It is studied whether the one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) tensor invariants really behave like invariants in the field that is whether their values are independent from the position of the current electrodes of the tensorial geoelectric configuration and what kind of results they can produce in numerical modelling and in field situations. It was shown that: 1. the invariants are "less and less" invariants with their increasing dimension number depending more and more on the position of the current electrodes. 2. They produced smaller and smaller anomalies both in their size and amplitude making the detection and interpretation of the anomalies more and more difficult. 3a. The 1D image produced for all models stable, reliable results which are ideal for creating a starting model. 3b. In spite of the uncertainty of the 2D data they improved the quality of the fault field image which has been received using the 1D data only. In the sites with building remnants and furnace however the 2D invariant was not able to give extra information to that obtained by the 1D invariant. 3c. Although the interpretation of the 3D results may be rather complicated it proved to be more useful than that of the 2D data both in the building remnants and furnace field studies. In special cases the 3D invariant may refine the 1D image. In summary the 2D invariant which is sensible to the two-dimensional changes of the subsurface (like that of the in the field practice most often used 2D ERT configuration) and which was expected to produce the best results proved to be almost the less useful in these investigations in spite of that the investigated models were more 2D/3D than 1D. Because even for such models the 1D invariant produced the best results its application is recommended. Regarding however that the 2D and 3D invariants may refine the 1D image even if their results are more uncertain joint interpretation of all dimensional invariants could also be worthwhile. Although the refined model is more risky it can be very useful e.g. in studies where the danger factor is high, e.g. because of filtrating of dangerous fluid or fissuring on the wall of a nuclear waste deposit. In such cases it is better to warn redundantly than eventually not recognize real danger. The results of these investigations should be taken into account in every research area, where tensorial measurements could be carried out, e.g. in magnetotelluric research.