considerations about some methodological concepts in highly precise gravimetric geoid determination...
TRANSCRIPT
Considerations about some methodological concepts in highly precise gravimetric geoid
determination
Bernhard HeckGeodetic Institute, University of Karlsruhe
Englerstr. 7D – 76128 Karlsruhe, Germany
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.