# How to compute the energy released by an earthquake.

To determine the total seismic energy radiated from an earthquake one would have to integrate the energy radiated at all frequencies over the entire focal sphere. The spectrum of the average radiation over the focal sphere can be approximated by a constant level at low frequencies (which is proportional to the moment, Mo) and a uniform decrease with increasing frequency above some corner frequency (Fc), so the seismic energy is a function of both Mo and Fc. For a given moment, the radiated energy will increase as Fc increases. Consider, for example, two earthquakes with the same displacement and rupture area that occur within rocks with the same shear modulus. They would have the same moment, which can be computed from:

```
Mo = u D A
where:
u = shear modulus (3 - 6 x 1011) dyn/cm2
D = average displacement
A = area of rupture
```
If one event were a "slow" earthquake with "more or less creep-like deformation" (Kanamori, H., 1972, Mechanism of Tusnami Earthquakes, Phys. Earth Planet. Interiors, v6, p. 346-359) while the other had a more typical rupture velocity near the shear wave velocity, much more energy would be radiated from the latter earthquake due to its rich high frequency radiation corresponding larger Fc than from the "slow" event.

Having said this, however, if only an earthquake's moment is known the radiated seismic energy can still be approximated because, if a large set of earthquakes is considered, the average corner frequency varies systematically with the moment. For the average earthquake, the seismic wave energy (E), moment (Mo) and moment magnitude (MW) are related by the following equations (Kanamori, H., 1977, The Energy Release in Great Earthquakes, Journal of Geophysical Research, v82, p. 2981- 2987):

```     E = Mo/(2 x 104) erg   (1 erg = 1 dyn cm)
log E = 1.5 MW + 11.8 (Gutenberg-Richter magnitude-energy relation)

Then:

log Mo - log(2 x 104) = 1.5 MW + 11.8
Mw = (log Mo - 16.1) / 1.5
```
The energy released by TNT (trinitrotoluene) and the TNT equivalent of the Hiroshima nuclear bomb (McGraw-Hill Encyclopedia of Science and Technology, 1992):
```     Energy per ton of TNT     = 4.18 x 109 Joules
= 4.18 x 1016 ergs
Energy per megaton of TNT = 4.18 x 1015 Joules
```

According to the Sandia National Laboratories' web site, the energy equivalent of the Hiroshima fission bomb was 15,000 tons of TNT.

Example -- consider an earthquake with moment magnitude Mw = 4.0

The total seismic energy radiated from the source, E(4), would be:

E(4) = 10**(1.5*4 + 11.8) = 10**17.8 ergs = 10**10.8 Joules = 6.3 x 1011 Joules

The moment, Mo(4), would be:

Mo(4) = E x (2 x 104) = 1.26 x 1016 Joules

It has been found that a 1 kton explosion will generate seismic waves approximately equivalent to a magnitude 4 earthquake.  Therefore, the amount of energy dissipated by TNT to yield seismic waves similar to a magnitude 4 is:

Energy of TNT(4) = 4.18 x 1012 Joules

Handy for units conversion:  http://www.thomasglobal.com/tools/