arXiv subjects

G. Molchan

Publications and source records attributed to G. Molchan.

24 records · Page 2Linked to original sources

arXiv2006

Seismic Interevent Time: A Spatial Scaling and Multifractality

The optimal scaling problem for the time t(LxL) between two successive events in a seismogenic cell of size L is considered. The quantity t(LxL) is defined for a random cell of a grid covering a seismic region G. We solve that problem in terms of a multifractal characteristic of epicenters in G known as the tau-function or generalized fractal dimensions; the solution depends on the type of cell randomization. Our theoretical deductions are corroborated by California seismicity with magnitude M>2. In other words, the population of waiting time distributions for L = 10-100 km provides positive information on the multifractal nature of seismicity, which impedes the population to be converted into a unified law by scaling. This study is a follow-up of our analysis of power/unified laws for seismicity (see PAGEOPH 162 (2005), 1135 and GJI 162 (2005), 899).

physics.geo-ph↗

arXiv2006

On the Spatial Scaling of Seismicity Rate

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 is some evidence in favor 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 c's are related to different types of spatial averaging applied to N(m,L) and what are the c's for which the distributions of a(L) best agree for small L. Theoretical analysis is tested using California data.

physics.geo-ph↗

arXiv2004

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.

physics.geo-ph↗

arXiv2004

Interevent time distribution in seismicity: a theoretical approach

This paper presents an analysis of the distribution of the time $τ$ between two consecutive events in a stationary point process. The study is motivated by the discovery of a unified scaling law for $τ$ for the case of seismic events. It is shown that the unified law must necessarily be exponential. We discuss the parameterization of the empirical unified law and the physical meaning of the parameters involved.

physics.geo-ph↗

arXiv2003

Unilateral Small Deviations for the Integral of Fractional Brownian Motion

We consider the paths of a Gaussian random process $x(t)$, $x(0)=0$ not exceeding a fixed positive level over a large time interval $(0,T)$, $T\gg 1$. The probability $p(T)$ of such event is frequently a regularly varying function at $\infty$ with exponent $θ$. In applications this parameter can provide information on fractal properties of processes that are subordinate to $x(\cdot)$. For this reason the estimation of $θ$ is an important theoretical problem. Here, we consider the process $x(t)$ whose derivative is fractional Brownian motion with self-similarity parameter $0<H<1$. For this case we produce new computational evidence in favor of the relations $\log p(T)=-θ\log T(1+o(1))$ and $θ=H(1-H)$. The estimates of $θ$ are to within 0.01 in the range $0.1\le H\le 0.9$. An analytical result for the problem in hand is known for the markovian case alone, i.e., for $H=1/2$. We point out other statistics of $x(t)$ whose small values have probabilities of the same order as $p(T)$ in the $\log$ scale.

math.PR↗

arXiv2003

On the uniqueness of the branching parameter for a random cascade measure

An independent random cascade measure is specified by a random generator, a vector of dimension c with non-negative components. The dimension c is called the branching cascade parameter. It is shown under certain restrictions that, if this measure has two generators with a.s. positive components, and the ratio ln c_1/ln c_2 for their branching parameters is an irrational number, then this measure is a Lebesgue measure. In other words, when c is a power of an integer number p and the p is minimal for c, then a cascade measure that has the property of intermittency specifies p uniquely.

math.PR↗