# Iteration Procedure for Evaluating High Degree Potential Coefficients from Gravity Data

## Abstract

Earlier a solution of the gravimetric free scalar boundary value (*BV*) problem was derived with respect to the potential coefficients *C* _{ n,m }, taking into account the correction terms due to both the earth’s ellipticity and surface topography. In the present paper the behavior of this solution is studied for large values of degree *n*. It is shown that the absolute value of the ellipsoidal correction to the spherical approximation \(
_{n,m}^{(0)}
\)
, having the sign opposite to the latter, becomes of the same order for *n* ≈ 360, is equal to it at *n* ≈ 600 and then exceeds it, increasing with growing *n*. A similar situation is observed for the ellipsoidal correction in{152-2} derived by other authors, in particular by Pellinen, Rapp and Cruz. In the present paper, proceeding from the same *BV* solution, an alternative iteration procedure for evaluating \(
_{n,m}
\)
is elaborated. The initial approximation in it is not \(
_{n,m}^{(0)}
\)
but a modified quantity \(
_{n,m,}^{(1)}
\)
depending on e^{2}. The new formulas are well-suited for evaluating the potential coefficients of any degree *n* and order *m*.

## Keywords

Iteration Procedure Harmonic Coefficient Spherical Harmonic Expansion Downward Continuation Spherical Harmonic Degree## Preview

Unable to display preview. Download preview PDF.

## References

- Brovar, V.V. (1964). Fundamental harmonic functions with a singularity on the segment and the solution of the exterior boundary value problem.
*Izv. vyssh. uchebn. zaved. Geodesiya i aerofo-tos’emka*, 3:51–61.Google Scholar - Cruz, J.Y. (1985).
*Disturbance vector in space from surface gravity anomalies using complementary models*. Dept. Geod. Sei and Surv., Rep. 366, Ohio State Univ.Google Scholar - Cruz, J.Y. (1986).
*Ellipsoidal corrections to potential coefficients obtained ¿from gravity anomaly data on the ellipsoid*. Dept. Geod. Sci and Surv., Rep. 371, Ohio State Univ.Google Scholar - Heck, B. (1990).
*On the linearized boundary value problems of physical geodesy*. Dept. Geod. Sci and Surv., Rep. 407, Ohio State Univ.Google Scholar - Jekeli, C. (1981).
*The downward continuation to the earth’s surface of truncated spherical and ellipsoidal harmonic series of the gravity and height anomaly*. Dept. Geod. Sci and Surv., Rep. 323, Ohio State Univ.Google Scholar - Jekeli, C. (1988). The exact transformation between ellipsoidal and spherical harmonic expansions.
*Manuscripta Geodaetica*, 13(2): 106–113. 164Google Scholar - Molodensky, M. S., Eremeev, V.F., and Yurk-ina, M.I. (1962).
*Methods for the study of the external gravitational field and figure of the Earth*. Israel Program for Scientific Translations, Jerusalem.Google Scholar - Moritz, H. (1980).
*Advanced physical geodesy*. Wich-mann Verlag Karlsruhe.Google Scholar - Pellinen, L.P. (1982). Effects of the earth’s ellipticity on solving geodetic boundary value problem.
*Bollettino di Geodesia Scienze Affini*, 41(1):89–103.Google Scholar - Petrovskaya, M.S. (1993).
*Solution of the geodetic boundary value problem in spectral form*. IAG Symposium no. 114, Geodetic Theory Today, III Hotine-Marussi Symp. on Mathematical Geodesy, L’Aquila, Italy, F. Sansò edition.Google Scholar - Petrovskaya, M.S. and Vershkov, A. N. (1993a).
*Formulas for determination of the disturbing potential at the Earth’s surface and in exterior space*. Preprint of Inst, of Theoretical Astronomy, no. 35, St. Petersburg. In Russian.Google Scholar - Petrovskaya, M.S. and Vershkov, A. N. (1993b).
*Solution of the boundary value problem for the geopotential with accounting for the Earth’s ellipticity and the surface topography*. Preprint of Inst, of Theoretical Astronomy, no. 32, St. Petersburg, in Russian.Google Scholar - Petrovskaya, M.S. and Vershkov, A. N. (1997a). Compact formulas for the disturbing potential at the earth’s surface and in exterior space.
*Artificial Satellites*, 32(l):39–48. Warsaw.Google Scholar - Petrovskaya, M.S. and Vershkov, A. N. (1997b). Construction of the solution of the scalar boundary value problem.
*Artificial Satellites*, 32(l):21–38. Warsaw.Google Scholar - Rapp, R.H. and Cruz, J. Y. (1986a).
*The representation of the earth’s gravitational potential in a spherical harmonic expansion to degree 250*. Dept. Geod. Sci., Rep. 372, Ohio State Univ.Google Scholar - Rapp, R.H. and Cruz, J. Y. (1986b).
*Spherical harmonic expansions of the earth’s gravitational potential to degree 360 using 30’ mean anomalies*. Dept. Geod. Sci., Rep. 376, Ohio State Univ.Google Scholar - Rapp, R.H. and Pavlis, N. K. (1990). The development and analysis of geopotential coefficient models to spherical harmonic degree 360.
*Journal of Geophisical Research*, 95(B13): 21885–21911.CrossRefGoogle Scholar