Wavelet method determination of long period tidal waves and polar motion in

superconducting gravity data

X.-G.. Hu1,2,*, L.T. Liu1, Ducarme. B.3, H.T. Hsu1, H.-P. Sun1

1Institute of Geodesy and Geophysics, Chinese Academy of Science, 340 Xu-Dong Road, Wuhan, China

2Graduate School of Chinese Academy of Sciences, Beijing, China3Royal Observatory of Belgium, Brussels

SUMMARY• Introduction• Wavelet filtering• Local atmospheric pressure effects on observation of the

LP tidal waves • Separation of eight long period tidal groups at periods

4~40 days • Separation of LP wave Ssa • Polar motion effect

– Fitting cosine functions to the observed pole tide – Fitting theoretical pole tide in the time domain– Fitting theoretical pole tide in the frequency domain

• Discussion and Conclusions

Introduction

• 5060-day uninterrupted gravity records, extending from 15, 11, 1986 up to 22, 9, 2000, from SG GWR T003 at the Royal Observatory of Belgium have been until now the longest continuous tidal gravity observations.

• We select 4800-day stable records for the study of pole tide. The initial phases for EOP and gravity observations are referred to 12, 03, 1987, 00.00 h UTC as initial epoch and 1, 05, 2000, 00.00 h UTC as final epoch.

Wavelet filtering

• A discrete signal fN(t) can be expressed as a wavelet expansion by orthonormal basis functions derived from two closely related functions: scaling function (t) and wavelet function (t).

where J, j and n are integer indices, N is the length of the discrete signal and suppose N=2M, 0<J< M.

• The above expansion means that scaling and translating (t) and (t) lead to an orthogonal basis for the analysis of signal.

2 22 2

0 1 0

1 1

2 (2 ) 2 (2 )( ) ( ) ( )J jN NJ

J J j j

J jn j n

N a t n d t nn nf t

Wavelet filtering

• The scaling function (t) and wavelet function (t) used to construct the orthonormal basis are derived from

• h(k), g(k) are called scaling coefficients and wavelets coefficients respectively.

( ) ( ) (2 )

( ) ( ) ( )

2

2 2

k

k

h k k

g k

t t

t t k

Wavelet filtering

• The 3-level wavelet analysis filter bank decomposes an input signal f(t) within Nyquist frequency band [0, F] into eight sub-band signals.

• Synthesis filter banks reconstruct the input signal only with w3,3, thus the output signal, which has the same length as the input signal, is located in sub-band [3F/23, 4F/23]

Wavelet filtering

Table 1. Sub-bands for the analysis of 12 LP tidal groups

p wavelet packet filter bank with a split factor of 3 or 2.

Tidal-groups Period interval

(days) Sampling interval (h)

Level of filtering bank

Sub-band (days)

Sa 291.96~411.79 24 8 256~512

Ssa 169 ~ 212.4 21 8(3)p 148.15~224

Sta 91.31~131.67 28 6 74.67~149.33

MSm, Mm 22.40 ~38.74 15 5 20~ 40

MSf, Mf 12.38 ~16.63 8 5 10.67~21.33

MStm, Mtm, 8.67~10.37 9 4(3)p 8~12

MSqm, MQM 6.64~7.66 9 4(2)p 6~8

OX3 5.48~5.90 19 3(3)p 4.22~6.33

Wavelet filtering

• The moment condition of a Daubechies wavelet can be expressed in terms of the scaling coefficients as

• It indicates that a wavelet filter has L/2 vanishing moments, which means wavelet filter can suppresses parts of the signal which are polynomial up to degree L/2-1.

• instrument drift in the signal can be suppressed thoroughly, whether in long-term or in short-term, harmonic components are smoothly represented.

1

0

( 1) ( ) 0 0,1,2,..., 2 1L

k m

k

h k k m L

Wavelet filtering

• The wavelet bandpass filter is able to filter LP tidal groups into very narrow frequency bands with good frequency response and without phase shift and Gibbs-like phenomena.

• Moreover, it can thoroughly suppress instrumental drift in SG records.

Example of observation equations

• tidal group MStm, Mtm can be simply modeled as the combination of a set of cosine functions and atmospheric noise

• There are 27 tidal constituents in the MStm group and 24 constituents in the Mtm one.

• C is the unknown pressure regression coefficient

27 24

1 1 1 1 1 2 2 2 2 21 1

( ) cos(2 ) cos(2 ) ( ) ri i i i i ii i

g t A f t A f t C P t

Local atmospheric pressure effects

• Pressure admittance for LP tidal bands

• We calculated frequency-domain admittances in these bands using the method proposed by Crossley et al. (1995).

Tidal-group Sub-band

(days) Admittance (nms-2hPa-1)

Correlation coefficient

Sa 291.96~411.8 -

Ssa 148.15~224 -1.892±0.047 0.602

Sta 74.67~149.33 -2.497±0.045 0.655

MSm, Mm 20~ 40 -3.178±0.013 0.950

MSf, Mf 10.67~21.33 -3.082±0.007 0.971

MStm, Mtm 8~12 -3.189±0.009 0.979

MSqm, MOM 6~8 -3.191±0.007 0.985

OX3 4.22~6.33 -3.279±0.009 0.980

Local atmospheric pressure effects

Separation of tidal groups at periods 4~40 days

Separation of tidal groups at periods 4~40 days

Separation of tidal groups at periods 4~40 days

Table 2. Tidal parameter for 9 tidal groups, obtained by wavelet method and program VAV; data from station Brussels (08.07.1987~29.08.2000)

Wavelet method VAV Tidal group Amplitude

factor Phase

difference (o)

Amplitude factor

Phase difference (o)

MSm 0.9022±0.0059 -1.1663±0.3725 1.12530±0.09149 -4.301±4.656 Mm 1.1418±0.0029 0.9003±0.1466 1.13835±0.01167 -0.540±0.743

MSf 1.1745±0.0095 2.2700±0.4626 1.13595±0.05115 0.013±2.578 Mf 1.1398±0.0007 0.1528±0.0373 1.14109±0.00284 -0.194±0.192

MStm 1.2334±0.0134 -4.4757±0.6205 1.19236±0.06965 -3.252±3.653 Mtm 1.1871±0.0076 -2.7856±0.3682 1.17591±0.01333 -0.520±0.649

MSqm 1.1142±0.0221 2.4644±1.1370 1.06604±0.06500 9.458±3.493 MQM 1.1134±0.0270 1.1589±1.3886 1.13522±0.07069 0.832±3.568

OX3 1.1076±0.1009 -6.1156±5.2202

Shallow water tides

• If the tidal forcing contains two frequencies ω1 and ω2, compound harmonic constituents at frequency ω1-ω2 and ω1+ω2 will be created.

• M2 (1.932274-day period) and S2 (2-day period) induce strong sea level oscillations at period of MS4 (6.103 hours=1/(ωM2+ωS2)) and MSf (14.77 days=1/(ωS2–ωM2)). The amplitude of the compound tide MSf is about 2.53 cm in Ostend (Belgium) and its amplification factor is 17 with respect to the static tide.

• a shallow water tide MSm is induced by μ2 and N2 (31.81 days=1/(ωN2–ωμ2) ), with a amplitude of about 1.75cm and amplification factor of 10.

Separation of LP wave Ssa

Separation of LP wave Ssa• There are environmental noises of unknown origin at frequency

about 0.0059 cpd and 0.0063 cpd. • Observed Ssa is modulated with a period of about 4.5 year

• The fitting result gives δ=1.0257±0.0067,κ=-9.4445±0.3725 and C=-2.2169±0.0513 nms-2hPa-1. • The value of amplitude factor at the Ssa tide is considerably smaller

than the theoretical value 1.158 by DDW99.

1

2

14

1 11

2 2

( ) cos(2 ) cos(2 )

cos(2 ) ( )

i i ii

r

g t A f t j N f t j

N f t j CgP t

Polar motion effect

Polar motion effect

In station Brussels, local atmospheric pressure correction did not reduce the variance of the residuals in the gravity signal at frequencies below 2 cycles/year.

Polar motion effect

• Theoretically for a rigid Earth, the gravity variations induced by polar motion can be predicted by

• An eight-level undecimated filter bank derived from Daubechies wavelet of order 127 is utilized to filter the gravity and local atmospheric pressure signal into a very narrow subband 0.5/28 ~ 0.5/27 cpd, i.e. 0.001953 ~ 0.003906 cpd (period 256 to 512 day). Thus instrument drift and all long-period tides, except the annual term Sa, are removed.

• The theoretical pole tide is also filtered into this sub-band.

2 sin 2 [ ( ) cos ( )sin ]rigidp R x t y t

Fitting cosine functions to the observed pole tide

• The spectral analysis of 4800-day predicted pole tide suggests a 435 days Chandler period and an annual period at 365.25 days.

• The SG records in the period band 256~512 day can be simply modeled as sum of three cosine functions (1=Chandler, 2=annual, 3=9 months)

• Similarly the theoretical pole tide is modeled as

= A1/B1 =a1-b1

3

1

cos( )i i ii

g A t a

2

1

cos( )i i ii

p B t b

Fitting cosine functions to the observed pole tide

• Our estimate yields the results δ=1.1681±0.0095, κ=-8.3607o±0.8484. The negative value of κ implies a phase advance between the observed pole tide and the theoretically predicted pole tide.

• With atmospheric pressure correction

δ=1.1784±0.0092, κ= -9.9873o±0.8266 and C= -2.842 nms-2 hPa-1.

Fitting cosine functions to the observed pole tide

• A tiny change of the selected Chandler frequency for harmonic analysis will cause a significant change of amplitude factor δ.– For example, at the period of 433 days

δ=1.1578±0.0095, κ=-7.9280o±0.9064; – at the period of 438 days

δ=1.1735±0.0092, κ= -8.5367o±0.8422.

Fitting theoretical pole tide in the time domain

• Since the Earth response to the long-period tide is not strongly frequency dependent, we may accept that amplitude factor δ at the Chandler period is equal to that at annual period

• The parameters A1, A2, b1, b2 and amplitude factor δ are estimated by fitting g(t) to p(t).

• The optimum time lag CH is determined by experimenting different values of CH in adjustment of A1, A2, b1, b2 and δ

1 1 1 2 2 2( ) ( ) cos( ) cos( ) g t p t t A t b A t b

Fitting theoretical pole tide in the time domain

At the optimum value of -3 days, this method yields δ=1.1638±0.0090.

Fitting theoretical pole tide in the frequency domain

• Usually the gravimetric factor is defined in the frequency domain as:

• where f represents frequency, G(f) and P(f) are the Fourier transform of observed pole tide g(t) and theoretical pole tide p(t),

• We try to minimize |G(f)|2 withover a narrow frequency range

f=1/454~1/418 cpd

( )( )

( )

G ff

P f

( ) if e

G(f)= G(f)–δ(f)P(f)

Fitting theoretical pole tide in the frequency domain

• Using a Hanning window, we obtain δ=1.1631±0.0045, κ= -2.2512o±0.1352.

• Note that this value of amplitude factor δ is very close to the value which we just directly obtained in the time domain, and the phase difference κ is near to -2 days.

Comparison of Results

Results from different methods

: amplitude factor, κ(o): phase difference (lag positive) Method δ κ(o)

a) this study Fitting at a Chandler period of 435 days Local pressure admittance -2.84 nms-2/hPa

1.1784±0.0092 -9.987±0.827

Fitting at a Chandler period of 435 days* 1.1681±0.0095 -8.361 ±0.848

Fitting in the time domain* 1.1638±0.0090 about -2.52 Fitting in the frequency domain* 1.1644±0.0045 -2.187±0.220 b) previous studies

Xu et al., 2004§ 1.1848±0.0504 -6.362.43

Harnisch and Harnisch, 2005§ 1.180.26 -8.311.0

Ducarme et al., 2005a¶ 1.1810±0.0083 about -3.36

* no local pressure correction

¶ fitting in the time domain

§fitting at a fixed Chandler period

CONCLUSIONS

• Wavelet filtering is used successfully to determine LP waves and polar motion amplitude factors and phase differences.

• Local pressure correction is reducing drastically the residues for periods up to 60 days.

• Shallow water tides loading effect is clearly seen at Msf frequency in Brussels.

• The determination of the pole tide amplitude factor depends strongly on the atmospheric correction model.

• If local pressure correction is applied the results are in good agreement with previous determinations.