Abstract
Meissl has derived weighting functions for converting point gravity anomaly degree variances into mean anomaly variances over a circular cap on a sphere. If the cap is sufficiently small so that the cap on a sphere degenerates into a circle on a plane, the problem may be considered that of the gain of a circular filter for a surface wave whose wave number depends on the spherical harmonic degree. The Meissl weights then become replaced by diffraction integrals of optical physics. The expected gain for a square filter for waves coming from random directions is derived and shown to be close to the gain of a circular filter with the same area. The expected gains and cross-gains for rectangular filters are also derived. When weighted by an anomaly degree variance model, these gains and cross-gains can be used to determine rectangular anomaly variances and covariances for arbitrary bandwidths. Using the Tscherning-Rapp model, analytic gravity anomaly variances and covariances are calculated for 1°×1° blocks.
Similar content being viewed by others
References
M. ABRAMOWITZ and I.A. STEGUN, eds. (1964):Handbook of Mathematical Functions. National Bureau of Standards Applied Mathematics Series, 55, U.S. Government Printing Office, Washington, DC.
R.B. BLACKMAN and J.W. TUKEY (1959):The Measurement of Power Spectra. Dover Publications, New York.
M. BORN and E. WOLF (1970):Principles of Optics. Fourth Edition, Pergamon Press, New York.
D.H. ECKHARDT (1968): Theory and interpretation of the electromagnetic impedance of the earth.J. Geophys. Res., 73, 5317–5326.
E.M. GAPOSCHKIN (1980): Averaging on the surface of a sphere.J. Geophys. Res., 85, 3187–3193.
G. HADGIGEORGE, G. BLAHA and T.P. ROONEY (1981): SEASAT altimeter reductions for detailed determinations of the oceanic geoid.Ann. Geophys., 37, 123–132.
W.A. HEISKANEN and H. MORITZ (1967): Physical Geodesy. W.H. Freeman and Co., San Francisco.
P. MEISSL (1971): A study of covariance functions related to the earth's disturbance potential. Dept. of Geod. Sci. Report No. 151, The Ohio State University, Columbus, Ohio.
H.N. POLLACK (1973): Spherical harmonic representation of the gravitational potential of a point mass, a spherical cap, and a spherical rectangle.J. Geophys. Res., 78, 1760–1768.
L. SJOBERG (1980): A reccurrence relation for the βn function.Bull. Geod., 54, 69–72.
C.C. TSCHERNING and R.H. RAPP (1974): Closed covariance expressions for gravity anomalies, geoid undulations, and deflections of the vertical implied by anomaly degree variance models. Dept. of Geod. Sci. Report No. 208, The Ohio State University, Columbus, Ohio.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Eckhardt, D.H. The gains of small circular, square and rectangular filters for surface waves on a sphere. Bull. Geodesique 57, 394–409 (1983). https://doi.org/10.1007/BF02520942
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02520942