GNSS in Practical Determination of Regional Heights - Global Navigation Satellite Systems: Signal,Theory and Applications - page 149

relationship between heterogeneous heights: h GNSS - H levelling - N model = 0 should have been
satisfied. However, because of physical realities and computational factors that cause
discrepancies among the heights, this equation cannot be realised at all in real world. As
such, this naturally affects the precision of transformation among the heights in practice.
Dealing with these disturbing factors, especially the element caused by the systematic errors
and datum inconsistencies as a part of geoid modelling, will reduce the discrepancies
among the three heights and hence improve the transformation precision of GNSS
ellipsoidal heights. As part of this chapter we therefore explain two methods, which are
aimed at minimizing the systematic differences of three heights in terms of optimal
combination of the heights, for the improvement of regional geoid models with limited
reference data in local areas. In the first approach, the height discrepancies are modelled
with a parametric equation, so called corrector surface model, which absorbs inconsistencies
of the height sets and allow a direct transformation of GNSS heights to the regional vertical
datum. The second method consists of the least squares adjustment of the orthometric
height differences, which are derived from ellipsoidal heights and regional geoid model, on
the base vectors. Hence the orthometric heights of the new points are derived using the
adjusted orthometric height differences. Brief descriptions of these height combination
approaches with formulations can be found in the sections below.

3.2.1 Corrector surface model

The corrector surfaces, determined according to combination of GNSS derived heights,
orthometric heights from the vertical datum and a gravimetric based geoid model, provides
an efficient and practical option to precise GNSS levelling in a local area (see e.g.,
Featherstone, 1998; Kotsakis & Sideris, 1999; Fotopoulos, 2003). The main idea of modelling
the corrector surface is to make the regional model estimate of the geoid coincident with the
valid vertical datum at GNSS/levelling benchmarks hence minimising the errors in the
regional geoid model and the observed heights at the benchmarks. This provides a practical
solution for GNSS users in order to accomplish a direct transformation from GNSS derived
ellipsoidal heights to orthometric heights, based on local vertical datum.

Determining an optimal parametric model for discrepancies of three heights follows the
similar steps as explained in section 3.1.2 for local GNSS/levelling geoid modelling. These
steps basically include: determining an appropriate type for model, selecting the optimum
extent (form) of the model, and finally assessing the performance of determined model.
Accordingly, although one can find numerous models suggested in the literature for
realizing corrector surfaces, selecting procedures of the parametric model is mostly arbitrary
and based on comparison of statistical test results that measure the accuracy and numerical
stabilities of the various models.

General expression of the discrepancies between GNSS/levelling derived geoid heights and
geoid heights from the regional geoid model as a function of geodetic position:

( )

=

(14)

h − HN F ϕλ − − , 0

GNSS lev . od el

that F (ϕ , λ) function can be presented in various forms in different levels of complexity (e.g.
having elements as only a bias, a bias and a tilt, or higher order polynomials), and multiple
regression equations generally as low-order polynomials (similar with Equation 5,