The downward continuation to the earth's surface of truncated spherical and ellipsoidal harmonic series of the gravity and height anomalies
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Date
1981-12
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Ohio State University. Division of Geodetic Science
Abstract
The problem of the divergence of the geopotential spherical harmonic series at the earth's surface is investigated from a numerical, rather than a theoretical, approach. It is shown that previous numerical evaluations, based on the expansion of the potential generated by the masses between the point of computation and the bounding sphere, are inconclusive with respect to the magnitude of the downward continuation error that is attributable to series divergence. A more representative model of the earth's potential is devised on the basis of a density layer, which, in the spherical approximation, generates a gravity field whose harmonic constituents decay according to an accepted degree variance model. This field, expanded to degree 300, and a topographic surface specified to a corresponding resolution of 67 km are used to compute the differences between truncated inner and outer series of the gravity and height anomalies at the surface of the earth model. Up to degree 300, these differences attain RMS values from 0.33 μgal to 86 μgal for the gravity anomaly and from 0.32 μm to 410 μm for the height anomaly, in areas ranging respectively from near the equator to the vicinity of the pole. In addition to these values, there is an expected truncation effect, caused by the neglect of higher degree components of the inner series, of about 30 mgal and 36 cm, respectively. The field is then subjected to a Gaussian filter which effectively cuts off information at degree 300 (at the 5% level). The RMS downward continuation error to degree 300 is thereby reduced by factors of 10 to 20, with a concomitant reduction in the truncation effect to about 0.3 mgal and 0.7 cm. Testing the downward continuation of ellipsoidal harmonic series on a similar model yields RMS values of the first 300 degrees of the error of 1x10-4 to 3x10-4 μgal for the (point) gravity anomaly, from pole to equator, respectively. [Some mathematical expressions are not fully represented in the metadata. Full text of abstract available in document.]