The estimation of b-value of the frequency-magnitude distribution and of
its confidence intervals from binned magnitude data

The estimation of the slope (*b*-value) of the frequency magnitude
distribution of earthquakes is usually based on a formula derived
decades ago under the hypothesis of continuous exponential distribution
of magnitudes. However, as the magnitude is provided with a limited
resolution (one decimal digit usually), its distribution is not
continuous but discrete. In the literature this problem is solved mostly
by applying an empirical correction to the minimum magnitude of the
dataset depending on the binning size, but a recent paper recalled that
this solution is only approximate and proposed an exact formula. The
same paper further showed that the *b*-value can be estimated also
by considering the positive magnitude differences (which are proven to
follow an exponential discrete Laplace distribution) and that in this
case the estimator is more resilient to the incompleteness of the
magnitude dataset. In this work we provide the complete theoretical
formulation including the derivation of i) the means and standard
deviations of the discrete exponential and Laplace distributions; ii)
the estimators of the decay parameter of the discrete exponential and
trimmed Laplace distributions by the methods of the mean as well as of
the maximum likelihood; and iii) the corresponding formulas for the
parameter *b*. We further deduce iv) the standard confidence limits
for the estimated *b*. Moreover, we are able v) to quantify the
error associated with the formula including the Utsu minimum-magnitude
correction. We tested such formulas on simulated synthetic datasets
including cases with a certain amount of incompleteness.

27 Sep 2023