A technical note on the bias in the estimation of the b-value and its uncertainty through the Least Squares technique

L. Sandri, W. Marzocchi

Abstract


We investigate conceptually,
analytically, and numerically the biases in the estimation of the
b-value of the Gutenberg-Richter law and of
its uncertainty made through the least
squares technique. The biases are introduced by the cumulation
operation for the cumulative form of the Gutenberg-Richter law, by
the logarithmic transformation, and by the measurement errors on
the magnitude.
We find that the least squares technique,
applied to the cumulative and binned form of
the Gutenberg-Richter law, produces strong bias in the b-value and
its uncertainty, whose amplitudes depend on the size of the sample.
Furthermore, the logarithmic transformation produces two different
endemic bends in the Log(N) versus M curve.
This means that this plot
might produce fake significant departures from the
Gutenberg-Richter law.
The effect of the measurement errors is negligible compared to those of
cumulation operation and logarithmic transformation.
The results obtained show that the least squares technique should
never be used to determine the slope of the Gutenberg-Richter law and
its uncertainty.

Keywords


b-value;Seismology;Least Squares technique;Synthetic-earthquake catalogs

Full Text:

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References


DOI: https://doi.org/10.4401/ag-4432
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Published by INGV, Istituto Nazionale di Geofisica e Vulcanologia - ISSN: 2037-416X