The magnitude scale is based on horizontal displacement, \[M_L=\log_{10}\left(\frac{A}{A_0}\right) \text{, where } A_0=\quantity{1}{\mu m} .\]
But the destructive power of an earthquake is better indicated by the amount of energy dissipated. It has been suggested that the energy, \(E\), is related to the displacement, \(A\), by a simple power law, \[E=E_0 \left(\frac{A}{A_0}\right)^p\] where \(E_0\) and \(p\) are constants.
Can you use the following data from the British Geological Survey and the US Geological Survey to verify this suggestion and find values for \(E_0\) and \(p\)?
| Earthquake | Magnitude, \(M_L\) | Energy, \(\quantity{E}{(GJ)}\) |
|---|---|---|
| San Francisco, USA, 1906 | \(7.8\) | \(6\times10^7\) |
| Chile, 1960 | \(9.5\) | \(1\times10^{10}\) |
| Kobe, Japan, 1995 | \(6.9\) | \(1\times10^6\) |
| Kent, 2007 | \(4.3\) | \(200\) |
| Lincolnshire, 2008 | \(5.2\) | \(3000\) |
Estimate the energy dissipation of the 2011 New Zealand earthquake, magnitude \(6.1\).
