New seismic attribute:Fractal scaling exponent based on gray detrended fluctuation analysis

Seismic attributes have been widely used in oil and gas exploration and development. However, owing to the complexity of seismic wave propagation in subsurface media, the limitations of the seismic data acquisition system, and noise interference, seismic attributes for seismic data interpretation have uncertainties. Especially, the antinoise ability of seismic attributes directly affects the reliability of seismic interpretations. Gray system theory is used in time series to minimize data randomness and increase data regularity. Detrended fluctuation analysis （DFA） can effectively reduce extrinsic data tendencies. In this study, by combining gray system theory and DFA, we propose a new method called gray detrended fluctuation analysis （GDFA） for calculating the fractal scaling exponent. We consider nonlinear time series generated by the Weierstrass function and add random noise to actual seismic data. Moreover, we discuss the antinoise ability of the fractal scaling exponent based on GDFA. The results suggest that the fractal scaling exponent calculated using the proposed method has good antinoise ability. We apply the proposed method to 3D poststack migration seismic data from southern China and compare fractal scaling exponents calculated using DFA and GDFA. The results suggest that the use of the GDFA-calculated fractal scaling exponent as a seismic attribute can match the known distribution of sedimentary facies.

Detrended analysis of Reynolds stress in a decaying turbulent flow in a wind tunnel with active grids

Multi-scale properties of Reynolds stress in decaying turbulence in a wind tunnel with high Reynolds number are investi-gated. Two filtering techniques i.e., the zeroth-order and first-order detrending methods are applied to the two velocity components, where the local mean value （resp. local linear trend） is removed in the former （latter） technique. Some basic statistics for thirty mea-surements show that the variation is very large at first two locations and relatively small at last two locations. Moderately good power law is found for the mean value of local Reynolds stress at last three measurement locations with scaling exponents approxi-mately being 1.0 and a dual power law exists for the mean value of standard deviation of local Reynolds stress at all four measureme-nt locations with scaling exponents being 0.53 and 0.58 for zeroth-and first-order filtering respectively. Present results about local Reynolds stress are useful to build and evaluate the model of sub-grid Reynolds stress in large eddy simulations.

Detrended cross-correlation analysis of electroencephalogram

In the paper we use detrended cross-correlation analysis （DCCA） to study the electroencephalograms of healthy young subjects and healthy old subjects. It is found that the cross-correlation between different leads of a healthy young subject is larger than that of a healthy old subject. It was shown that the cross-correlation relationship decreases with the aging process and the phenomenon can help to diagnose whether the subject＇s brain function is healthy or not.

Detrended Fluctuation Analysis on Correlations of Complex Networks Under Attack and Repair Strategy

We analyze the correlation properties of the Erd6s-Rdnyi random graph （RG） and the Barabdsi-Albert scale-free network （SF） under the attack and repair strategy with detrended fluctuation analysis （DFA）. The maximum degree kmax, representing the local property of the system, shows similar scaling behaviors for random graphs and scale-free networks. The fluctuations are quite random at short time scales but display strong anticorrelation at longer time scales under the same system size N and different repair probability pre. The average degree 〈k〉, revealing the statistical property of the system, exhibits completely different scaling behaviors for random graphs and scale-free networks. Random graphs display long-range power-law correlations. Scale-free networks are uncorrelated at short time scales; while anticorrelated at longer time scales and the anticorrelation becoming stronger with the increase of pre.

Relationships of exponents in multifractal detrended fluctuation analysis and conventional multifractal analysis

Multifractal detrended fluctuation analysis （MF-DFA） is a relatively new method of multifractal analysis. It is extended from detrended fluctuation analysis （DFA）, which was developed for detecting the long-range correlation and the fractal properties in stationary and non-stationary time series. Although MF-DFA has become a widely used method, some relationships among the exponents established in the original paper seem to be incorrect under the general situation. In this paper, we theoretically and experimentally demonstrate the invalidity of the expression r（q） = qh（q） - 1 stipulating the relationship between the multifractal exponent T（q） and the generalized Hurst exponent h（q）. As a replacement, a general relationship is established on the basis of the universal multifractal formalism for the stationary series as .t-（q） = qh（q） - qH - 1, where H is the nonconservation parameter in the universal multifractal formalism. The singular spectra, a and f（a）, are also derived according to this new relationship.

Correlation between detrended fluctuation analysis and the Lempel-Ziv complexity in nonlinear time series analysis

We study the correlation between detrended fluctuation analysis（DFA） and the Lempel-Ziv complexity（LZC） in nonlinear time series analysis in this paper.Typical dynamic systems including a logistic map and a Duffing model are investigated.Moreover,the influence of Gaussian random noise on both the DFA and LZC are analyzed.The results show a high correlation between the DFA and LZC,which can quantify the non-stationarity and the nonlinearity of the time series,respectively.With the enhancement of the random component,the exponent α and the normalized complexity index C show increasing trends.In addition,C is found to be more sensitive to the fluctuation in the nonlinear time series than α.Finally,the correlation between the DFA and LZC is applied to the extraction of vibration signals for a reciprocating compressor gas valve,and an effective fault diagnosis result is obtained.

A Multifractal Detrended Fluctuation Analysis of the Ising Financial Markets Model with Small World Topology

We present a multifractal detrended fluctuation analysis （MFDFA） of the time series of return generated by our recently-proposed Ising financial market model with underlying small world topology. The result of the MFDFA shows that there exists obvious multifractal scaling behavior in produced time series. We compare the MFDFA results for original time series with those for shuffled series, and find that its multifractal nature is due to two factors： broadness of probability density function of the series and different correlations in small- and large-scale fluctuations. This may provide new insight to the problem of the origin of multifractality in financial time series.

Detrended Fluctuation Analysis of the Human EEG during Listening to Emotional Music

A nonlinear method named detrended fluctuation analysis （DFA） was utilized to investigate the scaling behavior of the human electroencephalogram （EEG） in three emotional music conditions （fear, happiness, sadness） and a rest condition （eyes-closed）. The results showed that the EEG exhibited scaling behavior in two regions with two scaling exponents β1 and β2 which represented the complexity of higher and lower frequency activity besides α band respectively. As the emotional intensity decreased the value of β1 increased and the value of β2 decreased. The change of β1 was weakly correlated with the ＇approach-withdrawal＇ model of emotion and both of fear and sad music made certain differences compared with the eyes-closed rest condition. The study shows that music is a powerful elicitor of emotion and that using nonlinear method can potentially contribute to the investigation of emotion.

Correlation analysis between the Aral Sea shrinkage and the Amu Darya River

The shrinkage of the Aral Sea,which is closely related to the Amu Darya River,strongly affects the sustainability of the local natural ecosystem,agricultural production,and human well-being.In this study,we used the Bayesian Estimator of Abrupt change,Seasonal change,and Trend(BEAST)model to detect the historical change points in the variation of the Aral Sea and the Amu Darya River and analyse the causes of the Aral Sea shrinkage during the 1950–2016 period.Further,we applied multifractal detrend cross-correlation analysis(MF-DCCA)and quantitative analysis to investigate the responses of the Aral Sea to the runoff in the Amu Darya River,which is the main source of recharge to the Aral Sea.Our results showed that two significant trend change points in the water volume change of the Aral Sea occurred,in 1961 and 1974.Before 1961,the water volume in the Aral Sea was stable,after which it began to shrink,with a shrinkage rate fluctuating around 15.21 km3/a.After 1974,the water volume of the Aral Sea decreased substantially at a rate of up to 48.97 km3/a,which was the highest value recorded in this study.In addition,although the response of the Aral Sea's water volume to its recharge runoff demonstrated a complex non-linear relationship,the replenishment of the Aral Sea by the runoff in the lower reaches of the Amu Darya River was identified as the dominant factor affecting the Aral Sea shrinkage.Based on the scenario analyses,we concluded that it is possible to slow down the retreat of the Aral Sea and restore its ecosystem by increasing the efficiency of agricultural water use,decreasing agricultural water use in the middle and lower reaches,reducing ineffective evaporation from reservoirs and wetlands,and increasing the water coming from the lower reaches of the Amu Darya River to the 1961–1973 level.These measures would maintain and stabilise the water area and water volume of the Aral Sea in a state of ecological restoration.Therefore,this study focuses on how human consumption of recharge runoff affects the Aral Sea and provides scientific perspective on its ecological conservation and sustainable development.

Crossover Phenomena in Detrended Fluctuation Analysis Used in Financial Markets

A systematic analysis of Shanghai and Japan stock indices for the period of Jan. 1984 to Dec. 2005 is performed. After stationarity is verified by ADF （Augmented Dickey-Fuller） test, the power spectrum of the data exhibits a power law decay as a whole characterized by 1/f^β processes with possible long range correlations. Subsequently, by using the method of detrended fluctuation analysis （DFA） of the general volatility in the stock markets, we find that the long-range correlations are occurred among the return series and the crossover phenomena exhibit in the results obviously.Further, Shanghai stock market shows long-range correlations in short time scale and shows short-range correlations in long time scale. Whereas, for Japan stock market, the data behaves oppositely absolutely. Last, we compare the varying of scale exponent in large volatility between two stock markets. All results obtained may indicate the possibility of characteristic of multifractal scaling behavior of the financial markets.

Effects of quantization on detrended fluctuation analysis

Detrended fluctuation analysis （DFA） is a method foro estimating the long-range power-law correlation exponent in noisy signals. It has been used successfully in many different fields, especially in the research of physiological signals. As an inherent part of these studies, quantization of continuous signals is inevitable. In addition, coarse-graining, to transfer original signals into symbol series in symbolic dynamic analysis, can also be considered as a quantization-like operation. Therefore, it is worth considering whether the quantization of signal has any effect on the result of DFA and if so, how large the effect will be. In this paper we study how the quantized degrees for three types of noise series （anti-correlated, uncorrelated and long-range power-law correlated signals） affect the results of DFA and find that their effects are completely different. The conclusion has an essential value in choosing the resolution of data acquisition instrument and in the processing of coarse-graining of signals.

Detrended Fluctuation Analysis of Heart Rate and SaO_2 in Hypoxia

Detrended fluctuation analysis （DFA） is fit for studies on the long-range exponential correlation of non-stationary time serial. In this paper, in order to find a hy- poxia adaptability evaluation criterion, the heart rate and SaO2 signals are analyzed by this method. The demarcate exponent about fit-good-group and fit-bad-group in hy- poxia and normal air are calculated and compared. The result shows a is different in different situation, the α in hypoxia is much higher than α of breath in normal air. And α of fit-good-group is higher than fit-bad-group. It shows that DFA could be a good criterion to analyze hypoxia adaptability, which is useful in the analysis of hypoxia phys- iology signal.

Temporal-spatial cross-correlation analysis of non-stationary near-surface wind speed time series

Temporal-spatial cross-correlation analysis of non-stationary wind speed time series plays a crucial role in wind field reconstruction as well as in wind pattern recognition.Firstly,the near-surface wind speed time series recorded at different locations are studied using the detrended fluctuation analysis(DFA),and the corresponding scaling exponents are larger than 1.This indicates that all these wind speed time series have non-stationary characteristics.Secondly,concerning this special feature( i.e.,non-stationarity)of wind signals,a cross-correlation analysis method,namely detrended cross-correlation analysis(DCCA) coefficient,is employed to evaluate the temporal-spatial cross-correlations between non-stationary time series of different anemometer pairs.Finally,experiments on ten wind speed data synchronously collected by the ten anemometers with equidistant arrangement illustrate that the method of DCCA cross-correlation coefficient can accurately analyze full-scale temporal-spatial cross-correlation between non-stationary time series and also can easily identify the seasonal component,while three traditional cross-correlation techniques(i.e.,Pearson coefficient,cross-correlation function,and DCCA method) cannot give us these information directly.

Multifractal Analysis of Human Heartbeat in Sleep

We study the dynamical properties of heart rate variability （HRV） in sleep by analysing the scaling behaviour with the multifractal detrended fluctuation analysis method. It is well known that heart rate is regulated by the interaction of two branches of the autonomic nervous system： the parasympathetic and sympathetic nervous systems. By investigating the multifractal properties of light, deep, rapid-eye-movement （REM） sleep and wake stages, we firstly find an increasing multifractal behaviour during REM sleep which may be caused by augmented sympathetic activities relative to non-REM sleep. In addition, the investigation of long-range correlations of HRV in sleep with second order detrended fluctuation analysis presents irregular phenomena. These findings may be helpful to understand the underlying regulating mechanism of heart rate by autonomic nervous system during wake-sleep transitions.

Long-range correlation analysis of urban traffic data

This paper investigates urban traffic data by analysing the long-range correlation with detrended fluctuation analysis. Through a large number of real data collected by the travel time detection system in Beijing, the variation of flow in different time periods and intersections is studied. According to the long-range correlation in different time scales, it mainly discuss the effect of intersection location in road net, people activity customs and special traffic controls on urban traffic flow. As demonstrated by obtained results, the urban traffic flow represents three-phase characters similar to highway traffic. Moreover, compared by the two groups of data obtained before and after the special traffic restrictions （vehicles with special numbered plates only run in a special workday） enforcement, it indicates that the rules not only reduce the flow but also avoid irregular fluctuation.

Application of long-range correlation and multi-fractal analysis for the depiction of drought risk

By using the multi-fractal detrended fluctuation analysis method, we analyze the nonlinear property of drought in southwestern China. The results indicate that the occurrence of drought in southwestern China is multi-fractal and long- range correlated, and these properties are indifferent to timescales. A power-law decay distribution well describes the return interval of drought events and the auto-correlation. Furthermore, a drought risk exponent based on the multi-fractal property and the long-range correlation is presented. This risk exponent can give useful information about whether the drought may or may not occur in future, and provide a guidance function for preventing disasters and reducing damage.

Distribution Patterns of Ground Moss Species and Its Relationships with Environmental Factors in Changbai Mountain, Northeast China

The relationships of 42 species of ground moss with six environmental factors in 41 sites on Changbai Mountain Biosphere Reserve were analyzed. Four site groups and four groups of ground moss ecological species were identified using the method of Two-way Indicator Species Analysis (TWINSPAN). The results of Detrended Canonical. Correspondence Analysis (DCCA) showed that altitude, soil sand content, soil acidity, forest canopy coverage and soil water content are the five major environmental factors influencing the distributional patterns of the moss species. The four groups of ecological species, which correspond well with the four site groups, are projected on the species-environment biplot of DCCA. Group 1 dominated in the bogs of Larix olgensis forest, group 2 in the alpine tundra, group 3 in the dense conifer forest, and group 4 mainly in the Betula ermanii community and the Betula ermanii-Larix olgensis forest in sub-alpine respectively.

Morphology and multifractal features of a guyot in specific topographic vicinity in the Caroline Ridge,West Pacific

Massive seamounts have been surveyed and documented in the last decades.However,the morphologies of seamounts are usually described in qualitative manners,yet few quantitative detections have been carried out.Here,based on the high-re solution multi-beam bathymetric data,we report a recentlysurveyed guy ot on the Caroline Ridge in the West Pacific,and the large-scale volcanic structures and smallscale erosive-depositional landforms in the guyot area have been identified.The multifractal features of the guyot are characterized for the first time by applying multifractal detrended fluctuation analysis on the surveyed bathymetric data.The results indicate that the multifractal spectrum parameters of the seafloor have strong spatial dependency on the fluctuations of local landforms.Both small-and large-scale components contribute to the degree of asymmetry of the multifractal spectrum(B),while the fluctuations of B are mostly attributed to the changes in small-scale roughness.The maximum singularity strength(α0)correlates well with the roughness of large-scale landforms and likely reflects the large-scale topographic irregularity.Comparing to traditional roughness parameters or monofractal exponents,multifractal spectra are able to depict not only the multiscale characteristics of submarine landforms,but also the spatial variations of scaling behaviors.Although more comparative works are required for various seamounts,we hope this study,as a case of quantifying geomorphological characters and multiscale behaviors of seamounts,can encourage further studies on seamounts concerning geomorphological processes,ocean bottom circulations,and seamount ecosystems.

Fractal-based weak target detection within sea clutter

The target on the sea surface is complex and difficult to detect due to the interference of backscattered returns from the sea surface illuminated by the radar pulse. Detrended fluctuation analysis （DFA） has been used successfully to extract the time-domain Hurst exponent of sea-clutter series. Since the frequency of the sea clutter mainly concentrates around Doppler center so that we consider to extract frequency-do- main fractal characterization and then detect a weak target within sea clutter by using the difference of frequency-domain fractal characterization. The generalized detrended fluctuation analysis （GDFA） is more flexible than traditional DFA owing to its smoothing action for the clutters. In this paper, we apply the GDFA to evaluate the generalized Hurst exponent of sea-clutter series in the frequency domain. The difference of generalized Hurst exponents between different sea-clutter range bins would be used to determine whether the target exists. Moreover, some simulations with the real IPIX radar data have also been demonstrated in order to suooort this conclusion.

Surface pollen in the east of Qaidam Basin

Pollen analysis of 23 surface samples in the east of Qaidam Basin reveals the characteristics of pollen assemblages and their relationships with vegetation and climate. In pollen assemblages, Chenopodiaceae and Artemisia are preponderant types in all the samples, and Ephedra, Gramineae and Compositae are common types. The results of DCA （Detrended Correspondance Analysis） and Correlation Analysis show different pollen assemblages indicate different vegetations, coincided with respective vegetation types. A/C （Artemisia/Chenopodiaceae） in the desert can indicate the aridity. Depending on the aridity, the vegetation communities are divided into four groups： severe drought group, moderate drought group, slight drought group and tropophilous group. A/C value is less 0.2 in the severe drought group, 0.2-0.5 in the moderate drought group, 1.63 in the slight drought group and 5.72 slight-wetness group.