Statistical Estimation of the Maximum Magnitude and its Uncertainty from a Catalogue Including Magnitude Errors

Statistical methods of the maximum magnitude (M_max) estimation are based usually on the assumption that the earthquake magnitudes are known exactly. Actually there are errors in earthquake magnitudes of modern and historic events. Taking into account the standard deviations of errors in earthquake magnitudes results in modified distribution of observed magnitudes. The new magnitude distribution slightly differs from the Gutenberg-Richter’s one for the large magnitudes and can explain non linear character of observed magnitude-frequency curves.On a basis of the new distribution the formulas to obtain M_max confidence limits for large samples are derived. Numerical method for calculating exact M_max confidence limits for arbitrary sample size is also proposed.Maximum likelihood estimates of M_max based on the new distribution are compared with the common estimates of maximum magnitude equal to the maximum of observed values. On the example of artificially generated catalogues the behaviour of the estimates for the different sample sizes and different levels of the magnitude errors is analysed. It is shown that the uncertainty in the M_max is usually much higher then the errors in initial magnitudes in catalogue.As an example the M_max and its uncertainty was estimated for Caucasus region. We show that b value estimated with and without considering magnitude uncertainty are almost the same.