# Is There a Correlation Between Earthquake Magnitude and the Deformation Observed in the Rock?

I've been doing a little geology research. One of the topics that I noticed was the existence of "deformations" that appear in the rock layers if an earthquake has occurred. Smaller deformations (on average) imply a smaller earthquake magnitude; larger deformations (on average) imply a larger earthquake magnitude.

However, I have not been able to find a mathematical equation saying that a deformation of size X implies an earthquake magnitude of size Y, and vice versa.
Does such an equation exist?

Generalizing such relationships is, as suggested above, hard to impossible - but for well studied and understood systems such correlations have been observed. For instance there is a clear, and almost linear, relationship between earthquake magnitude and displacement on Mt Etna. Earlier work with a global focus has observed similar relationships - but with, as would be expected, much more variance between regions.

There is no mathematical equation relating geological deformation with earthquake magnitude.

Different faults move differently. Some move horizontally (strike-slip faults). Other move vertically (strike-dip faults), while others have a combination of the two (oblique slip faults).

The friction along the fault, and resistance to movement, is also a significant factor in how much a fault will move. Additionally the location of the earthquake and direction of the forces from the earthquake relative to the fault will affect the size of the movement along a fault.

The size of movement along a fault may or may not be uniform.

In addition to friction along a fault being a factor, the strength of protuberances along the fault will affect size of movement. Stronger protuberances can resist larger forces before they break. This can limit movement and it can determine the way a fault will move - smoothly or in an abrupt jerking manner.

You are probably thinking of pseudotachylite veins, which are interpreted as "fossil earthquakes" (among other things). To my knowledge, the first who tried to quantify some seismic parameters from pseudotachylite veins geometry was Sibson (1975). Notably, he determined a relationship between the vein thickness $$a$$ and the displacement (slip) along the fault $$d$$:

$$d = 436 a^2$$ (in cm)

Later on he also discussed shear stress estimates. But going all the way back from stress to magnitude (i.e., energy) does not seem trivial. You could dig in the literature citing Sibson, there have probably been some works trying to improve this in the last 40 years.

Edit $$-$$ I just found one of such studies: Ferrand et al. (2020). They found that pseudotachylite thickness scales with moment magnitude $$M_W$$:

Pseudotachylyte thickness $$a$$ (data) and dynamic molten zone width $$w$$ (calculations) vs. earthquake magnitude $$M_W$$.

### Surface Displacement vs. Earthquake Magntidue

This answer presumes that you are looking for a relationship between earthquake displacement and earthquake magnitude. I think what you are looking for the following:

From Wells and Coppersmith, 1994. This is a well cited, though still controversial reference. https://pubs.geoscienceworld.org/ssa/bssa/article/84/4/974/119792/New-empirical-relationships-among-magnitude

So yes, there is some relationship between displacement and earthquake magnitude, albeit with a lot of uncertainty.

## Some important notes and caveats

The word deformation carries a broader meaning than displacement. Deformation describes spatial gradients of displacement everywhere, not just on a fault. Displacement has a more specific definition: the relative motion measured in meters, for example, across a fault.

This relationship above is averaging displacement over the fault. So it may not be easy to reconstruct a magnitude from a single observation in the geological record.

It can be difficult or impossible to know from the geological record whether displacement occured in a single earthquake or many. Relatedly, not all fault displacement results in an earthquake.

The is some physical basis for these relationships related to the definition of earthquake moment in terms of fault displacement, area and shear modulus. You can refer to Kanamori and Anderson, 1975 for a very good (albeit detailed) explanation: https://pubs.geoscienceworld.org/ssa/bssa/article/84/4/974/119792/New-empirical-relationships-among-magnitude

No fixed equation, partly because no two faults will be the same but largely because faults don't necessarily move in the same direction during a given earthquake as it did last time. In New Zealand the last really large, magnitude 8.2, earthquake in Wellington, in 1885, raised the harbour seabed and lowered the southern end of the Ruahine Ranges, historically speaking quakes have usually done the opposite. You can make an approximation of the average rate and direction of movement along a particular fault but you even if you had records of the exact timing and magnitude of every earthquake on that fault, and no-one has been around long enough for that, you couldn't generalise enough to have such an equation.