algorithm theoretical basis document

Issue 2.2, 20.04.2015 Algorithm Theoretical Basis Document 1

Algorithm Theoretical Basis Document

GOCE+ GeoExplore

Document: DD-GOCE+-DNT-04

Issue: Issue 2.2

Date: 20 April 2015

Deutsches Geodätisches Forschungsinstitut (DGFI), Munich, Germany

Geological Survey of Norway (NGU), Trondheim, Norway

TNO, the Netherlands

Christian-Albrechts-Universität zu Kiel


Document Information Sheet

Document Name

Algorithm Theoretical Basis Document

Document ID Issue Date

DD-GOCE+-DNT-04 2.2 20/04/2015

Author Institute

J. Bouman J. Ebbing M. Schmidt V. Lieb M. Fuchs

DGFI NGU DGFI DGFI DGFI

Distribution

Person Institute

R. Haagmans Study Team

ESA DGFI, NGU, TNO

Document Change Record

Issue Date Reason for change Changed pages / paragraphs

Draft 27/04/2012 Draft issue for MTR All

1.0 13/02/2013 First issue for PM#4 All

1.1 04/06/2013 Issue for Final Review (July 2013) All

2.0 21/10/2014 Paragraph added on regional gravi-ty field analysis using tesseroids Corrections in Section 6.3

Sections 6.3 – 6.8 merged in one Section, new Section 6.4 has been added

2.1 21/11/2014 Updated after Final Meeting (Oc-tober 2014): definition of gradient grids above ellipsoid has been add-ed

Section 5.2

2.2 20/04/2015 Grid description updated to be-come more accurate

Section 5.2


Contents 1 Introduction ................................................................................................................................5

1.1 Purpose ...............................................................................................................................5

1.2 Applicability .........................................................................................................................5

1.3 Definitions ...........................................................................................................................5

2 Applicable and reference documents ..........................................................................................5

2.1 Applicable documents .........................................................................................................5

2.2 Reference documents ..........................................................................................................5

3 Background .................................................................................................................................6

4 Numerical standards ...................................................................................................................6

5 Reference systems ......................................................................................................................7

5.1 Reference System Definitions ..............................................................................................7

5.2 Major Reference Frames for Gradiometry ......................................................................... 10

5.3 GRF to MRF rotation .......................................................................................................... 12

5.3.1 GRF to LNOF .............................................................................................................. 12

5.3.2 LNOF to MRF.............................................................................................................. 13

6 Gravitational field ..................................................................................................................... 15

6.1 Functionals of the gravity field ........................................................................................... 15

6.1.1 Geoid height N ........................................................................................................... 15

6.1.2 Gravity anomaly Δg .................................................................................................... 15

6.1.3 Gravity disturbance δg ............................................................................................... 15

6.1.4 Deflection of the vertical (ξ,η) .................................................................................... 15

6.2 Derivations of the disturbing potential T ............................................................................ 16

6.3 Regional Gravity Field Modelling Using MSR ...................................................................... 17

6.3.1 Gravitational potential in geocentric polar coordinates .............................................. 19

6.3.2 GOCE gradients in Cartesian coordinates ................................................................... 21

6.3.3 GOCE gradients in the GRF ......................................................................................... 23

6.3.4 Comparison: series expansion in spherical harmonics versus series expansion in spherical scaling functions ......................................................................................................... 24

6.3.5 Regional combined gravity field solutions .................................................................. 27

6.4 Regional gravity field modelling using tesseroids ............................................................... 28

6.4.1 Tesseroid-voxel method ............................................................................................. 28

6.4.2 Rotations ................................................................................................................... 32

6.4.3 Lookup table .............................................................................................................. 34

6.4.4 Grids at satellite altitude ............................................................................................ 34

7 Gravity reduction and calculation .............................................................................................. 35

7.1 Free-air and Bouguer anomalies, terrain corrections ......................................................... 35

7.2 Spherical and planar calculations ....................................................................................... 36

7.2.1 Tesseroids .................................................................................................................. 37


7.2.2 Gravitational Fields of a Tesseroid (taken from Uieda 2011) ....................................... 38

8 References ................................................................................................................................ 39

Appendix A: Matlab scripts ............................................................................................................... 40


1 Introduction

1.1 Purpose

The purpose of this document is to provide detailed description of the final version of the algorithms including related data sources, processing steps and output data. In addition, the standards and con-ventions that are being used in the GOCE+ GeoExplore study are part of the document.

1.2 Applicability

This document is part of the List of Deliverables [AD-1] and is to be delivered in draft at MTR and in final form at FR [RD-1].

1.3 Definitions

The term “Agency” is used to indicate the European Space Agency.

The term “Study Team” is used to indicate the persons of the three institutes (DGFI, NGU and TNO) that carry out the STSE – GOCE+ Theme 2 study.

2 Applicable and reference documents

2.1 Applicable documents

[AD-1] EOP-SM/2048: Statement of Work, Support to Science Element (STSE) GOCE+, Issue 1.0, March 2010

2.2 Reference documents

[RD-1] PR-GOCE-DGFI-10/02: Tender, Heterogeneous gravity data combination for Earth interior and geophysical exploration research (Theme 2), Issue 1.0, 14 July 2010

[RD-2] DD-GOCE+-DNT-01: Requirement Baseline, Issue 2.1, 20 April 2012

[RD-3] DD-GOCE+-DNT-02: Preliminary Analysis Report, Issue 1.1, 24 April 2012

[RD-4] DD-GOCE+-DNT-03: Development and Validation Plan, Issue 1.0, 20 April 2012

[RD-5] DD-GOCE+-DNT-04: Algorithm Theoretical Basis Document, Issue 2.1, 21 November 2014

[RD-6] RP-GOCE+-DNT-05: Product Validation Report, Issue 2.1, 21 November 2014

[RD-7] MA-GOCE+-DNT-06: Data Set User Manual, Issue 1.2, 21 November 2014

[RD-8] PP-GOCE+-DNT-07: Promotion Plan, Issue 1.4, 31 May 2013

[RD-9] RP-GOCE+-DNT-08: Impact Assessment Report, Issue 1.1, 21 October 2014

[RD-10] RP-GOCE+-DNT-09: Scientific Roadmap, Issue 1.2, 21 November 2014

[RD-11] TN-GOCE+-DNT-10, LNOF to MRF gradient rotation, Issue 1.1, 08 February 2013


3 Background

GOCE data may improve the understanding and modelling of the Earth’s interior and its dynamic processes, contributing to gain new insights into the geodynamics associated with the lithosphere, mantle composition and rheology, uplift and subduction processes. However, to achieve this chal-lenging target, GOCE should be used in combination with additional data sources: e.g. magnetic, gravity and seismology in situ, airborne and satellite data sets.

The overall objective of the study is to combine GOCE gravity gradients with heterogeneous other satellite gravity information to arrive at a combined set of gravity gradients complementing (near)-surface data sets spanning all together scales from global down to 5 km useful for various geophysi-cal applications and demonstrate their utility to complement additional data sources (e.g., magnetic, seismic) to enhance geophysical modelling and exploration.

4 Numerical standards

The numerical standards defined for GOCE+ GeoExplore are summarized in the Table below. Values are equal to the GOCE HPF standards and conform with the WGS84 parameters.

Parameter Value Unit

GM, Geocentric Gravitational Constant including the mass of the Earth’s atmosphere

3.986004415 × 1014 m3 s-2

Flattening factor of the Earth 1/f 298.2577223563 -

Nominal mean angular velocity 7.292115 × 10-5 rad s-1

Earth’s semi-major axis 6378137.0 m

Some derived parameters are the semi-minor axis

b = a(1-f)

and the eccentricity squared

e2 = 2f – f2.

The mean height above the ellipsoid or plane that is to be used in the two study areas North-East Atlantic Margin and Rub’ al Khali is chosen close to the mean height of the GOCE satellite in those regions. The mean height is

NEA: 270 km;

RAK: 260 km.

The coordinates for the gravity computations are given in the Table below. These coordinates differ from the coordinates of the study areas, which are defined in the Data Set User Manual [RD-7]. The areas for the gravity computations are larger to avoid edge effects.

Coordinate North East Atlantic Mar-gin

Saudi Arabia Remark

λ -10 30 35 65 Spherical coordinates in degrees

φ 50 80 10 35

λ -10 30 35 65 Geodetic coordinates in degrees

𝜑 50.1894 80.0656 10.0660 35.1810

N 5865638 8925983 1129634 4081992 Northing (Y) and Easting (X) in me-ters E -1272723 785799 -601028 2333140


5 Reference systems

5.1 Reference System Definitions

Geocentric and geographic coordinates

Geocentric coordinates (𝑟, 𝜙, 𝜆) are used for a sphere whereas geographic (or geodetic) coordinates (ℎ, 𝜑, 𝜆) are used for the ellipsoid. The longitude 𝜆 is the same in both coordinate systems, whereas geodetic and geocentric latitude differ. This is shown in Figure 5-1.

Figure 5-1: Geocentric and geodetic latitude

The transformation of spherical coordinates to geocentric Cartesian coordinates is given as

𝑋 = 𝑟 cos 𝜙 cos 𝜆𝑌 = 𝑟 cos 𝜙 sin 𝜆𝑍 = 𝑟 sin 𝜙

and the transformation from geodetic coordinates to geocentric Cartesian coordinates is given as

𝑋 = (𝑁 + ℎ) cos 𝜑 cos 𝜆

𝑌 = (𝑁 + ℎ) cos 𝜑 sin 𝜆

𝑍 = (𝑁(1 − 𝑒2) + ℎ) sin 𝜑

with 𝑁 = 𝑎 (1 − 𝑒2 sin2 𝜑)1

2. See Appendix A for an implementation in Matlab of the geodetic to Cartesian transformation. The transformation of spherical to Cartesian coordinates can be done us-ing standard Matlab scripts.

The inverse transformations are given by

𝑟 = √𝑋2 + 𝑌2 + 𝑍2

𝜆 = atan𝑌

𝑋

𝜙 = asin𝑍

𝑟

and (see Appendix A)


𝜑 = atan𝑍 + 𝑒′2𝑏 sin3 𝜇

𝑑 − 𝑒2𝑎 cos3 𝜇

𝜆 = atan𝑌

𝑋

ℎ =𝑑

cos 𝜑 − 𝑁

for spherical and geodetic coordinates respectively with

𝑑 = √𝑋2 + 𝑌2

𝜇 = atan𝑍𝑎

𝑑𝑏

𝑒′2 =𝑎2 − 𝑏2

𝑏2=

𝑒2

1 − 𝑒2=

2𝑓 − 𝑓2

(1 − 𝑓)2.

MRF – Model Reference Frame

The Model Reference Frame (MRF) is a local Cartesian frame with the x-axis East, the y-axis North and the z-axis Up. For the planar coordinates the Universal Transversal Mercator (UTM) projection is used. In UTM different zones are defined. For the NEA Margin zone 33N is being used, for RAK zone 38N is being used. The table below gives the main parameters for both zones.

Zone False Easting [m] False Northing [m] Scale factor Central meridian Origin latitude

33N 500,000 0 0.9996 150 00

38N 500,000 0 0.9996 450 00

The UTM-projection is a transversal cylindrical projection, see Figure 5-2. In the plane the x-axis points East (Easting) and the y-axis points North (Northing). North and East in the plane do not coin-cide with North and East of the ellipsoid except for the central meridian. The corrections for the azi-muth are given in Section 5.3.

Figure 5-2: Transversal cylindrical projection

From geodetic longitude and latitude to Northing and Easting

We follow (Karney 2011) here, who based his equations on (Krüger 1912). See Appendix A for an implementation in Matlab. UTM coordinates are computed with

𝑁 = 𝑘𝐴𝜉 + 𝑁0

𝐸 = 𝑘𝐴𝜂 + 𝐸0


where N0, E0 and k are given the table above and

𝐴 =𝑎

1 + 𝑛(1 +

1

4𝑛2 +

1

64𝑛4)

with 2𝜋𝐴 the circumference of a meridian and third flattening 𝑛 =𝑓

2−𝑓. The variables 𝜉 and 𝜂 are

computed as

𝜉 = 𝜉′ + ∑ 𝛼𝑗 sin 2𝑗𝜉′ cosh 2𝑗𝜂′

4

𝑗=1

𝜂 = 𝜂′ + ∑ 𝛼𝑗 cos 2𝑗𝜉′ sinh 2𝑗𝜂′

4

𝑗=1

where

𝛼1 = 1

2𝑛 −

2

3𝑛2 +

5

16𝑛3 +

41

180𝑛4

𝛼2 = 13

48𝑛2 −

3

5𝑛3 +

557

1440𝑛4

𝛼3 =61

240𝑛3 −

103

140𝑛4

𝛼4 =49561

161280𝑛4

and

𝜉′ = tan−1𝜏′

cos 𝜆

𝜂′ = sinh−1sin 𝜆

√𝜏′2 + cos2 𝜆

with

𝜏′ = 𝜏√1 + 𝜎2 − 𝜎√1 + 𝜏2

where

𝜏 = tan 𝜑

𝜎 = sinh (𝑒 tanh−1𝑒𝜏

√1 + 𝜏2) .

From Northing and Easting to geodetic longitude and latitude

From the forward projection we have (see Appendix A for an implementation in Matlab):

𝑁 − 𝑁0

𝑘𝐴= 𝜉

𝐸 − 𝐸0

𝑘𝐴= 𝜂.

The variables 𝜉′ and 𝜂′ are computed as


𝜉′ = 𝜉 − ∑ 𝛽𝑗 sin 2𝑗𝜉 cosh 2𝑗𝜂

4

𝑗=1

𝜂′ = 𝜂 − ∑ 𝛽𝑗 cos 2𝑗𝜉 sinh 2𝑗𝜂

4

𝑗=1

where

𝛽1 = 1

2𝑛 −

2

3𝑛2 +

37

96𝑛3 −

1

360𝑛4

𝛽2 = 1

48𝑛2 +

1

15𝑛3 −

437

1440𝑛4

𝛽3 =17

480𝑛3 −

37

840𝑛4

𝛽4 =4397

161280𝑛4.

The geodetic longitude then is

𝜆 = tan−1sinh 𝜂′

cos 𝜉′

and the geodetic latitude is then obtained by Newton iteration using the initial value

𝜏′ =sin 𝜉′

√sinh2 𝜂′ + cos2 𝜉′

in

𝜏𝑖 = {𝜏′ for 𝑖 = 0𝜏𝑖−1 + 𝛿𝜏𝑖−1 otherwise,

𝜏𝑖′ = 𝜏𝑖√1 + 𝜎𝑖

2 − 𝜎𝑖√1 + 𝜏𝑖2

𝛿𝜏𝑖 =𝜏′ − 𝜏𝑖

′

√1 + 𝜏𝑖′2

1 + (1 − 𝑒2)𝜏𝑖2

(1 − 𝑒2)√1 + 𝜏𝑖2

.

The geodetic latitude now is

𝜑 = tan−1 𝜏.

5.2 Major Reference Frames for Gradiometry

LORF – Local Orbit Reference Frame

Origin OLORF located at the actual (nominal) satellite centre of mass;

XLORF axis (roll) parallel to instantaneous direction of the orbital velocity vector (V ) with the same sign as this vector.

YLORF axis (pitch) axis parallel to instantaneous direction of the orbital angular momentum

(N), with the same sign as N (V and N are orthogonal by definition, since N = R V, where R is the vector from the Earth centre to the origin).

ZLORF (yaw) axis parallel to V N, with the same sign as V N


GRF - Gradiometer Reference Frame

This is the coordinate system in which the gravity gradients are measured by GOCE. The GRF repre-sents the Three-Axis Gradiometer common reference for the mutual positioning and alignment of the three One Axis Gradiometers and for the positioning and orientation of the whole instrument with respect to external reference frames. Details of the other reference frames mentioned below are specified in (Gruber et al. 2010).

Origin OGRF located at the origin of the OAGRF3

XGRF ,YGRF ,ZGRF axes are parallel to the corresponding axes of OAGRF3 with the same sign.

Nominally the origins of all OAGRF’s coincide in one intersection point. The corresponding axes of each of the 3 OAGRF’s are parallel and point in the same directions. The corresponding 6 ARF’s are nominally parallel and point in the same direction.

LNOF – Local North Oriented Frame

The Local North Oriented Frame (LNOF) is a right-handed North-West-Up frame with the X-axis point-ing North, the Y-axis pointing West and the Z-axis Up. The calibrated gravity gradients of the EGG_TRF_2 products are provided in this system.

The origin OLNOF is located at the actual (or nominal) satellite centre of mass

ZLNOF is defined as the vector from the geocenter to the origin OLNOF, pointing radially out-ward,

YLNOF is parallel to the normal vector to the plane of the geocentric meridian of the satellite center of mass, pointing westward,

XLNOF is parallel to the normal vector to the plane defined by YLNOF and ZLNOF and forms a right-handed system.

In geocentric latitude and East longitude (φ, λ) of the GOCE center of mass in the Earth-Fixed Refer-ence Frame (EFRF) the 3 axes are defined as follows:

𝑍𝐿𝑁𝑂𝐹 = (cos 𝜙 cos 𝜆cos 𝜙 sin 𝜆

sin 𝜙) ; 𝑌𝐿𝑁𝑂𝐹 = (

sin 𝜆− cos 𝜆

0) ; 𝑋𝐿𝑁𝑂𝐹 = (

− sin 𝜙 cos 𝜆− sin 𝜙 sin 𝜆

cos 𝜙)

Grids of gravity gradients are provided in the LNOF above the ellipsoid. The gradient grids are given on a homothetic ellipsoid that has the same eccentricity as the WGS84 ellipsoid and a semi-major axis aH = aWGS84 + H, where aWGS84 = 6378.137 km and H is 225 km or 255 km. In practice this means that the height h above the WGS84 ellipsoid slightly varies from equator to the poles. For the lower grids, for example, the height is h = 225 km at the equator and h ≈ 224.25 km at the poles.


Figure 5-3: Definition of fundamental Reference Systems for GOCE. Ω corresponds to the right ascension of the ascending node of the GOCE orbital plane; u corresponds to the argument of latitude of the GOCE space-craft at a specific time.

Figure 5-3 shows how the fundamental reference frames are oriented with respect to each other. Because the satellite attitude is controlled with magneto-torquers, the GRF does not coincide with the LORF. In science mode the satellite will operate in drag-free mode for the flight direction only.

5.3 GRF to MRF rotation

The rotation of the gravity gradients in the instrument frame, the GRF, to the model reference frame is done in two steps. First, the gradients are rotated to the LNOF. Next, the rotation from LNOF to MRF is performed.

5.3.1 GRF to LNOF There are three steps:

1. Rotation from GRF to IRF using the inertial attitude quaternions from GOCE 2. Rotation from IRF to EFRF using the quaternions from GPS orbit determination 3. Rotation from EFRF to LNOF using the relation between the two frames (Section 5.2)

These steps are implemented in the standard GOCE data processing software.


5.3.2 LNOF to MRF In order to compare the gravity gradients from GOCE with model gravity gradients we need to trans-form spherical coordinates to model Cartesian coordinates. The GOCE gradients are given in spherical coordinates (𝜙, 𝜆, 𝑟) whereas the model gradients are given in (𝑥, 𝑦, 𝑧). The model coordinates fol-low from the transversal Mercator projection and with each (𝑥, 𝑦, 𝑧 = 0) ellipsoidal coordinates (𝜑, 𝜆, ℎ = 0) are associated. These ellipsoidal coordinates can be transformed to spherical coordi-nates.

5.3.2.1 Coordinates of one system in the other Forward transformation

To compute the coordinates of a point Q one can follow the procedure below (see Section 5.1 for the relevant equations):

1. For a (x,y)-pair compute ellipsoidal coordinates (𝜑, 𝜆) from (x, y) 2. Convert ellipsoidal coordinates (𝜑, 𝜆, ℎ) to (𝑋ℎ , 𝑌ℎ , 𝑍ℎ); these are the Cartesian coordinates

in the ECEF frame of point Q 3. Once the Cartesian ECEF coordinates are known, one can compute geocentric coordinates.

Inverse transformation

Suppose spherical coordinates are given. Then (see Section 5.1 for the relevant equations):

1. Convert spherical coordinates to Cartesian coordinates 2. Convert Cartesian coordinates to ellipsoidal coordinates (𝜑, 𝜆, ℎ) 3. Use these ellipsoidal coordinates with h = 0 to compute (𝑥, 𝑦)

5.3.2.2 Correction for difference in orientation The angle between the true North and the projected North is given by (Strang van Hees 2006)

𝛾 = Δ𝜆 sin 𝜑 +Δ𝜆3

3sin 𝜑 cos2 𝜑 (1 + 3𝑛2 + 2𝑛4) +

Δ𝜆5

15sin 𝜑 cos4 𝜑 (2 − 𝑡2)

with Δ𝜆 = 𝜆 − 𝜆0 the longitude difference between the central meridian and the computation point. This angle is shown for the NEA margin in Figure 5-4. See Appendix A for an implementation in Matlab.


Figure 5-4: Deviation of map North from geographic North in degree

The rotation from the x’-system (LNOF) to the x-system (MRF) is given as

(𝑥𝑦𝑧

) = (cos 𝛾 sin 𝛾 0

− sin 𝛾 cos 𝛾 00 0 1

) (𝑥′

𝑦′

𝑧′

).

If we denote the total rotation as Rtot then the gravity gradients transform as

𝑉𝑖𝑗 = 𝑅𝑡𝑜𝑡𝑉𝑖𝑗′ 𝑅𝑡𝑜𝑡

𝑇

and the inverse transformation is

𝑉𝑖𝑗′ = 𝑅𝑡𝑜𝑡

𝑇 𝑉𝑖𝑗𝑅𝑡𝑜𝑡 .

The coordinate transformation from LNOF to MRF and vice versa, as well as the rotation between LNOF and MRF are described in detail in Bouman (2013) [RD-11] and Bouman et al. (2013).


6 Gravitational field

6.1 Functionals of the gravity field

The global gravity field of the Earth is mainly expressed as gravity potential V or disturbing potential T after subtracting the normal potential U. Next to this parameter there are a lot of other functionals of the gravity field. In case of determining the shape of the Earth in terms of calculating a geoid, the functionals geoid height N, gravity anomaly Δg, gravity disturbance δg and deflection of the vertical (ξ,η) play an important role. They all can be derived from the disturbing potential T.

6.1.1 Geoid height N Surfaces with a constant potential are defined as equipotential surfaces. They are often used as height reference surfaces. Thus, a geoid means the equipotential surface in the gravity field of the Earth, which coincides with the undisturbed mean sea level extended continuously [eur-lex.europa.eu]. The geoid heights N are defined as vertical distances from the equipotential surface of a normal potential (rotational ellipsoid). Following Bruns’s formula

𝑁 =𝑇

𝛾 (6 − 1)

the disturbing potential T has to be computed on the geoid, whereas the theoretical gravity of the normal potential γ has to be computed at the corresponding point at the ellipsoid.

6.1.2 Gravity anomaly Δg Further, gravity anomalies Δg are the deviations between measured and computed gravity g at point P on the geoid and the theoretical gravity γ of the normal potential at the corresponding point Q on the ellipsoid:

Δ𝑔 = 𝑔𝑃 − 𝛾𝑄 . (6 − 2)

For given disturbing potential T, the gravity anomalies can be computed from the so called fun-damental equation of the physical geodesy (Heiskanen & Moritz 1967, p. 95):

Δ𝑔 = −𝜕𝑇

𝜕ℎ+

1

𝛾

𝜕𝛾

𝜕ℎ𝑇 . (6 − 3)

Therefore the parameter h is considered as height in the normal direction.

6.1.3 Gravity disturbance δg In contrast to gravity anomalies Δg, gravity disturbances δg describe the difference between meas-ured gravity g and normal gravity γ at one and the same point P on the geoid:

𝛿𝑔 = 𝑔𝑃 − 𝛾𝑃 . (6 − 4)

The small angle (further defined as deflection of the vertical) between the directions of the corre-sponding gravity vectors is in this case neglected.

6.1.4 Deflection of the vertical (ξ,η) Deflections of the vertical are the differences in the direction between the actual vertical and the normal on the ellipsoid of the normal potential. Thus, one distinguishes the deflection of the vertical in longitude ξ (north-south direction) and in latitude η (east-west direction). With the known disturb-ing potential T they can be expressed as

𝜉 = −1

𝑅 𝛾

𝜕𝑇

𝜕𝜑 ,


𝜂 = −1

𝑅𝛾 cos 𝜑

𝜕𝑇

𝜕𝜆 . (6 − 5)

Hereby R is an average radius of the Earth and (𝜆, 𝜑) are the geographic coordinates.

6.2 Derivations of the disturbing potential T

On the one hand derivations of the disturbing potential T are needed for the synthesis of series ex-pansions in spherical harmonics in terms of global gravity field determination. On the other hand the second derivatives correspond to the measured GOCE gravity gradients of the potential V, reduced by a background model. The use of these gradiometer observations is one of the main aspects han-dling with GOCE data.

In terms of spherical harmonics the disturbing potential T can be expressed as a series expansion

𝑇(𝑟, 𝜆, 𝜙) =𝐺𝑀

𝐴∑ (

𝐴

𝑟)

𝑙+1𝑙𝑚𝑎𝑥

𝑙=0

∑ [𝐶𝑙𝑚𝑐𝑜𝑠𝑚𝜆 + 𝑆𝑙𝑚𝑠𝑖𝑛𝑚𝜆] 𝑃𝑙𝑚(𝑐𝑜𝑠𝜃) , (6 − 6)

𝑙

𝑚=0

where

𝑟, 𝜆, 𝜙 spherical coordinates: radius r, longitude λ, latitude φ, 𝜃 spherical co-latitude, 𝐺𝑀 product of gravitational constant G and mass of the Earth M, 𝐴 radius of the Earth, 𝑙, 𝑚 degree l and order m, 𝐶𝑙𝑚 , 𝑆𝑙𝑚 spherical harmonic coefficients, 𝑃𝑙𝑚 with 4π normalized Legendre functions.

Further the term can be split into a scale portion 𝑓0, a summation part 𝑓𝑙(𝑟), which only depends on the radius r and an element f(λ,θ), which only depends on the longitude λ and the co-latitude θ. So a more general expression for the disturbing potential and its first and second derivatives (key) can be formulated as

𝑘𝑒𝑦 = 𝑓0 ∑ 𝑓𝑙(𝑟)

𝑙𝑚𝑎𝑥

𝑙=0

∑ 𝐶𝑙𝑚�̅�𝑙𝑚

𝑙

𝑚=−𝑙

(𝜆, 𝜃) . (6 − 7)

Finally fl(r) can be split into eigenvalues g(l) and an attenuation component of the field continuation Xl+α:

𝑓𝑙(𝑟) = 𝑔(𝑙)𝑋𝑙+𝛼 .

Table 6-1 gives an overview of the components of the disturbing potential and all the first and sec-ond derivatives.

According to these relations the functionals of the gravity field can be formulated as:

𝑁 =𝑇

𝛾 … geoid height,

Δ𝑔 = −𝑇𝑟 +1

𝛾

𝜕𝛾

𝜕ℎ𝑇 … gravity anomaly with the assumption

𝜕𝑇

𝜕ℎ≈

𝜕𝑇

𝜕𝑟 ,

𝛿𝑔 = Δ𝑔 −1

𝛾

𝜕𝛾

𝜕ℎ𝑇 … gravity disturbance,

𝜉 = −1

𝑅 𝛾𝑇𝜑 , 𝜂 = −

1

𝑅𝛾 cos 𝜑𝑇𝜆 … deflections of the vertical, with φ ≈ Φ = 90° - θ .


Table 6-1: First and second derivatives of the disturbing potential of the Earth

key f0 g(l) Xl+α α co(m) si(m) Legendre

T GM

A 1 A

r 1 cosmλ sinmλ Plm(cosθ)

Tr

rA

GM

- (l+1)

A


Tλ

GM

A 1 A

r 1 -m·sinmλ m·cosmλ Plm(cosθ)

Tθ

GM

A 1 A

r 1 cosmλ sinmλ

θ

θPlm

cos

Trr 2rA

GM

(l+1)(l+2)

A


Trλ

rA

GM

- (l+1)

A

r 2 -m·sinmλ m·cosmλ Plm(cosθ)

Trθ

rA

GM

- (l+1)

A

r 2 cosmλ sinmλ

θ

θPlm

cos

Tλλ

GM

A 1 A

r 1 -m2·cosmλ -m2·sinmλ Plm(cosθ)

Tλθ

GM

A 1 A

r 1 -m·sinmλ m·cosmλ

θ

θPlm

cos

Tθθ

GM

A 1 A

r 1 cosmλ sinmλ

2

2 cos

θ

θPlm

6.3 Regional Gravity Field Modelling Using MSR

One tool for the regional gravity field representation and the combination of input data from differ-ent observation techniques is the multi-scale representation (MSR). The MSR means viewing on a signal under different resolutions (microscope effect). In other words the MSR provides approxima-tions of the signal under different resolution levels. These approximations are representable in series expansions in so-called scaling functions; furthermore the differences between the approximations of two adjacent resolution levels are called detail signals. These detail signals are also representable by series expansions, this time in so-called wavelet functions. Scaling and wavelet functions are re-lated to each other by linear equation systems. The detail signals are band-passed filtered signals of the signal under investigation, e.g. the gravitational potential. Following this concept the geopoten-tial

𝑉(𝒙) ≈ 𝑉𝐽+1(𝒙) = ∑ 𝑑𝐽,𝑞

𝑁

𝑞=1

𝜙𝐽+1(𝒙, 𝒙𝑞) (6 − 8)

at highest resolution level J+1 and spatial position (vector) 𝒙 can be decomposed into the low-pass filtered approximation

𝑉𝑗0(𝒙) = ∑ 𝑑𝑗0,𝑞

𝑁

𝑞=1

𝜙𝑗0(𝒙, 𝒙𝑞) (6 − 9)


at lowest level 𝑗 = 𝑗0 and the detail signals

𝐺𝑗(𝒙) = ∑ 𝑑𝑗,𝑞

𝑁

𝑞=1

𝜓𝑗(𝒙, 𝒙𝑞) (6 − 10)

with 𝑗 = 𝑗0, … , 𝐽 according to

𝑉𝐽+1(𝒙) = 𝑉𝑗0(𝒙) + ∑ 𝐺𝑗(𝒙)

𝐽

𝑗=𝑗0

. (6 − 11)

Each detail signal is related to a specific frequency band, i.e. it means a band-passed filtered version of the geopotential. In the equations (6-8), (6-9) and (6-10) we introduced the spherical scaling func-tion

𝜙𝑗(𝒙, 𝒙𝑞) = ∑2𝑙 + 1

4𝜋 (

𝑅

𝑟)

𝑙+1

Φ𝑗(𝑙) 𝑃𝑙(𝒓𝑇𝒓𝑞) ,

2𝑗−1

𝑙=0

(6 − 12)

and the spherical wavelet function

𝜓𝑗(𝒙, 𝒙𝑞) = ∑2𝑙 + 1

4𝜋 (

𝑅

𝑟)

𝑙+1

Ψ𝑗(𝑙) 𝑃𝑙(𝒓𝑇𝒓𝑞) ;

2𝑗+1−1

𝑙= 2𝑗−1

(6 − 13)

𝑃𝑙 means the Legendre polynomial of degree l. Furthermore, 𝒙 = 𝑟 𝒓. The Legendre coefficients Φ𝑗(𝑙) in Eq. (6-12) and Ψ𝑗(𝑙) = Φ𝑗+1(𝑙) − Φ𝑗(𝑙) in Eq. (6-13) describe the frequency behaviour of

the radial base functions 𝜙𝑗(𝒙, 𝒙𝑞) and 𝜓𝑗(𝒙, 𝒙𝑞) and define their shapes. Figure 6-1 shows as an

example the Blackman scaling function 𝜙𝑗(𝒙, 𝒙𝑞) for level 𝑗 = 4. In contrast to spherical harmonics,

spherical base functions defined by the Eqs. (6-12) and (6-13) are highly localizing, i.e. they are char-acterized by a sharp peak. The sharpness of the peak depends on the chosen level. The higher this value is set the sharper is the function and the finer are the structures which can be modelled by Eq. (6-8).

Figure 6-1: Spherical Blackman scaling function of level 𝒋 = 𝟒. Thus, this function extracts signal parts until degree 15 from the signal according to the Eqs. (6-8) and (6-12).

The K position vectors 𝒙𝑞 with 𝑞 = 1, … , 𝑁 in the Eqs. (6-8) to (6-10) define a grid on a sphere with

radius R within the region of interest. The total number N of grid points is directly related to the size of the region and to the chosen value 𝐽 or 𝐽 + 1 for the maximum level in Eq. (6-6). For more details concerning the MSR as well as spherical base functions see Schmidt et al. (2007) and the references listed in this publication.


6.3.1 Gravitational potential in geocentric polar coordinates

The gravitational potential is in general given in geocentric polar coordinates:

𝜃 spherical pole distance 𝜆 geographic longitude 𝑟 radial distance

Coordinate system Cartesian 𝑺 Transformed �̅�

- Origin

- Axes

- Coordinates

- Center of the sphere

- Base vectors (𝒆1, 𝒆2, 𝒆3) 𝒆1 = (1,0,0) 𝒆2 = (0,1,0) 𝒆3 = (0,0,1)

- Spherical (𝜆, 𝛽, 𝑟) Longitude 𝜆: 0 … 2𝜋 Latitude 𝛽: − 𝜋

2⁄ … 𝜋2⁄

Radius 𝑟 > 0 Co-latitude 𝜃 = 𝜋

2⁄ − 𝛽

- Center of the sphere

- Base vectors (�̅�1, �̅�2, �̅�3)

- Spherical (𝜆′, 𝛽′, 𝑟′)

Transformation matrix 𝑨

Position vector 𝒙 = 𝑟 𝒓 𝒙 = 𝑨 𝒙

Unit vector 𝒓 = (cos 𝛽 cos 𝜆cos 𝛽 sin 𝜆

sin 𝛽) , |𝒓| = 1

𝒓

Vector to computation point P

𝒓𝒑 𝒓𝒑

Spherical distance 𝜓

𝒓𝒑𝑇

𝒓 = cos 𝜓

cos 𝜓 = cos 𝜃 cos 𝜃𝑞 + sin 𝜃 sin 𝜃𝑞 cos(𝜆 − 𝜆𝑞)

Inserting Eq. (6-12) into Eq. (6-8) yields with cos 𝜓 = 𝒓𝑇𝒓𝑞 and 𝑙𝐽′ = 2𝐽 − 1 the gravitational poten-

tial

𝑉(𝒙) = ∑ ∑2𝑙 + 1

4𝜋𝑑𝐽,𝑞𝜙𝐽+1,𝑙 (

𝑅

𝑟)

𝑙+1

𝑃𝑙(cos 𝜓)

𝑙𝐽′

𝑙=0

𝑁

𝑞=1

. (6 − 14)

Its first and second derivatives with respect to the spherical coordinates read

𝜕𝑉

𝜕𝑟= ∑ ∑

2𝑙 + 1


𝑅

𝑟)

𝑙+1

𝑃𝑙(cos 𝜓)

𝑙𝐽′

𝑙=0

𝑁

𝑞=1

⋅ (−𝑙 + 1

𝑟)

𝜕2𝑉

𝜕𝑟2= ∑ ∑

2𝑙 + 1


𝑅

𝑟)

𝑙+1

𝑃𝑙(cos 𝜓)

𝑙𝐽′

𝑙=0

𝑁

𝑞=1

⋅(𝑙 + 1)(𝑙 + 2)

𝑟2

𝜕𝑉

𝜕𝜃= ∑ ∑

2𝑙 + 1


𝑅

𝑟)

𝑙+1𝑙𝐽

′

𝑙=0

𝑁

𝑞=1

⋅𝜕𝑃𝑙(cos 𝜓)

𝜕𝜃


𝜕2𝑉

𝜕𝜃2= ∑ ∑

2𝑙 + 1


𝑅

𝑟)

𝑙+1𝑙𝐽

′

𝑙=0

𝑁

𝑞=1


𝜕𝜃2

𝜕𝑉

𝜕𝜆= ∑ ∑

2𝑙 + 1


𝑅

𝑟)

𝑙+1𝑙𝐽

′

𝑙=0

𝑁

𝑞=1


𝜕𝜆 (6 − 15)

𝜕2𝑉

𝜕𝜆2= ∑ ∑

2𝑙 + 1


𝑅

𝑟)

𝑙+1𝑙𝐽

′

𝑙=0

𝑁

𝑞=1


𝜕𝜆2

𝜕2𝑉

𝜕𝑟 𝜕𝜃= ∑ ∑

2𝑙 + 1


𝑅

𝑟)

𝑙+1𝑙𝐽

′

𝑙=0

𝑁

𝑞=1

⋅ (−𝑙 + 1

𝑟) ⋅

𝜕𝑃𝑙(cos 𝜓)

𝜕𝜃

𝜕2𝑉

𝜕𝑟 𝜕𝜆= ∑ ∑

2𝑙 + 1


𝑅

𝑟)

𝑙+1𝑙𝐽

′

𝑙=0

𝑁

𝑞=1

⋅ (−𝑙 + 1

𝑟) ⋅


𝜕𝜆

𝜕2𝑉

𝜕𝜆 𝜕𝜃= ∑ ∑

2𝑙 + 1


𝑅

𝑟)

𝑙+1𝑙𝐽

′

𝑙=0

𝑁

𝑞=1


𝜕𝜆 𝜕𝜃

The derivatives of the Legendre polynomials read


𝜕𝜃=


𝜕 cos 𝜓⋅

𝜕 cos 𝜓

𝜕𝜃


𝜕 cos 𝜓=

𝑙

(cos 𝜓)2 − 1(cos 𝜓 𝑃𝑙(cos 𝜓) − 𝑃𝑙−1(cos 𝜓))

𝜕 cos 𝜓

𝜕𝜃= ((− sin 𝜃 cos 𝜃𝑞) + (cos 𝜃 sin 𝜃𝑞 cos(𝜆 − 𝜆𝑞)))


𝜕𝜆=


𝜕 cos 𝜓⋅

𝜕 cos 𝜓

𝜕𝜆

𝜕 cos 𝜓

𝜕𝜆= − sin 𝜃 sin 𝜃𝑞 sin(𝜆 − 𝜆𝑞)

𝜕2𝑃𝑙(cos 𝜓)

𝜕𝜃2=


𝜕 cos 𝜓2

𝜕 𝑐𝑜𝑠 𝜓

𝜕𝜃⋅

𝜕 cos 𝜓

𝜕𝜃+


𝜕 cos 𝜓 ⋅

𝜕2 cos 𝜓

𝜕𝜃2 (6 − 16)

𝜕2 cos 𝜓

𝜕𝜃2= − cos 𝜃 cos 𝜃𝑞 − (sin 𝜃) sin 𝜃𝑞 cos(𝜆 − 𝜆𝑞)


𝜕𝜆2=


𝜕 cos 𝜓2


𝜕𝜆⋅

𝜕 cos 𝜓

𝜕𝜆+


𝜕 cos 𝜓 ⋅

𝜕2 cos 𝜓

𝜕𝜆2

𝜕2 cos 𝜓

𝜕𝜆2= − sin 𝜃 sin 𝜃𝑞 cos(𝜆 − 𝜆𝑞)


𝜕𝜆 𝜕𝜃 =


𝜕 cos 𝜓2


𝜕𝜃⋅

𝜕 cos 𝜓

𝜕𝜆+


𝜕 cos 𝜓 ⋅

𝜕2 cos 𝜓

𝜕𝜆 𝜕𝜃


𝜕2 cos 𝜓

𝜕𝜆 𝜕𝜃= − cos 𝜃 sin 𝜃𝑞 sin(𝜆 − 𝜆𝑞)

Appropriate to the first and second derivatives of the disturbing potential in table 6-1, the first and second derivatives of the gravitational potential can be arranged in a table (see table 6-2). Hereby akoeff and w(i) depend on the radial distance r and the Legendre polynomials P(l) depend on the spherical distance ψ, thus on the co-latitude θ and longitude λ of the observation point.

Table 6-2: First and second derivatives of the gravitational potential of the Earth w.r.t. spherical coordinates and Legendre Polynomials w.r.t. spherical distance

key akoeff w(l) P(l)

f0 Xl+α α g(l) Legendre

V 4

12 l

r

R 1 1 Pl(cosψ)

Vr

4

12 l

r

R 1

r

l 1 Pl(cosψ)

Vλ

4

12 l

r

R 1 1

)(coslP

Vθ

4

12 l

r

R 1 1

θ

Pl

)(cos

Vrr

4

12 l

r

R 1

²

)2)(1(

r

ll Pl(cosψ)

Vrλ

4

12 l

r

R 1

r

l 1

)(coslP

Vrθ

4

12 l

r

R 1

r

l 1

θ

Pl

)(cos

Vλλ

4

12 l

r

R 1 1

²

)(cos²

lP

Vλθ

4

12 l

r

R 1 1

θ

Pl )(cos²

Vθθ 4

12 l

r

R 1 1

2

2 )(cos

θ

Pl

6.3.2 GOCE gradients in Cartesian coordinates

To compute the potential derivatives in the local north-oriented frame (LNOF), we need the first and second derivatives of the gravitational potential with respect to the polar coordinates, which are given in Section 6.3 (GOCE gradients in GRF). Thus a transformation from polar to Cartesian coordi-nates has to be done.

Besides the spherical (geocentric polar) coordinates we define the Cartesian coordinates within the LNOF: 𝑥 = north; 𝑦 = west; 𝑧 = up.

The transformed derivatives (from geocentric polar to LNOF coordinates) read


𝜕2𝑉

𝜕𝑥2=

1

𝑟

𝜕𝑉

𝜕𝑟+

1

𝑟2

𝜕2𝑉

𝜕𝜃2

𝜕2𝑉

𝜕𝑥 𝜕𝑦=

1

𝑟2 sin 𝜃

𝜕2𝑉

𝜕𝜆 𝜕𝜃−

cos 𝜃

𝑟2 sin2 𝜃 𝜕𝑉

𝜕𝜆

𝜕2𝑉

𝜕𝑥 𝜕𝑧=

1

𝑟2

𝜕𝑉

𝜕𝜃−

1

𝑟

𝜕2𝑉

𝜕𝑟 𝜕𝜃 (6 − 17)

𝜕2𝑉

𝜕𝑦2=

1

𝑟

𝜕𝑉

𝜕𝑟+

1

𝑟2 tan 𝜃

𝜕𝑉

𝜕𝜃+

1

𝑟2 sin2 𝜃

𝜕2𝑉

𝜕𝜆2

𝜕2𝑉

𝜕𝑦 𝜕𝑧=

1

𝑟2 sin 𝜃

𝜕𝑉

𝜕𝜆−

1

𝑟 sin 𝜃

𝜕2𝑉

𝜕𝑟 𝜕𝜆

𝜕2𝑉

𝜕𝑧2=

𝜕2𝑉

𝜕𝑟2.

Inserting the first and second derivatives of the gravitational potential from chapter 6.4 into these formulas offers the observation equations for the 6 gradiometer measurements of GOCE w.r.t. Carte-sian coordinates in LNOF:

Table 6-3: Observation equations for the single components of the observed GOCE gravity gradients.

observation

observation equation (right side)

Modified scaling function �̃�𝐽+1

Vxx ∑ 𝑑𝐽,𝑞�̃�𝐽+1(𝒙𝒊, 𝒙𝒒)

𝑁𝐽

𝑞=1

∑2𝑙 + 1

4𝜋𝛷𝐽+1,𝑙 (

𝑅

𝑟)

𝑙+1𝑙𝐽

′

𝑙=0

[1

𝑟⋅ 𝑃𝑙(𝑐𝑜𝑠 𝜓) (−

𝑙 + 1

𝑟) +

1

𝑟2⋅


𝜕𝜃2]

Vxy ∑ 𝑑𝐽,𝑞�̃�𝐽+1(𝒙𝒊, 𝒙𝒒)

𝑁𝐽

𝑞=1

∑2𝑙 + 1


𝑅

𝑟)

𝑙+1𝑙𝐽

′

𝑙=0

[1

𝑟2 sin 𝜃⋅


𝜕𝜆 𝜕𝜃 −

cos 𝜃

𝑟2 𝑠𝑖𝑛2 𝜃⋅


𝜕𝜆]

Vxz ∑ 𝑑𝐽,𝑞�̃�𝐽+1(𝒙𝒊, 𝒙𝒒)

𝑁𝐽

𝑞=1

∑2𝑙 + 1


𝑅

𝑟)

𝑙+1𝑙𝐽

′

𝑙=0

[1

𝑟2⋅


𝜕𝜃−

1

𝑟⋅ (−

𝑙 + 1

𝑟) ⋅


𝜕𝜃 ]

Vyy ∑ 𝑑𝐽,𝑞�̃�𝐽+1(𝒙𝒊, 𝒙𝒒)

𝑁𝐽

𝑞=1

∑

2𝑙 + 1


𝑅

𝑟)

𝑙+1𝑙𝐽

′

𝑙=0

[1

𝑟⋅ 𝑃𝑙(𝑐𝑜𝑠 𝜓) ∙ (−

𝑙 + 1

𝑟) +

1

𝑟2 tan 𝜃⋅


𝜕𝜃+

1

𝑟2𝑠𝑖𝑛2𝜃


𝜕𝜆2]

Vyz ∑ 𝑑𝐽,𝑞�̃�𝐽+1(𝒙𝒊, 𝒙𝒒)

𝑁𝐽

𝑞=1

∑2𝑙 + 1


𝑅

𝑟)

𝑙+1𝑙𝐽

′

𝑙=0

[1

𝑟2 𝑠𝑖𝑛 𝜃⋅

𝜕𝑃𝑙(𝑐𝑜𝑠 𝜓)

𝜕𝜆−

1

𝑟 𝑠𝑖𝑛 𝜃⋅ (−

𝑙 + 1

𝑟) ⋅

𝜕𝑃𝑙(𝑐𝑜𝑠 𝜓)

𝜕𝜆 ]

Vzz ∑ 𝑑𝐽,𝑞�̃�𝐽+1(𝒙𝒊, 𝒙𝒒)

𝑁𝐽

𝑞=1

∑2𝑙 + 1

4𝜋𝜙𝐽+1,𝑙 (

𝑅

𝑟)

𝑙+1

𝑃𝑙(cos 𝜓)

𝑙𝐽′

𝑙=0

⋅(𝑙 + 1)(𝑙 + 2)

𝑟


6.3.3 GOCE gradients in the GRF

The aim is to use the original non-rotated and unaffected GOCE gravity gradients in the GRF. As the gravitational potential V (see Eq. 6-14) is defined in an Earth-fixed reference frame (in this case TRF) several steps have to be processed to set up observation equations which enable the use of the orig-inal measurements:

(1) Computation:

- 1st and 2nd derivatives of potential w.r.t. spherical distance 𝜓

- needs: 1st and 2nd derivatives of Legendre polynomials

- Eq. 6-15

- Eq. 6-16

(2) Transformation:

- spherical TRF -> Cartesian LNOF

- Eq. 6-17

(3) Setting up observation equations

- using modified scaling functions

�̃�𝐽+1,𝑖𝑗

- Table 6-3

(4) Rotation:

- LNOF -> GRF

- Eq. 6-21

(3) The observation equations from Table 6-3 are arranged in the order of a 3x3 tensor. The ten-sor elements read:

𝑉𝑎𝑏 = ∑ 𝑑𝐽,𝑞 �̃�𝐽+1,𝑎𝑏(𝒙𝒊, 𝒙𝒒)

𝑁𝐽

𝑞=1

. (6 − 18)

Inserting the modified scaling functions �̃�𝐽+1,ij gives:

[

𝑉𝑥𝑥 𝑉𝑥𝑦 𝑉𝑥𝑧

𝑉𝑦𝑥 𝑉𝑦𝑦 𝑉𝑦𝑧

𝑉𝑧𝑥 𝑉𝑧𝑦 𝑉𝑧𝑧

] = ∑ 𝑑𝐽,𝑞

𝑁𝐽

𝑞=1

[

�̃�𝐽+1,xx(𝒙𝒊, 𝒙𝒒) �̃�𝐽+1,𝑥𝑦(𝒙𝒊, 𝒙𝒒) �̃�𝐽+1,xz(𝒙𝒊, 𝒙𝒒)

�̃�𝐽+1,𝑦𝑥(𝒙𝒊, 𝒙𝒒) �̃�𝐽+1,yy(𝒙𝒊, 𝒙𝒒) �̃�𝐽+1,yz(𝒙𝒊, 𝒙𝒒)

�̃�𝐽+1,𝑧𝑥(𝒙𝒊, 𝒙𝒒) �̃�𝐽+1,𝑧𝑦(𝒙𝒊, 𝒙𝒒) �̃�𝐽+1,zz(𝒙𝒊, 𝒙𝒒)

] . (6 − 19)

(4) For the transformation of the observation equations from the LNOF to the GRF we use the 3-step transformation given in chapter 5.3.1. These steps are summarized within a rotation matrix R

𝑅 = [

𝑟11 𝑟12 𝑟13

𝑟21 𝑟22 𝑟23

𝑟31 𝑟32 𝑟33

] (6 − 20)

and finally applied to the gravitational potential 𝑉𝐿𝑁𝑂𝐹:

𝑉𝐺𝑅𝐹 = 𝑅 𝑉𝐿𝑁𝑂𝐹𝑅𝑇 . (6 − 21)

The components 𝑟𝑖𝑗 of the rotation matrix 𝑅 are delivered within the GOCE products.


Within an analysis step the observation equations for each tensor element are set up (see table 6-3) using reproducing kernels 𝛷𝐽+1,𝑙 (deterministic part). Transforming and rotating all equations into

the GRF enables to use the original GOCE gravity gradients. The unknown coefficients 𝑑𝐽,𝑞 then are

estimated by relative weighting of all observations using variance components (stochastic part), which is explained in detail in chapter 6.8.

Within the synthesis step, the estimated coefficients �̂�𝐽,𝑞 are finally used to compute gravity gradient

grid values. Therefore we model the gravitational potential (6-14) using Blackman scaling functions within the LNOF. The procedure in Figure 6-2 summarizes the transformation steps that are needed to model the gravitational potential in the LNOF from GOCE gradients measured in the GRF.

Figure 6-2: Transformation scheme to model the gravitational potential with GOCE gradients within the LNOF.

6.3.4 Comparison: series expansion in spherical harmonics versus series expansion in spherical scaling functions

Within the regional gravity field modelling we introduce the geopotential difference

∆𝑉(𝒙) = 𝑉(𝒙) − 𝑉𝑏𝑎𝑐𝑘(𝒙) (6 − 22)

as a series expansion

Δ𝑉(𝒙) ≈ Δ𝑉𝐽+1(𝒙) = ∑ 𝑑𝐽,𝑞

𝑁

𝑞=1

𝜙𝐽+1(𝒙, 𝒙𝑞) (6 − 23)

in terms of the spherical base functions (6-12) for 𝑗 = 𝐽 + 1. Table 6-4 gives recipes on the applica-tion of a spherical harmonic and the spherical scaling function approach according to the Eqs. (6-6) and (6-23).

For the determination of the gravity potential of the Earth V, there are different computational ap-proximation techniques. In global tasks modelling the potential via series expansion in spherical har-monics is the most common method whereas for regional models a series expansion in spherical scaling functions has essential advantages. In both cases we first reduce the well-known main com-ponents (i.e. the long-wavelength parts) of the potential to get potential differences whose struc-tures can be estimated more accurately. The subtraction of the normal potential U results in the dis-turbing potential T and the subtraction of a background model Vback (e.g. EGM2008) results in ΔV according to the Eqs. (6-6) and (6-22). The choice of the background model and its maximum degree of development depends on the input data. To ensure an uncorrelated difference the background


model must not contain any data contained in the input data. To analyse GOCE data for example, the ITG-Grace2010 model can be used. The maximum degree should always be smaller than the request-ed maximum degree of the regional gravity field, but not too small in case of filling up data gaps with this model information. On the other hand it should not cover the whole frequency spectrum of the input data to ensure a realistic result within the solution approach with variance component estima-tion (VCE) where the background model is treated as prior information.

Table 6-4: Comparison of series expansion in spherical harmonics with series expansion in spherical scaling functions

spherical harmonics spherical scaling functions

potential difference

𝑇 = 𝑉 − 𝑈 𝛥𝑉 = 𝑉 − 𝑉𝑏𝑎𝑐𝑘

Modeling global spherical harmonic functions localizing spherical scaling (base) functions

series expansion

Δ … average point distance (global)

𝑙𝑚𝑎𝑥 ≤𝜋⋅𝑅[𝑘𝑚]

𝛥[𝑘𝑚]

lmax … maximum degree 𝑁 = (𝑙𝑚𝑎𝑥 + 1)2 N … maximum number of parameters

𝐹(𝜆, 𝜑) = ∑ ∑ 𝐶𝑙𝑚𝑙𝑚=−𝑙

𝑙𝑚𝑎𝑥𝑙=0 �̅�𝑙𝑚(𝜆, 𝜑)

Δ … average point distance (regional)

𝑙𝐽+1′ ≤𝜋⋅𝑅[𝑘𝑚]

𝛥[𝑘𝑚]

lJ+1' … maximum degree lmax 𝑁𝐽 ≥ (𝑙𝐽+1′ + 1)2

NJ … maximum number of scaling functions

𝐹𝐽+1(𝒙) = ∑ 𝑑𝐽,𝑞𝑁𝐽

𝑞=1 𝜙𝐽+1(𝒙, 𝒙𝑞)

Clm Slm … coefficients Ylm … spherical harmonics

dJ,q … coefficients ΦJ+1 … scaling functions

lmin … minimum degree = 0 lmin … minimum degree, depends on expansion of the area

parameter estimation

N … number of unknown coefficients 𝑁 = (𝑙𝑚𝑎𝑥 + 1)2

NJ … number of unknown coefficients 𝑁𝐽 ≥ (𝑙𝐽+1′ + 1)2

(λ,φ) … observation locations x = xi … observation locations

MRR (not possible)

FJ+1 … approximation of target signal

𝐹𝐽+1 = 𝐹𝑗′ + ∑ 𝐺𝑗𝐽𝑗=𝑗′

Fj' … signal + Gj … detail signals

𝑙𝐽′ = 2𝐽 − 1

lj' … maximum degree of resolution level j 𝐽 = ⌊log2(𝑙𝐽′ + 1)⌋

J+1 … maximum level 𝑙𝐽+1 = 2𝐽+1 − 1

lJ+1 … maximum degree of target signal

𝑗′ = ⌊𝜋⋅𝑅[𝑘𝑚]

2⋅𝛱[𝑘𝑚]⌋

j' … minimum resolution

𝑙𝑗′ = 2𝑗′ − 1

lj' … maximum degree of resolution level j'

resolution r … minimum spatial resolution (half wavelength)

𝑟[𝑘𝑚] =𝜆

2[𝑘𝑚] =

20000[𝑘𝑚]

𝑙𝑚𝑎𝑥

r … minimum spatial resolution (half wavelength)

𝑟[𝑘𝑚] =𝜆

2[𝑘𝑚] =

20000[𝑘𝑚]

𝑙𝐽′

The target functions can be approximated within a series expansion by estimating the unknown pa-rameters. Therefore in global spherical functions the coefficients Clm and Slm have to be determined,


which are together with the Legendre functions Plm responsible for the shape of the global model. The number of coefficients N depends on the maximum degree lmax of the series expansion and thus fixes the spectral and spatial resolution.

Equivalent to that, regional gravity models can be approximated within a series expansion using lo-calizing spherical wavelet (base) functions. The unknown coefficients dJ,q vary the shape of the scaling functions ΦJ+1 and together with the number N of unknown parameters the model resolution is de-termined.

While the maximum degree of spherical harmonics has to be selected corresponding to the average global resolution of observation data sets, the maximum degree lJ' of spherical scaling functions can be derived from the average point distance Δ of the observations within the area of interest. So the coverage and density of the data is in most cases much higher.

The mathematical relation between maximum degree and number of coefficients is identical for both methods of series expansion developing global solutions. But in regional applications only a marginal extract is needed so that the smaller number of coefficients accelerates the computational process-es.

In different tasks and applications of gravity field models different spectral and spatial resolutions are demanded. Spherical harmonics can be developed up to the maximum degree which corresponds to the resolution of the (combined) observation techniques (see Figure 6-3). So the resolution r of the output signal corresponds to the resolution of the input signal.

As shown in Eq. (6-11) spherical scaling functions offer the additional possibility to split the target signal into different resolution levels. Hereby the value J+1 means the maximum resolution level of the target signal FJ+1 which can be displayed as the sum of the signal Fj' at the minimum resolution level j' and the detail signals Gj up to the maximum MRR-level J. The relationship between the degree lj' of the series expansion and the corresponding resolution level j is shown in Figure 6-2. Further an appropriate spatial resolution r and the measurement techniques which provide this resolution are given. Figure 6-3 focuses on the relation between the lower degrees lj' and the spatial resolution r, whereas Figure 6-4 highlights the correlations between the levels j and the different measurement techniques.

The difference of two levels results in wavelet (basis) functions and thus the method of splitting a signal into multiple detail signals is acting as a band-pass filter which assigns the detail signals to pre-defined frequency bands. The spectral combination of different observation datasets enables to ex-ploit the highest degree of information out of these different measurement techniques.

j 1 5 7 7 7 8 9 11

l

r [km]

2

10000

50

400

180

111

200

100

250

80

360

56

600

33

…

…

4000

5

satellite gravitmetry

altimetry

terrestrial gravimetry

Figure 6-2: Relation between degree of development l in series expansions, spatial resolution r [km], the corresponding measurement techniques and the resolution levels j within a MRR.

j 1 1 2 2 2 2 3 3 3 3

l r [km]

1 20000

2 10000

3 6667

4 5000

5 4000

6 3333

7 2857

8 2500

9 2222

10 2000


altimetry

Figure 6-3: Relation between degree of development l = 1...10 in series expansions, spatial resolution r [km], the corresponding measurement techniques and the resolution levels j within a MRR.


j 1 2 3 4 5 6 7 8 9 10 11 12

l r [km]

1 20000

3 6667

7 2857

15 1333

31 645

63 317

127 157

255 78

511 39

1023 20

2047 10

4095 5


altimetry

terrestrial gravimetry

Figure 6-4: Relation between resolution levels j = 1...12 within a MRR, degree of development l in series ex-pansions, spatial resolution r [km] and the corresponding measurement techniques.

One of the essential advantages of scaling functions compared to spherical harmonics is the possibil-ity to close data gaps by a complementary combination of the observations. The maximum resolution level thus depends on the largest gaps. The minimum resolution level depends on the expansion of the area of interest and determines the lowest degree of the series expansion. The average cross-section dimension Π corresponds to the wavelength of a spherical harmonic function. So the mini-mum level is defined as integer which contains the frequency spectrum with the largest wavelength that fits to this area. The final spatial resolution r of the modeled gravity fields, as mentioned before – for series expansions in spherical harmonics as well as in base functions – depends on the maxi-mum degree.

6.3.5 Regional combined gravity field solutions

Here we discuss how different observations types such as different GOCE gravity gradients can be evaluated together. We assume that each observation type provides a general observation equation of the kind

𝑦(𝑡) + 𝑒(𝑡) = 𝒂(𝑡)𝑇𝒅𝐽 (6 − 19)

for any time moment 𝑡We assume now that two different observation types shall be combined. With the index “1” for the first type of observations 𝑦1(𝑡𝑖) and the index “2” for the second type of obser-

vations 𝑦2(𝑡𝑗) we define the combined model

[𝑦1(𝑡𝑖)

𝑦2(𝑡𝑗)] + [

𝑒1(𝑡𝑖)

𝑒2(𝑡𝑗)] = [

𝒂1(𝑡𝑖)𝑇

𝒂2(𝑡𝑗)𝑇] 𝒅𝐽

with 𝐷 ([𝑦1(𝑡𝑖)

𝑦2(𝑡𝑗)]) = [

𝜎12 𝑉(𝑦1(𝑡𝑖)) 0

0 𝜎22 𝑉 (𝑦2(𝑡𝑗))

] , (6 − 20)

where 𝑉(∙) means the variance. Collecting all observations 𝑦1(𝑡𝑖) in the 𝑁1 × 1 vector 𝒚1 and all

bservations 𝑦2(𝑡𝑗) in the 𝑁2 × 1 vector 𝒚2 and considering the prior information 𝐸(𝒅𝐽) = 𝝁𝑑 and

𝐷(𝒅𝐽) = 𝚺𝑑 for the expectation vector and the covariance matrix of the vector 𝒅𝐽 of the unknown

parameters, the linear model

[

𝒚1

𝒚2

𝝁𝑑

] + [

𝒆1

𝒆2

𝒆𝑑

] = [𝑨1

𝑨2

𝑰

] 𝒅𝐽 with 𝐷 ([

𝒚1

𝒚2

𝝁𝑑

]) = [

𝜎12𝑷1

−1 𝟎 𝟎

𝟎 𝜎22𝑷2

−1 𝟎

𝟎 𝟎 𝜎𝑑2𝚺𝑑

] (6 − 21)

can be formulated, wherein besides the vector 𝒅𝐽 the three unknown variance factors 𝜎12, 𝜎2

2 and 𝜎𝑑2

have to be estimated; furthermore 𝑷1 and 𝑷2 are the given positive definite weight matrices of the observation vectors 𝒚1 and 𝒚2 with error vectors 𝒆1 and 𝒆2. The normal equations following from the model (6-21) read

( 1

𝜎12 𝑨1

𝑇𝑷1𝑨1 + 1

𝜎22 𝑨2

𝑇𝑷2𝑨2 + 1

𝜎𝑑2 𝚺𝒅

−1) �̂�𝑗


= ( 1

𝜎12 𝑨1

𝑇𝑷1𝒚1 + 1

𝜎22 𝑨2

𝑇𝑷2𝒚2 + 1

𝜎𝑑2 𝝁𝒅 ) . (6 − 22)

Since the unknown variance factors are calculated by variance component estimation, the normal equations (6-22) are solved iteratively. In the convergence point we obtain the solution

�̂�𝑗 = (1

�̂�12 𝑨1

𝑇𝑷1𝑨1 + 1

�̂�22 𝑨2

𝑇𝑷2𝑨2 + 1

�̂�𝑑2 𝚺𝒅

−1 )−1 ∙

( 1

�̂�12 𝑨1

𝑇𝑷1𝒚1 + 1

�̂�22 𝑨2

𝑇𝑷2𝒚2 + 1

�̂�𝑑2 𝝁𝒅 ) . (6 − 23)

6.4 Regional gravity field modelling using tesseroids This section is largely based on (de Oliveira, 2014).

6.4.1 Tesseroid-voxel method

The fundamental idea behind tesseroid-voxel method is solving a linear system 𝐴𝑥 = 𝑏 in order to estimate the densities correspondent to each voxel. In this system A represents the design matrix, x represents the densities of the voxels, and b represents the measurements. In our case there are four gradients used (the GRACE/GOCE VXX, VYY, VZZ, VXZ gradients in the GRF). It is important to note that all the adjustments are made in the design matrix and in the densities x, whereas the measure-ments remain as the original data. These adjustments are basically the application of the concept of the tesseroid-voxel approach and furthermore one works with certain rotations in order to improve the grid and therefore the estimated results, which are explained in more detail in Section 6.4.2. The system is solved using least squares where the a priori weights are adjusted using VCE as explained earlier.

In Figure 5 the geometric characteristics of the tesseroids are depicted. With six parameters one can specify the geometry of a tesseroid. The parameters are the two radii, the two lambdas for setting the longitudes and the two phis for setting the latitudes of the tesseroid. It is important to highlight that even though the geometrical representation of the tesseroid in spherical coordinates is more intuitive than the Cartesian coordinates, the last has certain advantages concerning computational effort, this fact is clearly demonstrated in a study made by [Grombein et al, 2013] where computa-tional advantages of using Cartesian coordinates with respect to spherical and also in respect to the prism approach are shown. This approach also has the advantage of not suffering from the polar singularity and can therefore be evaluated for any position on the globe.


Figure 5: Geometry of a tesseroid [Heck and Seitz, 2007]; the spherical coordinates are referred to the geo-centric Earth-fixed equatorial reference system defined by the base vectors.

In order to demonstrate the equations of the gradients based on the tesseroid approach, one needs initially the first Newton’s model for gravitational potential:

Equation 6-a

Next, the derivation of the elements of the Marussi tensor can be done following the basic idea of converting the integral kernels from spherical to Cartesian derivatives [Wild –Pfeiffer (2007, 2008)] using [Tscherning, 1976]. In the following equations the left-hand system is used as in the original documents. The relations between the acceleration and the potential are

Equation 6-b

The relations between the gradients and the potential are


Equation 6-c

By applying the Leibniz integral to Equation 6-a the second and first order derivatives can be ob-tained, both in spherical coordinates:

Equation 6-d

where

Equation 6-b and Equation 6-c cannot be solved analytically. Therefore, by following the suggestions of [Wild-Pfeiffer, (2007, 2008)], a two-step formalism can be set up:

1. Numerical evaluation of the spherical derivatives in the two groups of equations above:


2. Conversion of the resulting numerical values into the local frame by applying the functional

relations according to Equation 6-b and Equation 6-c.

In spherical coordinates the gravity gradients can derived based on the tesseroids as:

Equation 6-e

The optimal approach consists of an apparently unimportant modification, which actually changes the number of arithmetic operations. This modification is to determine the integral kernel of ele-ments of the Marussi tensor in Cartesian coordinates rather than with spherical coordinates. Howev-er, the integration domain is still bounded by spherical coordinates [Grombein et al., 2013]. In order to obtain the optimal integral elements, Equation 6-c must be applied, which results in

Equation 6-f

where

We can also write in a compact notation


Equation 6-g

6.4.2 Rotations

In order to apply the tesseroids as a standard voxel element, we must take into account the grid dis-tortions existing due to the spherical shape of the Earth. For this reason, a single strategy must be chosen in order to define the best grid for the areas of interest. The problem is clearly defined in Figure 6, which shows a tesseroid mesh in the southern part of the southern hemisphere, in which a clear distortion in the size of the spherical rectangles is apparent. The problem with this distortion is that, in further steps of the computation, this mesh will be used for evaluating the number of meas-urement points. Due to the distortion, regions at high latitude and lower latitude do not have a neg-ligible difference in the number of measurements. Therefore, if we adopt this mesh, an erroneously attribution of number of measurements would be given to different regions, which means a smaller contribution of the smaller regions in comparison to the larger regions. We use tesseroids of 0.5°, which at the equator corresponds to 56 km × 56 km. It would be 28 km × 56 km at 60° latitude and 7 km × 56 km at 83° degree latitude (the maximum latitude for GOCE). Although closer to the poles GOCE provides more measurements this resolution is not achievable. We therefore decided to use an equi-distributed mesh for each region of interest. The strategy adopted is to use the tesseroids a priori defined for the equatorial region, because that is the region with the least distortion.

Figure 6: The natural tesseroid distribution placed on the antarctic region


Figure 7: The natural tesseroid distribution placed on the equator (left panel). The adopted tesseroid distribu-tion consists of the tesseroid originally from the equatorial region applied to other regions, here the Antarctic region (right panel).

The final configuration of the grid is depicted in Figure 7 representing the tesseroid from the equato-rial region applied to the Antarctic region. The implication of this strategy is a sequence of rotations, which will be explained in detail in this section.

The first group of rotations are rotations of the tesseroids from the equatorial region to other re-gions. For this one uses the spherical coordinates in Earth Centered Earth Fixed Frame (ECEF) and places it in a local reference. The only input is the coordinates of the central point of the chosen re-gion (𝜆𝑐, 𝜑𝑐). The matrix applied is:

𝑅 = 𝑅2 ⋅ 𝑅3

where

𝑅2 = (cos(−𝜑𝑐) 0 − sin(−𝜑𝑐)

0 1 0sin(−𝜑𝑐) 0 cos(−𝜑𝑐)

)

𝑅3 = (cos(𝜆𝑐) − sin(−𝜆𝑐) 0

sin(−𝜆𝑐) cos(−𝜆𝑐) 00 0 1

).

The angle between the true North and the projected North is given by (Strang van Hees 2006)

𝛾 = Δ𝜆 sin 𝜑 +Δ𝜆3

3sin 𝜑 cos2 𝜑 (1 + 3𝑛2 + 2𝑛4) +

Δ𝜆5

15sin 𝜑 cos4 𝜑 (2 − 𝑡2)

with Δ𝜆 = 𝜆 − 𝜆0 the longitude difference between the central meridian and the computation point – in our case is the center of a block of tesseroids – and

𝑡 = tan 𝜑 , 𝑛2 =𝑒2 cos2 𝜑

1 − 𝑒2

and n is equal to zero once we use a model assuming the earth as sphere and not an ellipsoid.

Because the difference between map North and true North differs for the equatorial region (Figure 7, left) and other regions (Figure 7, right), this correction is applied for both positions, before being rotated 𝛾2 and after being rotated 𝛾1. The difference between them is 𝛾 = 𝛾1 − 𝛾2 and the corre-sponding rotation matrix is 𝑅3(𝛾).


After those steps, the GRF to LNOF rotation matrix can be applied in order to convert between the Local North Oriented Frame (LNOF) and the Gradiometer Reference Frame (GRF).

6.4.3 Lookup table For the re-computation of the gravity from the estimated density vector, usually the development of the full tesseroid series expansion would be necessary, which is computational expensive. Because the gravity field is a superposition of the contribution of all tesseroids, a computed gravity or gravity gradient field corresponds to the sum of the individual tesseroid elements scaled by the density. For this reason, a lookup table can be computed since the computational grid does not change for a fixed tesseroid size. Each tesseroid element is being developed for the spatial grid and stored with unitary density. This computation has to be performed only once. Afterwards, the lookup table can be read and used to compute the gravity or gradient field by simply computing the sum for each grid point of the scaled elements. The consequence of this application is a huge gain in computation time.

6.4.4 Grids at satellite altitude The nominal phase of the GOCE mission lasted until July 2012 in which the satellite had a perigee height of 255 km above the Earth. From August 2012 onward a number of orbit lowerings were car-ried out until the satellite had a perigee height of 225 km in April 2013. Thus, the so-called lower orbit phase contains data from August 2012 until October 2013, and the nominal phase from No-vember 2009 until July 2012. The data from the nominal phase are used to compute gravity gradient grids at 255 km altitude, whereas the data from the lower orbit phase are used to compute grids at 225 km. Here we discuss the computation procedure, whereas the validation of the grids and results for selected regions are presented in the Product Validation Report.

The GOCO03s global gravity field model (Mayer-Gürr et al., 2012) is used to compute along the GOCE orbit reference gradients in the GRF with which the measured along-track gravity gradients are re-duced. The reduced VXX, VYY, VXZ and VZZ gradients are used to estimate densities in pseudo-equal area blocks of 15° × 15° as explained above. The tesseroids have a resolution of 0.5° and there are therefore 900 tesseroids for each regional solution. The tesseroids are located at 100 km above the surface, which avoids downward continuation to the Earth’s surface and associated numerical insta-bilities. Locating the tesseroids even closer to the observation points leads to numerical instabilities as well. A global grid is obtained by a patchwork of regional 15° × 15° grids shifted by steps of 10° in latitude and longitude. The minimum overlap between adjacent grids is therefore 2.5°.

The accurate GRF gradients – TXX, TYY, TXZ and TZZ – in the region above the tesseroids are used to es-timate densities, which are then used to predict the vertical gravity gradient TZZ in 255 km (or 225 km) altitude in the LNOF. The altitude is above the sphere because of the tesseroid definition. The vertical gradients are interpolated on an equiangular 0.2° grid in 255 km (or 225 km) using natural neighbour interpolation. To avoid edge effects, blocks of 12° times 12° are used, discarding 1.5° on all sides. Next, Stokes coefficients are estimated from the interpolated TZZ values using spherical har-monic analysis, which are finally used to compute all six gravity gradients in the LNOF frame at a height of 255 km (or 225 km) above the reference ellipsoid.


7 Gravity reduction and calculation

7.1 Free-air and Bouguer anomalies, terrain corrections

For geophysical interpretation, typically a free-air or Bouguer anomaly is used (e.g. Blakely 1995). The free-air anomaly is defined as:

gFA=gOBS- go- gFA

The free-air anomaly corrects for the height of station with respect to the reference level, where normal gravity is defined. The simple Bouguer anomaly corrects for the masses between the station height and reference level:

gBSB =gFA -gSB

For gravity measurements over water, the Bouguer correction amounts to replacing the water (den-sity =1030 kg/m3) with a slab of rock density and thickness equal to bathymetric depth:

gSB

= 2𝜋𝛾ℎ ∗ 𝜌𝐵

The simple Bouguer anomaly ignores the differences between the real topography and the Bouguer slab (Figure 7-1), and therefore the complete Bouguer anomaly includes the terrain correction:

gCB= gFA -gSB-gT

Ideally, both Bouguer and terrain correction include a term for the curvature of the Earth (e.g. LaFehr 1991). For stations on the topography, it is the near-surface topography which has a large effect in the terrain correction.

It is possible to combine Bouguer and terrain correction in one step which we can refer to as topo-graphic correction. The main reason for the division in a terrain and Bouguer correction is that topo-graphic models have normally a higher resolution than the station density. Hence the Bouguer anomaly can be calculated quickly, just by knowing the station height, while terrain correction re-quires a high-resolution DEM.

For the satellite data we calculate the exact spherical solution of a topography model, and hence combine all calculations. This is possible as due to the large distance between the satellite and the topography (255-270 km), the gravity data are not sensitive to local changes in the terrain and we can use a topographic model with similar resolution as station density.


Figure 7-1: A) Stations are located either on topography or sea-surface. B) Bouguer correction reduces the station data for the gravity effect of a slab with constant height h. The height of the slab is the height of the station above the reference level. C) Terrain correction adjusts for difference between Bouguer slab and real topography.

7.2 Spherical and planar calculations

Traditionally, modelling software programs treat the Earth as flat. The structure of the crust is pro-jected from a geographical coordinate system to a flat Earth without adjusting for the changes in the difference between station and source outside the vertical axis (Figure 7.2). E.g. this means that no height correction is performed, and no correction for the change in the orientation of the coordinate system.

Here, we obviously make an error, but in the vertical direction the distance between interfac-es/masses and station location stays constant. The error is related to the masses in x, y direction. For gravity this is usually neglected, if the study area is small enough. In the Bouguer correction the spherical correction is done for masses outwards from 167 km distance. However, satellite data have already a distance of > 255 km from the surface, and we should therefore calculate the gravity field for a spherical Earth.


Figure 7-2: Flat Earth versus spherical Earth

7.2.1 Tesseroids

Topographic corrections for satellite data can be done with the tesseroids-software (Uieda 2011). Tesseroids is a software package for the direct modeling of gravitational fields in spherical coordi-nates. It can model the gravitational potential, acceleration and gradient tensor. The geometric ele-ment used in the modelling processes is a spherical prism, also called a tesseroid (Figure 7.3). Tes-seroids is coded in the C programming language, making it portable to GNU/Linux and Windows sys-tems. This software is developed by Leonardo Uieda in cooperation with Carla Braitenberg with some funding from GOCE Italy.

Figure 7-3: View of a tesseroid, the computation point P and the local coordinate system (Uieda 2011)


7.2.2 Gravitational Fields of a Tesseroid (taken from Uieda 2011)

The gravitational attraction of a tesseroid can be calculated using the formula (Grombein et al., 2010):

𝑔𝛼(𝑟𝑝, 𝜙𝑝𝜆𝑝) = 𝐺𝜌 ∫ ∫ ∫Δ𝑥𝛼

𝑙3𝜅

𝑟2

𝑟1

𝑑𝑟′𝜙2

𝜙1

𝑑𝜙′𝑑𝜆′𝜆2

𝜆1

𝛼 ∈ {1,2,3}.

The gravity gradients can be calculated using the general formula (Grombein et al., 2010):

𝑔𝛼𝛽(𝑟𝑝, 𝜙𝑝𝜆𝑝) = 𝐺𝜌 ∫ ∫ ∫ 𝐼𝛼𝛽

𝑟2

𝑟1

𝑑𝑟′𝜙2

𝜙1

𝑑𝜙′𝑑𝜆′𝜆2

𝜆1

𝛼, 𝛽 ∈ {1,2,3}

𝐼𝛼𝛽 = (3Δ𝑥𝛼Δ𝑥𝛽

𝑙5−

𝛿𝛼𝛽

𝑙3) 𝜅 𝛼, 𝛽 ∈ {1,2,3}

where 𝜌 is density, the subscripts 1, 2, and 3 should be interpreted as the x, y, and z axis, 𝛿𝛼𝛽 is the

Kronecker delta function, and

Δ𝑥1 = 𝑟′𝐾𝜙

Δ𝑥2 = 𝑟′ cos 𝜙 sin(𝜆′ − 𝜆𝑝)

Δ𝑥3 = 𝑟′ cos 𝜓 − 𝑟𝑝

𝑙 = √𝑟′2 + 𝑟𝑝2 − 2𝑟′𝑟𝑝 cos 𝜓

cos 𝜓 = sin 𝜙𝑝 sin 𝜙′ + cos 𝜙𝑝 cos 𝜙′ cos(𝜆′ − 𝜆𝑝)

𝐾𝜙 = cos 𝜙𝑝 sin 𝜙′ − sin 𝜙𝑝 cos 𝜙′ cos(𝜆′ − 𝜆𝑝)

𝜅 = 𝑟′2 cos 𝜙′

𝜙 is latitude, 𝜆 is longitude, 𝑟 is radius. The subscript 𝑝 is for the computation point.

The above integrals are solved using the Gauss-Legendre Quadrature rule (Asgharzadeh et al., 2007):

where 𝑊𝑟, 𝑊𝜙, and 𝑊𝜆 are weighting coefficients and 𝑁𝑟 , 𝑁𝜙, and 𝑁𝜆 are the number of quadra-ture nodes, i.e. the order of the quadrature.


8 References

Asgharzadeh, M. F., Von Frese, R. R. B., Kim, H. R., Leftwich, T. E. and Kim, J. W. (2007), Spherical prism gravity effects by Gauss-Legendre quadrature integration. Geophysical Journal Interna-tional, 169: 1–11. doi: 10.1111/j.1365-246X.2007.03214.x

Blakely, R.J., 1995. Potential Theory in Gravity & Magnetic Applications. Cambridge University Press Bouman J (2013): LNOF to MRF gradient rotation, TN-GOCE+-DNT-10, Issue 1.1, February 2013 Bouman J., Ebbing J., Fuchs M.J. (2013): Reference frame transformation of satellite gravity gradients

and topographic mass reduction. J. Geophys. Res., doi:10.1029/2012JB009747, in press Grombein, T.; Seitz, K.; Heck, B. (2010): Untersuchungen zur effizienten Berechnung topographischer

Effekte auf den Gradiententensor am Fallbeispiel der Satellitengradiometriemission GOCE. KIT Scientific Reports 7547, ISBN 978-3-86644-510-9, KIT Scientific Publishing, Karlsruhe, Germany. (http://digbib.ubka.uni-karlsruhe.de/volltexte/documents/1336300)

Grombein, T., Seitz, K., Heck, B. (2013), Topographic-isostatic reduction of GOCE gravity gradients, Earth on the edge: Science for a sustainable planet, Proceedings of the IAG General Assem-bly, Melbourne, Australia, June 28 - July 2, 2011.

Gruber, T., O. Abrikosov, and U. Hugentobler (2010), GOCE Standards, GO-TN-HPF-GS-0111, issue 3.2 Heck, B., Seitz, K. (2007) A comparison of the tesseroid, prism and pointmass approaches for mass

reductions in gravity field modelling, J. Geod. 81(2):121–136, doi:10.1007/s00190-006-0094-0.

Heiskanen WA, Moritz H (1967) Physical Geodesy, Freeman Karney CFF (2011) Transverse Mercator with an accuracy of a few nanometers, Journal of Geodesy,

85, 475-485, doi:10.1007/s001900-011-0445-3 Krüger JHL (1912) Konforme Abbildung des Erdellipsoids in der Ebene. New Series 52, Royal Prussian

Geodetic Institute, Potsdam. doi:10.2312/GFZ.b103-krueger28 LaFehr, T.R., 1991. An exact solution for the gravity curvature (Bullard B) correction. Geophysics, 56,

1179-1184 Marcus V.C.C.P. de Oliveira (2014) A voxel based approach in the Amundsen Sea Sector using GOCE

and GRACE measurements, Master Thesis, TU München Schmidt, M., S.-C. Han, J. Kusche, L. Sanchez and C.K. Shum (2006): Regional high-resolution spatio-

temporal gravity modeling from GRACE data using spherical wavelets. Geophys. Res. Lett., 33, L08403, doi:10.1029/2005GL025509

Schmidt, M., M. Fengler, T. Mayer-Gürr, A. Eicker, J. Kusche, L. Sanchez and S.-C. Han (2007): Regional Gravity Modelling in Terms of Spherical Base Functions. Journal of Geodesy, 81, 17-38, doi: 10.1007/s00190-006-0101-5

Strang van Hees G (2006) Globale en lokale geodetische systemen, Nederlandse Commissie voor Geodesie 30

Tscherning, C.C. (1976), Computation of the second-order derivatives of the normal potential based on the representation by a Legendre series, Manuscr. Geod. 1:71.

Uieda, L., 2011. Tesseroids 1.0: User Manual and API Documentation https:// tessero-ids.googlecode.com

Wild-Pfeiffer, F. (2007), Auswirkungen topographisch-isostatischer Massen auf die Satellitengradio-metrie, C 604, Deutsche Geodätische Kommission, München.

Wild-Pfeiffer, F. (2008), A comparison of different mass elements for use in gravity gradiometry, J Geod 82(10):637–653, doi:10.1007/s00190-008-0219-8.

http://digbib.ubka.uni-karlsruhe.de/volltexte/documents/1336300


Appendix A: Matlab scripts

function [lam,phi,h] = cart2ell(X,Y,Z) % From Cartesian to geodetic coordinates using Bowring

% Define constants (WGS84)

a = 6378137; % semi-major axis

f = 1/298.257223563; % flattening

e2 = 2*f - f^2; % eccentricity squared

b = a*(1-f); % semi-minor axis

% Geographic longitude

lam = atan2(Y,X);

% Auxiliary values

p = sqrt(X.^2 + Y.^2);

mu = atan(a*Z./(b*p));

ea2 = e2/(1-e2);

% Geographic latitude

phi = atan2(Z + ea2*b*sin(mu).^3, p - e2*a*cos(mu).^3);

% Height above ellipsoid

h = p./cos(phi) - a./sqrt(1-e2*sin(phi).^2);

% from radians to degrees

lam = lam*180/pi;

phi = phi*180/pi;

function [X,Y,Z] = ell2cart(lam_e,phi_e,h) % from ellipsoidal to Cartesian



ecc = 0.0818191908426215; % eccentricity

e2 = ecc^2; % eccentricity squared

% phi and lam from degrees to radians

phi_e = phi_e/180*pi;

lam_e = lam_e/180*pi;

% N

N = a./sqrt(1-e2*sin(phi_e).^2);

X = (N+h).*(cos(phi_e).*cos(lam_e));

Y = (N+h).*(cos(phi_e).*sin(lam_e));

Z = (N*(1-e2)+h).*sin(phi_e);


function [r,phi_s,lam_s,N]=elltosph(phi_e,lam_e,h) % Script to convert ellipsoidal coordinates to spherical coordinates

% Input are latitude and longitude in degrees and height in m

% Output is r, phi, lam, N with

% - r radial distance in m

% - phi, lam: geocentric latitude, longitude in degrees

% - N in m

%

% Johannes Bouman

% Upated 17 January 2012



ecc = 0.0818191908426215; % eccentricity

e2 = ecc^2; % eccentricity squared

% phi and lam from degrees to radians

phi_e = phi_e/180*pi;

lam_e = lam_e/180*pi;

% N

N = a./sqrt(1-e2*sin(phi_e).^2);

% from ellipsoidal to Cartesian

X = (N+h).*(cos(phi_e).*cos(lam_e));

Y = (N+h).*(cos(phi_e).*sin(lam_e));

Z = (N*(1-e2)+h).*sin(phi_e);

% from Cartesian to spherical

lam_s = atan2(Y,X);

phi_s = atan(Z./sqrt(X.^2+Y.^2));

r = sqrt(X.^2+Y.^2+Z.^2);

% from radians to degrees

lam_s = lam_s*180/pi;

phi_s = phi_s*180/pi;

end


function [Nort,East] = ell2utm(phi,lam,lam_0) % Function to compute Northing and Easting of Transverse Mercator Projec-

tion from

% geodetic coordinates phi and lambda.

%

% Coordinates are given in degrees and converted here

% For scale factor k and semi-major axis standard value are used.

%

% February 7, 2012

% Johannes Bouman

k = 0.9996; % Scale factor

a = 6378137; % Semi-major axis (WGS84)

x_0 = 500000; % False Easting

f = 1/298.257223563; % Flattening

e2 = (2-f)*f; % squared eccentricity

% Convert to radians and subtract lam_0

phi = phi*pi/180;

lam = cos(phi).*(lam-lam_0)*pi/180;

% Compute N, tan(phi) and n2

N = a./sqrt(1-e2*(sin(phi)).^2);

t = tan(phi);

n2 = e2/(1-e2)*(cos(phi)).^2;

% Auxiliary variables

A = 1 + 3/4*e2 + 45/64*e2^2 + 175/256*e2^3 + 11025/16384*e2^4;

B = 3/8*e2 + 15/32*e2^2 + 525/1024*e2^3 + 2205/4096*e2^4;

C = 15/256*e2^2 + 105/1024*e2^3 + 2205/16384*e2^4;

D = 35/3072*e2^3 + 105/4096*e2^4;

% Meridian arc

y = a*(1-e2)*(A*phi - B*sin(2*phi) + C*sin(4*phi) - D*sin(6*phi));

% More auxiliary variables

p = 1/12*(5 - t.^2 + 9*n2 + 4*n2.^2);

q = 1/360*(61 - 58*t.^2 + t.^4 + 270*n2 - 330*(t.^2).*n2);

r = 1/6*(1 - t.^2 + n2);

s = 1/120*(5 - 18*t.^2 + t.^4 + 14*n2 - 58*(t.^2).*n2);

u = 1/5040*(61 - 479*t.^2 + 179*t.^4 - t.^6);

% x and correction to y

x = lam.*N.*(1 + r.*lam.^2 + s.*lam.^4 + u.*lam.^6);

dy = 0.5*lam.^2.*N.*t.*(1 + p.*lam.^2 + q.*lam.^4);

% Norting and Easting

Nort = k*(y+dy);

East = k*x + x_0;


function [phi,lam] = utm2ell(Nort,East,lam_0) % Function to compute geodetic coordinates phi and lambda

% from Northing and Easting of Transverse Mercator Projection.

%

% For scale factor k and semi-major axis standard value are used.

%

% February 7, 2012

% Johannes Bouman

k = 0.9996; % Scale factor

a = 6378137; % Semi-major axis (WGS84)

x_0 = 500000; % False Easting

f = 1/298.257223563; % Flattening


% From Northing and Easting to x and ydy

ydy = Nort/k;

x = (East - x_0)/k;


A = 1 + 3/4*e2 + 45/64*e2^2 + 175/256*e2^3 + 11025/16384*e2^4;

B = 3/8*e2 + 15/32*e2^2 + 525/1024*e2^3 + 2205/4096*e2^4;

C = 15/256*e2^2 + 105/1024*e2^3 + 2205/16384*e2^4;

D = 35/3072*e2^3 + 105/4096*e2^4;

% Iterate phi

dphi = 1;

phi_old = zeros(size(Nort));

while max(abs(dphi)) > 1e-9

phi_new = ydy/(a*(1-e2))/A + B/A*sin(2*phi_old) - C/A*sin(4*phi_old) +

D/A*sin(6*phi_old);

dphi = phi_new - phi_old;

phi_old = phi_new;

end

% Compute N, tan(phi) and n2

N = a./sqrt(1-e2*(sin(phi_new)).^2);

t = tan(phi_new);

n2 = e2/(1-e2)*(cos(phi_new)).^2;

% More auxiliary variables

p = 1/12*(5 + 3*t.^2 + n2 - 9*t.^2.*n2);

q = 1/360*(61 + 90*t.^2 + 45*t.^4 + 107*n2 - 162*(t.^2).*n2 - 45*t.^4.*n2);

r = 1/6*(1 + 2*t.^2 + n2);

s = 1/120*(5 + 28*t.^2 + 24*t.^4 + 6*n2 + 8*(t.^2).*n2);

u = 1/5040*(61 + 662*t.^2 + 1320*t.^4 + 720*t.^6);

% phi and lamnda correction

x = x./N;

dlam = x./cos(phi_new).*(1 - r.*x.^2 + s.*x.^4 - u.*x.^6);

dphi = 0.5*x.^2.*t.*(1+n2).*(1 - p.*x.^2 + q.*x.^4);

phi = phi_new - dphi;

% Convert to degrees and add lam_0

phi = phi/pi*180;

lam = dlam/pi*180 + lam_0;


function gamma = meridian_conv(lam,phi,lam_0) % Function to compute meridian convergence

f = 1/298.257223563; % Flattening


% Convert to radians

phi = phi*pi/180;

lam = (lam - lam_0)/180*pi;


t = tan(phi);

n2 = e2/(1-e2)*(cos(phi)).^2;

% Gamma

t1 = lam.*sin(phi);

t2 = (1/3*lam.^2).*sin(phi).*(cos(phi).^2).*(1+3*n2+2*n2.^2);

t3 = (1/15*lam.^5).*sin(phi).*(cos(phi).^4).*(2-t.^2);

gamma = (t1 + t2 + t3)/pi*180;

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