Local earth gravity/potential modeling using ASCH
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ORIGINAL PAPER
Local earth gravity/potential modeling using ASCH Ghadi Younis
Received: 17 July 2014 / Accepted: 19 December 2014 # Saudi Society for Geosciences 2015
Abstract In Geodesy, the heights of points are normally orthometric heights measured above the geoid (an equipotential surface created by the earth masses and rotation which approximately coincides with the mean sea level) or the normal heights. It is necessary to transform the GNSS/GPS measured ellipsoidal heights (h) to classical physical heights (orthometric H/Normal H). The total gravity potential of the earth (W) is the summation of two components; gravitational potential (V) by earth masses and the centrifugal potential (Ω). The centrifugal potential is directly calculated, while the gravitational potential (V) needs to be modeled globally or locally using given measurements. The global models of the earth gravitational potential/gravity models (or so-called geoid models) are mostly given using spherical harmonics (SH). A modified approach of SH was defined to fit the use of SH for regional gravity/potential modeling called spherical cap harmonics (SCH). Due to the numerical difficulties of SCH, a simplified approach of SCH is selected to be used for a combined modeling of the earth potential using a variety of observations. This approach is called the Adjusted Spherical Cap harmonics.
Keywords Earth potential (W) . Gravitational potential (V) . Centrifugal potential (Ω) . Ellipsoidal heights (h) . Physical/ orthometric height (H) . Normal height (HN) . Geoid . Spherical harmonics . Spherical cap harmonics . Adjusted spherical cap harmonics
G. Younis (*) Surveying & Geomatics Engineering, Palestine Polytechnic University, Hebron, Palestine e-mail: [email protected]
Earth gravity potential There are two types of forces (accelerations) affecting a point P on the Earth’s surface, see Fig. 1. These types are the grav* itational acceleration g 1 due to the Earth’s mass M and the * centrifugal acceleration z due to the Earth’s rotation. The total * acceleration g , representing the actual gravity vector, is the vector summation of both gravitational and centrifugal accelerations (Fan 2004): *
*
*
g ¼ g 1þ z
ð1Þ *
The total earth potential (W) as result of the earth gravity g is the summation of potential related to the two acceleration components. These potential components are the gravitational potential V and the centrifugal potential Ω. This total gravity potential is given by: W ¼V þΩ
ð2Þ
As the angular velocity ω of the Earth around its minor axis is 0.7292115×10−4s−1as defined by the GRS80 or WGS84 (Hofmann-Wellenhof and Moritz 2005), the centrifugal potential reads: Ω ¼ 0:5 ω2 r2 cosϕ ¼
1 2 2 ω X þ Y2 2
ð3Þ
In Eq. (3), ϕ is spherical latitude of the point, r is the radial distance between the point P and the center of the earth. The centrifugal potential at a point is directly calculated. In the other hand, the gravitational potential needs to be modeled using variety of observations. Most of global gravity models are using the s
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