On the spatial scaling of seismicity rate

Abstract

S U M M A R YScaling analysis of seismicity in the space-time-magnitude domain very often starts from the relation λ(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 is some evidence in favour of multifractality being present in seismicity. In this case, the optimal choice of the scale exponent c is not unique. It is shown how different values of c are related to different types of spatial averaging applied to λ(m, L) and what are the values of c for which the distributions of a L best agree for small L. Theoretical analysis is tested using the California data.

Seismic Interevent Time: A Spatial Scaling and Multifractality

“…The utmost to be derived from observable data is a statement like this: seismicity looks as a multifractal in a range of scale ∆L. We have inferred this for M ≥2 California events in the range ∆L=10−100 km; a similar inference for M≥3 seems questionable (Molchan and Kronrod, 2005).…”

“…Values of practical interest for τ (p) and Table 1. Since (6) holds in the range ∆L and τ(p), 0≤p≤3 is nonlinear, we infer that the rate of M≥2 events in G looks as a multifractal in the scale range 10-100 km (Molchan and Kronrod, 2005 (p=0) is not considered because of the great effect of half-empty L×L cells for which the distributions of t(L×L) are poorly determined.…”

“…Tau-function for M ≥ 2 California events (solid line) and its bi-fractal approximation (broken line); τ (p) is based on the range of scales L=10-100 km. Molchan and Kronrod (2005) estimated the function τ (p), 0≤ p ≤ 3 and its derivatives from M ≥ 2 California seismicity for the scale range 10−100 km (Fig. 1).…”

“…However, treating the laws as approximate ones, one can look for the optimal form of scaling for the statistic ξ(L×L) that aspires to be the unified law. This problem has been solved for ξ =λ(L×L) by Molchan and Kronrod (2005). We use the multifractal formalism to find theoretically the index d f in (1) for small L. It is a function of the weights involved in the contributions of different L×L cells.…”

The fractal description of seismicity

“…Actually, however, the situation with the estimation of generalized dimensions is not as dramatic, provided one deals accurately enough with problems and above. Molchan & Kronrod (2005, 2007) used a multifractal analysis of southern California seismicity for optimal scaling of two random quantities: the interevent time and the number of events in a randomly selected cell Δ L of size L . An independent verification has confirmed the predicted scale exponents for the range of scales 10–100 km; thereby, we got an indirect test of d q and a useful quantitative consequence of the multifractal theory.…”