Multifractal of spatial distribution of seismicity in Liaoning area

Making use of multifractal theory and corresponding computational method and according to the feature of evolution of spatial distribution with respect to seismicity by earthquake data in Liaoning area, earthquake activity of the area has been studied in detail. The results show that the evolution of increase in seismicity and distributive process in space are a multifractal structure. Whole characteristic of evolution in fractal increasing process of seismicity is described by obvious variation in regard toτ(q)-q curve,f(α) spectrum and other parameters before and after moderate and strong earthquakes.     

Multifractal and long memory of humidity process in the Tarim River Basin

Based on the daily data of relative humidity from 23 meteorological stations in the Tarim River Basin of northwest China during the period from 1961 to 2010, this paper analyzed the multifractal and long memory property of humidity process. Main findings are as follows: (1) The processes present scaling and multifractal property. (2) The left-skewed multifractal spectrum f(α) indicates that the time series of relative humidity is predominated by small fluctuations. (3) There exists long memory with the δ ∈ (0, 0.5) in the processes, except for Kalpin and Aksu’s exhibiting non-stationary long memory with the parameter δ being 0.67 and 0.69 respectively. (4) We found that on the whole, the degree of multifractality exhibits a strengthening trend with the longitude and latitude increasing, but decreasing trend with elevation rising; For length of long memory, we investigated that on the whole, the δ values increased with the longitude and latitude increasing, which indicates that the bigger the longitude and latitude is, the longer the memory of humidity process is, but the higher the elevation is, the shorter the memory of humidity process is.     

Seismic Interevent Time: A Spatial Scaling and Multifractality

The optimal scaling problem for the time t(L × L) between two successive events in a seismogenic cell of size L is considered. The quantity t(L × L) 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 Pure and Applied Geophysics 162 (2005), 1135 and GJI 162 (2005), 899).     

Global and Local Multiscale Analysis of Magnetic Susceptibility Data

Geophysical well-logs often show a complex behavior which seems to suggest a multifractal nature. Multifractals are highly intermittent signals, with distinct active bursts and passive regions which cannot be satisfactorily characterized in terms of just second-order statistics. They need a higher-order statistical analysis. In contrast with monofractals which have a homogeneous scaling, multifractals may include singularities of many types. Here we describe how a multiscale analysis can be used to describe the magnetic susceptibility data scaling properties for a deep well (KTB, Germany), down to about 9000 m. A multiscale analysis describes the local and global singular behavior of measures or distributions in a statistical fashion. The global analysis allows the estimation of the global repartition of the various Holder exponents. As such, it leads to the definition of a spectrum, D(), called the singularity spectrum. The local analysis is related to the possibility of estimating the Lipschitz regularity locally, i.e., at each point of the support of a multifractal signal. The application of both approaches to the KTB magnetic susceptibility data shows a meaningful correlation between the sequence of Holder exponents vs. depth and the lithological units. The Holder exponents reach the highest values for gneiss units, intermediate ones for amphibolite units and the lowest values for variegated units. Faults are found to correspond to changes for H also when they are of intra-lithological type.     

Scaling and correlation of atmospheric acid deposition evolutions

Detrended fluctuation analysis (DFA) and power spectrum density are applied to verify the presence of temporal scaling behavior and long-range persistence (LRP) of weekly hydrogen ion deposition (WHD), NO₃^- and SO₄^2- deposition series in National Atmospheric Deposition Program, USA, for the period 1978–2001. The results show a common scaling behavior for all sites analyzed. Two distinct scaling regions are identified by DFA1, one corresponding to 1 month to 1 year and the other to 1 year to 5 years. The WHD series obey power-law in two temporal regimes respectively with mean DFA1 scaling exponents α ₁≈0.68 and α ₂≈0.45, implying the presence of LRP in the acid deposition series and there is a tendency for a large deposition event to be followed by another large event, and vice versa. For DFA2–DFA4, however, the annual crossover, which divides the temporal scale into two regimes, disappears gradually with the order q of DFAq increasing, and the two scaling regimes turn to share the same scaling exponent close to α ₁. The result indicates that the scaling behavior exits in the two regimes with the same scaling exponent α ₁, and LRP prevails during the examined 1-month to 5-years scale. NO₃^- and SO₄^2- deposition evolve the same way as WHD does, implying the pollutants involving in acid deposition may share some prominent mechanism controlling their evolutions. We ascribe the long-range power-law scaling of acid deposition evolutions to the self-organized critical behavior of atmosphere under pollution stress and it should be considered in the trend prediction of acid deposition as an important factor.     

Selecting the best IDF model by using the multifractal approach

In this work, the multifractal properties of hourly rainfall data recorded at a location in Southern Spain have been related to the scale properties of the corresponding intensity–duration–frequency (IDF) curves. Four parametric models for the IDF curves have been fitted to the quantiles of rainfall obtained using the generalized Pareto frequency distribution function with the extreme data series obtained for the same place. The scaling of the rainfall intensity moments has been analysed, and the empirical moments scaling exponent function has been obtained. The corresponding values of q₁ and γ₁ have been empirical and theoretically calculated and compared with some characteristics of the different IDF models. Thus, the scaling behaviour of IDF curves has been analysed, and the best model has been selected. Copyright © 2012 John Wiley & Sons, Ltd.     

Multifractal scaling of the kinetic energy flux in solar wind turbulence

The geometrical and scaling properties of the energy flux of the turbulent kinetic energy in the solar wind have been studied. Using present experimental technology in solar wind measurements we cannot directly measure the real volumetric dissipation rate, <varepsilon>(t), but are constrained to represent it by its surrogate the energy flux near the dissipation range at the proton gyro scale. There is evidence for the multifractal nature of the so defined dissipation field <varepsilon>(t), a result derived from the scaling exponents of its statistical moments. The generalized dimension D _q has been determined and reveals that the dissipation field has a multifractal structure, which is not compatible with a scale-invariant cascade. The related multifractal spectrum f(<alpha>) has been estimated for the first time for MHD turbulence in the solar wind. Its features resemble those obtained for turbulent fluids and other nonlinear multifractal systems. The generalized dimension D _q can for turbulence in high-speed streams be fitted well by the functional dependence of the p-model with a comparatively large parameter p ₁=0.87, indicating a strongly intermittent multifractal energy cascade. The experimental value for D _p/3 used in the scaling exponent s(p) of the velocity structure function gives an exponent that can describe some of the observations. The scaling exponent <mu> of the autocorrelation function of <varepsilon>(t) has also been directly evaluated, being 0.37. Finally, the mean dissipation rate was determined, which could be used in solar wind heating models.     

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 unified scaling laws for τ for the case of seismic events. We demonstrate that these laws cannot exist simultaneously in a seismogenic area. Under very natural assumptions we show that if, after rescaling to ensure Eτ =1, the interevent time has a universal distribution F, then F must be exponential. In other words, Corral’s unified scaling law cannot exist in the whole range of time. In the framework of a general cluster model we discuss the parameterization of an empirical unified law and the physical meaning of the parameters involved. An erratum to this article is available at .     

Multifractal behaviour of long‐term karstic discharge fluctuations

Karstic watersheds are highly complex hydrogeological systems that are characterized by a multiscale behaviour corresponding to the different pathways of water in these systems. The main issue of karstic spring discharge fluctuations consists in the presence and the identification of characteristic time scales in the discharge time series. To identify and characterize these dynamics, we acquired, for many years at the outlet of two karstic watersheds in South of France, discharge data at 3‐mn, 30‐mn and daily sampling rate. These hydrological records constitute to our knowledge the longest uninterrupted discharge time series available at these sampling rates. The analysis of the hydrological records at different levels of detail leads to a natural scale analysis of these time series in a multifractal framework. From a universal class of multifractal models based on cascade multiplicative processes, the time series first highlights two cut‐off scales around 1 and 16 h that correspond to distinct responses of the aquifer drainage system. Then we provide estimates of the multifractal parameters α and C₁ and the moment of divergence q_D corresponding to the behaviour of karstic systems. These results constitute the first estimates of the multifractal characteristics of karstic spingflows based on 10 years of high‐resolution discharge time series and should lead to several improvements in rainfall‐karstic springflow simulation models. Copyright © 2012 John Wiley & Sons, Ltd.     

Energetics of the magnetospheric superstorm on November 20, 2003

The magnetospheric storm on November 20, 2003 was one of two greatest events in 1957–2003. The D _st^* index reached −472 nT, the polar cap potential drop exceeded 200 kV, the polar cap boundary expanded up to Φ = 60°, the plasma layer density in the synchronous orbit reached 5 cm⁻³, and the inner edge of the plasma sheet penetrated up to L ∼ 1.5R _E. The sequence of disturbance modes including some previously unknown is described. The distribution of the total power input into the magnetosphere between the ionosphere (power Q _i) and the ring current (Q _DR), as well as the relative roles of the spontaneous substorms and the driven disturbances in the creation of the DR current, is analyzed. The values of the parameter α = Q _DR/Q _i are calculated with a step of 5 min. It is shown that intervals with α ≪ 1 and with maximums α ≫ 1 were observed in the events under consideration. These results contradict the dominant opinion that the energy input into the magnetosphere during disturbances is primarily dissipated in the ionosphere. The two types of α maximums are observed: one in the mode of a prevailing spontaneous substorm and the other in the mixed mode of the substorm and driven disturbance. It is concluded that both types of the maximums and corresponding enhancements of the DR current appeared due to the plasma turbulization processes in the substorm current wedge. The parameter α appears to slowly increase from α ≪ 1 to α > 1 with increasing activity level; this trend supports the driven model of creating the DR current due to an increase in the electric field of the solar wind.     

Multifractal modelling and spectrum analysis: Methods and applications to gamma ray spectrometer data from southwestern Nova Scotia, Canada

Multifractal theory was developed for handling scale invariant fields instead of geometry only[1―4]. From a multifractal point of view, some fractal models, ordinary physical processes and relevant probability distribution types can be considered as special cases of multifractal models which provides new insight into the interrelationships between systems and subjects. For example, the low order moment exponents τ (0), τ (1), τ (2) or τ ″(1) obtained by means of the moment method determi…     

Numerical investigation of apparent multifractality of samples from processes subordinated to truncated fBm

We investigate numerically apparent multi‐fractal behavior of samples from synthetically generated processes subordinated to truncated fractional Brownian motion (tfBm) on finite domains. We are motivated by the recognition that many earth and environmental (including hydrological) variables appear to be self‐affine (monofractal) or multifractal with Gaussian or heavy‐tailed distributions. The literature considers self‐affine and multifractal types of scaling to be fundamentally different, the first arising from additive and the second from multiplicative random fields or processes. It has been demonstrated theoretically by one of us that square or absolute increments of samples from Gaussian/Lévy processes subordinated to tfBm exhibit apparent/spurious multifractality at intermediate ranges of separation lags, with breakdown in power‐law scaling at small and large lags as is commonly exhibited by real data. A preliminary numerical demonstration of apparent multifractality by the same author was limited to Gaussian fields having nearest neighbor autocorrelations and led to rather noisy results. Here, we adopt a new generation scheme that allows us to investigate apparent multifractal behaviors of samples taken from a broad range of processes including Gaussian with and without symmetric Lévy and log‐normal (as well as potentially other) subordinators. Our results shed new light on the nature of apparent multifractality, which has wide implications vis‐a‐vis the scaling of many hydrological as well as other earth and environmental variables. Copyright © 2011 John Wiley & Sons, Ltd.     

Estimating design probable maximum precipitation using multifractal methods and comparison with statistical and synoptically methods case study: Basin of Bakhtiari Dam

The probable maximum precipitation which is defined as the maximum precipitation at a particular location for a given duration is used as a design criterion for major dams. The assumptions of deterministic consideration and an upper limit to probable maximum precipitation have been repeatedly criticized by hydrologists. Nowadays, multifractal method which strongly contains physical bases can be used to improve the probable maximum precipitation. In this research, the universal multifractal model was used to estimate the design probable maximum precipitation for specified exceedence probability in basin of Bakhtiari Dam, southwest Iran, and its results were compared with statistical and synoptically methods. The results revealed that the return period of statistical and synoptically probable maximum precipitation, estimated for the different durations, are about 10⁹ and 10³–10⁴ years, respectively; also, over periods ranging from 1 to 7 days, the ratios of design probable maximum precipitations, estimated based on multifractal method for return period of 10³–10⁹ years, to statistical and synoptically probable maximum precipitation estimates ranged from 0.61 to 1.1 and 1.33 to 2.37, respectively. These results indicated that the multifractal method can be used to reasonably estimate the probable maximum precipitation.     

Multifractal comparison of the outputs of two optical disdrometers

ABSTRACT

In this paper a universal multifractals comparison of the outputs of two types of collocated optical disdrometers installed on the roof of the Ecole des Ponts ParisTech is performed. A Campbell Scientific PWS100 which analyses the light scattered by the hydrometeors and an OTT Parsivel² which analyses the portion of occluded light are deployed. Both devices provide a binned distribution of drops according to their size and velocity. Various fields are studied across a range of scales: rain rate (R), liquid water content (ρ), polarimetric weather radar quantities such the horizontal reflectivity (Z_h) and the specific differential phase (K_dp), and drop size distribution (DSD) parameters such as the total drop concentration (N_t) and the mass-weighted diameter (D_m). For both devices, good scaling is retrieved on the whole range of available scales (2?h–30?s), except for the DSD parameters for which the scaling only holds down to few minutes. For R, the universal multifractal parameters are found to equal 1.5 and 0.2 for α and C₁, respectively. Results are interpreted with the help of the classical Z_h–R and R–K_dp radar relations.

Editor D. Koutsoyiannis; Associate editor E. Volpi     

A multifractal formalism for vector-valued random fields based on wavelet analysis: application to turbulent velocity and vorticity 3D numerical data

Extreme atmospheric events are intimately related to the statistics of atmospheric turbulent velocities. These, in turn, exhibit multifractal scaling, which is determining the nature of the asymptotic behavior of velocities, and whose parameter evaluation is therefore of great interest currently. We combine singular value decomposition techniques and wavelet transform analysis to generalize the multifractal formalism to vector-valued random fields. The so-called Tensorial Wavelet Transform Modulus Maxima (TWTMM) method is calibrated on synthetic self-similar 2D vector-valued multifractal measures and monofractal 3D vector-valued fractional Brownian fields. We report the results of some application of the TWTMM method to turbulent velocity and vorticity fields generated by direct numerical simulations of the incompressible Navier–Stokes equations. This study reveals the existence of an intimate relationship between the singularity spectra of these two vector fields which are found significantly more intermittent than previously estimated from longitudinal and transverse velocity increment statistics.

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Apparent/spurious multifractality of data sampled from fractional Brownian/Lévy motions

Many earth and environmental variables appear to be self‐affine (monofractal) or multifractal with spatial (or temporal) increments having exceedance probability tails that decay as powers of − α where 1 < α ≤ 2. The literature considers self‐affine and multifractal modes of scaling to be fundamentally different, the first arising from additive and the second from multiplicative random fields or processes. We demonstrate theoretically that data having finite support, sampled across a finite domain from one or several realizations of an additive Gaussian field constituting fractional Brownian motion (fBm) characterized by α = 2, give rise to positive square (or absolute) increments which behave as if the field was multifractal when in fact it is monofractal. Sampling such data from additive fractional Lévy motions (fLm) with 1 < α < 2 causes them to exhibit spurious multifractality. Deviations from apparent multifractal behaviour at small and large lags are due to nonzero data support and finite domain size, unrelated to noise or undersampling (the causes cited for such deviations in the literature). Our analysis is based on a formal decomposition of anisotropic fLm (fBm when α = 2) into a continuous hierarchy of statistically independent and homogeneous random fields, or modes, which captures the above behaviour in terms of only E + 3 parameters where E is Euclidean dimension. Although the decomposition is consistent with a hydrologic rationale proposed by Neuman (2003), its mathematical validity is independent of such a rationale. Our results suggest that it may be worth checking how closely would variables considered in the literature to be multifractal (e.g. experimental and simulated turbulent velocities, some simulated porous flow velocities, landscape elevations, rain intensities, river network area and width functions, river flow series, soil water storage and physical properties) fit the simpler monofractal model considered in this paper (such an effort would require paying close attention to the support and sampling window scales of the data). Parsimony would suggest associating variables found to fit both models equally well with the latter. Copyright © 2010 John Wiley & Sons, Ltd.     

Identification of trends in Malaysian monthly runoff under the scaling hypothesis

Abstract

Statistical tests have been widely used for several decades to identify and test the significance of trends in runoff and other hydrological data. The Mann-Kendall (M-K) trend test is commonly used in trend analysis. The M-K test was originally proposed for random data. Several variations of the M-K test, as well as pre-processing of data for use with it, have been developed and used. The M-K test under the scaling hypothesis has been developed recently. The basic objective of the research presented in this paper is to investigate the trends in Malaysian monthly runoff data. Identification of trends in runoff data is useful for planning water resources projects. Existence of statistically significant trends would also lead to identification of possible effects of climate change. Monthly runoff data for Malaysian rivers from the past three decades are analysed, in both five-year segments and entire data sequences. The five-year segments are analysed to investigate the variability in trends from one segment to another in three steps: (1) the M-K tests are conducted under random and correlation assumptions; (2) the Hurst scaling parameter is estimated and tested for significance; and (3) the M-K test under the scaling hypothesis is conducted. Thus the tests cover both correlation and scaling. The results show that the number of significant segments in Malaysian runoff data would be the same as those found under the assumption that the river flow sequences are random. The results are also the same for entire sequences. Thus, monthly Malaysian runoff data do not have statistically significant trends. Hence there are no indications of climate change in Malaysian runoff data.

Citation Rao, A. R., Azli, M. & Pae, L. J. (2011) Identification of trends in Malaysian monthly runoff under the scaling hypothesis. Hydrol. Sci. J. 56(6), 917–929.     

On the Occurence of Extreme Events in Long-term Correlated and Multifractal Data Sets

We review recent studies of the statistics of return intervals (i) in long-term correlated monofractal records and (ii) in multifractal records in the absence (or presence) of linear long-term correlations. We show that for the monofractal records which are long-term power-law correlated with exponent γ, the distribution density of the return intervals follows a stretched exponential with the same exponent γ and the return intervals are long-term correlated, again with the same exponent γ. For the multifractal record, significant differences in scaling behavior both in the distribuiton and correlation behavior of return intervals between large events of different magnitudes are demonstrated. In the absence of linear long-term correlations, the nonlinear correlations contribute strongly to the statistics of the return intervals such that the return intervals become long-term correlated even though the original data are linearly uncorrelated (i.e., the autocorrelation function vanishes). The distribution density of the return intervals is mainly described by a power law.