Determination of Vertical Deflections Using FFT Technique

The fast Fourier transform (FFT) technique is a very powerful tool for the efficient evaluation of gravity field convolution integrals.At present, there exist three types of convolution formulae in use, i.e.the planar 2D convolution, the spherical 2D convolution and the spherical 1D convolution.Up to now, many people are still used to applying the planar and spherical 2D FFT methods, due to the consideration of their gains in computer time, to perform the convolution evaluations in physical geodesy.It means that it is worthwhile discussing the question about making any possible improvement on the conventional 2D FFT approaches.As we know, the largest drawback of both the planar and the spherical 2D FFT methods is that, due to the approximations in the kernel function, only non-exact results can be achieved.Apparently, the reason is the meridian convergence at higher latitudes.As the meridians converge, the Δφ, Δλ blocks do not form a rectangular grid, as is assumed in 2D FFT methods.It should be pointed out that the meridian convergence not only leads to an approximation error in the kernel function, but also causes an approximation error during the implementation of 2D FFT in computer.In order to meet the increasing needs for precise determination of the vertical deflections, this paper derives a more precise planar 2D FFT formula for the computation of the vertical deflections.After having made a detailed comparison between the planar and the spherical 2D FFT formulae, we find out the main source of errors causing the loss in accuracy by applying the conventional spherical 2D FFT method.And then, a modified spherical 2D FFT formula for the computation of the vertical deflections is developed in this paper.A series of numerical tests have been carries out to illustrate the improvement made upon the old spherical 2D FFT.The second part of this paper is to discuss the influences of the spherical harmonic reference field, the limited capsize, the limited area extend, the discretization, the edge effects and the singular integral on the computation of the vertical deflections.The results of the vertical deflections over China by applying the spherical 1D FFT formula with different integration radii have been compared to the astro-observed vertical deflections in the South China Sea to obtain a set of optimum deflection computation parameters.