Performance of different regression procedures on the magnitude conversion problem

Through an extensive set of simulations we investigate the performance of different linear regression procedures commonly used to convert magnitudes from one type into another one, an operation that also has strong influence on the slope of the frequency-magnitude (the b-value of the Gutenberg-Richter) distribution. It has already been demonstrated that a general orthogonal regression provides the most reliable results. However, questions arise when the ratio between the variances of the magnitudes to be related (the knowledge of which is required to apply the general orthogonal regression) cannot be computed. We therefore systematically investigate the biases introduced by the classical standard least-squares regressions and the orthogonal regressions (or similar procedures) as a function of the true slope between magnitudes, of the ratio eta between magnitude variances, and of the absolute variances of magnitudes. We compute such biases through,h simulations very close to the real cases inferred from the German and Chinese broadband networks. We observe that for 0.7 < root eta(true) < 1.8 the orthogonal regression under the eta = 1 assumption performs better than standard regressions. For values outside this interval neither procedure is capable of correct estimates. Therefore it is recommended to estimate the absolute errors and their ratio from empirical data and apply the general orthogonal regression. This requires that a seismological data center publish average estimates of event magnitudes and also their related standard deviations. Regrettably, this is not yet a common practice, thus impeding the derivation of optimal magnitude conversion relations.