This paper investigates the terrain-aliasing effects on geoid determination using different gravimetric reduction schemes. The high resolution of digital terrain model (DTM), if available, should be used for every gravimetric reduction scheme since it can precisely map the details of the terrain. The reduction methods used in this study are the Rudzki inversion method, Helmert's second method of condensation, the residual terrain model (RTM) method, and the Pratt-Hayford (PH) topographic-isostatic reduction technique. The effect of using different DTM grid resolutions of 6", 15", 30", 45", I' and 2' on gravity anomalies and absolute geoid undulations is studied for each of these reduction schemes. A rugged area in the Canadian Rockies bounded by latitude between 49°N and 54°N and longitude between 236°E and 246°E is selected to conduct numerical tests. Our results suggest that a DTM grid resolution of 6" or higher is required for precise geoid determination with an accuracy of a decimetre or higher for any gravimetric reduction method chosen to treat the topographical masses above the geoid in rugged areas. The most precise geoid models obtained in this test are the ones obtained using Rudzki, Helmert, and RTM methods with 6" DTM resolution.

In this paper, it has been shown that the spectrum aliasing and folding effects occur only in the case of non-ideal signal sampling. When the duration of the signal sampling is equal to zero, these effects do not occur at all. In other words, the absolutely necessary condition for their occurrence is just a nonzero value of this time. Periodicity of the sampling process plays a secondary role.

A new model of ideal signal sampling operation is developed in this paper. This model does not use the Dirac comb in an analytical description of sampled signals in the continuous time domain. Instead, it utilizes functions of a continuous time variable, which are introduced in this paper: a basic Kronecker time function and a Kronecker comb (that exploits the first of them). But, a basic principle behind this model remains the same; that is it is also a multiplier which multiplies a signal of a continuous time by a comb. Using a concept of a signal object (or utilizing equivalent arguments) presented elsewhere, it has been possible to find a correct expression describing the spectrum of a sampled signal so modelled. Moreover, the analysis of this expression showed that aliases and folding effects cannot occur in the sampled signal spectrum, provided that the signal sampling is performed ideally.

In this paper, a new proof of ambiguity of the formula describing the aliasing and folding effects in spectra of sampled signals is presented. It uses the model of non-ideal sampling operation published by Vetterli et al. Here, their model is modified and its black-box equivalent form is achieved. It is shown that this modified model delivers the same output sequences but of different spectral properties. Finally, a remark on two possible understandings of the operation of non-ideal sampling is enclosed as well as fundamental errors that are made in perception and description of sampled signals are considered.

In this paper, the problem of aliasing and folding effects in spectrum of sampled signals in view of Information Theory is discussed. To this end, the information content of deterministic continuous time signals, which are continuous functions, is formulated first. Then, this notion is extended to the sampled versions of these signals. In connection with it, new signal objects that are partly functions but partly not are introduced. It is shown that they allow to interpret correctly what the Whittaker– Shannon reconstruction formula in fact does. With help of this tool, the spectrum of the sampled signal is correctly calculated. The result achieved demonstrates that no aliasing and folding effects occur in the latter. Finally, it is shown that a Banach–Tarski-like paradox can be observed on the occasion of signal sampling.