Table Of ContentLecture Notes in Earth Sciences 73
Editors:
.S Brooklyn Bhattacharji,
Troy and Brooklyn Friedman, M. G.
H. .J Bonn Neugebauer,
and Tuebingen A. Seilacher, elaY
Zden~kM artinec
Boundary-Value
Problems
for Gravimetric
Determination
of a Precise Geoid
With 15 Figures and 3 Tables
regnirpS
Author
Zden~k Martinec
Department of Geophysics
Faculty of Mathematics and Physics
Charles University
V Hole~ovi~kfich 2, 180 00 Prague 8, Czech Republic
e-mail, zdenek @ hervam.troja.mff.cuni.cz
"For all Lecture Notes in Earth Sciences published till now please see final pages of
the book"
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Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Martinet, Zden~k:
Boundary value problems for gravimetric determination in a precise
geoid : with 3 tables / Zden~k Martinet. - Berlin ; Heidelberg ; New
York ; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Santa
Clara ; Singapore ; Paris ; Tokyo : Springer, 1998
(Lecture notes ni earth sciences ; 73)
ISBN 3-540-64462-8
ISSN 0930-0317
ISBN 3-540-64462-8 Springer-Verlag Berlin Heidelberg New York
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Contents
List of symbols ix
Acknowledgements xii
Introduction 1
The Stokes two-boundary-value problem for geoid
determination 6
1A Formulation of the boundary-value problem ............ 6
1.2 Compensation of topographical masses ............... 7
1.3 Anomalous potential ......................... 9
1.4 Bruns's formula ............................ 10
1.5 Linearization of the boundary condition ............... 11
1.6 The first-degree spherical harmonics ................. 13
1.7 Numerical investigations ....................... 15
1.7.1 An example: constant height ................. t6
1.7.2 Axisymmetric geometry ................... t9
1.8 Different approximations leading to the fundamental equation of
physical geodesy ............................ 23
1.9 Conclusion ............................... 26
The zeroth- and first-degree spherical harmonics in the Helmert
2nd condensation technique 28
2.1 Principle of mass conservation .................... 28
2.2 Principle of mass-center conservation ................ 31
2.3 Conclusion ............................... 32
3 Topographical effects 34
3.1 Approximations used for 5V . . . . . . . . . . . . . . . . . . . . . 34
3.2 A weak singularity of the Newton kernel .............. 35
3.3 The Pratt-Hayford and the Airy-Heiskanen isostatic compensation
models ................................. 37
3.4 Helmert's condensation layer ..................... 39
3.5 The direct topographical effect on gravity ............. 40
3.6 The primary indirect topographical effect on potential ....... 42
vi Contents
3.7 The secondary indirect topographical effect on gravity ....... 44
3.8 Analytical expressions for integration kernels of Newton's type . . 44
3.8.1 The singularity of the kernel L-l(r, ¢, r') at the point ¢ = 0 45
3.9 Numerical tests ............................ 46
Planar approximation 50
4.1 Constant density of topographical masses .............. 50
4.2 Restricted integration ......................... 51
4.3 Planar approximation of distances .................. 15
4.4 The difference between spherical and planar approximation of
topographical effects ......................... 53
4.5 Conclusion ............................... 54
Taylor series expansion of the Newton kernel 56
5.1 The problem of the convergence of Taylor series expansion .... 57
5.2 The Taylor expansion of the terrain roughness term ........ 59
5.3 Numerical computations ....................... 60
5.3.1 The Taylor kernels Ki .................... 60
5.3.2 The primary indirect topographical effect on potential . . 62
5.4 Conclusion ............................... 64
A.5 Integration kernels Mi(r, ~, R) .................... 65
A.5.1 Spectral form ......................... 65
A.5.2 Recurrence formula ...................... 66
A.5.3 Spatial form .......................... 66
A.5.4 Singularity at the point ¢ = 0 ................ 67
A.5.5 Angular integrals ....................... 68
A.5.6 Proofs of eqns.(A.5.11) and (A.5.12) ............ 69
6 The effect of anomalous density of topographical masses 72
6.1 Topographical effects ......................... 73
6.2 One particular example: a lake ................... 74
6.3 Numerical results for the lake Superior ............... 77
6.4 Another example: the Purcell Mountains .............. 80
6.5 Conclusion ............................... 82
Formulation of the Stokes two-boundary-value problem with a
higher-degree reference field 84
7.t A higher-degree reference gravitational potential .......... 85
7.2 Reference gravity anomaly ...................... 87
7.3 Formulation of the two-boundary-value problem .......... 88
7.4 Numerical results for V • ~'j~.~. - Vj.~ ~'~ ................... 91
7.5 Conclusion ............................... 95
A.7 Spherical harmonic representation of ~V .............. 96
Contents vii
8 A discrete downward continuation problem for geoid
determination 99
8.1 Formulation of the boundary-value problem ............ 102
8.2 Poisson's integral ........................... 103
8.3 A continuous downward continuation problem ........... 105
8.4 Discretization ............................. 106
8.5 Jacobi's iterations ........................... 108
8.6 Numerical tests ............................ 109
8.6.1 Analysis of conditionality .................. 109
8.6.2 Analysis of convergency ................... 114
8.6.3 Power spectrum analysis of gravity anomalies ....... 116
8.6.4 Downward continuation of gravity anomalies ........ t17
8.7 Conclusion ............................... 123
A.8 Spherical radius of the near-zone integration cap .......... 125
B.8 Poisson's integration over near- and far-zones ........... 126
B.8.1 Near-zone contribution .................... 127
B.8.2 Truncation coetficients .................... 129
B.8.3 Far-zone contribution ..................... 131
B.8.4 Summary ........................... 131
The Stokes boundary-value problem on an ellipsoid of
revolution 132
9.1 Formulation of the boundary-value problem ............ 133
9.2 The zero-degree harmonic of T .................... 135
9.3 Solution on the reference ellipsoid of revolution .......... 136
9.4 The derivative of the Legendre function of the 2nd kind ...... 137
9.5 The uniqueness of the solution ...... . ............. 138
9.6 The approximation up to O(eg) ................... 139
9.7 The ellipsoidal Stokes function .................... 142
9.8 Spatial forms of functions Ki(cos )X ................. 143
9.9 Conclusion ............................... 147
A.9 Power series expansion of the Legendre functions ......... 148
B.9 Sum of the series (9.49) ........................ 150
10 The external Dirichlet boundary-value problem for the Laplace
equation on an ellipsoid of revolution 155
10.1 Formulation of the boundary-value problem ............ 156
10.2 Power series representation of the integral kernel .......... 157
t0.3 The approximation up to O(eo )2 ................... 160
10.4 The ellipsoidal Poisson kernel .................... 162
10.5 Residuals Ri(t,x) . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
10.6 The behaviour at the singularity ................... 166
10.7 Conclusion ............................... 167
A.10 Some sums of infinite series of Legendre polynomials ....... 168
viii Contents
B.10 Program KERL ............................ 169
11 The Stokes boundary-value problem with ellipsoidal corrections
in boundary condition 170
11.1 Formulation of the boundary-value problem ............ 172
11.2 The O(e~)-approximation ....................... 174
11.3 The 'spherical-ellipsoidal' Stokes function .............. 177
11.4 Spatial forms of functions M~(cos ¢) ................. 179
11.5 Conclusion ............................... 183
A.11 SpectrM form of etlipsoidal corrections ............... 183
B.11 An approximate solution to tridiagonal system of equations .... 185
C.11 Different forms of the addition theorem for spherical harmonics 186
12 The least-squares solution to the discrete altimetry-gravimetry
boundary-value problem for determination of the global gravity
model 192
12.1 Formulation of the boundary-value problem ............ 194
12.2 Paxametrization and discretization ................. 197
12.3 A least-squares estimation ...................... 198
12.4 The axisymmetric geometry ..................... 199
12.50verdetermination .......................... 205
12.6 Numerical examples .......................... 207
12.7 Conclusion ............................... 208
Summary 210
References 213
Index 221
List of symbols
Hbrmander's constant
alm
A gravitationaI attraction of spherical Bouguer shell
~ A gravitational attraction of compensed/condensed masses
~ A gravitational attraction of topographical masses
5A direct topographical effect on gravity
ob minor semi-axis of the reference ellipsoid best-fitting the geoid
D depth of compensation
first eccentricity
e
first eccentricity of reference elIipsoid
eo
E linear eccentricity of set of confocal ellipsoidal coordinate
surfaces u =const.
F free-air reduction
F hypergeometric function
g gravity of the Earth
@
gravity disturbance
Ag E free-air gravity anomaly
fgxz
low-frequency part of Ag F
high-frequency part of Ag F
G Newton's gravitational constant
h ellipsoidal height
H topographical height
jre] cut-off degree of reference potential ,
~,amj cut-off degree of anomalous potential
K spherical Poisson's kernel
Kell
ellipsoidal Poisson's kernel
e K spheroidal Poisson's kernel
distance between points (r, )~f and (R, a')
0e distance between points (R, )~f and (R, 0')
L distance between points (r, O) and (r', fY)
2L space of square-integrable functions on sphere
M mass of the Earth
c M mass of compensation/condensation layer
g M masses below the geoid
~ M topographical masses
N geoidal height
x List of symbols
jq truncation coefficients of Poisson's kernel
high-degree part of jq
P point on the Earth's surface
point on the geoid
Legendre polynomial of degree j
Legendre functions of the 1st kind
Q point on reference ellipsoid
Legendre functions of the 2nd kind
r radial distance
% radius of the geoid
R radius of mean sphere best-fitting the geoid
Paul's coefficients
S spherical Stokes's function
ocleS
sphericM-ellipsoidal Stokes's function
sell ellipsoidal Stokes's function
~S secondary indirect topographical effect on gravity
T anomalous gravitational potential harmonic outside the Earth
,T low-degree reference part of T
Te high-degree part of T
spherical harmonic coefficients of T
T h anomalous gravitational potential harmonic outside the geoid
Th,* high-degree part of T h
ellipsoidal coordinate
U
U normal gravity potential
V gravitational potential of the Earth
V B gravitational potential of spherical Bouguer shell
V c gravitational potential of compensated/condensed masses
V g gravitational potential of masses below the geoid
V R terrain roughness term
~ V gravitational potential of topographical masses
yt, e spherical harmonic coefficients of ~ V in the space external
~jm
to topographical masses
vjt,i spherical harmonic coefficients of t V in the space internal
m
to topographical masses
V o gravitational potential of compensated masses
V ~ gravitational potential of condensed masses
V ~ centrifugal potential
~V residual topographical potential
low-degree part of ~V
,v ~ high-degree part of ~V
~w gravity potential disturbance
W gravity potential of the Earth
oW gauge value of gravity potential on the geoid
Cartesian coordinates
X~ y~ Z
List of symbols xi
surface spherical harmonic of degree j and order m
azimuth
~O
reduced co-latitude
normal/reference gravity
F gamma function
Kronecker symbol
ellipsoidal correction
£h
ellipsoidal correction
Riemann zeta function
co-latitude
condition number
longitude
eigenvalue
density of topographical masses
density 0 averaged along topographical column
density of compensated masses
o~ mean density of topographical masses
density contrast at the Moho
ohoMO~A
"O density of condensed masses
X angular distance in ellipsoidM coordinates
¢ angular distance in spherical coordinates
angular velocity of the Earth's rotation
pair of angular spherical coordinates
fl pair of angular ellipsoidal coordinates
~o full solid angle
spherical cap of radius 0b~
