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.