Molodensky's truncation coefficients for cap integration in spectral gravity forward modelling

Important note: Do NOT use the \(Q_{np}^{1,1,j}\), \(Q_{np}^{2,1,j}\) and \(Q_{np}^{2,2,j}\) coefficients from this version (v1), as they are wrong for \(p > 3\).  Use the new version (v2) published on Aug 28, 2025 at https://www.doi.org/10.5281/zenodo.16983295.

Provided are Molodensky's truncation coefficients for cap-modified spectral gravity forward modelling from the Bucha et al. (2019) study.  The coefficients are evaluated for

the spherical distance of \(\psi_0 = 100000\ \mathrm{m} / 6378137\ \mathrm{m}\) (100 km integration radius from the evaluation point),

the reference sphere having the radius \(R = 6378137\ \mathrm{m}\),

the radius of the evaluation point \(r = 6378137\ \mathrm{m} + 7000\ \mathrm{m}\),

harmonic degrees \(n=0,\dots,21600\),

topography powers \(p=1,\dots,30\),

radial derivatives \(k=0,\dots,40\), and

the first- and second-order horizontal derivatives.

The coefficients were computed using 256 significant digits, ensuring 24-digit accuracy or better. After the evaluation, the coefficients were converted to double precision with 16 significant digits. Importantly, in some cases, the loss of significance errors may be encountered during the spherical harmonic synthesis when using the coefficients (see the reference below).

Bucha, B., Hirt, C., Kuhn, M., 2019. Cap integration in spectral gravity forward modelling up to the full gravity tensor. Journal of Geodesy, https://doi.org/10.1007/s00190-019-01277-3.