Lars E. Sjöberg

Reformulation of Stokes's theory for higher than second‐degree reference field and modification of integration kernels

An argument is put forward in favor of using a model gravity field of a degree and order higher than 2 as a reference in gravity field studies. Stokess approach to the evaluation of the geoid from gravity anomalies is then generalized to be applicable to a higher than second-order reference spheroid. The effects of truncating Stokess integration and of modifying the integration kernels are investigated in the context of the generalized approach. Several different modification schemes, starting with a Molodenskij-like modification and ending with the least squares modification, are studied. Particular attention is devoted to looking at both global and local biases and mean square errors of the individual schemes.

Higher-degree reference field in the generalized Stokes-Helmert scheme for geoid computation

In this paper we formulate two corrections that have to be applied to the higher-degree reference spheroid if one wants to use it in conjunction with the Stokes-Helmert scheme for geoid determination. We show that in a precise geoid determination one has to apply the correction for the residual topographical potential and the correction for the earth ellipticity. Both these corrections may reach several decimetres; we show how their magnitudes vary within Canada and we give their global ranges.

The new gravimetric quasigeoid model KTH08 over Sweden

Abstract The least squares modification of Stokes formula has been developed in a series of papers published in Journal of Geodesy between 1984 and 2008. It consists of a least squares (stochastic) Stokes kernel modification with additive corrections for the topography, downward continuation, the atmosphere and the ellipsoidal shape of the Earth. The method, developed at the Royal Institute of Technology (KTH) will here be denoted by the abbreviated name the KTH method. This paper presents the computational results of a new gravimetric quasigeoid model over Sweden (the KTH08 model) by employing the KTH method. Traditionally the Nordic Geodetic Commission (NKG) has computed gravimetric quasigeoid models over Sweden and other Nordic countries; the latest model being NKG 2004. Another aim of this paper is therefore to compare KTH08 and NKG 2004 quasigeoid models and to evaluate their accuracies using GNSS/levelling height anomalies. The rms fit of KTH08 in 196 GNSS data points distributed over Sweden by using a 1(4)-parameter transformation is 22 (20) mm. It is concluded that KTH08 is a significant step forward compared to NKG 2004.

DETERMINATION OF AREAS ON THE PLANE, SPHERE AND ELLIPSOID

Abstract This paper shortly reviews various methods to determine the area of a closed polygon on the plane, sphere and ellipsoid. A new method is derived for calculating the area of a geodetic polygon, i.e. a polygon on the ellipsoid limited by sections of geodesics. By a recursive procedure the area can be determined to any desired order of the eccentricity of the ellipsoid. Finally we also present a direct method for numerical integration of the area under the geodesic.

Glacial rebound near Vatnajokull, Iceland, studied by GPS campaigns in 1992 and 1996

Abstract Since about 1920 the Vatnajokull ice cap in Iceland has experienced a significant retreat, corresponding to a volume reduction of more than 180 km 3 . With two GPS campaigns in 1992 and 1996 along the southern border of the glacier preliminary results reveal land uplift rates of 1–6 mm/yr, after a one-parameter (bias) fit with recent earth rheology models. The best fit model suggests that the lithosphere in the area is about 30 km thick and the viscosity of the asthenosphere 5 × 10 18 Pa s. The rms fit of uplift rate at all GPS sites is ±1.4 mm/yr. As the GPS data alone cannot provide the absolute uplift rates, the one-parameter fit to the theoretical modelling implies that the absolute rates were estimated by the matching of the GPS data and model. The resulting uplift rate at station Hofn (1 mm/yr) is not consistent with two independent sources, and we therefore conclude that further GPS epoch and permanent GPS site data are needed to confirm the present geodynamic processes near Vatnajokull.

Precise Determination of the Clairaut Constant in Ellipsoidal Geodesy

Abstract The Clairaut constant, the cosine of the maximum latitude of the geodesic, is used in a number of applications in ellipsoidal geodesy. This study provides formulas to precisely determine the Clairaut constant from the coordinates of two given points on the geodesic.

Terrain effects in the atmospheric gravity and geoid corrections

In view of the smallness of the atmospheric mass compared to the mass variations within the Earth, it is generally assumed in physical geodesy that the terrain effects are negligible. Subsequently most models assume a spherical or ellipsoidal layering of the atmosphere. The removal and restoring of the atmosphere in solving the exterior boundary value problems thus correspond to gravity and geoid corrections of the order of 0.9 mGal and -0.7 cm, respectively.We demonstrate that the gravity terrain correction for the removal of the atmosphere is of the order of 50µGal/km of elevation with a maximum close to 0.5 mGal at the top of Mount Everest. The corresponding effect on the geoid may reach several centimetres in mountainous regions. Also the total effect on geoid determination for removal and restoring the atmosphere may contribute significantly, in particular by long wavelengths. This is not the case for the quasi geoid in mountainous regions.

On the Pratt and Airy models of isostatic geoid undulations

Abstract Usually the topographic-isostatic geoid undulation is derived by a downward continuation of the external potential to the geoid. Unfortunately, such an approach merely yields a fictitious potential at the continental geoid within the topographic masses. We show that the related bias may range to about 10m based on the classical Pratt and Airy models for isostatic compensation. The above geoid contribution is generated by the topographic masses along the vertical to the computation point, i.e. it is a local contribution. Our second conclusion is that the isostatic geoid undulation also experiences a regional contribution from the surrounding topographic masses. This contribution is significant for extended high mountains and deep oceans.

A recurrence relation for the βn-function

A recurrence relation is presented for the smoothing function, βn, which is used in geodesy to relate spherical harmonics to their mean values over circular areas (caps). The proposed formula does not require the computation of the Legendres polynomials. Moreover, it is numerically more stable than the formulas ofPellinen (1969) andMeissl (1971).

Solutions to the ellipsoidal Clairaut constant and the inverse geodetic problem by numerical integration

Abstract We derive computational formulas for determining the Clairaut constant, i.e. the cosine of the maximum latitude of the geodesic arc, from two given points on the oblate ellipsoid of revolution. In all cases the Clairaut constant is unique. The inverse geodetic problem on the ellipsoid is to determine the geodesic arc between and the azimuths of the arc at the given points. We present the solution for the fixed Clairaut constant. If the given points are not(nearly) antipodal, each azimuth and location of the geodesic is unique, while for the fixed points in the ”antipodal region”, roughly within 36”.2 from the antipode, there are two geodesics mirrored in the equator and with complementary azimuths at each point. In the special case with the given points located at the poles of the ellipsoid, all meridians are geodesics. The special role played by the Clairaut constant and the numerical integration make this method different from others available in the literature.