. HAL—Starting time is 1977, 231, 14, 07, 20. Length of record is 138.9 h. There were no defects or splices in this record. SUR—Starting time is 1977, 232, 10, 20, 20. This is considerably later than most other records as the mass of the gravimeter stuck for several hours after the earthquake. Length of record is 138.9 h. CMO—Starting time is 1977, 231, 11, 42, 20. Length of record is 138.9 h. There were no defects or splices in this record. RAR—Starting time is 1977, 231, 11, 43, 40. Length of record is 138.9h. There were no defects or splices in this record. NNA—Starting time is 1977, 231, 17, 21, 00. Length of record is 138.9 h. This time series required splicing data from two cassettes in order to obtain the same length of data as all other IDA stations. There is an 81 mm splice beginning 92.45 h into the record. GAR—Starting time is 1977, 231, 14, 49, 20. Again, a length of record of 138.9 h is achieved by splicing data from two cassettes. A line of length 5 mm and 20s is inserted after 62.1 h. BDF—Starting time is 1977, 232, 00, 23, 20. A splice of length 8 mm and 20 s was inserted 115.5 h into the record in order to obtain a total length of 138.9 h. 3.3.
West Irian earthquake
Again, just the regional network was considered for this event. Only JAS and WDC recorded usable time series as the noise level of the BKS system was such that there was no usable signal above the noise level for the size of the event. The starting time for JAS and WDC is 1977, 255, 08, 40, 00 and the record length is 45.51 h. The records were uncontaminated so that no special adjustments were required.
32
Colombian earthquake
3.4.
The starting time for all records is 1979, 346, 14, 00, 00. JAS—Lemgth of record is 45.51 h. There was one aseismic spike 10.9h into the record that was removed and replaced with a line of duration 34 mm. WDC—Length of uncontaminated record was 45.51h. BKS vertical, northwest and northeast—Length of uncontaminated record was 27.3 h. Beyond this time was a substantial section of contaminated record, so the time series was cut short at this point. The fundamental modes observed in these time series (for this less energetic event) all decayed to the noise level by this time so that no resolution was lost. Finally, the horizontal component recordings obtained following the Philippine, Indonesian and Colombian earthquakes were rotated vectorially by computer using great circle paths to produce radial and transverse directions of ground motion. Figures 2—7 show, for example, the spectra along
20
-
‘OO , 2
00
.
.
00 092
FPEOUENCY
~ ~
.
00 88
9090
(HZ.
26
-
2
. 0
TIME
8~
(H
PS.)
Fig. 3. Transverse component seismogram recorded at BKST following the Indonesian earthquake and its amplitude spectral density corrected for instrument response and filtering.
20
~
FREQUENCY
0
_5690
(HZ.)
FREQUENCY
~
(HZ.)
‘
22.05
TIME
15.51
(HAS.)
Fig. 2. Vertical component seismogram recorded at WDC following the Indonesian earthquake and its amplitude spectral density corrected for instrument response and filtering.
~‘0
22.75
TIME
‘5.51
IMPS.)
Fig. 4. Vertical component seismogram recorded at WDC following the Philippine earthquake and its amplitude spectral density corrected for instrument response and filtering.
33 20
20
o
019
01
.
H
1~~I2
UJ 8
.00072
.00119
.00265
.09392
FREOUENCY
.00498
.0060
~
.00092
.00498
.0060
(HZ.)
914
0
ZD
TIME
0
~
.
IMPS. I
,; S 5 TIME
Fig. 5. Vertical component seismogram recorded at WDC following the Colombian earthquake and its amplitude spectral density corrected for instrument response and filtering.
I
91
0
(MRS.)
Fig. 7. Vertical component seismogram recorded at NNA following the Indonesian earthquake and its amplitude spectral density corrected for instrument response and filtering.
with the prepared time series of each WDC time series, the BKS transverse component for the Indonesian earthquake, and the NNA time series of the Indonesian earthquake for comparison. In
20
computing the spectra the response of the seismometer and filters were compensated for yielding true ground displacement amplitude spectral density. All spectral plots are plotted at the same scale for comparison.
I 6 *
I
.00285
FPEOUENCY
886
- 60,6
.00179
(HZ. I
2
I—, 9
.00072
.00479
.00285 .00392 FREOUENCY 1HZ.
1 .00491.0018
4. 4.1.Data Introduction analysis The techniques of eigenvibration data analysis
_____________________________________ 09.91 TIME INNs. I Fig. 6. Vertical component seismogram recorded at WDC following the West Irian earthquake and its amplitude spectral density corrected for instrument response and filtering.
described in Paper 1 were applied to the 24 time series of ground motion following the four large earthquakes. Representative examples of the successes and problems encountered in the analysis will be presented in this section. All final estimates, 0 of period, Q, amplitude, and phase (accompanied with their respective uncertainties) for modes from 0S4 through 0S48 are listed in Tables Il—V. Tables II and V contain all estimates
34 TABLE II MODE
JASZ I
WDCZ I
0~9
IC 633.69 0.563 243.0 105.0 21.83 6.963 2.887 0.319
IA 633.95 0.161 244.0 30.0 31.20 3.752 3.082 0.120
lB 579.61 0.258 232.0 48.0 31.24 6.251 3.086 0.200
IC 579.74 0.459 167.0 44.0 32.43 8.364 3.428 0.258
T Q
a 6
T Q
a 6 o5~~ T Q
a 6
0~I2
T Q
a 6
0~I3
T Q
a 6
05(4
T Q
a
BKSZ I
JASZ P
WDCZ P
BKSZ P
JASZ C
IC 579.34 0.464 323.0 167.0 19.00 6.385 2.755 0.336
IC 579.28 0.287 263.0 69.0 30.18 6.251 2.613 0.207
lB 580.30 0.282 210.0 43.0 20.52 2.918 5.426 0.142
IA 502.01 0.152 296.0 53.0 22.30 3.370 5.293 0.151
IC 502.37 0.266 552.0 323.0 17.23 5.795 5.317 0.336
IC 537.43 0.780 220.0 140.0 31.70 15.948 0.857 0.503 lB 502.82 0.186 218.0 35.0 21.88 3.401 1.309 0.156
IA 502.29 0.246 288.0 82.0 18.02 4.328 0.850 0.240
IC 502.81 0.205 441.0 158.0 21.92 5.765 0.809 0.263
lB 473.71 0.244 264.0 72.0 16.29 4.457 1.858 0.274
IC 472.76 0.095 497.0 99.0 9.99 1.682 2.319 0.169 lB 448.03 0.089 357.0 50.0 18.55 2.612
IC 448.23 0.094 534.0 120.0 20.89 4.041
IA 447.63 0.137 295.0 53.0 16.75 2.914
lB 472.65 0.142 393.0 93.0 13.38 2.020 3.600 0.151 IC 447.79 0.144 557.0 199.0 16.50 3.670
IA 447.81 0.060 331.0 29.0 18.23 1.117
35
WDCZ C
BKSZ C
BKSR C
JASZ W
WDCZ W
R AVG
IC 634.48 0.842 260.0 180.0 10.89 5.188 4.753 0.476
3A 633.95 0.23 244.0 35.0
IA 580.11 0.205 256.0 46.0 21.08 2.307 5.017 0.109
6A 579.88 0.274 231.0 48.0
IC 537.43 0.780 220.0 140.0
5A 502.37 0.195 255.0 53.0
lB 472.82 0.118 318.0 50.0 20.89 2.245 3.941 0.107 lB 447.71 0.123 226.0 28.0 15.58 1.411
IC 473.18 0.400 147.0 37.0 15.32 2.933 1.560 0.191 IC 447.66 0.935 126.0 66.0 42.02 16.625
lB 473.32 0.212 261.0 61.0 6.70 1.129 0.565 0.169
IA 473.26 0.106 253.0 29.0 6.47 0.534 0.677 0.082
7A 473.00 0.137 264.0 45.0
IC 450.24 0.593 205.0 111.0 2.21 0.779
8A 447.88 0.101 293.0 39.0
36 TABLE II (continued) MODE
JASZ I
8
T Q
a 8
MODE
WDCZ I
0.383 0.193
lB 426.32 0.126 263.0 41.0 18.69 2.940 —0.99 0.157
IC 425.69 0.210 145.0 21.0 28.56 4.261 5.713 0.149
IC 426.23 0.261 190.0 44.0 27.06 6.256 —0.309 0.231
IC 426.25 0.346 215.0 75.0 18.13 5.407 1.639 0.298
JASZ I
WDCZ I
BKSZ I
BKSR I
IA 389.20 0.052 350.0 33.0 22.23 2.089 1.361 0.094
IC 407.10 0.335 366.0 220.0 15.48 7.779 —1.571 0.502 IC 389.26 0.065 572.0 109.0 16.34 2.582 —4.784 0.158
IC 405.80 0.353 299.0 155.0 14.39 5.873 —4.697 0.408 IC 389.48 0.209 148.0 24.0 38.89 6.413 —5.275 0.165
lB 373.80 0.130 350.0 85.0 10.05 2.220 0.650 0.277
lB 373.58 0.195 208.0 45.0 20.03 4.591 5.090 0.229
IC 373.71 0.121 282.0 51.0 24.27 4.311 —0.750 0.178
IC 374.09 0.195 388.0 57.0 17.26 6.055 — 1.032 0.351
IA 359.98 0.067 275.0 28.0 22.95 2.302 3.222 0.100
3A 359.96 0.043 243.0 14.0 27.65 1.153 3.719 0.042
IA 359.89 0.096 224.0 27.0 35.40 4.303 —3.135 0.122
IC 359.67 0.292 272.0 120.0 11.52 4.677 —1.008 0.406
T Q
a 8 0~I7
T Q
a 8
T Q
a 8
0~I9
T Q
a 8 0S20
T
JASZ P
0.611 0.141
0~I6
0~I8
BKSZ I
2B 347.29 0.054
WDCZ P
BKSZ P
2.494 0.174
2.663 0.222
JASZ C 1.542 0.061 lB 425.87 0.194 174.0 28.0 18.56 2.107 4.474 0.114
JASZ P
WDCZ P
BKSZ P
IA 406.40 0.083 244.0 24.0 30.58 3.121 3.570 0.102
IA 406.48 0.124 326.0 64.0 26.78 4.938 —2.735 0.184 IA 389.12 0.187 197.0 37.0 32.61 6.023 —4.426 0.185
IC 373.22 0.268 219.0 69.0 17.01 5.670 1.438 0.333
lB 373.06 0.186 290.0 84.0 16.35 4.588 —4.838 0.281
IA 389.14 0.126 232.0 35.0 30.59 4.610 1.905 0.151
IC 359.72 0.251 144.0 29.0 25.48 5.366 —0.629 0.211
37
WDCZ I
BKSZ C
1.494 0.091
1.215 0.396
BKSR C
JASZ W
JASZ C
WDCZ C
BKSZ C
lB 406.21 0.052 289.0 22.0 16.83 0.918 3.664 0.055
IA 426.00 0.060 507.0 72.0 4.41 0.344 2.741 0.078 JASZ W
WDCZ W
R AVG
3C 406.43 0.123 272.0 45.0 12.81 1.595 —5.200 0.125
IA 406.62 0.193 226.0 48.0 9.09 1.428 1.746 0.158
lB 406.55 0.254 198.0 49.0 6.80 1.342 1.821 0.197
8A 406.33 0.104 260.0 34.0
lB 389.24 0.109 221.0 27.0 5.95 0.554 3.585 0.093
9A 389.11 0.083 247.0 29.0
lB 373.50 0.107 171.0 17.0 9.08 0.688 2.487 0.076
IlB 373.50 0.128 177.0 28.0
2A 388.96 0.062 249.0 20.0 17.20 1.011 0.038 0.059
IC 373.91 0.139 125.0 12.0 27.98 1.995 2.026 0.071
lB 373.09 0.089 385.0 71.0 5.54 0.727 2.881 0.131
IC 374.09 0.225 164.0 33.0 15.46 2.426 —3.636 0.157
IC 359.91 0.142 134.0 14.0 29.77 2.603 0.123 0.087
2C 359.81 0.116 289.0 54.0 9.95 2.227 —2.598 0.224
IC 347.42 0.103
IC 388.91 0.232 340.0 138.0 20.16 5.134 —0.126 0.255
lB 347.22 0.157
7A 425.95 0.104 223.0 32.0
BKSR C
2B 388.87 0.085 253.0 28.0 18.75 1.506 5.810 0.080
IC 347.94 0.069
R AVG
1.263 0.352 2B 425.68 0.086 231.0 21.00 18.44 1.154 —1.839 0.063
—
WDCZ W
IC 373.50 0.079 184.0 14.0 10.53 0.660 2.423 0.063
7A 359.94 0.075 230.0 20.0
IC 347.17 0.118
lB 347.27 0.078
6B 347.41 0.080
38 TABLE II (continued) MODE
JASZ 1
a 8
T
Q a 8
0~22
T Q
a 6
MODE
T Q
a 8
IC 334.92 0.253 262.0 104.0 9.98 4.142 —5.549 0.415
lB 335.60 0.075 312.0 43.0 15.60 2.180 —4.639 0.140
lB 335.40 0.094 354.0 70.0 18.05 3.305 —4.856 0.183
IC 325.01 0.149 269.0 66.0 26.47 6.133 —5.606 0.232
1A 324.99 0.106 256.0 43.0 16.91 2.945 —2.018 0.174
IC 324.79 0.378 227.0 120.0 9.96 5.361 — 1.330 0.538
IA 324.99 0.181 245.0 67.0 17.43 4.704 — 1.827 0.270 BKSZ P
JASZ I
WDCZ I
BKSZ I
BKSR I
JASZ P
WDCZ P
lB 314.85 0.183 247.0 71.0 17.85 5.442 —4.300 0.305
IA 314.66 0.141 215.0 42.0 13.23 2.687 —3.859 0.203
lB 315.20 0.270 244.0 102.0 23.37 9.744 —2.652 0.417
IC 314.63 0.284 266.0 128.0 6.79 3.293 —6.083 0.485
IC 315.34 0.147 225.0 47.0 10.99 2.407 —3.526 0.219
2B 306.18 0.083 350.0 66.0 19.16 2.562 —0.465 0.133
5A 306.07 0.057 221.0 18.0 36.11 2.230 —0.450 0.062
lB 306.24 0.264 285.0 140.0 15.38 7.537 — 1.028 0.490
IC 306.04 0.182 165.0 32.0 21.10 4.418 — 1.873 0.209
IC 305.86 0.229 224.0 75.0 12.70 4.517 — 1.623 0.356
8
8
JASZ C
IA 325.47 0.062 289.0 32.0 14.60 1.238 —1.463 0.085
a
a
BKSZ P
IC 325.13 0.085 476.0 119.0 13.21 2.961 —2.007 0.224
Q
Q
WDCZ P
3A 335.89 0.043 237.0 15.0 23.55 1.065 —5.155 0.045
T
T
JASZ P
lA 335.77 0.056 237.0 19.0 22.81 1.830 —5.584 0.080
0~23
0~24
BKSZ I
355.0 39.0 14.23 1.120 —3.090 0.079
Q
0~2I
WDCZ I
1A 305.85 0.102 236.0 37.0 16.12 2.630 —0.366 0.163 2A 297.48 0.062 230.0 22.0 22.62 1.630 —4.880 0.072
IA 335.76 0.156 204.0 39.0 39.70 7.691 —5.523 0.194
5A 297.45 0.053 245.0 21.0 31.16 2.015 —5.278 0.065
lB 305.92 0.114 201.0 30.0 23.23 3.674 — 1.759 0.158
39
JASZ C
WDCZ C
BKSZ C
178.0 13.0 20.44 1.117 —4.421 0.055
178.0 19.0 23.13 1.807 —4.993 0.078
229.0 47.0 26.60 4.083 —4.857 0.154
IA 335.76 0.050 242.0 17.0 23.40 1.233 —3.908 0.053
2C 335.17 0.145 258.0 58.0 9.10 1.486 —4.792 0.163
lB 335.68 0.119 300.0 63.0 19.89 3.000 —3.958 0.151
2B 324.65 0.098 231.0 32.0 14.85 1.575 —3.511 0.106
BKSR C
2B 335.93 0.275 252.0 104.0 10.38 4.561 —3.207 0.442
JASZ W
WDCZ W
R AVG
353.0 85.0 4.55 0.824 —3.782 0.181
206.0 19.0 8.60 0.589 —3.884 0.069
213.0 25.0
IC 335.64 0.138 174.0 25.0 9.90 1.107 —5.791 0.112
IC 335.98 0.254 181.0 49.0 4.39 0.975 0.213 0.222
12A 335.75 0.076 239.0 25.0
2B 325.04 0.039 244.0 14.0 7.68 0.332 —4.352 0.043
9A 325.11 0.079 251.0 29.0
JASZ W
WDCZ W
R AVG
2B 315.28 0.063 223.0 20.0 5.11 0.342 —0.392 0.067
IOA 315.29 0.072 221.0 20.0
IC 324.85 0.259 202.0 65.0 18.89 4.294 —2.981 0.227
BKSR P
JASZ C
WDCZ C
BKSZ C
IC 315.04 0.202 157.0 31.0 19.45 4.195 —3.155 0.216
2C 315.51 0.110 182.0 23.0 20.78 2.171 —6.018 0.104
IA 315.31 0.032 222.0 10.0 29.34 0.981 0.063 0.033
lB 315.46 0.094 237.0 34.0 27.41 2.904 0.415 0.106
lB 306.17 0.069 214.0 21.0 28.78 2.102 —5.488 0.073
2B 305.83 0.077 203.0 21.0 18.53 1.454 0.358 0.078
lB 305.97 0.065 245.0 25.0 29.83 2.257 —5.495 0.076
lB 297.25 0.135 213.0 41.0 21.99 3.195 —5.760 0.145
BKSR C
lOA 306.02 0.085 220.0 27.0
lB 297.51 0.056 222.0 18.0 19.83 1.201 —5.280 0.061
lB 297.20 0.192 220.0 62.0 7.96 1.788 —2.087 0.225
5A 297.46 0.069 231.0 24.0
40 TABLE II (continued) MODE
JASZ I
WDCZ I
BKSZ I
BKSR I
0~26
4B 289.61 0.038 347.0 31.0 20.19 1.328 —3.180 0.066
2B 289.30 0.051 333.0 39.0 17.03 1.432 —5.262 0.084
SC 289.42 0.039 427.0 49.0 18.09 1.455 —4.116 0.080
IC 289.43 0.114 384.0 116.0 13.97 4.020 0.408 0.288
lB 282.57 0.287 194.0 76.0 14.02 5.958 —1.923 0.425
2B 282.11 0.07 1 303.0 46.0 13.21 1.514 —3.686 0.115
SB 282.30 0.077 215.0 25.0 23.57 2.124 —2.979 0.090
lB 275.11 0.136 325.0 105.0 11.78 3.786 — 1.616 0.321
IC 275.16 0.307 262.0 154.0 10.98 6.394 —2.854 0.583
IC 274.88 0.297 381.0 314.0 11.71 8.185 —2.535 0.699
T Q
a 8
0~27
T Q
a 8
T
Q a 8
2B 268.30 0.062 329.0 50.0 14.62 1.643 —2.515 0.112
T
Q a 8
MODE
JASZ I
OS
30
T Q
a 6
T
Q a
lB 256.32 0.203 239.0 91.0 16.28 6.550
JASZ P
WDCZ P
BKSZ P IC 289.62 0.170 187.0 41.0 15.32 3.489 —1.949 0.228
IC 267.92 0.179 204.0 56.0 13.04 3.736 —2.796 0.287
WDCZ 1
BKSZ 1
2B 262.48 0.099 219.0 36.0 16.43 2.072 —0.737 0.126
IC 262.31 0.149 316.0 113.0 13.95 4.904 —0.843 0.351 lB 256.27 0.192 254.0 97.0 16.14 6.49 I
JASZ P
WDCZ P
BKSR P
JASZ C
IC 262.10 0.285 305.0 202.0 6.17 3.786 —3.614 0.614 lB 256.02 0.123 221.0 47.0 15.13 2.393
41
BKSR P
JASZ C
WDCZ C
BKSZ C
2A 289.63 0.047 194.0 12.0 24.35 1.144 —2.357 0.047 2A 282.17 0.056 249.0 25.0 13.06 0.977 —2.167 0.075
BKSR C
JASZ W
WDCZ W
2B 289.85 0.074 239.0 29.0 13.46 1.264 —4.886 0.094 IC 282.18 0.078 254.0 36.0 13.78 1.436 — 1.978 0.104
7A 289.54 0.052 220.0 20.0
lB 281.79 0.149 277.0 81.0 5.02 1.129 —3.624 0.225
lB 282.01 0.094 187.0 23.0 8.68 0.832 —3.302 0.096
IC 275.23 0.088 236.0 36.0 12.23 1.387 —2.740 0.113
BKSZ C
2A 262.11 0.056 234.0 23.0 13.78 1.054 —0.292 0.077
lB 262.20 0.074 181.0 19.0 19.30 1.523 —0.103 0.079 lB 255.71 0.214 195.0 64.0 15.26 3.930
7A 282.15 0.081 227.0 31.0
4B 275.17 0.139 263.0 75.0
IA 268.57 0.049 238.0 21.0 14.39 0.939 0.253 0.065 WDCZ C
R AVG
BKSR C
IC 268.42 0.078 240.0 34.0 7.42 0.78 1 —2.664 0.105 JASZ W
WDCZ W
IC 262.05 0.102 154.0 18.0 10.86 1.024 —1.166 0.094
lB 268.28 0.044 184.0 11.0 10.61 0.494 —5.045 0.047 R AVG 6A 262.18 0.083 202.0 24.0
lB 255.72 0.050 193.0 15.0 7.97 0.467
SB 255.81 0.096 198.0 31.0
5A 268.39 0.057 204.0 18.0
42 TABLE II (continued) MODE 6
JASZ I —5.656 0.402
WDCZ 1
BKSZ I
JASZ P
WDCZ P
—0.024 0.402
BKSR P
JASZ C —0.984 0.158
osn
T Q
a 8 OS
33
IC 244.81 0.153 369.0 170.0 7.75 3.413 —0.955 0.441
T Q
a 8
2B 239.76 0.078 223.0 32.0 13.95 1.532 —5.321 0.110
O~34
T
Q a 6 0S35 T
Q a 8
T
Q a 8
IC 234.64 0.181 266.0 109.0 8.52 3.664 —4.438 0.430
IC 234.51 0.136 147.0 25.0 13.13 2.425 — 1.096 0.185 lB 229.66 0.055 242.0 28.0 8.15 0.989 —0.620 0.12 I
2B 234.52 0.096 248.0 50.0 9.59 1.481 0.036 0.155
43 TABLE II WDCZ C
BKSZ C
BKSR C
JASZ W
—1.046 0.258
WDCZ W —1.018 0.059
IC 250.24 0.058 182.0 15.0 8.13 0.546 —3.487 0.067 LB 245.33 0.257 260.0 142.0 10.20 4.039 —4.349 0.396 lB 239.48 0.112 230.0 49.0 15.57 2.608 0.557 0.168
IC 239.14 0.132 278.0 85.0 4.14 0.955 0.396 0.231
2A 250.36 0.108 197.0 34.0 4.96 0.672 —3.451
3C 229.65 0.096 219.0 40.0 6.03 0.924 —0.430 0.153
2A 250.28 0.078 189.0 26.0
0.136 2A 244.60 0.089 243.0 43.0 2.97 0.402 —2.644 0.135
3A 244.69 0.121 246.0 57.0
2B 239.47 0.058 175.0 15.0 7.74 0.535 —5.623 0.069
4B 239.53 0.080 188.0 24.0
3C 234.80 0.093 186.0 27.0 9.04 1.186 1.046 0.131 IA 229.86 0.068 151.0 13.0 17.09 1.181 —3.012 0.069
R AVG
4B 234.62 0.112 193.0 38.0
IC 229.65 0.044 164.0 10.0 7.44 0.379 —1.208 0.051
4A 229.71 0.059 164.0 15.0
44 TABLE III (continued) MODE
WDCZ P
O~37
IC 224.97 0.119 230.0 56.0 7.25 1.928 —2.568 0.266
T
Q a 6
T Q
a 6
JASZ C
IC 220.90 0.149 241.0 78.0 . 8.53 2.155 —4.021 0.253
Q a 8
Q a
8
lA 208.77 0.060 189.0 21.0 6.67 0.563 —1.998 0.084
Q
a 8
T
Q a
WDCZ W
R AVG
IC 187.70 0.101 281.0 85.0 . 3.67 0.830
3B 216.45 0.118 224.0 54.0 6.53 1.462 —4.528 0.224
2C 220.44 0.058 156.0 13.0
2B 216.44 0.150 220.0 59.0
IA 208.77 0.060 189.0 21.0
IC 204.79 0.081 185.0 27.0 8.77 0.997 0.283 0.114
T
O~47
JASZ W
IC 220.40 0.043 155.0 9.00 8.85 0.432 —3.530 0.049 IC 216.36 0.345 177.0 100.0 9.58 4.133 —1.971 0.432
T
T
BKSR C
IC 224.97 0.119 230.0 56.0
O~39
O~41
WDCZ C
IC 204.79 0.081 185.0 27.0
IC 187.70 0.101 281.0 85.0
45 TABLE II (continued) —
MODE
WDCZ P
8
JASZ C
WDCZ C
BKSR C
JASZ W
—4.962 0.226
0S48 T Q
a 6
of 0 obtained from the regional network while Table III corresponds to the estimates obtained from the IDA network. Table IV lists estimates of overtone modes obtained from both networks. Table V lists torsional oscillations recorded at BKS. As will become apparent later in this section, it was necessary to develop a method of assigning reliability weights to each eigenvibration mode analyzed. These weights aid in designating the reliability of the estimates used to describe the mode and in developing weights for averaging estimates of period and estimates of Q. The reliability weight and the variance of each estimate were used to develop formal weights; then averages of period and of Q were computed for the modes estimated within the regional network. These averages were then averaged with the estimates obtained from the IDA network to give a global average of period and of Q for a large number of fundamental spheroidal oscillations. 4.2. Reliability weights Much emphasis was given in Paper I to understanding biases introduced by the regression used in obtaining final estimates of 1. In practice, it is not always possible to eliminate totally such biases. Therefore, it is important to recognize when such biases are present and to weigh the estimates obtained accordingly. For example, compare two different estimates of Q for the mode 0S18 (see Tables II and III). A value of 261 with a standard
WDCZ W
R AVG -
IC 184.08 0.155 214.0 77.0 4.97 1.304 —4.483 0.263
IC 184.08 0.155 214.0 77.0
error of 11 was obtained from the RAR recording of the Indonesian earthquake. A value of 125 with a standard error of 12 was obtained from the JAS recording of the Colombian earthquake. The difference in these two estimates is so large that the correctness of one or both estimates is doubtful. By looking again at the complex demodulation graphs of RAR and JAS it is found that the oscillation recorded at RAR follows the model exemplified in Fig. 2 of Paper 1 very closely while the oscillation at JAS does not. Judged only by the standard errors, these two estimates should be given equal weight. Clearly, from scrutiny of the complex demodulation graphs, the estimates are not of equal reliability and the estimate of Q from the JAS recording should be given less weight than that from the RAR recording. For the above reason, each mode analyzed is assigned a letter grade of A, B, C or D which is the ‘reliability weight’ for that mode. The following procedure is used in assigning the reliability of each mode. A few examples will be taken from the NNA recording of the 1977 Indonesian earthquake. The uncorrected spectra used for analysis is given in Fig. 8. The time series is then demodulated at each frequency that corresponds to a peak in the spectrum. Each mode is evaluated for starting values of 8 and the appropriate length of time series is selected for application to the non-linear regression as described in Paper 1. When it was unclear how many modes should be included in the regression, the
46 TABLE III MODE
NNA I
T Q
a 8
O~6
T Q
a 6
0S7 T Q
a 8
Q
a 6
O~9
T
lB 962.18 0.218 725.0 239.0 8.93 1.019 0.700 0.114
lB 962.28 0.305 250.0 40.0 71.77 7.319 5.077 0.102
IA 811.03 0.046 286.0 9.0 65.79 1.384 0.884 0.021
2B 810.35 0.363 248.0 55.0 6.80 1.166 2.502 0.171
IA 811.12 0.190 268.0 34.0 68.22 5.692 5.261 0.083
Q a 8
T -
‘
,
lB 707.64 0.204 385.0 85.0 33.74 4.729 —2.704 0.140
IC 633.10 0.188 215.0 28.0 43.82 3.590 0.430 0.082
IA 633.17 0.060 327.0 20.0 38.08 1.590 —0.626 0.042
IC 579.32 0.110 388.0 57.0 23.43 2.270
lB 579.03 0.107 349.0 45.0 27.43 2.424
GAR I
RAR I
BDF I
SUR I
R AVG
IC 1544.19 1.114 337.0 164.0 55.26 11.487 1.033 0.208
lB 962.25 0.090 418.0 33.0 39.87 1.581 —0.687 0.040
IA 706.80 0.056 260.0 11.0 61.48 1.731 —0.304 0.028
T
a
HAL I
lB 1547.03 0.657 293.0 73.0 25.52 2.986 —2.526 0.117
O~4
Q
CMO I
IA 579.03 0.135 339.0 54.0 48.51 5.225
lB 962.26 0.205 354.0 54.0 47.76 3.505 5.428 ‘0.073
G AVG 2B 1546.44 0.774 295.0 81.0
IC 694.74 0.468 619.0 371.0 17.41 4.060 — 1.593 0.233
IC 960.46 0.350 263.0 50.0 19.35 2.371 1.119 0.123 2B 812.36 0.238 341.0 68.0 7.16 0.987 — 1.402 0.138
.‘
IA 707.58 0.117 369.0 45.0 46.05 3.612 —5.944 0.078
6B 962.28 0.171 349.0 48.0
4A 811.06 0.083 285.0 16.0
lB 706.76 0.232 255.0 43.0 21.59 2.544 —1.259 0.118
4A 706.97 0.092 267.0 19.0
lB 633.75 0.223 229.0 37.0 30.17 3.203 —5.027 0.106
IA 635.00 0.130 268.0 29.0 36.00 2.868 0.858 0.080
IC 633.86 0.044 456.0 29.0 16.76 0.648 0.172 0.039
3A 633.95 0.231 244.0 35.0
6A 633.66 0.085 293.0 27.0
lB 579.69 0.117 331.0 44.0 35.17 3.263
lB 580.28 0.135 377.0 66.0 21.69 2.552
IA 579.41 0.063 384.0 32.0 20.34 1.122
6A 579.88 0.274 231.0 48.0
7A 579.43 0.105 342.0 44.0
47 TABLE III (continued) MODE
NNA I
8
—2.113 0.097
8
O~13
T Q
a 8
O~I4
T Q
a 8
O~I5
T Q
a 8
O~l6
T
—0.218 0.118
—0.093 0.055
SUR I
R AVG
G AVG
IC 537.43 0.780 220.0 140.0
SB 538.04 0.202 215.0 28.0
lB 537.43 0.167 193.0 23.0 50.24 4.353 1.376 0.087
2B 539.02 0.164 242.0 36.0 6.62 0.923 0.968 0.968
3A 502.00 0.038 285.0 12.0 59.89 1.988 — 1.708 0.033
2C 501.19 0.079 293.0 27.0 25.62 2.122 2.489 0.083
IA 501.39 0.060 345.0 29.0 64.16 3.720 — 1.832 0.058
IA 502.90 0.037 290.0 12.0 32.31 0.994 —1.115 0.031
IC 501.40 0.065 265.0 18.0 30.59 1.514 3.257 0.050
1C 502.34 0.191 205.0 32.0 4.67 0.488 2.830 0.105
SA 502.24 0.197 297.0 68.0
7A 502.17 0.054 290.0 17.0
lB 472.88 0.054 290.0 19.0 69.87 3.353 0.793 0.048
lB 472.17 0.117 304.0 46.0 29.37 3.165 2.123 0.108
IA 472.50 0.064 333.0 30.0 66.63 4.239 —2.390 0.064
IA 473.42 0.055 301.0 21.0 58.73 2.923 —0.416 0.050
IC 472.66 0.086 280.0 28.0 29.05 2.155 0.213 0.074
IC 472.70 0.071 276.0 23.0 4.00 0.242 2.646 0.061
7A 473.00 0.137 264.0 45.0
7A 472.89 0.071 298.0 27.0
lB 448.05 0.074 301.0 30.0 52.78 3.733 2.895 0.071
IA 446.82 0.054 220.0 12.0 45.13 1.764 1.938 0.039
lB 447.71 0.099 291.0 38.0 53.77 5.007 —3.724 0.093
lB 448.17 0.057 297.0 23.0 42.92 2.328 —2.230 0.054
lB 448.74 0.076 317.0 34.0 38.23 2.960 1.383 0.077
8A 447.88 0.101 293.0 39.0
7A 447.76 0.072 248.0 21.0
IC 426.45 0.177 201.0 34.0 29.13 5.939 —2.745 0.121
IC 425.16 0.115 251.0 34.0 27.49 2.7 18 0.509 0.099
lB 425.54 0.058 351.0 34.0 59.08 4.034 —5.666 0.068
lB 426.21 0.054 478.0 58.0 15.28 1.187 — 1.399 0.078
3A 426.73 0.054 272.0 19.0 21.01 1.084 — 1.671 0.052
IC 426.61 0.127 166.0 16.0 2.92 0.228 — 1.545 0.078
7A 425.95 0.104 223.0 32.0
7A 426.18 0.072 262.0 27.0
IC 404.95 0.372
2C 406.21 0.078
IA 405.98 0.052
IA 406.83 0.017
3A 407.31 0.024
IA 406.48 0.028
8A 406.33 0.104
8A 406.81 0.031
8
a
BDF I
2B 537.98 0.152 234.0 31.0 59.57 8.685 —4.757 0.146
a
—5.147 0.093
RAR I
2B 537.53 0.217 218.0 38.0 32.96 6.38 1 —3.045 0.194
Q
Q
GAR I
—0.373 0.108
T
T
HAL I
—2.087 0.088
O~II
O~I2
CMO I
2C 448.40 0.103 248.0 28.0 15.82 2.719 2.781 0.172
IC 406.26 0.066
48 TABLE III (continued) MODE
NNA I
CMO I
HAL I
GAR I
RAR I
BDF 1
SUR I
R AVG
G AVG
Q
238.0 105.0 12.83 4.227 2.614 0.330
187.0 13.0 50.83 2.697 —0.932 0.053
301.0 23.0 61.73 3.441 — 1.846 0.056
285.0 7.0 64.94 1.162 —5.122 0.018
273.0 9.0 42.01 1.046 1.254 0.025
568.0 105.0 7.33 0.901 — 1.326 0.123
324.0 14.0 3.90 0.123 —0.579 0.032
260.0 34.0
284.0 12.0
IA 389.56 0.055 316.0 28.0 33.06 2.184 —0.235 0.066
2C 388.91 0.097 153.0 12.0 51.60 3.002 3.565 0.058
IC 388.68 0.190 168.0 28.0 36.42 4.620 —4.589 0.127
IC 389.68 0.117 349.0 73.0 17.21 2.543 —2.907 0.148
2B 390.41 0.077 345.0 47.0 19.40 1.886 3.940 0.097
IC 388.60 0.065 480.0 77.0 5.08 0.565 2.429 0.111
IA 389.34 0.061 325.0 33.0 2.23 0.163 4.477 0.073
9A 389.11 0.083 247.0 29.0
8A 389.39 0.074 257.0 29.0
lB 374.15 0.028 350.0 18.0 33.02 1.239 —1.195 0.038
lB 373.40 0.089 220.0 23.0 21.66 1.673 1.868 0.077
IIB 373.50 0.128 177.0 28.0
4A 374.25 0.040 266.0 16.0
a 8
O~I7
T Q
a 6
O~I8
T Q
a 8
lB 360.16 0.090 222.0 25.0 31.32 2.576 —2.3 15 0.082
O~I9
T Q
a 8
O~2O
T Q
a 8
O~2I
T Q
a 8
IA 374.47 0.032 261.0 11.0 50.73 1.639 —0.283 0.032
4A 347.78 0.016 266.0 7.0 61.33 1.105 — 1.039 0.018 4A 336.00 0.012 276.0 6.0 70.18 1.041 2.234 0.015
IA 335.44 0.039 293.0 20.0 45.68 2.282 1.588 0.050
lB 335.10 0.057 241.0 20.0 40.60 2.488 —3.778 0.061
IC 360.28 0.229 230.0 67.0 33.83 7.021 —2.600 0.207
IA 360.18 0.160 256.0 58.0 25.65 4.473 0.309 0.174
IA 360.08 0.077 257.0 28.0 12.80 1.037 — 1.173 0.081
7A 359.94 0.075 230.0 20.0
5A 360.06 0.094 236.0 27.0
lB 347.66 0.113 259.0 44.0 31.64 3.926 —0.784 0.124
2A 347.92 0.048 272.0 21.0 40.85 2.255 —2.348 0.055
IA 347.42 0.034 261.0 13.0 18.60 0.710 0.170 0.038
6B 347.41 0.080 213.0 25.0
5A 347.72 0.029 263.0 12.0
IA 336.00 0.023 259.0 9.0 53.99 1.384 —4.487 0.026
IA 336.05 0.055 237.0 18.0 61.98 3.554 3.716 0.057
IA 335.75 0.031 355.0 23.0 14.66 0.705 0.227 0.048
12A 335.75 0.076 239.0 25.0
8A 335.92 0.026 272.0 12.0
lB 335.90 0.056 417.0 58.0 1.16 0.108 0.734 0.093
49 TABLE III (continued) MODE
NNA I
o~22
4A 325.23 0.021 245.0 8.0 45.37 1.081 — 1.517 0.024
T Q a 8
o~23
T Q a 8
O~24
T
Q a 8
O~25
T Q a 8
O~26
T
Q a 8
O~27
T Q a
4B 315.86 0.037 212.0 11.0 30.31 1.148 1.246 0.038
CMO I
IA 314.92 0.019 288.0 10.0 34.51 0.902 —1.892 0.026
HAL I
GAR I
IA 324.45 0.067 319.0 42.0 45.98 4.468 — 1.678 0.097
2B 325.09 0.045 311.0 27.0 39.96 2.5 15 —3.859 0.063
RAR I
lB 314.86 0.102 269.0 47.0 45.36 5.965 0.339 0.131
3C 306.30 0.092 269.0 45.0 11.46 1.763 —0.202 0.157
BDF I
SUR I
R AVG
G AVG
IC 325.01 0.059 470.0 81.0 7.86 0.974 —0.636 0.124
lB 325.05 0.055 270.0 25.0 1.36 0.090 1.058 0.066
9A 325.11 0.079 251.0 29.0
6A 325.13 0.039 253.0 15.0
lB 315.64 0.044 297.0 25.0 48.00 2.908 —0.120 0.061
IC 314.95 0.030 371.0 26.0 8.74 0.450 3.443 0.051
IOA 315.29 0.072 221.0 20.0
6A 315.12 0.033 259.0 15.0
IC 305.96 0.209 439.0 264.0 8.87 4.029 —1.423 0.454
IC 305.66 0.073 263.0 33.0 12.38 1.171 1.302 0.095
lOA 306.02 0.085 220.0 27.0
4A 305.96 0.091 228.0 31.0
5A 297.46 0.069 231.0 24.0
7A 297.44 0.051 256.0 23.0
SB 297.79 0.035 326.0 25.0 28.30 1.589 1.745 0.056
2A 297.03 0.031 251.0 13.0 27.46 1.044 0.029 0.038
IA 298.22 0.110 227.0 38.0 30.52 3.872 —3.874 0.127
lB 297.84 0.074 320.0 51.0 32.58 3.757 —0.642 0.115
2B 298.25 0.086 266.0 41.0 34.56 3.920 2.106 0.113
IC 297.15 0.085 223.0 28.0 13.01 1.251 —0.758 0.096
lB 289.84 0.056 270.0 28.0 33.15 2.555 3.443 0.077
2A 289.22 0.032 332.0 24.0 16.05 0.872 2.670 0.054
7B 289.22 0.039 323.0 28.0 27.61 1.836 —5.141 0.067
lB 290.24 0.130 198.0 35.0 29.26 3.978 0.696 0.136
IC 289.88 0.065 314.0 44.0 26.88 2.849 0.722 0.106
lB 289.44 0.070 221.0 24.0 9.06 0.741 3.359 0.082
SC 282.33 0.031 341.0 26.0 25.88 1.475
2B 281.42 0.050 235.0 19.0 14.85 0.946
7B 281.58 0.048 272.0 25.0 31.70 2.597
‘
7A 289.54 0.052 220.0 20.0
7A 289.40 0.048 262.0 25.0
7A 282.15 0.081 227.0 31.0
4A 281.93 0.047 250.0 24.0
50 TABLE III (continued) MODE 8
T
Q a 8
0S29 T
Q a 8
NNA I
CMO I
1.993 0.057
1.299 0.064
—4.690 0.082
SC 275.41 0.037 249.0 17.0 33.57 1.738 —0.912 0.052
2B 274.76 0.022 295.0 14.0 24.85 0.970 3.457 0.039
—
SC 268.79 0.051 189.0 14.0 36.34 2.050 0.251 0.056
Q a 8 3C 256.15 0.063 291.0 42.0 5.53 0.638 —0.363 0.115
4B 255.62 0.031 255.0 16.0 19.77 0.934 —0.665 0.047
3C 250.54 0.066 249.0 33.0 8.63 1.168 1.191 0.135
4B 249.91 0.03 8 240.0 17.0 16.93 0.961 3.073 0.057
MODE
NNA I
CMO I
T
IC 245.01 0.042
4B 244.64 0.052
Q a 8
T
Q a
8
GAR I
RAR 1
2B 274.43 0.055 213.0 18.0 37.35 2.935 —4.259 0.079
2B 275.06 0.044 273.0 24.0 21.95 1.669 —0.459 0.076
lB 275.51 0.028 281.0 16.0 22.22 0.956 2.073 0.043
7B 267.76 0.096 237.0 40.0 27.95 5.963 —3.326 0.213
lB 268.71 0.133 259.0 66.0 17.63 3.3 16 1.107 0.188
4B 261.45 0.033 296.0 22.0 15.17 0.852 3.894 0.056
T
T
HAL I
IC 261.93 0.077 222.0 29.0 31.73 3.152 0.323 0.099
GAR I
IC 262.40 0.068 191.0 19.0 22.96 1.802 2.403 0.079
2C 249.95 0.092 324.0 77.0 6.18 1.181 0.343 0.191
IC 250.77 0.045 262.0 25.0 21.03 1.499 3.287 0.071
R AVG
G AVG
3A 244.69 0.121
3A 244.82 0.058
BDF I
R AVG
G AVG
4B 275.17 0.139 263.0 75.0
6B 275.00 0.034 270.0 19.0
5A 268.39 0.057 204.0 18.0
4A 268.47 0.069 206.0 24.0
6A 262.18 0.083 202.0 24.0
4A 261.70 0.052 236.0 23.0
SB 255.81 0.096 198.0 31.0
3B 255.71 0.045 245.0 21.0
2A 250.28 0.078 189.0 26.0
5A 250.24 0.054 226.0 23.0
SI TABLE III (continued) MODE
NNA I
CMO I
Q
214.0 16.0 13.28 0.772 —0.440 0.058
224.0 21.0 12.99 0.976 —0.479 0.075
a 8
O~34
T Q a 8
O~35
T Q a 6
O~37
T
Q a 8
O~38
T
Q a 8 O~48
T Q a 6
R AVG
G AVG
246.0 57.0
224.0 27.0
4B 239.53 0.080 188.0 24.0
3B 239.63 0.104 191.0 29.0
4B 234.62 0.112 193.0 38.0
2B 234.67 0.130 195.0 42.0
2B 224.50 0.054 268.0 35.0 9.64 1.000 —1.109 0.104
IC 224.97 0.119 230.0 56.0
2B 224.56 0.066 264.0 38.0
2C 220.54 0.089 177.0 25.0 12.76 1.574 1.846 0.123
2C 220.44 0.058 156.0 13.0
2C 220.47 0.069 160.0 16.0
IC 184.08 0.155 214.0 77.0
2B 184.36 0.106 202.0 43.0
IC 239.82 0.124 298.0 92.0 11.03 2.534 0.493 0.230
GAR I
IC 239.82 0.165 229.0 72.0 16.06 4.016 —0.334 0.250
IC 234.84 0.179 251.0 96.0 11.36 3.330 1.332 0.293
lB 184.43 0.090 201.0 39.0 11.00 1.705 —1.008 0.155
52 TABLE IV MODE
T Q a
8
T Q a 8
T
Q a 8
2~4
T Q a 6 2~5
T Q a 8
T Q a
NNA I
CMO I
HAL I
GAR I
SUR I
BKSR C
lB 1059.75 0.578 300.0 98.0 7.78 1.472 3.182 0.189 lB 853.01 0.290 417.0 118.0 12.84 1.971 —2.147 0.154
MODE
NNA I
2B 535.27 0.26 349.0 119.0 14.56 5.398 3.001 0.371
2~7
T Q a 8 2B 849.22 0.710 216.0 78.0 22.27 5.565 0.434 0.250
IC 849.44 0.543 127.0 21.0 36.25 4.274 0.297 0.118
1~9
T Q a 8
2B 802.96 0.316 391.0 120.0 4.85 0.982 2.221 0.203
I~8
T Q a & 2B 724.54 0.472 289.0 109.0 10.48 2.686 0.109 0.256
3C 509.42 0.347 278.0 106.0 6.96 2.006 — 1.749 0.289 3C 486.88 0.182 166.0 21.0 25.79 2.493 1.234 0.097
2~9
T Q a 8 IC 660.39 0.187 509.0 147.0 2.99 0.407 0.442 0.136
2~IO
T Q a 8 IC 593.77 0.568 144.0 40.0 30.51 6.307
2~I5
T Q a
CMO I
3C 309.41 0.303 200.0 75.0 5.47 1.974
53
HAL I
RAR I
2B 535.11 0.143 216.0 25.0 70.53 8.932 —2.214 0.127
BDF I
BKSR I
BKSR C
2B 535.25 0.057 237.0 12.0 19.68 0.943 —0.861 0.048
lB 535.91 0.310 254.0 75.0 32.71 7.522 —2.193 0.230
IC 535.95 0.293 148.0 24.0 30.04 3.184 —1.731 0.106
MODE
NNA I
Q a
8 lB 269.01 0.149 237.0 62.0 19.06 6.392 —3.798 0.336
Q a
8
T
Q a
8
3B 415.94 0.129 185.0 21.0 15.11 1.354 3.276 0.090
2C 415.99 0.186 214.0 41.0 9.55 1.339 —0.478 0.140
GAR I 2C 273.46 0.152 552.0 337.0 2.24 1.173 —1.556 0.522
T
T
2C 446.98 0.079 239.0 20.0 21.48 2.725 0.662 0.127
HAL I
3C 252.15 0.071 245.0 34.0 8.29 1.195 1.521 0.144
54 TABLE IV (continued) MODE
NNA I
8
CMO I
HAL I
GAR I
SUR I
-
I~8
IC
T
554.64 0.314 197.0 44.0 20.59 3.279 3.134 0.159
Q a &
BKSR C
MODE
—1.234 0.207
6
3C 303.30 0.112 377.0 112.0 5.18 1.268 1.186 0.252
Q a 6
modes in question are fitted as individual peaks and as groups. Consider the groups of peaks between 0.00285 and 0.00325 Hz in Fig. 8 (0S20— oS23). For these modes estimating the group simultaneously proved to be superior in minimizing the variances of the estimates. (This is similar to the example, case C, of non-linear regression given in Paper 1.)
CMO I
3.129 0.355
4S7 T
,
NNA I
When final estimates have been found, all the modes are summed together with the use of eq. 2 of Paper 1 to form a synthetic time series. The amplitude spectra of this synthetic seismogram is computed and compared to that computed from the data. Figure 9 is the synthetic spectra to compare with the NNA spectra of Fig. 8. The shapes of the peaks in synthetic spectra are useful as
00
*
0
~~~001 5
97
~ FREQUENCY
.00029
1HZ.)
~
.00444
~1JiJ ~ .00257
.00373
FREQUENCY
(HZ.)
.00459
i II
—~~~0
•5.09
TIME
90.19
(((P5.1
Fig. 8. Vertical component seismogram recorded at NNA following the Indonesian earthquake and its Fourier amplitude spectral density.
—~o
os.o.
TIME
90.49
(HRS.I
Fig. 9. Synthetic seismograms computed by summing the modes estimated from the NNA recording of the Indonesian earthquake and its Fourier amplitude spectral density.
55 TABLE V MODE
BKST I
BKST C
R AVG
1C 736.43 0.525 295.0 124.0 52.64 15.783 1.737 0.299
IC 736.43 0.525 295.0 124.0
OTII
lB
lB
T
576.65 0.443 230.0 81.0 41.47 10.379 0.304 0.250
576.65 0.443 230.0 81.0
IA 537.70 0.277 213.0 47.0~ 53.48 11.087 —0.128 0.207
IA 537.70 0.277 213.0 47.0
T Q a 6
Q a &
T Q a 6
T
Q a 6 0TI4
T Q a &
T Q a
T Q a 8
0T20 T Q a & O~’23
T Q a 6
lB 504.95 0.35 1 205.0 59.0 41.48 6.455 3.426 0.156
0~’13
MODE
lB 504.95 0.351 205.0 59.0
T Q a 8
lB 476.70 0.395 270.0 121.0 30.39 10.938 4.497 0.359
lB 476.70 0.395 270.0 121.0
IC 451.60 0.970 170.0 124.0 23.85 17.109
IC 451.60 0.970 170.0 124.0
0T35 T Q a 6
T
Q a
BKST I
BKST C
R AVG
IC 408.78 0.288 340.0 162.0 16.11 6.290 —4.785 0.390 IC 359.01 0.360 175.0 61.0 27.01 9.086 —2.577 0.336
IC 408.78 0.288 340.0 162.0
lB 360.87 0.124 330.0 74.0 17.83 2.833 —2.262 0.159
2B 360.72 0.157 284.0 70.0
IC 324.48 0.265 150.0 37.0 32.10 8.175 —0.411 0.255
IC 324.48 0.265 150.0 37.0
IC 309.53 0.239 320.0 159.0 8.15 3.764 —3.924 0.462
IC 309.53 0.239 320.0 159.0
IA 225.29 0.063 244.0 33.0 13.60 1.920 —4.899 0.141 IC 181.28 0.142 360.0 202.0 3.32 1.964
6
2B 225.75 0.130 295.0 100.0 4.67 1.910 —3.798 0.409
2A 225.36 0.078 248.0 41.0
IC 181.28 0.142 360.0 202.0
56 TABLE V (continued) MODE &
0T6 T Q a 8
BKST I
BKST C
R AVG
4.382 0.717 IC 429.19 0.506 168.0 67.0 26.97 10.47 —3.697 .0388
MODE
BKST I
6
—4.053 0.591
BKST C
R AVG
IC 429.19 0.506 168.0 67.0
decision criteria for two possible solutions obtamed for an oscillation. The synthetic spectra displays the character of groups (such as 0S20—0S23) remarkably well. At this time each mode is individually reevaluated in order to assign its reliability weight. If the following criteria are met the reliability of a mode will be assigned an A grade. (1) Complex demodulation graphs behave as the model predicts (see fig. 2 of Paper 1). (2) Regression estimates differ by only two standard deviations from estimates of complex demodulation (especially Q values). (3) The shapes of the synthetic spectra overlay the signal spectra very closely. An example of a mode given an A quality factor is 0S22 as recorded at NNA. This mode corresponds to the peak at 0.00307 Hz in Figs.8 and 9. The complex demodulates for this mode are shown in Fig. 10. A line fit to the first 36 h of the instantaneous amplitude plot yields a Q value of 235 which can be compared to the regression estimate in Table III of 245±8. A mode that deviates from linear decay in logarithms of instantaneous amplitude or in linear flat phase is not assigned an A grade. Only modes that fit the model very well are assigned an A quality factor. B-grade quality factors are assigned to modes that have small deviations from one of the above three critera. An example of this, again from Figs. 8 and 9, is the mode o~Ig at 0.00267 Hz. The complex demodulates for this mode are
graphed in Fig. 11. The slightly rounded decay of the instantaneous amplitude yields a Q value of 240. This value is quite different from the value of 350 given in Table III. The shape of the synthetic peak in Fig. 9 closely follows that of Fig. 8, al-
I N STANTAN EQ US
.
AMP L
325. IS
PER I 00
AT
I
-.
- I -o -
40
- 20 -
-2.
7
0
(5.17
30.34
60.68
45.54
80600074
~
75.95
91.02
.001299
INSTANTANEOUS
PHASE
__________________________________ 628
~ 5
I .26
-I
.•
26 ~
-
~.
-6
-
....
.• .
2
~
s.
I7
30.
34
4
T I ME
5. 5
I
5—....
.
60
.
68
75
- 85
9
I
- 02
(H P S)
Fig. 10. Complex demodulates of 0S22 recorded at NNA following the Indonesian earthquake.
57 INSTANTANEOUS
o
15.12
AMPL.
30.34 1690421070
AT
15.51 IS
60.68
INSTANTANEOUS
.
75.85
77
~
. .
..—..-—--,...
3
75.85
.2
PHASE
~-.
<1.26
.
I
~-l.2
.
30.34
45.51
TIME
60.68
91.02
C90165400L19—T
.~
z
....-
...
...~.
15.17
60.68
IS .004288
426.33
_,J..
—3 .
0
45.51
PERIOD
-INSTANTANEOUS
.~ .
30.34 169091078
_./_...
<1.26
AT
AMPL.
6.28
~•
~-l.Z6
15.17
PHASE
‘.-
6
0
•.
z
-
91.92
000L(590C616—T
.004209
6.28
3
INSTANTANEOUS
374.06
PERIOO
—
...~
.
- 6.
.... 75.85
91.02
IMPS)
Fig. 11. Complex demodulates of oSi
8 recorded at NNA fol-
lowing the Indonesian earthquake.
though the synthetic peak is taller. Amplitude and Q values are more uncertain than for the previous example of 0S22, and should be weighted accordingly. A grade of C is assigned to a mode with significant deviations in at least two of the three selection criteria. The mode 0S~shown at frequency 0.00234 Hz in Figs. 8 and 9 is an example of a grade C estimate. The complex demodulates for this mode, graphed in Fig. 12, exhibit a changing phase, and two sections of different amplitude decay rate. Because of this unusual decay, the regression estimate of Q differs significantly from a demodulate estimate. Even though comparison of spectra in Figs. 8 and 9 shows fair agreement, a reliability weight of C is assigned to this mode. For other modes the choice of grade C is made based on poor comparison of the synthetic oscillation as well as differences in Q estimates. The assignment of a grade D reliability weight is to decide that a peak in a spectrum fails all three
~
...
.
2 0
15.17
30.34
48.51
TIME
Fig. 12. Complex demodulates of lowing the Indonesian earthquake.
60.68
— 75.95
91.02
IMPS) O~I5 recorded
at NNA fol-
criteria and is unusable. In Fig. 8 two such examples are the small peaks for 0S19 (0.0028 Hz) and 0S30 (0.0038 Hz). 4.3. Methods of averaging It is shown below that there are significant differences in periods of free oscillations of the same order estimated from the global network of recorders. For this reason it is of interest to apply the diagonal sum rule that follows from selection rule (iii) described earlier. For a sufficiently distributed network of worldwide stations, the diagonal sum rule predicts that an average of all estimated periods of an eigenvibration mode will be an estimate of the period, ~ for a spherically-symmetric Earth model. This average for T0 is calculated for the IDA network observations plus the regional network estimates. However, in some cases the global average of all stations will not be sufficiently representative for
58
the diagonal sum rule to apply, for the following reasons. First, for many of the modes listed in Tables II and III, estimates of 0 were not ob tained for all stations. This average could possibly introduce a bias to T0 due to restricted world-wide coverage. Second, since all estimates of 0 are not of equal reliability and equal variance, they cannot be given equal weight in averaging. Instead a weighted average is made. If the weights were all nearly equal, sufficient coverage would yield an unbiased estimate of T0. Since the regional network represents a small localized region in a worldwide average, the average of 0, from the regional network is used in averaging 0 worldwide. We now use a theorem (Lindley, 1965) for combining estimates of known uncertainties. Given a group of estimates x distributed as N(X, ~~2) where the average ~ is unknown and the variances are known, estimates of ~ and var are given by .~
~
=
I
I
//
~ w2
(1)
TABLE VI Values for wr, for various reliability grades
-_______________________________________________ Grade
wr
A B
1.0 0.8 0.7 0.6 0.2
C D
1
for penod and Q for penod for Q for period for ~
0.0 for penod and Q
to the values averaged. Values for wr1 are given in Table VI. When averaging period, larger values were assigned to wr1 than when averaging Q. This was done because it was found in Paper 1 that the values of Q were far more drastically affected by interference by neighboring modes than the values of period. Therefore, period, of a C grade mode, say, is more reliable then the value of Q of that same mode. The valuestheforauthor’s wr, are opinion arbitraryof and were chosen to reflect the relative reliability of the measurements.
—I
varx(~wi)
(2)
4.4.
1
Overtone modes
where — I / 2 (3\ WI — / ~ ‘ / In the present case, reliability weights are included with the variance in producing the overall weights w1 in eq. 1. A reliability weight is assigned, wi, 0 ~ wr ~ 1 depending on the grade, D to A. The weights, w0., to use in eq. 1 are then computed from — ( 2 / 2~ (4) — ~ / ~ WrE 2. The var S~ where is the smallest of theaverage a is thena~ computed from avalue weighted of the a2 —
Many peaks in the spectra (see Fig. 8, for example) correspond to overtone modes. All of these oscillations were analyzed and 23 estimates of 0 were found for 17 modes. These estimates are shown in Table IV. When overtone modes in the spectrum are isolated peaks between fundamental modes they can be treated one at a time as are the fundamental modes. In a few cases, overtone oscillations had frequencies close towas neighboring fundamentals that specialsoattention required (see Case B in Paper 1). To illustrate the use of non-linear regression to simultaneously estimate 8 for close neighbors, consider two examples of
~ (wr, /a 12)
var ~
=
Wri
I
(5)
Equations 1,4 and 5 are used to compute weighted averages of period and Q for each order 1 of eigenvibration. A reliability grade is then assigned to each average equal to the highest grade assigned
oscillations following the 1977 Indonesian earthquake. First ~thepair of 0S7 and 2S3 (Tables III and IV) recorded at CMO, and second the pair ~ and 2S7 recorded at HAL. Figure 13 shows the Fourier amplitude spectra for the pair of modes 0S7 and 2S3 and the synthetic modes generated with the regression estimates. In this case, it is clear there is more than
59
S Y N THE T I C
DATA
INSTANTANEOUS
AMPL.
AT
PERIOD
810.64
~
I
‘~
0
15.17
30.34 8flN0W1016
45.51
60.68
INSTANTANEOUS
W
.00125
.00105
6 .28 3.77
.00125
,... —~....
Fig. 13. Amplitude spectral density of
0S7—2S3 pair.
—j
I .26
-I
one mode present. This interference is well exhibited by the complex demodulates shown in Fig. 14. The regression estimates (see Tables III and IV) for Q for these two modes differ by nearly 150. The variance of Q for the smaller amplitude peak 2 5 3 , is much greater than for 0 S~ as might be expected for closely separated unequal amplitude peaks. It can be seen that a higher value of Q for 2S3 is correct, as this peak loses energy less quickly than 0S7. This difference in Q is found by computing different amplitude spectra with successively later starting times. As in Case B of Paper 1, there are a number of significant non-zero linear correlation terms computed from the covariance matrix (see Table VII) indicating trade-offs
.
75.65
PHASE
-.~
/
-
./
~
26 .
~
.‘~—.
-
~
.~/
:‘ ....~.
-6. 0
IS. 17
30.34 ME
p5
68
78.85
a1 a2
82 Tt 72
Q1
O~7 2
91 .02
Fig. 14. Complex (atearthquake. 810.64 s) of the CMO seismogram following demodulates the Indonesian
may be occurring between 0 of each mode. For this reason it seems appropriate that reliability weights of grade A cannot be assigned to such closely separated oscillations. The Fourier amplitude spectra of the second example is shown in Fig. 15 on the left with the
TABLE VII Correlations for
91.02
CYCILSIO(LTA—T
IS .004249
S3 regression
a1
a2
81
82
71
72
Q1
—0.5092 0.0 0.3973 0.0 —0.2628 —0.8045 0.4170
—0.3973 0.0 0.2147 0.0 0.4374 —0.8348
—0.5092 —0.8047 0.4174 0.0 0.2634
0.4369 —0.8347 —0.2138 0.0
—0.3 190 0.0 —0.1152
0.1141 0.0
—0.3191
60
HAL DATA
____ .00105
CMO SYNTHETIC
.00205
.00185
DATA
SYNTHETIC
__~J.
.00205
.00185
.00205
.00205
.00185
Fig. IS. Amplitude spectral density of O~II~2S7 pair.
synthetic modes again shown on the right for comparison. Although this appears as a single peak in the Fourier spectrum for HAL, the complex demodulates in Fig. 16 show a very pronounced interference effect indicating the presence of two oscillations. (Spectra from CMO were included in Fig. 15 as an example of a case in which two closely spaced modes are clearly present.) Two modes were fitted quite well by non-linear regression to this single peak. Again, there were large linea correlations between parameters of adjacent
INSTANTANEOUS
AT
.
PER 100
536.39
~ 85
-
~ ~
.20
-.
- I II
-
I
.
0
modes (see Table VIII), so even though the uncertainties obtained were quite low formally indicating a close fit, a reliability weight of grade B must be assigned. Certainly, one of the modes is 0S11, but the identification of the close neighbor is not as clear. It may simply correspond to the overtone mode 2~7 as predicted by numerical calculations for various Earth models (Derr, 1968). However, an alternative identification may be appropriate. This corresponds to the results of studying a realistic Earth model with the theory of quasi-degeneracy (Dahlen, 1969; Luh, 1973, 1974). The results show that Coriolis coupling (McDonald and Ness, 1961) may occur between torsional and spheroidal oscil-
AMPL
so
I
15.17
30.34 •*8061076
45.51 IS
60.68
I N S TAN TAN E 0 U S 6.28
75.85
SI .02
.004209
P HAS E
.
-
~
~
-
.
....‘
-
26
I
.
~—..
-~. ..
-.6
.
2~
I ~.
I,
30. 34
45. 5 I
T I ME
60
. 08
~
.
I
.
02
I HRS)
Fig. 16. Complex demodulates (at 536.39 s) of the HAL seismogram following the Indonesian earthquake.
61 TABLE VIII Correlations for
O~II ~2~7
a a1 a2 81 82
li 12
regression
1
a2
81
0.9006 0.0 0.1706 0.2974 —0.4629 —0.8475 —0.6417
—0.1706 0.0 0.4353 —0.3215 —0.6592 —0.8378
0.9006 —0.8482 —0.6406 —0.2992 0.4614
—0.6601 —0.8371 —0.4367 0.3195
lations only closely separated frequency. Such is 5I1 in and the case for the modes 0 0T12 whose difference in periods is less than 1 s (Derr, 1969). Luh (1974) points out that usual mode designation loses its meaning because of strong coupling, and any analysis of normal mode Fourier spectra for modes that are strongly coupled must incorporate information on the extent of coupling. The inference here is that the torsional oscillation 0T12 could, through Coriolis coupling, produce vertical motion with a period only slightly different from 0S11. Virtually all ten vertical component time series following the 1977 Indonesian earthquake contained energy at these periods although none were resolvable as single oscillations, and in fact all exhibited similar patterns in their complex demodulates as in Fig. 16. It is interesting to note that the largest amplitude oscillation recorded on the BKS transverse component seismometer has a period corresponding to the single oscillation of 0TI2 (see Table V). This shows the energy in the mode 0T12 is present for coupling to possibly occur. In this study (primarily concerned with regional variations of well-resolved fundamental modes), no theoretical analysis of quasi-degenerate coupling is made. For this reason, the modes will be designated as 0S~1 and 2S7 in the tables although these are not certain identifications. 4.5. Tables of eigenvibration parameters Tables Il—V contain all the estimates and uncertainties for the modes resolved using the optimal methods outlined. These oscillations were gen-
Q2
12
&2
0.3243 0.0 —0.5614
0.5628 0.0
0.3244
erated by the four large earthquakes described earlier. Contained also in the tables, just above the estimate of period, is a number designating the number of modes included in the regression analysis and a letter corresponding to the reliability weight assigned to that estimate. In the columns for regional and global averages of period and Q, the number designates the number of modes used in the average. The following abbreviations are used in the tables. T represents period in seconds. Amplitude, a, is in microns, and phase, 8, is radians. The four earthquakes are designated I— Indonesian, P—Philippine, C—Colombian, and W— West Irian. Under each estimate of period, amplitude, Q, and phase is its corresponding formal standard deviation.
5. Results from network analysis 5.1. Introduction An initial goal of this study was to assess what variability in period and damping factors Q of eigenvibrations may exist in a limited region such as northern California. Extensive periodogram analysis of world-wide seismographic station network (WWSSN) shows considerable scatter in both period and Q of eigenvibrations (Madariaga, 1971; Dziewonski and Gilbert, 1972, 1973; Bolt and Currie, 1975; Jobert and Roult, 1976; Anderson and Hart, 1978; Hansen, 1978, 1979, l981a; Geller and Stein, 1979; Hansen and Bolt 1980). For example, a representative scatter is for the mode
62
0S14, the published values range through 447.17— 449.41 s for period and 183—435 for Q. At the present there is much interest in tracing the source of this variation and separating measurement errors, measurement bias, and various complexities of the Earth. In this study eigenspectra have been systematically analyzed from the four large earthquakes in order to determine to what extent each effect may be detected. By using the network of three stations in northern California described in Section 2, some isolation has been achieved for such variability as: (1) crustal structure beneath each receiver; (2) positions of nodal lines of eigenfunctions; (3) different locations of earthquake sources. The following sections address these questions.
5.3. Effects of local structure beneath receiver sites
5.2. Basis for statistical comparison
About 350 km north, the WDC seismographic station is located in the southeast portion of the Klamath mountains just northwest of the Central Valley. The seismometer is sited on Devonian meta-volcanic rock. The crustal thickness here is about 30 km. There is evidence that suggests the lithosphere in northern California is an anomolous region. The upper mantle P-wave velocity structure has been investigated by analyzing the azimuthal variation of teleseismic P-wave residuals recorded by the University of California at Berkeley seismographic stations (Bolt and Nuttli, 1966; Simila, 1980). The relative residual variations at a single station in the northern part of the state exceed ±l.Os. Simila has modelled the residual variations by lateral variations of the P-wave velocity in the upper mantle. A low-velocity zone is inferred beneath the Gorda plate to account for the positive residual pattern along the coast. A paleosubduction zone is modelled beneath northern California to represent a region of high velocity which explains the negative residual values. This structure may extend into the lithosphere to a depth of 150—200 km. It is now convenient to restate the first question of Section 5.1. Given that the whole Earth is oscillating with a set of eigenvibrations, can the different crustal and upper mantle structures under the three receivers cause measureable changes in 0 for each mode? Since the separations of these seismometers (<350 km) are less than half the
Because each time series is considered as a superposition of signal upon stationary noise (see Paper 1) it is appropriate to use a statistical criteria for assessing the significance of differences in measured eigenvibration parameters. The estimation procedures of complex demodulation and non-linear regression were developed in Paper 1 for this purpose. By estimating the statistical uncertainties for each parameter, confidence intervals can be estimated for the quantity OjJ — 0~.(0~~ denotes the] ~ estimate of the oh parameter in 0.) Given the parameters X1 and x2 with variances and ~2’ the quantity var(x~— x2) is given by ~
var(x~— x2)
=
.~
+
~
(6)
To test the hypothesis that x1 x2 we form the N% confidence interval and test if =
I x1
—
> CN
(s? +
(7)
2)1/2
If eq.7 holds, x1 and x2 are significantly different at the N% confidence level. That is, a difference greater than CN(s~+ s~)I/2 can occur by chance alone only (100 — N)% of the time. For N 95, CN 1.96, and for N 99, CN 2.5758. Equation 7 gives a quantitative measure of testing for significant differences in eigenspectra parameters in the following sections of this paper. =
=
=
California is a region of complex geologic structures with tectonic properties changing quickly with distance. The recording stations BKS, JAS and WDC are each situated in quite different tectonic regions; the coastal range, the Sierra Nevada mountains, and the Klamath mountains, respectively. The BKS seismographic station is located in the Berkeley Hills in the Coast Range. Here the crustal thickness H is approximately 22 km. The JAS seismographic station is sited east of the Central Valley in the foothills of the Sierra Nevada mountain range. Geologically, the Sierra consists of granite overlain by remnants of older metamorphic rocks. The foundation rock at the JAS seismometer is metamorphic serpentine.
63
wavelength of a mode of order up to oS~o~ it is expected that any differences in parameters obtamed from the three receivers would be due to local effects of lithosphere structure rather than the global changes due to large scale lateral heterogeneities. (The question of distance separation of stations required to see significant shifts in multiplet period due to lateral heterogeneities is explored in Section 5.4.) To estimate the order of magnitude of the response of a thin layer of crust to a forced oscillation at the base of the lithosphere, let us assume the layer can be represented as a spring and dashpot system as in a linear damped harmonic oscillator being driven by a harmonic force. The result for reasonable rock properties suggests that if the receivers are located on bedrock, such as basalt or granite, there should not be any detectable changes in period, Q, or phase of eigenvibrations between the seismometers in the regional network. Ten or twenty percent changes in the properties of the bedrock again cause no detec-
there are over 60 modes for which comparisons are made to determine if indeed there are observable, systematic shifts in period, Q, or phase of an eigenvibration mode within a small region due to the tectonic differences in the region. A statistical analysis of the relevant values in Table II indicates that there are no detectable systematic shifts in period, Q, or phase within the California regional array for any eigenvibration modes excited by any one of the four sources considered. At the 95% confidence level, there are 24 cases where two estimates of period differ significantly. However, at a 99% confidence level the number of cases reduces to 13. Of these 13 cases, 11 involve C grade estimates while the other 2 cases both are between B grade estimates and the differences are on the order of 0.34 s. Such small differences would not appear to be significant in the light of the probably underestimated uncertainties of B and C grade estimates 0 (see Section 4 and Paper 1). A particular illustration is for the mode 0S24 re-
table changes in the results. Even if the differences extend for 200 km as Simila’s model suggests, there is no detectable shift.
corded following the Colombian earthquake. The JAS recording, reliability B, and the WDC recording, reliability B, differ in period by 0.34 s. Given standard errors of 0.069 and 0.077 s, respectively, the values calculated from eq.7 indicate this difference is significant at a 99% confidence level. It is judged that such a small difference may in fact not be significant for estimates assigned a B grade reliability. There is only one example of two values of Q differing significantly at either the 95 or 99% confidence level. This difference occurred for the mode 0S14 following the Colombian earthquake. Analysis of the JAS recording yielded an A grade estimate while that of the WDC recording was a B grade. The 99% value of CN(a? + ~~/2 is 103.8 while the difference in Q is 105. Because this Q difference involves a grade B estimate it cannot be considered significant. The conclusion is that the decay rate of eigenvibrations does not change significantly due to local crustal differences within a region whose dimensions are less than a wavelength. The parameter for which the largest number of significant differences occur is phase. At a 95% confidence level there are 53 cases where the phase
5.3.1. Case] The following comparison, hereafter referred to as ‘case 1’, bears on the initial question of Section 5.1. To eliminate any source characteristics, case 1 will only compare estimates of 0 for spheroidal modes obtained from the three different receiver sites of the California regional network for each of the four earthquakes independently. Therefore, for example, a JAS OSI4 estimate of 0 from the Columbian earthquake is not compared to a WDC 0S14 estimate of 0 from the Philippine earthquake. At BKS, the comparisons were made for both the vertical and longitudinal components of ground motion. From the four earthquakes recorded by the California regional network there were 184 resolvable observations of eigenvibrations in the range of o~9through 0S48 (see Table III). Of these 184 observations, there are 59 cases where two or more estimates of 0 were obtained from the regional network for the same mode excited by the same earthquake. Because some of these 59 cases had three or four observations of the same mode,
64
of the same oscillation differed significantly at two different sites. At a 99% confidence level the number of cases reduced to 43, still a large number. Of these 43 cases, 27 of them involved comparison of a vertical component recording with the longitudinal component recording. The magnitude of the difference varies from mode to mode and in some cases is as large as iT. This indicates that in general the semi-major axis of the elliptical particle motion of spheroidal modes is inclined from the vertical. Furthermore, although most estimates of 0 from the longitudinal component are grade C, it seems the amount of inclination is a function of order number, Of the remaining 16 cases of significant phase differences from vertical recordings, the magnitude of the shift, though significant, is small, less than IT/6 in all four cases involving A-grade estimates. The direction of phase shift is random both in order number and between different earthquake sources. Since no systematic pattern emerges, it is concluded that local geological differences within this small region create no detectable shift in the phase of an eigenvibration. 5.4. Comparison with theory This section will address the second two questions of Section 5.1. To do this, comparison will be made between the observations in Tables Il—V and theoretical and numerical results. There are three cases to be tested using observations from the California regional and IDA networks. These three cases will be called case 2, case 3, and case 4, respectively, Early theories of Earth eigenvibrations were made assuming spherically-symmetric Earth models. Theoretical descriptions of the Earth’s vibrations using these models are solved with spherical harmonic expansions about the earthquake source. This serves to define eigenfunctions, with specific well-defined nodal patterns that are a function of source position. In the early I 960s the effects of the Earth’s rotation and ellipticity were investigated by a number of people. It was shown that such effects remove the degeneracy of the problem giving rise to splitting of the eigenfrequencies. Rotation further complicates the problem by in-
troducing Coriolis coupling between nearby modes (see Section 4.4) and westward drift of the nodal lines (McDonald and Ness, 1961). A method, making use of the specific nodal patterns, for identifying the order m of a mode ,~Sf1was proposed by Nowroozi (1965). By plotting the spherical harmonics of different order m as a function of / for a particular recording site it is found that the zeros of the functions for different m occur for different values of I. Therefore, certain values of m can be ruled out for each large amplitude oscillation. Different results have been obtained from more recent studies of realistic Earth models. The work of Stifler (1979) is in contradiction with the above. His finite element calculations led to three principle theoretical results on the Earth’s free oscillations. (I) The general effect of a lateral heterogeneity on a sphere is to remove the degeneracy of the multiplet eigenfrequencies. The non-degenerate eigenfunctions are then completely defined with respect to the inhomogeneities that have removed the sphere’s symmetry. For the lowest-order torsional oscillation, the Earth’s ellipticity of figure governs the positioning of the eigenfunctions. For higher-order modes, the relative positions of the continents determine the orientation of the eigenfunctions. (2) The combined effect of the Earth’s continents and its ellipticity on the eigenfrequency perturbations is found not to be the sum of the individual effects. Thus, the eigenfrequency perturbations due to lateral inhomogeneities are nonlinear functions of the perturbations to the starting model. (3) For torsional oscillations o7’2 — 0T10 the effects of the Earth’s ellipticity are greater than the effects of the structural contrasts between continents and oceans. The finite element computations show that the range of the ellipticity-induced eigenfrequency splitting is approximately 0.2% of the degenerate eigenfrequency; whereas, the range of the continent-induced splitting increases from 0.01 to 0.1%. The first result states that the eigenfunctions of an inhomogenous body are unique and fixed in position for all time independent of the source
65
position. Mote (1972) in a discussion of the effects of notches on rotating disks, argues that the presence of an azimuthal heterogeneity ‘locks’ the eigenfunctions onto the body and prevents them from behaving as travelling waves. The westward movement of nodal lines is then not a valid conclusion. However, it is not possible to test this with observations as the period of motion of these nodal lines is longer than the time required for a mode to decay it into the noise level. For example, this period for torsional oscillations is 1(1 + 1)/fl (144 h for o~’2) where fi is the period of the rotation of the Earth. The passage of a nodal line of the eigenvibrations studied in this paper through a regional array is too slow to observe. The by discrimination procedure order in m light suggested Nowroozi (1965) is also for incorrect of Stifler’s first result. In fact, the amplitude of a recorded singlet would be governed by the relative positions of both the source and the receiver with respect to the fixed nodes of the eigenfunction and not with each other. Graphs of amplitudes of multiplet eigenfunctions presented by Stifler (1979) suggest that virtually all multiplets consist of multiple singlets of comparable amplitude. If this is the case in general, the question occurs as to why so many modes presented in Tables Il—V were resolvable as single oscillations? The answer to this may lie in the analysis by Dahlen (1 979a, b, 1980) of the influence of lateral heterogeneities on the splitting of single station seismic spectra. Consider the lateral heterogeneity of the Earth expressed as a perturbation expanded in spherical harmonics of orders s~ through 5m~ where the perturbation, ôm, i5 slight, i.e. symbolically 18m/mol<< 1. Dahien states that under the condition Sm~<< 1< < I 6m/m and n <<1, there is cancellation of m — 1 0 2/ adjacent lines, and the mode ~5f~ would behave as a single resonance peak broadened by attenuation alone. He further states that the degree of cancellation varies as a function of separation, ~, of the source and receiver and that maximum cancellation should occur at ‘antinodes’ where cos[(/ + 1/2) i~— iT/4] = 1. Right at the antipode from the source, all amplitudes of the singlets are in phase yielding no cancellation. A test of these theories would be to compare 5~j~
1
=
L
—
eigenvibrations recorded by the California regional network excited by sources located at different azimuths and distances from the array. If the sources could be shown to supply the same relative energy to the singlet modes, then, according to Stifler’s conclusion, the estimates of 0 would not change. However, since the position of the source determines the relative excitement of the singlets, such a comparison is not crucial. As Dahlen (1980) points out, it is not a simple matter to determine the range of validity of his theory for the Earth. It certainly must fail for low 5max’ s~, and 1, say n -~1, as the condition ôm/m 0 are in fact depth dependent, and different modes sample the Earth down to 5min’ different 5max’ and depth. As a result the quantities Iôm/moI depend upon the mode under consideration. A way of determining the validity of this theory over a range of I is to examine multiplet spectra for deviations from single-peak behaviour. The methods of complex demodulation and nonlinear regression developed in Paper 1 are ideal for such an examination.
I
I
Case 2 In this case comparisons are made of eigenvibration parameters estimated within the California regional network excited by the four different earthquake sources. These four earthquakes represent sources of different azimuths and distances from northern California. It can be seen from Table III that there are significant shifts in period from stations distributed world-wide from a single earthquake source. (For example see the HAL and RAR estimates for the mode o~I6’ The difference in period is 1.33 s, which is significant at the 99% confidence level.) If eigenfunctions are locked in place, much less variation would in general be expected in period for modes excited by different sources but recorded in the same regional locality, because the variation in the singlet amplitudes is a function of source and receiver positions with respect to the eigenfunctions. By fixing the receiver positions to a small region the only variation in singlet amplitudes is due to the source. There still could be shifts in period from source mechanism variation. However, if a source excites well a singlet whose nodal line lies nearby the 5.4.1.
66
recorder it will not affect the regional recording, while world-wide there is likely a station near an antinode whose recording would be much influenced by this singlet. Comparisons are made in this case for period, Q, and amplitudes. Period and Q comparisons are straightforward following the procedure in Section 5.3. Comparisons of amplitude are made in a different sense as a test for positions of nodal lines. If nodal line positions do not change in the region of Northern California for different earthquake sources, the relative amplitudes between JAS, WDC, and BKS for each eigenvibration should remain constant. There are 13 modes between ~ and 0S3~where relative amplitudes can be compared between JAS, WDC, and BKS. Unfortunately, not all 13 modes were estimated from all three sites, and, of the estimates made, 19 were assigned A grades, 34 were assigned B grades, and 30 were assigned C grades. Furthermore, there were no combinations of multiple A grade modes estimated from different earthquake sources. For example, 0S19 was well recorded at JAS, WDC and BKS following the 1977 Indonesian earthquake and all three estimates were assigned A reliabilities. The other three earthquake sources also generate this mode although estimates were all assigned C grade reliabilities. For this example of ØSj9~ it is interesting to note that even though there is some difficulty in making reliable estimates of 9 following the latter three earthquakes, the periods only differ at most by 0.26 s which is not significant at the 99% confidence level. If we disallow the C grade estimates of amplitude in the comparison we are left with only five modes for comparison. Of these five cases there are three (0S21, O~23’ and o~31) that show agreement between relative amplitudes and two modes (0S24 and 0S27) that show disagreement between relative amplitudes, Comparisons were made between periods and between Q values for 25 modes from 0S10 to O~36 recorded following two or more of the four earthquakes studied. Of these 25 modes, there were only 5 modes (o~I7,22,23,3o,3l) where periods differed significantly and only 2 modes where Q values differed significantly. It was concluded that even though there is scatter in Q data, the uncertainties
‘
indicate that differences in Q are not significant. In all five cases the A grade period estimates, though significant at the 99% confidence level, differ by less than 0.65 s. Though it is not conclusive, the above compari. sons for period, Q, and amplitude indicate that there are differences in eigenspectra excited from different sources and recorded by the same regional stations. The eigenspectra tend to compliment each other, each exhibiting different wellresolved modes following the different earthquakes. Even though shifts in period are small relative to observed shifts between IDA stations, the fact that the eigenspectra are complimentary to each other suggests that the source characteristics, including location of the earthquake, are important in determining the patterns of singlet eigenfunctions. Since the effects on 0 are small relative to global characteristics the results lend support to the theory of fixed eigenfunctions. 5.4.2. Case 3 This case will address the question of variations of Q values of spheroidal eigenvibrations as a function of position on the Earth following the Indonesian earthquake. Comparing Q values of A grade modes whose periods are significantly different will distinguish whether attenuation characteristics of the singlet eigenfunctions vary with a multiplet. The global array used for this comparison consists of the seven IDA stations and the California regional network average values for period and Q. In all there are 114 A or B-grade estimates of Q following the Indonesian earthquake. (C grade modes do not yield reliable estimates of Q for comparisons.) A minimum of two estimates was obtained for 26 different modes of oscillation. Of these 26 modes, significant differences were found in 11 different modes if B grade modes are ineluded, and in five different modes if only A grade modes are included. (Note, again that modes assigned reliability factors other than A are not strictly following the model of a single decaying oscillation. See Section 4.) Because of the considerable scatter in measurements of Q, usually due to unknown biases, the 99% confidence level is used for the comparisons.
67
To illustrate the variation in Q, consider the mode 0S16 in Table III. Here we have eight estimates of grade A. Estimates of period for this mode distributes fairly evenly from 405.98 to 407.31 s, a maximum difference of 1.33 s. The Q values range from 260 ± 34 to 324 ± 14 for the A grade estimates with a weighted average of 284 ± 12. The largest differences do not appear to be associated with the modes whose periods differ the most. Since the number of observed significant differences is small and no systematic pattern emerges concerning shifts in Q values, it appears that measurements of Q of eigenvibrations are not a function of the position on the Earth of the recording instrument, or of the great circle paths for modes in this period range. 5.4.3. Case 4 This final observational analysis addresses two aspects of the theory of cancellation of eigenvibration singlets to produce a single decaying osciliation for each mode S,. First, a test of the validity of Dahlen’s asymptotic theory (presented above) of single-peak approximation. Second, using the California regional array and the IDA network inferences are drawn on spatial separations of receivers required to see significant shifts in penods of eigenvibrations. Since each mode resolved in Tables Il—V was closely scrutinized for single-peak behavior, the eigenvibration estimates obtained in my study are ideal for Dahlen’s suggestion for determining the range of values of / for which his asymptotic theory is valid. My observations range from O~4 through 0S48 and the results for A grade modes indicate that the assumption of a single oscillation is valid from 0S7 through approximately 0S30. There are, however, still some inconsistencies between the observations and the theory. It was computed, for each station, which modes would theoretically lie at an asymptotic antinode (see expression above). When a station is sited at an asymptotic antinode maximum cancellation of singlets is theoretically obtained for producing a single oscillation. By considering the well-excited peaks from 0S7 to 0S30 in all the spectra that correspond to asymptotic antinodes it was con,,
cluded that there are about an equal number of cases where stations at asymptotic antinodes record single oscillations and record oscillations with interference. For example, the mode 0S8 excited by the 1977 Indonesian earthquake has an asymptotic antinode at NNA. This mode is well recorded and exhibits a sharp peak in the spectrum (see Fig. 8, 0.0014 Hz) and linear complex demodulates. Now consider the example of 0S15 in Section 4.2 (Figs. 8 and 12). NNA again lies at an asymptotic antinode for 0S15 and the mode shows up well in the spectra. The behaviour of this oscillation, as exhibited by the complex demodulates, is not that of a single oscillation. There are also modes recorded at sites that do not correspond to asymptotic antinodes that very closely follow the behavior of a single oscillation. Apparently, the degree of cancellation of singlets, though observed from 057 through ca. 0S30, is not completely described by the asymptotic theory. The distance separation of stations required to observe shifts in period of a mode 5, due to different relative amplitudes of the m singlets is variable. Furthermore, for two given world-wide stations, the shift in period of a mode is dependent upon degree / of the oscillation, and probably dependent upon the excitation position. To illustrate this point consider the modes 0S14 and 0S21 in Table III. Differences in period vary between all the possible pairs of stations. For 0S21 the estimate in California differs by 0.36 s at a separation of approximately 30° from CMO and by 0.65 s at a separation of approximately 450 from HAL while there is no difference at all from BDF at a separation of approximately 95°.RAR and NNA have no significant difference although separated by approximately 80°across the Pacific Ocean. On the other hand, the mode 0S14 shows a difference between the California estimate and CMO of 1.06 s, and between California and HAL only 0.19 s. NNA and RAR have a difference of 0.69 s for 0S14. In summary, there emerged no systematic pattern for the direction of a shift in period as a function of 1, nor was there any one station that consistently yielded the highest or lowest estimate of period as a function of 1.
68
6. ConcLusions
0S48 show that the assumption for a multiplet S, to behave as a single oscillation is valid from 0S7 through approximately 0S30. This result supports Dahlen’s theory on singlet cancellation. However, subtle differences between observation and theory still exist suggesting that the degree of singlet cancellation is not completely described by the asymptotic theory (see Section 5.4.3.) (5) No systematic pattern emerged for the direction or amount of shift of eigenperiod as a function of order / or position of recording station following the Indonesian earthquake. No single regional or IDA station consistently recorded the highest or lowest estimate of period as a function of / (see Section 5.4.3). The first conclusion suggests a possible method of summing seismograms to improve the signal-tonoise ratio. One problem with ‘stacking’ procedures is that estimates of Q become biased due to interference of different relative amplitude singlets. Since parameters of 0 are seen not to change within a small region, a network, such as in northem California, can be used to sum its seismograms without creating interferences between singlets. Finally, more observational analysis needs to be made with an expanded data base to compare with the theories of ‘locked’ eigenfunctions and singlet cancellation. The latter (Dahlen, 1980) predicts that near the epicenter and antipode of an earthquake source there is much less cancellation of singlets. This phenomenon should be an observable feature with use of complex demodulation on the records of broad-band stations positioned correctly with respect to an earthquake source. Such observations should provide a crucial test between the normal mode theories of distinct eigenfunctions locked in permanent positions by Earth structure and standing wave patterns positioned relative to the source. ,,
The main objectives of this investigation were to improve the resolution of observations of terrestrial eigenvibrations. It is now clear that such an improvement is needed if the resolution of inverse procedures in computing reference Earth models (Dziewonski and Anderson, 1981) is to be significantly sharpened. Improved observations of o obtained by these methods from the California regional network and the global IDA network were used to trace the~source of variations of spectral parameters that arise due to various cornplexities of the Earth (e.g. lateral heterogeneities). (1) An analysis of eigenvibration parameters obtained within the California regional network following each earthquake source indicates that there are no detectable systematic shifts in period, Q, or phase within a region that is small with respect to the wavelength of the oscillation. Use of both vertically and longitudinally oriented seismometers in the analysis indicated that in general the semi-major axis of the elliptical particle motion of spheroidal modes is inclined from the vertical (see Case 1 in Section 5.3.) (2) Comparison of periods, Qs, and amplitudes of eigenvibrations excited by earthquake sources of varying distance and azimuth to the recording sites in northern California indicate that there are differences in eigenspectra due to the source function and position. Though not conclusive, slight systematic shifts in period (<0.65 s) and relative amplitudes between the California regional stations lend support to the theory of ‘locked’ eigenfunctions. Differences in Q values are not significant (see Section 5.4.1.). (3) Differences of Q values obtained world-wide from both the California regional network and the IDA network are found not to be significant, even though their corresponding periods varied significantly. This observation indicates that measurements of Q of eigenvibrations are not a function of the position of the recorder on the Earth. The conclusion is that all singlet eigenfunctions have the same damping characteristics independent of great circle paths for this period range (see Section 5.4.2.). (4) Observations ranging from periods of 0S4 —
Acknowledgements The author would like to thank Bruce A. Bolt for his suggestions in this work, and Robert Uhrhammer, Don Michniuk and Richard Lee for many helpful discussions. Thanks are also due to Freeman Gilbert for
69
supplying the IDA data of the 1977 Indonesian earthquake, and Guy Masters for a computer program that computes the response of the IDA gravimeters, and to CIRES for the use of their computing facilities in preparing this manuscript. This work was supported by NSF grant EAR 19694.
References Agnew, D., Berger, J., Buland, R., Farrell, W. and Gilbert, F., 1976. International deployment of accelerometers: a network for very long-period seismology. Eos, Trans. Am. Geophys. Union, 57: 180—188. AId, K. and Richards, P.G., 1980. Quantitative Seismology: Theory and Methods. Freeman. Anderson, D.L. and Hart, R.S., 1978. Attenuation models of the Earth. Phys. Earth Planet. Inter., 16: 289—306. Bolt, B.A. and Currie, R.G., 1975. Maximum entropy estimates of Earth torsional eigenperiods from 1960 Trieste data, Geophys. J.R. Astron. Soc., 40: 107—I 14. Bolt,. B.A. and Nuttli, OW., 1966. P-wave residuals as a function of azimuth. I. Observations. J. Geophys. Res., 71: 5977—5985. Dahlen, F.A., l979a. The spectra of unresolved split normal mode multiplets. Geophys. J.R. Astron. Soc., 58: 1—34. Dahlen, F.A., I979b. Exact and asymptotic synthetic multiplet spectra on an ellipsoidal Earth. Geophys. J.R. Astron. Soc., 59: 19—42. Dahlen, F.A., 1980. A uniformly valid asymptotic representation of normal mode multiplet spectra on a laterally heterogeneous Earth. Geophys. JR. Astron. Soc., 62: 225—247. Dahlen, F.A., 1981. The free oscillations of an anelastic aspherical Earth. Geophys. J.R. Astron. Soc., 66: 1—22. Derr, iS., 1969. Free oscillation observations through 1968. Bull. Seismol. Soc. Am., 59: 2079—2099. Dziewonski, A.M. and Anderson, D.L., 1981. Preliminary reference Earth model. Phys. Earth Planet. Inter., 25: 297—356. Dziewonski, A.M. and Gilbert, F., 1972. Observations of normal modes from 84 recordings of the Alaskan earthquake of March 28, 1964. Geophys. J.R. Astron. Soc., 27: 393—446. Dziewonski, A.M. and Gilbert, F., 1973. Observations of normal modes from 84 recordings of the Alaskan earthquake of March 28, 1964, II. Further remarks based on new spheroidal overtone data. Geophys. J.R. Astron. Soc., 35: 401—437. Geller, Ri. and Stein, S., 1979. Time domain attenuation measurements for fundamental spheroidal modes (
0S6—
0S22) for the 1977 Indonesian earthquake (abstract). Eos, Trans. Am. Geophys. Union, 59: 1140. Gilbert, F., 1971. The diagonal sum rule and averaged eigenfrequencies. Geophys. JR. Astron. Soc., 23: 119—123.
Hansen, R.A., 1978. Complex demodulation estimates of eigenspectral amplitudes and damping factors from the August 1977 Indonesian earthquake Eos, Trans. Am. Geophys. Ha~’~. 1979. Estimates of free oscillation periods, amplitudes and damping factors excited by different earthquake sources (abstract). Earthquake Notes, 49: 106. Hansen, R.A., 1980. Simultaneous estimation of terrestrial eigenvibrations. Eos, Trans. Am. Geophys. Union, 61: 298 (abst.). Hansen, R.A., l98la. Observational Study of Terrestrial Eigenvibrations. Ph.D. thesis, Univ. California, Berkeley, CA. Hansen, R.A., l98Ib. Simultaneous estimation of terrestrial eigenvibrations, accepted by Geophys. J.R. Astron. Soc. Hansen, R.A. and Bolt, B.A., 1980. Variations between Q values estimated from damped terrestrial eigenvibrations: J. Geophys. Res., 85: BlO: 5237—5243. Jobert, N. and Roult, G., 1976. Periods and damping of free oscillations observed in France after sixteen earthquakes. Geophys. JR. Astron. Soc., 45, 155—176. Lindley, D.V., 1965. Introduction to Probability and Statistics from a Bayesian Viewpoint, Part 2, Inference, Cambridge University Press, Cambridge, Ma. — Luh, P.C., 1973. Free oscillations of the laterally inhomogeneous Earth: quasidegenerate multiplet coupling. Geophys. JR. Astron. Soc., 32: 197—202. Luh, P.C., 1974. Normal modes of a rotating, self-gravitating inhomogeneous Earth. Geophys. J.R. Astron. Soc., 38: 187— 224. MacDonald, G.J.F. and Ness, N.F., 1961. A study of the free oscillations of the Earth. Geophys. Res., 66: 1865—1911. Maderiaga, R., 1971. Free Oscillations of the Laterally Heterogeneous Earth. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA. Meichior, P., 1966. The Earth Tides, Pergamon Press. Mendiguran, J., 1972. Source mechanism of a Deep Earthquake from Analysis of Worldwide Observations of Free Oscillations. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA. Mote, C.D., 1972. Stability control analysis of rotating plates by finite element: emphasis on slots and holes. Am. Soc. Mech. Eng., Pap. No. 71-WA/Aut-2, pp. 64—70. Nowroozi, A.A., 1965. Terrestrial Eigenvibrations Following the Great Alaskan Earthquake, March 1964. Ph.D. thesis, University of California, Berkeley, CA. Simila, G.W., 1980. Seismic velocity structure and associated tectonics of northern California. Ph.D. thesis, Univ. California, Berkeley, CA. Stifler, iF., I979. Finite element modelling of torsional eigenvibrations of the Earth. Ph.D. thesis, Univ. California, Berkeley, CA. Tukey, J.W., 1961. Discussion emphasizing the connection between analysis of variance and spectrum analysis, Technometrics, 3: 1—29.
27
Observational study of terrestrial eigenvibrations Roger A. Hansen Seismographic Station, University of California at Berkeley, Berkeley, CA Q4720((J.S.A.) (Received August 24, 1981; revision accepted November 24, 1981)
Hansen, R.A., 1982. Observational study of terrestrial eigenvibrations. Phys. Earth Planet. Inter.. 28: 27—69. A comprehensive analysis has been made of analog and digital recordings of eigenvibration ground motion obtained following four great earthquakes; August 1976 (Philippines), August 1977 (Indonesia), September 1979 (West Irian), and December 1979 (Colombia). The time series (ranging in length from —28 to 140 h) are assumed to be linear combinations of damped harmonics in the presence of noise. Tables are calculated from values of the four parameters: 8, used in describing eigenvibrations, period of oscillation, amplitude, damping factor Q, and phase together with their statistical uncertainties (53 spheroidal modes, 054 to 0~48’ and 13 torsional modes. 0T8 to 0T45). The estimation procedures are by the methods of complex demodulation and non-linear regression that specifically incorporate into the basic model the decaying aspect of the oscillations. These methods, extended to simultaneous estimations of groups of modes, help to eliminate measurement error and measurement bias from estimations of 8. The result is that overtone modes veiy near in frequency to fundamental modes can, under certain conditions, be resolved through a non-linear regression technique, although parameter uncertainties are underestimated in general. Of the time series analyzed, 17 were from a northern California regional network of ultra-long period seismographs at Berkeley (three components), Jamestown (vertical component), and Whiskeytown (vertical component) following the four listed earthquakes. The other 7 time series were recorded digitally by the worldwide IDA network following the 1977 Indonesian earthquake. Weighted regional and worldwide averages were made for period and Q of each eigenvibration mode. From the theoretical viewpoint, comparisons of measured period, Q, amplitude, and phase for all modes analyzed led to five conclusions. First, there are no detectable systematic shifts in period. Q, or phase of eigenvibrations within a region whose dimensions are less than a wavelength. Second, though not conclusive, there may be slight systematic shifts in period (<0.65 s) and relative amplitudes within the California regional network due to different source positions and mechanisms. Differences in Q values are not statistically significant. Third, even though differences in period obtained worldwide were as great as 1.33 s (~‘0.33%),differences between Q values (as great as 20%) for the same mode were not significant. The conclusion is that the damping characteristics of singlet eigenfunctions are not observed to be significantly different. Fourth, the assumption that a multiplet ,, 5, behaves as a single oscillation is valid from at least 0~7 through 0S30. Fifth, no systematic pattern emerged for the shift of eigenperiod as a function of order / or position on the Earth.
1. Introduction This study consists of a statistical analysis of 24 time series of ground motion obtained following four large earthquakes. The analysis yields tables of the four parameters (denoted by 8) used in describing eigenvibrations, period of oscillation, *
Present address: NRC Research Associate, NOAA/ CIRES/NGSDC, Boulder, CO 80303, U.S.A.
amplitude, damping factor Q, and starting phase together with their statistical uncertainties. The estimation procedures are by the methods of cornplex demodulation (Tukey, 1961) and non-linear regression that specifically incorporates in the basic model the decaying aspect of the oscillations (penodogram analysis does not). Complex demodulation displays are used not only to locate as precise a value of eigenfrequency as possible, but also to ascertain the nature of an oscillation that corre-
0031-9201 /82/0000—0000/$02.75 © 1982 Elsevier Scientific Publishing Company
28
sponds to a particular spectral peak in a periodogram. These displays exhibit, as a function of time, the amplitude and phase of a particular harmonic oscillation isolated in frequency. This is used to determine whether difficulties in resolution arise from such physical causes as multiple energy sources, interference of neighboring fundamental and overtone modes, or splitting of peaks due to Earth inhomogeneities and rotation. The instantaneous phase is especially useful in making decisions on resolution. Final estimates of the above four parameters are found through a non-linear regression scheme in the frequency domain. Small frequency intervals which contain the spectral peak or peaks of interest are fitted to a model of exponentially decaying cosines. The statistical assumptions made in the formulation of this procedure allow estimates of formal uncertainties to be made. In this way, more reliable estimates than previously available have been made for a set of both spheroidal and torsional eigenvibrations. A description of the analysis methods is given in Hansen (1981 b) hereafter referred to as Paper 1. Using the vanances, statistical comparisons are made between each of the estimates of the four parameters. Complex demodulation and non-linear regression are used on two different data sets in order to systematically assess the effects of the Earth’s regional variations on its eigenvibrations. One of the data sets provides information regarding the differences in the eigenvibrations as recorded within a region whose spatial dimension is small relative to the wavelength of the oscillation. This set relates to oscillations from four different great earthquakes recorded by a network of three broad-band seismographic stations (one three-component and two vertical components) located in northern California. The stations are separated by distances of only 200—350 km, yet each station samples motions in different tectonic settings. The observed spectral variations may be related to different tectonic regions which change quickly with distance. In addition, by considering four earthquake sources, variations may be assessed for different source positions, orientations, and time functions as observed within this regional network of receivers. The results aid in aswering the follow-
ing question: given that the eigenfunctions fix the positions of nodal lines, do different source parameters cause shifts in multiplet locations at a particular receiver site due to different relative excitement of singlet amplitudes? A special feature of the discussion is a comparison of parameters made with statistical significance tests which use the calculated formal uncertainties. The second data set comes from a global network of receivers which recorded the large 1977 Indonesian earthquake. This earthquake is the most efficient event since 1964 for exciting free oscillations of the Earth. Records from the three California stations are compared with those obtained from a global network of seven gravimeters. These gravimeters are from the IDA Network (International Deployment of Accelerometers operated by the University of California at San Diego) and have proved most valuable in gathering spheroidal mode data. The IDA gravimeters are not sensitive to torsional oscillations so that no comparisons could be made with the spectra based on measurements by the horizontal pendulums at northern California stations (i.e. at Berkeley). The IDA records proved to be rich in overtone data as well as the fundamental modes of spheroidal oscillations. When interference between closely spaced modes were recognized with the complex demodulation, non-linear regression was successfully used to solve for groups of spectral peaks yielding estimates for previously unresolved modes. Direct comparisons of frequency shifts of multiplets at different receiver sites are made for a single source. Implications on the spatial separations of receivers required to see frequency shifts are explored with these two different scale networks of receivers. As a by-product, this study of free oscillation modes with the above complexities in mind also provides some measurements of values of period, amplitude, and Q for modelling source mechanisms and a laterally varying and attenuating Earth.
2. Use of networks The California regional network consists of a set of three-component ultra-long period seismom-
29
eters in Berkeley (free pendulum period 7~= 100 s), a vertical component seismometer at Jamestown (7~= 40 s), and a vertical component seismometer at Whiskeytown (7~ 40 s). Coordinates of these stations are given in Table I. For all stations the signals are continuously recorded on analog magnetic tape and for the present study were digitized to produce ±2048 counts. Response curves converting counts to microns of ground motions for the frequency range of interest are shown in Fig. 1. The distances between these three sites range from 200—350 km. This distance is much less than half a wavelength of 0S~,for example. At these separations, each site is located on a different geologic structure within the rapidly changing tectonics of northern California. A theory developed by Mendiguren (1972) for treating the free oscillations of a laterally heterogeneous earth provides some selection rules between degree of oscillation 1, and order s of a spherical harmonic expansion of the lateral heterogeneity (see Aki and Richards, 1980, p. 771): (1) s is even; (2) s < 21; (3) m’ t + m. Rule (1) indicates that lateral inhomogeneity expressed by spherical harmonics of odd-order number does not show an observable effect on freeoscillation periods. Rule (2) indicates that free oscillations of order 1 are affected only by the lateral inhomogeneity harmonics of orders lower
than 2!. Rule (3) indicates that the coupling between the unperturbed eigenfunctions ~U,m and ~U,m’occurs only through inhomogeneity with the azimuthal order number t equal to m’ m. From rule (3) it follows that the average of the singlet frequencies comprising a multiplet is equal to the frequency for the unperturbed, spherically symmetric Earth model. This averaging is called the diagonal sum rule, and was first applied to the perturbations of free oscillations by Gilbert (1971). Of particular interest is rule (2) which states that those heterogeneities whose scale length is shorter than half the wavelength of the free oscil—
lation do not affect the eigenfrequency. Specifically, the aim of the regional comparison is to determine if significant differences in observed eigenspectra can be detected within the regional network. Support for no difference in period is found from comparisons of observations of 0 obtained following each earthquake. The question raised concerning shifts in multiplet frequencies due to different source parameters is addressed by the comparison of all observations of 0 following the four earthquakes. In the second part of the study, the global network of accelerometers called IDA is used. Information concerning this network can be found in Agnew et al. (976). A list of the stations used and their coordinates are given in Table I. Response curves converting counts to microns of ground motion are also found in Fig. I. Five of the stations have such similar response curves that
TABLE I Seismographic stations Location
Code
Latitude
Longitude
Elevation (m)
Jamestown, California Whiskeytown, California Berkeley, California Nana, Peru Sutherland, S. Africa Halifax, N.S., Canada Rarotonga, Cook Islands Garm, USSR Brasilia, Brazil College, Alaska
JAS WDC BKS NNA SUR HAL RAR GAR BDF CMO
37°95’N 40°58’N 37°88’N ll099~S 32°38’S 44°64’N 2l°2l’S 39°00’N I 5°66’S 64°86’N
120°44’W 122°54’W l22°24W 76°84’W 20°80’E 63°59’W 159°77’W 70°32’E 47°90’W l47°84’W
457 300 276 575 1770 38 28 1300 1260 183
30 o o.
______________________________________ JAS woC -
_____
—
-
—
—
/ ,/. / /.~1/
55 NNA US
/:
/.‘ /“ ,~I ,,.‘ /“,‘/ /•‘‘~‘/
GA R I 0.
/ // /
i..’ /‘I
/.. /
,7’ / / /
/.;‘)‘ f/I
ft/I ,)~/// f//f
I.
,f/,// ~‘2/ i/ ,‘//.~,~f
/ p.1/ // j
,///.J
.1
/I ~
/ / I/i /~/,/ 1/ f / ./
~/J. // oI 000I
,!/
~
00 FR ~ 15 U EN C V (H Z I
Fig. I. Response curves of seismometers and filters.
only one curve in Fig. I represents them, The separation of stations in the IDA network are on the order of a few wavelengths of normal modes up to the order of interest (0S~).Analysis is made of recordings from seven stations (see Table I) following the 1977 Indonesian earthquake. Comparisons of frequency shifts of multiplets at different receiver sites are made for this single source. Spatial separations of receivers, required to see frequency shifts, are explored with these two different scale networks of receivers.
3. Preparation of seismograms In this study of long-period seismograms, data were collected from two different networks of seismographic stations. Seventeen vertical and horizontal seismograms were recorded on a regional network of broadband seismometers in northern California (operated and maintained by the U.C. Berkeley Seismographic Station) following four large shallow earthquakes: the Philippine earthquake of August 16, 1976 (M5 = 7.9 BRK); the Indonesian earthquake of August 19, 1977 (M~= 8.0 BRK); the West Irian earthquake of September 12, 1979 (M~= 7.9 BRK); and the Columbian earthquake of December 12, 1979 (M5 = 7.7 BRK). Seven seismograms were obtained corresponding to the vertical ground motions on the world-wide IDA network of gravimeters following’ the Indonesian earthquake of August 19, 1977. Spectral preparation of each of these seismograms is necessary to obtain a usable time series of the form of eq. 1 of Paper 1. The data obtained from the Berkeley network were recorded continuously on analog tape. Thus, it was necessary to use an analog-to-digital converter to obtain the digital time series for use on a high-speed computer. A sample interval of 10 s was the largest available with the existing machine. A digital interval of 20s was obtained by low-pass anti-alias filtering (at 100 s) the digital time series with an eight-pole Butterworth filter and taking every other point. The IDA data are recorded in digital form with a time interval of 20 s so no analog-to-digital conversion, filtering, or decimation was necessary. In each case the usable time series was begun just after the instrument was no longer clipping, which varied from 4 to 11 h. For the regional network, all starting times are the same for each earthquake. Since eq. 1 of Paper 1 represents only decaying cosinusoids in the presence of random noise, any irregularities such as glitches, aftershocks, or nondecaying harmonic components (e.g. long-period tides) must be removed. The four largest amplitude tides are M2, S2, 0, and K with periods of 12.42, 12.00, 25.829, and 23.934h, respectively (Melchior, 1966). Cosine and sine waves of these
31
periods were fitted in a least squares sense for the amplitudes. The time series X( t) is then formed by subtracting the fitted tides from the observed data at each point. Since glitches in the data can affect the fit of these tides, any aseismic irregularities must be removed before the least-squares fit is made. This is done by removing the contaminated section of data and replacing it with a straight line of approximately the same slope. Simply adding zeros or a constant in a non-zero or sloping region is not as satisfactory because this would also affect the fit of the tides, It was necessary to make these replacements in seven of the seismograms. Following is a list of what was replaced in those seismograms identified by earthquake and station name. 3.1. Philippine earthquake The starting time for all records is Year 1976, Day 229, Hour 22, Minute 20, Second 00.0. The length of each of five records (JASZ, WDCZ, BKSZ, BKST, BKSR) is 45.51 h. The only contaminated record was WDC. There were four aseismic spikes within the 45.51 h record. The first occurred 21 h into the record for a length of 54 mm. The other three were at 35.7, 39.3, and 42.6 h into the record and had durations of 24, 14, and 10 mm., respectively. 3.2. Indonesian earthquake JAS vertical component—Starting time is 1977, 231, 14, 36, 40. Length of record is 45.51 h. This case calls for special attention. A large section of record, 15.1 h from the start, was contaminated by a large offset, 58% of full scale, which then returned to the mean position in about 11 h with what appears to be an exponential decay. Rather than discard 11 h of data, the offset was removed in the following manner. A function of the form A exp( — bi) was fitted to this 11 h stretch of data in a least-squares sense and then subtracted point by point from the original data. All oscillations present in the data were preserved, except for a short section 26 mm. long at the beginning of the offset. This section was replaced with a straight line, WDC vertical component—Starting time is 1977, 231, 14, 36, 40. Length of record is 45.51 h.
One and three quarters hours into the record a 0.5 h glitch was repaired, and 13.5 h into the rëcord, a 22 mm section was repaired. BKS vertical northwest and northeast—Starting time is 1977, 231, 14, 36, 40. Length of record is 45.51 h. There were no defects requiring treatment in these records. HAL—Starting time is 1977, 231, 14, 07, 20. Length of record is 138.9 h. There were no defects or splices in this record. SUR—Starting time is 1977, 232, 10, 20, 20. This is considerably later than most other records as the mass of the gravimeter stuck for several hours after the earthquake. Length of record is 138.9 h. CMO—Starting time is 1977, 231, 11, 42, 20. Length of record is 138.9 h. There were no defects or splices in this record. RAR—Starting time is 1977, 231, 11, 43, 40. Length of record is 138.9h. There were no defects or splices in this record. NNA—Starting time is 1977, 231, 17, 21, 00. Length of record is 138.9 h. This time series required splicing data from two cassettes in order to obtain the same length of data as all other IDA stations. There is an 81 mm splice beginning 92.45 h into the record. GAR—Starting time is 1977, 231, 14, 49, 20. Again, a length of record of 138.9 h is achieved by splicing data from two cassettes. A line of length 5 mm and 20s is inserted after 62.1 h. BDF—Starting time is 1977, 232, 00, 23, 20. A splice of length 8 mm and 20 s was inserted 115.5 h into the record in order to obtain a total length of 138.9 h. 3.3.
West Irian earthquake
Again, just the regional network was considered for this event. Only JAS and WDC recorded usable time series as the noise level of the BKS system was such that there was no usable signal above the noise level for the size of the event. The starting time for JAS and WDC is 1977, 255, 08, 40, 00 and the record length is 45.51 h. The records were uncontaminated so that no special adjustments were required.
32
Colombian earthquake
3.4.
The starting time for all records is 1979, 346, 14, 00, 00. JAS—Lemgth of record is 45.51 h. There was one aseismic spike 10.9h into the record that was removed and replaced with a line of duration 34 mm. WDC—Length of uncontaminated record was 45.51h. BKS vertical, northwest and northeast—Length of uncontaminated record was 27.3 h. Beyond this time was a substantial section of contaminated record, so the time series was cut short at this point. The fundamental modes observed in these time series (for this less energetic event) all decayed to the noise level by this time so that no resolution was lost. Finally, the horizontal component recordings obtained following the Philippine, Indonesian and Colombian earthquakes were rotated vectorially by computer using great circle paths to produce radial and transverse directions of ground motion. Figures 2—7 show, for example, the spectra along
20
-
‘OO , 2
00
.
.
00 092
FPEOUENCY
~ ~
.
00 88
9090
(HZ.
26
-
2
. 0
TIME
8~
(H
PS.)
Fig. 3. Transverse component seismogram recorded at BKST following the Indonesian earthquake and its amplitude spectral density corrected for instrument response and filtering.
20
~
FREQUENCY
0
_5690
(HZ.)
FREQUENCY
~
(HZ.)
‘
22.05
TIME
15.51
(HAS.)
Fig. 2. Vertical component seismogram recorded at WDC following the Indonesian earthquake and its amplitude spectral density corrected for instrument response and filtering.
~‘0
22.75
TIME
‘5.51
IMPS.)
Fig. 4. Vertical component seismogram recorded at WDC following the Philippine earthquake and its amplitude spectral density corrected for instrument response and filtering.
33 20
20
o
019
01
.
H
1~~I2
UJ 8
.00072
.00119
.00265
.09392
FREOUENCY
.00498
.0060
~
.00092
.00498
.0060
(HZ.)
914
0
ZD
TIME
0
~
.
IMPS. I
,; S 5 TIME
Fig. 5. Vertical component seismogram recorded at WDC following the Colombian earthquake and its amplitude spectral density corrected for instrument response and filtering.
I
91
0
(MRS.)
Fig. 7. Vertical component seismogram recorded at NNA following the Indonesian earthquake and its amplitude spectral density corrected for instrument response and filtering.
with the prepared time series of each WDC time series, the BKS transverse component for the Indonesian earthquake, and the NNA time series of the Indonesian earthquake for comparison. In
20
computing the spectra the response of the seismometer and filters were compensated for yielding true ground displacement amplitude spectral density. All spectral plots are plotted at the same scale for comparison.
I 6 *
I
.00285
FPEOUENCY
886
- 60,6
.00179
(HZ. I
2
I—, 9
.00072
.00479
.00285 .00392 FREOUENCY 1HZ.
1 .00491.0018
4. 4.1.Data Introduction analysis The techniques of eigenvibration data analysis
_____________________________________ 09.91 TIME INNs. I Fig. 6. Vertical component seismogram recorded at WDC following the West Irian earthquake and its amplitude spectral density corrected for instrument response and filtering.
described in Paper 1 were applied to the 24 time series of ground motion following the four large earthquakes. Representative examples of the successes and problems encountered in the analysis will be presented in this section. All final estimates, 0 of period, Q, amplitude, and phase (accompanied with their respective uncertainties) for modes from 0S4 through 0S48 are listed in Tables Il—V. Tables II and V contain all estimates
34 TABLE II MODE
JASZ I
WDCZ I
0~9
IC 633.69 0.563 243.0 105.0 21.83 6.963 2.887 0.319
IA 633.95 0.161 244.0 30.0 31.20 3.752 3.082 0.120
lB 579.61 0.258 232.0 48.0 31.24 6.251 3.086 0.200
IC 579.74 0.459 167.0 44.0 32.43 8.364 3.428 0.258
T Q
a 6
T Q
a 6 o5~~ T Q
a 6
0~I2
T Q
a 6
0~I3
T Q
a 6
05(4
T Q
a
BKSZ I
JASZ P
WDCZ P
BKSZ P
JASZ C
IC 579.34 0.464 323.0 167.0 19.00 6.385 2.755 0.336
IC 579.28 0.287 263.0 69.0 30.18 6.251 2.613 0.207
lB 580.30 0.282 210.0 43.0 20.52 2.918 5.426 0.142
IA 502.01 0.152 296.0 53.0 22.30 3.370 5.293 0.151
IC 502.37 0.266 552.0 323.0 17.23 5.795 5.317 0.336
IC 537.43 0.780 220.0 140.0 31.70 15.948 0.857 0.503 lB 502.82 0.186 218.0 35.0 21.88 3.401 1.309 0.156
IA 502.29 0.246 288.0 82.0 18.02 4.328 0.850 0.240
IC 502.81 0.205 441.0 158.0 21.92 5.765 0.809 0.263
lB 473.71 0.244 264.0 72.0 16.29 4.457 1.858 0.274
IC 472.76 0.095 497.0 99.0 9.99 1.682 2.319 0.169 lB 448.03 0.089 357.0 50.0 18.55 2.612
IC 448.23 0.094 534.0 120.0 20.89 4.041
IA 447.63 0.137 295.0 53.0 16.75 2.914
lB 472.65 0.142 393.0 93.0 13.38 2.020 3.600 0.151 IC 447.79 0.144 557.0 199.0 16.50 3.670
IA 447.81 0.060 331.0 29.0 18.23 1.117
35
WDCZ C
BKSZ C
BKSR C
JASZ W
WDCZ W
R AVG
IC 634.48 0.842 260.0 180.0 10.89 5.188 4.753 0.476
3A 633.95 0.23 244.0 35.0
IA 580.11 0.205 256.0 46.0 21.08 2.307 5.017 0.109
6A 579.88 0.274 231.0 48.0
IC 537.43 0.780 220.0 140.0
5A 502.37 0.195 255.0 53.0
lB 472.82 0.118 318.0 50.0 20.89 2.245 3.941 0.107 lB 447.71 0.123 226.0 28.0 15.58 1.411
IC 473.18 0.400 147.0 37.0 15.32 2.933 1.560 0.191 IC 447.66 0.935 126.0 66.0 42.02 16.625
lB 473.32 0.212 261.0 61.0 6.70 1.129 0.565 0.169
IA 473.26 0.106 253.0 29.0 6.47 0.534 0.677 0.082
7A 473.00 0.137 264.0 45.0
IC 450.24 0.593 205.0 111.0 2.21 0.779
8A 447.88 0.101 293.0 39.0
36 TABLE II (continued) MODE
JASZ I
8
T Q
a 8
MODE
WDCZ I
0.383 0.193
lB 426.32 0.126 263.0 41.0 18.69 2.940 —0.99 0.157
IC 425.69 0.210 145.0 21.0 28.56 4.261 5.713 0.149
IC 426.23 0.261 190.0 44.0 27.06 6.256 —0.309 0.231
IC 426.25 0.346 215.0 75.0 18.13 5.407 1.639 0.298
JASZ I
WDCZ I
BKSZ I
BKSR I
IA 389.20 0.052 350.0 33.0 22.23 2.089 1.361 0.094
IC 407.10 0.335 366.0 220.0 15.48 7.779 —1.571 0.502 IC 389.26 0.065 572.0 109.0 16.34 2.582 —4.784 0.158
IC 405.80 0.353 299.0 155.0 14.39 5.873 —4.697 0.408 IC 389.48 0.209 148.0 24.0 38.89 6.413 —5.275 0.165
lB 373.80 0.130 350.0 85.0 10.05 2.220 0.650 0.277
lB 373.58 0.195 208.0 45.0 20.03 4.591 5.090 0.229
IC 373.71 0.121 282.0 51.0 24.27 4.311 —0.750 0.178
IC 374.09 0.195 388.0 57.0 17.26 6.055 — 1.032 0.351
IA 359.98 0.067 275.0 28.0 22.95 2.302 3.222 0.100
3A 359.96 0.043 243.0 14.0 27.65 1.153 3.719 0.042
IA 359.89 0.096 224.0 27.0 35.40 4.303 —3.135 0.122
IC 359.67 0.292 272.0 120.0 11.52 4.677 —1.008 0.406
T Q
a 8 0~I7
T Q
a 8
T Q
a 8
0~I9
T Q
a 8 0S20
T
JASZ P
0.611 0.141
0~I6
0~I8
BKSZ I
2B 347.29 0.054
WDCZ P
BKSZ P
2.494 0.174
2.663 0.222
JASZ C 1.542 0.061 lB 425.87 0.194 174.0 28.0 18.56 2.107 4.474 0.114
JASZ P
WDCZ P
BKSZ P
IA 406.40 0.083 244.0 24.0 30.58 3.121 3.570 0.102
IA 406.48 0.124 326.0 64.0 26.78 4.938 —2.735 0.184 IA 389.12 0.187 197.0 37.0 32.61 6.023 —4.426 0.185
IC 373.22 0.268 219.0 69.0 17.01 5.670 1.438 0.333
lB 373.06 0.186 290.0 84.0 16.35 4.588 —4.838 0.281
IA 389.14 0.126 232.0 35.0 30.59 4.610 1.905 0.151
IC 359.72 0.251 144.0 29.0 25.48 5.366 —0.629 0.211
37
WDCZ I
BKSZ C
1.494 0.091
1.215 0.396
BKSR C
JASZ W
JASZ C
WDCZ C
BKSZ C
lB 406.21 0.052 289.0 22.0 16.83 0.918 3.664 0.055
IA 426.00 0.060 507.0 72.0 4.41 0.344 2.741 0.078 JASZ W
WDCZ W
R AVG
3C 406.43 0.123 272.0 45.0 12.81 1.595 —5.200 0.125
IA 406.62 0.193 226.0 48.0 9.09 1.428 1.746 0.158
lB 406.55 0.254 198.0 49.0 6.80 1.342 1.821 0.197
8A 406.33 0.104 260.0 34.0
lB 389.24 0.109 221.0 27.0 5.95 0.554 3.585 0.093
9A 389.11 0.083 247.0 29.0
lB 373.50 0.107 171.0 17.0 9.08 0.688 2.487 0.076
IlB 373.50 0.128 177.0 28.0
2A 388.96 0.062 249.0 20.0 17.20 1.011 0.038 0.059
IC 373.91 0.139 125.0 12.0 27.98 1.995 2.026 0.071
lB 373.09 0.089 385.0 71.0 5.54 0.727 2.881 0.131
IC 374.09 0.225 164.0 33.0 15.46 2.426 —3.636 0.157
IC 359.91 0.142 134.0 14.0 29.77 2.603 0.123 0.087
2C 359.81 0.116 289.0 54.0 9.95 2.227 —2.598 0.224
IC 347.42 0.103
IC 388.91 0.232 340.0 138.0 20.16 5.134 —0.126 0.255
lB 347.22 0.157
7A 425.95 0.104 223.0 32.0
BKSR C
2B 388.87 0.085 253.0 28.0 18.75 1.506 5.810 0.080
IC 347.94 0.069
R AVG
1.263 0.352 2B 425.68 0.086 231.0 21.00 18.44 1.154 —1.839 0.063
—
WDCZ W
IC 373.50 0.079 184.0 14.0 10.53 0.660 2.423 0.063
7A 359.94 0.075 230.0 20.0
IC 347.17 0.118
lB 347.27 0.078
6B 347.41 0.080
38 TABLE II (continued) MODE
JASZ 1
a 8
T
Q a 8
0~22
T Q
a 6
MODE
T Q
a 8
IC 334.92 0.253 262.0 104.0 9.98 4.142 —5.549 0.415
lB 335.60 0.075 312.0 43.0 15.60 2.180 —4.639 0.140
lB 335.40 0.094 354.0 70.0 18.05 3.305 —4.856 0.183
IC 325.01 0.149 269.0 66.0 26.47 6.133 —5.606 0.232
1A 324.99 0.106 256.0 43.0 16.91 2.945 —2.018 0.174
IC 324.79 0.378 227.0 120.0 9.96 5.361 — 1.330 0.538
IA 324.99 0.181 245.0 67.0 17.43 4.704 — 1.827 0.270 BKSZ P
JASZ I
WDCZ I
BKSZ I
BKSR I
JASZ P
WDCZ P
lB 314.85 0.183 247.0 71.0 17.85 5.442 —4.300 0.305
IA 314.66 0.141 215.0 42.0 13.23 2.687 —3.859 0.203
lB 315.20 0.270 244.0 102.0 23.37 9.744 —2.652 0.417
IC 314.63 0.284 266.0 128.0 6.79 3.293 —6.083 0.485
IC 315.34 0.147 225.0 47.0 10.99 2.407 —3.526 0.219
2B 306.18 0.083 350.0 66.0 19.16 2.562 —0.465 0.133
5A 306.07 0.057 221.0 18.0 36.11 2.230 —0.450 0.062
lB 306.24 0.264 285.0 140.0 15.38 7.537 — 1.028 0.490
IC 306.04 0.182 165.0 32.0 21.10 4.418 — 1.873 0.209
IC 305.86 0.229 224.0 75.0 12.70 4.517 — 1.623 0.356
8
8
JASZ C
IA 325.47 0.062 289.0 32.0 14.60 1.238 —1.463 0.085
a
a
BKSZ P
IC 325.13 0.085 476.0 119.0 13.21 2.961 —2.007 0.224
Q
Q
WDCZ P
3A 335.89 0.043 237.0 15.0 23.55 1.065 —5.155 0.045
T
T
JASZ P
lA 335.77 0.056 237.0 19.0 22.81 1.830 —5.584 0.080
0~23
0~24
BKSZ I
355.0 39.0 14.23 1.120 —3.090 0.079
Q
0~2I
WDCZ I
1A 305.85 0.102 236.0 37.0 16.12 2.630 —0.366 0.163 2A 297.48 0.062 230.0 22.0 22.62 1.630 —4.880 0.072
IA 335.76 0.156 204.0 39.0 39.70 7.691 —5.523 0.194
5A 297.45 0.053 245.0 21.0 31.16 2.015 —5.278 0.065
lB 305.92 0.114 201.0 30.0 23.23 3.674 — 1.759 0.158
39
JASZ C
WDCZ C
BKSZ C
178.0 13.0 20.44 1.117 —4.421 0.055
178.0 19.0 23.13 1.807 —4.993 0.078
229.0 47.0 26.60 4.083 —4.857 0.154
IA 335.76 0.050 242.0 17.0 23.40 1.233 —3.908 0.053
2C 335.17 0.145 258.0 58.0 9.10 1.486 —4.792 0.163
lB 335.68 0.119 300.0 63.0 19.89 3.000 —3.958 0.151
2B 324.65 0.098 231.0 32.0 14.85 1.575 —3.511 0.106
BKSR C
2B 335.93 0.275 252.0 104.0 10.38 4.561 —3.207 0.442
JASZ W
WDCZ W
R AVG
353.0 85.0 4.55 0.824 —3.782 0.181
206.0 19.0 8.60 0.589 —3.884 0.069
213.0 25.0
IC 335.64 0.138 174.0 25.0 9.90 1.107 —5.791 0.112
IC 335.98 0.254 181.0 49.0 4.39 0.975 0.213 0.222
12A 335.75 0.076 239.0 25.0
2B 325.04 0.039 244.0 14.0 7.68 0.332 —4.352 0.043
9A 325.11 0.079 251.0 29.0
JASZ W
WDCZ W
R AVG
2B 315.28 0.063 223.0 20.0 5.11 0.342 —0.392 0.067
IOA 315.29 0.072 221.0 20.0
IC 324.85 0.259 202.0 65.0 18.89 4.294 —2.981 0.227
BKSR P
JASZ C
WDCZ C
BKSZ C
IC 315.04 0.202 157.0 31.0 19.45 4.195 —3.155 0.216
2C 315.51 0.110 182.0 23.0 20.78 2.171 —6.018 0.104
IA 315.31 0.032 222.0 10.0 29.34 0.981 0.063 0.033
lB 315.46 0.094 237.0 34.0 27.41 2.904 0.415 0.106
lB 306.17 0.069 214.0 21.0 28.78 2.102 —5.488 0.073
2B 305.83 0.077 203.0 21.0 18.53 1.454 0.358 0.078
lB 305.97 0.065 245.0 25.0 29.83 2.257 —5.495 0.076
lB 297.25 0.135 213.0 41.0 21.99 3.195 —5.760 0.145
BKSR C
lOA 306.02 0.085 220.0 27.0
lB 297.51 0.056 222.0 18.0 19.83 1.201 —5.280 0.061
lB 297.20 0.192 220.0 62.0 7.96 1.788 —2.087 0.225
5A 297.46 0.069 231.0 24.0
40 TABLE II (continued) MODE
JASZ I
WDCZ I
BKSZ I
BKSR I
0~26
4B 289.61 0.038 347.0 31.0 20.19 1.328 —3.180 0.066
2B 289.30 0.051 333.0 39.0 17.03 1.432 —5.262 0.084
SC 289.42 0.039 427.0 49.0 18.09 1.455 —4.116 0.080
IC 289.43 0.114 384.0 116.0 13.97 4.020 0.408 0.288
lB 282.57 0.287 194.0 76.0 14.02 5.958 —1.923 0.425
2B 282.11 0.07 1 303.0 46.0 13.21 1.514 —3.686 0.115
SB 282.30 0.077 215.0 25.0 23.57 2.124 —2.979 0.090
lB 275.11 0.136 325.0 105.0 11.78 3.786 — 1.616 0.321
IC 275.16 0.307 262.0 154.0 10.98 6.394 —2.854 0.583
IC 274.88 0.297 381.0 314.0 11.71 8.185 —2.535 0.699
T Q
a 8
0~27
T Q
a 8
T
Q a 8
2B 268.30 0.062 329.0 50.0 14.62 1.643 —2.515 0.112
T
Q a 8
MODE
JASZ I
OS
30
T Q
a 6
T
Q a
lB 256.32 0.203 239.0 91.0 16.28 6.550
JASZ P
WDCZ P
BKSZ P IC 289.62 0.170 187.0 41.0 15.32 3.489 —1.949 0.228
IC 267.92 0.179 204.0 56.0 13.04 3.736 —2.796 0.287
WDCZ 1
BKSZ 1
2B 262.48 0.099 219.0 36.0 16.43 2.072 —0.737 0.126
IC 262.31 0.149 316.0 113.0 13.95 4.904 —0.843 0.351 lB 256.27 0.192 254.0 97.0 16.14 6.49 I
JASZ P
WDCZ P
BKSR P
JASZ C
IC 262.10 0.285 305.0 202.0 6.17 3.786 —3.614 0.614 lB 256.02 0.123 221.0 47.0 15.13 2.393
41
BKSR P
JASZ C
WDCZ C
BKSZ C
2A 289.63 0.047 194.0 12.0 24.35 1.144 —2.357 0.047 2A 282.17 0.056 249.0 25.0 13.06 0.977 —2.167 0.075
BKSR C
JASZ W
WDCZ W
2B 289.85 0.074 239.0 29.0 13.46 1.264 —4.886 0.094 IC 282.18 0.078 254.0 36.0 13.78 1.436 — 1.978 0.104
7A 289.54 0.052 220.0 20.0
lB 281.79 0.149 277.0 81.0 5.02 1.129 —3.624 0.225
lB 282.01 0.094 187.0 23.0 8.68 0.832 —3.302 0.096
IC 275.23 0.088 236.0 36.0 12.23 1.387 —2.740 0.113
BKSZ C
2A 262.11 0.056 234.0 23.0 13.78 1.054 —0.292 0.077
lB 262.20 0.074 181.0 19.0 19.30 1.523 —0.103 0.079 lB 255.71 0.214 195.0 64.0 15.26 3.930
7A 282.15 0.081 227.0 31.0
4B 275.17 0.139 263.0 75.0
IA 268.57 0.049 238.0 21.0 14.39 0.939 0.253 0.065 WDCZ C
R AVG
BKSR C
IC 268.42 0.078 240.0 34.0 7.42 0.78 1 —2.664 0.105 JASZ W
WDCZ W
IC 262.05 0.102 154.0 18.0 10.86 1.024 —1.166 0.094
lB 268.28 0.044 184.0 11.0 10.61 0.494 —5.045 0.047 R AVG 6A 262.18 0.083 202.0 24.0
lB 255.72 0.050 193.0 15.0 7.97 0.467
SB 255.81 0.096 198.0 31.0
5A 268.39 0.057 204.0 18.0
42 TABLE II (continued) MODE 6
JASZ I —5.656 0.402
WDCZ 1
BKSZ I
JASZ P
WDCZ P
—0.024 0.402
BKSR P
JASZ C —0.984 0.158
osn
T Q
a 8 OS
33
IC 244.81 0.153 369.0 170.0 7.75 3.413 —0.955 0.441
T Q
a 8
2B 239.76 0.078 223.0 32.0 13.95 1.532 —5.321 0.110
O~34
T
Q a 6 0S35 T
Q a 8
T
Q a 8
IC 234.64 0.181 266.0 109.0 8.52 3.664 —4.438 0.430
IC 234.51 0.136 147.0 25.0 13.13 2.425 — 1.096 0.185 lB 229.66 0.055 242.0 28.0 8.15 0.989 —0.620 0.12 I
2B 234.52 0.096 248.0 50.0 9.59 1.481 0.036 0.155
43 TABLE II WDCZ C
BKSZ C
BKSR C
JASZ W
—1.046 0.258
WDCZ W —1.018 0.059
IC 250.24 0.058 182.0 15.0 8.13 0.546 —3.487 0.067 LB 245.33 0.257 260.0 142.0 10.20 4.039 —4.349 0.396 lB 239.48 0.112 230.0 49.0 15.57 2.608 0.557 0.168
IC 239.14 0.132 278.0 85.0 4.14 0.955 0.396 0.231
2A 250.36 0.108 197.0 34.0 4.96 0.672 —3.451
3C 229.65 0.096 219.0 40.0 6.03 0.924 —0.430 0.153
2A 250.28 0.078 189.0 26.0
0.136 2A 244.60 0.089 243.0 43.0 2.97 0.402 —2.644 0.135
3A 244.69 0.121 246.0 57.0
2B 239.47 0.058 175.0 15.0 7.74 0.535 —5.623 0.069
4B 239.53 0.080 188.0 24.0
3C 234.80 0.093 186.0 27.0 9.04 1.186 1.046 0.131 IA 229.86 0.068 151.0 13.0 17.09 1.181 —3.012 0.069
R AVG
4B 234.62 0.112 193.0 38.0
IC 229.65 0.044 164.0 10.0 7.44 0.379 —1.208 0.051
4A 229.71 0.059 164.0 15.0
44 TABLE III (continued) MODE
WDCZ P
O~37
IC 224.97 0.119 230.0 56.0 7.25 1.928 —2.568 0.266
T
Q a 6
T Q
a 6
JASZ C
IC 220.90 0.149 241.0 78.0 . 8.53 2.155 —4.021 0.253
Q a 8
Q a
8
lA 208.77 0.060 189.0 21.0 6.67 0.563 —1.998 0.084
Q
a 8
T
Q a
WDCZ W
R AVG
IC 187.70 0.101 281.0 85.0 . 3.67 0.830
3B 216.45 0.118 224.0 54.0 6.53 1.462 —4.528 0.224
2C 220.44 0.058 156.0 13.0
2B 216.44 0.150 220.0 59.0
IA 208.77 0.060 189.0 21.0
IC 204.79 0.081 185.0 27.0 8.77 0.997 0.283 0.114
T
O~47
JASZ W
IC 220.40 0.043 155.0 9.00 8.85 0.432 —3.530 0.049 IC 216.36 0.345 177.0 100.0 9.58 4.133 —1.971 0.432
T
T
BKSR C
IC 224.97 0.119 230.0 56.0
O~39
O~41
WDCZ C
IC 204.79 0.081 185.0 27.0
IC 187.70 0.101 281.0 85.0
45 TABLE II (continued) —
MODE
WDCZ P
8
JASZ C
WDCZ C
BKSR C
JASZ W
—4.962 0.226
0S48 T Q
a 6
of 0 obtained from the regional network while Table III corresponds to the estimates obtained from the IDA network. Table IV lists estimates of overtone modes obtained from both networks. Table V lists torsional oscillations recorded at BKS. As will become apparent later in this section, it was necessary to develop a method of assigning reliability weights to each eigenvibration mode analyzed. These weights aid in designating the reliability of the estimates used to describe the mode and in developing weights for averaging estimates of period and estimates of Q. The reliability weight and the variance of each estimate were used to develop formal weights; then averages of period and of Q were computed for the modes estimated within the regional network. These averages were then averaged with the estimates obtained from the IDA network to give a global average of period and of Q for a large number of fundamental spheroidal oscillations. 4.2. Reliability weights Much emphasis was given in Paper I to understanding biases introduced by the regression used in obtaining final estimates of 1. In practice, it is not always possible to eliminate totally such biases. Therefore, it is important to recognize when such biases are present and to weigh the estimates obtained accordingly. For example, compare two different estimates of Q for the mode 0S18 (see Tables II and III). A value of 261 with a standard
WDCZ W
R AVG -
IC 184.08 0.155 214.0 77.0 4.97 1.304 —4.483 0.263
IC 184.08 0.155 214.0 77.0
error of 11 was obtained from the RAR recording of the Indonesian earthquake. A value of 125 with a standard error of 12 was obtained from the JAS recording of the Colombian earthquake. The difference in these two estimates is so large that the correctness of one or both estimates is doubtful. By looking again at the complex demodulation graphs of RAR and JAS it is found that the oscillation recorded at RAR follows the model exemplified in Fig. 2 of Paper 1 very closely while the oscillation at JAS does not. Judged only by the standard errors, these two estimates should be given equal weight. Clearly, from scrutiny of the complex demodulation graphs, the estimates are not of equal reliability and the estimate of Q from the JAS recording should be given less weight than that from the RAR recording. For the above reason, each mode analyzed is assigned a letter grade of A, B, C or D which is the ‘reliability weight’ for that mode. The following procedure is used in assigning the reliability of each mode. A few examples will be taken from the NNA recording of the 1977 Indonesian earthquake. The uncorrected spectra used for analysis is given in Fig. 8. The time series is then demodulated at each frequency that corresponds to a peak in the spectrum. Each mode is evaluated for starting values of 8 and the appropriate length of time series is selected for application to the non-linear regression as described in Paper 1. When it was unclear how many modes should be included in the regression, the
46 TABLE III MODE
NNA I
T Q
a 8
O~6
T Q
a 6
0S7 T Q
a 8
Q
a 6
O~9
T
lB 962.18 0.218 725.0 239.0 8.93 1.019 0.700 0.114
lB 962.28 0.305 250.0 40.0 71.77 7.319 5.077 0.102
IA 811.03 0.046 286.0 9.0 65.79 1.384 0.884 0.021
2B 810.35 0.363 248.0 55.0 6.80 1.166 2.502 0.171
IA 811.12 0.190 268.0 34.0 68.22 5.692 5.261 0.083
Q a 8
T -
‘
,
lB 707.64 0.204 385.0 85.0 33.74 4.729 —2.704 0.140
IC 633.10 0.188 215.0 28.0 43.82 3.590 0.430 0.082
IA 633.17 0.060 327.0 20.0 38.08 1.590 —0.626 0.042
IC 579.32 0.110 388.0 57.0 23.43 2.270
lB 579.03 0.107 349.0 45.0 27.43 2.424
GAR I
RAR I
BDF I
SUR I
R AVG
IC 1544.19 1.114 337.0 164.0 55.26 11.487 1.033 0.208
lB 962.25 0.090 418.0 33.0 39.87 1.581 —0.687 0.040
IA 706.80 0.056 260.0 11.0 61.48 1.731 —0.304 0.028
T
a
HAL I
lB 1547.03 0.657 293.0 73.0 25.52 2.986 —2.526 0.117
O~4
Q
CMO I
IA 579.03 0.135 339.0 54.0 48.51 5.225
lB 962.26 0.205 354.0 54.0 47.76 3.505 5.428 ‘0.073
G AVG 2B 1546.44 0.774 295.0 81.0
IC 694.74 0.468 619.0 371.0 17.41 4.060 — 1.593 0.233
IC 960.46 0.350 263.0 50.0 19.35 2.371 1.119 0.123 2B 812.36 0.238 341.0 68.0 7.16 0.987 — 1.402 0.138
.‘
IA 707.58 0.117 369.0 45.0 46.05 3.612 —5.944 0.078
6B 962.28 0.171 349.0 48.0
4A 811.06 0.083 285.0 16.0
lB 706.76 0.232 255.0 43.0 21.59 2.544 —1.259 0.118
4A 706.97 0.092 267.0 19.0
lB 633.75 0.223 229.0 37.0 30.17 3.203 —5.027 0.106
IA 635.00 0.130 268.0 29.0 36.00 2.868 0.858 0.080
IC 633.86 0.044 456.0 29.0 16.76 0.648 0.172 0.039
3A 633.95 0.231 244.0 35.0
6A 633.66 0.085 293.0 27.0
lB 579.69 0.117 331.0 44.0 35.17 3.263
lB 580.28 0.135 377.0 66.0 21.69 2.552
IA 579.41 0.063 384.0 32.0 20.34 1.122
6A 579.88 0.274 231.0 48.0
7A 579.43 0.105 342.0 44.0
47 TABLE III (continued) MODE
NNA I
8
—2.113 0.097
8
O~13
T Q
a 8
O~I4
T Q
a 8
O~I5
T Q
a 8
O~l6
T
—0.218 0.118
—0.093 0.055
SUR I
R AVG
G AVG
IC 537.43 0.780 220.0 140.0
SB 538.04 0.202 215.0 28.0
lB 537.43 0.167 193.0 23.0 50.24 4.353 1.376 0.087
2B 539.02 0.164 242.0 36.0 6.62 0.923 0.968 0.968
3A 502.00 0.038 285.0 12.0 59.89 1.988 — 1.708 0.033
2C 501.19 0.079 293.0 27.0 25.62 2.122 2.489 0.083
IA 501.39 0.060 345.0 29.0 64.16 3.720 — 1.832 0.058
IA 502.90 0.037 290.0 12.0 32.31 0.994 —1.115 0.031
IC 501.40 0.065 265.0 18.0 30.59 1.514 3.257 0.050
1C 502.34 0.191 205.0 32.0 4.67 0.488 2.830 0.105
SA 502.24 0.197 297.0 68.0
7A 502.17 0.054 290.0 17.0
lB 472.88 0.054 290.0 19.0 69.87 3.353 0.793 0.048
lB 472.17 0.117 304.0 46.0 29.37 3.165 2.123 0.108
IA 472.50 0.064 333.0 30.0 66.63 4.239 —2.390 0.064
IA 473.42 0.055 301.0 21.0 58.73 2.923 —0.416 0.050
IC 472.66 0.086 280.0 28.0 29.05 2.155 0.213 0.074
IC 472.70 0.071 276.0 23.0 4.00 0.242 2.646 0.061
7A 473.00 0.137 264.0 45.0
7A 472.89 0.071 298.0 27.0
lB 448.05 0.074 301.0 30.0 52.78 3.733 2.895 0.071
IA 446.82 0.054 220.0 12.0 45.13 1.764 1.938 0.039
lB 447.71 0.099 291.0 38.0 53.77 5.007 —3.724 0.093
lB 448.17 0.057 297.0 23.0 42.92 2.328 —2.230 0.054
lB 448.74 0.076 317.0 34.0 38.23 2.960 1.383 0.077
8A 447.88 0.101 293.0 39.0
7A 447.76 0.072 248.0 21.0
IC 426.45 0.177 201.0 34.0 29.13 5.939 —2.745 0.121
IC 425.16 0.115 251.0 34.0 27.49 2.7 18 0.509 0.099
lB 425.54 0.058 351.0 34.0 59.08 4.034 —5.666 0.068
lB 426.21 0.054 478.0 58.0 15.28 1.187 — 1.399 0.078
3A 426.73 0.054 272.0 19.0 21.01 1.084 — 1.671 0.052
IC 426.61 0.127 166.0 16.0 2.92 0.228 — 1.545 0.078
7A 425.95 0.104 223.0 32.0
7A 426.18 0.072 262.0 27.0
IC 404.95 0.372
2C 406.21 0.078
IA 405.98 0.052
IA 406.83 0.017
3A 407.31 0.024
IA 406.48 0.028
8A 406.33 0.104
8A 406.81 0.031
8
a
BDF I
2B 537.98 0.152 234.0 31.0 59.57 8.685 —4.757 0.146
a
—5.147 0.093
RAR I
2B 537.53 0.217 218.0 38.0 32.96 6.38 1 —3.045 0.194
Q
Q
GAR I
—0.373 0.108
T
T
HAL I
—2.087 0.088
O~II
O~I2
CMO I
2C 448.40 0.103 248.0 28.0 15.82 2.719 2.781 0.172
IC 406.26 0.066
48 TABLE III (continued) MODE
NNA I
CMO I
HAL I
GAR I
RAR I
BDF 1
SUR I
R AVG
G AVG
Q
238.0 105.0 12.83 4.227 2.614 0.330
187.0 13.0 50.83 2.697 —0.932 0.053
301.0 23.0 61.73 3.441 — 1.846 0.056
285.0 7.0 64.94 1.162 —5.122 0.018
273.0 9.0 42.01 1.046 1.254 0.025
568.0 105.0 7.33 0.901 — 1.326 0.123
324.0 14.0 3.90 0.123 —0.579 0.032
260.0 34.0
284.0 12.0
IA 389.56 0.055 316.0 28.0 33.06 2.184 —0.235 0.066
2C 388.91 0.097 153.0 12.0 51.60 3.002 3.565 0.058
IC 388.68 0.190 168.0 28.0 36.42 4.620 —4.589 0.127
IC 389.68 0.117 349.0 73.0 17.21 2.543 —2.907 0.148
2B 390.41 0.077 345.0 47.0 19.40 1.886 3.940 0.097
IC 388.60 0.065 480.0 77.0 5.08 0.565 2.429 0.111
IA 389.34 0.061 325.0 33.0 2.23 0.163 4.477 0.073
9A 389.11 0.083 247.0 29.0
8A 389.39 0.074 257.0 29.0
lB 374.15 0.028 350.0 18.0 33.02 1.239 —1.195 0.038
lB 373.40 0.089 220.0 23.0 21.66 1.673 1.868 0.077
IIB 373.50 0.128 177.0 28.0
4A 374.25 0.040 266.0 16.0
a 8
O~I7
T Q
a 6
O~I8
T Q
a 8
lB 360.16 0.090 222.0 25.0 31.32 2.576 —2.3 15 0.082
O~I9
T Q
a 8
O~2O
T Q
a 8
O~2I
T Q
a 8
IA 374.47 0.032 261.0 11.0 50.73 1.639 —0.283 0.032
4A 347.78 0.016 266.0 7.0 61.33 1.105 — 1.039 0.018 4A 336.00 0.012 276.0 6.0 70.18 1.041 2.234 0.015
IA 335.44 0.039 293.0 20.0 45.68 2.282 1.588 0.050
lB 335.10 0.057 241.0 20.0 40.60 2.488 —3.778 0.061
IC 360.28 0.229 230.0 67.0 33.83 7.021 —2.600 0.207
IA 360.18 0.160 256.0 58.0 25.65 4.473 0.309 0.174
IA 360.08 0.077 257.0 28.0 12.80 1.037 — 1.173 0.081
7A 359.94 0.075 230.0 20.0
5A 360.06 0.094 236.0 27.0
lB 347.66 0.113 259.0 44.0 31.64 3.926 —0.784 0.124
2A 347.92 0.048 272.0 21.0 40.85 2.255 —2.348 0.055
IA 347.42 0.034 261.0 13.0 18.60 0.710 0.170 0.038
6B 347.41 0.080 213.0 25.0
5A 347.72 0.029 263.0 12.0
IA 336.00 0.023 259.0 9.0 53.99 1.384 —4.487 0.026
IA 336.05 0.055 237.0 18.0 61.98 3.554 3.716 0.057
IA 335.75 0.031 355.0 23.0 14.66 0.705 0.227 0.048
12A 335.75 0.076 239.0 25.0
8A 335.92 0.026 272.0 12.0
lB 335.90 0.056 417.0 58.0 1.16 0.108 0.734 0.093
49 TABLE III (continued) MODE
NNA I
o~22
4A 325.23 0.021 245.0 8.0 45.37 1.081 — 1.517 0.024
T Q a 8
o~23
T Q a 8
O~24
T
Q a 8
O~25
T Q a 8
O~26
T
Q a 8
O~27
T Q a
4B 315.86 0.037 212.0 11.0 30.31 1.148 1.246 0.038
CMO I
IA 314.92 0.019 288.0 10.0 34.51 0.902 —1.892 0.026
HAL I
GAR I
IA 324.45 0.067 319.0 42.0 45.98 4.468 — 1.678 0.097
2B 325.09 0.045 311.0 27.0 39.96 2.5 15 —3.859 0.063
RAR I
lB 314.86 0.102 269.0 47.0 45.36 5.965 0.339 0.131
3C 306.30 0.092 269.0 45.0 11.46 1.763 —0.202 0.157
BDF I
SUR I
R AVG
G AVG
IC 325.01 0.059 470.0 81.0 7.86 0.974 —0.636 0.124
lB 325.05 0.055 270.0 25.0 1.36 0.090 1.058 0.066
9A 325.11 0.079 251.0 29.0
6A 325.13 0.039 253.0 15.0
lB 315.64 0.044 297.0 25.0 48.00 2.908 —0.120 0.061
IC 314.95 0.030 371.0 26.0 8.74 0.450 3.443 0.051
IOA 315.29 0.072 221.0 20.0
6A 315.12 0.033 259.0 15.0
IC 305.96 0.209 439.0 264.0 8.87 4.029 —1.423 0.454
IC 305.66 0.073 263.0 33.0 12.38 1.171 1.302 0.095
lOA 306.02 0.085 220.0 27.0
4A 305.96 0.091 228.0 31.0
5A 297.46 0.069 231.0 24.0
7A 297.44 0.051 256.0 23.0
SB 297.79 0.035 326.0 25.0 28.30 1.589 1.745 0.056
2A 297.03 0.031 251.0 13.0 27.46 1.044 0.029 0.038
IA 298.22 0.110 227.0 38.0 30.52 3.872 —3.874 0.127
lB 297.84 0.074 320.0 51.0 32.58 3.757 —0.642 0.115
2B 298.25 0.086 266.0 41.0 34.56 3.920 2.106 0.113
IC 297.15 0.085 223.0 28.0 13.01 1.251 —0.758 0.096
lB 289.84 0.056 270.0 28.0 33.15 2.555 3.443 0.077
2A 289.22 0.032 332.0 24.0 16.05 0.872 2.670 0.054
7B 289.22 0.039 323.0 28.0 27.61 1.836 —5.141 0.067
lB 290.24 0.130 198.0 35.0 29.26 3.978 0.696 0.136
IC 289.88 0.065 314.0 44.0 26.88 2.849 0.722 0.106
lB 289.44 0.070 221.0 24.0 9.06 0.741 3.359 0.082
SC 282.33 0.031 341.0 26.0 25.88 1.475
2B 281.42 0.050 235.0 19.0 14.85 0.946
7B 281.58 0.048 272.0 25.0 31.70 2.597
‘
7A 289.54 0.052 220.0 20.0
7A 289.40 0.048 262.0 25.0
7A 282.15 0.081 227.0 31.0
4A 281.93 0.047 250.0 24.0
50 TABLE III (continued) MODE 8
T
Q a 8
0S29 T
Q a 8
NNA I
CMO I
1.993 0.057
1.299 0.064
—4.690 0.082
SC 275.41 0.037 249.0 17.0 33.57 1.738 —0.912 0.052
2B 274.76 0.022 295.0 14.0 24.85 0.970 3.457 0.039
—
SC 268.79 0.051 189.0 14.0 36.34 2.050 0.251 0.056
Q a 8 3C 256.15 0.063 291.0 42.0 5.53 0.638 —0.363 0.115
4B 255.62 0.031 255.0 16.0 19.77 0.934 —0.665 0.047
3C 250.54 0.066 249.0 33.0 8.63 1.168 1.191 0.135
4B 249.91 0.03 8 240.0 17.0 16.93 0.961 3.073 0.057
MODE
NNA I
CMO I
T
IC 245.01 0.042
4B 244.64 0.052
Q a 8
T
Q a
8
GAR I
RAR 1
2B 274.43 0.055 213.0 18.0 37.35 2.935 —4.259 0.079
2B 275.06 0.044 273.0 24.0 21.95 1.669 —0.459 0.076
lB 275.51 0.028 281.0 16.0 22.22 0.956 2.073 0.043
7B 267.76 0.096 237.0 40.0 27.95 5.963 —3.326 0.213
lB 268.71 0.133 259.0 66.0 17.63 3.3 16 1.107 0.188
4B 261.45 0.033 296.0 22.0 15.17 0.852 3.894 0.056
T
T
HAL I
IC 261.93 0.077 222.0 29.0 31.73 3.152 0.323 0.099
GAR I
IC 262.40 0.068 191.0 19.0 22.96 1.802 2.403 0.079
2C 249.95 0.092 324.0 77.0 6.18 1.181 0.343 0.191
IC 250.77 0.045 262.0 25.0 21.03 1.499 3.287 0.071
R AVG
G AVG
3A 244.69 0.121
3A 244.82 0.058
BDF I
R AVG
G AVG
4B 275.17 0.139 263.0 75.0
6B 275.00 0.034 270.0 19.0
5A 268.39 0.057 204.0 18.0
4A 268.47 0.069 206.0 24.0
6A 262.18 0.083 202.0 24.0
4A 261.70 0.052 236.0 23.0
SB 255.81 0.096 198.0 31.0
3B 255.71 0.045 245.0 21.0
2A 250.28 0.078 189.0 26.0
5A 250.24 0.054 226.0 23.0
SI TABLE III (continued) MODE
NNA I
CMO I
Q
214.0 16.0 13.28 0.772 —0.440 0.058
224.0 21.0 12.99 0.976 —0.479 0.075
a 8
O~34
T Q a 8
O~35
T Q a 6
O~37
T
Q a 8
O~38
T
Q a 8 O~48
T Q a 6
R AVG
G AVG
246.0 57.0
224.0 27.0
4B 239.53 0.080 188.0 24.0
3B 239.63 0.104 191.0 29.0
4B 234.62 0.112 193.0 38.0
2B 234.67 0.130 195.0 42.0
2B 224.50 0.054 268.0 35.0 9.64 1.000 —1.109 0.104
IC 224.97 0.119 230.0 56.0
2B 224.56 0.066 264.0 38.0
2C 220.54 0.089 177.0 25.0 12.76 1.574 1.846 0.123
2C 220.44 0.058 156.0 13.0
2C 220.47 0.069 160.0 16.0
IC 184.08 0.155 214.0 77.0
2B 184.36 0.106 202.0 43.0
IC 239.82 0.124 298.0 92.0 11.03 2.534 0.493 0.230
GAR I
IC 239.82 0.165 229.0 72.0 16.06 4.016 —0.334 0.250
IC 234.84 0.179 251.0 96.0 11.36 3.330 1.332 0.293
lB 184.43 0.090 201.0 39.0 11.00 1.705 —1.008 0.155
52 TABLE IV MODE
T Q a
8
T Q a 8
T
Q a 8
2~4
T Q a 6 2~5
T Q a 8
T Q a
NNA I
CMO I
HAL I
GAR I
SUR I
BKSR C
lB 1059.75 0.578 300.0 98.0 7.78 1.472 3.182 0.189 lB 853.01 0.290 417.0 118.0 12.84 1.971 —2.147 0.154
MODE
NNA I
2B 535.27 0.26 349.0 119.0 14.56 5.398 3.001 0.371
2~7
T Q a 8 2B 849.22 0.710 216.0 78.0 22.27 5.565 0.434 0.250
IC 849.44 0.543 127.0 21.0 36.25 4.274 0.297 0.118
1~9
T Q a 8
2B 802.96 0.316 391.0 120.0 4.85 0.982 2.221 0.203
I~8
T Q a & 2B 724.54 0.472 289.0 109.0 10.48 2.686 0.109 0.256
3C 509.42 0.347 278.0 106.0 6.96 2.006 — 1.749 0.289 3C 486.88 0.182 166.0 21.0 25.79 2.493 1.234 0.097
2~9
T Q a 8 IC 660.39 0.187 509.0 147.0 2.99 0.407 0.442 0.136
2~IO
T Q a 8 IC 593.77 0.568 144.0 40.0 30.51 6.307
2~I5
T Q a
CMO I
3C 309.41 0.303 200.0 75.0 5.47 1.974
53
HAL I
RAR I
2B 535.11 0.143 216.0 25.0 70.53 8.932 —2.214 0.127
BDF I
BKSR I
BKSR C
2B 535.25 0.057 237.0 12.0 19.68 0.943 —0.861 0.048
lB 535.91 0.310 254.0 75.0 32.71 7.522 —2.193 0.230
IC 535.95 0.293 148.0 24.0 30.04 3.184 —1.731 0.106
MODE
NNA I
Q a
8 lB 269.01 0.149 237.0 62.0 19.06 6.392 —3.798 0.336
Q a
8
T
Q a
8
3B 415.94 0.129 185.0 21.0 15.11 1.354 3.276 0.090
2C 415.99 0.186 214.0 41.0 9.55 1.339 —0.478 0.140
GAR I 2C 273.46 0.152 552.0 337.0 2.24 1.173 —1.556 0.522
T
T
2C 446.98 0.079 239.0 20.0 21.48 2.725 0.662 0.127
HAL I
3C 252.15 0.071 245.0 34.0 8.29 1.195 1.521 0.144
54 TABLE IV (continued) MODE
NNA I
8
CMO I
HAL I
GAR I
SUR I
-
I~8
IC
T
554.64 0.314 197.0 44.0 20.59 3.279 3.134 0.159
Q a &
BKSR C
MODE
—1.234 0.207
6
3C 303.30 0.112 377.0 112.0 5.18 1.268 1.186 0.252
Q a 6
modes in question are fitted as individual peaks and as groups. Consider the groups of peaks between 0.00285 and 0.00325 Hz in Fig. 8 (0S20— oS23). For these modes estimating the group simultaneously proved to be superior in minimizing the variances of the estimates. (This is similar to the example, case C, of non-linear regression given in Paper 1.)
CMO I
3.129 0.355
4S7 T
,
NNA I
When final estimates have been found, all the modes are summed together with the use of eq. 2 of Paper 1 to form a synthetic time series. The amplitude spectra of this synthetic seismogram is computed and compared to that computed from the data. Figure 9 is the synthetic spectra to compare with the NNA spectra of Fig. 8. The shapes of the peaks in synthetic spectra are useful as
00
*
0
~~~001 5
97
~ FREQUENCY
.00029
1HZ.)
~
.00444
~1JiJ ~ .00257
.00373
FREQUENCY
(HZ.)
.00459
i II
—~~~0
•5.09
TIME
90.19
(((P5.1
Fig. 8. Vertical component seismogram recorded at NNA following the Indonesian earthquake and its Fourier amplitude spectral density.
—~o
os.o.
TIME
90.49
(HRS.I
Fig. 9. Synthetic seismograms computed by summing the modes estimated from the NNA recording of the Indonesian earthquake and its Fourier amplitude spectral density.
55 TABLE V MODE
BKST I
BKST C
R AVG
1C 736.43 0.525 295.0 124.0 52.64 15.783 1.737 0.299
IC 736.43 0.525 295.0 124.0
OTII
lB
lB
T
576.65 0.443 230.0 81.0 41.47 10.379 0.304 0.250
576.65 0.443 230.0 81.0
IA 537.70 0.277 213.0 47.0~ 53.48 11.087 —0.128 0.207
IA 537.70 0.277 213.0 47.0
T Q a 6
Q a &
T Q a 6
T
Q a 6 0TI4
T Q a &
T Q a
T Q a 8
0T20 T Q a & O~’23
T Q a 6
lB 504.95 0.35 1 205.0 59.0 41.48 6.455 3.426 0.156
0~’13
MODE
lB 504.95 0.351 205.0 59.0
T Q a 8
lB 476.70 0.395 270.0 121.0 30.39 10.938 4.497 0.359
lB 476.70 0.395 270.0 121.0
IC 451.60 0.970 170.0 124.0 23.85 17.109
IC 451.60 0.970 170.0 124.0
0T35 T Q a 6
T
Q a
BKST I
BKST C
R AVG
IC 408.78 0.288 340.0 162.0 16.11 6.290 —4.785 0.390 IC 359.01 0.360 175.0 61.0 27.01 9.086 —2.577 0.336
IC 408.78 0.288 340.0 162.0
lB 360.87 0.124 330.0 74.0 17.83 2.833 —2.262 0.159
2B 360.72 0.157 284.0 70.0
IC 324.48 0.265 150.0 37.0 32.10 8.175 —0.411 0.255
IC 324.48 0.265 150.0 37.0
IC 309.53 0.239 320.0 159.0 8.15 3.764 —3.924 0.462
IC 309.53 0.239 320.0 159.0
IA 225.29 0.063 244.0 33.0 13.60 1.920 —4.899 0.141 IC 181.28 0.142 360.0 202.0 3.32 1.964
6
2B 225.75 0.130 295.0 100.0 4.67 1.910 —3.798 0.409
2A 225.36 0.078 248.0 41.0
IC 181.28 0.142 360.0 202.0
56 TABLE V (continued) MODE &
0T6 T Q a 8
BKST I
BKST C
R AVG
4.382 0.717 IC 429.19 0.506 168.0 67.0 26.97 10.47 —3.697 .0388
MODE
BKST I
6
—4.053 0.591
BKST C
R AVG
IC 429.19 0.506 168.0 67.0
decision criteria for two possible solutions obtamed for an oscillation. The synthetic spectra displays the character of groups (such as 0S20—0S23) remarkably well. At this time each mode is individually reevaluated in order to assign its reliability weight. If the following criteria are met the reliability of a mode will be assigned an A grade. (1) Complex demodulation graphs behave as the model predicts (see fig. 2 of Paper 1). (2) Regression estimates differ by only two standard deviations from estimates of complex demodulation (especially Q values). (3) The shapes of the synthetic spectra overlay the signal spectra very closely. An example of a mode given an A quality factor is 0S22 as recorded at NNA. This mode corresponds to the peak at 0.00307 Hz in Figs.8 and 9. The complex demodulates for this mode are shown in Fig. 10. A line fit to the first 36 h of the instantaneous amplitude plot yields a Q value of 235 which can be compared to the regression estimate in Table III of 245±8. A mode that deviates from linear decay in logarithms of instantaneous amplitude or in linear flat phase is not assigned an A grade. Only modes that fit the model very well are assigned an A quality factor. B-grade quality factors are assigned to modes that have small deviations from one of the above three critera. An example of this, again from Figs. 8 and 9, is the mode o~Ig at 0.00267 Hz. The complex demodulates for this mode are
graphed in Fig. 11. The slightly rounded decay of the instantaneous amplitude yields a Q value of 240. This value is quite different from the value of 350 given in Table III. The shape of the synthetic peak in Fig. 9 closely follows that of Fig. 8, al-
I N STANTAN EQ US
.
AMP L
325. IS
PER I 00
AT
I
-.
- I -o -
40
- 20 -
-2.
7
0
(5.17
30.34
60.68
45.54
80600074
~
75.95
91.02
.001299
INSTANTANEOUS
PHASE
__________________________________ 628
~ 5
I .26
-I
.•
26 ~
-
~.
-6
-
....
.• .
2
~
s.
I7
30.
34
4
T I ME
5. 5
I
5—....
.
60
.
68
75
- 85
9
I
- 02
(H P S)
Fig. 10. Complex demodulates of 0S22 recorded at NNA following the Indonesian earthquake.
57 INSTANTANEOUS
o
15.12
AMPL.
30.34 1690421070
AT
15.51 IS
60.68
INSTANTANEOUS
.
75.85
77
~
. .
..—..-—--,...
3
75.85
.2
PHASE
~-.
<1.26
.
I
~-l.2
.
30.34
45.51
TIME
60.68
91.02
C90165400L19—T
.~
z
....-
...
...~.
15.17
60.68
IS .004288
426.33
_,J..
—3 .
0
45.51
PERIOD
-INSTANTANEOUS
.~ .
30.34 169091078
_./_...
<1.26
AT
AMPL.
6.28
~•
~-l.Z6
15.17
PHASE
‘.-
6
0
•.
z
-
91.92
000L(590C616—T
.004209
6.28
3
INSTANTANEOUS
374.06
PERIOO
—
...~
.
- 6.
.... 75.85
91.02
IMPS)
Fig. 11. Complex demodulates of oSi
8 recorded at NNA fol-
lowing the Indonesian earthquake.
though the synthetic peak is taller. Amplitude and Q values are more uncertain than for the previous example of 0S22, and should be weighted accordingly. A grade of C is assigned to a mode with significant deviations in at least two of the three selection criteria. The mode 0S~shown at frequency 0.00234 Hz in Figs. 8 and 9 is an example of a grade C estimate. The complex demodulates for this mode, graphed in Fig. 12, exhibit a changing phase, and two sections of different amplitude decay rate. Because of this unusual decay, the regression estimate of Q differs significantly from a demodulate estimate. Even though comparison of spectra in Figs. 8 and 9 shows fair agreement, a reliability weight of C is assigned to this mode. For other modes the choice of grade C is made based on poor comparison of the synthetic oscillation as well as differences in Q estimates. The assignment of a grade D reliability weight is to decide that a peak in a spectrum fails all three
~
...
.
2 0
15.17
30.34
48.51
TIME
Fig. 12. Complex demodulates of lowing the Indonesian earthquake.
60.68
— 75.95
91.02
IMPS) O~I5 recorded
at NNA fol-
criteria and is unusable. In Fig. 8 two such examples are the small peaks for 0S19 (0.0028 Hz) and 0S30 (0.0038 Hz). 4.3. Methods of averaging It is shown below that there are significant differences in periods of free oscillations of the same order estimated from the global network of recorders. For this reason it is of interest to apply the diagonal sum rule that follows from selection rule (iii) described earlier. For a sufficiently distributed network of worldwide stations, the diagonal sum rule predicts that an average of all estimated periods of an eigenvibration mode will be an estimate of the period, ~ for a spherically-symmetric Earth model. This average for T0 is calculated for the IDA network observations plus the regional network estimates. However, in some cases the global average of all stations will not be sufficiently representative for
58
the diagonal sum rule to apply, for the following reasons. First, for many of the modes listed in Tables II and III, estimates of 0 were not ob tained for all stations. This average could possibly introduce a bias to T0 due to restricted world-wide coverage. Second, since all estimates of 0 are not of equal reliability and equal variance, they cannot be given equal weight in averaging. Instead a weighted average is made. If the weights were all nearly equal, sufficient coverage would yield an unbiased estimate of T0. Since the regional network represents a small localized region in a worldwide average, the average of 0, from the regional network is used in averaging 0 worldwide. We now use a theorem (Lindley, 1965) for combining estimates of known uncertainties. Given a group of estimates x distributed as N(X, ~~2) where the average ~ is unknown and the variances are known, estimates of ~ and var are given by .~
~
=
I
I
//
~ w2
(1)
TABLE VI Values for wr, for various reliability grades
-_______________________________________________ Grade
wr
A B
1.0 0.8 0.7 0.6 0.2
C D
1
for penod and Q for penod for Q for period for ~
0.0 for penod and Q
to the values averaged. Values for wr1 are given in Table VI. When averaging period, larger values were assigned to wr1 than when averaging Q. This was done because it was found in Paper 1 that the values of Q were far more drastically affected by interference by neighboring modes than the values of period. Therefore, period, of a C grade mode, say, is more reliable then the value of Q of that same mode. The valuestheforauthor’s wr, are opinion arbitraryof and were chosen to reflect the relative reliability of the measurements.
—I
varx(~wi)
(2)
4.4.
1
Overtone modes
where — I / 2 (3\ WI — / ~ ‘ / In the present case, reliability weights are included with the variance in producing the overall weights w1 in eq. 1. A reliability weight is assigned, wi, 0 ~ wr ~ 1 depending on the grade, D to A. The weights, w0., to use in eq. 1 are then computed from — ( 2 / 2~ (4) — ~ / ~ WrE 2. The var S~ where is the smallest of theaverage a is thena~ computed from avalue weighted of the a2 —
Many peaks in the spectra (see Fig. 8, for example) correspond to overtone modes. All of these oscillations were analyzed and 23 estimates of 0 were found for 17 modes. These estimates are shown in Table IV. When overtone modes in the spectrum are isolated peaks between fundamental modes they can be treated one at a time as are the fundamental modes. In a few cases, overtone oscillations had frequencies close towas neighboring fundamentals that specialsoattention required (see Case B in Paper 1). To illustrate the use of non-linear regression to simultaneously estimate 8 for close neighbors, consider two examples of
~ (wr, /a 12)
var ~
=
Wri
I
(5)
Equations 1,4 and 5 are used to compute weighted averages of period and Q for each order 1 of eigenvibration. A reliability grade is then assigned to each average equal to the highest grade assigned
oscillations following the 1977 Indonesian earthquake. First ~thepair of 0S7 and 2S3 (Tables III and IV) recorded at CMO, and second the pair ~ and 2S7 recorded at HAL. Figure 13 shows the Fourier amplitude spectra for the pair of modes 0S7 and 2S3 and the synthetic modes generated with the regression estimates. In this case, it is clear there is more than
59
S Y N THE T I C
DATA
INSTANTANEOUS
AMPL.
AT
PERIOD
810.64
~
I
‘~
0
15.17
30.34 8flN0W1016
45.51
60.68
INSTANTANEOUS
W
.00125
.00105
6 .28 3.77
.00125
,... —~....
Fig. 13. Amplitude spectral density of
0S7—2S3 pair.
—j
I .26
-I
one mode present. This interference is well exhibited by the complex demodulates shown in Fig. 14. The regression estimates (see Tables III and IV) for Q for these two modes differ by nearly 150. The variance of Q for the smaller amplitude peak 2 5 3 , is much greater than for 0 S~ as might be expected for closely separated unequal amplitude peaks. It can be seen that a higher value of Q for 2S3 is correct, as this peak loses energy less quickly than 0S7. This difference in Q is found by computing different amplitude spectra with successively later starting times. As in Case B of Paper 1, there are a number of significant non-zero linear correlation terms computed from the covariance matrix (see Table VII) indicating trade-offs
.
75.65
PHASE
-.~
/
-
./
~
26 .
~
.‘~—.
-
~
.~/
:‘ ....~.
-6. 0
IS. 17
30.34 ME
p5
68
78.85
a1 a2
82 Tt 72
Q1
O~7 2
91 .02
Fig. 14. Complex (atearthquake. 810.64 s) of the CMO seismogram following demodulates the Indonesian
may be occurring between 0 of each mode. For this reason it seems appropriate that reliability weights of grade A cannot be assigned to such closely separated oscillations. The Fourier amplitude spectra of the second example is shown in Fig. 15 on the left with the
TABLE VII Correlations for
91.02
CYCILSIO(LTA—T
IS .004249
S3 regression
a1
a2
81
82
71
72
Q1
—0.5092 0.0 0.3973 0.0 —0.2628 —0.8045 0.4170
—0.3973 0.0 0.2147 0.0 0.4374 —0.8348
—0.5092 —0.8047 0.4174 0.0 0.2634
0.4369 —0.8347 —0.2138 0.0
—0.3 190 0.0 —0.1152
0.1141 0.0
—0.3191
60
HAL DATA
____ .00105
CMO SYNTHETIC
.00205
.00185
DATA
SYNTHETIC
__~J.
.00205
.00185
.00205
.00205
.00185
Fig. IS. Amplitude spectral density of O~II~2S7 pair.
synthetic modes again shown on the right for comparison. Although this appears as a single peak in the Fourier spectrum for HAL, the complex demodulates in Fig. 16 show a very pronounced interference effect indicating the presence of two oscillations. (Spectra from CMO were included in Fig. 15 as an example of a case in which two closely spaced modes are clearly present.) Two modes were fitted quite well by non-linear regression to this single peak. Again, there were large linea correlations between parameters of adjacent
INSTANTANEOUS
AT
.
PER 100
536.39
~ 85
-
~ ~
.20
-.
- I II
-
I
.
0
modes (see Table VIII), so even though the uncertainties obtained were quite low formally indicating a close fit, a reliability weight of grade B must be assigned. Certainly, one of the modes is 0S11, but the identification of the close neighbor is not as clear. It may simply correspond to the overtone mode 2~7 as predicted by numerical calculations for various Earth models (Derr, 1968). However, an alternative identification may be appropriate. This corresponds to the results of studying a realistic Earth model with the theory of quasi-degeneracy (Dahlen, 1969; Luh, 1973, 1974). The results show that Coriolis coupling (McDonald and Ness, 1961) may occur between torsional and spheroidal oscil-
AMPL
so
I
15.17
30.34 •*8061076
45.51 IS
60.68
I N S TAN TAN E 0 U S 6.28
75.85
SI .02
.004209
P HAS E
.
-
~
~
-
.
....‘
-
26
I
.
~—..
-~. ..
-.6
.
2~
I ~.
I,
30. 34
45. 5 I
T I ME
60
. 08
~
.
I
.
02
I HRS)
Fig. 16. Complex demodulates (at 536.39 s) of the HAL seismogram following the Indonesian earthquake.
61 TABLE VIII Correlations for
O~II ~2~7
a a1 a2 81 82
li 12
regression
1
a2
81
0.9006 0.0 0.1706 0.2974 —0.4629 —0.8475 —0.6417
—0.1706 0.0 0.4353 —0.3215 —0.6592 —0.8378
0.9006 —0.8482 —0.6406 —0.2992 0.4614
—0.6601 —0.8371 —0.4367 0.3195
lations only closely separated frequency. Such is 5I1 in and the case for the modes 0 0T12 whose difference in periods is less than 1 s (Derr, 1969). Luh (1974) points out that usual mode designation loses its meaning because of strong coupling, and any analysis of normal mode Fourier spectra for modes that are strongly coupled must incorporate information on the extent of coupling. The inference here is that the torsional oscillation 0T12 could, through Coriolis coupling, produce vertical motion with a period only slightly different from 0S11. Virtually all ten vertical component time series following the 1977 Indonesian earthquake contained energy at these periods although none were resolvable as single oscillations, and in fact all exhibited similar patterns in their complex demodulates as in Fig. 16. It is interesting to note that the largest amplitude oscillation recorded on the BKS transverse component seismometer has a period corresponding to the single oscillation of 0TI2 (see Table V). This shows the energy in the mode 0T12 is present for coupling to possibly occur. In this study (primarily concerned with regional variations of well-resolved fundamental modes), no theoretical analysis of quasi-degenerate coupling is made. For this reason, the modes will be designated as 0S~1 and 2S7 in the tables although these are not certain identifications. 4.5. Tables of eigenvibration parameters Tables Il—V contain all the estimates and uncertainties for the modes resolved using the optimal methods outlined. These oscillations were gen-
Q2
12
&2
0.3243 0.0 —0.5614
0.5628 0.0
0.3244
erated by the four large earthquakes described earlier. Contained also in the tables, just above the estimate of period, is a number designating the number of modes included in the regression analysis and a letter corresponding to the reliability weight assigned to that estimate. In the columns for regional and global averages of period and Q, the number designates the number of modes used in the average. The following abbreviations are used in the tables. T represents period in seconds. Amplitude, a, is in microns, and phase, 8, is radians. The four earthquakes are designated I— Indonesian, P—Philippine, C—Colombian, and W— West Irian. Under each estimate of period, amplitude, Q, and phase is its corresponding formal standard deviation.
5. Results from network analysis 5.1. Introduction An initial goal of this study was to assess what variability in period and damping factors Q of eigenvibrations may exist in a limited region such as northern California. Extensive periodogram analysis of world-wide seismographic station network (WWSSN) shows considerable scatter in both period and Q of eigenvibrations (Madariaga, 1971; Dziewonski and Gilbert, 1972, 1973; Bolt and Currie, 1975; Jobert and Roult, 1976; Anderson and Hart, 1978; Hansen, 1978, 1979, l981a; Geller and Stein, 1979; Hansen and Bolt 1980). For example, a representative scatter is for the mode
62
0S14, the published values range through 447.17— 449.41 s for period and 183—435 for Q. At the present there is much interest in tracing the source of this variation and separating measurement errors, measurement bias, and various complexities of the Earth. In this study eigenspectra have been systematically analyzed from the four large earthquakes in order to determine to what extent each effect may be detected. By using the network of three stations in northern California described in Section 2, some isolation has been achieved for such variability as: (1) crustal structure beneath each receiver; (2) positions of nodal lines of eigenfunctions; (3) different locations of earthquake sources. The following sections address these questions.
5.3. Effects of local structure beneath receiver sites
5.2. Basis for statistical comparison
About 350 km north, the WDC seismographic station is located in the southeast portion of the Klamath mountains just northwest of the Central Valley. The seismometer is sited on Devonian meta-volcanic rock. The crustal thickness here is about 30 km. There is evidence that suggests the lithosphere in northern California is an anomolous region. The upper mantle P-wave velocity structure has been investigated by analyzing the azimuthal variation of teleseismic P-wave residuals recorded by the University of California at Berkeley seismographic stations (Bolt and Nuttli, 1966; Simila, 1980). The relative residual variations at a single station in the northern part of the state exceed ±l.Os. Simila has modelled the residual variations by lateral variations of the P-wave velocity in the upper mantle. A low-velocity zone is inferred beneath the Gorda plate to account for the positive residual pattern along the coast. A paleosubduction zone is modelled beneath northern California to represent a region of high velocity which explains the negative residual values. This structure may extend into the lithosphere to a depth of 150—200 km. It is now convenient to restate the first question of Section 5.1. Given that the whole Earth is oscillating with a set of eigenvibrations, can the different crustal and upper mantle structures under the three receivers cause measureable changes in 0 for each mode? Since the separations of these seismometers (<350 km) are less than half the
Because each time series is considered as a superposition of signal upon stationary noise (see Paper 1) it is appropriate to use a statistical criteria for assessing the significance of differences in measured eigenvibration parameters. The estimation procedures of complex demodulation and non-linear regression were developed in Paper 1 for this purpose. By estimating the statistical uncertainties for each parameter, confidence intervals can be estimated for the quantity OjJ — 0~.(0~~ denotes the] ~ estimate of the oh parameter in 0.) Given the parameters X1 and x2 with variances and ~2’ the quantity var(x~— x2) is given by ~
var(x~— x2)
=
.~
+
~
(6)
To test the hypothesis that x1 x2 we form the N% confidence interval and test if =
I x1
—
> CN
(s? +
(7)
2)1/2
If eq.7 holds, x1 and x2 are significantly different at the N% confidence level. That is, a difference greater than CN(s~+ s~)I/2 can occur by chance alone only (100 — N)% of the time. For N 95, CN 1.96, and for N 99, CN 2.5758. Equation 7 gives a quantitative measure of testing for significant differences in eigenspectra parameters in the following sections of this paper. =
=
=
California is a region of complex geologic structures with tectonic properties changing quickly with distance. The recording stations BKS, JAS and WDC are each situated in quite different tectonic regions; the coastal range, the Sierra Nevada mountains, and the Klamath mountains, respectively. The BKS seismographic station is located in the Berkeley Hills in the Coast Range. Here the crustal thickness H is approximately 22 km. The JAS seismographic station is sited east of the Central Valley in the foothills of the Sierra Nevada mountain range. Geologically, the Sierra consists of granite overlain by remnants of older metamorphic rocks. The foundation rock at the JAS seismometer is metamorphic serpentine.
63
wavelength of a mode of order up to oS~o~ it is expected that any differences in parameters obtamed from the three receivers would be due to local effects of lithosphere structure rather than the global changes due to large scale lateral heterogeneities. (The question of distance separation of stations required to see significant shifts in multiplet period due to lateral heterogeneities is explored in Section 5.4.) To estimate the order of magnitude of the response of a thin layer of crust to a forced oscillation at the base of the lithosphere, let us assume the layer can be represented as a spring and dashpot system as in a linear damped harmonic oscillator being driven by a harmonic force. The result for reasonable rock properties suggests that if the receivers are located on bedrock, such as basalt or granite, there should not be any detectable changes in period, Q, or phase of eigenvibrations between the seismometers in the regional network. Ten or twenty percent changes in the properties of the bedrock again cause no detec-
there are over 60 modes for which comparisons are made to determine if indeed there are observable, systematic shifts in period, Q, or phase of an eigenvibration mode within a small region due to the tectonic differences in the region. A statistical analysis of the relevant values in Table II indicates that there are no detectable systematic shifts in period, Q, or phase within the California regional array for any eigenvibration modes excited by any one of the four sources considered. At the 95% confidence level, there are 24 cases where two estimates of period differ significantly. However, at a 99% confidence level the number of cases reduces to 13. Of these 13 cases, 11 involve C grade estimates while the other 2 cases both are between B grade estimates and the differences are on the order of 0.34 s. Such small differences would not appear to be significant in the light of the probably underestimated uncertainties of B and C grade estimates 0 (see Section 4 and Paper 1). A particular illustration is for the mode 0S24 re-
table changes in the results. Even if the differences extend for 200 km as Simila’s model suggests, there is no detectable shift.
corded following the Colombian earthquake. The JAS recording, reliability B, and the WDC recording, reliability B, differ in period by 0.34 s. Given standard errors of 0.069 and 0.077 s, respectively, the values calculated from eq.7 indicate this difference is significant at a 99% confidence level. It is judged that such a small difference may in fact not be significant for estimates assigned a B grade reliability. There is only one example of two values of Q differing significantly at either the 95 or 99% confidence level. This difference occurred for the mode 0S14 following the Colombian earthquake. Analysis of the JAS recording yielded an A grade estimate while that of the WDC recording was a B grade. The 99% value of CN(a? + ~~/2 is 103.8 while the difference in Q is 105. Because this Q difference involves a grade B estimate it cannot be considered significant. The conclusion is that the decay rate of eigenvibrations does not change significantly due to local crustal differences within a region whose dimensions are less than a wavelength. The parameter for which the largest number of significant differences occur is phase. At a 95% confidence level there are 53 cases where the phase
5.3.1. Case] The following comparison, hereafter referred to as ‘case 1’, bears on the initial question of Section 5.1. To eliminate any source characteristics, case 1 will only compare estimates of 0 for spheroidal modes obtained from the three different receiver sites of the California regional network for each of the four earthquakes independently. Therefore, for example, a JAS OSI4 estimate of 0 from the Columbian earthquake is not compared to a WDC 0S14 estimate of 0 from the Philippine earthquake. At BKS, the comparisons were made for both the vertical and longitudinal components of ground motion. From the four earthquakes recorded by the California regional network there were 184 resolvable observations of eigenvibrations in the range of o~9through 0S48 (see Table III). Of these 184 observations, there are 59 cases where two or more estimates of 0 were obtained from the regional network for the same mode excited by the same earthquake. Because some of these 59 cases had three or four observations of the same mode,
64
of the same oscillation differed significantly at two different sites. At a 99% confidence level the number of cases reduced to 43, still a large number. Of these 43 cases, 27 of them involved comparison of a vertical component recording with the longitudinal component recording. The magnitude of the difference varies from mode to mode and in some cases is as large as iT. This indicates that in general the semi-major axis of the elliptical particle motion of spheroidal modes is inclined from the vertical. Furthermore, although most estimates of 0 from the longitudinal component are grade C, it seems the amount of inclination is a function of order number, Of the remaining 16 cases of significant phase differences from vertical recordings, the magnitude of the shift, though significant, is small, less than IT/6 in all four cases involving A-grade estimates. The direction of phase shift is random both in order number and between different earthquake sources. Since no systematic pattern emerges, it is concluded that local geological differences within this small region create no detectable shift in the phase of an eigenvibration. 5.4. Comparison with theory This section will address the second two questions of Section 5.1. To do this, comparison will be made between the observations in Tables Il—V and theoretical and numerical results. There are three cases to be tested using observations from the California regional and IDA networks. These three cases will be called case 2, case 3, and case 4, respectively, Early theories of Earth eigenvibrations were made assuming spherically-symmetric Earth models. Theoretical descriptions of the Earth’s vibrations using these models are solved with spherical harmonic expansions about the earthquake source. This serves to define eigenfunctions, with specific well-defined nodal patterns that are a function of source position. In the early I 960s the effects of the Earth’s rotation and ellipticity were investigated by a number of people. It was shown that such effects remove the degeneracy of the problem giving rise to splitting of the eigenfrequencies. Rotation further complicates the problem by in-
troducing Coriolis coupling between nearby modes (see Section 4.4) and westward drift of the nodal lines (McDonald and Ness, 1961). A method, making use of the specific nodal patterns, for identifying the order m of a mode ,~Sf1was proposed by Nowroozi (1965). By plotting the spherical harmonics of different order m as a function of / for a particular recording site it is found that the zeros of the functions for different m occur for different values of I. Therefore, certain values of m can be ruled out for each large amplitude oscillation. Different results have been obtained from more recent studies of realistic Earth models. The work of Stifler (1979) is in contradiction with the above. His finite element calculations led to three principle theoretical results on the Earth’s free oscillations. (I) The general effect of a lateral heterogeneity on a sphere is to remove the degeneracy of the multiplet eigenfrequencies. The non-degenerate eigenfunctions are then completely defined with respect to the inhomogeneities that have removed the sphere’s symmetry. For the lowest-order torsional oscillation, the Earth’s ellipticity of figure governs the positioning of the eigenfunctions. For higher-order modes, the relative positions of the continents determine the orientation of the eigenfunctions. (2) The combined effect of the Earth’s continents and its ellipticity on the eigenfrequency perturbations is found not to be the sum of the individual effects. Thus, the eigenfrequency perturbations due to lateral inhomogeneities are nonlinear functions of the perturbations to the starting model. (3) For torsional oscillations o7’2 — 0T10 the effects of the Earth’s ellipticity are greater than the effects of the structural contrasts between continents and oceans. The finite element computations show that the range of the ellipticity-induced eigenfrequency splitting is approximately 0.2% of the degenerate eigenfrequency; whereas, the range of the continent-induced splitting increases from 0.01 to 0.1%. The first result states that the eigenfunctions of an inhomogenous body are unique and fixed in position for all time independent of the source
65
position. Mote (1972) in a discussion of the effects of notches on rotating disks, argues that the presence of an azimuthal heterogeneity ‘locks’ the eigenfunctions onto the body and prevents them from behaving as travelling waves. The westward movement of nodal lines is then not a valid conclusion. However, it is not possible to test this with observations as the period of motion of these nodal lines is longer than the time required for a mode to decay it into the noise level. For example, this period for torsional oscillations is 1(1 + 1)/fl (144 h for o~’2) where fi is the period of the rotation of the Earth. The passage of a nodal line of the eigenvibrations studied in this paper through a regional array is too slow to observe. The by discrimination procedure order in m light suggested Nowroozi (1965) is also for incorrect of Stifler’s first result. In fact, the amplitude of a recorded singlet would be governed by the relative positions of both the source and the receiver with respect to the fixed nodes of the eigenfunction and not with each other. Graphs of amplitudes of multiplet eigenfunctions presented by Stifler (1979) suggest that virtually all multiplets consist of multiple singlets of comparable amplitude. If this is the case in general, the question occurs as to why so many modes presented in Tables Il—V were resolvable as single oscillations? The answer to this may lie in the analysis by Dahlen (1 979a, b, 1980) of the influence of lateral heterogeneities on the splitting of single station seismic spectra. Consider the lateral heterogeneity of the Earth expressed as a perturbation expanded in spherical harmonics of orders s~ through 5m~ where the perturbation, ôm, i5 slight, i.e. symbolically 18m/mol<< 1. Dahien states that under the condition Sm~<< 1< < I 6m/m and n <<1, there is cancellation of m — 1 0 2/ adjacent lines, and the mode ~5f~ would behave as a single resonance peak broadened by attenuation alone. He further states that the degree of cancellation varies as a function of separation, ~, of the source and receiver and that maximum cancellation should occur at ‘antinodes’ where cos[(/ + 1/2) i~— iT/4] = 1. Right at the antipode from the source, all amplitudes of the singlets are in phase yielding no cancellation. A test of these theories would be to compare 5~j~
1
=
L
—
eigenvibrations recorded by the California regional network excited by sources located at different azimuths and distances from the array. If the sources could be shown to supply the same relative energy to the singlet modes, then, according to Stifler’s conclusion, the estimates of 0 would not change. However, since the position of the source determines the relative excitement of the singlets, such a comparison is not crucial. As Dahlen (1980) points out, it is not a simple matter to determine the range of validity of his theory for the Earth. It certainly must fail for low 5max’ s~, and 1, say n -~1, as the condition ôm/m 0 are in fact depth dependent, and different modes sample the Earth down to 5min’ different 5max’ and depth. As a result the quantities Iôm/moI depend upon the mode under consideration. A way of determining the validity of this theory over a range of I is to examine multiplet spectra for deviations from single-peak behaviour. The methods of complex demodulation and nonlinear regression developed in Paper 1 are ideal for such an examination.
I
I
Case 2 In this case comparisons are made of eigenvibration parameters estimated within the California regional network excited by the four different earthquake sources. These four earthquakes represent sources of different azimuths and distances from northern California. It can be seen from Table III that there are significant shifts in period from stations distributed world-wide from a single earthquake source. (For example see the HAL and RAR estimates for the mode o~I6’ The difference in period is 1.33 s, which is significant at the 99% confidence level.) If eigenfunctions are locked in place, much less variation would in general be expected in period for modes excited by different sources but recorded in the same regional locality, because the variation in the singlet amplitudes is a function of source and receiver positions with respect to the eigenfunctions. By fixing the receiver positions to a small region the only variation in singlet amplitudes is due to the source. There still could be shifts in period from source mechanism variation. However, if a source excites well a singlet whose nodal line lies nearby the 5.4.1.
66
recorder it will not affect the regional recording, while world-wide there is likely a station near an antinode whose recording would be much influenced by this singlet. Comparisons are made in this case for period, Q, and amplitudes. Period and Q comparisons are straightforward following the procedure in Section 5.3. Comparisons of amplitude are made in a different sense as a test for positions of nodal lines. If nodal line positions do not change in the region of Northern California for different earthquake sources, the relative amplitudes between JAS, WDC, and BKS for each eigenvibration should remain constant. There are 13 modes between ~ and 0S3~where relative amplitudes can be compared between JAS, WDC, and BKS. Unfortunately, not all 13 modes were estimated from all three sites, and, of the estimates made, 19 were assigned A grades, 34 were assigned B grades, and 30 were assigned C grades. Furthermore, there were no combinations of multiple A grade modes estimated from different earthquake sources. For example, 0S19 was well recorded at JAS, WDC and BKS following the 1977 Indonesian earthquake and all three estimates were assigned A reliabilities. The other three earthquake sources also generate this mode although estimates were all assigned C grade reliabilities. For this example of ØSj9~ it is interesting to note that even though there is some difficulty in making reliable estimates of 9 following the latter three earthquakes, the periods only differ at most by 0.26 s which is not significant at the 99% confidence level. If we disallow the C grade estimates of amplitude in the comparison we are left with only five modes for comparison. Of these five cases there are three (0S21, O~23’ and o~31) that show agreement between relative amplitudes and two modes (0S24 and 0S27) that show disagreement between relative amplitudes, Comparisons were made between periods and between Q values for 25 modes from 0S10 to O~36 recorded following two or more of the four earthquakes studied. Of these 25 modes, there were only 5 modes (o~I7,22,23,3o,3l) where periods differed significantly and only 2 modes where Q values differed significantly. It was concluded that even though there is scatter in Q data, the uncertainties
‘
indicate that differences in Q are not significant. In all five cases the A grade period estimates, though significant at the 99% confidence level, differ by less than 0.65 s. Though it is not conclusive, the above compari. sons for period, Q, and amplitude indicate that there are differences in eigenspectra excited from different sources and recorded by the same regional stations. The eigenspectra tend to compliment each other, each exhibiting different wellresolved modes following the different earthquakes. Even though shifts in period are small relative to observed shifts between IDA stations, the fact that the eigenspectra are complimentary to each other suggests that the source characteristics, including location of the earthquake, are important in determining the patterns of singlet eigenfunctions. Since the effects on 0 are small relative to global characteristics the results lend support to the theory of fixed eigenfunctions. 5.4.2. Case 3 This case will address the question of variations of Q values of spheroidal eigenvibrations as a function of position on the Earth following the Indonesian earthquake. Comparing Q values of A grade modes whose periods are significantly different will distinguish whether attenuation characteristics of the singlet eigenfunctions vary with a multiplet. The global array used for this comparison consists of the seven IDA stations and the California regional network average values for period and Q. In all there are 114 A or B-grade estimates of Q following the Indonesian earthquake. (C grade modes do not yield reliable estimates of Q for comparisons.) A minimum of two estimates was obtained for 26 different modes of oscillation. Of these 26 modes, significant differences were found in 11 different modes if B grade modes are ineluded, and in five different modes if only A grade modes are included. (Note, again that modes assigned reliability factors other than A are not strictly following the model of a single decaying oscillation. See Section 4.) Because of the considerable scatter in measurements of Q, usually due to unknown biases, the 99% confidence level is used for the comparisons.
67
To illustrate the variation in Q, consider the mode 0S16 in Table III. Here we have eight estimates of grade A. Estimates of period for this mode distributes fairly evenly from 405.98 to 407.31 s, a maximum difference of 1.33 s. The Q values range from 260 ± 34 to 324 ± 14 for the A grade estimates with a weighted average of 284 ± 12. The largest differences do not appear to be associated with the modes whose periods differ the most. Since the number of observed significant differences is small and no systematic pattern emerges concerning shifts in Q values, it appears that measurements of Q of eigenvibrations are not a function of the position on the Earth of the recording instrument, or of the great circle paths for modes in this period range. 5.4.3. Case 4 This final observational analysis addresses two aspects of the theory of cancellation of eigenvibration singlets to produce a single decaying osciliation for each mode S,. First, a test of the validity of Dahlen’s asymptotic theory (presented above) of single-peak approximation. Second, using the California regional array and the IDA network inferences are drawn on spatial separations of receivers required to see significant shifts in penods of eigenvibrations. Since each mode resolved in Tables Il—V was closely scrutinized for single-peak behavior, the eigenvibration estimates obtained in my study are ideal for Dahlen’s suggestion for determining the range of values of / for which his asymptotic theory is valid. My observations range from O~4 through 0S48 and the results for A grade modes indicate that the assumption of a single oscillation is valid from 0S7 through approximately 0S30. There are, however, still some inconsistencies between the observations and the theory. It was computed, for each station, which modes would theoretically lie at an asymptotic antinode (see expression above). When a station is sited at an asymptotic antinode maximum cancellation of singlets is theoretically obtained for producing a single oscillation. By considering the well-excited peaks from 0S7 to 0S30 in all the spectra that correspond to asymptotic antinodes it was con,,
cluded that there are about an equal number of cases where stations at asymptotic antinodes record single oscillations and record oscillations with interference. For example, the mode 0S8 excited by the 1977 Indonesian earthquake has an asymptotic antinode at NNA. This mode is well recorded and exhibits a sharp peak in the spectrum (see Fig. 8, 0.0014 Hz) and linear complex demodulates. Now consider the example of 0S15 in Section 4.2 (Figs. 8 and 12). NNA again lies at an asymptotic antinode for 0S15 and the mode shows up well in the spectra. The behaviour of this oscillation, as exhibited by the complex demodulates, is not that of a single oscillation. There are also modes recorded at sites that do not correspond to asymptotic antinodes that very closely follow the behavior of a single oscillation. Apparently, the degree of cancellation of singlets, though observed from 057 through ca. 0S30, is not completely described by the asymptotic theory. The distance separation of stations required to observe shifts in period of a mode 5, due to different relative amplitudes of the m singlets is variable. Furthermore, for two given world-wide stations, the shift in period of a mode is dependent upon degree / of the oscillation, and probably dependent upon the excitation position. To illustrate this point consider the modes 0S14 and 0S21 in Table III. Differences in period vary between all the possible pairs of stations. For 0S21 the estimate in California differs by 0.36 s at a separation of approximately 30° from CMO and by 0.65 s at a separation of approximately 450 from HAL while there is no difference at all from BDF at a separation of approximately 95°.RAR and NNA have no significant difference although separated by approximately 80°across the Pacific Ocean. On the other hand, the mode 0S14 shows a difference between the California estimate and CMO of 1.06 s, and between California and HAL only 0.19 s. NNA and RAR have a difference of 0.69 s for 0S14. In summary, there emerged no systematic pattern for the direction of a shift in period as a function of 1, nor was there any one station that consistently yielded the highest or lowest estimate of period as a function of 1.
68
6. ConcLusions
0S48 show that the assumption for a multiplet S, to behave as a single oscillation is valid from 0S7 through approximately 0S30. This result supports Dahlen’s theory on singlet cancellation. However, subtle differences between observation and theory still exist suggesting that the degree of singlet cancellation is not completely described by the asymptotic theory (see Section 5.4.3.) (5) No systematic pattern emerged for the direction or amount of shift of eigenperiod as a function of order / or position of recording station following the Indonesian earthquake. No single regional or IDA station consistently recorded the highest or lowest estimate of period as a function of / (see Section 5.4.3). The first conclusion suggests a possible method of summing seismograms to improve the signal-tonoise ratio. One problem with ‘stacking’ procedures is that estimates of Q become biased due to interference of different relative amplitude singlets. Since parameters of 0 are seen not to change within a small region, a network, such as in northem California, can be used to sum its seismograms without creating interferences between singlets. Finally, more observational analysis needs to be made with an expanded data base to compare with the theories of ‘locked’ eigenfunctions and singlet cancellation. The latter (Dahlen, 1980) predicts that near the epicenter and antipode of an earthquake source there is much less cancellation of singlets. This phenomenon should be an observable feature with use of complex demodulation on the records of broad-band stations positioned correctly with respect to an earthquake source. Such observations should provide a crucial test between the normal mode theories of distinct eigenfunctions locked in permanent positions by Earth structure and standing wave patterns positioned relative to the source. ,,
The main objectives of this investigation were to improve the resolution of observations of terrestrial eigenvibrations. It is now clear that such an improvement is needed if the resolution of inverse procedures in computing reference Earth models (Dziewonski and Anderson, 1981) is to be significantly sharpened. Improved observations of o obtained by these methods from the California regional network and the global IDA network were used to trace the~source of variations of spectral parameters that arise due to various cornplexities of the Earth (e.g. lateral heterogeneities). (1) An analysis of eigenvibration parameters obtained within the California regional network following each earthquake source indicates that there are no detectable systematic shifts in period, Q, or phase within a region that is small with respect to the wavelength of the oscillation. Use of both vertically and longitudinally oriented seismometers in the analysis indicated that in general the semi-major axis of the elliptical particle motion of spheroidal modes is inclined from the vertical (see Case 1 in Section 5.3.) (2) Comparison of periods, Qs, and amplitudes of eigenvibrations excited by earthquake sources of varying distance and azimuth to the recording sites in northern California indicate that there are differences in eigenspectra due to the source function and position. Though not conclusive, slight systematic shifts in period (<0.65 s) and relative amplitudes between the California regional stations lend support to the theory of ‘locked’ eigenfunctions. Differences in Q values are not significant (see Section 5.4.1.). (3) Differences of Q values obtained world-wide from both the California regional network and the IDA network are found not to be significant, even though their corresponding periods varied significantly. This observation indicates that measurements of Q of eigenvibrations are not a function of the position of the recorder on the Earth. The conclusion is that all singlet eigenfunctions have the same damping characteristics independent of great circle paths for this period range (see Section 5.4.2.). (4) Observations ranging from periods of 0S4 —
Acknowledgements The author would like to thank Bruce A. Bolt for his suggestions in this work, and Robert Uhrhammer, Don Michniuk and Richard Lee for many helpful discussions. Thanks are also due to Freeman Gilbert for
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supplying the IDA data of the 1977 Indonesian earthquake, and Guy Masters for a computer program that computes the response of the IDA gravimeters, and to CIRES for the use of their computing facilities in preparing this manuscript. This work was supported by NSF grant EAR 19694.
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