Least squares modification of Stokes formula with application to the Estonian geoid

Sammanfattning: Many applications in geodesy,geophysics and engineering require physically defined heightsrelated to an equipotential surface of the Earth, thegeoid.During the last two decades theincreased need for refined geoid models has been driven by thedemands of users of GPS, who nowadays must transformGPS-derived geodetic heights into orthometric heights, makingthem compatible with the local vertical datum. Therefore, forthe conversion and combination of these fundamentally differentheight systems, the geoid model must be known with an accuracycomparable to the accuracy of GPS and traditional levelling,i.e. a few centimetres.Stokes formula enables thedetermination of the geoidal height from the global coverage ofgravity anomalies. In practice, the area of integration islimited, thus implying a truncation error of the estimatedgeoidal height. The modification of Stokes formula allows theuser to compensate the lack of a global coverage of gravitydata by a combination of terrestrial gravity with a globalgeopotential model (here EGM96). The minimization of thetruncation error, the influence of erroneous gravity data andpotential coefficients could be treated using the least squaresmodification of the Stokes formula, proposed by Sjöberg in1984.The aim of this thesis is todetermine the Estonian geoid by the biased least squaresmodification of Stokes formula. The significant improvement ofthe data coverage in Baltic Sea region in last few years iscomprised in the research.Geoid determination by Stokesformula requires that there are no masses outside the geoid,and the classical approach is to apply a so-calledremove-restore technique.As proposed by Sjöberg in1994, corrections related to topography (and atmosphere) couldbe made on the geoid directly. Thus the combined correction oftopographic (and atmospheric) masses, as a sum of direct andindirect effects, can replace the traditional approaches.The expressions for the totaltopographic (and atmospheric) effects for the biased leastsquares modification of Stokes formula are derived and appliedin the numerical computations.The geoid is calculated bydifferent sets of modification coefficients. The preference isgiven to the geoid model called EST-01 which is calculated bythe following initial conditions: modification degree 360,terrestrial gravity anomaly variance 9mGal2(correlation length 0.1° ), truncationcap radius 2° .The model EST-01 is validated by26 high-precision GPS/levelling points.The mean bias of -30cmand a small tilt between EST-01 and GPS/levellingpoints were detected, most likely caused by long wavelengtheffects. A root mean square error of 3.0cmfor post-fitting residuals was obtained using afour-parameter fitting.The accuracy of the relativeheight determination was evaluated by a set of lower orderGPS/levelling points. For baselines shorter than 12kma root mean square error of 2.3cmwas obtained, which is satisfactory for the mostpractical applications.Key words: geoid, Stokes formula, least squaresmodification, total topographic correction, total atmosphericcorrection, GPS/levelling.