20320130406015 2-3-4

International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308

(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 6, November – December (2013), © IAEME

145

ELLIPSOIDAL APPROXIMATION FOR TOPOGRAPHIC-ISOSTATIC

MASSES EFFECTS ON AIRBORNE AND SATELLITE GRAVITY

GRADIOMETRY

A.A. MAKHLOOF

Civil Engineering Department, Faculty of Engineering-Minia University

ABSTRACT

The topographic-isostatic masses represent an important source of gravity field information

especially in the high-frequency band of the gravity field spectrum, even if the detailed density func-

tion inside the topographic masses is unknown. If this information is used within a remove-restore

procedure, then the instability problems related to the downward continuation of gradiometer from

airplane or satellite altitude can be reduced. In this paper, integral formulae are derived for the de-

termination of gravitational effects of topographic-isostatic masses of the second order derivatives of

the gravitational potential for various topographic-isostatic models. The application of these formu-

lae is useful especially for airborne gradiometry and satellite gravity gradiometry. The computation

formulae are presented in ellipsoidal approximation by separating the three-dimensional integration

in an analytical integration in ellipsoidal element direction and integration over the unit area. There-

fore, in the numerical evaluation procedure the ellipsoidal volume elements can be considered as

being approximated by mass-lines, located in the centre of the discretization compartments (the mass

of this element is condensated mathematically along its ellipsoidal normal axis). The formulae are

applied to various scenarios of satellite gradiometry measurement campaigns. The gravitational ten-

sor in the ellipsoidal normal direction component at a satellite altitude of 230 km for ESA’s gravity

satellite mission GOCE (Gravity Field and Steady-State Ocean-Circulation Explorer) has been com-

puted. The numerical computations are based on digital elevation models with five arc-minute reso-

lutions for gravity gradiometry effects at satellite altitude.

Keywords: Topographic-Isostatic Models, Ellipsoidal Approximation, Mass-Lines, Satellite Gravity

Gradiometry, Downward Continuation, Regularization, GOCE.

INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND

TECHNOLOGY (IJCIET)

ISSN 0976 – 6308 (Print)

ISSN 0976 – 6316(Online)

Volume 4, Issue 6, November – December, pp. 145-159

© IAEME: www.iaeme.com/ijciet.asp

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IJCIET

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146

1. INTRODUCTION

The determination of the gravity field from observations at aircraft or satellite altitudes is an

improperly posed problem in the sense that small changes in the observations at flight level produce

large effects in the gravity field parameters on the Earth’s surface or geoid level. This holds espe-

cially for the high-frequency constituents of the observation spectrum. To prevent the results from

unrealistic oscillations in the parameters, regularization techniques are usually applied in very poorly

conditioned cases (e.g. Ilk 1993). Most of the regularization methods represent a filtering procedure

and the filtering property can be controlled by a regularization parameter (e.g. Ilk 1998). This is

critical in those cases where the signal shows similar spectral characteristics as the observation noise.

The topographic-isostatic masses represent a gravity field information especially in the high fre-

quency band of the gravity field spectrum which can be superposed with the measurement noise in

aircraft or satellite altitude. Therefore, it is helpful if those signal parts are reduced before the down-

ward continuation and restored afterwards. In this case, it can be assumed that the high-frequency

part in the observations is mainly caused by the observation noise, which can be filtered without

loosing gravity field information. This procedure is only a first step to process airborne or satellite

gravity field information in an integrated computation environment by involving all available Earth

system information as sketched in Ilk (2000).

In global applications as envisaged here, the frequently applied planar approximations of the

topographic-isostatic models cannot be used anymore (see Novák el al. 2001). Therefore, the very

efficient fast Fourier transformation (FFT) techniques cannot be applied for the present computations

as demonstrated by Schwarz et al. (1990) in case of airborne gravimetry applications. Also, spherical

approximation can not give the to be applied, especially for global or large-scale regional applica-

tions. There are two principal possibilities for calculating the effects of the topographic-isostatic

masses on gravitational functionals in ellipsoidal approximation: the representation of the topog-

raphic masses by any ellipsoidal discretisation in form of ellipsoidal compartments (e.g. defined by

geographical coordinate lines) and a subsequent integration (Abd-Elmotaal 1995b, Smith et al. 2001;

Tenzer et al. 2003; Heck 2003) or the representation of the Newton’s integral by a spherical har-

monic expansion (e.g. Sünkel 1985; Rummel et al. 1988; Tsoulis 1999, 2001). Sjöberg (1998) im-

plemented the formulae for the exterior Airy-Heiskanen topographic/isostatic gravitational potential

and the corresponding gravity anomalies. Geoid determinations with density hypotheses from

isostatic models based on geological information have been studied by Kuhn (2003).

The investigations performed thus far are limited to the determination of the second deriva-

tives of the gravitational potential of the topographic-isostatic masses, necessary for Satellite Gravity

Gradiometry (SGG) are not treated for the general case. Only the topographic-isostatic effects on the

vertical component of the gravitational tensor have been studied by Wild and Heck (2004a,b) and

Heck and Wild (2005). The effects of topographic-isostatic masses on satellite-to-satellite tracking

(SST) data and SGG functionals based on spherical harmonic series are investigated by Makhloof

and Ilk (2004). This procedure is very efficient but limited to an upper spherical harmonic degree of

about 2700 which corresponds to a 4 arc-minute resolution. Beyond this degree numerical computa-

tion problems concerning the stability of the recursive computation of Legendre’s polynomials occur

(see e.g. Holmes and Featherstone 2002).

In this paper integral formulae in ellipsoidal approximation and based on mass-lines, ap-

proximating the ellipsoidal volume elements of the topographic-isostatic masses are presented which

can be evaluated numerically based on global digital elevation/bathymetry models over land and

oceans. The formulae are derived for the gravitational potential of topographic-isostatic masses itself,

as well as the first and second derivatives. Different topographic-isostatic models have been investi-

gated such as the Airy-Heiskanen, Pratt-Hayford the formulae for Helmert’s first and second conden-

sation method are derived as well. The computation formulae are applied to the gravitational ele-



147

ments at satellite altitude envisaged for European Space Agency’s (ESA’s) gravity satellite mission

GOCE (Gravity Field and Steady-State Ocean-Circulation Explorer). The effect of the topographic

and isostatic masses

2. THE EFFECT OF THE TOPOGRAPHIC AND COMPENSATING MASSES

In the following, the geoid used as reference surface for the heights given by the DEMs is

approximated by a geocentric reference ellipsoid of major radius (a=6378 km). The geocentric

heights of the computation and the integration points are given from DEM heights, interpreted here

as orthometric heights. Therefore, the Cartesian coordinates of the points can be determined from the

ellipsoidal coordinates as follows (Fig. 1):

[ ]

[ ]2

( ) ( , ) cos cos

( ) ( , ) cos sin

( )(1 ) ( , ) cos ,

= +

= +

= − +

x N h

y N h

z N E h

ϕ ϕ λ ϕ λ

ϕ ϕ λ ϕ λ

ϕ ϕ λ ϕ

(1)

Fig. 1: Geometry of the topographical masses in ellipsoidal approximation

where ( , )h ϕ λ is the ellipsoidal height, refers the topographical surface to the surface of the

geocentric biaxial ellipsoid used in geodesy as a reference body for geometric and the ellipsoidal

prime vertical radius of curvature

2 2 1 2( )

(1 sin )=

−

aN

eϕ

ϕ (2)

where e is the first eccentricity of the reference ellipsoid.

Reference ellipsoid hP`

P`

ξ

l

hQ

hP l1

Surface of the earth



148

The potential of the topographical masses can be computed from Newton integral in ellip-

soidal coordinates as follows (Novák and Grafarend 2005, Fig. 1):

22

0 2 0

( , , ) ( , , )( ) ( ( ) )( ( ) )

= =− =

= = + +∫∫∫ ∫ ∫ ∫Qh

T

Q Q

v

V p G dv G N M d dl l

ππ

λ ϕ π ξ

ρ ϕ λ ξ ρ ϕ λ ξϕ ξ ϕ ξ ξ σ , (3)

with dσ = cos Q Q Qd dϕ ϕ λ , n is the geoid undulation and the ellipsoidal meridian of curvature is given

by:

2 2 3 2( )

(1 sin )=

−Q

Q

aM

Eϕ

ϕ (4)

and the distance between the computation and the integrated point is

1 2

2 2

12l u lξ ξ = + + (5)

with

0 0 0cos cos cos sin sin

= = =

= − + − + − h h hQ P Q P Q Pu x x y y z zξ ξ ξ ξ ξϕ λ ϕ λ ϕ (6)

and

{ }1 2

2 2 2

1 0 0 0P P Pl x x y y z zξ ξ ξ= = =

= − + − + − . (7)

Eq. (3) can be written as the sum of spherical effect and ellipsoidal correction to the spherical ap-

proximation (in case of constant density):

( )T Ts Te

V p V V= +

Where

( , )

2

1 2 30

2 = + + ∫∫Qh

cr

TsV G a k ak k dϕ λ

σ

ρ σ and (8)

( , )2 2 2

1 20

2 (2sin 1) = + − ∫∫Qh

cr

TeV G E a k k d

ϕ λ

σ

ρ ϕ σ (9)

with

1

2 1

1 2 2

3 1 12

ln ,

,

( 3 ) (3 ) .

= + +

= −

= − + −

k u l

k l uk

k u l u l k

ξ

ξ

(10)



149

The direct topographical effect on gravity (the first derivatives of the potential of the topog-

raphical masses with respect to the ellipsoidal surface normal) can be given by:

( , )

( ( ) )( ( ) )

=

+ + ∂ ∂=

∂ ∂ ∫∫ ∫

QhTQ Q

cr

n

N MVG d d

h h l

ϕ λ

σ ξ

ϕ ξ ϕ ξρ ξ σ . (11)

Also, the first derivative of potential can be separated into spherical and ellipsoidal correction

to the spherical term as follow:

∂ ∂ ∂= +

∂ ∂ ∂

T T TV V V

h h hs e

(12)

Where

( , )2

1 2 3

0

2∂

= + + ∂ ∫∫Q

Th

cr

VG a P aP P d

hs

ϕ λ

σ

ρ σ (13)

( , )2 2

1 20

2 (2sin 1)∂

= + − ∂ ∫∫Q

Th

cr

VG a P P d

he

ϕ λ

σ

ρ ϕ σ (14)

With

2

11 2 2 2 2

1 1( ) ( )

Ru Sl R SuP

l u l l u lξ

− −= +

− −, (15)

2 2 2 2

1 1 12 12 2 2 2

1 1

2

( ) ( )

Rl Sl u Sl Ru SuP Sk

l u l l u lξ

− + −= − +

− −, (16)

4 2 2 2 2 2 3 2

21 1 1 1 13 12 2 2 2

1 1

2 3 2 5 6( 3 )

( ) ( )

Sl Rl u Sl u Ru Sl u Su Rl SP R Su k

l u l l u l lξ ξ

+ − + − −= + + + −

− −, (17)

and

0 0 0cos cos cos sin sin

= = =

= − + − + − h h hQ P P P Q P P P Q P PR x x y y z zϕ λ ϕ λ ϕ , (18)

cos cos cos cos cos sin cos sin sin sin= + +P P Q Q P P Q Q P Q

S ϕ λ ϕ λ ϕ λ ϕ λ ϕ ϕ . (19)



150

Then, the effect of the topographic masses at airborne or satellite altitude (the second deriva-

tives of the potential of the topographic masses with respect to ellipsoidal height) can be given by:

( , )2 2

2 2

0

( ( ) )( ( ) ) ∂ ∂ + +=

∂ ∂ ∫∫ ∫

QhT

cr

V N MG d d

h h l

ϕ λ

σ

ϕ ξ ϕ λρ ξ σ (20)

This equation can be transformed to the following equation:

( , )2 2

2 2

0

( , ) ( , ) ( , )2 2 2 2

2 2 2

0 0 0

( ( ) )( ( ) )

1( ) ( ) ( ( ) ( )) .

∂ ∂ + +=

∂ ∂

∂ ∂ ∂= + + +

∂ ∂ ∂

∫∫ ∫

∫∫ ∫ ∫ ∫

Q

Q Q Q

hT

cr

h h h

cr

V N MG d d

h h l

G N M d N M d d dh l h l h l

ϕ λ

σ

ϕ λ ϕ λ ϕ λ

σ

ϕ ξ ϕ λρ ξ σ

ξ ξρ ϕ ϕ ξ ϕ ϕ ξ ξ σ

(21)

Three integral of Eq. (21) can be estimated separately; the first term

( , ) ( , )2 2 2 2

2 3 5

0 0

2 2

2 2

1

1

1 1 ( 2 )( ) ( ) ( ) ( ) 3 ,

( )( ) ( ) 3 6 3

( )

: ( ) ( )

∂ + += − +

∂

− += + + +

−

=

∫ ∫Q Qh h

R RS SN M d N M d

h l l l

uN M R A RSB S C

l u

N M w

ϕ λ ϕ λξ ξ

ϕ ϕ ξ ϕ ϕ ξ

ξϕ ϕ

ϕ ϕ

(22)

Where

( , )

5 2 2 3 2 2

1 10

1 ( ) 2( )

3( ) 3(

+ += = +

− − ∫

Qh

u uA d

l l u l l u l

ϕ λξ ξ

ξ (23)

( , )

5 3

0

1

3

− = = + ∫

Qh

B d uAl l

ϕ λξ

ξ (24)

( , ) 22

1

5 3

0

.2 2 2

−= = − +

∫

Qh

luC d B A

l l

ϕ λξ ξ

ξ (25)

The second integral is given by:

[ ] [ ]

[ ]

[ ]

( , ) ( , )2 2 2 2 3

2 3 5

0 0

22 21

2 2

1

2

1 ( 2 )( ) ( ) ( ) ( ) 3 ,

( )( ) ( ) 3 6 3 ,

( )

: ( ) ( )

∂ + ++ = + − +

∂

+= + + + +

−

= +

∫ ∫Q Qh h

R RS SN M d N M d

h l l l

l uN M R B RSC S D

l u l

N M w

ϕ λ ϕ λξ ξ ξ ξ

ϕ ϕ ξ ϕ ϕ ξ

ξϕ ϕ

ϕ ϕ

(26)



151

where

( , ) 3 3

2

15 32 .

= = + +

∫

Qh

o

D d uC l Bl l

ϕ λξ ξ

ξ (27)

The third integral of Eq. ( 21 ) is given by:

( , ) ( , )2 2 2 2 2 3 2 4

2 3 5

0

2 22 21

12 2

1

3

( 2 )3

( 2 )3 6 3 ,

( )

:

∂ + += − +

∂

−= − + + +

−

=

∫ ∫Q Qh h

n

R RS Sd d

h l l l

l uk R C RSD S F

l u l

w

ϕ λ ϕ λξ ξ ξ ξ ξ

ξ ξ

ξ (28)

where

( , ) 2 24 3

2 11 15 3 2 2

10

( 2 )2 .

3 ( )

−= = − + − +

− − ∫

Qh

l uF d uD l B k

l l l u

ϕ λξξ ξ

ξ (29)

Using the binomial expansion

1

221 3

(1 ) 1 ..........2 8

x x x−

− = + + + (30)

that can be successfully be truncated for 1px , both radii of curvature can be written in the follow-

ing form:

2 21( ) 1 sin

2

= +

N a Eϕ ϕ (31)

2 2 23( ) 1 sin

2

= + −

M a E Eϕ ϕ (32)

Then, Eq. (21) can be written in the following formula:

2 2 2 2

2

1 2 32 2 2 2 2 3 2

(1 ) 2 (1 sin )

(1 sin ) (1 sin )

∂ − − += + +

∂ − − ∫∫ ∫∫ ∫∫

TV E E

a w d a w d w dh E Eσ σ σ

ϕσ σ σ

ϕ ϕ. (33)

The effect of the topographic masses can be written also in two terms: one of the spherical

effect and the other is the correction to spherical term.

Eq. (33) are used to compute the effect of the topographic masses for the case of Bouguer

model, Airy-Heiskanen model. For calculating the effect of condensation masses in case of Helmert



152

second method of condensation and generalized Helmert method of condensation (Heck 2003), the

effect of the condensation masses can be given as follow:

1 1

2

( ) ( )( )= − −∫∫CM k

V p G N D M D dlσ

σ (34)

where k is he surface layer density and given by (Novak and Grafarend (2005), 1D is the

Helmert condensation depth and it equals zero in Helmert second method of condensation and 32 km

in case of generalized Helmert of condensation , 2l here denotes the distance between the computa-

tion point and the integrated point at the condensation surface. In case of constant mass density of the

topographical masses, the surface mass density is given by (Novák and Grafarend 2005)

3

1( , ) 1

2 *M 3 *

+= + +

Q Q Q Q

Q Q

Q Q Q Q

h N M hk h

N N Mϕ λ ρ (35)

Eq. (34) can be transformed also in two terms: one for the spherical approximation and the

other is the ellipsoidal correction to the spherical approximation. The first and second derivative of

the condensated topography can be determined as follows:

1 1

2

2 2

1 12 2

2

( ) 1( )( )

( ) 1( )( )

∂ ∂= − −

∂ ∂

∂ ∂= − −

∂ ∂

∫∫

∫∫

CM

CM

V pG k N D M D d

h h l

V pG k N D M D d

h h l

σ

σ

σ

σ

(36)

In case of the Airy-Heiskanen model the topographic masses of constant density ρ float on a

mantle of constant but larger density mρ . An elevation column of height h is compensated by a cor-

responding root of thickness t. The higher the topographic features are, the deeper they sink. Thus

the thickness of a root column under a mountain column with height h can be determined by the for-

mula

( )( ) ( ) ( )0

−

= =− −

+ + = ∆ + +∫∫ ∫ ∫∫ ∫Q

Q

h T

T t

N M d d N M d dσ ξ σ ξ

ρ ξ ξ ξ σ ρ ξ ξ ξ σ . (37)

After Performing the numerical integration for finding the root for height larger than 10 km

and comparing the result with results from spherical formula and the error was lesser than 0.01%.

Then the spherical formula for computing the roots is applied in the present investigation. The root is

given by (Khun 2000)

3 3

33

( )( ) 1 1

( )

+ − = − − − − ∆

R h Rt R T

R T

ρ

ρ, (38)

and the anti-roots of thickness ′t in case of oceanic water columns of height ′h by,



153

3 3

33

( ) ( )( ) 1 1

( )

′ − − − ′ = − + − − ∆

wR R h

t R TR T

ρ ρ

ρ, (39)

with the water density wρ and the mean radius of the Earth R (R=6371 km). The density difference

∆ = −mρ ρ ρ in the roots and anti-roots produces a buoyancy so that hydrostatic equilibrium is

reached.

The above formulae can be simplified if the heights of the Earth’s surface are expressed by

the rock-equivalent topographic heights eqh (Rummel et al. 1988). It can be expressed in planar ap-

proximation as follows:

land: h

ocean: h

=

= − ′=

eq

eq weq

h

hh

ρ ρ

ρ

, (40)

The potential of the isostatic masses at the computation point P can be determined analogous-

ly as the potential of the topographic masses by Newton’s integral,

22

0 2

( , , ) ( , , )( ) ( ( ) )( ( ) )

−

= =− =− −

∆ ∆= = + +∫∫∫ ∫ ∫ ∫

T

I

Q Q

v T t


ππ

λ ϕ π ξ

ρ ϕ λ ξ ρ ϕ λ ξϕ ξ ϕ ξ ξ σ . (41)

This integration can be written as the sum

( )I Is IeV p V V= + (42)

where

2

1 2 32−

− − = + + ∫∫

Q

T

crT t

IsV G a k ak k d

σ

ρ σ (43)

2 2 2

1 22 (2sin 1)

−

− − = + − ∫∫

Q

T

crT t

IeV G E a k k dσ

ρ ϕ σ (44)

The combined effects of the topographic-isostatic masses on the different gravity functionals

are the differences between the effect of the topographic masses and the effect of the isostatic com-

pensation masses. It reads e.g.,

= == −

P P P

T C

TI h h h h hV V V (45)

In case of the Pratt-Hayford model a certain adjustment surface is defined, in case of a ellip-

soidal approximation an ellipse in a constant depths 2D ( 2 100=D km ). At this ellipsoidal surface

hydrostatic equilibrium is anticipated, i.e. the pressure of any topography column is identical at this



154

ellipse which requires constant mass but different density depending on the elevation of the surface

of the Earth. The effect of the topographic masses according to the Pratt-Hayford model is calculated

for two cases: one for the land areas and the other for the ocean areas. The same has to be done for

the effects of the isostatic compensation masses.

Then, the effect of the topographic masses is given by

( )− = +P T land Ocean

T TV p V V (46)

where 22

0 2 0

( , , )( ) ( ( ) )( ( ) )

= =− =

= = + +∫∫∫ ∫ ∫ ∫Qh

land L LT Q Q

v


ππ

λ ϕ π ξ

ρ ρ ϕ λ ξϕ ξ ϕ ξ ξ σ , (47)

22 0

0 2

( ) ( )( ) ( ( ) )( ( ) )

′= =− =

− −= = + +∫∫∫ ∫ ∫ ∫

Ocean w w

T Q Q

v h


ππ

λ ϕ π ξ

ρ ρ ρ ρϕ ξ ϕ ξ ξ σ (48)

The effect of the compensation masses for land areas is given by

22

0 2 0

( ) ( )( ) ( ( ) )( ( ) )

= =− =

− −= = + +∫∫∫ ∫ ∫ ∫

Qh

land L LC Q Q

v


ππ

λ ϕ π ξ

ρ ρ ρ ρϕ ξ ϕ ξ ξ σ , (49)

and for ocean areas by:

22

0 2

( ) ( )( ) ( ( ) )( ( ) )

−

= =− =−

− −= = + +∫∫∫ ∫ ∫ ∫

Qh

Ocean O OC Q Q

v D


ππ

λ ϕ π ξ

ρ ρ ρ ρϕ ξ ϕ ξ ξ σ (50)

where density Lρ in land areas and density under ocean Oρ can be determined from Kuhn (2000).

3 3

3 3

( )

( ) ( )

− −=

+ − − L

R R D

R h R Dρ ρ , (51)

3 3 3 3

3 3

( ) ( )

( ) ( )

′ − − − − − =′ − − −

w

O

R R D R R h

R h R D

ρ ρρ , (52)



155

3. TEST REGIONS

The numerical tests are based on the ETOPO5 with five arc-minute resolution. One typical

test regions have been selected: This test area is covering the Himalaya region (Fig. 2), shall be used

to visualize the damping of the topographic-isostatic effects in the component of the gravity gradi-

ometry in the ellipsoidal normal direction at 230km above sea surface. In this case resolution for the

elevation compartments has been used. The accuracy of this DEM is sufficient to demonstrate the

topographic-isostatic effects at satellite altitude.

Fig. 2: Topography of the test area (ETOPO5)

The densities of crust and topography are considered to be constant and equal to 2670 kg/m3.

As density of sea water a value of 1030 kg/m3 has been taken and as density of the mantle the fre-

quently used value of 3270 kg/cm3 has been assumed. The Airy-Heiskanen depth of compensations

is considered to be 30 km, The depth of the condensation surface of Helmert’s first condensation

method is 21 km and 32 km for the generalized Helmert’s condensation model.

4. NUMERICAL ANALYSIS

To give an impression of the size of the topographic-isostatic effects on the gravity

gradiometry at satellite altitude, the topographic-isostatic effects on the of the gravitational tensor at

a satellite altitude of 230 km are computed. It can be found that, the structure of the topographic-

isostatic effects at a satellite altitude of 230 km shows still steep changes with a gradient of 0.80

Eötvös per 100km in north-south direction and of 0.60 Eötvös per 100 km in east-west direction. If it

is possible to remove this sort of roughness from the observations with a noise level of approximate-

ly 5 mE for GOCE the downward continuation can be considerably simplified. The GOCE mission is

designed to derive the static part of the gravity field with an extremely high precision. Therefore, it is

very important to filter the observations by the topographic-isostatic gravity field effects to ease the

requirements for the downward continuation. An additional regularization might be avoided in that

case; but this depends on the envisaged resolution of the gravity field model and on the error level of

the observations. Fig. 3 gives an impression of the size of these effects for the hh-component of the

gravitational tensor.



156

5. CONCLUSIONS

In this paper the formulae for the calculation of the first and second derivatives of the gravita-

tional potential of the topographic-isostatic masses are derived for various frequently applied topo-

graphic-isostatic models in ellipsoidal approximation by approximating the ellipsoidal volume ele-

ments by mass-lines located in the middle of the compartments. Only the formula for the determina-

tion of the first derivatives in the ellipsoidal approximation has been derived by one author. The situ-

ation is different for the second derivatives of the potential; the second derivatives of the potential in

the ellipsoidal normal direction have not been studied till now.

The formulae can be used to determine the topographic-isostatic effects at aircraft altitudes

for airborne applications or for satellite altitudes to reduce the observation functionals of airborne or

satellite gravity gradiometry missions. The integral formulae presented here allow to use DEMs with

an - in principle - arbitrary high resolution depending on the numerical integration method. Obvious-

ly such a high resolution is necessary for airborne altitudes. In these cases it might not be to apply

formulae which are based on the expansion of Newton’s integral in spherical harmonics at least for

global applications (see e.g., Makhloof 2007). As the higher resolution of the DEMs is indispensable

and the computation of the corresponding spherical harmonic coefficients can be critical because of

numerical problems (Holmes and Featherstone 2002). Therefore only a maximum degree of 2700

corresponding to a compartment size of 4 arc-minutes resolution can be selected. Due to this limita-

tion a spherical harmonic expansion of the topographic-isostatic masses cannot be used for exact

determinations of the geoid (see Kuhn and Seitz 2005).

a) Bouguer method (topography) b) Generalized Helmert (D=32km)

b) Helmert first method (D=21 km) b) Airy-Heiskanen method



157

f) Helmert first method

Fig. 1: Effects of the topographic-isostatic masses on the tensor component zzV at an altitude of 230

km for different topographic-isostatic models (Eötvös).

Now the question arises which model should preferably be used for the filtering of satellite-

borne observations such as gravity gradients as preparation for the subsequent downward continua-

tion procedure? While Helmert’s second condensation method might be useful for geoid computa-

tions, its usefulness for the processing of satellite in-situ observations cannot be answered in such a

simple generally valid way. Indeed, the task of filtering topographic-isostatic effects in the satellite

observations is helpful only if these quantities have a significant magnitude larger than the observa-

tion noise. Obviously, this fact may depend on the validity of the topographic-isostatic hypothesis in

specific geographic regions. This can be decided only after a careful analysis of the specific gravity

field features within the various geographical regions of the Earth to find out which model describes

the reality in these regions in the best possible way. It is well-known that the Earth is isostatically

compensated by an amount of approximately 90% (Heiskanen and Moritz 1967), but it is difficult to

decide which model fits best. Although seismic measurement results indicate the validity of an Airy-

Heiskanen type of topographic-isostatic compensation, but in some parts of the Earth the isostatic

compensation seems to follow anther model (Heiskanen and Moritz 1967). The change of the con-

densation level by using a sort of a generalized Helmert model (Heck 2003) could be used to fit the

topographic-isostatic model to the reality. Very promising seems to be to introduce geophysical

models in coincidence with modern models of plate tectonics.

If the specific topographic-isostatic model holds more or less uniformly for a larger region

then this model can be used to filter the satellite observations before the application of the regional

gravity field recovery procedure. The situation is more complicated in case of regionally varying

deviations of the reality from a specific model; further investigations are necessary to consider these

cases. Because of the varying effects of the topographic-isostatic models depending on the type of

observables such as gravity vectors or tensor components the frequently expressed argument that a

high resolution gravity field model might be sufficient to reduce observations at aircraft or satellite

altitude is not valid. Therefore, additional investigations are necessary to demonstrate the benefit of a

remove-restore procedure taking into account individually selected topographic-isostatic models for

the processing of airborne measurements and SGG observables. Finally, the results computed here

are computed with the results computed using spherical approximation and it is found that the ellip-

soidal approximation gives exact results.



158

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