Chi-square regression for seismic strength parameter relations, and their uncertainties, with applications to an M-w based earthquake catalogue for central, northern and northwestern Europe

A generalized chi-square regression approach to establish empirical relations between different types of seismic strength parameters with uncertainties in all input data is presented and discussed in comparison with standard least-squares techniques. The chi-square technique can consider errors of individual entries but can also be applied when errors are not exactly known and only weaker quantitative constraints can be made. It can preserve the symmetry of the derived relations and is preferred for complex regression models. Results for three types of regression models are presented for (1) a linear relation between M-S and m(b) for events in the North Atlantic Ocean; (2) a quadratic relation between M-w and M-L for events in central Europe; (3) linear relations between M-L and I-0, with logarithmic dependency of the focal depth, for several regions in central and northern Europe.