Compact Approximation of the Geoid Height on the Base of Modified High Resolution Global Geopotential Models

Abstract

For description of various characteristics of the Earth’s gravitational field truncated expansions of the geopotential

$${V^{({N_{\max }})}} = \frac{{{\text{fM}}}}{r}\{ 1 + {\sum\limits_{n = 2}^{{N_{\max }}} {\sum\limits_{m = 2}^n {(\frac{r}{R})} } ^n}{\bar P_{nm}}(\cos \vartheta )[{\bar C_{nm}}\cos m\lambda + {\bar S_{nm}}\sin m\lambda ]\}$$

(1)

are widely used. Here the following designations are accepted: fM = geocentric gravitational constant; R = equatorial Earth’s radius; r, ϑ, λ = radius-vector, polar angle and longitude of a point of external space; ${\overline p _{nm}}$ = fully normalized adjoint Legendre functions. The fully normalized coefficients $\{ {\overline C _{nm}},{\overline S _{nm}}\}$ are practically represented by a geopotential model constructed on the base of observational data as the best approximation in the mean square metric L₂(∑), where ∑ is the reference sphere. Contemporary high resolution geopotential models contain harmonics up to 360 degrees and orders, the amount of them exceeding hundred of thousands. The construction and practical application of such bulky models entail technical difficulties and financial waste. Besides, high degree harmonics involve large errors. Therefore an actual problem is to obtain more compact (economized) geopotential models closer to the best uniform approximations and hence, allowing to reduce the number of spherical harmonics in conventional models and increase the accuracy of approximation for a fixed N_max.