# How to calculate cumulative moment magnitude?

What is the proper way to calculate cumuluative moment magnitude if I know the moment magnitude of several earthquakes?

• I've no experience in calculating cumulative magnitudes, but if the question is just about how to add exponentials it would be: $log_{10}(10^{M_1}+10^{M_2} ... 10^{M_n})$. – Tactopoda May 20 '17 at 14:06

## 1 Answer

The moment magnitude ${\displaystyle M_{\mathrm {w} }}$ is a dimensionless number defined by Hiroo Kanamori as:

$$M_{\mathrm {w} }={\frac {2}{3}}\log _{10}(M_{0})-10.7$$

To sum several moment magnitudes we need to first convert them to ${\displaystyle M_{\mathrm {0} }}$ (seismic moment) which you can simply sum and then convert it back

1. for each of the magnitudes you have - insert the magnitude into ${\displaystyle M_{\mathrm {w} }}$, solve for ${\displaystyle M_{\mathrm {0} }}$ and write down the answer.

2. Sum all the answers

3. Insert the sum back into ${\displaystyle M_{\mathrm {0} }}$ and solve for ${\displaystyle M_{\mathrm {w} }}$