Gutenberg–Richter law

Remarks

This is an observed regularity, not a theorem. It holds over many orders of magnitude of $N$, with departures at the largest magnitudes because of small samples and finite fault size. Equivalently, the magnitudes above $M_c$ follow an exponential distribution with rate $\beta = b \ln 10$: the probability that $M \ge m$ is $10^{-b(m - M_c)}$. Its maximum likelihood estimate from magnitudes $M_1, \dots, M_k$ is $\hat b = \log_{10} e / (\bar M - M_c)$, where $\bar M$ is their mean. Example: with $b = 1$, each step of one unit in magnitude reduces the number of earthquakes tenfold. A region with $1000$ events of magnitude at least $3$ per year has about $10$ of magnitude at least $5$. Since $M_w = \frac23(\log_{10} M_0 - 9.1)$, the seismic moments then follow a Pareto distribution: $N(M_0) \propto M_0^{-2b/3}$.

Going deeper

For a student

In the language of probability, magnitudes are exponentially distributed and moments are power-law distributed, which makes earthquake size scale-invariant. The $b$-value is estimated from catalogues and is a basic input of probabilistic seismic hazard analysis. Its variations in space and time are studied as possible signs of changes in stress. Combined with Omori's law for aftershocks, it forms the core of statistical earthquake models such as the ETAS model.