Positioning America for the Future NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION National Ocean Service National Geodetic Survey On the Solutions of

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





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





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.





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





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





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





Exterior/interior BVPs (continued)

where

is the space between the Earth’s surface and the reference ellipsoid

ellell

I

xxx

xxx

)()(

)()(

'

I





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

'





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





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





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)(





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

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Positioning America for the Future NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION National Ocean Service National Geodetic Survey On the Solutions of