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![Page 1: Positioning America for the Future NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION National Ocean Service National Geodetic Survey On the Solutions of.](https://reader035.fdocuments.in/reader035/viewer/2022071709/56649cf65503460f949c5a1e/html5/thumbnails/1.jpg)
Positioning America for the Future
NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION
National Ocean Service
National Geodetic Survey
On the Solutions of the Geodetic Exterior/Interior Fixed Boundary Value Problems
Yan Ming WangNational Geodetic Survey
Hotine-Marussi SymposiumMay 29 – June 2, 2006
![Page 2: Positioning America for the Future NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION National Ocean Service National Geodetic Survey On the Solutions of.](https://reader035.fdocuments.in/reader035/viewer/2022071709/56649cf65503460f949c5a1e/html5/thumbnails/2.jpg)
Positioning America for the Future
NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION
National Ocean Service
National Geodetic Survey
Overview
• Background
• The exterior and interior boundary value problem (BVP)
• Harmonic and non-harmonic solution of the interior BVP inside the topographic mass
• Non-harmonic solution for constant density of topographic mass
• Conclusions
![Page 3: Positioning America for the Future NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION National Ocean Service National Geodetic Survey On the Solutions of.](https://reader035.fdocuments.in/reader035/viewer/2022071709/56649cf65503460f949c5a1e/html5/thumbnails/3.jpg)
Positioning America for the Future
NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION
National Ocean Service
National Geodetic Survey
Background
• Earth’s surface can be considered as known; gravity disturbances can replace gravity anomalies
• Geoid over land areas can be computed by solving interior BVP (Poisson equation)
• Current solutions of interior BVP requires knowledge of global mass density (impossible task)
• We are seeking disturbing potentials above the geoid under the assumption that the topographic mass density is known.
![Page 4: Positioning America for the Future NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION National Ocean Service National Geodetic Survey On the Solutions of.](https://reader035.fdocuments.in/reader035/viewer/2022071709/56649cf65503460f949c5a1e/html5/thumbnails/4.jpg)
Positioning America for the Future
NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION
National Ocean Service
National Geodetic Survey
Fixed geodetic boundary value problems (BVP)
• The exterior BVP:
where is known.
• The interior BVP:
SggradW
W
E
EE
xxx
xx
)()(
2)( 2
SggradW
GW
I
II
xxx
xxx
)()(
2)(4)( 2
S
![Page 5: Positioning America for the Future NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION National Ocean Service National Geodetic Survey On the Solutions of.](https://reader035.fdocuments.in/reader035/viewer/2022071709/56649cf65503460f949c5a1e/html5/thumbnails/5.jpg)
Positioning America for the Future
NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION
National Ocean Service
National Geodetic Survey
Simplification of boundary values
• Replacing the norm of the gravity by the vertical derivatives:
where
• Introducing interior reference potential that satisfies:
000u sssxxg ggggradW u )()(
2222 1)()()( xggggxg u
22)(4)( xx ellI GU
222
![Page 6: Positioning America for the Future NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION National Ocean Service National Geodetic Survey On the Solutions of.](https://reader035.fdocuments.in/reader035/viewer/2022071709/56649cf65503460f949c5a1e/html5/thumbnails/6.jpg)
Positioning America for the Future
NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION
National Ocean Service
National Geodetic Survey
Exterior/interior BVPs (continued)
• The exterior BVP for disturbing potential
• The interior BVP for disturbing potential
SOhghu
T
T
E
EE
xxxx
xx
))(()()(
0)(
211
SOhghu
T
GT
I
II
xxxx
xxx
))(()()(
)(4)(
211
![Page 7: Positioning America for the Future NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION National Ocean Service National Geodetic Survey On the Solutions of.](https://reader035.fdocuments.in/reader035/viewer/2022071709/56649cf65503460f949c5a1e/html5/thumbnails/7.jpg)
Positioning America for the Future
NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION
National Ocean Service
National Geodetic Survey
Exterior/interior BVPs (continued)
where
is the space between the Earth’s surface and the reference ellipsoid
ellell
I
xxx
xxx
)()(
)()(
'
I
![Page 8: Positioning America for the Future NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION National Ocean Service National Geodetic Survey On the Solutions of.](https://reader035.fdocuments.in/reader035/viewer/2022071709/56649cf65503460f949c5a1e/html5/thumbnails/8.jpg)
Positioning America for the Future
NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION
National Ocean Service
National Geodetic Survey
Solutions of the interior BVP
• Decompose solution in the space of topography into harmonic and non-harmonic
which are defined by:
tHI TTT
Su
T
GT
t
It
xx
xxx
0)(
)(4)(
ShOghu
T
T
H
IH
xx
xx
)()(
0)(
211
'
![Page 9: Positioning America for the Future NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION National Ocean Service National Geodetic Survey On the Solutions of.](https://reader035.fdocuments.in/reader035/viewer/2022071709/56649cf65503460f949c5a1e/html5/thumbnails/9.jpg)
Positioning America for the Future
NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION
National Ocean Service
National Geodetic Survey
Solutions of the interior BVP
• Harmonic solution can be obtained by using the method of analytical downward continuation
• Non harmonic potential satisfies the compatibility condition and the solutions of BVP exist and unique.
tT
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Positioning America for the Future
NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION
National Ocean Service
National Geodetic Survey
Strategy of non-harmonic solutions
• Two step strategy for non-harmonic solutions:
1. Solution of Bouguer sphere:
2. Corrector field that satisfies:)()()( 0 xxx TTT tC
Sr
TT
GT
xx
x
xxx
0)(
,0)(
)(4)(
00
0
Sr
T
r
TTT
TC
C
IC
xxx
xx
xx
)(~
)(),(
~)(
0)(
00
![Page 11: Positioning America for the Future NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION National Ocean Service National Geodetic Survey On the Solutions of.](https://reader035.fdocuments.in/reader035/viewer/2022071709/56649cf65503460f949c5a1e/html5/thumbnails/11.jpg)
Positioning America for the Future
NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION
National Ocean Service
National Geodetic Survey
A special case: constant density of topography
1. Solution of the Bouguer sphere:
2. BVP of the corrector field:
)3
21(2)( 2
0 PPP hR
hGhT
ShhGT
T
SPC
IC
xx
xx2
'
)(2)(
0)(
![Page 12: Positioning America for the Future NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION National Ocean Service National Geodetic Survey On the Solutions of.](https://reader035.fdocuments.in/reader035/viewer/2022071709/56649cf65503460f949c5a1e/html5/thumbnails/12.jpg)
Positioning America for the Future
NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION
National Ocean Service
National Geodetic Survey
Conclusions
• Geoid can be computed by solving interior BVP
• Solutions of the interior BVP are decomposed into harmonic and non-harmonic components
• Harmonic solution can be obtained by using analytical downward continuation
• Non-harmonic solution can be split into a solution of Bouguer sphere and the corrector filed
• Other method can be used, e.g. finite element method