Prof. Artu Ellmann • Native of Estonia

• MSc degree at the Moscow State University of Geodesy & Cartography (1986-1993)

• Work in industry and Estonian National Land Board (1993-2000)

• 2000-2004 PhD studies (supervisor prof Lars Sjöberg) at the Royal Institute of Technology, Stockholm

• 2004-2006 Post-doctorate research (supervisor prof. Petr Vanicek) at the University of New Brunswick, Canada

• 2006….present Tallinn University of Technology, Estonia, Head of Chair of Geodesy

Research interests: geodetic networks, gravity field and geoid modelling

Geoid models so far:

• Estonia,

• Baltic countries,

• Canada,

• Taiwan,

• Australia,

• Brazil (Amazon area),

• Konya basin

Stokes-Helmert method of the gravity field modelling

by

Artu Ellmann

in Selcuk University, Konya, May 17, 2012

• Formulation of the appropriate BVP • Determination of the boundary values • Rigorous treatment of topoeffects + DWC • Global geopotential models •Modified Stokes’s formula for regional geoid modelling •Stochastic and deterministic modifications • Selection of the modification limits • Conclusions

Outline

Recapitulation - anomalous g-quantities

• Disturbing potential: T (r,) = W(r,) – U(r,)

• Gravity disturbance

• Gravity anomaly

If the disturbing potential T can be determined, then all other quantities can be derived from T , incl. geoid

(… and vice versa!)

.

, 1 2, , ,

g

g e gspherapprox

T r Tg g r r T r T

n n r r

.

, , ,, ,

' spherapprox

W r U r T rg g r r

n n r

0

( , )( )

gT rN

6

Harmonicity of T and Boundary Value

Problems in potential theory

2 2 2

2 2 20ext

T T TT

x y z

2 2 2

int 2 2 204

T T TT G

x y z

2 2 2

2 2 2x y z

Laplacian operator

External potential (for a domain outside of masses) is a

harmonic function, i.e.

Internal (inside of masses) potential is a NOT harmonic

function, i.e.

A boundary value problem is finding the

harmonic functions (i.e. solutions

satisfying the partial diffferential

equations, such as Laplace's equation)

An example • Given the boundary values (on some boundary)

the corresponding harmonic function needs to be found/solved

• In physical geodesy - given the gravity anomaly or disturbance values on the geoid (reference surface of the disturbing potential!) the shape of the geoid needs to (and can!) be determined

• To satisfty the harmonicity condition of T all the external (with respect to the geoid) masses (e.g. atmosphere, topography) need to be (mathematically!) removed!!!

8

Boundary value problems in geodesy

Given 0 (outside geoid)

Sought ??? (outside geoid)

T

Given T N onthe geoid

T

Given 0 (outside geoid)

Given

Sought ??? outside geoid

T

Tg on the geoid

n

T on the geoid and

(1) Dirichlet

(2) Neumann

(3) mixed

Given 0 (outside geoid)

2Given

Sought ??? outside geoid

T

TT g onthe geoid

n R

T onthe geoid and

9

Ideal world

2 , 0T r

3lim ,r T r O r

,gg r ,gg r ,gg r

Geoid

Real world

2 , 4 ,T r G r

,tg r

,tg r

,tg r ,tg r

Unknown function Region of interest

0, , , , , .gT r W r U r r r

0

,gT r

N(Ω)

Reduction of external masses

Ho()

g(rt,)

Geoid

() = Ho()

Terrain

Helmert’s second condensation method

, ,

, , ,

t ct

h FA t ctV r V r

g r g r g r A Ar r

Helmertization of the disturbing potential

, , , , , ,h t ct a caT r T r V r V r V r V r

Conversion of free-air anomaly to Helmert anomaly

, , , ,, ,

2 2, , , ,

t ct a ca

h FA

t ct a ca

g n

V r V r V r V rg r g r

r r

V r V r V r V rr r

, ,1, ,

,

t

t t

T r rg r T r

n r n

Fundamental gravimetric equation

2 , 0hT r

2 , 0h

gr g r r r Harmonic!!!

Real space Helmert space

geoid

co-geoid

ellipsoid

topography

HO (HO)h

N Nh

PITE < 2 m

,h

tg r ,FA

tg r

Do

wn

war

d c

on

tin

uat

ion

,h

gg r

,hD g r

“Spherical” topographic effects

2, 4 /tA r G H O H R

Spherical

3, 4 /t

gV r G RH O H R

H

1, 4 /t

gV r G RH O H R

, 0gA r

Planar

,t

tV r

,t

gV r H

, 2tA r G H

, 2gA r G H

Disqualifies for the harmonization of the Earth gravity field!!!

Rigorous estimation of the far-zone contribution

14

The real world is even more complicated…

Excesses and deficiencies of the topographical

masses (with respect to the Bouguer shell)

Evaluation of the direct topographic effect (by 3D + 2D integration)

O

O

O

O

O

2O O2

O

o 2

1R

2

oR

1R

2

R

, R 14πG 1

R 3 R

, , ,G d d

, , ,G d d

t

t

t

t

tr r

H

r H

r r

H

r

r r

V r H HH

r r

l r rr r

r

l r rr r

r

O

O

2O O2

O

o 2

13 3

o

( )

13 3

( )

, 14πG 1

R 3 R

, , ,RG d

3

, , ,RRG d

3

t

t

t

ct

tr r

t t

r r

t

r r

V r H HRH

r r

l rr r

r

l rr

r

Con

den

se

d m

asses

Topogr.

masses

AT

TR

AC

TIO

N o

f

16

Numerical estimation of topographical effects –

closer to the computation point finer integration

elements are needed.

Final expression of the direct topogr. effect

O O

O

1 13 3'

2

o o

( ) ( )

1'

2

( )

, ,,

, , , , , ,( ) ( ')G d d G d

3

, , ,+G ( ') d d G

t t

t

t ct

t tt

t

R Ht

R H

r r r r

R H

R

r r

V r V rA r

r

l r r l r Rr rr r

r r

l r rr r

r

O

13 3

( )

, , ,( )( ') d

3t

t

r r

l r Rr R

r

, , , ,, ,

2 2, , , ,

t ct a ca

h FA

t ct a ca

g n

V r V r V r V rg r g r

r r

V r V r V r V rr r

Expression of the Helmert

gravity anomaly

d´

R2d´

(r,)

r l

R

Unit sphere, r = 1

Initial surface

(radius R)

18

Downward continuation of harmonic Helmert gravity anomaly

Poisson’s formula – upward

continuation of a harmonic function

(Helmert anomaly)

Downward continuation is an inverse

problem!!!

O

2 2 2

3, ,

4

h hR r Rg r g R d

r l

Conclusions (1)

• Rigorously, geoid determination by Stokes’ formula holds

only on a spherical boundary, assuming also the masses

outside the geoid to be absent.

• This necessitates for some correction terms (for the Earth’s

ellipticity, downward continuation, the contribution of

topographic and atmospheric masses).

• Helmert (topographically corrected) gravity anomalies are

appropriate boundary values for geoid modelling in regions

with significant topographical masses

20 40 60 80 100 120 140 160 180-5

0

5

10

15

20

25

30

KAUGUS [kaarekraadides °]

ST

OK

ES

'I F

UN

KT

SIO

ON

I V

ÄÄ

RT

US

Stokes’s solution to the mixed BVP

00 0

,, ' '

4, '

g hT r R

N g R d PITES

Combination of a high-degree reference field and EGM

Requires global coverage of gravity data

020

2

, ' '4

2

2

'

1

,M

h h

n

n

Mh

n

n

LRN g R g d

RPITE g

S

n

Near and far-zone contributions 0

R

S() (0,0,0)

R g(R,´)

d´=d´d´cos´

N()

The quality of the reference EGM is important in the regional geoid modelling!

0

2 20

2, ' , ' '

4 2 1

Mh h

n n

n

Mh L

n

R RN g R S dg

ng

Due to availability, quality, and type of data, the accuracy of a global EGM vary regionally. Hence, the performance of any EGM needs to be validated in a regional scale.

where

22 0

( , , ) ( 1) { cos sin } (cos )

n n

nm nm nm

n m

GM ag r n C m S m P

r r

Modified Stokes’s: combination of regional terrestrial data and a reference EGM

SPHERICAL-HARMONIC COEFFICIENTS of the Earth Gravitational Model

nmC

2 0 -0.484165371736E-03 0.000000000000E+00 0.35610635E-10 0.00000000E+00

2 1 -0.186987635955E-09 0.119528012031E-08 0.10000000E-29 0.10000000E-29

2 2 0.243914352398E-05 -0.140016683654E-05 0.53739154E-10 0.54353269E-10

3 0 0.957254173792E-06 0.000000000000E+00 0.18094237E-10 0.00000000E+00

3 1 0.202998882184E-05 0.248513158716E-06 0.13965165E-09 0.13645882E-09

3 2 0.904627768605E-06 -0.619025944205E-06 0.10962329E-09 0.11182866E-09

3 3 0.721072657057E-06 0.141435626958E-05 0.95156281E-10 0.93285090E-10

4 0 0.539873863789E-06 0.000000000000E+00 0.10423678E-09 0.00000000E+00

4 1 -0.536321616971E-06 -0.473440265853E-06 0.85674404E-10 0.82408489E-10

4 2 0.350694105785E-06 0.662671572540E-06 0.16000186E-09 0.16390576E-09

4 3 0.990771803829E-06 -0.200928369177E-06 0.84657802E-10 0.82662506E-10

4 4 -0.188560802735E-06 0.308853169333E-06 0.87315359E-10 0.87852819E-10

5 0 0.685323475630E-07 0.000000000000E+00 0.54383090E-10 0.00000000E+00

5 1 -0.621012128528E-07 -0.944226127525E-07 0.27996887E-09 0.28082882E-09

5 2 0.652438297612E-06 -0.323349612668E-06 0.23747375E-09 0.24356998E-09

5 3 -0.451955406071E-06 -0.214847190624E-06 0.17111636E-09 0.16810647E-09

5 4 -0.295301647654E-06 0.496658876769E-07 0.11981266E-09 0.11849793E-09

5 5 0.174971983203E-06 -0.669384278219E-06 0.11642563E-09 0.11590031E-09

nmSn m nmC nmS

Altogether 4.7 million coefficients EGM08

23

Space-borne mapping of the Earth’s gravity field

SPUTNIK 1957

• LAGEOS

• CHAMP (2000)

• GRACE (2005)

• GOCE (2009)

Modification of Stokes Formula (Molodensky et al. 1960)

( )4

RN S gd

Sir Gabriel Stokes, 1849

02

ˆ ˆ( )4 2

ML

n n

n

R RN S gd b g

0

2 1( ) ( ) (cos )

2

LL

k k

k

kS S s P

modif. coef.

How to

minimize?

Errors

* Truncation (cap)

*Terrestrial data

* GGM coefficents

Deterministic and stochastic

modifications

Truncation bias only Truncation bias

Errors of terrestrial data

Errors of geopotential model

Philosophical dilemma – either one uses (possibly doubtful)

error models, or these are completely neglected…

02 2

2ˆ ˆ ˆ( )

4 1

M ML

n n

n n

RN S g g d c g

n

Wong and Gore (1969)

Vanicek and Kleusberg (1987) Wenzel 1983 (EGG97 computations)

Sjöberg (1984, 1991, 2003)

Stochastic (LS) modifications Deterministic modifications Para-

meters Biased Optimum Unbiased Wong-Gore Vaníček-Kleusberg Simple

sn = 2

, 2,3... .L

kn n k

n

a s h k L

2

1n

2

1nt

n

0

bn = sn L

n n n

n n

Q s c

c dc

L

n nQ s

L

n nQ s Qn

0

2

2

ˆ ˆ( )4

ML

n n

n

RN S gd c b g

2

22 2 2 2 2 * * * 2

2

2 2

1 2( )

4 1

ML L

n n n n n n n n nNn n

m E N N d c b dc c b Q s c Q sn

Complete gravity anomaly instead of residual anomaly

GLOBAL MEAN SQUARE ERROR OF THE GEOID ESTIMATOR

*

2

2 2

2ˆ ˆ

1

ML T S

n n n n n n n

n n

N c Q s g c b gn

Spectral form of geoid estimator with data errors....

2

2

1n

n

N c gn

Modified Stokes’s function

1 2 3 4 5 60

50

100

150

200

250

Modif. Stokes function across 6° integr. cap

Spherical distance [°]

Un

itle

ss v

alu

e o

f (m

od

ifie

d)

Sto

ke

s fu

nctio

n

Stokes's original functionModified Stokes's

2

2 1( ) (cos )

1n

n

nS P

n

2 2

2 1 2 1(co( ) s ) (cos )

1 2

L

n n n

n n

L n nP s PS

n

2

1ns

n

LS parameters sk vary, and depend on:

Local gravity data quality

Selected radius of the integration cap

Characteristics of a GGM (noise, modif. degree)

Deterministic parameters are

a priori defined, e.g. Truncation bias smaller when SL() passes

through zero at the integration cap edge!

2

1ns

n

• Integration is often limited to a few hundred kilometres, implying thus that a relatively high modification degree should counterbalance this

• On the other hand, the EGM error grows with increasing degree, which provides a rationale for choosing a compromise modification limit.

• The improved accuracy of recent EGM-s allows the user to safely increase the modification degree (up to 100 or even beyond, with GOCE M=250).

Selection of modification limits

Conclusions – based on geoid modelling results

• The numerical tests involved five different modification methods

• the discrepancies between any pair of the geoid models remain within 9 cm (in the target area).

• The deviations among the recent geopotential models are more crucial than the numerical discrepancies among the tested modification methods.

• The accuracy of the five tested methods is the same to the accuracy of the control (GPS-levelling) points

• LS methods provide more superior accuracy than that of deterministic methods

Thanks for your attention!

Back-up slides

• This new EGM08 takes advantage of updated satellite, terrestrial gravity, elevation and altimetry data,

• The project is sponsored by the National Geospatial-Intelligence Agency (formerly NIMA, DMA) of the USA.

• The resolution of the EGM08 is ~5´ arc-minutes (corresponding to 9 km, i.e. to the degree of 2160),

• The EGM08 accuracy is expected to be superior (but not entirely errorless) over earlier EGM-s. •In addition to the geodetic applications the EGM08 will also contribute to other geosciences

Earth Gravitational Model EGM08