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Inverse Vening Meinesz formula and deflection-geoid formula: applications to the predictions of gravity and geoid over the South China Sea

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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: hwang@geodesy.cv.nctu.edu.tw

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¯ection-geoid 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 em-ployed 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.

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

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

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 representa-tion approach. The de¯ecrepresenta-tion-geoid formula, which con-verts 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.

2 Fundamentals

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¯ection-geoid formula. The earth's disturbing potential T can be expanded into a series of spherical harmonics as T …r; /; k† ˆGMr X1 nˆ2 R r  nXn mˆ0 X1 aˆ0 Ca nmYnma …/; k† …1†

where r; /; k are the spherical coordinates (geocentric distance, geocentric latitude and longitude), R is the earth's mean radius, Ca

nmare the harmonic coecients and Ynma are

the fully normalized spherical harmonics de®ned as Ya

nm…/; k†ˆ RSnm…/; k†ˆ Pnm…sin /† cos mk; a ˆ 0 nm…/; k†ˆ Pnm…sin /† sin mk; a ˆ 1



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with Pnm…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† ˆT c0jrˆRˆ R X1 nˆ2 Xn mˆ0 X1 aˆ0 Ca nmYnma …/; k† …3†

and gravity anomaly as Dg…/; k† ˆ ÿoTor ÿ2rT   jrˆRˆ c0 X1 nˆ2 …n ÿ 1† Xn mˆ0 X1 aˆ0 Ca nmYnma …/; k† …4† where c0ˆGMR2 …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

z…/; k† ˆ …n g† ˆ ÿ1 RrN…/; k† ˆ ÿX1 nˆ2 Xn mˆ0 X1 aˆ0 Ca nmrYnma …/; k† …6† where n and g are the north-south component and west-east component of the de¯ection vector, respectively, and r is the gradient operator on the sphere de®ned as:

r ˆ o o/ o cos /ok   …7† Thus the basis functions of the de¯ection vector are rYa

nm, rather than Ynma .

Finally we recall a variant of Green's formula (Meissl 1971, p. 12) ZZ rrf  rgdr ˆ ÿ ZZ rf D gdr ˆ ÿZZ rgD f dr …8†

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 Dˆcos /1 o/o cos /o/o   ‡cos12/ o2 ok2   …9† For the surface spherical harmonics, we have

DYa

nm…/; k† ‡ n…n ‡ 1†Ynma …/; k† ˆ 0 …10†

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

ZZ rrY a nmrYsrbdr ˆ 4pn…n‡1†; if n ˆ s and m ˆ r and a ˆ b 0; if n6ˆ 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 H…wpq† ˆ X1 nˆ2 …2n ‡ 1†…n ÿ 1† n…n ‡ 1† Pn…cos wpq† …12† where p and q are two points on the unit sphere with a spherical separation of wpq, so that (see Fig. 1)

cos wpqˆsin /psin /q‡ cos /pcos /qcos…kqÿ kp† …13†

The Legendre polynomial Pn …cos wpq† in Eq. (12) can be

further decomposed into the series: Pn…cos wpq†ˆ2n ‡ 11 Xn mˆ0 X1 aˆ0 Ya nm…/p; kp†Ynma …/q; kq† …14†

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 getZZ

rrqH…wpq†  rqN…q†drq ˆ R ZZ r X1 nˆ2 n ÿ 1 n…n ‡ 1† Xn mˆ0 X1 aˆ0 Ya nm…p†rqYnma …q† " #  X1 nˆ2 Xn mˆ0 X1 aˆ0 Ca nmrqYnma …q† " # drq ˆ RX1 nˆ2 n ÿ 1 n…n ‡ 1† Xn mˆ0 X1 aˆ0 Ca nmYnma …p†  ZZ r rqY a nm…q†  rqYnma …q†   drq ˆ 4pRX1 nˆ2 …n ÿ 1†Xn mˆ0 X1 aˆ0 Ca nmYnma …p† …15†

Fig. 1. Spherical distance wpqbetween points p and q, and components

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Comparing Eqs. (4) and (15), we have Dg…p† ˆ c0

4pR ZZ

rrqH…wpq†  rqN…q†drq …16†

As shown in the Appendix, the closed form of H is H…wpq† ˆ 1 sinwpq 2 ‡ log sin3 w2pq 1 ‡ sinwpq 2 ! …17† Furthermore, rqH…wpq† ˆdwdH pqrqwpq ˆ dH dwpq owpq o/q owpq cos /qokq ! …18† With Eq. (17) the derivative of H with respect to wpq is

H0ˆ dH dwpqˆ ÿ coswpq 2 2 sin2 wpq 2 ‡cos wpq 2 …3 ‡ 2 sinw2pq† 2 sinwpq 2 …1 ‡ sinw2pq† …19† Di€erentiating Eq. (13) with respect to /q and kq, we

have (cf. Heiskanen and Moritz 1967, p. 113) ÿ sin wpqowo/pq

q ˆ cos /qsin /pÿ sin /qcos /pcos…kqÿ kp†

ÿ sin wpqowokpq

q ˆ ÿ cos /pcos /qsin…kqÿ kp†

…20† Referring to the spherical triangle in Fig. 1, the following relationships hold:

sin wpqcos aqpˆ cos /qsin /pÿ sin /qcos /pcos…kqÿ kp†

sin wpqsin aqpˆ ÿ cos /psin…kqÿ kp† …21†

Thus owpq

o/q ˆ ÿ cos aqp;

owpq

okq ˆ ÿ cos /qsin aqp …22†

Inserting Eqs. (18) and (22) into Eq. (16) we ®nally get the inverse Vening Meinesz formula

Dg…p† ˆ ÿ4pc0 ZZ rH 0 owpq o/q owpq cos /okq !  …nqgq†drq ˆc0 4p ZZ rH 0 n qcos aqp‡ gqsin aqp ÿ  drq ˆ4pc0 ZZ rH 0e qpdrq …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 H0 and the

longitudinal de¯ection components over the unit sphere.

Figure 2 shows the function H0, which changes

rap-idly as w approaches zero. The ®rst zero crossing of H0

occurs at w ˆ 43. When w is small, we have the

as-ymptotic representation: H0…w†  ÿ 2

w2 …24†

It is noted that the asymptotic representation of H0 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†  2

w 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 X1 nˆ2 2n ‡ 1 n…n ‡ 1†Pn…cos wpq† …25† Then, ZZ rrqC…wpq†  rqN…q†drq ˆ R ZZ r X1 nˆ2 1 n…n ‡ 1† Xn mˆ0 X1 aˆ0 Ya nm…p†rqYnma …q† " #  X1 nˆ2 Xn mˆ0 X1 aˆ0 Ca nmrqYnma …q† " # drq

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ˆ RX1 nˆ2 1 n…n ‡ 1† Xn mˆ0 X1 aˆ0 Ca nmYnma …p† ZZ r‰rqY a nm…q†  rqYnma …q†Šdrq ˆ 4pRX1 nˆ2 Xn mˆ0 X1 aˆ0 Ca nmYnma …p† …26†

Thus, the geoidal undulation at point p can be obtained by integrating the scalar products of rH and rN over the unit sphere:

N…p† ˆ4p1 ZZ

rrqH…wpq†  rqN…q†drq …27†

Using Eqs. (22) and (27) we obtain the de¯ection-geoid formula N…p† ˆ4pR ZZ r dC dwpqeqpdrq …28†

According to the Appendix, the closed form of C is C…wpq† ˆ ÿ2 log sinw2pqÿ32cos wpqÿ 1 …29†

and C0ˆ dC dwpq ˆ ÿ cot wpq 2 ‡ 3 2sin wpq …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 C0.

The ®rst zero crossing of C0 occurs at w ˆ 70:5. The

asymptotic representation of C0 when w is small is:

C0…w†  ÿ2

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 for-mula 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) Dg/p…kp† N/p…kp† ( ) ˆ c0 R   D/Dk 4p X/n /qˆ/1 Xkn kqˆk1 H0…Dkqp† C0…Dk qp†  

 …ncoscos aqp‡ gcossin aqp†

ˆ c0 R   D/Dk 4p Fÿ11  ( X/n /qˆ/1  F1…H0…Dkqp† cos aqp† F1…C0…Dkqp† cos aqp†  F1…ncos† ‡ F1…H0…Dkqp† sin aqp† F1…C0…Dkqp† sin aqp†ŠF1…gcos† ) … where ncosˆ n cos /; gcosˆ g cos /; Dkqpˆ kqÿ kp; D/

and Dk are grid intervals in the directions of latitude and longitude, and F1is 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† where i ˆpÿ1. Let Y …k† be the Fourier transform of y…k†, we have H…0† ˆ Re…Y …0††; and H…k† ˆ1 2 Re …Y …k† ‡ Y …n ÿ k†† ‡1 2i Im …Y …k† ‡ Y …n ÿ k†† for k ˆ 1; . . . ; n ÿ 1 G…0† ˆ Im …Y …0††; and G…k† ˆ1 2Re …Y …k† ‡ Y …n ÿ k†† ÿ1 2i 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 ncosand gcos, and the

complex array holding H0…Dk

qp† cos aqp and

H0…Dk

qp† sin aqp (or C0…Dkqp† cos aqp and C0…Dkqp†

sin aqp† are Fourier transformed. Taking advantage of

the gridded data, azimuth and spherical distance can be calculated as

tan aqpˆ ÿ cos /psin Dkqp

ÿ sin…/qÿ /p† ‡ 2 sin /qcos /psin2 Dk2qp

…35† sin2 wqp 2   ˆ sin2 D/qp 2   ‡ sin2 Dkqp 2   cos /qcos /p …36† where D/qp ˆ /qÿ /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 approximated as R2dr

qˆ dxqdyq and wqp ˆqRqp, with

qqp being the planar distance. With the asymptotic

representation in Eq. (24), the inverse Vening Meinesz formula becomes

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Dg…xp; yp† ˆ4pc0 ZZ D ÿ2R2 q3 qp nq…qqpcos aqp†  ‡gq…qqpsin aqp† dxRqdy2 q ˆÿc0 2p ZZ D ypÿ yq ‰…xpÿ xq†2‡ …ypÿ yq†2Š 3 2nqdxqdyq ( ‡ZZ D xpÿ xq ‰…xpÿ xq†2‡ …ypÿ yq†2Š 3 2gqdxqdyq ) ˆÿc2p0 y …x2‡ y2†32 ! n ‡ x …x2‡ y2†32 ! g ( ) …37† where * is the convolution operator and D is the data domain. Schwarz et al. (1990) show that

F xy   …x2‡ y2†ÿ3=2   ˆ ÿ2pi uv   …u2‡ v2†ÿ3=2 …38†

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

DG…u; v† ˆ ic0

u2‡ v2

p ‰vX…u; v† ‡ uE…u; v†Š …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 N…xp; yp† ˆ4pR ZZ D ÿ2R q2 qp nq…qqpcos aqp†  ‡gq…qqpsin aqp† dxRqdy2 q ˆÿ12p ZZ D ypÿ yq …xpÿ xq†2‡ …ypÿ yq†2nqdxqdyq ( ‡ZZ D xpÿ xq …xpÿ xq†2‡ …ypÿ yq†2gqdxqdyq ) ˆÿ12p x2‡ yy 2   n ‡ x2‡ yx 2   g   …40† The Fourier transforms of x

x2‡y2and x2‡yy 2, 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 0 ‰log…1 ‡  a2‡ r2 p † ÿ log rŠJ0…br†rdr ˆ 1 b2…1 ÿ eÿab†; a > 0; b > 0 …41†

Let a in Eq. (41) approach in®nity so that 1 ‡pa2‡ r2 ˆ K remains a constant for any r, and

eÿabˆ 0. Then, Z 1 0 log K r   J0…br†rdr ˆb12 …42†

This means that the Hankel transform of log…K

r† is 2pq12, with q2ˆ u2‡ v2. Furthermore, x y   1 x2‡ y2ˆ xy   1 r2ˆ ÿ o ox o oy ! log Kr   …43† By the di€erentiation theorem of Fourier transform (see, e.g., Mesko 1984), we have

F xy   1 x2‡ y2   ˆ ÿi uv   1 u2‡ v2 …44†

With Eqs. (40) and (44), we obtain the relationship between geoidal undulation and de¯ection components in the frequency domain

N…u; v† ˆ2p…u2i‡ 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 relation-ship [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 H0and C0

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 compo-nents at the neighbourhood of point p can be expanded into the Taylor series (see Fig. 3)

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

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nqˆ np‡ xnx‡ yny‡12ÿx2nxx‡ 2xynxy‡ y2nyy‡    gqˆ gp‡ xgx‡ ygy‡12ÿx2gxx‡ 2xygxy‡ y2gyy‡    …46† where nxˆonox; ny ˆonoy; nxxˆo 2n ox2; nyyˆo 2n oy2 and nxy ˆ o 2n oxoy.

Retaining only the linear terms in Eq. (46) and assuming that the innermost zone is circular, with Eq. (24) we have Dgiˆ4pc0 Z 2p aˆ0 Z s0 sˆ0 ÿ2 s2 ‰…np‡ s sin a nx‡ s cos a ny†

 cos…a ‡ p† ‡ …gp‡ s sin a gx‡ s cos a gy†

 sin…a ‡ p†Šsdsda ˆc0 2p ny Z 2p aˆ0cos 2adaZ s0 sˆ0ds  ‡gx Z 2p aˆ0sin 2adaZ s0 sˆ0ds  ˆs02c0…ny‡ gx† …47†

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

Niˆs 2 0

4…ny‡ gx† …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 s0ˆ  DxDy p r …49†

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

and Parsons (1995) to grid the de¯ections of di€erent azimuths into the north-south and west-east compo-nents at a 20 20interval. 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¯ection-geoid formula. First, over the South China Sea we generated gridded north-south and west-east compo-nents 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. Be-cause 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-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 9.90 2.8

1D FFT/no IE 10.06 2.4

2D FFT 10.11 1.0

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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€er-ence 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 con-siderable attention from the oceanographers in South-east Asia.

9 Conclusion

In this paper we showed the detailed derivations of the inverse Vening Meinesz formula and the de¯ection-geoid formula using the spherical harmonic representa-tions 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 Fig. 4. Grey-shaded relief map of the pre-dicted gravity anomalies over the South China Sea, with illumination from the north-west

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;

aIE: innermost zone e€ect)

Case mean min. max. std. dev. RMS

EGM96 deg 181 to 360 0.000 )2.998 3.812 0.542 0.542

EGM96 - 1D FFT/IEa 0.038 )0.010 0.100 0.014 0.041

EGM96 - 1D FFT/no IE 0.038 )0.055 0.169 0.021 0.043

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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 hwang@geodesy.cv.nctu.edu.tw.

Acknowledgements. This research is funded by the National Science Council of Republic of China, under contract NSC86- 2611-M-009-001-OS, with the project title: Surveying gravity and bathy-metry over the South China Sea using satellite altibathy-metry. The au-thor is grateful to Roger Haagamans and an anonymous reviewer for their important suggestions.

Appendix: derivations of closed forms of H and C functions

Consider the generating function of Legendre poly-nomials (Hobson 1965) f…k† ˆ 1 1 ÿ 2k cos w ‡ k2 p ˆX1 nˆ0 knP n…cos w† …A1†

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

X …w† ˆX1 nˆ0 Pn…cos w† ˆ 1 2 ÿ 2 cos w p ˆ 1 2 sinw2 …A2†

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

f …k† ÿ 1 k ˆ X1 nˆ1 knÿ1P n…cos w† …A3†

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 X1 nˆ1 Z k 0 k nÿ1dkP n…cos w† ˆ X1 nˆ1 1 nknPn…cos w† ˆ Z k 0 f …k† ÿ 1 k dk …A4†

ˆ ÿ log…2p1 ÿ 2k cos w ‡ k2‡ 2 ÿ 2k cos w† ‡ log 4

Setting K=1, we get Y …w† ˆX1

nˆ1

1

nPn…cos w† ˆ ÿ log sinw2 1 ‡ sinw2

 

 

…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

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

(9)

X1 nˆ0 Z k 0 k ndkP n…cos w† ˆ X1 nˆ0 1 n ‡ 1kn‡1Pn…cos w† ˆZ k 0 f…k†dk ˆ log…2p1 ÿ 2k cos w ‡ k2 ‡ 2k ÿ 2 cos w†

ÿ log…2 ÿ 2 cos w† …A6† Setting k ˆ 1, we have Z…w† ˆX1 nˆ0 1 n ‡ 1Pn…cos w† ˆ log 1 ‡ sinw 2 sinw 2 ! …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† ˆX1 nˆ2 …2n ‡ 1†…n ÿ 1† n…n ‡ 1† Pn…cos w† ˆX1 nˆ2 2 ÿ1 nÿ 2 n ‡ 1   Pn…cos w† ˆ 2…X …w† ÿ P0ÿ P1† ÿ …Y …w† ÿ P1† ÿ 2…Z…w† ÿ P0ÿ12P1† ˆ 1 sinw2‡ log sin3 w 2 1 ‡ sinw2 ! …A8† and C…w† ˆX1 nˆ2 2n ‡ 1 n…n ‡ 1†Pn…cos w† ˆX1 nˆ2 1 n‡ 1 n ‡ 1   Pn…cos w† ˆ Y …w† ÿ P1‡ Z…w† ÿ P0ÿ12P1 ˆ ÿ2 log sinw 2ÿ32cos w ÿ 1 …A9† where P0 ˆ 1; P1ˆ cos w. References

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數據

Fig. 1. Spherical distance w pq between points p and q, and components
Figure 2 shows the function H 0 , which changes rap-
Fig. 3. Local rectangular coordinates for the innermost zone e€ect in a cap of radius s 0
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;
+2

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