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.