Stokes-Helmert's Scheme for Precise Geoid Determination

The Stokes-Helmert’s scheme for the precise geoid determination is a work developed in the University of New Brunswick, Canada, by more than ten years of investigation without interruption, in which outstanding scientists have participated. This work, presents in synthetic form the steps to obtain a centimeter geoide, as well as the mathematical formulation on which east scheme is based. Stokes-Helmert’s geoid software Stokes-Helmert’s geoid software (SHGEO) is a scientific software for precise geoid determination based on the Stokes-Helmert theory of determination of the gravimetric geoid. The software has been developed during more then 10 years period under leadership of professor Petr Vanicek at the Department of Geodesy and Geomatics Engineering, University of New Brunswick. Authors of particular programs are: M. Najafi, P. Novak, J. Huang, J. Janak and R. Tenzer. We also have to mention Z. Martinec, A. Kleusberg, L.E. Sjoberg, W.E. Featherstone, W. Sun whose research presented in their papers was incorporated into the SHGEO software. SHGEO soft* University of New Brunswick. Email: rtenzer@unb.ca ** Department of Theoretical Geodesy, Faculty of Civil Engineering, Slovak University of Technology, Radlinskeho 11, 81368 BRATISLAVA, Slovak Republic. Email: janak@aries.svf.stuba.sk ˆ 136 Robert Tenzer et al. Revista Cartografica 74-75 ware uses various global models (e.g. TUG87, GRIM4-S4, EGM96). These global models play an important role in the geoid computation scheme. Therefore we acknowledge the contribution of all research teams that have developed these or other global models. Reference manual was compiled by R. Tenzer and J. Janak. Stokes-Helmert’s scheme for precise geoid determination Introduction This part of the manual gives a brief theoretical overview of the precise geoid determination process. The details can be found in references. The Stokes-Helmert scheme for determination of the precise geoid can be summarized to the following steps: 1. Formulation of the boundary value problem on the Earth surface 2. Evaluation of the Helmert gravity anomalies on the Earth surface 3. Downward continuation of the Helmert gravity anomalies onto the geoid 4. Stokes’s integration (solution to the Stokes’s boundary value problem) 5. Transformation of geoidal heights from the Helmert space to the real space. Formulation of the boundary value problem The quantity to be solved is the earth’s gravity potential ( ) Ω , r W on and outside the geoid and the geoid itself. The geoid is the equipotential surface that approximates the mean sea level most closely. The gravity potential on the geoid is denoted by ( ) . , const r Wo = Ω In order to solve this problem a normal gravity potential ( ) Ω , r U generated by the mean geocentric ellipsoid of revolution is introduced. The normal gravity potential o U on the mean geocentric ellipsoid is chosen to be equal to the earth’s potential on the geoid: o o W U = . The difference of the gravity potential ( ) Ω , r W and the normal gravity potential ( ) Ω , r U defines the disturbing potential ( ) Ω , r T , ( ) ( ) ( ) Ω − Ω = Ω , r U , r W , r T (1.1) When atmospheric attraction is neglected, ( ) Ω , r T is harmonic outside the Earth and it satisfies the Laplace equation ( ) 0 , r T 2 = Ω ∇ (1.2) enero-diciembre 2002 Stokes-Helmert’s Scheme for Precise Geoid Determination 137 Once ( ) Ω , r T has been solved, the gravity potential ( ) Ω , r W can be obtained at any point by adding ( ) Ω , r U , which can be computed from existing models. Also when ( ) Ω , r T is known on the geoid, the vertical separation between the reference ellipsoid and the geoid can be obtained by the Bruns formula ( ) ( ) ( ) ( ) Ω γ Ω = Ω o g r T N (1.3) where ( ) ( ) Ω g r T is the disturbing potential on the geoid, and ( ) Ω γ o is the normal gravity on the mean geocentric ellipsoid. The problem is now reduced to the determination of ( ) Ω , r T on and outside the geoid. However, the disturbing potential ( ) Ω , r T does not satisfy the Laplace equation inside of topographical masses where the geoid is often located. Therefore in order to satisfy Laplace’s equation, all atmospheric and topographical masses have to be removed or condensed on or beneath the geoid. In Helmert’s second condensation method the atmospheric and topographical masses are condensed directly onto the geoid. When atmospheric and topographical masses are condensed as a single layer that is located on the geoid, the Earth gravity field will slightly change. The space obtained after such a condensation is the Helmert space. The quantities given in the Helmert space are denoted by superscript H . Helmert’s gravity potential is defined as follows ( ) ( ) ( ) ( ) Ω δ − Ω δ − Ω = Ω , r V , r V , r W , r W a t H (1.4) The residual topographical potential ( ) Ω δ , r V t is defined as a difference of the gravitational potential ( ) Ω , r V t of topographical masses and the gravitational potential ( ) Ω , r V ct of condensed topographical masses ( ) ( ) ( ) Ω − Ω = Ω δ , r V , r V , r V ct t t (1.5) Similarly, the residual atmospheric potential ( ) Ω δ , r V a is defined as a difference of the gravitational potential ( ) Ω , r V of atmospheric masses and the gravitational potential ( ) Ω , r V of condensed atmospheric masses 138 Robert Tenzer et al. Revista Cartografica 74-75 ( ) ( ) ( ) Ω − Ω = Ω δ , r V , r V , r V ca a a (1.6) By subtracting the normal gravity potential ( ) Ω , r U from eqn. (1.4), the disturbing potential ( ) Ω , r T H in Helmert’s space becomes ( ) ( ) ( ) Ω − Ω = Ω , r U , r W , r T H H (1.7) Helmert’s disturbing potential ( ) Ω , r T H is harmonic above the geoid, so that it satisfies the Laplace equation ( ) 0 , r T 2 = Ω ∇ H (1.8) To determine ( ) Ω , r T H , the boundary value problem of the third kind outside the geoid has to be solved. Therefore, the boundary values on the a-priori unknown geoid are needed. In this problem, the Helmert gravity anomalies on the geoid serve as the boundary values. To find a relation between the disturbing potential and the Helmert gravity anomalies, let us introduce the radial derivative of the Helmert disturbing potential ( ) ( ) ( ) r , r U r , r W r , r T ∂ Ω ∂ − ∂ Ω ∂ = ∂ Ω ∂ H H (1.9) The negative radial derivative of the Helmert disturbing potential ( ) Ω , r T H defines the Helmert gravity disturbance ( ) Ω δ , r g , i.e., ( ) ( ) ( ) Ω e − Ω δ = ∂ Ω ∂ − δ , r , r g r , r T g H H (1.10) The second term on the right hand side of eqn. (1.10) is the ellipsoidal correction to the gravity disturbance (Vanicek et al., 1999). Since the geoidal height above the ellipsoid are not usually available, the gravity disturbance is not considered to be a measurable quantity on the surface of the Earth. Therefore Helmert’s gravity disturbance ( ) Ω , H r g δ has to be transformed to more commonly available quantity, which is the Helmert gravity anomaly ( ) Ω ∆ , H r g . This transformation is achieved by adding a term ( ) Ω Γ , r to the gravity disturbance. This term accounts for the change in normal gravity due to the difference between the geodetic height ( ) Ω h and the commonly available orthometric height ( ) Ω O H . This expression can be written with a sufficient accuracy as ˆ enero-diciembre 2002 Stokes-Helmert’s Scheme for Precise Geoid Determination 139

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