# Calculating the displacement of a fault

In the calculation of scalar moment magnitude of an earthquake we have the formula

$$M_0=\mu AD$$

where:

• $\mu$ is the shear modulus of the rocks involved in the earthquake (in Pa)
• $A$ is the area of the rupture along the geologic fault where the earthquake occurred (in m2), and
• $D$ is the average displacement on $A$ (in m).

In many cases the displacement can occur in the subsurface.In order to predict the moment magnitude of such earthquakes the value of $D$ must be estimated. How is this process carried out ?

## 1 Answer

The process is carried out by solving an "inverse problem" and there are many ways to estimate the moment depending on the observable. For example if you have some measurements of ground deformation following the earthquake (using GPS/InSAR) then by combing a physics based model, with an optimizer, you can estimate the area/slip distribution that best explains the observations. Once you know the area/slip distribution then you can estimate the moment.

The physical models can be simple or complex (e.g., semi-analytical or fully numerical). Same goes for the optimizer (local or global). For computationally expensive numerical models global optimization is typically not feasible. For local optimization you need a 'reasonable' initial model.

• my doubt was that in cases where the faulting has occoured underground then how can we estimate the amount of displacement in the fault or the total area of rupture.without the knowledge of these parameters how can magnitude be determined ? – shrey Jan 27 '15 at 10:53
• No matter where the earthquake occurs it will still result in some surface deformation (geodetic or seismic). How else would you know that the earthquake even occurred? – stali Jan 27 '15 at 12:23