What does the concave multifractal spectrum shape physically mean?

[Meryka]

TL;DR

I investigate the damage to the metal surface with box-counting method and some surface images give a convex multifractal spectrum f for a result - which is also widespread in the literature, but some other surface images form a concave multifractal spectrum. Not much information is available on this concave fractal spectrum - how could it be interpreted in physically?

I investigate the damage to the metal surface with box-counting method and some surface images give a convex multifractal spectrum f for a result - which is also widespread in the literature, but some other surface images form a concave multifractal spectrum. Not much information is available on this concave fractal spectrum - how could it be interpreted in physically?

 

[Meryka]

I just add a picture of one graph - the Hook on the left is not counted in.