SUMMARY A ‘gravity anomaly’ is essentially the difference between the gravitational acceleration caused by the Earth’s masses and that generated by some reference mass distribution. However, there are numerous subtleties to the definition and, moreover, to the practical realization of a ‘gravity anomaly’. An attempt is made here to clarify the definition of a ‘gravity anomaly’ from the geodetic and geophysical perspectives, point out some of the key differences in terminology and philosophy and to identify some of the problems remaining in its practical realization from a variety of observation types. It is argued that if the ‘gravity anomaly’ is defined and realized in a rigorous and consistent manner, this may lead to the improvement of its use in both geodesy and geophysics.

Geodetic versus geophysical perspectives of the ‘gravity anomaly’

SUMMARY This paper presents a number of new concepts concerning the gravity anomaly. First, it identifies a distinct difference between a surface (2-D) gravity anomaly (the difference between actual gravity on one surface and normal gravity on another surface) and a solid (3-D) gravity anomaly defined in the fundamental gravimetric equation. Second, it introduces the ‘no topography’ gravity anomaly (which turns out to be the complete spherical Bouguer anomaly) as a means to generate a quantity that is smooth, thus suitable for gridding, and harmonic, thus suitable for downward continuation. It is understood that the possibility of downward continuing a smooth gravity anomaly would simplify the task of computing an accurate geoid. It is also shown that the planar Bouguer anomaly is not harmonic, and thus cannot be downward continued.

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New views of the spherical Bouguer gravity anomaly

Fast evaluation of coIIvolution iIItegrals on the sphere using1D FFT and a comparison with existing methods for stokes’ integral

Helmert’s first and second method of condensation are reviewed and generalized in two respects: First, the point at which the effects of topographical and condensation masses are calculated may be situated on or outside the topographical surface; second, the depth of the condensation layer below the geoid is arbitrary. While the first extension permits the application of the generalized model to the evaluation of airborne and satellite data, the second one gives an additional degree of freedom which can be used to provide a smooth gravity field after reducing the observation data. The respective formulae are derived for the generalized condensation model in both planar and spherical approximation. A comparison of the planar and the spherical model shows some structural differences, which are primarily visible in the out-of-integral terms. Considering the respective formulae for the combined topographic–condensation reduction on the background of the density structure of the Earth’s lithosphere, the consequences for the residual gravity field are investigated; it is shown that the residual field after applying Helmert’s second model of reduction is very rough, making this procedure unfavourable for downward continuation. Further considerations refer to the question of which sets of formulae should be used in geoid and quasigeoid determination. It is concluded that for high-precision applications the generalized spherical model, involving a depth of the condensation layer of between 20 and 30 km, should be superior to Helmert’s second model of condensation, although it requires the direct calculation of the indirect effect, which is larger than in the case of Helmert’s second method of condensation.

On Helmert's methods of condensation

The main problem of the rigorous definition of the orthometric height is the evaluation of the mean value of the Earth's gravity acceleration along the plumbline within the topography. To find the exact relation between rigor- ous orthometric and Molodensky's normal heights, the mean gravity is decomposed into: the mean normal gravity, the mean values of gravity generated by topographical and atmo- spheric masses, and the mean gravity disturbance generated by the masses contained within geoid. The mean normal gravity is evaluated according to Somigliana-Pizzetti's the- ory of the normal gravity field generated by the ellipsoid of revolution. Using the Bruns formula, the mean values of gravity along the plumbline generated by topographical and atmospheric masses can be computed as the integral mean between the Earth's surface and geoid. Since the disturb- ing gravity potential generated by masses inside the geoid is harmonic above the geoid, the mean value of the gravity disturbance generated by the geoid is defined by applying the Poisson integral equation to the integral mean. Numeri- cal results for a test area in the Canadian Rocky Mountains show that the difference between the rigorously defined or- thometric height and the Molodensky normal height reaches ∼0.5 m.

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The rigorous determination of orthometric heights

The definition of the mean Helmert anomaly is reviewed and the theoretically correct procedure for computing this quantity on the Earth's surface and on the Helmert co-geoid is suggested. This includes a discussion of the role of the direct topographical and atmospherical eAects, primary and secondary indirect topographical and atmospherical eAects, ellipsoidal corrections to the gravity anomaly, its downward continuation and other eAects. For the rigorous deriva- tions it was found necessary to treat the gravity anomaly systematically as a point function, defined by means of the fundamental gravimetric equation. It is this treat- ment that allows one to formulate the corrections necessary for computing the 'one-centimetre geoid'. Compared to the standard treatment, it is shown that a 'correction for the quasigeoid-to-geoid separation', amounting to about 3 cm for our area of interest, has to be considered. It is also shown that the 'secondary indirect eAect' has to be evaluated at the topography rather than at the geoid level. This results in another diAerence of the order of several centimetres in the area of interest. An approach is then proposed for determin- ing the mean Helmert anomalies from gravity data observed on the Earth's surface. This approach is based on the widely-held belief that complete Bouguer anom- alies are generally fairly smooth and thus particularly useful for interpolation, approximation and averaging. Numerical results from the Canadian Rocky Mountains for all the corrections as well as the downward contin- uation are shown.

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Determination of the Boundary Values for the Stokes-Helmert Problem

Stokes's kernel used for the evaluation of a gravimetric geoid is a function of the spherical distance between the point of interest and the dummy point in the integration. Its values thus are obtained from the positions of pairs of points on the geoid. For the integration over the near integration zone (near to the point of interest), it is advantageous to pre-form an array of kernel values where each entry corresponds to the appropriate locations of the two points, or equivalently, to the latitude and the longitude-difference between the point of interest and a dummy point. Thus, for points of interest on the same latitude, the array of the Stokes kernel values remains the same and may only be evaluated once. Also, only one half of the array need be evaluated because of its longitudinal symmetry: the near zone can be folded along the meridian of the point of interest.
 Numerical tests show that computation speed improves significantly after this algorithm is implemented. For an area of 5 by 10 arc-degrees with the grid of 5 by 5 arc-minutes, the computation time reduces from half an hour to about 1 minute. To compute the geoid for the whole of Canada (20 by 60 arc-degrees, with the grid of 5 by 5 arc-minutes), it takes only about 17 minutes on a 400MHz PC computer.
 Compared with the Fast Fourier Transform algorithm, this algorithm is easier to implement including the far zone contribution evaluation that can be done precisely, using the (global) spectral description of the gravity field.

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J. Huang

Papers

Determination of the Boundary Values for the Stokes-Helmert Problem

An Alternative Algorithm to FFT for the Numerical Evaluation of Stokes's Integral