Figure 8. Signal content of gravity anomaly based on Kaula’s
rule of thumb vs. sampling error (by L1-norm and L2-norm as well)
according to a sampling rate, \(T\) of 10 km.
The sampling error is estimated by L1-norm and L2-norm using equation
(15). The value of the sampling rate has been set to \(T=10km\), and
the values of \(N\) were determined by using \(T\) as the independent
variable. The resulted curves are shown in Figure 8. The maximal value
of \(N\) was 0.5 according to the Nyquist criterion, which is at the
distance of 20 km. The \(N=1\) abscissa is displayed on the figure by
a vertical dotted line, which refers to the \(T=T=10km\) value. By
any mean, signal content below \(T=10km\) appears as noise in the
signal, as the corresponding frequencies are overlooked (not sampled) by
the \(T=10km\) sampling rate. The omission of the signal content of
the short wavelength gravity (i.e. at less than \(10km\) resolution) may
particularly be relevant at mountainous regions, where the gravity may
change more sharply, the gradients of the gravity are larger. Basically,
omission error of the short wavelength gravity is the primer source of
100 mGal large, very localized outliers in Figure 7.
Beyond the omission error, the commission error due to the sampling is
also large: according to Figure 8, the sampling generated error reaches
the signal at 28.1 km (L1-norm) and 26.8 km (L2-norm). A signal is
reliable up to that point where its noise content is at least one order
of magnitude smaller than the signal content, i.e. the Signal-to-Noise
ratio is 10. In the case of Figure 8, this is reached at 73.8 km
(L1-norm) and 70.7 km (L2-norm), so any smaller scale gravity
information can be gained only quite uncertainly due to the commission
error. As in an actual case both the omission and commission errors
affect the solution, the gravity field with 10 km sampling may describe
properly the gravity field features up to 100 km resolution only.
Summary
Sampling of continuous signals cannot be avoided for practical
applications therefore sampling errors are contaminating the knowledge
of the continuous signal. In order to check whether the sampling is
sufficiently fine for a certain application, a simple test on the
sampling rate can be performed. In this study, analytic formulation for
L1-norm sampling error estimate of a periodic signal has been delivered
in a closed form. Also, the L2-norm error estimate has been derived
making use of a symbolic programming module of Matlab, which has not
been presented here but applied for the tests. According to the results,
the 5%, 1% and 0.1% errors can be reached by N=0.177, 0.078 and
0.025, equivalent to 6, 13 and 41 samples per period, respectively.
As an example, GRACE-borne gravity anomaly time series are analysed for
the region of the Amazonas basin. It was found that the unmodelled
non-periodic components and observation errors can be handled
independently from the annual component, and that the annual component
can be described by an error of 2.49% of the signal content due to the
sampling and an error of 1.15% of the signal content due to the
averaging.
As a spatial example, the case of the Hungarian gravity network is
analysed. With stations located about 10 km distance from each other,
both omission and commission errors were found to be relevant. Fine
features of the gravity field are largely contaminated by the omission
of the gravity signal over less than 10 km scales, resulting in huge,
local peaks in the gravity anomaly error map in Figure 6. Commission
error was found to be relevant up to a resolution of approximately 70
km. As in an actual case both the omission and commission errors affect
the solution, the gravity field with 10 km sampling may describe
properly the gravity field features up to 100 km resolution only. For
reducing the spatial resolution, the sampling density should be
increased further.