Considerations about some methodological concepts in highly precise gravimetric geoid determination...

Considerations about some methodological concepts in highly precise gravimetric geoid

determination

Bernhard HeckGeodetic Institute, University of Karlsruhe

Englerstr. 7D – 76128 Karlsruhe, Germany

heck@gik.uni-karlsruhe.de

Contents

Historical remarks

Geodetic boundary value problems

Geoid determination and GBVP

Towards a mathematically rigorous concept of

geoid determination

Errors in gravity anomaly data sets

Questions – instead of final conclusions

Historical remarks

Isaac Newton (1642 – 1727) :

Front page of the first edition of the Principia (1686)

Physical arguments:

Earth in hydrostatic equilibrium

Ellipsoid of revolution

The Earth as a geoid:

C.F. Gauß (1843/1846)

G.G. Stokes (1849)

J.B. Listing (1873)

New concept:

Reference surface =

equipotential surfaceOn the Variation of Gravity at the Surface of the Earth. By G.G. Stokes, M.A., Fellow of Pembroke College, Cambridge

The Earth with irregular boundary surface

Boundary surface = topographic surface of the Earth

Solution of a GBVP

M.S. Molodenskii (1945 – 1960)Molodenskii, M.S.; Eremeev, V.F.; Yurkina, M.I:Methods for Study of the External Gravitational Field and Figure of the Earth. Transl. from Russian by the Israel Program for Scientific Translations for the Office of Technical Services, Jerusalem 1962

Further developments: Hirvonen, Moritz, Krarup, Sanso, Hörmander, Holota, Grafarend, …

Geodetic Boundary Value Problems

Geodetic boundary value problems

Gravity field: gravity potential W

W = V + Z

V gravitational potential

Z centrifugal potential

Differential equation

Lap V = - 4G

Laplace-Poisson equationGBVP:Given: W - Wo and gravity vector grad W on the boundary surface SUnknown: W in external space of S and eventually geometry of S

s

x

s

y

xx’

xS

z

s

x

s

y

xx’

xS

z

(1)„Fixed“ GBVP S known (GPS positioning)Given: grad W on SUnknown: W in space external of S

(2) „Free“ GBVPa) Vectorial free GBVP“

S completely unknown Given: W - Wo and grad W on S

Unknown: W in space external of Sand position vector of S

b) „Scalar free“ GBVP , known (horizontal coordinates)

Given: W - Wo and on S

Unknown: W in space external of S and vertical coordinate (h)

Classification of the GBVP: „free“, „non-linear“, „oblique“

s

x

s

y

xS

z

s

x

s

y

xS

z

)P(gradW)P(

Solution scheme for the GBVP

Approximations: normal potential U, telluroid

Approximation:

Approximation: l X l ~ R = const.

Analytical solution (integral formula)

r/h/

Free, non-linear GBVP

Linearisation

Linear GBVP

Spherical approximation

Linear GBVP inspherical approximation

Constant radiusapproximation

Spherical GBVP

The scalar free GBVP

„Geodetic“ variant of the Molodensky problem

Given on S: W(P) - Wo: Levelling + gravity

Unknown: W (X) in space external of S W = V + Z Lap V = 0

h=HN+

)P(gradW)P(

The scalar free GBVP

Reference for linearisation:

U Normal potential of a level ellipsoid HN Normal height; postulate:

Wo - W(P) = Uo - U(Q)(telluroid mapping)HN (P) = h(Q)

Decomposition

W = U + T T: Disturbing potential h = HN+ : height anomaly

(=quasigeoidal height)

h=HN+

The scalar free GBVP

Linearisation: = T(Q)/ (Q) Bruns formula

Fundamental equation

Analytical solution (series expansion)

~ terrain correction

h=HN+

)gradU(

Q

Tr

2

h

T-

)Q()P(g:g

d)hh(

RG2

1C

d)(S...)Cg(4

R

3

22

Analytical approximation errors in the GBVP

Linearisation

Non-linear terms in the boundary condition

Spherical approximation

ellipsoidal terms in the boundary condition

topographical terms in the boundary condition

Planar approximation

omission of terms of order (h/R) ~ 10-3

Constant radius approximation

~ downward continuation effect, Molodensky‘s series terms

Evaluation of the non-linear boundary condition (North America)

True field ~ EIGEN_GL04C; Nmax = 360Topography model: GTOPO30Runtime: user 1d 18h (K. Seitz)Output:

non-linear BC non-linear effects in the BC Coordinates of the telluroid points (input for ellipsoidal effects)

Statistics [mGal] Min Max Mean L1 L2Linear BC -244.885 229.076 -8.246 20.011 25.884Non-linear BC -245.197 229.235 -8.246 20.018 25.895Non-linear effect -0.326 0.259 0.000 0.011 0.018

Zeta (P-Q) [m] -62.631 13.516 -29.888 29.967 32.066

Ellipsoidal correction δNE = δTE(rE(φ,λ), φ, λ))/γ(φ) in m, 0 ≤ m ≤ n ≤ 360 (Hammer equal-area projection)

Heck, B. and Seitz, K. (2003): Solutions of the linearized geodetic boundary value problemfor an ellipsoidal boundary to order e3. JGeod, 77, 182-192. DOI 10.1007/s00190-002-0309-y.

Power spectrum of δNE (in m2) and T (in m4s-4)

Heck, B. and Seitz, K. (2003): Solutions of the linearized geodetic boundary value problemfor an ellipsoidal boundary to order e3. JGeod, 77, 182-192. DOI 10.1007/s00190-002-0309-y.

Numerical approximation errors

Evaluation of surface integrals: - Stokes integral - Terrain correction - Molodensky‘s series terms of higher order - Poisson integral and derivatives - ………

Truncation error Integration over spherical cap, neglection of outer zone Modified integral kernels

Numerical evaluation by FFT (gridded data) Finite region - boundary effects, periodic continuation (zero padding) 2D FFT - neglection of sphericity (1D FFT for large regions) Aliasing, etc.