GNSS in Practical Determination of Regional Heights - Global Navigation Satellite Systems: Signal,Theory and Applications - page 130

is applied using the geoidal heights ( N ) from a geoid model that must be known with
sufficient accuracy (Fotopoulos et al., 2001; Fotopoulos, 2005). The computation methods of
geoid models are many (Schwarz et al., 1987; Featherstone, 1998; Featherstone, 2001; Hirt
and Seeber, 2007; Erol et al., 2008; Erol et al., 2009). The most commonly used methods for
geoid surface construction are described in textbooks like Heiskanen & Moritz (1967),
Vaniček & Krakiwsky (1986), Torge (2001). The so-called remove-restore (R-R) procedure is
one of these methods; where a global geopotential model and residual topographic effects
are subtracted (and later added back) (see Equations 1 and 2). The smooth resulting data set
is then suitable for interpolation or extrapolation using for example least squares collocation
with parameters (Sideris, 1994). According to R-R method, the reduced gravity anomaly is:

Δ g = Δ g FA − Δ g GM − Δ g H (1)

and the computed geoid height is:

N = N GM − N Δ g − N ind (2)

where Δ g GM is the effect of the global geopotential model on gravity anomalies, Δ g H is the
terrain effect on gravity, N Δg is the residual geoid height, which is calculated using Stokes
integral (see Equation 3), N ind is the indirect effect of the terrain on the geoid heights and
N GM is the contribution of the global geopotential model (expressed by the Equation 4),
(Heiskanen & Moritz, 1967; Sideris, 1994). The residual geoid height, computed from
Stokes's equation is;

R

()

∫∫

N = Δ Ψσ gSd (3)

Δ g

4 π

σ

where σ denotes the Earth's surface, Δ g is the reduced gravity anomaly (Equation 1) and
S (Ψ) is the Stokes kernel function where Ψ is the spherical distance between the
computation and running points (Haagmans et al., 1993; Sideris; 1994).

Th e globa l geopotential model derived geoid height using spherical harmonic coefficients,
m

C A and S A , is;

m

A max A

∑∑

( )

NR ≈ P sin θ ⎡ C mS cos λ + sin m λ ⎤ (4)

⎣ ⎦

GM A m A m A m

A == 2 m 0

where R is the me a n radius of the Earth, ( θ , λ ) are co-latitude and longitude of the
computation point, P A are fully normalized Legendre functions for degree ℓ and order m ,

and ℓ max is the maximum degree of the global geopotential model (Heiskanen & Moritz,
1967).

m

Following the Equation 2, it is obvious that the accuracy of the computed geoid heights
depends on the accuracy of the three height components, namely N GM , N Δ g and N H
(Fotopoulos, 2003). The global geopotential model not only contributes to the long
wavelength geoid information but also introduces long-wavelength errors that originate
from insufficient satellite tracking data, lack of terrestrial gravity data and systematic errors
in satellite altimetry. The two main types of errors can be categorized as either omission or
commission errors. Omission errors occur from the truncation of the spherical harmonic