A MATLAB-based graphical user interface program for computing functionals of the geopotential up to ultra-high degrees and orders: Efficient computation at irregular surfaces

Fast spherical harmonic synthesis (SHS) at multiple points based on the lumped coefficients approach is a very well-established technique. However, this method cannot be applied to SHS at irregular surfaces, as the points must be regularly spaced and refer to a regular surface such as the sphere or the ellipsoid of revolution. In this paper we present a MATLAB®-based graphical user interface software for ultra-high degree (e.g. tens of thousands or even higher) SHS on grids at irregular surfaces, like the Earth surface. This software employs the highly efficient lumped coefficient approach for SHS at regular surfaces and the Taylor series expansions to continue the functionals to the irregular surfaces, e.g. the Earth surface. The generalized idea of continuing functionals using the Taylor series was presented by Hirt (2012) (J. Geod. 86, 729-744). We took  advantage of the software GrafLab (Bucha and Janák, 2013. Comput. Geosci. 56, 186-196), which employs the lumped coefficients approach, and developed a new software isGrafLab (Irregular Surface GRAvity Field LABoratory). Compared to the commonly used “two loops” approach, the factor of increased computational speed can reach a value of several hundreds. isGrafLab allows accurate evaluation of 38 functionals of the geopotential on grids at irregular surfaces. High orders of the Taylor series can be used for the continuation. The new software offers all the other options available in GrafLab, such as the employment of three different approaches to compute the fully normalized associated Legendre functions, the graphical user interface or the possibility to depict data on a map.