^aSection Mathematics, University of Halle, Halle G.D.R.

The “u-net” is composed of a grid subdivided into degrees or degree-grid (“Gradnetz”) and an equal-area-grid (“flächengleiches Netz”). Having replaced the direction σ of strike by its normal δ, the cracks are registered into the meshes of the degree-grid and are there counted. Then they are transmitted into the equal-area-grid according to the percent values of the u-table. From the pattern of frequency numbers (“Häufigkeitszahlen”) in the equal-area-grid the sphere of crack locations is obtained.

Particular specifications regulate the procedure for special measure values, i.e., those of the integral multiples of 5°, especially the angles of inclination τ = 0° and τ = 90° (see 1.4). With greater inaccuracies in measurements, one changes, by means of a given table (see Table V), to a degree-grid of 10°. With very small inaccuracies, on the other hand, the procedure may be simplified, the degree-grid becoming unnecessary (see 1.5). The meshes near the centre, being too long, may be avoided by an additional circle (“Zusatzkreis”—see 1.3).

The “u-net” was constructed in such a way that the spheres of crack locations report the real frequency distribution at all times, free from systematic errors. This is achieved by the method that all calculations follow the principle of area equality or area proportionality on the hemisphere (see 2.1). The procedure using the “u-net” can be adapted to differential accuracies of measuring. It is especially suitable for large numbers of cracks; it is simple in calculation and may easily be programmed for digital computers. Thus the “u-net” is advantageous for all applications in which a large number of cracks has to be dealt with. Such applications are very frequent in rock mechanics, in engineering geology for the purpose of foundation of large hydraulic buildings (dams, caverns), in petrography, tectonics and in geophysical investigations such as the determination of crack structures with a view to explaining micromagnetic occurrences, for instance. Furthermore, the “u-net” is applicable not only to crack statistics but also to other similar statistical methods, e.g., to the statistics of cristal axes or to geographic frequency analyses.