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Wissenschaften 
GEOID ESTIMATION THROUGH GPS OBSERVATIONS 
R. Barzaghi, A. Borghi 
DIIAR Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano 
D. Dominici 
Dipartimento di Architettura ed Urbanistica, Università di L’Aquila, Monteluco di Roio, 67100 L’Aquila 
S. Gandolfi 
DISTART, Facoltà di Ingegneria, Università di Bologna, Viale Risorgimento 2, 40136 Bologna 
Commission VI, Working Group 3 
KEY WORDS: GPS, geoid, interpolation 
ABSTRACT 
GPS observations are more and more used for different purposes. Monitoring deformations caused by 
landslide, earthquakes and volcanism has nowadays become possible on a large scale using GPS 
techniques. Furthermore, GPS is used in connection with photogrammetry, both for field survey and the 
determination of the camera position at shutter release time. Kinematic positioning with GPS is also applied 
to terrestrial navigation, in the perspective of its fully automatic control. On a more geodetic side, GPS is a 
powerfull tool for checking the precision of a gravimetric geoid estimate. By combinig GPS derived height 
and spirit leveling, geoid undulations can be obtained and compared with geoid estimates coming from e.g. 
Stokes formula. Besides, this method can be used to derive directly the geoid without knowing gravity, by 
directly interpolating a set of observed undulations. In this paper, this way of getting the geoid is revised and 
critically analyzed, to possibly define a procedure to get reliable estimates. Numerical tests on simulated data 
are presented and discussed and different interpolation methods are compared. 
using S-Plus. 
'ariate Statistik. 
Methods. Me 
1. INTRODUCTION 
1.1 GPS derived undulations 
The basic equation which allows geoid estimation 
through GPS observations is the well known formula 
(Heiskanen and Moritz, 1967) 
N = h - H (1) 
orthometric height 
where h is the ellipsoidal height from GPS, H is the 
orthometric height from spirit leveling and N is the 
geoid undulation. This equation holds in an 
approximate way but its accuracy is in any case 
sufficient for practical purposes. 
Hence, by performing GPS observations on spirit 
leveling benchmarks, pointwise geoid undulation 
values can be obtained. 
This values can be then interpolated to get a local 
estimate of the geoid to be used in connection with 
GPS measuremets to derive orthometric heights. By 
simlpy rewriting equation (1) in the following way 
H = h - N (2) 
the so called GPS leveling method can be applied to 
get H. Although the precision of H obtained in this 
way is by no means less than the one obtained by 
spirit leveling, GPS levelling is much faster than 
spirit leveling. So, for standard medium precision 
height surveys, this new techinque is preferred. 
The weak point of this procedure applied to get H, is 
basically related to the precision of the geoid 
estimate that one can get by interpolating the point 
estimate of N. There are still many open problems in 
doing such an operation and many misleading ways 
of performing it. The worst way of estimating a geoid