Terrain corrections in gravimetry and gradiometry...

Institut National des Sciences Appliquées de Strasbourg

M. Sc. DissertationSection Surveying and Geomatics

Terrain corrections

in gravimetry and gradiometry for

GRACE and GOCE satellite missions

Presented in June 2011 by Nathalie VANNES

Carried out in : Dublin Institute for Advanced Studies5 Merrion Square, Dublin 2Ireland

Supervision : Evaluation :Zdeněk Martinec Gilbert Ferhat

Pascal Bonnefond

English version

Acknowledgements

I would like to express my deepest gratitude to my supervisor Zdeněk Martinec who of-fered me this fascinating subject, and guided me throughout my work with such enthusiasm.He was always available to give attention to my work despite his busy schedule. He is a sourceof information and knowledge and knows how to share it.

I would like to thank Gilbert Ferhat and Pascal Bonnefond who prepared me for this workand followed my progress. Their opinions and advices have been essential for my development.

I am greatly appreciative of all the staff at the Dublin Institute for Advanced Studies,who were here to bring me solutions on the specific scientific softwares. My gratitude goesespecially to Celine Tirel and Joanne Adam who spend their time and vigour to give mesuggestions and comments on my work in French.

I acknowledge as well my former classmates who encouraged me during this project andreviewed my work.

Lastly, I offer my regards and blessings to Marcin Kałęcki who supported and believed inme, and to all of those who helped me in one way or another during the completion of theproject.

i

Contents

Introduction 1

1 State of the art and definitions 31.1 Newton integral for gravitational potential . . . . . . . . . . . . . . . . . . . . 31.2 Topographical masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Residual potential of topographical masses . . . . . . . . . . . . . . . 41.2.2 Total potential of topographical masses . . . . . . . . . . . . . . . . . 41.2.3 Bouguer plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.4 Terrain roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Compensation of the gravitational effects of topographical masses . . . . . . . 61.3.1 Isostatic compensation models . . . . . . . . . . . . . . . . . . . . . . 61.3.2 Helmert condensation layer . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Satellites missions GRACE and GOCE . . . . . . . . . . . . . . . . . . . . . . 71.5 The Remove-Compute-Restore procedure (RCR) . . . . . . . . . . . . . . . . 9

2 Numerical studies over Ireland, France and Iran 112.1 The study areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Direct topographical effect on gravity . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 On the Earth’s surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 At satellite altitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Direct topographical effect on gradiometry . . . . . . . . . . . . . . . . . . . . 172.3.1 Residual effect δE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.2 Total effect V t

rr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 Results of the direct effect on gradiometry over the three areas . . . . . . . . 202.5 Conclusions on the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Conclusions and future works 23

iii

List of symbols 25

List of acronyms 27

List of figures 28

List of tables 30

References 31

A Appendix 35A.1 Work Package ESA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35A.2 Bruns’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

A.2.1 Proof of Bruns’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . 38A.2.2 Expressions of the undulation N and the indirect effect δN . . . . . . 39

A.3 Maps of the direct topographical effect on surface gravity over France . . . . 40A.4 Maps of the direct topographical effect on satellite gravity over Ireland . . . . 40

iv

Introduction

Determination of the figure of the earth

The mathematical figure of the Earth was first described by Carl Friedrich Gauss (Gauss,1828 ; Hofmann-Wellenhof & Moritz, 2006). Johann Benedict Listing (1873) later named it thegeoid.

Earth'splumb line

geoid

ellipsoid

Equipotential

ocean

W=W0

H

N

ellipsoidal normal

surface

surfaces

surface

~~

~~

~~

Fig. 1 Earth’s surface, geoid, ellipsoid and theheights separating them.

The geoid is defined as an equipotentialsurface along which the Earth’s gravity po-tential W is constant, equal to a referencevalue W0 (Fig.1). This datum is chosen suchthat the geoid coincides with a mean level ofthe oceans, and can be mathematically ex-tended over the continents. As a result ofunequal distribution of masses in the Earth’sinterior, the geoid is irregularly shaped. Itdescribes the figure of the Earth by a phys-ical quantity, the gravity potential, in con-trast to the idealized geometrical figure of a reference ellipsoid. The separations between thetwo surfaces is called the geoid undulation N , or geoidal heights.

The distance between the levelling point at the Earth’s surface and the geoid countedalong the plumb line 1 is the so-called orthometric height H. Hence, the geoid is geometricallyconsidered as the reference surface, or the "level 0" of the orthometric heights.

Since the geoid cannot be measured directly, George Gabriel Stokes (1849) derived aformula for computing the geoid from gravity measurements 2 Stokes assumed that• there are no masses outside the geoid,

1. A plumb line is a curve to which the vertical axis of the instrument, when adjusted, is tangential.2. Gravimetry is the method of measuring gravity and the instrument used is called a gravimeter . In the

past, gravity data were exclusively provided by terrestrial surveys. Later transportable relative gravimeters weredesigned for the use on ship and airborne, however, the data accuracy was very variable and geographicallyuneven distributed. Recently, satellites gravimetry has emerged providing the global coverage of repeatedmeasurements.

1

• the gravity measurements are referred to the geoid.

However, the presence of the topography violates these assumptions because• the geoid is a surface extended through the continents so there are masses of continents

outside the geoid and below the topography,• the geoid is a mathematical surface where the gravity measurements cannot directly be

carried out.

Motivation of the study

To adjust Stokes’s requirements, the Remove-Compute-Restore (RCR) procedure (Fors-berg & Tscherning, 1981) is used :

Remove step : the gravitational effect of the residual topographical masses is first sub-tracted from the data,

Compute step : the so-called residual geoid, or co-geoid is computed,

Restore step : the contribution of the topography is added to the solution.

The aim of this work is to study and compute the gravitational effect of the residualtopographical masses, which is needed in the first step of the RCR procedure. The point-wise computation method was chosen as it has never been applied in processing of satellitedata. This study focuses on GRACE and GOCE satellite missions because they are the twolatest satellites for very precise, long-wavelength geoid determination from gravity (GRACEmission) and gradiometry 3 (GOCE mission) measurements, respectively. Note that there is aESA project on processing and interpretation of GOCE observations that should have start inMay 2011. The results of this study will be used as a first step for the GOCE data processingunder Work Package WP6 (see WP6.2 in Apendix A.1) of which the Dublin Institute forAdvanced Studies (DIAS) is scheduled to be the contractor.

First, the basic notions needed for a good understanding of the study will be described,such as the Newton integral for gravitational potential, the measurement concepts and de-scription of the two satellites missions and the Remove-Compute-Restore procedure.

Then the mathematical expressions for the gravitational effects of topographical massesare derived by using Newton’s integral and the Helmert second condensation layer approach.

After a preparatory step for programming and numerical implementation, the results ofthe computation of the topographical effects will be analyzed.

3. Gravity gradiometry is the study of variations in the acceleration due to gravity. It is the measurementof the rate of change of gravitational acceleration called gravity gradient is the spatial.

2

Chapter 1

State of the art and definitions

1.1 Newton integral for gravitational potential

#

'

P

P

#

'

L

O

rr

z

x y

Fig. 1.1 Spherical coordinates ofthe computation point P (r,Ω), anintegration point P ′(r′,Ω′), the

distance L and angular distance ψbetween P and P ′.

The Newton’s volume integral for the gravitational po-tential V at a computation point P (r,Ω) of radius r andgeocentric direction Ω defined by the (co-latitude ϑ, longi-tude ϕ),Ω = (ϑ, ϕ), is

V (r,Ω) = G

∫Ω′∈Ω0

∫ rt(Ω′)

r′=0

%(r′,Ω′)L(r, ψ, r′)r

′2dr′dΩ′ , (1.1)

where G is the Newton’s gravitational constant, G =6.67 · 10−11 m3.kg−1.s−2, rt is the radius of the surface ofthe Earth, %(r′,Ω′) is the mass density inside the Earth’sinterior located at P ′(r′,Ω′), L(r, ψ, r′) is the distance be-tween P and P ′ and ψ is the angular distance betweenthe geocentric directions Ω = (ϑ, ϕ) and Ω′ = (ϑ′, ϕ′), seeFig.1.1,

cosψ = cosϑ cosϑ′ + sinϑ sinϑ′ cos(ϕ− ϕ′) . (1.2)

The integration of Ω′ (Eqn.(1.1)) is taken over the full solid angle Ω0. In the text, the abbre-viation ∫

Ω0=∫ 2π

ϕ=0

∫ π

ϑ=0(1.3)

is used for the integral over the full solid angle Ω0, and dΩ′ = sinϑ′dϑ′dϕ′.

3

Chap 1. State of the art and definitions

1.2 Topographical masses

1.2.1 Residual potential of topographical masses

The topographical masses are the masses outside the geoid of radius rg, and below thetopographical surface of radius rt.

Since geophysical observations show that there must exist a compensation mechanismwhich reduces the gravitational effect of topographical masses (Martinec (1998) ; Hofmann-Wellenhof & Moritz (2006)), we thus introduce the gravitational potential of the compensatedmasses V c (see later, section 1.3) as an approximation of the gravitational potential of thetotal masses V t (see later, section 1.2.2).

The residual topographical potential of the topographical masses is the difference betweenV t and V c,

δV := V t − V c. (1.4)

1.2.2 Total potential of topographical masses

The gravitational potential V t generated by the topographical masses is

V t(r,Ω) = G

∫Ω0

∫ rt(Ω′)

r′=rg(Ω′)

%(r′,Ω′)L(r, ψ, r′)r

′2dr′dΩ′ . (1.5)

To abbreviate notations, Martinec (1998) introduces the symbol L−1(r, ψ, r′) for an indef-inite radial integral of the Newton kernel,

L−1(r, ψ, r′) :=∫r′

r′2

L(r, ψ, r′)dr′ . (1.6)

Assuming that the density of the topographical masses does not vary in radial direction, thatis %(r′,Ω′) = %(Ω′), and subsituting Eqn.(1.6) in Eqn.(1.5), the Newton’s volume integral forthe gravitational potential V t becomes

V t(r,Ω) = G

∫Ω0%(Ω′) L−1(r, ψ, r′)

∣∣∣rt(Ω′)

r′=rg(Ω′)dΩ′ . (1.7)

1.2.3 Bouguer plate

The Bouguer plate, used as an approximate model in gravity and gravity anomaly com-putations account for the bulk of topographical effects.

In spherical geometry Fig.1.2(b), the Bouguer plate is regarded as a spherical layer ofthickness HP and density %0. The gravitational potential of the spherical Bouguer layer is

V B(r,Ω) = G%0

∫Ω0L−1(r, ψ, r′)

∣∣∣rt(Ω)

r′=rg(Ω)dΩ′ . (1.8)

4

1.2. Topographical masses

P

P

H

g

P%

t

0

geoid

topography

(a) Infinite Bouguer plate

P

P

H

g

P

geoid

topographyt

%0

(b) Spherical Bouguer layer

Fig. 1.2 The Bouguer plate in cartesian (a) and spherical (b) geometry.

For evaluating this integral, the radius of the geoid rg(Ω) is approximated by a sphere ofconstant radius R = 6371 km which is the mean radius of the Earth. The integral Eqn.(1.8)can be evaluated analytically, e.g Wichiencharoen (1982). We will use the spherical Bouguerlayer model since it gives more accurate results than the planar approximation (Martinec &Vaníček, 1994b ; Rózsa, 1998). Moreover, this approximation of the geoid is good to about0.5% according to Martinec & Vaníček (1994a).

1.2.4 Terrain roughness

Since the actual Earth’s surface deviates from the Bouguer sphere, there are deficienciesand abundances of topographical masses with respect to the mass of the Bouguer plate Fig.1.3.These contribute to the topographical potential V t through the term V R as

P

P

H

g

P

geoid

Bouguer shellt

topography

deficiences

abundances

Fig. 1.3 Roughness of the terrain.

V t(r,Ω) = V B(r,Ω) + V R(r,Ω). (1.9)

The terrain roughness term V R (Martinec & Vaníček, 1994a) is expressed by the Newtonintegral

V R(r,Ω) = G%0

∫Ω0

[L−1(r, ψ, r′)

∣∣∣rt(Ω′)

r′=R− L−1(r, ψ, r′)

∣∣∣rt(Ω)

r′=R

]dΩ′ . (1.10)

5


1.3 Compensation of the gravitational effects of topographicalmasses

1.3.1 Isostatic compensation models

Two extremely idealized isostatic compensation models were proposed in the past to ap-proximate the effect of topographical abundances from surface gravity observations. ThePratt-Hayford model, outlined by J.H Pratt in 1854, assumes that the mountains have risenfrom the underground somewhat like a fermenting dough (Hofmann-Wellenhof & Moritz,2006, sect.3).The Airy-Heiskanen model, proposed by G.B Airy in 1855, assumes that the mountains arefloating on a fluid lava of higher known density (somewhat like an iceberg floating on water)(Hofmann-Wellenhof & Moritz, 2006, sect.3).

1.3.2 Helmert condensation layer

In the limiting case, the topographical masses may be compensated by a thin mass layerlocated on the geoid (somewhat like a glass sphere made over very thin but very robust glass(Hofmann-Wellenhof & Moritz, 2006)).

H

topography

Pg

Pt

condensationlayer

topographical masses

Helmert'sgeoid

%0

%0

Fig. 1.4 Helmert condensation layer of density σ

As shown Fig.1.4, the topographical masses are condensed as a surface mass layer on thegeoid. This kind of compensation is called the Helmert 2nd condensation (Helmert, 1884),and will be used for this study. It approximates the actual potential of the topographicalmasses V t by the potential of a single layer V c which is expressed by Newton’s integral as

V c(r,Ω) = GR2∫

Ω0σ(Ω′)L−1(r, ψ,R)dΩ′ , (1.11)

where σ(Ω) is a surface density of the Helmert’s condensation layer, and L−1 is the reciprocaldistance 1/L. Analogously to Eqn.(1.9), we can rewrite Eqn.(1.11) as

V c(r,Ω) = V σ,B(r,Ω) + V σ,R(r,Ω) , (1.12)

6

1.4. Satellites missions GRACE and GOCE

whereV σ,R(r,Ω) = GR2

∫Ω0

[σ(Ω′)− σ(Ω)

]L−1(r, ψ,R)dΩ′ . (1.13)

The symbol V σ,B(r,Ω) denotes the gravitational potential of a spherical layer with densityσ(Ω) and radius R. The condensation density σ(Ω) can be chosen in a variety of ways. Forthis study, it will be chosen as in Martinec & Vaníček (1994a) and Martinec (1998) such thatthe principle of conservation of topographical masses (Wichiencharoen, 1982) is respected :

σ(Ω) = %0τ(Ω) , (1.14)

withτ(Ω) = H(Ω)

(1 + H(Ω)

R+ H2(Ω)

3R2

). (1.15)

1.4 Satellites missions GRACE and GOCE

Satellites gravity missions are designed to provide global, regular and dense gravity datasets of high and homogenous quality. A satellite is considered as a mass in free fall in theEarth’s gravitational field and the gravitational field is deduced from its orbital motion aroundthe Earth. Four fundamental criteria have to be fulfilled to overcome the limitations of groundtracked satellites (Rummel et al., 2002) :

1. Uninterrupted tracking in three spatial directions,

2. Orbit as low as possible for a strong gravity signal,

3. Measurement or compensation of the effect of non-gravitational forces (air drag, radia-tion pressure),

4. Counteract gravity field attenuation with the altitude.

As shown in Fig.1.5, the two concepts of Satellite-to-Satellite Tracking in low-low mode(SST-ll) and Satellite Gravity Gradiometry (SGG) meet the four criteria when combined withthe concept of Satellite-to-Satellite Tracking in high-low mode (SST-hl) (that only meets thethree first criterias).

In SST-hl, a Low Earth Orbiter (LEO) is tracked by high orbiting satellites, and a 3-Daccelerometer placed in the center of its mass measures or compensate the effect of non-gravitational forces.

The GRACE mission

The Gravity Recovery and Climate Experiment (GRACE) mission 1 realizes the measure-ment concept of SST-ll (Fig.1.6(a)).

1. GRACE is a joint project between NASA and the German Space Agency, DLR launched in March 2002and expected to end in 2015.

7


•3-D accelerometry

•gravity acceleration of 1 LEO

•measurement : grad V

SST-hl

•inter-satellite link

•acceleration differences between 2 LEOs•measurement : d(grad V )/dt

SST-llcombined with SST-hl

•gradiometry

•acceleration gradients within 1 LEO

•measurement : grad grad V

SGG combined with SST-hl

4. Counteract gravity field attenuation

3. Measure of non-gravitational

forces

2. Low orbit

1. Continuous tracking

GPS receiver,

+Laser retroreflector

LEO

3-D accelerometer

additionnal LEO,

+ 2 accelerometer

Gradiometer

(3 pairs of accelerometers)

CH

AM

P

mis

sion

GR

AC

E

mis

sion

GR

AC

E

mis

sion

CH

AM

P

mis

sion

GO

CE

m

issi

on

Fig. 1.5 Satellite-measurement concepts meeting the four fundamental criteria.

In SST-ll, two LEOs are placed in the same orbit of 400 km in this case, separated byseveral hundred kilometers (D = 220 km here). The range D between both spacecrafts ismeasured with the highest possible accuracy of the order of the µm.

Global gravity is mapped by measuring the Earth’s mass variations from place to placeas the twin satellites pass over. The speeding up and slowing down of the satellites providesthe measure of the distance between them and, therefore, a mapping of the Earth’s gravityfield with a spatial resolution of 200 km.

Acceleration differences measured by this technique are mathematically expressed by thedifference of the first derivatives of the gravitational potential δV . The differenciation ofobservables provides a much higher sensitivity than the hl-technique.

(a) SST-ll for GRACE (b) SGG for GOCE

Fig. 1.6 Measurement concepts of ’SST-ll’ and ’SGG’ (Seeber (2003), p 471 ; Hofmann-Wellenhof& Moritz (2006), p 278-279).

8

1.5. The Remove-Compute-Restore procedure (RCR)

The GOCE mission

The Gravity fields and steady-state Ocean Circulation Explorer (GOCE) mission 2 is thefirst mission that realizes the measurement concept of SGG (Fig.1.6(b)).

GOCE is equipped by three pair of accelerometers called gradiometer, placed in one LEO.The objective of the mission is measuring the Earth’s gravity field and modelling the geoid,with extremely high accuracy and a spatial resolution of 100 km. The orbit altitude of theLEO is 250 km.

In SGG in situ measurement of acceleration gradients is mathematically expressed bythe second derivatives of the gravitational potential. The advantage, compared with the SSTtechnique, is that non-gravitational accelerations vanish by differencing.

1.5 The Remove-Compute-Restore procedure (RCR)

The RCR procedure is a method that fulfills Stokes’s requirements 3 for computing thegeoid.

In Fig.1.7, the procedure for processing gravity and gradiometry data is summarized.The measurement of gravity (GRACE) corresponds to g = |grad V | and the measurement

of the components of the gradiometric tensor (GOCE) corresponds to Vrr = (grad grad V )rrfor the rr component.

Subtracting the gravitational effect of the residual topographical masses, δV , from theactual anomalous gravitational potential T creates potential T h that is harmonic outside thegeoid as :

T h = T − δV . (1.16)

The topographical effects δA and δE correspond respectively to the first and second radialderivatives of the residual potential of the topographical masses δV . To make a potentialharmonic in a space above the geoid, these effects have to be calculated and removed fromthe observations ∆gobs and ∆V obs

rr , respectively :

∆gh = ∆gobs − δA , (1.17)

∆V hrr = ∆V obs

rr − δE. (1.18)

The computation step starts with the downward continuation (DWC) of the values ob-tained after removing the effects, this method is described in Martinec (1998, chap 8). Then,the computation of the co-geoidal height Nh can be performed, using Stokes’s function S(ψ)(Hofmann-Wellenhof & Moritz, 2006, p 104) or Green’s function with the kernel Krr(ψ)(Martinec, 2002).

2. GOCE is a project of the European Space Agency (ESA) launched in March 2009 and expected to finishend of 2012.

3. No masses outside the geoid, and the measurements are referred to the geoid (see Introduction p.1).

9


RESTORE

Restoration of the indirect effect on geoid

REMOVE

Removing the effects of the topographical masses

COMPUTE

DWC of the data to the co-geoid

Computation of a co-geoid

Gravity Gradiometry

Direct topographical effect

on gravity dA

Primary indirect topographical

effect on geoid dN

Direct topographical effect

on gradiometry dE

Fig. 1.7 Summary of the RCR procedure for gravity and gradiometry.

The last step of the RCR procedure is the determination of the geoidal height N by restor-ing the indirect topographical effect δN to the solution obtained with the computation Nh

(Fig 1.8), using Bruns’s formula 4 (Martinec & Vaníček, 1994a).

N co-geoid

Pg

ellipsoid

geoid

NN

Q

h

Fig. 1.8 Indirect effect δN , undulation Nh of the co-geoid and undulation N of the geoid.

Prior to the first step of the Remove-Compute-Restore procedure and independently fromthe measured data, the topographical effects δA and δE have to be determined, which is thetopic of this study.

4. See Bruns’s formula in Appendix A.2

10

Chapter 2

Numerical studies over Ireland,France and Iran

This chapter shows numerical investigations over Ireland, France and Iran. The data usedfor numerical computations are the topographic heights given by Etopo 1-, 2- and 5-arc-minute topographical models corresponding to the resolutions of about 1.8 km, 3.7 km and9 km, respectively. A fortran program was written and executed over the three areas and forthe three Etopo topographical models of different resolutions.

For the topographical effects on gravity δA, we will call δAsurf and δAsat the directtopographical effects on surface and satellite gravity, respectively. The topographical effecton gradiometry δE was computed and our results will be compared with the studies on theSGG 1 by Eshagh (2009) and Bagherbandi (2011) who computed the topographical effects onSGG in Iran.

2.1 The study areas

Three areas were chosen because of their various type of topography (low and flat, high andrough, both)(Fig.2.1), and their size is reasonable according to the time needed for pointwisecomputation :• Ireland with a relatively flat terrain surrounded by ocean. The square area is limited

between longitudes -10.7 and -5.4 and latitudes 51.4 and 55.4,• France with a rough topography in the south east, but flat elsewhere. The area limited

between longitudes -4.8 and 8.3 and latitudes 42.3 and 51.1,• Iran with the roughest topography compared with the other areas. The area is limited

between longitudes 44 and 64 and latitudes 25 and 40.

1. Satellite Gravity Gradiometry.

11

Chap 2. Numerical studies over Ireland, France and Iran

44˚ 48˚ 52˚ 56˚ 60˚ 64˚

28˚

32˚

36˚

40˚

0 1000 2000 3000 4000

m

−4˚ 0˚ 4˚ 8˚

44˚

46˚

48˚

50˚

0 1000 2000 3000 4000

m

−10˚ −8˚ −6˚

52˚

53˚

54˚

55˚

0 200 400 600 800

m

Ireland France Iran

Fig. 2.1 Topographic heights over Ireland, France and Iran computed for Etopo 1 model. Themaps were generated by GMT. Unit : 1 m

2.2 Direct topographical effect on gravity

2.2.1 On the Earth’s surface

Residual effect δAsurf

The computation of the residual direct topographical effect on gravity on the Earth’s sur-face δAsurf uses this expression (Martinec, 1998, chap 3, Eqn.(3.45) modified) :

δA(r,Ω) = G%0

∫Ω0

∂L−1(r, ψ, r′)∂r

∣∣∣∣∣rt(Ω′)

r′=R− ∂L−1(r, ψ, r′)

∂r

∣∣∣∣∣rt(Ω)

r′=R

− R2[τ(Ω′)− τ(Ω)]∂L−1(r, ψ,R)∂r

]r=rt(Ω)

dΩ′ , (2.1)

Table 2.1 shows the minimum, mean, maximum and root mean square (rms) of δAsurf inmGal 2.

Table 2.1 Direct topographical effect δA on surface gravity over Ireland, France and Irancomputed for Etopo 1, 2 and 5 models. Unit : 1 mGal

δAsurf on surface gravityIreland France Iran

Grid min mean max rms min mean max rms min mean max rms1′ -17.4 0.006 5.7 ± 0.7 -120.6 0.3 118.3 ± 6.7 -142.1 0.3 86.0 ± 7.32′ -11.1 0.002 3.5 ± 0.5 -101.1 0.3 122.9 ± 6.4 -155.6 0.2 73.3 ± 7.45′ -2.4 0.00005 0.5 ± 0.2 -78.6 0.02 26.2 ± 3.1 -85.6 0.02 30.8 ± 3.6

2. unit of gravity : 1 mGal=10−5 m.s−2.

12

2.2. Direct topographical effect on gravity

δAsurf is much smaller (almost ten times) over Ireland in comparison with France and Iran,according to the minimum, mean, maximum and rms values of Table 2.1. This is explained bythe lower topography in Ireland. This shows that the flatter and lower the terrain, the smallerthe reduction of the surface gravity observations in the remove step of the RCR 3 procedureis.

The rms values in Table 2.1 are much smaller for Etopo 5 than for Etopo 1 for the threeareas 4, as also illustrated in Fig.2.2 for Ireland. The rms values of of δAsurf are smaller withsparse griding. It is thus recommended to use as finer gridding as possible.

−10˚ −8˚ −6˚

52˚

53˚

54˚

55˚

−10˚ −8˚ −6˚

52˚

53˚

54˚

55˚

−10˚ −8˚ −6˚

52˚

53˚

54˚

55˚

−0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 0.5

mGal

from Etopo 1 from Etopo 2 from Etopo 5

Direct topographical effect on surface gravity

Fig. 2.2 Comparison of the resolution of the direct topographical effect on surface gravity inIreland from Etopo 1, 2 and 5.

Total effect Atsurf

The computation of the total direct topographical effect Atsurf uses the following expres-sion :

At(r,Ω) = G%0

∫Ω0

∂L−1(r, ψ, r′)∂r

∣∣∣∣∣rt(Ω′)

r′=RdΩ′ , (2.2)

where r = rt(Ω).The total direct topographical effect on gravity Atsurf which is the case when the Helmert

condensation is not employed, is larger in amplitude than the residual effect δAsurf , andmostly negative. The values of Atsurf are prevailingly distributed around 0 mGal in flat areas.

The total effect Atsurf on surface gravity is highly correlated with the topography (seeFig.2.3(b)).

3. Remove-Compute-Restore procedure, Sect.1.5, p 9.4. See also maps of δAsurf computed over France, using Etopo 1 and Etopo 5 in Appendix A.3.

13


Table 2.2 Total direct topographical effect At on gravity for Ireland, France and Iran computedwith Etopo 1, 2 and 5. Unit : 1 mGal

Total effect Atsurf on surface gravityIreland France Iran

Grid min mean max rms min mean max rms min mean max rms1′ -27.5 -0.6 2.6 ± 1.5 — — — — — — — — *2′ -9.8 -0.3 1.7 ± 0.7 -238.8 -7.4 23.2 ± 19.0 -300.9 6.4 -31.4 ± 35.15′ -2.6 -0.2 0.4 ± 0.2 -117.8 -3.9 8.2 ± 9.0 -137.0 0.6 -17.4 ± 16.3

*The symbol — stands for values not yet computed.

−10˚ −8˚ −6˚

52˚

53˚

54˚

55˚

−5 −4 −3 −2 −1 0 1mGal

Total effect (surf)

(a)

−200

−100

0

100

200

300

Top

ogra

phy

(m)

−4

−2

0

2

4

6

Res

idua

l effe

ct (

mG

al)

−4

−2

0

2

4

6

Tot

al e

ffect

(m

Gal

)

−10 ° −9 ° −8 ° −7 ° −6 °longitude

Profile θ=52°35′N

(b)

Fig. 2.3 Total topographical effect Atsurf on surface gravity for Ireland from Etopo 1.(a) At

surf

and cross-section of the profile. (b) Longitudinal profile of latitude 5235′N. curve of the residualeffect δAsurf (black) and total effect At

surf (dashed) with the topography (grey).

2.2.2 At satellite altitudes

Residual effect δAsat

The computation of the residual direct topographical effect on satellite gravity δAsat usesEqn.(2.1) replacing r = rt(Ω) by r = rsat = rGRACE = R+ 400 km.

Table 2.3 Direct topographical effect δA on satellite gravity for Ireland, France and Iran computedwith Etopo 1, 2 and 5. Unit : 10−3 mGal

δAsat on satellite gravityIreland France Iran

Grid min mean max rms min mean max rms min mean max rms1′ -0.3 -0.0002 13.9 ± 0.7 — — — — — — — —2′ -0.3 -0.0001 11.9 ± 0.7 -38.3 -0.02 504.8 ± 26.9 -100.2 -0.07 1151.9 ± 51.05′ -0.3 -0.0002 6.9 ± 0.5 -38.5 -0.02 453.5 ± 27.2 -104.9 -0.06 688.7 ± 47.0

Table 2.3 shows that the residual effect on gravity δAsat computed at satellite altitude isof the order of 10−5 mGal, that is significantly smaller in comparison with δAsurf which is ofthe order of 10’s to 100’s of mGals.

14

2.2. Direct topographical effect on gravity

Let us now compare the residual effect δAsat with the total effect Atsat. 5

Total effect Atsat

The computation of the total direct topographical effect on satellite gravity Atsat usesEqn.(2.2) for r = rGRACE .

Table 2.4 Total direct topographical effect At on satellite gravity for Ireland, France and Irancomputed with Etopo 1, 2 and 5. Unit : 1 mGal

Total effect Atsat on satellite gravityIreland France Iran

Grid min mean max rms min mean max rms min mean max rms1′ -0.87 -0.74 -0.51 ± 0.08 — — — — — — — —2′ -0.93 -0.79 -0.55 ± 0.08 -18.1 -10.5 -2.7 ± 4.0 -78.1 -47.4 -13.8 ± 16.45′ -0.83 -0.70 -0.48 ± 0.08 -18.8 -2.8 -10.8 ± 4.1 -81.0 -48.8 -14.3 ± 16.9

The Table 2.4 shows that the computation of the total effect Atsat using Etopo 1, 2 and 5gives invariant rms values.

A coarser gridding does not affect the results of the computation of the total topographicaleffect on satellite gravity Atsat.

Although the residual effect δAsat is small, we can easily recognize the topographicalpattern in Fig.2.4(b)(right), whereas we see in Fig.2.4(b)(left) that the total effect Atsat has along wavelength feature. This is also observed along the profile in Fig.2.5. The gravitationalsignal of the topography is attenuated when going to satellite altitudes in such a way that shortwavelengths of the gravitation are attenuated faster than long wavelengths. The fact that thetotal effect Atsat is of the order of 10’s of mGals and the residual effect δAsat almost vanishesshows that the compensation of the masses by the Helmert condensation is an efficient wayto process satellite gravity data. However, this is not the case for surface gravity data (seeFig.2.4(a)).

5. Note that the Appendix A.3 shows a map of the total effect and a map of the residual effect at satellitealtitude.

15


−4˚ −2˚ 0˚ 2˚ 4˚ 6˚ 8˚

44˚

46˚

48˚

50˚

−200 −150 −100 −50 0mGal

−4˚ −2˚ 0˚ 2˚ 4˚ 6˚ 8˚

44˚

46˚

48˚

50˚

−6 −4 −2 0 2 4 6 8mGal

Total effect (surf) Residual effect (surf)

(a) Atsurf and δAsurf on the Earth’s surface

−4˚ −2˚ 0˚ 2˚ 4˚ 6˚ 8˚

44˚

46˚

48˚

50˚

−18−16−14−12−10 −8 −6 −4 −2mGal

−4˚ −2˚ 0˚ 2˚ 4˚ 6˚ 8˚

44˚

46˚

48˚

50˚

−0.04 −0.02 0.00 0.02 0.04 0.06mGal

Total effect (sat) Residual effect (sat)

(b) Atsat and δAsat at satellite altitude and axis of the

profiles of total and residual effects Fig.2.5Fig. 2.4 Comparison between the total and residual effect on surface (a) and satellite (b) gravity

over France. Maps generated from Etopo 2.

−3000

−2000

−1000

0

1000

2000

3000

4000

5000

Top

ogra

phy

(m)

−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

Res

idua

l effe

ct (

mG

al)

−30

−20

−10

0

10

20

30

40

50

Tot

al e

ffect

(m

Gal

)

−2 ° −1 ° 0 ° 1 ° 2 ° 3 ° 4 ° 5 ° 6 ° 7 ° 8 °longitude


Fig. 2.5 Profile of the direct effects on gravity at satellite altitude, comparing the behaviour ofresidual effect δAsat (plain dark line) and the total effect At

sat (dashed line) with the topography(grey line). Longitudinal profile of latitude 4555′ over France, generated from Etopo 2.

16

2.3. Direct topographical effect on gradiometry

2.3 Direct topographical effect on gradiometry

2.3.1 Residual effect δE

The expression of the direct topographical effect on gradiometry δE is , (Martinec, 1998,derived from Eqn.(3.45), chap 3)

δE(r,Ω) = G%0

∫Ω0

∂2L−1(r, ψ, r′)∂r2

∣∣∣∣∣rt(Ω′)

r′=R− ∂2L−1(r, ψ, r′)

∂r2

∣∣∣∣∣rt(Ω)

r′=R

− R2[τ(Ω′)− τ(Ω)]∂2L−1(r, ψ,R)

∂r2

]r=rsat

dΩ′ , (2.3)

for rsat = rGOCE = R + 250 km. Table 2.5 shows the minimum, mean, maximum and rmsvalues of δE in Eötvös 6.

Table 2.5 Direct topographical effect δE on gradiometry over Ireland, France and Iran computedwith Etopo 1, 2 and 5. Unit : mE

δE on gradiometryIreland France Iran

Grid min mean max rms min mean max rms min mean max rms1′ -2.13 -0.00001 0.07 ± 0.10 -38.6 -0.01 10.2 ± 1.6 -40.1 0.003 13.5 ± 3.92′ -1.99 -0.000004 0.07 ± 0.10 -27.5 -0.01 10.2 ± 1.7 -40.1 0.003 13.5 ± 3.95′ -1.06 -0.00001 0.06 ± 0.07 -23.7 -0.01 9.7 ± 1.6 -24.4 0.002 14.6 ± 3.9

The values of rms for δE (Table 2.5) differ by less than 0.1 mE when computed withthe different grid resolution. This shows that the results of the topographical effect δE ongradiometry are not affected by whether the resolution of the grid is 1′ or 5′.

δE has very small values over Ireland compared with France and Iran. This topographicaleffect on gradiometry is much smaller over low and flat topography.

The effect is more important when the terrain is more high an rough, up to - 40 mE inIran and France. According to the measurement accuracy announced for GOCE gradiometer(10 mE), the effect of δE should still be taken into account in the RCR procedure, becausethe data are affected by this effect, especially over mountainous area.

2.3.2 Total effect V trr

The total direct topographical effect V trr on gradiometry is computed by

V trr(r,Ω) = G%0

∫Ω0

∂2L−1(r, ψ, r′)∂r2

∣∣∣∣∣rt(Ω′)

r′=RdΩ′ , (2.4)

6. unit of gradiometry : 1 E = 10−9 m/s2/m.

17


44˚ 48˚ 52˚ 56˚ 60˚ 64˚

28˚

32˚

36˚

40˚

−0.015

−0.010

−0.005

0.000

0.005

0.010

0.015E

Residual effect

Fig. 2.6 Residual topographical effect on gradiometry δE over Iran generated from Etopo 2, andthe cross-section shown in Fig.2.7.

where r = rGOCE .

Table 2.6 Total effect on gradiometry V trr over Ireland, France and Iran computed with Etopo 1, 2and 5. Unit : 1 E

Total effect V trr on gradiometry

Ireland France IranGrid min mean max rms min mean max rms min mean max rms

1′ 0.01 0.07 0.11 ± 0.02 -0.05 0.37 1.52 ± 0.37 -0.38 1.01 2.85 ± 0.892′ 0.01 0.07 0.12 ± 0.02 -0.05 0.37 1.52 ± 0.37 -0.38 1.01 2.85 ± 0.895′ 0.01 0.06 0.11 ± 0.02 -0.06 0.38 1.58 ± 0.39 -0.38 1.03 2.97 ± 0.91

Table 2.6 shows that the values of the total effect V trr are much larger in magnitude than

the values of the residual effect δE (two orders in magnitude).This shows that the compensation of the masses by the Helmert condensation is an effi-

cient way to process satellite gravity gradient data (as well as the satellite gravity data).

The distribution of the total effect V trr between the maximum and minimum values is

much more homogeneous compared to the distribution of the residual effect δE (with mostvalues around 0 E).

Fig.2.8(a) shows a smooth gradual shading and the curve of V trr Fig.2.7 confirms a long-

wavelength feature of the total topographical effect on gradiometry V trr.

Comparison with studies on SGG over Iran

Eshagh (2009) and Bagherbandi (2011) recently calculated the total topographical ef-fect, representing the topography by a spherical harmonic series truncated at degree andorder 360 which corresponds to a 0.5 × 0.5 (56 km × 56 km) spatial resolution. In their

18

2.3. Direct topographical effect on gradiometry

−2000

−1000

0

1000

2000

3000

4000T

opog

raph

y (m

)

−0.02

−0.01

0.00

0.01

0.02

0.03

0.04

Res

idua

l effe

ct (

E)

−2

−1

0

1

2

3

4

Tot

al e

ffect

(E

)

44 ° 46 ° 48 ° 50 ° 52 ° 54 ° 56 ° 58 ° 60 ° 62 ° 64 °longitude


Fig. 2.7 Profile of the direct effects on gradiometry, comparing the behaviour of residual effect δE(plain dark line) and the total effect V t

rr (dashed line) with the topography (grey line). Longitudinalprofile of latitude 2945′ over Iran, generated from Etopo 2.

study, they computed the six components of the gradiometric tensor in cartesian coordinates(xx, yy, zz, xy, xz, yz), with the zz component V t

zz (shown in Fig.2.8(b)) being the equivalentto V t

rr in spherical coordinates.Table 2.7 compares the results of these studies with our values computed with the point-

wise method over Iran using Etopo 5.

44˚ 48˚ 52˚ 56˚ 60˚ 64˚

28˚

32˚

36˚

40˚

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0E

Total effect

(a) (b)

Fig. 2.8 Comparison of the total topographical effect on gradiometry from two methods and twospatial resolutions. (a) V t

rr by our pointwise computations in a spatial domain using a 2′ × 2′ grid,in spherical coordinates. Cross-section used in Fig.2.7 by dash-dotted line. (b) V t

zz computed in thefrequency domain by Eshagh (2009) employing harmonic expansion to degree and order 360 with

corresponding 0.5 × 0.5 resolution, in cartesian coordinates.

Fig.2.8 shows the comparison between the results of the total effect from our study to thatfrom the study of Eshagh (2009). It is clear that the results are similar in terms of resolutionand amplitude. The results are also similar with those of Bagherbandi (2011), given in Table2.7.

This shows that both methods of computation, however different, give nearly the sameresults. The gridding is thus not significantly modifying the long-wavelength behaviour of V t

rr.

19


Table 2.7 Total effect on gradiometry V tzz over Iran. Unit : 1 E

V tzz(P ) - Iran

Grid min mean max rms0.5 -0.81 0.64 2.63 ± 0.94 Eshagh (2009)0.5 -0.96 0.20 3.21 ± 0.91 Bagherbandi (2011)5′ -0.38 1.03 2.97 ± 0.91 Table 2.6

2.4 Results of the direct effect on gradiometry over the threeareas

The results for the total effect V trr and the residual effect δE over Ireland, France and Iran

are shown in Fig.2.9.Over Ireland Fig (a), the total effect is rather homogenous and the residual effect is weak

in magnitude.Over France Fig (b), the total effect is strong over the Alps whereas is close to 0 elsewhere,

where the terrain is much flatter and lower. The residual effect shows long wavelengths overmountainous areas and short wavelength elsewhere.

Over Iran Fig (c), the total effect has a large amplitude and the residual effect shows shortwavelengths over Zagros mountains.

20

2.4. Results of the direct effect on gradiometry over the three areas

−10˚ −8˚ −6˚

52˚

53˚

54˚

55˚

0.00 0.02 0.04 0.06 0.08 0.10Eotvos

−10˚ −8˚ −6˚

52˚

53˚

54˚

55˚

−0.0004 −0.0002 0.0000Eotvos

Total effect Residual effect

(a)

−4˚ −2˚ 0˚ 2˚ 4˚ 6˚ 8˚

44˚

46˚

48˚

50˚

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4Eotvos

−4˚ −2˚ 0˚ 2˚ 4˚ 6˚ 8˚

44˚

46˚

48˚

50˚

−0.002 0.000 0.002 0.004Eotvos


(b)

44˚ 48˚ 52˚ 56˚ 60˚ 64˚

28˚

32˚

36˚

40˚

0 1 2 3Eotvos

44˚ 48˚ 52˚ 56˚ 60˚ 64˚

28˚

32˚

36˚

40˚

−0.01 0.00 0.01Eotvos


(c)

Fig. 2.9 Total direct topographical effect V trr (left) and residual effect δE (right) on gradiometry.

Over Ireland (a), France (b) and Iran (c) using Etopo 1. Unit : 1E

21


2.5 Conclusions on the results

Topographical effects were computed pointwise on surface or at satellite altitude, usingthe Helmert condensation over Ireland, France and Iran from Etopo 1-, 2- and 5- arc-minute.The interpretation of the results gave the following conclusions :

– When the terrain is flat and low, as it is in Ireland, the topographical effects are muchsmaller than over rough terrain like Iran and the South-East of France.

– After computing the topographical effects over each area but using models of differentresolution, the results at satellite altitude were shown not to depend on the grid spacing.This is not the case for the topographical effects on surface measurements.

– At satellite altitude, the total effect of the topographical masses (not compensated)produces a long wavelength signal.

– The fact that the residual topographical effects are much smaller than the total topo-graphical effects, shows that the Helmert condensation is an efficient way to compensatethe topographical masses.

– The method of pointwise computation of the total topographical effect on gradiometryyields the same results as approach by spherical harmonics. This is a validation of ourpointwise computation method, introduced here for analysis of gradiometric measure-ments (GOCE).

22

Conclusions and future works

By this study, I present a method for evaluation of terrain corrections that have to be de-duced during the first step of the Remove-compute-Restore procedure of geoid determinationwith gravitational data from the recent satellite missions, GRACE and GOCE.

The first step of the RCR procedure comprises correction of gravitational data thanks tothe effect of the topographical masses (the focus of this study project).

For surface measurements, topographical effects have already been studied. Currently,these effects are the focus of new studies on satellite measurements. The recent and futureGRACE and GOCE requires the development of reliable computation methods for the deter-mination of the geoid.

In this study, the topographical effects are expressed using the Newton integral. TheHelmert condensation layer is the model used to compensate the topographical masses. Afterhaving mathematically defined the effects, I created a program that enables a method ofpointwise computation using digital elevation models of topography. This method of pointwisecomputation has not been applied previously to topographical effect computations at satellitealtitude.

After estimating the computation time required for a global study, test areas have beenchosen such that the computation has been possible according to the computer ressourcesavailable where this study took place.

The results show that the topographical effect on satellite measurements are much smallerthan the effect on surface measurement. This means that the correction terrain deduced duringthe RCR procedure, will be smaller when processing satellite measurement. Thus, it is moreadvantageous to process gravitational measurements from satellites. It was also shown thatthe compensation of the topographical masses by the Helmert second condensation methodis gratifying.

Comparing the method of pointwise computation with a different method, the results weresimilar, meaning that the method developed provides reliable results.

Hence, the pointwise computation method is validated and the results of the tests areconclusive and promising .

23


As future work, a study of the isostatic compensation of the topographic masses couldbe made and compared with the method of Helmert on condensated masses. The Helmertmethod is efficient, but is not the only way to compensate the topographic masses.

The computation at a global scale can be tested, for example if using multi-processingcomputation or parallel computing as this will reduce significantly the duration of computa-tion. Also, to optimize the duration of computation, further study should be made to find agood compromise between the resolution of the topographic model and the accuracy of themeasurements.

The results of end-of-study project will be used as a starting point of the processing andinterpretation of GOCE data. The Remove-Compute-Restore procedure can then be appliedin full.

24

List of symbols

AB gravitational attraction of spherical Bouguer shellAc gravitational attraction of compensed/ condensed massesAt gravitational attraction of topographical massesAR gravitational attraction of roughness terrainδA direct topographical effect on surface gravityδE direct topographical effect on surface gradiometryg gravity of the EarthG Newton’s gravitational constantH topographical heightHP Bouguer shell thicknessKrr kernel of Green’s functionL spatial distanceL−1 reverse of the distance LL−1 indefinite radial integral of the Newton kernelN geoidal height or undulationNh undulation of the co-geoidδN indirect topographical effect on geoidP computation point of coordinates (r,Ω)P ′ integration point of coordinates (r′,Ω′)Pg point on the geoidPt point on the Earth surfaceQ point on reference ellipsoidr radial distancerg radius of the geoidrt radius of the Earth surfaceR radius of mean sphere best-fitting the geoid

25

S Stokes’s functionSg surface of the geoidSt surface of the topographyT anomalous gravitational potential harmonic outside the EarthT h anomalous gravitational potential harmonic outside the geoidU normal gravity potentialV gravitational potential of the EarthV B gravitational potential of BouguerV c gravitational potential of compensated/condensed massesV R gravitational potential ofV t gravitational potential of topographical massesδV residual topographical potential/ ndirect topographical effect on potentialVrr composante rr du tenseur gradiométriqueW gravity potential of the EarthW0 reference value of gravity potential on the geoid

γ normal/reference gravityϑ co-latitudeλ latitudeϕ longitude% density of topographical masses%0 mean density of topographical masses%c density of compensated massesσ density of condensated massesψ angular distance in spherical coordinatesΩ pair of angular spherical coordinatesΩ0 full solid angle

26

List of acronyms

DEM Digital Elevation ModelDWC Downward continuationESA European Space AgencyFortran FORmula TRANslatorGMT Generic Mapping ToolsGOCE Gravity fields and steady-state Ocean Circulation ExplorerGPS Global Positioning SystemGRACE Gravity Recovery and Climate ExperimentLEO Low Earth OrbiterNASA National Aeronautics and Space AdministrationRCR Remove-Compute-Restorerms root mean squareSGG Satellite Gravity GradiometrySST-hl Satellite to Satellite Tracking in high-low modeSST-ll Satellite to Satellite Tracking in low-low modeWP Work Package

27

List of figures

1 Earth’s surface, geoid, ellipsoid and the heights separating them. . . . . . . . 1

1.1 Spherical coordinates of the computation point P (r,Ω), an integration pointP ′(r′,Ω′), the distance L and angular distance ψ between P and P ′. . . . . . 3

1.2 Optional caption for list of figures . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Roughness of the terrain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Helmert condensation layer of density σ . . . . . . . . . . . . . . . . . . . . . 6

1.5 Satellite-measurement concepts meeting the four fundamental criteria. . . . . 8

1.6 Optional caption for list of figures . . . . . . . . . . . . . . . . . . . . . . . . 8

1.7 Summary of the RCR procedure for gravity and gradiometry. . . . . . . . . . 10

1.8 Indirect effect δN , undulation Nh of the co-geoid and undulation N of the geoid. 10

2.1 Topographic heights over Ireland, France and Iran computed for Etopo 1 model.The maps were generated by GMT. Unit : 1 m . . . . . . . . . . . . . . . . . 12

2.2 Comparison of the resolution of the direct topographical effect on surface grav-ity in Ireland from Etopo 1, 2 and 5. . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Total topographical effect Atsurf on surface gravity for Ireland from Etopo1.(a) Atsurf and cross-section of the profile. (b) Longitudinal profile of lati-tude 5235′N. curve of the residual effect δAsurf (black) and total effect Atsurf(dashed) with the topography (grey). . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Comparison between the total and residual effect on surface (a) and satellite(b) gravity over France. Maps generated from Etopo 2. . . . . . . . . . . . . . 16

2.5 Profile of the direct effects on gravity at satellite altitude, comparing thebehaviour of residual effect δAsat (plain dark line) and the total effect Atsat(dashed line) with the topography (grey line). Longitudinal profile of latitude4555′ over France, generated from Etopo 2. . . . . . . . . . . . . . . . . . . . 16

2.6 Residual topographical effect on gradiometry δE over Iran generated fromEtopo 2, and the cross-section shown in Fig.2.7. . . . . . . . . . . . . . . . . . 18

29

2.7 Profile of the direct effects on gradiometry, comparing the behaviour of residualeffect δE (plain dark line) and the total effect V t

rr (dashed line) with the topog-raphy (grey line). Longitudinal profile of latitude 2945′ over Iran, generatedfrom Etopo 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.8 Comparison of the total topographical effect on gradiometry from two methodsand two spatial resolutions. (a) V t

rr by our pointwise computations in a spatialdomain using a 2′ × 2′ grid, in spherical coordinates. Cross-section used inFig.2.7 by dash-dotted line. (b) V t

zz computed in the frequency domain byEshagh (2009) employing harmonic expansion to degree and order 360 withcorresponding 0.5 × 0.5 resolution, in cartesian coordinates. . . . . . . . . . 19

2.9 Total direct topographical effect V trr (left) and residual effect δE (right) on

gradiometry. Over Ireland (a), France (b) and Iran (c) using Etopo 1. Unit : 1E 21

A.1 Description of the Work Package 6. This work will probably will used for theWP6.2 which is : "To model the gravity and gradiometric effects generated by[...] the surface topography". . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

A.2 Schedule of the differents Works packages for the ESA-GOCE project. As writ-ten on top of the page, "all dates in the diagram should be treated as relativeonly and might be shifted". Note that the project didn’t start yet. . . . . . . 37

A.3 Geoid and reference ellipsoid (Hofmann-Wellenhof & Moritz, 2006, Fig 2.12, p91). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

A.4 Comparison of the direct effect on surface gravity from Etopo 1 and 5. . . . . 40A.5 Comparison of the total and residual direct effect on satellite gravity from

Etopo 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

30

List of tables

2.1 Direct topographical effect δA on surface gravity over Ireland, France and Irancomputed for Etopo 1, 2 and 5 models. Unit : 1 mGal . . . . . . . . . . . . . 12

2.2 Total direct topographical effect At on gravity for Ireland, France and Irancomputed with Etopo 1, 2 and 5. Unit : 1 mGal . . . . . . . . . . . . . . . . . 14

2.3 Direct topographical effect δA on satellite gravity for Ireland, France and Irancomputed with Etopo 1, 2 and 5. Unit : 10−3 mGal . . . . . . . . . . . . . . . 14

2.4 Total direct topographical effect At on satellite gravity for Ireland, France andIran computed with Etopo 1, 2 and 5. Unit : 1 mGal . . . . . . . . . . . . . . 15

2.5 Direct topographical effect δE on gradiometry over Ireland, France and Irancomputed with Etopo 1, 2 and 5. Unit : mE . . . . . . . . . . . . . . . . . . . 17

2.6 Total effect on gradiometry V trr over Ireland, France and Iran computed with

Etopo 1, 2 and 5. Unit : 1 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.7 Total effect on gradiometry V t

zz over Iran. Unit : 1 E . . . . . . . . . . . . . . 20

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References

Bagherbandi, M. (2011). An Isostatic Earth Crustal Model and its Applications. Doctoraldissertation in geodesy, Royal Institute of Technology (KTH), Stockolm, Sweden. 207 p.

Eshagh, M. (2009). On Satellite Gravity Gradiometry. Doctoral dissertation in geodesy, RoyalInstitute of Technology (KTH), Stockolm, Sweden. 241 p.

Forsberg, R., & Tscherning, C. (1981). The use of Height Data in Gravity Field Approximationby Collocation, vol. 86 , No. B9.

Gauss, C. (1828). Bestimmung des Breitenunterscchiedes zwischen den Sternwarten von Got-tingen und Altona. Gottingen.

Helmert, F. (1884). Die matematischen und physikalischen Theorien der höheren Geodäsie,vol. 2. Leipzig (reprinted in 1962 by Minerva GMBH, Frankfurt/Main) : B.G. Teubner.572 p.

Hofmann-Wellenhof, B., & Moritz, H. (2006). Physical Geodesy. Heidelberg : Springer-Verlag,second ed. 404 p.

Listing, J. (1873). Uber unsere jetzige Kenntnis der Gestalt und Grosse der Erde. Nachr. K

Martinec, Z. (1998). Boundary-Value Problems for Gravimetric Determination of a PreciseGeoid. Heidelberg : Springer-Verlag. Lecture Notes in Earth Sciences. 223 p.

Martinec, Z. (2002). Green’s function solution to spherical gradiometric boundary-value prob-lems. Journal of Geodesy, 77 , 41–49.

Martinec, Z. (2011). The Earth gravity field. Lectures at the Dublin Institute for AdvancedStudies.

Martinec, Z., & Vaníček, P. (1994a). The indirect effect of stokes-helmert’s technique for aspherical approximation of the geoid. Man. Geod., 19 , 213–219.

Martinec, Z., & Vaníček, P. (1994b). Direct topographical effect of helmert’s condensationfor a spherical geoid. Man. Geod., 19 , 257–268.

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Rózsa, S. (1998). Determination of terrain correction in hungary and the surrounding area.published in Reports of the Finnish Geodetic Institute, 98 , 4.

Rummel, R., Balmino, G., Johannessen, J., Visser, P., & Woodworth, P. (2002). Dedicatedgravity field missions - principles and aims. J. of Geodynamics, 33 , 3–20.

Seeber, G. (2003). Satellite Geodesy. Berlin. New York. : Walter de Gruyter., 2nd ed. 589 p.

Stokes, G. (1849). On the variation of gravity on the surface of the Earth, vol. 8. Transactionsof the Cambridge Philosophical Society. 672-695.

Wichiencharoen, C. (1982). The indirect effects on the computation of geoids undulations.Rep 336, Department of Geodetic Science, The Ohio State University, Columbus.

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Appendix A

Appendix

A.1 Work Package ESA

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AO/1-6367/10/NL/AF STSE – GOCE+

UWB/GOCE-GDC

Towards a better understanding of the Earth’s interior and geophysical exploration research

“GOCE-GDC” Financial, management and administrative proposal

Page A-6 July 14, 2010 Issue 1rev0

PROJECT:GOCE-GDC PHASE: 1

WP REF: WP6

WP Title: Geophysical test scenario B – continental small-scale regions covered by exploration geophysical data

Contractor: DIAS

Start event: MTR (T0 + 12 m) End event: PM4 (T0 + 20 m)

WP Manager: Z. Martinec

Sheet 1 of 1 Issue: 1 Issue Date: 23.6.2010

Participants: AUT,TUD

Objectives: To select two or more application areas where the data sets shall be employed complementing

others that are already used for geophysical modelling (like topography, seismic, magnetotelluric data etc.)

To demonstrate and assess the benefits of the combined (satellite) data set with near surface data in simulation experiments and modelling based upon real data in a well understood setting.

Inputs: Satellite gravity gradient data along the GOCE orbit (WP4 output) Combined gravity gradient data near or at the Earth Surface (WP4 output)

Activities:

WP6.1 To derive an apriori information on the lithospheric density from seismic and magnetotelluric structure parameters by applying a thermomechanical model. To combine the expertise in magnetic and seismic exploration geophysics available at the DIAS with the GOCE data in small-scale regions.

WP6.2 To model the gravity and gradiometric effects generated by the apriori density model and by the surface topography.

WP6.3 To test the sensitivity of gravity and gradiometric data on density parameters. WP6.4 To improve the apriori density model by assimilating satellite and surface gravity and

gradiometry data. WP6.5 To understand the benefits of the new gravity field for exploration geophysical

applications.

Outputs / deliverables: WP6 report

Fig. A.1 Description of the Work Package 6. This work will probably will used for the WP6.2which is : "To model the gravity and gradiometric effects generated by [...] the surface topography".

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AO/1-6367/10/NL/AF STSE GOCE+

UWB/GOCE-GDC

geophysical exploration research -

Financial, management and administrative proposal

Page 23 July 14, 2010 Issue 1rev0

date will depend on negotiation between the contractor and ESA so all dates in the diagram should be treated as relative only and might be shifted.

Figure 3 - The GANTT diagram

Figure 4 shows the workload per institution during the project duration. All WPs start and end with a project meeting, so the whole period can be divided to 6 sub-periods. Dark colors mean, that the institution works as a WP manager in the particular sub-period, light colors mean its participation on some WP. The numbers denote WP numbers in respective sub-periods, bold numbers imply the WP management.

Figure 4 - Workload per institution.

Fig. A.2 Schedule of the differents Works packages for the ESA-GOCE project. As written on topof the page, "all dates in the diagram should be treated as relative only and might be shifted". Note

that the project didn’t start yet.

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A.2 Bruns’s formula

A.2.1 Proof of Bruns’s formula

Notes on the lecture of Martinec (2011) on tuesday, March 22nd 2011.

Fig. A.3 Geoid and reference ellipsoid (Hofmann-Wellenhof & Moritz, 2006, Fig 2.12, p 91).

The gauge value of gravity potential on the geoid W0 is defined by

W0 = UPg + TPg , (A.1)

where T is the anomalous gravitational potential harmonic outside the Earth taken at a pointon the geoid Pg and U is the normal gravity potential as

UPg = UQ + ∂U

∂h

∣∣∣∣QN + 1

2∂2U

∂h2

∣∣∣∣∣Q

N2 +O(N3) . (A.2)

The normal gravity γ := |grad U | so :

UPg = UQ − γ|QN −12∂γ

∂h

∣∣∣∣QN2 +O(N3) . (A.3)

Then we substitute (A.3) into (A.1) :

W0 = UQ − γ|QN −12∂γ

∂h

∣∣∣∣QN2 +O(N3) + TPg , (A.4)

but the potential UQ on geoid is equal to W0, so

0 = −γQN −12∂γ

∂h

∣∣∣∣QN2 +O(N3) + TPg , (A.5)

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and we admit that 12∂γ∂h

∣∣∣QN2 is very small, so (A.5) becomes

0 = −γQN + TPg , (A.6)

which is the BRUNS’S FORMULA :

N =TPg

γQ. (A.7)

Note : If we consider W0−UQ is not exactly 0 but a small value ±∆W , then we can writeBruns’s formula as

N =TPg + ∆W

γQ. (A.8)

A.2.2 Expressions of the undulation N and the indirect effect δN

To find the expression for δN , we start with geoidal height N , derived from Bruns’ formulaas Eqn.(A.7).

As δV = T − T h, we obtain

N = T h + δV

γQ. (A.9)

Applying Bruns’ formula to the undulation of the co-geoid, Nh = T h/γQ, the geoidalheight N is

N = Nh + δN , (A.10)

whereδN = δV

γQ, (A.11)

and δV must be taken at a point on the geoid Pg.

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A.3 Maps of the direct topographical effect on surface gravityover France

−4˚ −2˚ 0˚ 2˚ 4˚ 6˚ 8˚

44˚

46˚

48˚

50˚

−6 −4 −2 0 2 4 6 8mGal

−4˚ −2˚ 0˚ 2˚ 4˚ 6˚ 8˚

44˚

46˚

48˚

50˚

−6 −4 −2 0 2 4 6 8mGal

From Etopo 2 From Etopo 5

Fig. A.4 Comparison of the direct effect on surface gravity from Etopo 1 and 5.

A.4 Maps of the direct topographical effect on satellite gravityover Ireland

−10˚ −8˚ −6˚

52˚

53˚

54˚

55˚

−1.0 −0.9 −0.8 −0.7 −0.6 −0.5mGal

−10˚ −8˚ −6˚

52˚

53˚

54˚

55˚

0.000 0.001 0.002 0.003 0.004mGal


Fig. A.5 Comparison of the total and residual direct effect on satellite gravity from Etopo 1.

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