A technical note on the bias in the estimation of the b-value and its uncertainty through the Least Squares technique
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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.
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.
Article Details
How to Cite
Sandri, L. and Marzocchi, W. (2007) “A technical note on the bias in the estimation of the b-value and its uncertainty through the Least Squares technique”, Annals of Geophysics, 50(3), pp. 329–339. doi: 10.4401/ag-4422.
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