On the Frequency-magnitude Law for Fractal Seismicity

Scaling analysis of seismicity in the space-time-magnitude domain very often starts from the relation N(m,L)=a(L)*10**(-bm)*L**c for the rate of seismic events of magnitude M>m in an area of size L. There are some evidences in favor of multifractal property of seismic process. In this case the choice of the scale exponent 'c' is not unique. It is shown how different 'c''s are related to different types of spatial averaging applied to lambda(m, L) and what are the 'c''s for which the distributions of a(L) best agree for small L. Theoretical analysis is supplemented with an analysis of California data for which the above issues were recently discussed on an empirical level.