Figure 7. Differences of interpolated gravity fields based on MGH-50 and on MGH-2000 gravimetric network data. The values of the colourbar are in mGal unit.
Using the formulations above, the adequacy of the resolution of the Hungarian gravity network can be tested. According to equation (15), with stations located about 10 km distance from each other, features of the gravity field can be observed with 5%, 1% and 0.1% sampling error up to a resolution of 0.18 km, 0.42 km and 1.33 km, respectively. Indeed, local gravity anomalies can be relevant with even finer resolutions, therefore it is obvious that the 10 km distance between the stations are capable only for capturing middle wavelength gravity field variations.
A more reliable and informative test can be provided if also the magnitude of the observable is considered. Since the gravity field sums up of gravity features at different scales (ranging from local to global), the spectral behaviour of the gravity field should be modelled. In our case it is approximated by the Kaula’s rule of thumb (Kaula 1966), which is expressed in equation (31) for gravity anomaly, \(g\):
\(\sigma\left(g\right)=\frac{\text{kM}}{R^{2}}\left(n-1\right)\bullet\frac{\sqrt{2n+1}}{n^{2}}10^{-5}\)(31).
The Kaula’s rule of thumb estimates the signal content of gravity as a function of the spherical harmonic degree, \(n\), for an Earth with an average mass \(M\) and an average radius \(R\). The ‘classical’ rule of thumb (referring to the potential) is converted to gravity anomaly,\(g\) by multiplying it with the transfer function of the spherical harmonic expansion of the gravity anomaly. (For detailed explanation on the physical content of the Kaula’s rule of thumb see Kaula (1992), Rummel (2004) and McMahon et al. (2016)).
The spherical harmonic degree, \(n\) can approximately be related to the scale of the gravity content, i.e. to the spatial resolution of the features of the gravity field (described by a wavelength \(T\)).
\(T\left(g\right)=\frac{2\pi R}{n}\) (32).
Using (31) and (32), Figure 8 shows an estimate of the gravity field content by the spatial resolution according to Kaula’s rule of thumb in logarithmic scale. In the discussion below, two terms highly used in geodesy are applied here. These are: omission error refers to those errors, which are committed due to omitting certain parts of the frequency spectrum, while commission error refers to the errors provided by the involved frequencies. Certainly, the total error is a combination of the omission and commission errors, and one can only attempt to separate them based on the spatial resolution of the observed total error.