G. Molchan - Academia.edu (original) (raw)

Papers by G. Molchan

Communications in Mathematical Physics, 1999

Let b (t), b (0) = 0 be fractional Brownian motion, i.e., a Gaussian process with the structure f... more Let b (t), b (0) = 0 be fractional Brownian motion, i.e., a Gaussian process with the structure function Ejb (t)?b (s)j 2 = jt?sj , 0 < < 2. We study the logarithmic asymptotics of P T = P fb (t) < 1; t 2 T g as T ! 1, where is either the interval (0; 1) or a bounded region that contains a vicinity of 0 for the case of multidimensional time. It is shown that log P T = ?D log T (1 + o(1)), where D is the dimension of zeroes of b (t) in the former case and the dimension of time in the latter.

$Research paper thumbnail of Scaling exponents and multifractal dimensions for independent random cascades$

Communications in Mathematical Physics, 1996

This paper is concerned with Mandelbrot's stochastic cascade measures. The problems of (i) scalin... more This paper is concerned with Mandelbrot's stochastic cascade measures. The problems of (i) scaling exponents of structure functions of the measure, τ(#), and (ii) multifractal dimensions are considered for cascades with a generator vector (wι w c ) of the general type. These problems were previously studied for independent strongly bounded variables w l : 0 < a < w/ ^ c. Consequently, a broad class of models used in applications, including Kolmogorov's log-normal model in turbulence, log-stable "universal" cascades in atmospheric dynamics, has not been covered. Roughly speaking, problems (i), (ii) are here solved under the condition that the scaling exists; the τ-function is calculated for all arguments (previously this was done for positive q) and a new effect emerges: the τ-function can generally involve discontinuities in the first derivative as well as in the second.

Geophysical Journal International, 2015

Recently A. Gusev suggested and numerically investigated the doubly stochastic earthquake source ... more Recently A. Gusev suggested and numerically investigated the doubly stochastic earthquake source model. The model is supposed to demonstrate the following features in the far-field body waves: 1) the omegasquare high-frequency (HF) behavior of displacement spectra; 2) lack of the directivity effect in HF radiation; and 3) a stochastic nature of the HF signal component. The model involves two stochastic elements: the local stress drop (SD) on a fault and the rupture time function (RT) with a linear dominant component. The goal of the present study is to investigate the Gusev model theoretically and to find conditions for (1, 2) to be valid and stable relative to receiver site. The models with smooth elements SD, RT are insufficient for these purposes. Therefore SD and RT are treated as realizations of stochastic fields of the fractal type. The local smoothness of such fields is characterized by the fractional (Hurst) exponent H, 0 < H < 1. This allows us to consider a wide class of stochastic functions without regard to their global spectral properties. We show that the omega-square behavior of the model is achieved approximately if the rupture time function is almost regular (H~1) while the stress drop is rough function of any index H. However, if the rupture front is linear, the local stress drop has to be function of minimal smoothness (H~0). The situation with the directivity effect is more complicated: for different RT models with the same fractal index, the effect may or may not occur. The nature of the phenomenon is purely analytical. The main controlling factor for the directivity is the degree of smoothness of the two dimensional distributions of RT random function.

Journal of Statistical Physics - J STATIST PHYS, 2002

Mandelbrot cascade measures were introduced to explain intermittency in fully developed turbulenc... more Mandelbrot cascade measures were introduced to explain intermittency in fully developed turbulence. They are defined by the scale hierarchy with a fixed branching parameter c and by the distribution of breakdown coefficients which are responsible for the transport of energy from larger to smaller scales. We show that the measures corresponding to both conservative and nonconservative cascades strongly depend on the parameter c. In particular, only Lebesgue measure can be generated by a cascade process with an arbitrary integer c.

$Research paper thumbnail of Anomalies in Multifractal Formalism for Local Time of Brownian Motion$

Journal of Statistical Physics - J STATIST PHYS, 1998

The Renyi function for the logical time measure µ of Brownian motion is found. It is shown that t... more The Renyi function for the logical time measure µ of Brownian motion is found. It is shown that this function, the Legendre transform of the multifractal spectrum of µ and the t-function derived by the reciprocal measure formalism are not identical. More examples of µ having similar anomalies are discussed.

Pure and Applied Geophysics PAGEOPH, 1997

... S(u, i) = min [c~j + fljA - :gruA + O(S(u + A,j)(1 -- ruA) + S(O,j)ruA)]. J ... FENG, DE YI, ... more ... S(u, i) = min [c~j + fljA - :gruA + O(S(u + A,j)(1 -- ruA) + S(O,j)ruA)]. J ... FENG, DE YI, JXNG PING Gu, MIN~ ZHOU LIN, SHAO XIE Xu, and HuE JUN Yu (1985), Assessment of Earthquake Hazard by Simultaneous Use of the Statistical Method and the Method of Fuzzy Mathemat-ics ...

Pure and Applied Geophysics, 2010

The quality of earthquake prediction is usually characterized by a two-dimensional diagram n vers... more The quality of earthquake prediction is usually characterized by a two-dimensional diagram n versus τ, where n is the rate of failures-to-predict and τ is a characteristic of space–time alarm. Unlike the time prediction case, the quantity τ is not defined uniquely. We start from the case in which τ is a vector with components related to the local alarm times and find a simple structure of the space–time diagram in terms of local time diagrams. This key result is used to analyze the usual 2-d error sets {n, τ w } in which τ w is a weighted mean of the τ components and w is the weight vector. We suggest a simple algorithm to find the (n, τ w ) representation of all random guess strategies, the set D, and prove that there exists the unique case of w when D degenerates to the diagonal n + τ w = 1. We find also a confidence zone of D on the (n, τ w ) plane when the local target rates are known roughly. These facts are important for correct interpretation of (n, τ w ) diagrams when we discuss the prediction capability of the data or prediction methods.

$Research paper thumbnail of Multifractal analysis of Brownian zero set$

Journal of Statistical Physics, 1995

ABSTRACT The multifractal structure of zerosZ of Brownian motion is considered. For different mea... more ABSTRACT The multifractal structure of zerosZ of Brownian motion is considered. For different measures onZ we find typical characteristics: the -function and the multifractal spectrumf(). A dimensional interpretation off() is also discussed.

Journal of Statistical Physics, 1997

ABSTRACT The statistical properties of solutions of the one-dimensional Burgers equation in the l... more ABSTRACT The statistical properties of solutions of the one-dimensional Burgers equation in the limit of vanishing viscosity are considered when the initial velocity potential is fractional Brownian motion (FBM). We establish the asymptotic power-law order for log-probability of large values, both velocity and shock (amplitude of velocity discontinuity). This confirms the conjecture of U. Frisch and his collaborators. Rigorous results for this problem were previously derived for the case of Brownian motion using Markov techniques. Our approach is based on the intrinsic properties of FBM and the theory of extreme values for Gaussian processes. Key WordsBurgers equation–fractional Brownian motion–tail probabilities

Journal of Statistical Physics, 2000

An independent random cascade measure µ is specified by a random generator (w 1 , ..., w c ), E w... more An independent random cascade measure µ is specified by a random generator (w 1 , ..., w c ), E w i = 1 where c is the branching parameter. It is shown under certain restrictions that, if µ 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 µ 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.

The quality of space-time earthquake prediction is usually characterized by a two-dimensional err... more The quality of space-time earthquake prediction is usually characterized by a two-dimensional error diagram (n,tau), where 'n' is the rate of failures-to-predict and 'tau' is the normalized measure of space-time alarm. The most interesting space measure for analysis of a prediction strategy is the rate of target events m(dg) in a sub-area 'dg'. In this case the quantity H=1-(n+tau) determines

Scaling analysis of seismicity in the space-time-magnitude domain very often starts from the rela... more 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.

The number of successes 'n' and the normalized measure of space-time alarm 'tau' ... more The number of successes 'n' and the normalized measure of space-time alarm 'tau' are commonly used to characterize the strength of an earthquake prediction method and the significance of prediction results. To evaluate better the forecaster's skill, it has been recently suggested to use a new characteristic, the gambling score R, which incorporates the difficulty of guessing each target event by using different weights for different alarms. We expand the class of R-characteristics and apply these to the analysis of results of the M8 prediction algorithm. We show that the level of significance 'alfa' strongly depends (1) on the choice of weighting alarm parameters, (2) on the partitioning of the entire alarm volume into component parts, and (3) on the accuracy of the spatial rate of target events, m(dg). These tools are at the disposal of the researcher and can affect the significance estimate in either direction. All the R-statistics discussed here corrob...

One of the most consequential issues of the earthquake prediction problem is reliable testing of ... more One of the most consequential issues of the earthquake prediction problem is reliable testing of hypothetical prediction methods. The danger of self-deception by data overfitting here is especially high due to both the scarceness of large earthquakes and the absence of a conventional wide-reaching theoretical framework. This talk gives an overview of the methods currently employed to test prediction algorithms and bridges the commonly accepted approaches to the problem. The main focus is on the two most widely used approaches to assessing prediction methods. Both of them evaluate the amount of new information revealed by the prediction method about the impending earthquake activity. The first one starts by estimating the expected spatio-temporal distribution of seismicity, and uses the classical likelihood paradigm to evaluate the prediction power. Accordingly, it uses the nomenclature of statistical estimation. The second one applies results of G. Molchan [Pure Appl. Geophys., 149:...

Pure and Applied Geophysics, 2002

Ð We present an attempt to compare modeled ground-motion acceleration ®elds with macroseismic obs... more Ð We present an attempt to compare modeled ground-motion acceleration ®elds with macroseismic observations. Two techniques for the representation of the observed intensities by isoseismals, a smoothing technique and one which visualizes the local uncertainty of an isoseismal, are tested with synthetic and observed data. We show how noise in the data and irregularities in the distribution of observation sites aect the resolution of the isoseismal's shape. In addition to``standard'' elongated shapes, we identify cross-like patterns in the macroseismic observations for two Italian earthquakes of strike-slip type; similar patterns are displayed by the theoretical peak acceleration ®elds calculated assuming the point source models given in the literature.

$Research paper thumbnail of Seismic Interevent Time: A Spatial Scaling and Multifractality$

Pure and Applied Geophysics, 2007

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Pure and Applied Geophysics, 2011

The site effect is usually associated with local geological conditions, which increase or decreas... more The site effect is usually associated with local geological conditions, which increase or decrease the level of shaking compared with standard attenuation relations. We made an attempt to see in the macroseismic data of Italy some other effects, namely, hot/cold spots in the terminology of Olsen , which are related to local fault geometry rather than to soil conditions. We give a list of towns and villages liable to amplify (+) or to reduce (-) the level of shaking in comparison with the nearby settlements. Relief and soil conditions cannot always account for the anomalous sites. Further, there are sites where both (+) and (-) effects are observed depending on the earthquake. The opposite effects can be generated by events from the same seismotectonic zone and along the same direction to the site. Anomalous sites may group themselves into clusters of different scales. All isolated anomalous patterns presented in this paper can be used in hazard analysis, in particular, for the modeling and testing of seismic effects.

Physics of the Earth and Planetary Interiors, 1990

ABSTRACT The premonitory seismicity pattern described as a ‘burst of aftershocks’ is analysed in ... more ABSTRACT The premonitory seismicity pattern described as a ‘burst of aftershocks’ is analysed in 13 regions of the world. Its high statistical significance is ascertained. The mathematical approach developed here may be useful in the analysis of other earthquake precursors.

Physics of the Earth and Planetary Interiors, 1999

ABSTRACT