Effects of magnitude uncertainties on estimating the parameters in the Gutenberg-Richter frequency-magnitude law

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Abstract
The evaluation of the parameters in the Gutenberg-Richter (GR) frequency-magnitude law log Nr = a − bm is shown to be strongly affected by magnitude uncertainties. If the magnitude errors are assumed to be distributed normally with standard deviation σ, the observed magnitude that we call the “apparent magnitude,” becomes a random variable, and the frequency-apparent magnitude law differs from the GR relation. We show that there is a range of magnitudes within which this law may be approximated as log Na = a − bm + γ2log(e). Here, γ2 = β2σ2/2, [β = b/log(e)], and Na stands for the apparent number of the earthquakes. As both the true and the apparent magnitude curves have the same slope on a logarithmic graph, the estimators usually employed for β remain valid. However, the usual estimators for a are biased by a quantity which is a quadratic function of the error standard deviation σ and of β. This implies that the apparent number, Na, of earthquakes exceeding a given magnitude tends to be larger than the real number Nr and for realistic values of a and b (or β), Na is expected to be even as large as twice Nr. The implications of the magnitude variability on the seismic risk analysis are also considered: in particular, the evaluations of the attenuation law parameters and of the exceedance probability of the ground motion peak acceleration are shown to be affected.