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The Geodetic Uses of Gravity Measurements and their Appropriate Reduction

J. de Graaff-Hunter

1951
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Proceedings of the Royal Society A
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In § 1 are stated the needs of geodesy which must be met to allow results to be expressed in one unique reference system. At present the great surveys are in disconnected systems and are partially spheroidal and partially geoidal. § 2 recalls a theorem of Stokes, relating geoid with spheroid. Thence are deduced expressions for the deviations of the vertical and the curvature of the geoid. All these are in the form of integrals of the gravity anomalies over the earth. § 3: this ideal earth is a
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... s ideal earth is a body bounded by a level surface, and to bring the actual earth into its scope, protuberances of the topography above the co-geoid-a surface differing slightly from the geoid-must be annulled. § 4: the relation between gravity at different levels is found in terms of the local mean geoidal curvature, instead of the customary mean of the whole earth. In §5 a practical observational method of finding this curvature is proposed, depending on reciprocal vertical angles observed at pairs of points. Atmospheric refraction is involved and the mode of dealing with this is discussed. Some general remarks in §6 terminate the paper. T h e g e o d e tic u ses o f g r a v ity m e a su r e m e n ts a n d th eir a p p r o p r ia te r e d u c tio n Voi. 206. A . (22 March 1951) [ 1 ] z 2 observations reduced for height above the geoid, given by spirit levelling. Usually sufficient observed values of g for the dynamic reduction have not been available, and either this has been neglected or formula values of g have been used. Except in highly disturbed regions this may serve for present-day accuracy. These regions have generally not been penetrated by spirit levelling, though they may well be especially interesting from the point of view of constitution and content, and call for precise height control. All angles of triangulation are measured in the geoidal horizontal, and rays which are sensibly elevated require corrections on account of deflexion of the vertical relative to the spheroid. After the origin deflexions have been assigned deflexions at other stations can be found by processes of (e). 1 • 3. It will be seen then that for strict computation of geodetic survey it is necessary to know the geoid's separation from the universal spheroid at each base-line of a survey; and also to compute the deflexions at the origin (or some related point). If these things are not done-and hitherto they have not been done-separate disconnected surveys will remain in unconnected reference systems and unfit for full exploitation. In addition, the observed geoidal angles of triangulation must be reduced to spheroidal angles for correct computation, failing which no amount of least-square adjustment will avail to make good this defect. There is accordingly much need for observing the direction of gravity as well as its amount. In the past there has not usually been nearly enough gravity data available; and even now data are often far from adequate. The possibilities have been profoundly modified by the recent development of various gravimeters, which allow gravity data to be acquired far more rapidly and easily than hitherto, and accordingly in much greater detail. These gravimeters have indeed been developed on account of their geophysical uses; but geodesists should not fail to take advantage of their possibilities for geodetic purposes. To this end, widespread gravity surveys by gravimeter will need to be undertaken, including those in regions not thought to have mineral or other resources, in order to get a sufficiently detailed knowledge of the gravity field. 1*4. In practice while heights of high precision are provided by spirit levelling, in many cases resort is had to heights determined by reciprocal vertical angles in the course of triangulation. The former are geoidal heights, the latter, if corrected for deflexions of the vertical at the triangulation stations, are spheroidal heights. It is clearly important to relate the two systems. For this the separation of geoid and spheroid should be traced and numerous observations of deflexions are needed. The supplementary process sketched in § 5 below will contribute to this. 2. Stokes's gravity theorem and some derivatives 21. In his classical paper Stokes (1849) considered a nearly spherical surface which bounded and was equipotential of a rotating gravitational interior. He showed that the elevation, N, at a point P of this surface above P0 vertically below on the reference spheroid could be expressed in the form J. de Graaff-Hunter (21) Vol. 206. A.

doi:10.1098/rspa.1951.0052
fatcat:mq7obyjx4jf77j2x26uq7ylbbq