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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