Oceanography from Space by Vittorio Barale J.F.R. Gower & L. Alberotanza

Author:Vittorio Barale, J.F.R. Gower & L. Alberotanza
Language: eng
Format: epub
Publisher: Springer Netherlands, Dordrecht

11.3 Upward Continuation

The potential of the anomaly field, T, obeys Laplace’s equation outside any volume enclosing its sources. If its sources are confined within a sphere of radius a, then T may be expanded in spherical harmonics:

(11.3)

In (11.3), are the spherical coordinates radius, geocentric colatitude, and longitude, G is the Newtonian gravitational constant, M the mass of the Earth, Y is a surface spherical harmonic of degree n and order m, and α is a dimensionless coefficient on Y.

In dealing with anomalies whose spatial extent is small compared with the mean radius of the Earth, a flat Earth approximation may be useful. Letting be coordinates in the east, north and upward directions, and assuming the sources of the field are confined to the half-space , the potential may be written:

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