Comparison between the theoretical and observed tidal gravimetric factors

192

Physics of the Earth and Planetary Interiors, 49(1987)192—212 Elsevier Science Publishers B.V., Amsterdam — Printed in The Netherlands

Comparison between the theoretical and observed tidal gravimetric factors V. Dehant Chargée de Recherche du Fonds National Beige de Ia Recherche Scientifique, Institut d’Astronomie et de Géophysique G. LemaItre, Université Catholique de Louvain, 2, Chemin du Cyclotron B]348 Louvain-La-Neuve (Belgium)

B. Ducarme Chercheur Qualifié du Fonds National Beige de Ia Recherche Scienrifique, Observatoire Royal de Belgique, 3, avenue Circulaire, B1180 Bruxelles (Belgium) (Received November 12, 1986; revision accepted January 29, 1987)

Dehant, V. and Ducarme, B., 1987. Comparison between the theoretical and observed tidal gravimetric factors. Phys. Earth Planet. Inter., 49: 192—212. At present there is a discrepancy of — 1.5% between the Wahr’s gravimetric factor and the observed one. The aim of this paper is to explain a part of this discrepancy. To compare correctly the theory and the observations, it is necessary to adopt a different definition of the tidal gravimetric factor 6. A new theoretical value is then computed at each tidal frequency, for an elliptical uniformly rotating Earth model. From the ICET data bank, we deduce an experimental model for five tidal waves by regression between all the observations currently available and the function of the latitude which enter in the theoretical form of 6. The latitude dependent part fits the theory and the discrepancy on the constant part is reduced by a factor of two. The remaining gap is 0.5% for 01, 0.4% for P 1, 0.8% for K1 and 0.8% for M2.

I. Introduction According to the conventions used for tidal data analysis, we adopt here the following defimtion of the gravimetric factor 8: ‘In the frequency domain, the tidal gravimetric factor is the transfer function between the tidal force exerted along the vertical and the tidal gravity changes measured by a gravimeter.’ We assume that the local vertical coincides with the normal to the ellipsoid. 8 is a frequency dependent coefficient which can be deduced from the observations at each tidal frequency. On the other hand, the surface displacement field and the eulerian potential can be calculated in the frequency domain. Their numerical values are computed by using seismic profiles 0031-9201/87/$03.50

C) 1987 Elsevier Science Publishers B.V.

of the rheological parameters (Lamé parameters or bulk and shear modulus) and by integrating the equations of motion together with the Poisson’s equation and the stress—strain relationship. From these numerical results, a theoretical gravimetric factor can be deduced for each frequency of the tidal external force. To compare the observed value of the tidal gravimetric factor with the computed one, a unique definition has to be used. Moreover, the observations must be corrected from instrumental, atmospheric and oceanic effects for comparison with the theory. Melchior (1981) compared the observed tidal gravimetric factors with the best existing model at that time, Wahr’s (1979) model. A discrepancy of

193

1.5% was found for the four main tidal waves: F1, K1 and M2 (Meichior and de Becker, 1983). In the present paper, the authors point out that the definition used by Wahr (cf. eq. 33) and that used by the International Center of Earth Tides (ICET) (cf. eq. 14) are different. They propose to adapt the theory, compare the new values and deduce a new difference which is smaller than the previous discrepancy by a factor of two. —

01,

2. General definition of the tidal gravimetric factor deduced from the observations The goal of any tidal measurement is to determine the response of the Earth to the tidal force F(t). In the physical system response, i.e. in the output of the gravimeter, a distinction must be made between what is due to the Earth and what is due to the instrument. The way to separate the Earth’s response from the gravimeter response is to use a modelling system for the instrument and to determine independently the amplitude as well as the phase of its transfer function. This operation is called calibration of the instrument. For most of the gravimeters, this calibration has to be checked regularly as the instrumental characteristics are changing with time, e.g. due to external perturbations. In the frequency domain, the transfer functions between the tidal force acting on the gravimeter (along the local vertical) and its response corrected for instrumental effects, is exactly the Earth transfer function. To compare it with the theory, the observations must be corrected for the atmospheric pressure effects and the oceanographic loading effects. It must be pointed out that the pressure principally affects S2 and 54 while the effects on M2, 01 and the other principal tides are lower than the instrumental precision except for the superconducting gravimeters and for few zero-method instruments such as the tidal La Coste (ET) gravimeters. Unfortunately, this correction is not always possible because the atmospheric pressure data are not always available in temporary stations. To compare these results with the Molodensky’s (1961) theoretical ones, we must correct them for

the inertial effect, i.e. the relative acceleration of the gravimeter which was not taken into account in this theory. This is not the case in Wahr’s (1979, 1981b) theory. The successive steps of the computations are (Melchoir, 1983): (1) correction of the atmospheric pressure effect (if the data are available); (2) harmonic analysis (Venedikov, 1966, at ICET); (3) correction for the transfer function of the gravimeter; (4) correction for the ocean effects using Schwiderski’s (1980) cotidal map. The results give the Earth’s response to the tidal force in the frequency domain, A0. If we divide A0 by the vertical tidal force at the same frequency, Ath, we obtain the transfer function of the Earth at this frequency w. In the frequency domain, the amplitude of the transfer function between the Earth response to the tidal force and the amplitude of this force along the vertical, defines the gravimetric tidal factor. It is usually referred to as 8(w). .

3. Theoretical computation of the tidal gravimetnc factor 3.1. Introduction of the variables in Wahr’s theory Wahr (1979, 1981a, b, c, 1982) evaluated the surface displacement field for an elliptical, urnformly rotating Earth using Smith’s (1974) equations. Smith and Wahr introduce the Earth’s ellipticity considering that the surfaces of equal rheological properties, the equipotential surfaces at the equilibrium state and the equal density surfaces, are ellipsoids. They choose a development in function of the ellipticity c using the Generalised Spherical Harmonic functions (GSH), D7’~(Phinney and Burridge, 1973) dp p = p0 + ~e r D~° (0) (1) d ~ = ~ + 4cr_L~D~0(0) (2) dr 2 dX0 20 X—X0+~r—--——D0 3 dr -~—~

—

194

dW W0 + ~er—~--~D020(0) D,~~(0,A*)

(4)

=

ment field, order (1, m) response reduces toto an external potential of ~+~÷

(9)

1+i11+a~2+~2 =

2’(l—n)! 1

(l)t+m

x(1

V

m,~1—2 and ~l±2 are spheroidal displacements and andand ~ Wahr are toroidal ones. in a where ff~ ~Smith Because are working

(l+m)!(1n)! 1(1— m)!(l+ n)!

~)(1/2X~~m)(

—

1 + ~—(1/2Xn±m)

x d,2~”((1

I+n

I—n

7”

)

(5)

(1 + ~a) e~’ where 0 is the colatitude, A” is the longitude and p. = cos 0. These GSH functions have the following useful property —

p.s)

D” ED” m’n

m”n”

=

I

(

~

~

x

~,,

, —

,

~‘

~

~

/

and by the tangential displacements 7—It

m)

(6) where appears the 3 is also noted II’+l” I

I

~

‘

fl

=

/=

—j

m’

i’—i”i

+ fl + m”

symbols of Wigner. This

00

~‘

1”

fl

12

m’

m’

I ,

+

+1

~

S7”(r)D,~(0, A*)e~

(7)

1=0 m=—1n—

where ê, are the three new base vectors defined by Smith (1974) (for further information see Dehant, 1986). The total eulenan potential is +1

00

vm = S,fll±+ s,” W m=S~—S m 1 1

(ii) (12)

3.2. Computation of the gravimetric factor

Smith (1974) introduced these relations in the vectorial equations for the Earth’s deformations and he developed the scalars, the vectorial and tensorial components using the GSH. The displacement i~,for example, is equal to ii =

(10)

=

\

~“

m

appear in the response of a 1= 2, m = 0 tidal potential. These effects have to be taken out to obtain the tides and their corresponding eulerian potential 47’, i.e. the potential due to the deformations. The deformations are represented by the radial displacements

F

(~‘~ +n

1

/

,,s

uniformly rotating frame, the nutations produce tidal potential in of the the response order I = of2, the m =Earth 1; thetoefdisplacements fects of the variations of the length of the daya

Let us compute the gravimetric factor corresponding to a fixed frequency, w, and a fixed order, 1 and m, of the external tidal potential

wfm

T,~n(r

) D,~,n(8,

A*)

(13)

The response of the Earth to the vertical component of the tidal force at the frequency w can be measured by a gravimeter. A tidal gravimeter is conceptually equivalent to a test mass falling from an Earth’s fixed point and kept continuously at this reference point by some antagonist force f. We can use the variations of f to measure the tidal gravity tides including the lunisolar attraction, the direct effects of the Earth’s surface displacements and the mass redistribution effects. By

~ ~7’(r) D,~,

V~=

0(8, A”)

(=0

m

(8)

~‘

In this way, Smith obtains an infinite set of scalar equations which are ordinary differential equations of the first order in d/ d r. He truncates at the first order in the ellipticity. It is easy to show (Dehant, 1985a, b, 1986) that the total displace-

direct effects of the Earth’s surface displacements, we mean the gradient of the gravitational potential along the deformation, the relative acceleration, the centrifugal and Coriolis forces. The gravimetric factor is defined in the present paper, as a proportionality coefficient between what is measured by the gravimeter and the ampli-

195

tude of the tidal force along the vertical, as a fixed frequency w. The effipsoidal normal is assumed to

‘~

+(

coincide with the local vertical. Let us write R7’(w) for what is measured at w, corresponding to an external potential defined by (13). The gravimetric factor 67’ is defined by m = 67’(w)F m(w) R7’(w) = 87’(w)vwf 1 (14)

2~ +

n

4

(

4 L’I /

T4,7

11 +~

0

mI

Im

“

x (4,7

2 —

0u,m

2%’~/ + —~,c 3r

2

kIm +101

+Iv,m

0

mI

~ +

(

0

4/ 3r

~,

—

1 I

0

m

(~7+ T~7) 11

2

1+2

—

+

0

nmi

2

II 0 0m~

0 rn

rn _4

i~I I

—

2

2

/

1 I

—

+1

0

in I

d

2

—

2

dr)~0H 0

—

I

2

i—il

-

+

0

mI

d

(47 + T,~rn)

in

1I

1

—

+

0

m

Vm

lW~~

1)

11—2 0 rn

11 02 o~m Oml

+

d4,7.2

dr

2 +goUi_2)D~1o

R~1(w)+ R7,(w)

(15) (16)

with T~7’ d4,7 Id + R7~(w)=~—~——dr

/

d Il0 20 r~._)go~ in 0 I ~

1

+Iv,m

rn

0

mI

—

+

0

m

—

~

—

/—1~

+

2 —g r 0Utm)D,~10

(17)

is proportional to the ellipticity (2) The gravimeter is on the ellipsoid at a distance r’ from the Earth’s centre. Using a Taylor R7~(w)

tm 011 U, m

_W2U,m_2W~O

i

m

0 m

11—2 0

=

m

1 —2wf~O m

+

d 1+ 2 r~_4&i 0 rn

—

2y~I 3r I~

+

\ rd ~)go

1—lI \ + IW,~1I

2

rn

—

1 I

—

—

2

~g

+ T4,7) +

1~1 xIIo

—

d

/

+1 d

—

1 +2 0

12 2 dr +

+

R7(w, r)

(d4,7’

.~

d4,7

The reduction of the observations is performed at the distance from the Earth’s centre corresponding to the position of the gravimeter, on the Earth’s surface. The theoretical value must then be computed on the ellipsoid. Let us develop all the terms (1) For a distance r from the Earth’s centre and for a fixed frequency, R7” is developed in the annexe. The results are at the first order in the ellipticity

—~

—~-,—

—

using the same notation as in the previous paragraph.

—

-~-

4~22U,m

development around the mean radius r, R has the form 2

d

R7’(r’, w)=R7’(r, w)—rcrT(R7’(r, w))D 00

\

(18)

IWI~1ID,~O

J

=R~,(w)+R7,(w)+R~1(w)

(19)

196

with

/ 21(1

RT 1(w)

=

d24,7

2

Tm

3r

~

—

1)

—

+ —r~

3

—

dr~

tm + ~ego—~—) DP~OD~O dU

If we enter (16?) and (20) in (19) and considering (21) and (22), we obtain the expression of the

4 —g r

(20)

Thebesecond the eulerian potential(3)can foundderivative from theof Poisson’s equation expressed by using the GSH (21) and the definition of a new variable (22) 2 2L’ 2 r r — +—~—4,7 dr 47TG~-p 1’7 — 47TG~-(p 0 0U1”) (21) —

left-hand side of (13). The sum of (25), (26) and right-hand of (13). Working the firstinorder (28) gives side the tidal force whichatappears the of the ellipticity, we get

(R~(w) + R7,(w) + R~(w) F 1 07 ( w)

i

—

F~’(w)R~(w

F27(w)

F

This tidal0T(w) gravimetric ~F0T(W)~tH factor 8,~, depends on (29) the

—

d4,7 g, 4ITGp0Uf’ (22) (4) The tidal force along the normal must also be expressed at a distance r’ from the Earth’s centre in order to compare the computed gravimetric factor with the observed one. F~m(w)must be expressed on the ellipsoid =

—

F,m(r’ w)

=

Fm(r

w)

mn

Using the Associated Legendre polynomials through the relation

w))D~

~er~(Ftm(r,

—

latitude. It is convenient to group the terms of (29) according to the GSH function order 1~2 D’2 (30) D D’

=

21 +1 Y

(23) Expressing the external potential in terms of the

87’

GSH,wefind w)=F 0’7(w)+F1’7(w)

87’=8~+6~-—+&

~m(r

dT4,77

,

with F0’7(w)

=

F1T ( w)

0/ 4L’ (II rn

=

I 1+ 2 +

I

I

~

+10 I 1— 2

I

.L T4,TD!O

dr D,~,0=

I

—

02

+ in ~‘

—

2

/

—

+

0

~

r’)

=

Fdr(w)

I

m

m I ~)

with F~(w)=

+

-~

(32)

?flfl

The numerical values of each term are expressed for 01 an example. These results, part for the constant partasand the latitude dependent

(26)

Cartwright and Tayler’s expansion for the potential. And, that expansion is for the potential at the mean equatorial radius’ (J.M. Wahr, personal communication, 1986). When one compares this

(27) (28)

with the observed gravimetric factors as defined at the ICET, this correction is not necessary. Wahr (1981b) uses the term ‘apparent gravi-

F

1T(w) + F2T(w) 21(1 1) ET4,TD!OD~ 3r —

— ______

~i+2

that we are working with the force along the normal and Wahr is working with the radial derivative of the mean spherical tidal potential. However, the three terms of the Table I are due to the fact that Wahrlast ‘expected reader to use

~

We then obtain at r’ F,m(W

can also be noted

these tables, they are are compared Wahr’s reare presented suits. The differences in Tables I mainly and II, with due respectively. to the fact In

I

I

2 0

(25)

I

D,~0 2~r +1 / D,~

—

m

(24)

(31)

197 TABLE I Comparison between the constant part of Wahr’s gravimetric factor (column ‘Wahr’) and the values deduced by using the definition of the present paper (column ‘present paper’) Wahr a (G

Present paper (8~)

0)

I

1

=

1+1 d~I’~

~g0U,’”with g0

--~7 4L1I1 2 11 —-~-~o +1 —

4L’I~

rnI -~-

-~-

dr th) ~—r~—

/1 +Iv,m

2 —

Im

0 2

1

g0Jo

d\

mI ~‘1I + m

—

0

Iw,’~1 I

II

2 0

II olu,’” 1

rn

0

m

_~2Un~~

1 0 1

rn

—2w~2IO

—

2 00

I

2

II

0 0 /

rnI

0.00016

rn

x[

4~L1I1

2

11

d

1/

mI

2

___(s_r_~goIo

=0.00002=

~/~/ 2

drj dE

rn 1/

fii~

drj

rn

—(s—r—’~gøIo

=0.01969= =0.00090=

_w2Up~~

—0M0008 =0.00069=

4 d~ Il 2 +~(E—r—~goIO rn 00 ‘, _2O2UII~ dr,

=—0.00027=

_2w~Io

1

1~

ol

=

0.00034

=0.00032=

~

—

Il

2

Il

E~O 0

01 rnI

2 r211

/1

0

2 0 0

E—IO ‘Irn

Ii rn

2 0 0

+

—

0.00045

X[_~87’+_i_4I7’_41TGPo 2 L’2

=

0.00009

1’

4r’’0 02 011 go—~-— dU/” rnOrnI 4L~1~ 2 0 — +I

=

iI

rnI

—

nil

1 dp +

—

tm 0

p

0dr Ui’”

Missing

mI

—

ol

=

I L’V,’”

0.00058

/l

2 0 0

rn

=

/1

oIU,m mI

/~d2~T~

21/

rnI

/1 0 Ig0U/”

0

0 rnd~’I

=-~Io

xI

~I0

+ m

—

mI

r

+

mJ

i—il

+

0

Im

0.00062

0 2

m /—li

—

2(1—1)

—

+Iv,m

Missing

oI

1J ——~r+--~--~TI r 2 2 2L’

il

—

I

0 i

—2w1110 rn

=0.04556=

2

—

rn

mIth2II~Xt

2 0 0

~ +1

=

—~~~-Io 0I~~I 2I/

~

Ii

11 m 0 I4~~Gp0(I + 1)U1 rnI

2(/—1) / ~ E 10 rn 21/ 2 0 Im 0

11

2

Ii

—

mI

0

1/ Im0

4L’I~

Iw~

—

rn — ~-~

l~r + m 1—lI +

—

1/

—~

=

~

1/ —2w~IO

=0.00048=

dW0 —-a-—

V~1~rn~

I~7’

rn

4/

—~g0U,”with g0=

mI

21 d~\ Il _Is_r_)golO

(

=0.60453=

Il

2

I~

—

2

1+1

=—0.51643=

0.00048

2

11

4r 0 ~rI 0 0 mOrn Missing

dtJ/”

dU, +

I

198 TABLE I (continued)

Present paper (6~)

~

4L’(l+l)

8L’

~I o

Il

2 0 — ~ rn 0 2(1—1) Il + ~ ~o

—fI

rn +

iI I~’T

2

+

—

Irn

1~

i/~~

Wahr

o

0.00025

Missing

—0.00029

Missing

0.00032

Missing

=0.00019

Missing

=

mI

11

~goU,’” ml 2 11 0 0 0 mI +

4(1—1) 1 2 3/ c~o~ 0 0 rn 0

=

I

=

/lI1~’,’”

0

(G

0)

mI

2(l+1)(l-1) I! 2 1I 0 0 3! I~’ 0 Missing Missing

/1 I~’

-

0

=

—0.00016

Missing

mI — —

Missing

0.00222 0.00134

0.00115

=

—

=

=

______

1.1551 a

a

1.1522

The Wahr’s results presented in the second column of this table must be multiplied by 2// in order to obtain the same numerical values

as those presented in

his thesis (1979).

I ,nt(ext)

means evaluated just at the interior (exterior).

metric factor’ to define G. He defines

4

R7(w)=G~W~(a) r I

( G0+G =

~+2

+~~)We(a)

(33)

2g

where

We(a)=H~i0

0

is the frequency dependent tidal potential amplitude in metres observed at the equator, as given by Cartwnght and Tayler.

\~‘-_‘,/ tL

—a-ET1 w480 296 m551 29

= standard devCataon of the loadu,g crnputatson 10% of I

= ntaniar~ideviation of the obserOatCos for M 2 C

4. The tidal observations 4.1. The tidal data bank At the International Centre for Earth Tides (ICET), all available tidal gravity observations are collected and re-analyzed using the standard procedure defined above. One finally obtains the so

at the eqoator,

at 45’. O.S6gal total standard deviation 2 2 2

=

‘I.’

Tidal loading vector Observad tidal vector correctad tidal vector Model

1 ugal

6th

~5)

== K,, I 60L , I I = X Ii 0 Ath , 5) =~ ‘ A~ , a,,)

Ath

= vertical tidal force

with ~ with a,,,

- L - 0 forth,, elastic =~

Fig. 1. Vectorial representation for the comparison between the theoretical tidal gravimetnc factor (from the model), ôm, and the corrected gravimetric factor, 6.~,deduced from the observed one, 6~.

199 TABLE II Comparison between Wahr’s latitude dependent part of the gravimetric factor and the corresponding values defined in the present paper Present paper (&~) 2/DL Coefficient of D,~ i—i

Wahr ~ (G1) /—i

~7~I-2

=

d~ ~g0U,’”~2 with g0

dW

—~j—

4L1~12

2

/

0 2 — 0

o

rn

d dr 4

mI

2

Iv,m

—

+

Im

0

ml

1/—2 0 rn

2 — 0

i—il + Iw1~,

2

dc

/—2

2

rn

nil

2 d /—2 ——~-(e—r~--)go~ 0 rn

=

Iu,m

—2w~2lo

— —

I

11 +

0

nil

2

i—it + IW/~

0

ml

II

~

0

Iu,’”

r

—

rn

ml

Il

0 0 OI4wGp0(i+1)U,m rn 0 rn~ 2(/—1) 11—2 2 / ~ ~0 001 rn O,,il 247’ 21/—2 rn 02 rn~dr2 / d L 0 0 0 0 21/—2 2 iii 2 ,,, 21)2 ~l 0 0 + 0 rnl\ r —i--+7’)

=—~~-l

Missing 2(/—1) 1/—2 =

2

2

,.2I/—2

2

r21z’2

2

oI(— g,

I

/ d24’”~ 2 ~ m~dr 0

—~~-~-l m 0 o0

—

/

~ ~io ool rn 0,,iI

—

2

0 0 ml 2L’2 —j—~7’ — 4irGp

m

—

2 —g7’

r (1./vim

iI

ol

=~1c~7I0

x(

+

0 I dp0

dU/”\\

x r 1/—2 0 rn

+

~I

—

4r 1/—2 0 rn

j~~I

~l-2

0 rn

2 0 0

jI 0 Ig0U/” rnI 2 0 0 2 —

o

Il

~2

+

2(/—1) ~

2 —

0 /—2 0 rn

dU/”

-

*(1)

=

—

4,’ 1/—2 0 I rn

~I

Missing

2 o—

Il

+ lgoU,m rnl 2 0 0

p0dr

dr))

Missing

0 nil i~ + I rnI

411(1+1) l1/—2 0 ~7 m 81) 1/—2 —1 0 %~I2Im

vim

—

—

4 d 1—2 2 +~(—r~—)g010 0 rn 0 1/—2 1 i—il

=

I

—

rnI

0

—-~-(—r~--)g0~0

=

+ Iw,_17 nil

o

—

+I~7’

—

1~Irn 2

m~

o

rn 1/—2

0 2

to

il

11 +11 m / I

—

4~JJI~2 =

-a——0

—

2

10

~‘Irn

1/—2 0

/—2

g0 =

4LhI~2

2 it 0 0 Irn UrnI 1 1—lI

d

~g0U,’”2 with

=

+I~~

2 dc 1—2 —~--(c—r~—)g 0Io 2

—

I

+1 rnI / I

—

rn 4EL~~2

=

+/ l~7’ ml

Missing

Missing

11

ol

mI

Missing

2 0 0

11 dU,’” 0 lgo~~ ml

200 TABLE II (continued) Present paper (~~) +

—

4(/—1) ~ tg

Wahr a (G’)

/—2

2

11

0 rn

0 0

ml

01

0 Ui”

2(/+I)(l—1) 1/—2 El 0 rn 2/D,,’,

Missing

iI

2 0 0

~7’

0

Missing

rnl

Coefficient 1+3 of ~

/+3

=—0.00244=

~7~/+2

dW

~g0U,’”~2 with

g0

—0.00209=

0 4LhI~2 V~1~rn

— 2 0

nil

=—0.00072=

4L1~1+2 ‘/il ~

2

/1

—

0

~goujtm÷2 with g0 = -~j-—

+ I ml

4~L/~!+2 2 11 0 — +I~7’ rn 0 nil 2 dE 11+2 2 / I 0 — + Irn 0 nil

I~’”

—

11+2 0

th

2 0 0

/t olu,”

rn rnl 1+2 2 jI 0 0 0 I4~~Gp0(l + 1)u,’” ni 0 nil 2(/—1) 11+2 2 ‘I

I

~

—

—0.00031

=

‘1

4L1~1÷2 2 0 — rn 0

4

=

I

0 0 0 m OmI 2 !ld2~~l 2 lent 0 0 ol—! 21/+2 0 2 rnI Ildr 0 0 01 rn 0 rnj 2 2L’2 \

=

=

—

dW0 =

—

+1 iI

m 0 ml 2 dE 11+2 —(c—r—)g01 0 dr rn

—0.00007=

2 — 0

1i!

=

=

li+2 4 dc _~(c_r~_)go~ 0 rn

—0.00026=

0.00041

—0.00104

rnl

~l

+

=

2

3 0 0Ornl 0 lm ~l!~-2 2 iId24,71l rn 0 rnl dr2 l~~ 211+2 2 Il 0 0 rn 0 ml I 2 2L’2

o

oI—l

=—~7I

I x~__~r+_i_tII7)

ol

~

—0.00132 + —0.00149

=

(LIV

1 dp0

1m +

/+2 0

lm 11+2

4r

—

~7I 0

rn

4L1,1!+2 0 rn

r

(I

2 0 0

0 g0U,’” nil 2 0 0

11 0

—

81)

1+2 0

I =

rnl

=

0.000291_*( —1)

2 0 —

2 0 —

it + ~goL~hi” rnI

1

I

rnlI~7’ +

=

=

p0dr

dr

Missing

=

=

—0.00029 -

—

4r 1/+2 -~jc~~

rn

0.00209

Missing

—0.00108

Missing

0.00126

Missing

ml

0

I

)I~/2lm

—0.00189

dLJ,’”

~

ii + I

2

4L’ (1+1) 11+2 Cl rn0 ,/j j2

—CI

=

cUJIm))

Ui” +

—

2 0 0

iI d(J/” lgo~~

0

nil

IV/”

il olui”

Il

2

I

rnj

Missing 2(/—1) 11+2

=

2 0 0

1 +

201 TABLE II (continued)

Present paper (8~)

Wahr ~ (G~)

2(i—1~ 1+2 2 1 / C 0 0 0 mOm 4(1—P ‘-

—

/

g0 1+2 0

2 0

1 tm 0 U,

2(/+1)(/_i)~~2

=

0.00104

Missing

=

0.00063

Missing

—0.00054

Missing

=

0.0023 —0.0017

0.0087 —0.0065

—

—

a The Wahr’s results presented in the second column of this table must be multiplied by 2/1 in order to obtain the same numerical

values as those presented in his thesis (1979).

I

~

means evaluated just at the interior (exterior).

called ‘observed gravimetric factor’ 6~,and its phase difference a 0 (see Fig. 1). In addition, the data bank contains 6~factors and phase differences a0 resulting directly from other authors’ computations. For these data we must first check if the inertial correction has been applied before inserting them in the data bank. For all the data, an internal consistency factor is then computed according to some well defined criteria (Chueca et a!., 1985). We obtain the so called ‘corrected tidal factors’, ôc~by subtracting the oceanic attraction and loading effects (see Fig. 1). This is done for the nine main tidal waves (Qi, Oi~F1, K1, N2, M2, S2, K2 and M1) by using the Schwiderski (1980) cotidal maps, according to the Farrell (1972) algorithm. This corrected factor is independent of any assumption concerning the body tides reference model and can thus be used for general compariSons between models (Meichior and De Becker, 1983). The inertial effect introduces a frequency dependent bias that can be easily corrected afterwards if it is necessary. 4.2. Modelling To determine the latitude effect from the observations, we apply again the procedure described in Meichior (1981) and Melchior and De Becker (1983). Following these authors, we lin-

earize the problem by computing for each station, the value of the generalized spherical hannonics entering in equation (38) (see Table III). We then compute a linear regression under the form ~ 6 +6 ~ (34) =

0

+

with y

6 ~

an ,.

and x ~

=_

Y,~(9, X*) yrn (9 / = 2,

x*)

,

~ 1 diurnal waves I. 2 semi-diurnal waves For that purpose, it is necessary to apply selection criteria on the 266 stations now available in the data bank. Using first the two internal consistency factors Q1 and Q2, we reject 73 stations for which Q1 <1 and Q2 <4. On the other hand, we do not allow the corrected phase a~to exceed 10 as this indicates strong perturbations on the sine component of the corrected tidal amplitude A~ ÔCAth. After a first computation, we rejected very few stations for which the residue exceeds 2.6a (a = standard deviation), following a statistical critenon for a gaussian distribution of the residues. In practice, the rejected stations are quite the same for all waves. In our case, a is of the order of 0.01 for all waves. In Table IV, we show the experimental results for the main tidal waves. =

=

202 TABLE III Gravixnetric factor for 1066A model of the Earth’s Interior (Gilbert and Dziewonski, 1975). 0 is the colatitude Tides

Wahr’s definition

Present paper

Constant part

Latitude dependent part

Constant part

Latitude dependent part

J1 001 M2

1.1516 1.1520 1.1520 1.1486 1.1470 1.1437 1.1321 1.2347 1.1668 1.1545 1.1540 1.1599

_O.0063*(1) _O.0063*(1) _O.0063*(1) _O.0063*(1) _O.0063*(1) _O.0062*(1) _0.0062*(1) _0.0069*(1) _0.0064*(1) _O.0063*(i) _0.0063*(1) _0.0045*(2)

1.1547 1.1551 1.1551 1.1516 1.1500 1.1467 1.1348 1.2393 1.1702 1.1577 1.1571 1.1613

_O.0014*(1) _0.0014*(1) _O.0014*(1) _0.0015*(1) _O.0015*(1) _0.0016*(1) _O.0019*(1) _O.0000*(1) _O.O0i0*(1) _O.O013*(1) _0.0013*(1) _0.0009*(2)

M,

1154 .

M

1

Q1 0~ iT1

P1 S1 K1

‘P1

M3

—O.O069~(O,+) \f +0.O045~(O,—)

1.0707

(0, +)

-

48—3Ocos2O+3 3 ___________________ 35cos3cos2O—1 4~

11573 . 1.0730

_O.O034*(3)

f

‘~

—O.0016~(O,+) +O.O053~(0,—)

_O.001O*(3)

—

(0,_)=—~. (1)’_~:.(7cOs283) (2)

4i.(7

20 —1) cos (3).%.’.i.(9cos2O_1) =

TABLE IV Values of the constant part Tides

and the latitude dependent part 8+ of the gravimetric factor deduced from the observations 8~±mse

Criteria

132

1.1611±0.0009

—0.0026±0.0009

dmin=O,

1

53

K1

129 146 127

1.1545±0.0016 1.1440±0.0011 1.1697±0.0017 1.1707±0.0017

—0.0033±0.0017 —0.0034±0.0012 —0.0021±0.0007 —0.0024±0.0008

dmin=0, a~”=1° dmin=O, arc=1° 0 dmin=O, ar’=1° dmin=1O, arced

p

nb st

&.,

~±mse

arced0

M 130 1.1694±0.0017 —0.0017±0.0008 dmin = 0, a~”~” = 0.75° 114 1.1705±0.0018 — 0.0022 ±0.0008 dmin = 10, = 0.75° 92 1.1708 ±0.0020 — 0.0023 ±0.0010 dmin = 10, ar’ = 0.5° S2 105 1.1742±0.0020 —0.0019±0.0009 dmin=i0, a~=1° a The relative acceleration of the gravimeter (so called ‘inertial effect’) is included. In order to compare with the results of Melchior 2

ar’

and Dc Becker (1983), it is necessary to subtract 0.009 for 0~,0.0010 for P1 and K1, 0.0039 for M2 and 0.0041 for ~2• The observations are selected according to the criterion of the quality factors plus those reported in the last column of the table, ‘nb st’ = number of stations which have been considered; ‘mse’ = mean square error.

203

For the M2 wave, we made several computations with supplementary and stronger criteria: (1) minimum distance to the sea: stations with a distance to the sea (dmin) lower than 10 km must be rejected; (2) maximum phase difference: 2~ = 0.50 or 0.75° insteadstaof tionsmust with bea~rejected. I > a~°’ The rejection of coastal 10, stations systematically increases the latitude coefficient &~which stabilizes around 0.0023. It is —

obviously due to residual effects of the oceanic tides in the coastal stations. The influence of the maximum phase criterion is less visible, especially when dmin = 10 km. Of course high residual phase lags occur generally in coastal areas. The best results of S2 are obtained for a maximum phase equal to 10 and a dmin = 10 km. But they differ systematically for the equivalent results of Al2. This is probably due to atmospheric pres-

~1.18

a

5. Comparison For each tidal wave, we compute the theoretical values using Wahr’s definition or the definition proposed here (Table III). The experimental val-

~1.18

a a

1 14

sure effects which generally are not corrected. It is more difficult to obtain the latitude dependent coefficient for the diurnal waves because these waves vanish at low latitudes. However, as diurnalthere oceanic tides havetogenerally a low amplitude, is no need reject systematically coastal stations. It is thus not to be wondered that our maximum phase criteria suppress all stations with a latitude lower than 100 for P 1 and K1, and 60 for 01. Finally we obtain inside the diurnal band, very coherent values for 6~.

a

~~~~t=:_

i 14

LATITUDE

LATITUDE

>-

I.. -~

‘

w

r

w

5

1 14

~__~4o____~_===io=_~_8;_

14

LATITUDE Fig. 2. Results for 01. In the first plot, points = observed tidal gravimetric factors and in the second plot, points = means computed for each span of 10° from the observed tidal gravimetric factors; dashed line = curve deduced from the observed gravimetric factors by regression; fine line = curve of Wahr’s model; thick line = curve of the gravimetric factor as defined in the present paper, for an elliptical uniformly rotating Earth.

LATITUDE Fig. 3. Results for P1. In the first plot, points = observed tidal gravimetric factors and in the second plot, points = means computed for each span of 100 from the observed tidal gravimetric factors; dashed line = curve deduced from the observed gravimetric factors by regression; fine line = curve of Wahr’s model; thick line curve of the gravimetric factor as defined in the present paper, for an elliptical uniformly rotating Earth.

204

cu <1 18

I—

0

-

1 12

——

~0•

14

00

__s=oo~);

~

LATITUDE

>-

LATITUDE

>-

<1.18 I— W

a-

W

1 12

a,

1 14

LATITUDE Fig. 4. Results for K

1. In the first plot, points = observed tidal gravimetric factors and in the second plot, points = means computed for each span of 10° from the observed tidal gravimetric factors; dashed line = curve deduced from the observed gravimetric factors by regression; fine line curve of Wahr’s model; thick line curve of the gravimetric factor as defined in the present paper, for an elliptical uniformly rotating Earth.

ues of the tidal gravimetric factors deduced from the regression, are presented in Table IV. Finally we draw in Figs. 2, 3, 4 and 5 those three curves with the selected stations, respectively, for 01, F1, K1 and M2. The dispersion of the observed 6~

LATITUDE Fig. 5. Results for M2. In the first plot, points = observed tidal gravimetric factors and in the second plot, points = means computed for each span of 10° from the observed tidal gravimetric factors; dashed line = curve of Wahr’s model; thick line = curve of the gravimetric factor as defined in the present paper, for an elliptical uniformly rotating Earth.

factors is obviously large. Therefore, we present also in these plots mean values for each span of 100 in latitude. These means show clearly the latitude dependence. A systematic difference also appears clearly in the 6~term as has been previously pointed out (Melchior and De Becker, 1983). It should be

TABLE V Gravinietric factor deduced from the observations and from the theories Tides

01 P1 K1 M2

Observations

part

cstt part

lat dep

1.1611 1.1545 1.1440 1.1707

_O.0026*(1) _O.0033*(1) _O.0034*(1) _O.0024*(2)

Present paper def.

Wahr’s definition

cstt part

lat dep part

cstt part

lat dep part

1.1551 1.1500 1.1348 1.1613

_O.0014*(1) _O.0016*(1) _O.0019*(1) _0.0009*(1)

1.1520 1.1470 1.1320 1.1599

_O.0063*(1) _0.0063*(1) _O.0062*(1) _0.0045*(2)

The functions of the latitude dependent term (‘lat dep part’) are the same as in Table III. ‘cstt part’ gravimetric factor.

=

constant part of the tidal

205

noted that to compare Table IV with the results of these authors, it is necessary to subtract the so called ‘inertial effect’ (the values of this effect are given in Table IV). Previously, the disagreement was of the order of 1.5% as shown by the results of the superconducting gravimeter normalized on the Brussels fundamental system (Ducarme et a!., 1985). The new definition of 8 as well as the new reduction of the observational data, reduce the discrepancy by a factor of two. The remaining gap is 0.5% for 01, 0.4% for F1, 0.8% for K1 and 0.8% for M2. Preliminary results presented at the 10th International Symposium on Earth Tides (van Ruymbeke, 1985; Edge et a!., 1985) showed that the calibration of the Brussel’s fundamental station based only on Askama gravimeters (Ducarme, 1975) should be modified. According to these authors, 8(01) should be decreased by 0.5 to 1%. If this is confirmed, all the stations linked to the Trans-World Tidal Gravity Profiles (Melchior et a!., 1984) should be re-evaluated and this should produce a systematic decrease of the observed 6 factors in these stations. As they represent almost two-thirds of the available data at middle latitude and 90% of the high and low latitude data, 6~will decrease accordingly.

6. Conclusions According to the definition used for tidal data analysis, we adopt here the following definition of 8: ‘In the frequency domain, the tidal gravimetric factor is the transfer function between the tidal force along the vertical and the tidal gravity changes measured by a gravimeter.’ We assume that the local vertical coincides with the normal to the Earth’s ellipsoid. This definition differs significantly from what Wahr has assumed when he developed his theory. We have thus recomputed the theoretical values of 6. The constant term (8~) in the representation of 6 is only slightly increased while the latitude dependent term (6 + or 6_) is divided by a factor of two (Table III). Using the ICET data bank, we recompute 6~and 6~ by linear regression, from all the available observations. From the numerical results (Tables III, IV

and V), we may conclude that the new value of the coefficient of the latitude dependent part of 6, 6~, fits the observations very well. However, the cornputed constant part, 6~,is still lower than the one deduced from the observations but the discrepancy has been strongly reduced and now reaches only 0.6%. Acknowledgements Prof. P. Melehior is gratefully acknowledged for his helpful comments. We thank Prof. J. Wahr who provided us with his programs and kindly explained some details of his work. The computations were performed at the Royal Observatory of Belgium.

References Chandrasekhar, S. and Roberts, P.H., 1963. The ellipticity of a slowly rotating configuration. Astrophys. J., 138: 801—808. Chueca, R.M., Ducarme, B. and Melchior, P., 1985. Preliininary investigations about a quality factor for tidal gravity stations. Bull. Inf. Mar6es Terrestres, 94: 6334—6337. Dehant, V., 1985a. Earth Models and Earth Rotation. In: A. Cazenave (Editor),solved Proceedings NATO working group on ‘Earth rotation: and unsolved problems.’, 1987, Bonas, France, Reidel, pp. 269—275. Dehant, V., 1985b. Body tides for an effiptical rotating Earth with an inelastic mantle. Proceedings 10th International Symposium on Earth tides, Madrid, Spain, pp. 367—377. Dehant, V., 1986. Integration des equations aux deformations d’une Terre elliptique, inélastique en rotation uniforme et noyau liquide. Ph.D. thesis, UniversitC Catholique de Louvain, 298 pp. Ducarme, B., 1975. A fundamental station for Trans-World Tidal Gravity Profiles. Phys. Earth Planet. Inter., 11: 119-127. Ducarme, B., van Ruymbeke, M. and Poitevin, C., 1985. Three years of registration with a superconducting gravimeter at the Royal Observatory of Belgium. Proceedings 10th International Symposium on Earth Tides, Madrid, Spain, pp. 113130. Edge, Ri., Baker, T.F. and Jeffnes, G., 1985. Improving the accuracy of tidal gravity measurements. Proceedings 10th International Symposium on Earth Tides, Madrid, Spain, ~ 213—222. Farrell, W.E., 1972. Deformation of the Earth by surface loads. Rev. Geophys. Space Phys., 10, 3: 761—796. Gilbert, F. mode and Dziewonski, 1975. An application of normal theory to the A.M., retrieval of structure parameters and source mechanisms for seismic spectra. Philos. Trans.

a

R. Soc. London A, 278: 187—269.

206 Melchior, P., 1981. The effects of the Earth ellipticity and inertial forces is visible from M 2 and 01 tidal gravity measutements in the Trans-World Profiles. Communication de l’Observatoire Royal de Belgique, 63A, Série Géophys., 141: 1—10. Melchior, P., 1983. The tides of the Planet Earth. Pergamon Press, Oxford, 2nd ed., 641 pp. Melchior, P., 1985. Report of the International Centre for Earth Tides. In: Consejo Superior de Investigaciones cientificas (Editor) Proceedings of 10th International Symposium on Earth Tides, 1986, Madrid, Espagne, pp. XXXIX—XLIX. Melchior, P. and Dc Becker, M., 1983. A discussion of worldwide measurements of tidal gravity with respect to oceanic interactions, lithosphere heterogeneities, Earth’s flattening and inertial forces. Phys. Earth Planet. Inter., 31: 27—53. Melchior, P., Ducarme, B., van Ruymbeke, M. and Poitevin, C., 1984. Trans-World Tidal Gravity Profiles. Bull. Ohservations Marécs Terrestres, Observatoire Royal de Belgique, 5, 1, 102 pp. Molodensky, M.S., 1961. The theory of nutation and diurnal Earth tides. Communication de l’Observatoire Royal de Belgique, 188, Série Géophys., 58: 25—56. Phinney, R.A. and Burridge, R., 1973. Representation of the Elastic-Gravitational Excitation of a Spherical Earth Model by Generalized Spherical Harmonics. Geophys. J. R. Astron. Soc., 34: 451—487. Schwiderski, E.W., 1980. On Charting Global Ocean Tides. Rev. Geophys. Space Phys., 18, 1: 243—268. Smith, M.L., 1974. The Scalar Equations of Infinitesimal Elas-

tic-Gravitational Motion for a Rotating, Slightly Elliptical Earth. Geophys. J. R. Astron. Soc., 37: 491—526. Van Ruymbeke, M., 1985. Calibration of Lacoste—Romberg gravimeters by the inertial force resulting from a vertical periodic movement. Proceedings 10th International Symposium on Earth Tides, Madrid, Spain, pp. 35—42. Venedikov, A.P., 1966. Une méthode pour l’analyse des marees terrestres partir d’enregistrements de longueur arbitraire. Communication de l’Observatoire Royal de Belgique, 250, SCrie Géophys., 71: 437—459. Wahr, J.M., 1979. The Tidal Motions of a Rotating, Effiptical, Elastic and Oceanless Earth. Ph. D. Thesis, University of Colorado, 216 pp. Wahr, J.M., 1981a. A normal mode expansion for the forced response of a rotating Earth. Geophys. J. R. Astron. Soc., 64: 651—675. Walir, J.M., 1981b. Body tides on an elliptical, rotating, elastic and oceanless Earth. Geophys. J. R. Astron. Soc., 64: 677—703. Walir, J.M., 1981c. The forced nutations of an elliptical, rotating, elastic and oceanless Earth. Geophys. J. R. Astron. Soc., 64: 705—727. Walir, J.M., 1982. Computing tides, nutations and tidally-induced variations in the Earth’s rotation rate for a rotating, elliptical Earth. In: H. Moritz and H. Sünkel (Editors), Lecture at the Third Int. Summer School in the Mountains, on Geodesy and Global Geodynasnics, Admont, Austria, Graz, 689 pp. 327—379.

a

207

Appendix Computation of the first part of the gravimetric factor We must compute: ñ~(ç7W1 + or

n1~.(~v1+

VWeT_

~

VWeT

—

i~

—

w~+ QA(f~Aiu)+ 2iwf2fliu)

w~i~ + 2iwIlAü)

—

with V1, the total eulerian potential and the total initial potential (1) Ao~VV1 Using the expression of the eulerian potential in function of the GSH (defined by eqs. 14 and 16) and computing the normal vector to the ellipsoid in the new base (defined by Smith, 1974), one obtains ~,

~F~~ 1-~D~l\ =hi~+i 0 II~ [_~7D~1 I~Lo]

=

[_%I~~~ED~+]J

~

V~2L’4~rn(D1D2+DID2))

1,m

By developing the product of two GSH, it comes out

~ ( ( d47

~/~4L’

=

r

3r

I

2 — 0

i/~4L’ 1+2 0 3r 1I m

2

11

+I47JD,,~o

—

1 ) rn 1 7D,~2 I~4L’ 1—2 + I 4) — 0 ml rn

Im

l,m —

1 10

0

1

Id4~”— _____ 10~

2

rn

0

I,m~

r

3r

I

~/~4L’2 Il 3r 1 ()

—

rn

0

2

—

2 0

—

‘~m V~4L’~2Il +14),— 10

ml

3r

rn

it

+

l4)7D,~2 j

2

1+21

(Al)

ml

14)1+2

—

+

0

m1

\

/—21

14)12 D,~, ml I 0 +

(A2)

(2) A0~VWT is developed in the same way

10 I rn

_~(I7v~4L11’ l,m(j dr 3r

—

V’i4L’ 3r 1I 1+2 rn 0 ,~.,

(dT

2 0—

_____

1

mlI +

or=~I—~-—— 4)7 ~/~4L’ 10 ~1 r 3r I rn

~i ml

02

—

4’i mO rmD!+2_

2 0

—

j 3r ~4L’

+l~,— Il‘Tm

ml

~

0 11—2 Irn

—

2 0

2 Il 10 ~/~4L’~ 3r I rn

______

+

I

l48IDrnO

Il ml 2 —

0

J

1+21 +

14)1+2

ml

(A3)

208

Ii

~4L’2

—

2

lo

3r

\

121T

0

rn

I

I

m

(A4)

4)72JD,,~0

+

—

J

(3) —ñ0~(üv~~~)

=

I

—h0• ~

Sr’D?2l Idi S~°D~

II

l~mL5m+Dl I

2r2

with

m+J

d~I(ddod+)(Wo+4)~+~r~D~) 2 dW0

2~

Ld+]I I

4~°~T~ =

(I!

2

11

_r~—)~ofjo —

I

d\

3r

Ii

~jm

l,m

Il

2

1—il +

0

ml

rn

/

/

—

d~

1—2

+

0

ml

0 2

l,m

Il

2

rn

l—lj

dc 3r \

+

Iw~

0

m

I

/

2

0

0

ii olu,mID,,~2l

rn

Oml

2 0 0

rnlI

rn

th\

Il I

2

1—21I

rn

0

rn

4 / de’ Ii Vi_2_~e_r~)goIo rn

2 0

3r

—

I

+ i+ij

2

-

rnl

0

,i~l I

0

dr

I’)

\

Il

2 0

+

0

0

1+21I 0 m I

\\

1—21 0

m

Iu1~2JjD~0

I

d\

or=~((~gouIm_2ulm+~1/

r— Jg~ dr,

l,m

I!

xlo rn

3r

it

2 —

0

ml

3r~ fIi+i d\ ii — r~—) g0 ~Im II 0

(~

Ii

de

2

r~)golo 0 rn

2

0 —

0

11

\

olu,mlD~o

ml

11 I ml + I Ø71~I~l +

(AS)

olu,mJ

rn

—

1

/—2

dr,

2

+(~(E_r_)goIo

3r

Il

rn

Il

mJ

it u,” lD~2 OmI ) 2

+lv1m+lo /1 tr 1

2 0

-

——(e—r—Jg0~o r~ drj

-

3r

d\

0

0

1/ 1+21I 4/ d\ I + IVi~2__(_r~)gol0 3r m I rn

d\

/

4/

m)

—

+

~

rn

de\ 3r\(-r—)g0IlO drj Illrn

or ~

lW~

—

o~u1mJD,~0 rnl I

0

3r\

Oml

1+11

\

11

~

—

m

+Iv m+lO ml 1 lm

11

2

rj_)go~ 0

3r ~

—

2

I

Il vjm__(_r~—)go~ d 1+2 0

2

—

Il

~)~o~0 rn

—

de /+2 +1(2~~ —( r~_)go~ 0 3r ~ m

+I(

I

I 41 lw,~1i—.~--i r d\

—

Il

/

li—i

2

11

I

0

ml + I w”D’

rn

—

1

1/

(A6)

209

(4)

1+2

or

=

~

0~Um)D~±2

2

~

0

O~Uim)D~~2

~/~w2

~

0

2

1

—

+

m

0

m

1 V,m+ 0 rn m

0

m

—4~/~w2 0

2

1+2

—

+

rn

0

rn

1

2 —

rn

0

1 V,m+ 0 rn rn

/

—

+

0

w,~+ ~

rn

m

1—2 V,tm D,~2— ~V~~w2

0

/+1

2

+

—

_w2U,m_~,/~w2

1

1

2

0 _w2u,m_4~/~w2rn

1, m

—

jt,m(~r~)g~~2

n~~w~ü 1

=~

~

0

2

1

—

+

rn

0

rn

/

2

1+1

+

—

+

0

rn

1

2

1—2

0

—

+

rn

0

rn

2

1—i

—

+

0

m

W,~ D,~

V~m D~2

(A7)

1 W,~+ 0 rn

2

i—i

—

+

0

rn

W,~,

VI+ 2—~I~W

V,~2 D,~0

(A8)

—uy (5) h0• 2iwczAu =

2iw~2ñ0

=

2iwWi0~

uz

0

If we express that u~= 0, u~= u~and u~= u, and if we consider the relation between the cartesian base vector and the new reference frame vectors (L, ê0, e~),we get —

iS,mn0D~~,o_J=~ sin

=2iw~1h0~

9

+

i

cos 9S~D,~

i(s~D~++s7D,~)-~ sin9 sin 0— i cos

iSfl0D,~o_j

iS7’°D~OD~ + iSf’~D~D~~J =2iw~2ñ0 ~ l,m

iSfD,~D—iS~D,~D~+ _iS~oD,~,oD~+_is,m+DL+D~

At the first order in the ellipticity and using the expression of products of GSH, it comes out

1

=(—2w~l)~ 0 1, m m

1

i

0

+ m

—

1 or =(—2w~2)~ 0 l,m m

m+ 0i m

V,

1

1

—

+

0

m

v”D’+

1 0

—

1+1 + W,~+ 0/ m

m

1 0

—

i—i + W,~ D~

(A9)

m

1+1 0

1

1

—

+

w,~D~1+ 0

rn

0

rn

rn

1—i

1

/

—

+

0

m (AlO)

210

Let us make the sum of all these terms

~

=

3r

((~

3r

II — +4) 1 (~~4)7’ V~4L’ 11+2 0 — 20 +lT4)rJD~o / Il4)TD~2 11—2 0~4L’ 2 10 lm +Il I4)7’Dj~, 02 ml ~ 3r 110 Ii rn 2 T4)7D,~2 11+ rnI 3r 1 I rn 0 2 — /~/~4L’ dr ~4L’ 02 ml it I 11—2 2I0rn ml II 0 ml / 1I11+2 0 + 3r rn0 — +

~ ~4L’ l,m 3r —

I

I

I

— —

—

~l

+(~goUim_2uim+c(_r~)~oIlo d [Il Ii

2

+Io Irn

i1~)

—

+

0

rn

3r

4(

dc

11+2 I m0

drj

~,

I

dr

(Ii

2

_(~w2I 0 Irn

Urn

~ (d4)7’ l,mk —

~4L’2

______

3r

I 1 lo I

v~4L~2Ii

______

3r

—

0

~4L’ ji 3r 10 Irn

rn —

0 1

Ii

+(—2w~2)IIO or

-

lo I

rn

2

II

4

d

20 0

it d 1—2 +Iv,m_ 3r ~ rj~).~oI0

20 0

(

~

0

miI

-2

it 1 +IV,m+lo

)

ml

—

rn

Irn

0

1

I /

~

~l~’

0 ml 1— 2 I —

~4L’~ 3r

II

2

10 Irn

—

_____ loI

1 /

dT

—

4)7~ %/~4L +I4)72+—j~--— ~

0

ml

2

/+21

—

0

+

I

rnl

r m

4),+2

~I~4L’2Io II

_______

3r

I

rn

Im 2 —

0

0

rn

I

i

/

i—il +

—

0

rn

2

11

lW,~1lD~o

ml

0

I

rn

+

+lv1mJD~2

-

lw,~1+lo

+

21—il

it

I /

1+11

—

m

rn

2

Im

11

it D’2 0l~’~ mO j

ml

1+~o-

I

m

1/—2

+l~mD~2_~v~w2l 0

+iv,m+~o ml rn 0

o‘I I /j mO 2 m~ 1UmID~+

i+ll + lw,~ I!

-2

ml

2

ml

0

1+2

—

~lrn 0 1+2

i-i-il + lWm l+1

2

-

+~v,m_~(r~)go~ m 0 ml

2

+(_W2uIm_~w2IIo

2

it

2 0

rn

Ii

dc

—

d li—2 IE_r_)goI 0

3r \

0

_r~-~)golo0 olu,mID~o Im 0 ml /

——

-

+lV,m+lO it Ii ml Irn it

2

-

kIm

m

lw~D’md1I I

I I

I

1+21 + l4)712 ml

0

it

2

IT

—

0 1—2 +

rn

I rn~

+

—g

m ~u,m —

0~,

(All)

211

2y’~/ +—lc 3r \

—

d\ (Ii r~—)g~ 0

d

4

th

/

4/

1+11

Irn

II Iw,~1+lo

+

ml

0

2

Irn

rnj

2 1—21 0 0 mO ml i 2 it

i—lI

lw~1~

—

+

0

ml

1+21 + IV,±2

—

ml

i—21

lu,~2+__-_(--r.~-~)golo 3r \

0

2

Im

~,

lv,_2 m

—

+

0

ml

II

d

~,

~l

+_w2U,m_~V~w2Il0 Urn —

2 —

Ii 20 olu1 it m~21~ de Il 2 3r (E_r_)go~o dr IrnOml m 0 Ii 2 1+21 2~/~ d I/

__Ic_r.~_,)golo 0 3r ~ Im 0 3r

ml

0

rn

4 / 3r\

it Il +lF~,m+lo

2 —

Il ~I~ew2l 0 ni

2

0

i+21

I

+

—

ml

0 i

1

+(—2w~tO

—

Il

2

+lv,m+lo ml Im

—

~±2

—

it

Ii

0

m

I

2

1—21

0

—

+

i

rn 0 i+il

—

+

1/

2

Iw,1~i+lo

+

Il

~I~w2l

+IV,m+l0

1+11

—

Irn

i—it

—

+

0

m

lw~1I I

I

ml i

Ii

lw~1+lo

i—il +

—

\\

lw~1IID~o

I

(A12)

0 m Irn 0 m At the first order in the ellipticity, the response of the Earth to an external potential of the order (1, rn) can be written T4)7 ~4L’ I 1 2 1 I lo (d4)7 d r I rn 0 ml

~lm

ml

0

rn

—

Ii

1

1

it

_2w~(lo

—

rn

0

+lv,m+to Im

ml

i

i—il +

0

ml

[Ii

~ + —Ic—r 2y’~ d~ +~goU,m_~2U, 3r \I ~)~o~l0

kIm __l_r~)golo 3r~ 41 ~ Ii rn

020

+

____

dr

2~/~ I 3r k

+—l—r

—

0U,~2 ~4L’ 3r —

d) 111+2 g dr Oh m 0

1I

i 2 —

0

it Ii 2 ~i~lv,m+lo —

ml

Irn

/—lI + lw,~1I

0

m

I

ml olu,m_w2u,mpD~o lI

2g

(d4)7!~2+

\

lw,~

—

2 0

—

11+2

1I 0

rn

II

2 —

0

‘I ml th

Ii+2

ml 3r~ I m +Iv,m__(_r~)go~ 0

2 00

it

2 ml ) olu,mpD~

212

(d4)7~2+ 2g0U dr — I—2 2%/s

+__(c

3r

/

—

r

d\

ii V~4L

111—2

-

4~ d Ii—2 —(e-r 3r\ ~:)~o~ 0

m

2 —

0

rn 2 0

2

2

rn

3r

~j~)~o~l0

~

it

11I(4)~+T4)7)

—

+

0

rnI

it 11—2 +IV,m+l 0 ml I m

2

i—il

—

+

Iwm

0

rn I

1—1

\

olu,mJD,~2 Ornl /

(A13)