Solution of Molodensky’s boundary-value problem for gravity disturbances with a relative error the Earth`s flattening square (second) order

Abstract

The solution of the geodetic boundary value problem for determining the anomalous potential from gravity measurements in the spherical approximation, taking into account the relief and compression, was developed in sufficient detail in 1960 and is also based on the results of G. G. Stokes. Flattening of the reference surface was taken into account by D. V. Zagrebin in several works dated 1940–1970; in 1956 M. S. Molodensky proposed a simpler method for an oblate ellipsoid, based on the already known one for a sphere. It turned out that at the very beginning of developing the mentioned solution, an error was made when obtaining the derivative of the reciprocal distance between two points on the ellipsoid in the direction of the outward normal; later it was repeated in issue 131 of the Proceedings of the TsNIIGAiK. That inaccuracy did not influence the conclusions of V. V. Brovar in 1996 and 1999, where the relief was added to the consideration of compression. This solution of Molodensky’s problem, taking the compression into account, was seldom used, and that is probably why the mistake went unnoticed. V. V. Popadyev saw it and found the correct expression for the mentioned derivative, I. I. Mezhenova adapted subsequent conclusions for the gravity disturbances.

National Report for the IAG of the IUGG 2019–2022

“…, where e' is the second eccentricity of the ellipsoid, r is the length of the chord connecting the points on the surface of the ellipsoid with latitudes B and B0, N and N0 are the corresponding radii of curvature of the first vertical, index 0 refers to the calculation point. In this expression, given by M. S. Molodensky without derivation, the authors of [Mezhenova, Popadyev, 2022] found an error, which, however, does not affect the solutions with preservation of terms of the order of the square of the eccentricity (but affects the deductions with preservation of terms with the fourth degree of eccentricity) , the correct expression is obtained for the derivative of the function of the reciprocal length of the chord along the direction of the outer normal to the ellipsoid: 𝜕1/𝑟 𝜕𝜈 = − 𝑁 − 𝑁 0 cos𝜓 Г − 𝑒 2 sin𝐵(𝑁sin𝐵 − 𝑁 0 sin𝐵 0 ) 𝑟 3 ,…”