Gutenberg–Richter Frequency–Magnitude Relation

Plot the logarithm of how many earthquakes a region produces above each magnitude against magnitude, and you get a straight line. Beno Gutenberg and Charles Richter published this in 1944, and $\log_{10} N = a - bM$ has held up remarkably well for eight decades across every tectonic setting anyone has tested it in. It is worth being clear about what kind of statement it is: it is a curve fit to a catalogue, not a law derived from physics. Nobody has produced a first-principles derivation, and the constants are not constants of nature.

The b-value is the slope, and with $b \approx 1$ it says that each step up in magnitude makes an earthquake about ten times rarer. It is genuinely close to 1 nearly everywhere, which is one of the more striking regularities in geophysics, but "close to 1" covers a real range of roughly 0.6 to 1.5. Low b-values appear in stable continental interiors and on locked plate-boundary segments where stress accumulates without releasing; high b-values appear in volcanic and geothermal areas, in aftershock sequences, and in swarms. Borrowing a b-value from a well-studied region to use in a poorly studied one is a standard mistake in seismic hazard work, and it biases the answer in whichever direction the borrowed regime happens to differ.

The a-value is worse behaved, and readers should be suspicious of any a quoted without its context. It is the intercept — the log of the number of events above magnitude zero — and it depends entirely on the catalogue that produced it: how many years it covers, how large an area, and crucially the MAGNITUDE OF COMPLETENESS below which small events were simply not detected by the network of the day. Fitting a line through data below the completeness magnitude flattens the slope and corrupts both constants, which is why a proper fit is a maximum-likelihood estimate over the complete part of the catalogue only. Two a-values from different studies of the same fault are not comparable unless both windows are stated.

The straight line also has a ceiling that the equation knows nothing about. A region cannot produce an earthquake larger than its faults can host, so the real distribution rolls off at the top — modelled either as a truncated or a tapered Gutenberg–Richter distribution — and extrapolating the bare line to magnitude 9 in a region whose longest fault is 80 km long returns a number with no physical meaning. Read alongside the energy relation, the line also explains a nice tension: small earthquakes dominate the COUNT ten to one per magnitude unit, but large ones dominate the ENERGY thirty-two to one, so with $b = 1$ the energy budget of a region is set almost entirely by its rarest events.