Comparative Evaluation of Statistical and Fractal Approaches for JRC Calculation Based on a Large Dataset of Natural Rock Traces

After the publication of the type-profiles for the estimation of the joint roughness coefficient (JRC) a discussion evolved about how to adequately use these traces. Based on the chart numerous researchers assembled mathematical correlations with various parameters seeking objectivity in the determination of JRC. Within these works differences concerning the database and the mathematical implementations exist. Consequently, each correlation, although predominantly the same parameters are used, leads to different JRC values. In theory, for any arbitrary profile, irrespective of the particular calculation approach, the same JRC should result. This is a requisite because of the referencing of all correlations to the 10 type-profiles. However, it is shown in this study that in most cases equal or even satisfactorily similar results are not obtained. The discrepancies are vast when non-standard profiles are evaluated, in this case, more than 40,000 traces from six different rock surfaces that cover a broad range of roughness categories. The simple intuitive parameter Z2 served as an agent for the statistical methods because of its broad use and consequently good comparability. On the part of the fractal approaches, three definitions were used. However, JRC inferred from fractal correlations are very much dependent on the particular calculation routine. In fact, the theory of fractals is overly complex for the sparse and low-resolution type-profiles. In summary, fractal approaches do not produce safer or more reliable estimates of roughness compared to simple statistical means and using Z2 perfectly suffices to determine the class of JRC.

Reflection and transmission of high-frequency acoustic, electromagnetic and elastic waves at a distinguished class of irregular, curved boundaries

Reflection and transmission phenomena associated with high-frequency linear wave incidence on irregular boundaries between adjacent acoustic or electromagnetic media, or upon the irregular free surface of a semi-infinite elastic solid, are studied in two dimensions. Here, an ‘irregular’ boundary is one for which small-scale undulations of an arbitrary profile are superimposed upon an underlying, smooth curve (which also has an arbitrary profile), with the length scale of the perturbation being prescribed in terms of a certain inverse power of the large wave-number of the incoming wave field. Whether or not the incident field has planar or cylindrical wave-fronts, the associated phase in both cases is linear in the wave-number, but the presence of the boundary irregularity implies the necessity of extra terms, involving fractional powers of the wave-number in the phase of the reflected and transmitted fields. It turns out that there is a unique perturbation scaling for which precisely one extra term in the phase is needed and hence for which a description in terms of a Friedlander–Keller ray expansion in the form as originally presented is appropriate, and these define a ‘distinguished’ class of perturbed boundaries and are the subject of the current paper.