Correcting the Gutenberg–Richter b-value for effects of rounding and noise

Abstract

The effects of magnitude rounding and of the presence of noise in the rounded magnitudes on the estimation of the Gutenberg-Richter b-value are explored, and the ways to correct for these effects are proposed. For typical values, b = 1 and rounding interval DM = 0.1, the rounding error is approximately-10-3 and it can be corrected to a negligible approximately-10-5. For the same typical values, the effect of noise can be larger, depending on the characteristics of the noise distribution; for normally distributed noise with standard deviation r = 0.1, the correct b-value may be underestimated by a factor *0.97. Keywords b-Value Á Noise Á Magnitude 2 Correcting the effects of rounding In what follows, we present our own version of the effects of rounding, in order to introduce the notation that will be used in proposing a correction for them.

Illuminating Faulting Complexity of the 2017 Yellowstone Maple Creek Earthquake Swarm

“…To calculate a Gutenberg-Richter b value, we used a cutoff magnitude of −0.5, somewhat more conservative than the suggestion to use a magnitude 0.2 units higher than the maximum frequency magnitude bin (Wiemer & Wyss, 2000;Woessner & Wiemer, 2005). This provided nearly 10,000 total events above the cutoff magnitude for the swarm, with an overall b value of 0.74 as determined by maximum likelihood (Tinti & Mulargia, 1987; Figure S3). However, as noted in section 2, we are using an M L -like magnitude scale at small magnitudes, with magnitude scaling as~1.0*log moment, rather than ⅔*log (moment) as with M w (Deichmann, 2017;Hanks & Kanamori, 1979).…”

Estimation of Earthquake Hazard Parameters from Incomplete Data Files. Part III. Incorporation of Uncertainty of Earthquake‐Occurrence Model

“…The methodology implemented here will refer only to the Soft Bound Model. Despite the complexity of implementing this model, it is undoubtedly the preferred model to account for magnitude uncertainty (Tinti and Mulargia, 1985;Rhoades, 1996;Rhoades and Dowrick, 2000;Marzocchi and Sandri, 2003;Márquez-Ramírez et al, 2015).…”

“…The KS-II procedure can also account for uncertainties in determining seismic-event magnitudes by considering the assumption that the observed magnitude is the true magnitude, subject to a random error. It is assumed that this random error follows a Gaussian distribution, having a zero mean and a known standard deviation (Tinti and Mulargia, 1985;McGuire, 2004;Márquez-Ramírez et al, 2015).…”

Short-term volcano-tectonic earthquake forecasts based on a moving mean recurrence time algorithm: the El Hierro seismo-volcanic crisis experience

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“…We picture the situation at El Hierro as voluminous bodies of dense magma intruding into the mantle under the crust from a deeper magma generation region (Martínez-Arevalo et al, 2013). That magma is dense enough to remain in almost isostatic equilibrium at the base of the crust in a process of magmatic underplating (Leahy et al, 2010).…”