Fast and accurate evaluation of geomagnetic field elements at arbitrary points in space

SUMMARY An algorithm and software are developed for fast and accurate evaluation of the elements of the geomagnetic field represented in high-degree (>720) solid spherical harmonics at many scattered points in the space above the surface of the Earth. The algorithm is based on representation of the geomagnetic field elements in solid ellipsoidal harmonics and application of tensor product needlets. Open source FORTRAN and MATLAB realizations of this algorithm that rely on data from the Enhanced Magnetic Models 2015, 2017 (EMM2015, EMM2017) have been developed and extensively tested. The capabilities of the software are demonstrated on the example of the north, east and down components of the geomagnetic field as well as the derived horizontal intensity, total intensity, inclination and declination. For the range from −417 m under the Earth reference ellipsoid up to 1000 km above it the FORTRAN and MATLAB versions of the software run 465 and 189 times faster than the respective FORTRAN and MATLAB versions of the software using the standard spherical harmonic series method, while the accuracy is less than 1 nT and the memory (RAM) usage is 9 GB.

A Constructive Proof of Helmholtz’s Theorem

Summary It is a known result that any vector field ${\boldsymbol{u}}$ that is locally Hölder continuous on an arbitrary open set $\Omega\subset \mathbb{R}^3$ can be written on $\Omega$ as the sum of a gradient and a curl. Should $\Omega$ be unbounded, no conditions are required on the behaviour of ${\boldsymbol{u}}$ at infinity. We present a direct, self-contained proof of this theorem that only uses elementary techniques and has a constructive character. It consists in patching together local solutions given by the Newtonian potential that are then modified by harmonic approximations—based on solid spherical harmonics—to assure convergence near infinity for the resulting series.

The topographic bias in Stokes’ formula vs. the error of analytical continuation by an Earth Gravitational Model - are they the same?

Geoid determination below the topographic surface in continental areas using analytical continuation of gravity anomaly and/or an external type of solid spherical harmonics determined by an Earth GravitationalModel (EGM) inevitably leads to a topographic bias, as the true disturbing potential at the geoid is not harmonic in contrast to its estimates. We show that this bias differs for the geoid heights represented by Stokes’ formula, an EGMand for the modified Stokes formula. The differences are due to the fact that the EGM suffers from truncation and divergence errors in addition to the topographic bias in Stokes’ original formula.