Improved Convergence Rates for the Truncation Error in Gravimetric Geoid Determination

Abstract

When Stokes's integral is used over a spherical cap to compute a gravimetric estimate of the geoid, atruncation error results due to the neglect of gravity data over the remainder of the Earth. Associated with the truncation error is an error kernel dfined over these two complementary regions. An important observation is that the rate of decay of the coefficients of the series expansion for the truncation error in terms of Legendre polynomials is determined by the smoothness properties of the error kernel. Previously published deterministic modifications of Stokes's integration kernel involve either a discontinuity in the error kernel or itsfirst derivative at the spherical cap radius. These kernels aregeneralised and extended by constructing error kernels whose derivatives at the spherical cap radius arecontinuous up to an arbitrary order. This construction is achieved by smoothly continuing the error kernel function into the spherical cap using a suitable degree polynomial. Accordingly, an improved rate of convergence of the spectral series representation of the truncation error is obtained.