Inverse Vening Meinesz formula and deflection-geoid formula

Journal of Geodesy (1998) 72: 304±312

Inverse Vening Meinesz formula and de¯ection-geoid formula: applications to the predictions of gravity and geoid over the South China Sea C. Hwang Department of Civil Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30050, Taiwan, ROC Fax: +886 3 5716257; e-mail: [email protected] Received: 7 April 1997 = Accepted: 7 January 1998

Abstract. Using the spherical harmonic representations of the earth's disturbing potential and its functionals, we derive the inverse Vening Meinesz formula, which converts de¯ection of the vertical to gravity anomaly using the gradient of the H function. The de¯ectiongeoid formula is also derived that converts de¯ection to geoidal undulation using the gradient of the C function. The two formulae are implemented by the 1D FFT and the 2D FFT methods. The innermost zone e€ect is derived. The inverse Vening Meinesz formula is employed to compute gravity anomalies and geoidal undulations over the South China Sea using de¯ections from Seasat, Geosat, ERS-1 and TOPEX//POSEIDON satellite altimetry. The 1D FFT yields the best result of 9.9-mgal rms di€erence with the shipborne gravity anomalies. Using the simulated de¯ections from EGM96, the de¯ection-geoid formula yields a 4-cm rms di€erence with the EGM96-generated geoid. The predicted gravity anomalies and geoidal undulations can be used to study the tectonic structure and the ocean circulations of the South China Sea.

marine gravity from satellite altimetry. Marine de¯ections of the vertical can be derived from altimeter-measured geoidal undulations (if the sea-surface topography is properly removed) and the use of de¯ection as data type can reduce many systematic errors in satellite altimetry (Hwang 1997). Indeed, a frequency-domain version of the inverse Vening Meinesz formula exists in the literature, e.g., Haxby et al. (1983), Hwang and Parsons (1996), Sandwell and Smith (1997). A space-domain version has also been derived in, e.g., Molodenskii et al. (1962, Eq. III. 2.11). This paper attempts to derive the inverse Vening Meinesz formula for all cases using a spectral representation approach. The de¯ection-geoid formula, which converts de¯ections of the vertical to geoidal undulations, will be also derived using the same approach. Practical methods for implementing the two formulae will be presented. As an example, the two formulae will be employed to compute the gravity anomalies and the geoidal undulations over the South China Sea using the de¯ections of the vertical from Seasat, Geosat, ERS-1 and TOPEX/POSEIDON satellite altimetry.

Key words De¯ection-geoid formula
Gravity anomaly
Inverse Vening Meinesz formula
Satellite altimetry
South China Sea

2 Fundamentals

1 Introduction Since the publication of the Vening Meinesz formula (Vening Meinesz 1928), little attention has been paid to its inverse formula, which converts de¯ections of the vertical to gravity anomalies. This is mainly because measurements of de¯ections are not widely available. With the advent of satellite altimetry, however, de¯ections of the vertical become available in the oceans and the inverse Vening Meinesz formula can be useful if one wishes to compute

First we brie¯y review some of the basic equations in physical geodesy necessary for the derivations of the inverse Vening Meinesz formula and the de¯ectiongeoid formula. The earth's disturbing potential T can be expanded into a series of spherical harmonics as 1  n X n X 1 GM X R C a Y a …/; k† …1† T …r; /; k† ˆ r nˆ2 r mˆ0 aˆ0 nm nm where r; /; k are the spherical coordinates (geocentric distance, geocentric latitude and longitude), R is the earth's a a are the harmonic coecients and Ynm are mean radius, Cnm the fully normalized spherical harmonics de®ned as  R …/; k† ˆ P nm …sin /† cos mk; a ˆ 0 a …/; k† ˆ nm …2† Ynm S nm …/; k† ˆ P nm …sin /† sin mk; a ˆ 1

305

with P nm …sin /† being the fully normalized associated Legendre function (Heiskanen and Moritz 1967). On the sphere of radius R, geoidal undulation can be expressed as N …/; k† ˆ

1 X n X 1 X T a a jrˆR ˆ R Cnm Ynm …/; k† c0 nˆ2 mˆ0 aˆ0

and gravity anomaly as   1 X oT 2 …n ÿ 1† ÿ T jrˆR ˆ c0 Dg…/; k† ˆ ÿ or r nˆ2 

n X 1 X mˆ0 aˆ0

…3†

…4†

a a Cnm Ynm …/; k†

GM R2

…5†

is the mean gravity. Spherical harmonic representations of other functionals of the earth's disturbing potential can be found in Rummel and van Gelderen (1995). Geoidal undulation and gravity anomaly are two scalar functions derived from the earth's disturbing potential. De¯ection of the vertical, however, is a vector function and can be expressed as 1 z…/; k† ˆ …n g† ˆ ÿ rN …/; k† R 1 X n X 1 X a a Cnm rYnm …/; k† ˆÿ

…6†

nˆ2 mˆ0 aˆ0

where n and g are the north-south component and westeast component of the de¯ection vector, respectively, and r is the gradient operator on the sphere de®ned as:   o o rˆ …7† o/ cos /ok Thus the basis functions of the de¯ection vector are a a , rather than Ynm . rYnm Finally we recall a variant of Green's formula (Meissl 1971, p. 12) ZZ ZZ ZZ  rf
rgdr ˆ ÿ f D gdr ˆ ÿ gD f dr …8† r

a rYnm
rYsrb dr r  4pn…n‡1†; if n ˆ s and m ˆ r and a ˆ b ˆ 0; if n 6ˆ s or m 6ˆ r or a 6ˆ b

…11†

3 The inverse Vening Meinesz formula The key to deriving the inverse Vening Meinesz formula is to look for a suitable kernel function for converting de¯ection of the vertical to gravity anomaly. Based on Meissl's (1971) approach, we introduce the kernel function H de®ned as 1 X …2n ‡ 1†…n ÿ 1† …12† Pn …cos wpq † H …wpq † ˆ n…n ‡ 1† nˆ2 where p and q are two points on the unit sphere with a spherical separation of wpq , so that (see Fig. 1)

where c0 ˆ

ZZ

r

cos wpq ˆ sin /p sin /q ‡ cos /p cos /q cos…kq ÿ kp †

…13†

The Legendre polynomial Pn …cos wpq † in Eq. (12) can be further decomposed into the series: n X 1 1 X a Y a …/ ; kp †Ynm …/q ; kq † …14† Pn …cos wpq † ˆ 2n ‡ 1 mˆ0 aˆ0 nm p which is termed decomposition formula by Heiskanen and Moritz (1967), see also Hobson (1965). Integrating the scalar products of rH and rN over the unit sphere and using Eq. (11), we get ZZ rq H …wpq †
rq N …q†drq r # ZZ "X 1 n X 1 nÿ1 X a a ˆR Ynm …p†rq Ynm …q† r nˆ2 n…n ‡ 1† mˆ0 aˆ0 " # 1 X n X 1 X a a  Cnm rq Ynm …q† drq nˆ2 mˆ0 aˆ0

1 n X 1 X nÿ1 X a a ˆR Cnm Ynm …p† n…n ‡ 1† nˆ2 mˆ0 aˆ0 ZZ   a a  rq Ynm …q†
rq Ynm …q† drq r

ˆ 4pR

1 X nˆ2

…n ÿ 1†

n X 1 X mˆ0 aˆ0

…15†

a a Cnm Ynm …p†

r

where f and g are two arbitrary functions de®ned on the unit sphere, and D is the Laplace surface operator (Courant and Hilbert 1953) de®ned as     1 o o 1 o2 D ˆ cos / …9† ‡ cos / o/ o/ cos2 / ok2 For the surface spherical harmonics, we have a a …/; k† ‡ n…n ‡ 1†Ynm …/; k† ˆ 0 D Ynm

…10†

Using Eqs. (8) and (10) and the orthogonality relationship of fully normalized spherical harmonics (Heiskanen and Moritz 1967, p. 31), we obtain the result

Fig. 1. Spherical distance wpq between points p and q, and components of the de¯ection of the vertical at q

306

Comparing Eqs. (4) and (15), we have ZZ c rq H …wpq †
rq N …q†drq Dg…p† ˆ 0 4pR r

…16†

As shown in the Appendix, the closed form of H is ! w sin3 2pq 1 H …wpq † ˆ ‡ log …17† w w 1 ‡ sin 2pq sin 2pq Furthermore, dH dH owpq owpq rq wpq ˆ rq H …wpq † ˆ dwpq dwpq o/q cos /q okq

!

…18† With Eq. (17) the derivative of H with respect to wpq is w

w

w

cos 2pq cos 2pq …3 ‡ 2 sin 2pq † dH ˆÿ ‡ H0 ˆ w w w dwpq 2 sin2 pq 2 sin pq …1 ‡ sin pq † 2

2

…19†

2

Di€erentiating Eq. (13) with respect to /q and kq , we have (cf. Heiskanen and Moritz 1967, p. 113) owpq ˆ cos /q sin /p ÿ sin /q cos /p cos…kq ÿ kp † o/q owpq ˆ ÿ cos /p cos /q sin…kq ÿ kp † ÿ sin wpq okq ÿ sin wpq

…20† Referring to the spherical triangle in Fig. 1, the following relationships hold: sin wpq cos aqp ˆ cos /q sin /p ÿ sin /q cos /p cos…kq ÿ kp † sin wpq sin aqp ˆ ÿ cos /p sin…kq ÿ kp †

…21†

Thus owpq owpq ˆ ÿ cos aqp ; ˆ ÿ cos /q sin aqp o/q okq

…22†

Inserting Eqs. (18) and (22) into Eq. (16) we ®nally get the inverse Vening Meinesz formula ! ZZ owpq owpq c0 0 H
…nq gq †drq Dg…p† ˆ ÿ 4p r o/q cos /okq ZZ ÿ
c ˆ 0 H 0 nq cos aqp ‡ gq sin aqp drq 4p r ZZ c0 ˆ H 0 eqp drq 4p r

Fig. 2. Function H 0 …w† and function C 0 …w†

…23†

where eqp is the de¯ection component at point q in the direction of the azimuth aqp (Heiskanen and Moritz 1967, p. 187), or simply the longitudinal de¯ection component. The meaning of the inverse Vening Meinesz formula is: assuming that everywhere on the unit sphere the north-south and west-east de¯ection components are known, the gravity anomaly at any given point can

be obtained by integrating the products of H 0 and the longitudinal de¯ection components over the unit sphere. Figure 2 shows the function H 0 , which changes rapidly as w approaches zero. The ®rst zero crossing of H 0 occurs at w ˆ 43 . When w is small, we have the asymptotic representation: H 0 …w†  ÿ

2 w2

…24†

It is noted that the asymptotic representation of H 0 is equal to that of dS=dw (the derivative of Stokes' function) and agrees with the asymptotic representation of the kernel function in Eq. (III.2.111) of Molodenskii et al. (1962). Furthermore, H …w†  w2 as w approaches zero, so the asymptotic representations of the function H and Stokes' function are identical. 4 The de¯ection-geoid formula Next we derive a formula for converting de¯ection of the vertical to geoidal undulation. The derivation is almost the same as that for the inverse Vening Meinesz formula. First, we introduce the kernel function C C…wpq † ˆ

1 X 2n ‡ 1 Pn …cos wpq † n…n ‡ 1† nˆ2

Then, ZZ rq C…wpq †
rq N …q†drq # ZZ "X 1 n X 1 X 1 a a ˆR Ynm …p†rq Ynm …q† r nˆ2 n…n ‡ 1† mˆ0 aˆ0 " # 1 X n X 1 X a a Cnm rq Ynm …q† drq  r

nˆ2 mˆ0 aˆ0

…25†

307

(

1 X

n X 1 X 1 a a ˆR Cnm Ynm …p† n…n ‡ 1† nˆ2 mˆ0 aˆ0 ZZ a a ‰rq Ynm …q†
rq Ynm …q†Šdrq  r

ˆ 4pR

1 X n X 1 X nˆ2 mˆ0 aˆ0



/q ˆ/1

Thus, the geoidal undulation at point p can be obtained by integrating the scalar products of rH and rN over the unit sphere: ZZ 1 rq H …wpq †
rq N …q†drq …27† N …p† ˆ 4p r Using Eqs. (22) and (27) we obtain the de¯ection-geoid formula ZZ R dC eqp drq …28† N …p† ˆ 4p r dwpq According to the Appendix, the closed form of C is wpq 3 ÿ cos wpq ÿ 1 2 2

…29†

and C0 ˆ

wpq 3 dC ˆ ÿ cot ‡ sin wpq 2 dwpq 2

…30†

which agrees with the result of Molodenskii et al. (1962, Eq. III.2.8) if the summation of the series in Eq. (25) starts from n ˆ 1. Figure 2 also shows the funciton C 0 . The ®rst zero crossing of C 0 occurs at w ˆ 70:5 . The asymptotic representation of C 0 when w is small is: C 0 …w†  ÿ

2 w

…31†

5 Computations by 1D FFT: rigorous implementations We propose two computational schemes for the inverse Vening Meinesz formula and the de¯ection-geoid formula when the north-south and west-east components of de¯ections are given on a regular grid. The ®rst scheme is based on the one-dimensional fast Fourier transform (1D FFT) method, which, given regularly gridded data, can rigorously implement a surface integral such as Eq. (23) or (28). In such a scheme, gravity anomalies or geoidal undulations at the same parallel are computed simultaneously by FFT as (cf. Haagmans et al. 1993) ( )    /n kn  0 X Dg/p …kp † H …Dkqp † c0 D/Dk X ˆ N/p …kp † 4p / ˆ/ k ˆk C 0 …Dkqp † R q 1 q

1

 …ncos cos aqp ‡ gcos sin aqp †   c0 D/Dk ÿ1 ˆ F 4p 1 R

F1 …C 0 …Dkqp † cos aqp †

F1 …ncos †



F1 …H 0 …Dkqp † sin aqp † ŠF1 …gcos † ‡ F1 …C 0 …Dkqp † sin aqp †

…26†

a a Cnm Ynm …p†

C…wpq † ˆ ÿ2 log sin

 /n  X F1 …H 0 …Dkqp † cos aqp †

)

…

where ncos ˆ n cos /; gcos ˆ g cos /; Dkqp ˆ kq ÿ kp ; D/ and Dk are grid intervals in the directions of latitude and longitude, and F1 is the 1D FFT. Since all quantities are real-valued, we can compute the Fourier transforms of two real-valued arrays simultaneously to save computer time (Hwang 1993). Speci®cally, let h…k† and g…k†, k ˆ 0; . . . ; n ÿ 1, be the two real-valued arrays to be Fourier transformed. We ®rst form the complex array y…k† as y…k† ˆ h…k† ‡ i g…k†; k ˆ 0; . . . ; n ÿ 1 …33† p where i ˆ ÿ1. Let Y …k† be the Fourier transform of y…k†, we have H …0† ˆ Re…Y …0††; and H …k† ˆ 12 Re …Y …k† ‡ Y …n ÿ k†† ‡ 12 i Im …Y …k† ‡ Y …n ÿ k†† for k ˆ 1; . . . ; n ÿ 1

G…0† ˆ Im …Y …0††; and G…k† ˆ 12 Re …Y …k† ‡ Y …n ÿ k†† ÿ 12 i Im …Y …k† ‡ Y …n ÿ k†† for k ˆ 1; . . . ; n ÿ 1

…34†

where H …k† and G…k† are the Fourier transforms of h…k† and g…k†; respectively, and Re(.) and Im(.) are the real and the imaginary parts of a complex number. In practice, the complex array holding ncos and gcos , and the and complex array holding H 0 …Dkqp † cos aqp H 0 …Dkqp † sin aqp (or C 0 …Dkqp † cos aqp and C 0 …Dkqp † sin aqp † are Fourier transformed. Taking advantage of the gridded data, azimuth and spherical distance can be calculated as tan aqp ˆ

sin2



wqp 2

ÿ cos /p sin Dkqp ÿ sin…/q ÿ /p † ‡ 2 sin /q cos /p sin2



ˆ sin2



D/qp 2



‡ sin2

Dkqp 2

…35†

  Dkqp cos /q cos /p 2 …36†

where D/qp ˆ /q ÿ /p . 6 Computations by 2D FFT: planar approximations The second computational scheme is based on the planar approximations of the two formulae, and hence the two-dimensional fast Fourier transform (2D FFT). In a local rectangular, x ÿ y coordinate system, the surface element and the spherical distance can be q approximated as R2 drq ˆ dxq dyq and wqp ˆ Rqp , with qqp being the planar distance. With the asymptotic representation in Eq. (24), the inverse Vening Meinesz formula becomes

308

ZZ

ÿ2R2  nq …qqp cos aqp † 3 D qqp  dxq dyq ‡gq …qqp sin aqp † R2 (ZZ yp ÿ yq ÿc ˆ 0 n dx dy 2 2 3 q q q 2p D ‰…xp ÿ xq † ‡ …yp ÿ yq † Š2 ) ZZ xp ÿ x q g dx dy ‡ 2 2 3 q q q D ‰…xp ÿ xq † ‡ …yp ÿ yq † Š2 ( ! ! ) ÿc0 y x n‡ g ˆ 3 3 2p …x2 ‡ y 2 †2 …x2 ‡ y 2 †2

Dg…xp ; yp † ˆ

c0 4p

…37† where * is the convolution operator and D is the data domain. Schwarz et al. (1990) show that      u x 2 2 ÿ3=2 ˆ ÿ2pi …u2 ‡ v2 †ÿ3=2 …38† F …x ‡ y † v y where F is the 2D FT, and u and v are the spatial frequencies. Thus the relationship between gravity anomaly and de¯ection components in the frequency domain is ic0  ‰vX…u; v† ‡ uE…u; v†Š DG…u; v† ˆ p 2 u ‡ v2

…39†

where DG; X and E are the Fourier transforms of Dg, n, g, respectively. Equation (39) can also be found in, for example, Hwang and Parsons (1996), Sandwell and Smith (1997), Haxby et al. (1983), who derived this formula using di€erent approaches. The planar approximation of the de¯ection-geoid formula reads ZZ R ÿ2R  n …q cos aqp † N …xp ; yp † ˆ 4p D q2qp q qp  dxq dyq ‡gq …qqp sin aqp † R2 (ZZ yp ÿ yq ÿ1 ˆ n dxq dyq 2 2 q 2p D …xp ÿ xq † ‡ …yp ÿ yq † ) ZZ x p ÿ xq g dx dy ‡ 2 2 q q q D …xp ÿ xq † ‡ …yp ÿ yq †      ÿ1 y x ˆ n ‡ g 2p x2 ‡ y 2 x2 ‡ y 2

Let pa in  Eq. (41) approach in®nity so that 1 ‡ a2 ‡ r2 ˆ K remains a constant for any r, and eÿab ˆ 0. Then,   Z 1 K 1 log …42† J0 …br†rdr ˆ 2 r b 0 1 This means that the Hankel transform of log…Kr † is 2pq 2, with q2 ˆ u2 ‡ v2 . Furthermore, !       o 1 K x 1 x ˆ ˆ ÿ ox …43† log o y r2 y x2 ‡ y 2 r oy

By the di€erentiation theorem of Fourier transform (see, e.g., Mesko 1984), we have      1 1 u x …44† ˆ ÿi F v u2 ‡ v2 y x2 ‡ y 2 With Eqs. (40) and (44), we obtain the relationship between geoidal undulation and de¯ection components in the frequency domain N…u; v† ˆ

i 2p…u2

‡ v2 †

‰vX…u; v† ‡ uE…u; v†Š

…45†

which can also be found in Olgiati et al. (1995, Eq. 6). Using Eq. (39) and the geoid-gravity spectral relationship [see, e.g., Schwarz et al. (1990)], one can also derive Eq. (45). 7 The innermost zone e€ects At zero spherical distance the kernel function H 0 and C 0 become singular and the azimuth is unde®ned. Thus we must account for the innermost zone e€ect (Heiskanen and Moritz 1967). First we consider such an e€ect in the inverse Vening Meinesz formula. The de¯ection components at the neighbourhood of point p can be expanded into the Taylor series (see Fig. 3)

…40† x x2 ‡y 2

y , x2 ‡y 2

The Fourier transforms of and which do not exist in the literature, are now derived. We begin with the de®nite integral found in Gradshteyn and Ryzhik (1994, p. 782): Z 1 p ‰log…1 ‡ a2 ‡ r2 † ÿ log rŠJ0 …br†rdr 0 …41† 1 ˆ 2 …1 ÿ eÿab †; a > 0; b > 0 b

Fig. 3. Local rectangular coordinates for the innermost zone e€ect in a cap of radius s0

309

ÿ
nq ˆ np ‡ xnx ‡ yny ‡ 12 x2 nxx ‡ 2xynxy ‡ y 2 nyy ‡


ÿ
gq ˆ gp ‡ xgx ‡ ygy ‡ 12 x2 gxx ‡ 2xygxy ‡ y 2 gyy ‡


…46† on ox ; ny

on oy ; nxx

2

o n ; nyy ox2

2

o n oy 2

o2 n oxoy .

ˆ ˆ ˆ and nxy ˆ where nx ˆ Retaining only the linear terms in Eq. (46) and assuming that the innermost zone is circular, with Eq. (24) we have Z 2p Z s0 c ÿ2 ‰…np ‡ s sin a nx ‡ s cos a ny † Dgi ˆ 0 4p aˆ0 sˆ0 s2  cos…a ‡ p† ‡ …gp ‡ s sin a gx ‡ s cos a gy †  sin…a ‡ p†Šsdsda  Z 2p Z s0 c cos2 ada ds ˆ 0 ny 2p aˆ0 sˆ0 Z 2p Z s0  2 sin ada ds ‡gx aˆ0

s0 c 0 …ny ‡ gx † ˆ 2

sˆ0

…47†

Using a similar derivation, the innermost zone e€ect in the case of the de¯ection-geoid formula is Ni ˆ

s20 …n ‡ gx † 4 y

…48†

Thus the innermost zone e€ects for gravity anomaly and geoidal undulation depend on the gradients of the de¯ection components. For discrete data ny and gx can be obtained by numerically di€erentiating n and g along the y and x directions, respectively. If the planar grid intervals are Dx and Dy, the radius of the innermost zone may be approximated by r DxDy s0 ˆ …49† p

8 Applications: gravity and geoid over the South China Sea from satellite altimetry As an application of the inverse Vening Meinesz formula, we computed marine gravity anomalies over the South China Sea (de®ned domain: 5  latitude  25 , 105  longitude  125 † using the 1D FFT and 2D FFT methods and the remove-restore procedure. The EGM96 geopotential model (Lemoine et al. 1997) to degree 360 was used as the reference ®eld. The altimeter data used are from Seasat, Geosat/ERM, Geosat/GM, ERS-1/35-day, ERS-1/GM and TOPEX/ POSEIDON. The sea-surface topography of Levitus (1982) is subtracted from the altimeter sea surface heights before generating the de¯ections of the vertical. At the centre of the South China Sea the average altimeter data density is 1560 points in 1  1 , and at the continental borders the densities drop sharply. We used the method of least-squares collocation and the covariance functions of de¯ections derived by Hwang

Table 1. RMS di€erences between the shipborne and the predicted gravity anomalies over the South China Sea and the CPU time ratios (aIE: innermost zone e€ect) Method

RMS di€erence (mgal)

CPU time ratio

1D FFT/IEa 1D FFT/no IE 2D FFT Sandwell and Smith (1997)

9.90 10.06 10.11 14.32

2.8 2.4 1.0 unavailable

and Parsons (1995) to grid the de¯ections of di€erent azimuths into the north-south and west-east components at a 20  20 interval. We used a 1 -border to avoid bad results at the edges. Further, 100% zero paddings were applied to data arrays and kernel arrays to avoid edge e€ects in convolutions by FFT. Table 1 shows the comparisons between the shipborne gravity anomalies and the gravity anomalies derived from the 1D FFT, the 2D FFT, and Sandwell and Smith's (1997) methods. The shipborne gravity anomalies were provided by the National Geophysical Data Center (NGDC). A total of 180297 shipborne gravity anomalies were used for the comparisons. The 1D FFT produces a slightly better result than the 2D FFT. Table 1 also shows the CPU time ratio between a given method and the 2D FFT method on a Sun Sparc 20 machine. The 1D FFT requires more than doubled computer time than the 2D FFT. Employing the innermost zone e€ect improves the result. We also did tests in other areas, and found that the innermost zone e€ect always improves the result. Furthermore, the result from the 1D FFT is about 30% better than the recently published altimeter-derived gravity anomalies from Sandwell and Smith (1997). Figure 4 shows the predicted gravity anomalies over the South China Sea (1D FFT plus innermost zone e€ect). In Fig. 4, the outline of the basin of the South China Sea is clearly visible. A median valley-like feature running from the southwest to the northeast is probably the spreading centre of the South China Sea, which has been identi®ed by, e.g., Briais et al. (1993), using geomagnetic data. The predicted gravity anomalies can be further used to interpret the tectonic structure of the South China Sea. Because generally there are no measured geoidal undulations at sea, we used the simulation approach of Tziavos (1996) to evaluate the performances of the 1D and 2D FFT methods that implement the de¯ectiongeoid formula. First, over the South China Sea we generated gridded north-south and west-east components of de¯ections, and geoidal undulations using harmonic coecients from EGM96 (Lemoine et al. 1997) at a 7:50  7:50 interval. Only harmonic coecients between degrees 181 and 360 were used, so that the remove-restore procedure need not be used. Because going from de¯ection to geoid is a smoothing process, we used a 5 border to avoid edge e€ects. The north-south and west-east de¯ection components were then used to compute geoidal undulations. The statis-

310

Fig. 4. Grey-shaded relief map of the predicted gravity anomalies over the South China Sea, with illumination from the north-west

tics of the di€erences between the EGM96-generated undulations, considered as the ``ground truth'', and the computed undulations from various methods are given in Table 2. From Table 2, the 1D FFT again gives a better result than the 2D FFT. Also, the use of the innermost zone e€ect signi®cantly reduces the di€erence between the `true' and the predicted geoids. Having done these tests, we then used the de¯ection data as used in predicting the gravity anomalies to compute the geoidal undulations over the South China Sea. The result is shown in Fig. 5. The predicted geoid can be used to study the ocean circulations over the South China Sea, which recently have received considerable attention from the oceanographers in Southeast Asia.

9 Conclusion In this paper we showed the detailed derivations of the inverse Vening Meinesz formula and the de¯ectiongeoid formula using the spherical harmonic representations of the functionals of the earth's disturbing potential, and for each we presented the 1D FFT and 2D FFT methods for computations. In all cases the 1D FFT yields better results than 2D FFT, but the former needs nearly doubled computer times. Over the South China Sea, by the inverse Vening Meinesz formula and the 1D FFT we derived a set of gravity anomalies better than that from Sandwell and Smith (1997) when comparing with the shipborne gravity anomalies. Using the simulated de¯ections from EGM96 over the South

Table 2. Statistics of the EGM96 geoid and the di€erences between the predicted and the EGM96 geoids over the South China Sea (unit: m; a IE: innermost zone e€ect) Case EGM96 EGM96 EGM96 EGM96

deg 181 to 360 - 1D FFT/IEa - 1D FFT/no IE - 2D FFT

mean

min.

max.

std. dev.

RMS

0.000 0.038 0.038 0.033

)2.998 )0.010 )0.055 )0.060

3.812 0.100 0.169 0.169

0.542 0.014 0.021 0.019

0.542 0.041 0.043 0.038

311

Fig. 5. The predicted geoid over the South China Sea, contour interval is 0.5 m

China Sea, the de¯ection-geoid formula yields a 4-cm accuracy. The predicted gravity anomalies and geoidal undulations over the South China Sea are freely available to all scientists. Interested readers please send e-mail to [email protected].

Using Eq. (A1) and the fact that P0 …cos w† ˆ 1, we have

Acknowledgements. This research is funded by the National Science Council of Republic of China, under contract NSC86- 2611-M009-001-OS, with the project title: Surveying gravity and bathymetry over the South China Sea using satellite altimetry. The author is grateful to Roger Haagamans and an anonymous reviewer for their important suggestions.

Integrating Eq. (A3) with respect to k between the limits k, 0, and using the result in Gradshteyn and Ryzhik (1994, p. 101), we have

Consider the generating function of Legendre polynomials (Hobson 1965) 1 X 1 k n Pn …cos w† …A1† f…k† ˆ p ˆ 1 ÿ 2k cos w ‡ k 2 nˆ0 The series in Eq. (A1) is absolutely and uniformly convergent if k < 1 (Hotine 1969, p. 310). The case k ˆ 1 will be of conditional convergence except at w ˆ 0. In the following derivations, we exclude the point w ˆ 0 in all results. Setting k ˆ 1 in Eq. (A1), we have 1 X

1 1 Pn …cos w† ˆ p ˆ 2 ÿ 2 cos w 2 sin w2 nˆ0

1 Z X nˆ1

Appendix: derivations of closed forms of H and C functions

X …w† ˆ

1 f …k† ÿ 1 X k nÿ1 Pn …cos w† ˆ k nˆ1

…A2†

k

0

k nÿ1 dkPn …cos w† ˆ

…A3†

1 X 1

k n Pn …cos w† n nˆ1 Z k f …k† ÿ 1 ˆ dk k 0

…A4†

p ˆ ÿ log…2 1 ÿ 2k cos w ‡ k 2 ‡ 2 ÿ 2k cos w† ‡ log 4 Setting K=1, we get Y …w† ˆ

   w w Pn …cos w† ˆ ÿ log sin 1 ‡ sin n 2 2 nˆ1

1 X 1

…A5† Furthermore, integrating Eq. (A1) with respect to k between the limits k, 0 and using the result in Gradshteyn and Ryzhik (1994, p. 99), we have

312 1 Z X nˆ0

k 0

1 X

1 k n‡1 Pn …cos w† n ‡ 1 nˆ0 Z k f…k†dk ˆ 0 p ˆ log…2 1 ÿ 2k cos w ‡ k 2

k n dkPn …cos w† ˆ

‡ 2k ÿ 2 cos w† ÿ log…2 ÿ 2 cos w† Setting k ˆ 1, we have 1 X

1 ‡ sin w2 1 Pn …cos w† ˆ log Z…w† ˆ n‡1 sin w2 nˆ0

…A6† ! …A7†

It is easy to see that H …w† and C…w† are linear combinations of the three basic in®nite series, X …w†; Y …w† and Z…w†, namely, H …w† ˆ ˆ

1 X …2n ‡ 1†…n ÿ 1† nˆ2 1  X nˆ2

n…n ‡ 1†

Pn …cos w†

 1 2 Pn …cos w† 2ÿ ÿ n n‡1

ˆ 2…X …w† ÿ P0 ÿ P1 † ÿ …Y …w† ÿ P1 † ÿ 2…Z…w† ÿ P0 ÿ 12 P1 † ! sin3 w2 1 ˆ ‡ log sin w2 1 ‡ sin w2

…A8†

and 1 X 2n ‡ 1 Pn …cos w† n…n ‡ 1† nˆ2  1  X 1 1 ˆ ‡ Pn …cos w† n n‡1 nˆ2

C…w† ˆ

…A9†

ˆ Y …w† ÿ P1 ‡ Z…w† ÿ P0 ÿ 12 P1

ˆ ÿ2 log sin w2 ÿ 32 cos w ÿ 1 where P0 ˆ 1; P1 ˆ cos w. References

Briaiis A, Patrait P, Tapponnier P (1993) Updated interpretation of magnetic anomalies and sea ¯oor spreading stages in the South China Sea: implications for the Tertiary tectonics of Southeast Asia. J Geophys Res 98: 6299±6328 Courant R, Hilbert D (1953) Methods of mathematical physics, vol I. Interscience, New York

Gradshteyn IS, Ryzhik IM (1994) Table of integrals, series, and products (5th edn). Academic Press, New York Haagmans R, de Min E, van Gelderen M (1993) Fast evaluation of convolution integrals on the sphere using 1D FFT, and a comparison with existing methods for Stokes integral. Manuscr Geod 18: 227±241 Haxby WF, Karner GD, Labreque JL, Weissel JK (1983) Digital images of combined oceanic and continental data sets and their used in tectonic studies. EOS Trans Am Geophys Un 64: 995± 1004 Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman WH, San Francisco Hobson EW (1965) The theory of spherical and ellipsoidal harmonics (2nd edn). Chelsea, New York Hotine M (1969) Mathematical geodesy. ESSA Monogr no 2, US Dept Commerce, Washington DC Hwang C (1993) Fast algorithm for the formation of normal equations in a least-squares spherical harmonic analysis by FFT. Manuscr Geod 18: 46±52 Hwang C (1997) Analysis of some systematic errors a€ecting altimeter-derived geoid gradient with applications to geoid determination over Taiwan. J Geod 71: 113±130 Hwang C, Parsons B (1995) Gravity anomalies derived from Seasat, Geosat, ERS-1 and TOPEX/POSEIDON altimetry and ship gravity: a case-study over the Reykjanes Ridge. Geophys J Int 122: 511±568 Hwang C, Parsons B (1996) An optimal procedure for deriving marine gravity from multi-satellite altimetry. Geophys J Int 125: 705±718 Lemoine FG, et al. (1997) The development of the NASA GSFC and NIMA joint geopotential model. In: Segawa J, Fwimoto H (eds) Proc Int Symp Gravity, geoid, and marine geodesy Levitus S (1982) Climatological atlas of the world ocean. NOAA professional paper 13, US Dept Commerce. Rockville, Md Meissl P (1971) A study of covariance functions related to the earth's disturbing potential. Rep no 151, Dept Geod Surv, Ohio State University, Columbus Mesko AM (1984) Digital ®ltering: applications in geophysical exploration for oil. Akademiai Kiado, Budapest Molodenskii MS, Eremeev VF, Yurkina MI (1962) Methods for study of the external gravitational ®eld and ®gure of the earth. Works of Central Research Institute of Geodesy, Aerial Photography and Cartography, no 131, Moscow Olgiati A, Balmino G, Sarrailh M, Green CM (1995) Gravity anomalies from satellite altimetry: comparison between computation via geoid heights and via de¯ections of the vertical. Bull Geod 69: 252±260 Rummel R, van Gelderen M (1995) Meissl scheme ± spectral characteristics of physical geodesy. Manuscr Geod 20: 379±385 Sandwell D, Smith WHF (1997) Marine gravity anomaly from Geosat and ERS-1 satellite altimetry. J Geophys Res 102: 10039±10054 Schwarz KP, Sideris MG, Forsberg R (1990) The use of FFT techniques in physical geodesy. Geophys J Int 100: 485±514 Tziavos IN (1996) Comparisons of spectral techniques for geoid computations over large regions. Manuscr Geod 70: 357±373 Vening Meinesz FA (1928) A formula expressing the de¯ection of the plumb-line in the gravity anomalies and some formulae for gravity ®eld and the gravity potential outside the geoid. Proc Koninkl Ned Akad Wettenschap. vol 31