Radiated Energy from Surface-Wave Magnitude

$\log_{10} E = 4.8 + 1.5 M_s$, with $E$ in joules, is Gutenberg and Richter's calibration of radiated seismic energy against surface-wave magnitude, published in the 1950s. It is the source of every "an earthquake released the energy of N atomic bombs" statement you have ever read, and it deserves to be used with more caution than it usually gets.

Start with what it does well. The slope of 1.5 is solid and is the structural fact worth memorising: a magnitude unit is $10^{1.5} \approx 31.6$ times the energy. It is also the slope that moment magnitude was constructed to preserve, which is why the two scales agree about ratios even where they disagree about absolute values.

Now the cautions, and they are substantial. This was calibrated with 1950s instruments against a magnitude scale that saturates near 8.3, and modern broadband estimates of radiated energy for the same event can differ from it by up to an order of magnitude. The reason is physical rather than instrumental: how much of an earthquake's energy actually radiates as waves depends on the stress drop and the rupture velocity, and both vary enormously between events. Two earthquakes of identical moment can radiate very different amounts of energy, which is exactly why the modern literature reports an independent energy magnitude $M_e$ rather than inferring energy from $M_w$.

Two more points of hygiene. The magnitude the relation wants is $M_s$; substituting $M_w$ is defensible between about 5 and 7.5 where the two scales track each other, and misleading outside that band. And the 4.8 assumes joules — the original was written in ergs, where the same relation reads $\log_{10} E = 11.8 + 1.5 M_s$, and $1\ \text{erg} = 10^{-7}\ \text{J}$. Finally, remember that this is RADIATED energy, the part that leaves the fault as waves. It is typically well under a tenth of the total energy released, with the rest going into fracturing rock and heating the fault plane — which is why the borehole temperature measurements after the Chi-Chi and Tohoku earthquakes were such a big deal.