IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

# Overview

Gravimetric quantities are commonly represented in terms of high degree surface or solid spherical harmonics. After EGM2008, such expansions routinely extend to spherical harmonic degree 2190, which makes the computation of gravimetric quantities at a large number of arbitrarily scattered points in space using harmonic synthesis, a very computationally demanding process.

We present here the development of an algorithm and its associated software for efficient, precise and stable evaluation of gravimetric quantities, represented in high degree solid spherical harmonics, at arbitrarily scattered points in the space exterior to the surface of the Earth. Our algorithm is based on representation of the quantities of interest in solid ellipsoidal harmonics and application of the tensor product trigonometric needlets. FORTRAN and MATLAB implementations of this algorithm have been developed and extensively tested. The capabilities of the code are demonstrated using as examples

• Disturbing potential $T$
• Height anomaly $\zeta$
• Gravity anomaly $\Delta g$
• Gravity disturbance $\delta g$
• North-south deflection of the vertical $\xi$
• East-west deflection of the vertical $\eta$
• Second radial derivative of the disturbing potential $T _ {rr}$.

In Global Gravitational Models like EGM2008 all these quantities are represented in solid spherical harmonics of degree 2190.

For the range from the surface of the Earth up to 544 kilometers above that surface the current version of the software runs on a standard PC at speeds between 20,000 and 40,000 point evaluations per second, depending on the gravimetric quantity being evaluated, while the relative error does not exceed $10^{-6}$ and the memory (RAM) use is 9.3 GB.

A detailed description of our algorithm and software is presented in the enclosed manual file and preprint.

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