which «as filtered electronically prior to recording to reduce the tidal amplitudes. The record (top) was processed to remove tides by twice subtracting three hour running averages. The resulting record (bottom) was then tapered at both ends for spectral analysis.
The amplitude and phase spectra are shown in Figure 2.4.
Frequencies are given in cycles per minutes, and some of the modes are identified. Hodes appear as peaks in the amplitude spectra, and as
"whitetl areas in the phase spectrum. (This is because of the n phase shift across a resonance peak [Narion, 1970]. The phase is incoherent except near a well developed peak, yielding the dark areas on the plot, and coherent across a well developed peak.) This can be seen best for
"clean" peaks, for example 0512 (.12 cpm). Noisier and more jagged peaks, for example 052' do not stand out as well in the phase spectrum.
(Each singlet should ideally have its own resonance curve, which interferes unless the singlets are well separated.) The poor quality of the longest period peaks proved a limiting factor in the later analysis. Some
modes (e.g., 050) are barely resolvable on this record, but can be resolved on the UCLA gravity record. Torsional modes are generally indistinct.
The tapered spectrum was filtered to isolate individual multiplets, as shown in the follm<ing set of figures . The filter windows used in the analysis are listed in Table 2.1. The filtered data show the characteristic beating patterns resulting from singlet interference.
The vertical scale on each figure gives the trace amplitude in digital units. The one unit level can usually be regarded as a nominal noise level. As discussed later, in some cases the true noise level
,s,
oS.. oSo0S5 oT] Of4 Figure 2.4.CHILE ISABELLA STRAIN <i3o oS, oS. oSt oSlO
os.
05.2 051] rA~ oS.5 Amplitude and phase spectra of the Isabella strain record. FrequencieH are in cycles per minute. The short period cutoff is 300 seconds.Table 2.1 FILTER PA55Bfu~D5
Node Nin. Frequency (cpm) Hax. Frequency (cpm)
05
2 0.01750 0.01950
OS3 0.02726 0.02882
05
4 0.03821 0.03935
OS5 0.04986 0.05070
OT3 0.03481 0.03566
OT4 0.04565 0.04640
OSO 0.04830 0.04950
i t" ~
-I I ~ iL~ ~ v.". W\~Jf\(VN v VVvJ~~
.t 1__ t._ .... .... I.a ,.", I." ,_ ..... I_a 1M J.G lao ."'" l.a .M .__ .... ,. 1._ ..... .... t._ ."'" ..... ",..,.,
Figure 2,5. Bandpass filtered data and Hilbert transform envelope for OS2' Time is in hours, amplitude in digital units."
.'
:[i
-t" '._-i:i.: 1 , , ,
CHILE ,5. ~ ... ~
. ..
Figure 2.6. Bandpass filtered data and Hilhert tran .. form envelope for OS3. Time is in hours. ampli tude in digital units.I i ... ,_ Figure 2.7. Bandpass for 084'
"
-_ . .
' ,.-/.'"..
.~ "j-... - "... "
filtered data and Hilbert transform envelope Time is in hours, amplitude in digital units.I
; r , L .. .. . v . ~ . v ~~ . ;- -~ . . .
1·1 I ...... ..., ......... I.JCIIl ,.., ,... I .... ,_ I,a , ... , ........ , .... ,"'"'."" ... _ .... Figure 2.8. Bandpass for aSy-
filtered data and Hilbert transform envelope Time is in hours, amplitude in digital units.~ !
~
I,a I .... I." I," ..... .... ..ND Figure 2.9 Bandpass filtered data and Hilbert transform envelope for 01"3. Time is in hours, amplitude in digital units.... '.100 •.• ..a 1.100 '.1IOlI , . ."" I .... CHILE oT. Figure 2.10. Bandpass for T
o
4'1.0'" 1 .... ---1 . .., I.'" I.d
I.
,._ '.11>1- ,
filtered data and Hilbert transform envelope Time is in hours, amplitude in digital units.appears to be higher. Some modes remain above the noise level for extended periods of time. In contrast, some modes, such as OT)
(Figure 2.9) and OT4 (Figure 2.10) are barely above the noIse level initially and soon decay below it.
The Im.er portions of each figure shm. the Hilbert transform envelopes of the time series. These have the information needed for studying the beat patterns and their time decay, and are much easier to store and manipulate than the entire time series. He use these in the later analysis.
The second data set used here is the UCLA gravity meter record of the 1964 Alaskan earthquake [Slichter, 1967}. This is a La Coste- Romberg tidal gravimeter, whose output was punched directly onto cards
[Ness ~ a1. , 1961}. 1\/0 such meters were operating in 1964, one of which (meter 7) yielded data far inferior to the other (meter 4). Only
the meter 4 record is used in the later analysis.
The raw record (Figure 2.11) contains a number of large amplitude glitches which have periods of tens of minutes. As this could
significantly bias the long period modes, these glitches were removed.
Wiggins and ~Iiller [1972} used a prediction error operator to isolate and remove glitches. I'e fit a straight line across the affected areas
(Figure 2.11, bottom), which yielded acceptable results.
The effects of deglitching are shm"" in Figure 2 .12 , >,hich shm,s the amplitude and phase spectra before deglitching. The amplitude and phase spectra after deglitching are given in Figure 2.13.
Filtered time series and envelopes (Figures 2.14-2.17) were obtained for the four spheroidal modes, using the same filter windows as for
t
ALASKAN EARTHOUAK E UClA METER 4
UNDEGUTCHED
DEGUTCHEO
Figure 2.11. UCLA gravity (meter 4) record of the Alaskan earthquake (top). The origin time of this figure, and all other Alaska ones, is 0336 hours, 28 1·larch 1964, the origin time of the main shock. The digitized record begins 256 minutes later. The middle trace shows the large glitches present after tide removal, the lower trace shows the final deglitched record.
ALASKA UCLA GRAVITY BEFORE DEGLITCHING oSz os, rf34 ,;;, 0'1. 05, oS. os, 050 Os" Figure 2.12. Amplitude and phase spect·ra of the undeglitched gravity meter record. Frequencies are given in cycles per minute. Several long period modes are identified. The short period cutoff is 300 seconds.
ALASKA UCLA GRAVITY DEGLITCHED ,s, ~ <P. ,;;,
,s.
oS. os, oSo os. Figure 2.13. Amplitude and phase spectra of the gravity meter record after deglitching. The noise level is reduced substantially. Frequencies are in cycles per minute, "ith a short period cutoff of 300 seconds.1 U I." ... I .... ALASKA oS. Figure 2.14. Bandpass for 052.
1.1<0 ',n '."" ... filtered data and Hilbert transform envelope Time is in hours, amplitude in digital units.
I ..... -i ... Figure 2.15. Bandpass for OS3'
, .. 1 . ..0 -I .... ,..0 1.'" --i~"" .-••
...
:-.... filtered data and Hilbert transform envelope Time is in hours, amplitude in digital units.• :r 1
!. . . ... . ... .... .. .
... I.-I." 1._ " ... , ... I," 1._ I .• , ... 1._-
Figure 2.16.oS. Bandpass for OS4'
filtered data and Hilbert transform envelope Time is in hours, amplitude in digital units.
.
.'
...
t .... --t._ 0.00' --------,.-.... I.D I.D '.n ..... t;;;;--t;,-j IaI t.... 1._ :._: I0Il :.... .._~~t...
:-~~. ..,
ALASKA 0% Figure 2.17. Bandpass for OSS'~, filtered data and Hilbert transform envelope Time is in hours, amplitude in digital units.
I
: .
U •. /011 ' . .00 ...... ..... i·._ 1._ ,.- Figure 2.18.ALASKA Bandpass for OSO'
"", ... :
.. . .. .
..-
.~ I..., I .... 1.0IIII '.G i,1OII .,." .... ~, oSo _._;_.", filtered data and Hilbert transform envelope Time is in hours, amplitude in digital units.the Chile data (Table 2.1). In addition, the fundamental radial mode, OSO' which was well recorded here (in contrast to the Chile record) is shmm in Figure 2.18.
ANALYSIS
To analyze these data, synthetic seismograms for each multiplet were generated using the method previously described. These were then compared to the data to provide an estimate of Q.
The synthetics were calculated, for the appropriate source mechanism, without including the effects of attenuation. A range of different
Q values were then applied to the "Q-1ess" synthetics, and the res,:,lting time series were then tapered and filtered in the same way as the data.
The eigenfrequency of each singlet is computed using Anderson and Hart's [1976] values for the unperturbed eigenfrequencies and Dahlen's
[1968] rotational splitting parameters. TIle elliptical splitting parameters are not used, due to the difficulties involved in previous calculations of these parameters [Ioloodhouse, 1976]. Dahlen [ personal communication1 states that more accurate recent computations show that
the spheroidal mode elliptical splitting parameters may be neglected for our purposes. Accurate values have not yet been published for the elliptical splitting or torsional modes, or for splitting due to lateral heterogeneities.
The Hilbert transform envelopes of both data and synthetics were smoothed with running averages, for ease of comparison. This is necessary since as time progresses the general decay of data and synthetics are similar but individual beats are not well in phase.
For the Chilean synthetics, we use the source mechanism determined by Kanamori and Cipar [1974] from long period surface waves. The
rupture was initiated at 38°5, 286.5°E and propagated at 3.5 km/sec to 46°5, 286.5°E, on a fault plane dipping 10° east and striking N 10°E. (I"e approximate the finite source by five point sources at a depth of 55 km.) The slip angle is 90°, a pure thrust motion. We are also including a precursory slip [Kanamori and Cipar, 1974; Kanamori and Anderson, 1975] at 41.5°5, 285.7°E <lith a rise time of 5 min starting 15 min before the main shock, and with a moment equal to that of the main shock.
For the Alaskan synthetics, the fault parameters used are those (p
=
114°, A=
90°, <I=
20°, as=
29.9, ~s=
212.4°, L=
500 Ian, VR=
3.5 km/sec) determined by Kanamori [1970].
All the figures show the smoothed Hilbert transforms of the data and synthetics, for six values of Q: 300, 400, 500, 600, 750, 1000. When necessary, Qs of 1250 and 1500 were added. The Chilean results for 052 - 055' OT3 and OT4 are presented first (Figures 2.19-2.24).
The noise levels marked on each plot are assigned somewhat subjectively. The highest noise level, four digital units for 052' seems appropriate in that the signal amplitude stabilizes at about this level. This is consistent with the fact that the noise level for the 052 multiplet (see the spectrum, Figure 2.4) seems higher than for the other modes. For the other Isabella records, the noise level is assumed to be the nominal one digital unit noise level. The smoothing intervals were also chosen empirically, to smooth out major irregularities and noise bursts in the data.
~ ~ ~ ~ ~
!
~ ~ i,
!, ,
0:0CHILE oS.
", -.
400 300 D.D! 0._ -'.100 0.Il00 1.000 I.'" 1.100 ... I.'" 1._ J.lIDII 1.0:10 1._ 1.I0Il J.OCII I.JIII 1._ J.IIJI I.a '.GOO •. _ '.ttll '.Il10 '.101 I.DOII"'"'"
Figure 2.19. Data and synthetics for eight different Q values, for OS2. The noise level is four digital units, and the smoothing window is 7S hours. Time is in hours since the origin time (1911 hours, 22 Nay 1960). All curves are normalized to begin at the same point. The time and normalization conventions are the same for all the Chile records.~ ~
:
~ I,
i,
~,
E ••
TI'CHILE oS, __ ,_,. ,(II 0'1000
-
700 400 300_ ' ..
,.. .. .
~.:IOO O.iICII 0.100 0.100 !.IlOl !.;!OD 1.Il00 l,toO "too 1.000 ".20:) j.'IClI 1.100 j,t:!) -,.Il0:l --'.:100 1.000 1.100 1.100 ',OCI! 1.2W •• Il00 '.IOCI t.'" 1.000"', .. "
Figure 2.20. Data and synthetics for OS3' The noise level is one digital unit, and the smoothing window is 100 hours long.~ !
. ..
0.:10': 0.'I0Il 0.'" 0.100CHILE oS. 0'1000 '00 JOe
.. -
,.=..
~ Figure 2,21.~ ,.C I.D }.OXIO J,JQl '.Im '.Il00 J.1ICIl I.IUI 1._ '.Q $.11111 1.1IXl ',lD) ',l'O:l '.Q ,.l1li 1.1m 5.Il00
""" '
Data and synthetics for OS4' The noise level is one digital unit, and the smoothing window is SO hours long,CHILE 05. Figure 2.22. Data and synthetics for OS. The noise level is one digital unit, and the smoo?hing "indo<' is 30 hours long.
~
•
a'I
I,
~ ~ ~ i,
~ ~!
- CHILE aT,
..
~..
~. . - ..
~.. - ..
~ ,.~ '.m U:..~S '/102 ,.-. .
~ '.m. .
~. . - . . - . .
~ •. m _.1iOO. . -
Figure 2.23. Data and synthetics for OT3' The noise level is one digital unit. and the smoothing I'Iindo\ol is 30 hours long.. . - ..
!
!
!, ! !
§,
~ ~ ~ u,
: 0:0CHILE oT. _ •• 00;, ... \0 0.100 0.'000 0.100 0.I0Il 1.000 l.'IOl 1.Il00 LlOO 1.100 UXX) l.:IW 1.100 1.«IC! l.bI 1._ 1.1m !.1iO) J.G 1,l1l;I '.Im •. l'CIl 0,Il1)0 ~.IIQO ~.fIXI HClI.IIS Figure 2.24. Data and synthetics for OT4' The noise level is one digital unit, and the smoothing window is 30 hours long.