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 based on a formula derived by Aki decades
ago, assuming a continuous exponential distribution. However, as the
magnitude is usually provided with a limited resolution, its
distribution is not continuous but discrete. In the literature this
problem was initially solved by an empirical correction (due to Utsu) to
the minimum magnitude, and later by providing an exact formula such as
that by Tinti and Mulargia, based on the geometric distribution theory.
A recent paper by van der Elst showed that the b-value can be estimated
also by considering the 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 i) the derivation of 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; and iii) the corresponding formulas for
the parameter b. We further deduce iv) the standard 1-sigma confidence
limits for the estimated b. Moreover, we are able v) to quantify the
error associated with the Utsu minimum-magnitude correction. We tested
extensively such formulas on simulated synthetic datasets including
complete catalogues as well as catalogues affected by a strong
incompleteness degree such as aftershock sequences where the
incompleteness is made to vary from one event to the next.

13 Apr 2023

16 Apr 2023