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 establishempirical relations between different types of seismic strengthparameters with uncertainties in all input data is presented anddiscussed in comparison with standard least-squares techniques.The chi-square technique can consider errors of individual entriesbut can also be applied when errors are not exactly known and onlyweaker quantitative constraints can be made. It can preserve thesymmetry of the derived relations and is preferred for complexregression models. Results for three types of regression modelsare presented for (1) a linear relation between M _Sand m _bfor events in the North Atlantic Ocean; (2) a quadratic relationbetween M _w and M _L forevents in central Europe; (3) linearrelations between M _L and I ₀,with logarithmic dependency ofthe focal depth, for several regions in central and northernEurope.