A search for the Slichter triplet in superconducting gravimeter data

transcript

ELSEVIER Physics of the Earth and Planetary Interiors 90 (1995) 183-195

Ptt'~ SICS ()1 ] t t l t ARl t t

ANI) PI .ANITAR~ IN-Ft RIORS

Jacques Hinderer a,*, David Crossley b, Olivia Jensen b a Institut de Physique du Globe, 5, rue Rend Descartes, 67084 Strasbourg Cedex, France

b Department of Earth and Planetary Sciences, McGill University, 3450 University Street, Montreal, Que. H3A 2AZ Canada

Received 11 October 1993; accepted 4 November 1994

Abstract

There have been several investigations in the past for weak harmonic signals in the residual spectrum of superconducting gravimeter (SG) data. Recently, a stack of four European SG records led to the claim of the detection of the Slichter mode of translation of the inner core (Smylie, Science, 1992, 255: 1678-1682). We attempt to verify the presence of the Slichter triplet by two methods. We re-analyse the four European SG records with a spectral method differing from that used by Smylie et al. (Phys. Earth Planet. Inter., 1993, 80: 135-157) and compare the results to two homogeneous 2 year data sets from the French and Canadian SG. We find three weak peaks in the European stack coincident with the claimed triplet using a product spectrum, but these disappear when a cross spectrum is used which takes into account phase information. Further we find no evidence of the triplet in the 2 year data sets, where the noise level in the core-mode band is one order of magnitude lower than in the European stack. Moreover various synthetic tests (e.g. pure harmonics in random white noise; damped harmonics in brown noise) lead to the conclusion that a simulated Slichter triplet with characteristics similar to that claimed, would be easily detectable in our less noisy data, especially in cross spectral estimates.

I. I n t r o d u c t i o n

Theoretical considerations indicate that geophysical signals originating in the Ear th 's outer and inner core possess a gravimetric signal that possibly can be seen by a superconducting gravimeter (SG) owing to its high sensitivity better than 1 ngal (one part in 1012 of the Earth 's surface gravity).

After investigations which involved only individual stations (Warburton and Goodkind, 1978; Richter, 1986; Melchior and Ducarme, 1986;

* Corresponding author.

Ducarme et al., 1986; Ziirn et al., 1987; Mansinha et al., 1990; Florsch et al., 1991), the first stack using several superconducting gravity meters (two records from Brussels (1982-1986, 1987-1991), one from Bad Homburg (1986-1988) and one from Strasbourg (1987-1991)) was done by Smylie et al. (1993a,b) in order to investigate common periodic or quasi-periodic gravity signals in intertidal frequency bands and in the subtidal band below 8 h period. Previously, the I D A network of Lacos t e -Romberg gravimeters has also been used in a search for core undertones in the intertidal frequency bands (between 1 and 2 cycle day - l) by Cummins et al. (1991); the gravity data following four large earthquakes were stacked according to

184 .L Hinderer et al. / Physics of the Earth and Planetary Interiors 90 (1995) 183-195

low-order spherical harmonic amplitude pattern and no signal could clearly be identified. The coincidence of three spectral peaks in the SG stack, marginally emerging from the ambient noise in the observations with theoretical conjectures (Smylie et al., 1992), was the basis of the claim by Smylie (1992) for the identification of the translational eigenmodes of the solid inner core split by rotation; this paper will be referred to hereafter as paper I. This set of three modes is usually called the Slichter triplet in honour of the scien- tist who first pointed out the possibility that the inner core may oscillate (Slichter, 1961). The claim has not gone unchallenged (Crossley et al., 1992) and there is continuing debate on the the- ory of core modes for a realistic earth model incorporating rotation and mantle deformation which concerns the frequency location of the Slichter triplet. Moreover, the European SG stack involves stations which are located close to each other and one might suspect a regional systematic effect of oceanic or atmospheric origin. We investigate again in this study the presence of the Slichter triplet in the subtidal band of SG data. The four European gravity residual signals used in paper I will be re-analysed with a different spectral method and the results compared with a common 2 year data set from the French and Canadian SG. We also generate various types of synthetic gravity signals in order to tackle the difficult problem of detection of harmonic lines buried in noise.

2. Residual gravity from the French and Cana- dian superconducting instruments

In this study we process two gravity records from the French and Canadian superconducting gravimeters over a common observing period of 2 years (14 December 1989-13 December 1991). The two instruments are located at comparable latitudes, 48.62°N for the French station near Strasbourg (J9) and 45.58°N for the Canadian station at Cantley, Quebec (CSGI for Canadian Superconducting Gravimeter Installation), and at longitudes 7.68°E and 75.80°W, respectively. Each data set is processed by following exactly the

same methodology starting from the original sampling rate of the digitized gravity (one value every 2 s for the Strasbourg instrument and one value every second of which only every second sample is kept for the Cantley meter. The pre-processing of the gravity data and the standard least-squares tidal fitting procedure are given in detail in (Hinderer et al., 1994) and we only indicate here the major steps involved. The data are first decimated from 2 s to 5 min using a low-pass filter. After editing, to locate major disturbances to the instrument (e.g. earthquakes, instrumental main- tenance, helium refill), the related gaps are filled with a local synthetic tide based on local gravimetric amplitude and phase factors (see Dehant and Ducarme, 1987; Hinderer et al., 1991) for the main tidal waves. The gravity data are then further decimated to hourly samples and a least- squares fit to lunisolar tides, instrumental drift and local air pressure changes is performed providing the so-called residual gravity signal (raw signal--all fitted parameters).

We used two different tidal analysis codes: HYCON (hybrid least squares frequency domain convolution) (Schfiller, 1986) which invokes the C a r t w r i g h t - T a y l e r - E d d e n tidal potential (Cartwright and Tayler, 1971; Cartwright and Ed- den, 1973) with 505 waves (major waves up to third order and degree for the Moon and Sun) and ETERNA (Wenzel, 1995) based on the more recent and accurate Tamura (1987) potential with 1200 waves complete up to the fourth degree and order for the Moon. Basically, we found that, for a given data set, the use of the Tamura potential leads to a reduction of the residual gravity noise in the tidal frequency bands, as previously shown by Wenzel and ZiJrn (1990), but essentially to no change in non-tidal bands. As we focus in this study on non-tidal bands, we only report results obtained with ETERNA.

Two major steps are able to reduce the effect of (non-geophysical) disturbances on the residual gravity noise level.

(1) The despiking procedure (Florsch et al., 1991) applied to data which have been first high- pass filtered (cut-off period of a few days); all spikes exceeding a specific threshold which is taken as 3~r (~zgal), where ~r is the standard

J. Hinderer et aL /Physics of the Earth and Planetary Interiors 90 (1995) 183-195 185

deviation of the gravity residuals, are removed by this process; such a procedure has been investigated on the same data sets by Hinderer et al. (1994) and has clearly shown a significant reduction in the noise level with respect to uncorrected data.

(2) The deslewing procedure applied to unfiltered data which removes all spikes in the residual gravity derivatives exceeding a fixed value of 3o- (now /zgal h- l ) ; this procedure not only removes spikes from the gravity data but also off- sets (contrary to the previous method) and has been investigated by Jensen et al. (1992a) and Crossley et al. (1993); an example of reduction of apparent noise level in the core-mode band using slew corrections is shown in Jensen et al. (1995).

Both methods lead to comparable residual noise levels, the second approach exhibiting slightly lower levels. We only consider slew-corrected gravity residuals here. Each individual residual gravity signal with N time samples is

10 , ,

J J9 I

5 ,t, i tt t!

i t t 1 00.0 5000.0 10000.0- 15000.0

_ . t , -1°oo 5o0o.0 10000.0 ISooo.o

Time (hour)

Fig. 1. Time fluctuations of the slew-corrected residual gravity signals for the Strasbourg station J9 (top) and the Cantley station CSGI (bottom).

tapered with a 10% cosine bell and padded from N to NP = 216 = 65536; the Fourier transform

provides power spectral densities

I f ( k ) l 2 S(k) N

Finally, spectral smoothing is introduced to in- crease the reliability of the spectral estimates, i.e. S ( k ) = S ( k ) * W ( k ) , where W(k) is a spectral window of pre-determined length (* denotes convolution).

When two data sets are available, the product of complex Fourier transforms leads to cross- spectral density estimates defined in the following way (see, e.g. Jenkins and Watts, 1968)

[Fl* ( k ) F2( k ) ] IC'2(k) I = N (1)

where N = N 1 = N 2 is the common number of time samples of the two sets relative to the same common observing period. For four data sets of non-equal lengths, a possible generalization of Eq. (1) would be an expression of the form

[Fl* ( k ) F2( k ) F3* ( k ) F4( k ) ] 1/2 ] C 1 2 3 4 ( k ) I =

where N = (NIN2N3N4)1/4. Notice that, in Eqs. (1) and (2), phase information is taken into account.

An example of the time changes (hourly values) of deslewed gravity residual signals from the French and Canadian superconducting gravimeters is given in Fig. 1. A station-dependent slew function exceeding the 3~r threshold (0.216 /xgal h - l and 0.282/zgal h -1 for CSGI and J9, respectively), where or is the standard deviation of the uncorrected residuals, was used to correct the raw hourly gravity values before a new least- squares tidal fit was performed with ETERNA leading to the residuals shown on Fig. 1. The corresponding power spectral densities (PSD) are given by Fig. 2 from 0 to 0.3 cycles per hour (c.p.h.) using an l l -point truncated Parzen win-

186 J. Hinderer et al. / Physics of the Earth and Planetary Interiors 90 (1995) 183-195

dow to smooth the raw spectra; frequencies between 0.3 and 0.5 c.p.h. (our Nyquist frequency for hourly samples) are not shown because their amplitude is reduced due to the decimation filter used in the preprocessing. The residual gravity spectra decreases in amplitude with increasing frequency and are close to a brownian noise ( f -2 shape) as first shown by Jensen et al. (1992a) and argued more extensively in Jensen et al. (1995).

The statistics of the PSD are indicated in Table 1. The stations have comparable signal levels though the CSGI level is lower than that of J9 except in the semi-diurnal tidal band (band 5) probably because of larger oceanic loading effects. The RMS normalised amplitude (/zgal) gives the true amplitude of a purely harmonic

signal The mean PSD (/zgal 2 c.p.h. -1) is the arithmetic mean of the PSD values per band. The integrated PSD is the power (~gal 2) integrated for each frequency band (to get the total power present in the signal, one has to multiply by a factor 2 because the PSD is symmetric for negative frequencies). Finally, the equivalent time standard deviation (SD) (/xgal) corresponds to a band-limited white-noise signal, the PSD of which would be constant in the selected frequency band: for such a signal, the power is simply SD 2. Typi- cally, in the subtidal band 10 (0.17-0.33 c.p.h.), we have a normalised amplitude of the order of 1 ngal (0.68 and 0.44 for J9 and CSGI, respectively) and standard deviations of about some tens of nanogal (45 and 29 for J9 and CSGI, respectively).

T a b l e 1

S t a t i s t i c s o n t h e s p e c t r a l c o n t e n t o f t h e s l e w - c o r r e c t e d r e s i d u a l g r a v i t y s i g n a l s in C a n t l e y ( C S G I ) a n d

c o m m o n 2 y e a r o b s e r v i n g p e r i o d

S t r a s b o u r g ( J 9 ) f r o m a

B a n d W i d t h r .m . s , n o r m a l i s e d M e a n P S D I n t e g r a t e d E q u i v a l e n t t i m e

# n o . ( cyc le h - i ) a m p l i t u d e P S D s t a n d a r d d e v i a t i o n

C a n t l e y

1 0 .000 - 0 .014 0 . 9 3 2 1 5 E - 01 38.255 0 .53644 6.1851

2 0 .014 - 0 .033 0 . 1 2 5 8 2 E - 01 0 .69337 0 . 1 3 1 8 3 E - 01 0 .83270

3 ( D ) 0 .033 - 0 .047 0 . 5 2 9 4 3 E - 02 0 .12277 0 . 1 7 1 9 7 E - 02 0 .35038

4 0 .047 - 0 .075 0 . 2 5 3 6 2 E - 02 0 . 2 8 1 7 4 E - 01 0 . 7 8 9 3 0 E - 03 0 .16785

5 (S) 0 .075 - 0 .089 0 . 4 4 0 9 0 E - 02 0 . 8 5 1 4 2 E - 01 0 . 1 1 9 3 9 E - 02 0 .29179

6 0 .089 - 0 .117 0 . 1 1 6 6 7 E - 02 0 . 5 9 6 2 0 E - 02 0 . 1 6 7 0 3 E - 03 0 . 7 7 2 1 5 E - 01

7 ( T ) 0 .117 - 0 .131 0 . 1 0 5 4 8 E - 02 0 . 4 8 7 2 8 E - 02 0 . 6 8 2 5 7 E - 04 0 . 6 9 8 0 6 E - 01

8 0.131 - 0 .158 0 . 7 5 8 9 8 E - 03 0 . 2 5 2 3 1 E - 02 0 . 6 8 1 8 3 E - 04 0 . 5 0 2 3 1 E - 01

9 ( Q ) 0 .158 - 0 .172 0 . 6 7 3 5 1 E - 03 0 . 1 9 8 6 8 E - 02 0 . 2 7 8 3 1 E - 04 0 . 4 4 5 7 4 E - 01

10 0 .172 - 0 .333 0 . 4 4 1 6 0 E - 03 0 . 8 5 4 1 4 E - 03 0 . 1 3 7 5 3 E - 03 0 . 2 9 2 2 6 E - 01

11 0 .333 - 0 .500 0 . 1 3 1 0 4 E - 03 0 . 7 5 2 1 4 E - 04 0 . 1 2 5 6 3 E - 04 0 . 8 6 7 2 6 E - 02

12 0 .000 - 0 .500 0 . 1 5 8 6 0 E - 01 1.1073 0 .55368 1.0523

S t r a s b o u r g

1 0 .000 - 0 .014 0 .16493 119.38 1.6740 10.926

2 0 .014 - 0 .033 0 . 1 5 3 4 2 E - 01 1 .0310 0 . 1 9 6 0 2 E - 01 1.0154

3 ( D ) 0 .033 - 0 .047 0 . 7 2 4 6 9 E - 02 0 .23003 0 . 3 2 2 2 2 E - 02 0 .47961

4 0 .047 - 0 .075 0 . 3 5 8 4 6 E - 02 0 . 5 6 2 7 9 E - 01 0 . 1 5 7 6 7 E - 02 0 .23723

5 (S ) 0 .075 - 0 .089 0 . 3 0 8 3 5 E - 02 0 . 4 1 6 4 4 E - 01 0 . 5 8 3 9 7 E - 03 0 .20407

6 0 .089 - 0 .117 0 . 1 6 6 1 8 E - 02 0 . 1 2 0 9 6 E - 01 0 . 3 3 8 8 8 E - 03 0 .10998

7 ( T ) 0 .117 - 0 .131 0 . 1 3 8 1 0 E - 02 0 . 8 3 5 3 4 E - 02 0 . 1 1 7 0 1 E - 03 0 . 9 1 3 9 7 E - 01

8 0.131 - 0 .158 0 . 1 1 2 1 0 E - 02 0 . 5 5 0 4 2 E - 02 0 . 1 4 8 7 4 E - 03 0 . 7 4 1 9 1 E - 01

9 ( Q ) 0 .158 - 0 .172 0 . 1 1 1 2 9 E - 02 0 . 5 4 2 4 6 E - 02 0 . 7 5 9 8 6 E - 04 0 . 7 3 6 5 2 E - 01

10 0 .172 - 0 .333 0 . 6 8 3 3 1 E - 03 0 . 2 0 4 5 0 E - 02 0 . 3 2 9 2 7 E - 03 0 . 4 5 2 2 2 E - 01

11 0 .333 - 0 .500 0 . 2 0 6 0 3 E - 03 0 . 1 8 5 9 2 E - 03 0 . 3 1 0 5 4 E - 04 0 . 1 3 6 3 6 E - 01

12 0 .000 - 0 .500 0 . 2 7 8 3 2 E - 01 3 .3995 1.6998 1.8438

T h e f r e q u e n c y b a n d s 3, 5, 7, 9 c o r r e s p o n d to t h e d i u r n a l , s e m i - d i u r n a l , t e r - d i u r n a l a n d q u a r t - d i u r n a l t i d a l b a n d s . T h e r .m . s .

n o r m a l i s e d a m p l i t u d e is e x p r e s s e d in m i c r o g a l s . T h e m e a n p o w e r s p e c t r a l d e n s i t y ( P S D ) is m i c r o g a l s s q u a r e d p e r cyc le p e r h o u r .

T h e p o w e r ( i n t e g r a t e d P S D ) is in m i c r o g a l s s q u a r e d . T h e e q u i v a l e n t t i m e s t a n d a r d d e v i a t i o n is in m i c r o g a l s .

Z Hinderer et al. / Physics o f the Earth and Planetary Interiors 90 (1995) 183 - 195 187

o 10 °

o_ 102 <

~ 1 0 ~

P a r z e n 11

10 ~ ~ _ - - ~ ~ ; 0.00 0.05 0.10 0.15 0.20 0.25 0.30

, - ~ - r r

~ 1 0 ° g ~ 1 0 .2

104 - ' " = . . . . . " ' t' f ' [~ r ' [ 'F !

Parzen 11

1 0 ~000 0.05 O. 10 0 , 1 5 0.20 0.25 0.30

Frequency (cycle/hour)

Fig. 2. Power spectral density estimates (tzgal 2) of the gravity

signals shown in Fig. 1. This spectrum has been smoothed

with an 1 l-point Parzen spectral window.

F o l l o w i n g a p r e v i o u s s t u d y w h i c h d e m o n -

s t r a t e d t h a t u n m o d e l l e d l o w - f r e q u e n c y c o n t r i b u -

t i o n s r a i s e t h e r e s i d u a l s i g n a l l e v e l s ( H i n d e r e r e t

a l . , 1994 ) , w e h a v e a l s o h i g h - p a s s f i l t e r e d t h e

o ~ _a E

g g_ lO ~

semi-diurnal

tlcles $3

High-pass filter attenuation

Parzen 11

1 0 5 . .

0.00 0,05 0.10 0.15 0,20 0.25 030 Frequency (cycle~hour)

F ig . 3. C r o s s - s p e c t r a l d e n s i t y e s t i m a t e s o f t h e s l e w - c o r r e c t e d

and high-pass filtered (cut-off period of 3 days) gravity data from Strasbourg and Cantley common 2 year observing pe-

s l e w - c o r r e c t e d h o u r l y g r a v i t y s i g n a l f o r C a n t l e y

a n d S t r a s b o u r g , p r i o r t o t i d a l f i t t i n g , u s i n g a f i l t e r

w i t h a c u t - o f f p e r i o d o f 3 d a y s . A c c o r d i n g t o E q .

(1) , w e h a v e t h e n c o m p u t e d t h e c r o s s - s p e c t r u m o f

t h e s e t w o d a t a s e t s a s s h o w n in F i g . 3. R e s i d u a l

t i d a l l i n e s ( e s p e c i a l l y i n t h e s e m i - d i u r n a l b a n d )

Table 2 Statistics of the cross-spectral estimates of slew-corrected and high-pass filtered gravity residual signals

Cantley for the 2 year common observing period between Strasbourg and

Band Width r.m,s, normalised Mean PSD Integrated Equivalent time

#no. (cycles h - I ) amplitude PSD standard deviation

l 0.000 - 0.014 0.25301E - 02 0.27462E - 01 0.38511E - 03 0.16572

2 0.014 - 0.033 0.60610E - 02 0.15759 0.29963E - 02 0.39699

3 (D) 0.033 - 0.047 0.39705E - 02 0.67630E - 01 0.94734E - 03 0.26006 4 0.047 - 0.075 0.19873E - 02 0.16942E - 01 0.47465 E - 03 0.13016

5 (S) 0.075 - 0.089 0.26899E - 02 0.31041E - 01 0.43528E - 03 0.17618

6 0.089 - 0.117 0.98292E - 03 0.41447E - 02 0.11611E - 03 0.64379E - 01 7 (T) 0.117 - 0.131 0.15336E - 02 0.10089E - 01 0.14133E - 03 0.10045

8 0.131 - 0.158 0.63646E - 03 0.17378E - 02 0.46962E - 04 0.41687E - 01

9 (Q) 0.158 - 0.172 0.65114E - 03 0.18189E - 02 0.25478E - 04 0,42648E - 01

l0 0.172 - 0.333 0,38854E - 03 0.64763E - 03 0.10428E - 03 0.25449E - 01 11 0.333 - 0.500 0.12487E - 03 0,66894E - 04 0.11173E - ()4 0,81790E - 02 12 0.000 - 0.500 0.16273E - 02 0.11361E - 01 0.56805E - 02 0.10659

188 J. Hinderer et al. / Physics of the Earth and Planetary Interiors 90 (1995) 183-195

and common harmonics of the solar tide S1 (see Haurwitz and Cowley, 1973) are enhanced with respect to the background noise; the low- frequency part on the left part of the spectrum has been filtered out as mentioned above. The statistics of the cross-spectrum are given in Table 2 and one can notice the decrease in noise levels in all frequency bands, especially in the non-tidal ones, with respect to unfiltered single station data (see Table 1).

3. Product spectra of SG data sets

In this section we investigate the spectral content of the subtidal frequency band for two residual gravity data sets: (1) the four European SG records used in paper I (two from Brussels (Bel- gium), one from Bad Homburg (Germany) and one from Strasbourg (France)) and (2) the Cant- ley and Strasbourg common 2 year SG records described in the former section.

The first set of SG records was processed by Smylie et al. (1993) and was the basis of the claim of observational identification of the Slichter triplet (see paper I). The power spectral density estimates for each individual record were obtained with the Welch method (Welch, 1967) by averaging the PSD from windowed overlapping time segments. Then taking the product of the (real) power spectral density estimates relative to each station yields the so-called product spectra. For the purpose of comparison of the two sets of data, we use a normalised form of the product spectra in order to keep true power spectral densities units (/zgal 2 c.p.h.-l) for any number m of individual spectra

PI2 . . . . . . ( k ) = [ S l ( k ) S 2 ( k ) . . . . . Sin(k)] 1/m (3 )

where Si(k)(i = 1, m) is a smoothed individual power spectral density. Notice that, in this method, phase information is lost contrary to the cross-spectral approach (see Eqs. (1) and (2)). The differences between a cross and product spectrum are shown in Fig. 4 for two synthetic data sets. We introduced a series of four harmonic signals (phase coherent) of amplitude 10

10 ° - - , ,

Parzen 37 PRODUCT SPECTRUM I

" 1=11[ . . . . . . . . . . . . 10 .2 ~ , ~ r r~p~,I~r~,.l~,~ m.,#,~.~ V.~.r,~¢.~. ° r~'r'm~rr"i~ i

104 l i

105 - • . 0.00 0.10

10 ° , , •

Parzen 37

o>, 102 o

< 1 0 3

E 1 0 4

1 , J , 1 J

0.20 0.30 0.40 0.50

CROSS SPECTRUM I

0.00 0.10 0.20 0.30 0.40 0.50 Frequency (cycle/hour)

Fig. 4. Product and cross-spectra relative to two synthetic data sets consisting of harmonic lines injected in random white noise.

ngal at frequencies 0.1, 0.2, 0.3 and 0.4 c.p.h, in one data set and similarly in the second one except for the harmonic at 0.4 c.p.h, which is missing; white random noise of standard deviation 0.1 /zgal was added to both records. The synthetic time sequences were then treated in the same way as the real data in the previous section. Both spectra were smoothed using a 37 point Parzen window. The three common harmonics (at 0.1, 0.2 and 0.3 c.p.h.) appear in both spectra with comparable amplitudes. The harmonic at 0.4 c.p.h, is marginally visible in the product spectrum but missing in the cross-spectrum; this is due to the incoherency of the phase information between the two time signals at 0.4 c.p.h. In another context, the importance of phase coherency in the search for weak harmonics in gapped gravity data relative to a single station has already been pointed out by Rydelek and Knopoff (1984). More interesting, the mean noise level in

J. Hinderer et a l./Physics o f the Earth and Planetary Interiors 90 (l 995) 183-195 1 8 9

10 ° - : - ~ . . . . . .

P R O D U C T S P E C T R U M

4 .015 h 3 .768 h 3 .582 h 5 10 ~ i

~0 ~ b ~

1 0 3 ! E

Parzen 37

01265 ~ 0 .276

. . . . ]

10 ; . 2 4 5 -" 0 .255

C R O S S S P E C T R U M

; lo ] x= i

I ~- 1 0 ~

o10 it "1, n!,,M,N,,rl , ' i

Parzen 37 i

104 , , 0 .245 0 .255 0 .265 0 ,275

F r e q u e n c y (cyc le /hour )

Fig. 5. Product and cross-spectra relative to the four Euro- pean SG stack analysed in paper I.

cross-spectral estimates is lower than in the product spectrum as expected from the incoherency of the random noise sequences. However, the scatter around the mean value is obviously larger in the cross-spectrum than in the product one.

We have computed the product spectrum for the four European SG records according to Eq. (3). Similarly, the cross-spectrum was estimated according to Eq. (2) with N = 27847, taking care to use a common time origin. The results are shown in Fig. 5 (this is the only figure in this paper where the data have been padded to 217 before Fourier transforming them to accommo- date the time interval of the four sets).

On one hand, the claimed triplet (4.015, 3.7677 and 3.5820 h) does coincide with weak peaks in the product spectrum with a PSD level close to 10 -2 /zgal 2 c.p.h. -L (almost identical to that quoted in paper I). This means that we have indeed redetected the same triplet in using a different spectral technique. This rises the ques- tion of the (geographic) origin of the signal pre-

sent in the combined data knowing that the mix- ing of better data with noisier ones does not improve in general detection capability (W. Zfirn, personal communication, 1994). Although the data sets used in the European SG stack have different noise levels, which themselves may vary in time, notice that none of the individual power spectra could reveal the triplet appearing in the product spectrum.

On the other hand, there is no evidence of such a triplet in the cross-spectrum although the mean noise level is reduced (as expected from the synthetic test shown in Fig. 4). This indicates that there is no coherent information associated with the claimed triplet in the four European records. One of the reasons is clearly the rather small time overlap: the longest common period bc- tween any three data sets is about 15 months and there is no overlap at all for the four sets (see Jensen et al., 1995). The high damping of the Slichter triplet, inferred by the bandwidth estimates in paper 1 (Q values around 100), naturally reduces even further any common phase information that might have been present in the Euro- pean stack. The fact that the entire spectral seg- ment in Fig. 5 is reduced in mean power when the cross-spectrum is used suggests that there is no coherency between the four European stations at these frequencies.

We have also compared the product spectra relative to the European stack and our homogeneous 2 year stack. The results, where an 11-point Parzen window is used to smooth the raw product spectra, are shown in Fig. 6. The product spectrum for the Cantley/Strasbourg SG records has a PSD level a factor of 10 lower than the Euro- pean stack (the mean PSD levels in the band 0.17-0.33 c.p.h, are 0.82 × 10 -~ and 0.51 × 10 2 /zgal 2 c.p.h. I, respectively) and, most impor- tantly, does not exhibit any significant peaks coincident with the triplet. A more heavily smoothed comparison of spectra is shown in Fig. 7 (these spectra are smoother than in Fig. 5 owing to the different padding). The results confirm those evi- dent in Fig. 6. The fact that the amplitude of the 6th harmonic of the solar day (S~,) is quite different in both spectra is merely the consequence of different estimates of pressure to gravity admit-

190 J. Hinderer et aL / Physics o f the Earth and Planetary Interiors 90 (1995) 183-195

10" E~Eurouropean SG stack

10 ~ : - - ,~ ~ . . . . . i 0.245 0.255 0,265 0.275 0.285

F ig . 6. Product spectra of SG data sets. The top curve is relative to the four European SG stack analysed in paper I

and the bottom curve to our 2 year homogeneous Cantley/Strasbourg stack.

i ' 2 ngal in one i 10 ° , Parzen 11 lo ngal in both / I

~ 10 z

.~ 10 ~ E

10 .6 [ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

10~ F Parzen 101 . . . . . . . . . .

~ 10 2

~ 10 4 E

10 s 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0,40 0,45 0.50

F ig . 8. Synthetic cross-spectra of undamped sinusoids injected in random white noise.

- i . . . . . .

European stack

4.015 h \ \ 3.768 h 3,582 h

Strasbourg/Canltey stack Parzen 37

10 s 0.245 0.255 0.265 0.275

F i g . 7. Same as in F i g . 6 but with more smoothing.

tances in the least squares fit which differ for each station. We have also verified that the triplet was not present in the product spectrum of the correcting functions used to deslew both records. The scatter in the four SG product spectrum is of course smaller than that for only two stations, owing to different degrees of freedom in each individual record (see, e.g. Jenkins and Watts, 1968). Besides, in the product spectral approach, there is no reduction in the average noise level with more smoothing, as for the cross-spectral estimates (see Fig. 8).

We therefore conclude from this first experi- ment that our product spectrum processing of the European data sets produces weak peaks at the frequencies and approximate signal levels as the claimed triplet found with the Welch method used in paper I. These peaks are clearly not an artefact of the details used to arrive at the product spectrum, but they disappear in a cross-spectral estimate of the same data sets. Moreover, the fact that with the same method the triplet does

Z Hinderer et al./Physics of the Earth and Planetary Interiors 90 (1995) 183-195 191

not emerge from a noise background which is by one order of magnitude lower than in the Euro- pean stack proves, in our view, that the triplet is not present in our data.

4. Cross-spectral tests using synthetic data

We consider now a series of synthetic tests dealing with the detection of (damped) harmonic signals in gravity records.

The first test consists of injecting undamped (quality factor Q = oo) sinusoids in two different random white noise records having the same length as the data (2 years of hourly values, i.e. 17520 samples): (1) 10 ngal signal at 0.10 c.p.h, in one record and at 0.20 c.p.h, in both; (2) 5 ngal signal at 0.25 c.p.h, in one record and at 0.30 c.p.h, in both; (3) 2 ngal signal at 0.35 c.p.h, in one record and at 0.40 c.p.h, in both.

The standard deviations of the white noise were taken identical to the residual noise (assumed to be band-limited white) observed in the subtidal frequency band (0.17-0.33 c.p.h.) in Cantley and Strasbourg, 0.026 /xgal and 0.042 /xgal, respectively. The cross-power spectral estimates are shown in Fig. 8 for two different amounts of smoothing. Note that the average noise level decreases with increasing smoothing because of the incoherency of the noise content in the individual records and this is a clear advan- tage of cross-spectra over product spectra. The three common harmonics are seen above the background noise. The 10 and 5 ngal signals which are present in only one of the two records are still visible but with lower amplitudes as expected. There is of course no strict proportional- ity between the amplitudes of these spectral lines because of the randomness of the noise in both records. However the 2 ngal signal which is injected in only one record cannot be distinguished from the noise. Other tests confirmed that a 1 ngal signal injected even in both records would not be visible (to an acceptable level of confi- dence like 95%) in the cross-spectrum. This is not in contradiction with the assumption of the nanogal instrumental sensitivity we mentioned previously: in fact, we simply think that the sensitivity

(ability of the instrument to detect weak gravity changes) is better than the noise levels we ob- serve in our SG records which come essentially from environmental contributions (e.g. from barometric changes). Obviously it is impossible to distinguish between instrumental and physical noise without running in parallel at the same station several instruments.

This test shows the limitation of detecting pure harmonic lines in a 2 year record of gravity data with white noise characteristics close to the observed ones in the subtidal band. Recent theoretical computations (Crossley et al., 1991; Hinderer and Crossley, 1993) have shown that the expected level to be reached by core modes excited even by large earthquakes is most of the time below the nanogal level at the Earth's surface. There are two ways to enhance the signal-to-noise ratio in order to detect undamped harmonic signal of 1 ngal or less: either increasing the length of common observations (the signal-to-noise ratio in PSD increases, for an individual record, proportional to T, where T is the duration) or increasing the number of stations with incoherent background noise contents.

The second test more specifically deals with the search of the Slichter triplet in SG gravity data and is a generalization of the previous one: we assume now that the sinusoids forming the Slichter triplet are damped (finite Q values) and injected in brown noise ( f 2 shape in PSD), this frequency dependence being closer (than that of white noise) to typical gravity residuals (sec Jensen et al., 1992a, 1995). In particular, we simulate a triplet having the same periods T and quality factors Q as observed in the stack of the four European SG records (see paper l) and scaled in amplitude to the observed value (= 6 ngal). Fig. 9 shows the noise-free triplet excited differently in each channel by a Poisson distribution of random excitations. Notice that the Pois- son distribution of the impulses in time is uniquc for both stations (see Fig. 10); what is changing is the amplitude (positive or negative) of the excitation impulses being taken from two different random Gaussian distributions. We therefore simulate a model of excitation processes happening at the same time but inducing surface gravity pat-

192 J. Hinderer et aL / Physics of the Earth and Planetary Interiors 90 (1995) 183-195

tern different in amplitude for each station (phase changes are either 0 or 7r in this model); it is to some extent similar to earthquake excitation where the Slichter triplet is excited differently depending on the the source model and the geographic location of the station (Crossley, 1988). However, we have not investigated here the detailed case of an earthquake with a specific source function, where each individual excited singlet (for different azimutal number m = 0 or m = + 1), has a well-defined theoretical amplitude and phase at a given location. In the latter case, a very powerful technique which can be applied is then the stacking method proposed by Cummins et al. (1991) and based on a spherical harmonic decomposition of the gravity changes.

The standard deviation of the additive brown noise (supposed to be incoherent between the two channels) was also scaled (tr = 0.04/xgal) to match the observed noise in our 2 year SG stack. Fig. 10 shows the time fluctuations of the two

to be injected into Strasbourg hourly gravity

Poisson distribution of random excitations

-0,20 . . . . . 0.0 5000.0 10000.0 15000.0

0.20 . - -

to be injected into Cantley hourly gravity

"0"200.0 5000.0 10000.0 15000.0 Time (hour)

Fig. 9. T ime f luc tuat ions o f a s imulated t r ip le t o f sinusoids

with period and damping similar to the ones reported in paper I and assumed to be randomly excited by a Poisson distribution of impulses.

channel 1 ~/'%

V 10 I random walk ,~ /

-lO 0.0

5000.0 10000.0 150oo.0

channel2

Poisson distribution of excitation impulses

-10 0.0 5000.0 10000,0 15000.0

Time (hour)

Fig. 10. T ime f luctuat ions o f two synthetic b rown noise chan-

nels where the simulated triplet of Fig. 9 was injected. A Poisson distribution (unique for both channels) provides the series of vertical spikes with random amplitudes.

synthetic channels essentially describing two different random walks, the common injected triplet being too small to be visible on the same scale. Also shown is the Poisson distribution of the time locations of the impulses having random (negative or positive) amplitudes assumed to be re- sponsible for the excitation of the triplet. The cross power spectral density estimates of these two channels are given by the top curve of Fig. 11 for the full frequency band (from 0 to 0.5 c.p.h.). The bottom curve is a magnification in the subtidal band where the triplet is located. Both curves allow us to distinguish the triplet from the ambient noise with signal-to-noise ratio higher than that in paper I (see Fig. 1). In our view, this proves that a triplet of damped sinusoids with characteristics similar to the claim of paper I could be seen in this synthetic test closely simu- lating our 2 year SG cross-power results. The Q values we introduced came from paper I and

J. Hinderer et al. / Physics of the Earth and Planetary Interiors 90 (1995) 183-195 193

were quite low, much lower than expected from either estimated viscous or magnetic damping in the core (Crossley and Smylie, 1975; Aldridge and Lumb, 1987) or possible anelasticity (Cross- ley et al., 1991). For anelasticity, the Q values are O(103) giving a damping time exceeding 1 year. Injecting the previous signals with increased Q values by 10, shows (Fig. 12) a much higher level than in Fig. 10.

In a last synthetic test, the simulated triplet of damped sinusoids (same T, Q values as in paper I, noise free) mentioned above and depicted by Fig. 9 is injected in real data. The cross-spectrum of the pure (noise-free) triplet is given by the top curve of Fig. 13. We injected the simulated triplet into the (slew-corrected and high-pass filtered) hourly raw SG data from Cantley and Strasbourg prior to the complete processing (least squares fit to tides, drift and pressure) and performed a cross-spectral density estimate of the resulting gravity residuals (bottom curve of Fig. 13). There is clear evidence of the injected signal well above

10 o , .

Parzen 11 simulated Slichter triplet

(T,IOQ)

.c: -~ tO ~

0 0.20 0.22 0 .24 0.26 0.28

Frequency (cycte/hour)

Fig. 12. Cross-spectrum in the same subtidal band as in Fig. 11 but assuming that the damping of the triplet is weaker by one order of magni tude (Q values larger by a factor 10).

. . . . . . I 1 /

10 4 • t Parzen 11

° t O ~ t Iocation~, of Slichter . . . . . triplet t

10 6 0.00 0.10 0.20 0.30 0.40 0.50

10 ° . . . . ~ . . . . ~ . . . . r . . . .

10' Magnification of upper box

~ 10 3

b : 10"

10 -s . . . . . . . . . . . . . . . ~ . . . .

0 . 2 0 0.23 0.25 0.28 0.30 Frequency (cycle/hour)

Fig. 11. Cross-spectra of the time signals shown in Fig. 10. The bot tom curve is a magnification of the top in the subtidal frequency band where the triplet is located.

the ambient noise level which is observed in our two SG data sets. We injected also the triplet directly into the post fitted gravity residuals and obtained similar results, demonstrating that the tidal analysis code does not absorb signals outside the tidal frequency bands. As already said, our model of random excitations of damped harmonics does not allow the use of the possible phase coherency information between the two records which helps a lot to improve signal-to-noise ratio in a smoothed cross-spectrum (see Fig. 4). This also means that we underestimate somehow the detection capability of two stations, and, by ex- trapolation, of a network of SG stations.

These tests establish once again that the triplet of damped sinusoids claimed in paper I would be detectable in our 2 year stack but is nowhere visible. Other attempts to confirm the Slichter triplet in the French/Canadian SG records with different spectral analysis methods have come actually to the same conclusion (Jensen et al., 1992b, 1995). Of course, one can argue that,

194 J. Hinderer et aL / Physics of the Earth and Planetary Interiors 90 (1995) 183-195

t 0 ° , - - - r - - ~

I noise free case simulated Slichter triplet

0 o . . . . . .

~ . . . . , ,hi , ~ 1 0 4 ,

o • ~ 10 6

I 108 !

0.20 0.22 0.24 0.26 0.28 0.30

1 0 ° !

after injection in CSGI and J9 gravity data

108 0.20

L .1 . . . . . . ± . . . .

0.22 0.24 0.26 0.28 0.30 Frequency (cycle/hour)

Fig. 13. Cross-spectrum of the noise-free simulated triplet shown in Fig. 9 (top curve) and after injection in Cantley and Strasbourg hourly data before least-squares fit to tides, local pressure and instrumental drift (bottom curve).

because of the extremely small overlap in time of the 2 year Cantley/Strasbourg stack and the data sets used in paper I, the triplet was excited in a very different way and this may explain the ab- sence of signal in our spectra.

5. Conclusion

We have confirmed the existence of weak spectral peaks in the product spectrum of residual gravity from the stack of four European SG data sets, coincident with the claimed Slichter triplet of Smylie (1992). These peaks are therefore not an artefact of a particular spectral analysis method. Unfortunately, these peaks do not appear in the corresponding cross-spectrum which should in principle enhance common harmonic signals, although the time overlap is quite limited and the harmonics may well be highly damped as

suggested by Smylie (1992). We have shown detailed residual gravity spectra from a common 2 year observing period of the French and Cana- dian superconducting gravimeters. These data sets were processed in an identical way from the raw 2 s sampling rate to the final cross-spectral density estimates. The spectral peaks mentioned above are not visible in the less noisy product and cross-spectra from the homogeneous Strasbourg/ Cantley SG stack. Moreover, all synthetic tests (pure sinusoids in white noise, damped sinusoids in brown noise or in observed SG residuals) show that a simulated Slichter triplet would be easily detectable in our cross-spectrum. Remembering that the detectability limit of surface gravity effects related to more general core modes is indeed 1 ngal, we also conclude that a clear identification of such effects is a very difficult task even in a stack of long high-quality SG data sets. We therefore strongly support the idea of having an observing period of 6 years of a significant number of world-wide distributed SG, as suggested by the GGP (Global Geodynamics Project), in order to make sensible progress in this field of research.

Acknowledgements

D. Crossley wishes to acknowledge funding through Canadian NSERC Operating and Infras- tructure Grants. J. Hinderer would like to thank NSERC for providing an International Research Fellowship during his sabbatical at McGill Uni- versity. This study has also been supported by INSU-CNRS (France). We kindly thank W. Zi.irn for his thorough and helpful review.

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A search for the Slichter triplet in superconducting gravimeter data

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