Arbitrary-order Hilbert Spectral Analysis Research Articles

Intermittency study of global solar radiation under a tropical climate: case study on Reunion Island

The arbitrary order Hilbert spectral analyses are applied to study the intermittency and multifractality of Global Horizonal Irradiation (GHI) based on one available high sampling rate of 1-year GHI records located at Saint-Denis (Moufia) over Reunion Island. The scaling exponents xi left( q right) is estimated through the arbitrary order Hilbert spectral analyses, and three parameters: Hurst exponent (H), the fractal co-dimension (C1), and Lévy parameter (alpha) are taken to study the multifractal process of the GHI in the sub-daily, daily fluctuations and also in seasonal variations. A power law behaviour with a spectral exponent β = 1.68 close to the Kolmogorov spectrum is detected through Fourier spectrum analysis, which indicates that the sub-daily fluctuations of GHI are nonstationary. The scaling exponent ζ(q) is then estimated by the arbitrary order Hilbert spectral analysis and the multifractal properties is detected. The log-stable model parameters H,{ }C_{1} and alpha characterize the concavity of the scaling exponent ζ(q) for analysing the intermittency of GHI. The classification method is applied to the daily GHI for analysing the distribution of the daily intermittency process and five classes with GHI and Kb are obtained.

Influence of Intermittent Turbulence on Air Pollution and Its Dispersion in Winter 2016/2017 over Beijing, China

With rapid urbanization in recent years, severe air pollution has emerged as a major issue for many regions of China, especially in some metropolises. A persistent pollution case during 6 December 2016–8 January 2017 was selected to investigate the relations between turbulent intermittency and frequent PM2.5 (particulate matters with diameter less than 2.5 μm) pollution events over the metropolitan region of Beijing, China. The accumulation of PM2.5 near the surface frequently occurred as a combined result of strong inversion layers, stagnant winds, high ambient humidity levels, and stable stratification during this case. Arbitrary-order Hilbert spectral analysis indicated that steep decreases in the PM2.5 concentration were simultaneous with the occurrence of intermittent turbulence and strong vertical mixing. A wind profiler observation revealed existence of low-level jets (LLJs) at the end of the polluted periods, suggesting that the upper-level turbulent mixing accompanied by the wind shear of LLJ was transported downward and enhanced the vertical mixing near the surface, which might have caused an abrupt reduction in PM2.5 and improvement in air conditions.

Scaling Properties of Atmospheric Wind Speed in Mesoscale Range

The scaling properties of turbulent flows are well established in the inertial sub-range. However, those of the synoptic-scale motions are less known, also because of the difficult analysis of data presenting nonstationary and periodic features. Extensive analysis of experimental wind speed data, collected at the Mauna Loa Observatory of Hawaii, is performed using different methods. Empirical Mode Decomposition, interoccurrence times statistics, and arbitrary-order Hilbert spectral analysis allow to eliminate effects of large-scale modulations, and provide scaling properties of the field fluctuations (Hurst exponent, interoccurrence distribution, and intermittency correction). The obtained results suggest that the mesoscale wind dynamics owns features which are typical of the inertial sub-range turbulence, thus extending the validity of the turbulent cascade phenomenology to scales larger than observed before.

Unravelling the scaling characteristics of daily streamflows of Brahmani river basin, India, using arbitrary-order Hilbert spectral and detrended fluctuation analyses

This paper presents arbitrary-order Hilbert spectral analysis (AOHSA) and multifractal detrended fluctuation analysis (MF-DFA) approaches to describe the multifractality of daily streamflows from four stations, namely Tilga, Jeraikela, Gomlai and Jenapur of Brahmani river basin in India. In the former method, the spectral slopes of Hilbert spectra for different moment orders depict the multifractality, and in this study, AOHSA method detected the scale invariance between synoptic to intra-seasonal scales (3 days–3 months approximately) in the daily streamflows of all the four stations. The MF-DFA method detected a crossover within 80–110 days (~ 3 months) in the four time series in addition to the crossover at annual scale. The robust estimates of Hurst exponents made by following the adaptive detrending method of pre-processing detrending operation ranged between 0.7 and 0.73 for the four time series which confirmed the universal multifractal properties within Brahmani river basin. The behaviour of spectral slope plot of AOHSA, the characteristics of scaling exponent plot, mass exponent plot and multifractal spectra confirmed that the highest multifractal degree is for the streamflow records of Tilga station which is having the smallest drainage area, and it may be attributed to the faster response of this sub-catchment to local precipitation events. The multifractality of all the four streamflow time series is found to be due to the dominant influence of correlation properties than due to the broadness of probability density function.

Arbitrary-order Hilbert Spectral Analysis and Intermittency in Solar Wind Density Fluctuations

Abstract The properties of inertial- and kinetic-range solar wind turbulence have been investigated with the arbitrary-order Hilbert spectral analysis method, applied to high-resolution density measurements. Due to the small sample size and to the presence of strong nonstationary behavior and large-scale structures, the classical analysis in terms of structure functions may prove to be unsuccessful in detecting the power-law behavior in the inertial range, and may underestimate the scaling exponents. However, the Hilbert spectral method provides an optimal estimation of the scaling exponents, which have been found to be close to those for velocity fluctuations in fully developed hydrodynamic turbulence. At smaller scales, below the proton gyroscale, the system loses its intermittent multiscaling properties and converges to a monofractal process. The resulting scaling exponents, obtained at small scales, are in good agreement with those of classical fractional Brownian motion, indicating a long-term memory in the process, and the absence of correlations around the spectral-break scale. These results provide important constraints on models of kinetic-range turbulence in the solar wind.

Diurnal characteristics of turbulent intermittency in the Taklimakan Desert

A case study is performed to investigate the behavior of turbulent intermittency in the Taklimakan Desert using an intuitive, direct, and adaptive method, the arbitrary-order Hilbert spectral analysis (arbitrary-order HSA). Decomposed modes from the vertical wind speed series confirm the dyadic filter-bank essence of the empirical mode decomposition processes. Due to the larger eddies in the CBL, higher energy modes occur during the day. The second-order Hilbert spectra $${\mathcal{L}}_{2} (\omega )$$ delineate the spectral gap separating fine-scale turbulence from large-scale motions. Both the values of kurtosis and the Hilbert-based scaling exponent $$\xi \left( q \right)$$ reveal that the turbulence intermittency at night is much stronger than that during the day, and the stronger intermittency is associated with more stable stratification under clear-sky conditions. This study fills the gap in the characteristics of turbulence intermittency in the Taklimakan Desert area using a relatively new method.

Intermittent particle dynamics in marine coastal waters

Abstract. Marine coastal processes are highly variable over different space scales and timescales. In this paper we analyse the intermittency properties of particle size distribution (PSD) recorded every second using a LISST instrument (Laser In-Situ Scattering and Transmissometry). The particle concentrations have been recorded over 32 size classes from 2.5 to 500 μm, at 1 Hz resolution. Such information is used to estimate at each time step the hyperbolic slope of the particle size distribution, and to consider its dynamics. Shannon entropy, as an indicator of the randomness, is estimated at each time step and its dynamics is analysed. Furthermore, particles are separated into four classes according to their size, and the intermittent properties of these classes are considered. The empirical mode decomposition (EMD) is used, associated with arbitrary-order Hilbert spectral analysis (AHSA), in order to retrieve scaling multifractal moment functions, for scales from 10 s to 8 min. The intermittent properties of two other indicators of particle concentration are also considered in the same range of scales: the total volume concentration Cvol-total and the particulate beam attenuation coefficient cp(670). Both show quite similar intermittent dynamics and are characterised by the same exponents. Globally we find here negative Hurst exponents (meaning the small scales show larger fluctuation than large scales) for each time series considered, and nonlinear moment functions.

Two-scale correlation and energy cascade in three-dimensional turbulent flows

In this paper, we propose a high-order harmonic-free methodology, namely arbitrary-order Hilbert spectral analysis, to estimate the two-scale correlation (TSC). When applied to fully developed turbulent velocity, we find that the scale-dependent Hilbert energy satisfies a lognormal distribution on both the inertial and dissipation ranges. The maximum probability density function of the logarithm of the Hilbert energy obeys a power law with a scaling exponent γ ≃ 0.33 in the inertial range. For the measured TSC, we observe a logarithmic correlation law with an experimental exponent α ≃ 0.37 on both the inertial and dissipation ranges. The correlation itself is found to be self-similar with respect to the distance between the two considered scales and a central frequency ωc in the logarithm space. An empirical nonlinear and nonlocal triad-scale interaction formula is proposed to describe the observed TSC. This triadic interaction can be interpreted as experimental evidence of a small-scale nonlinear and nonlocal coupling inside the self-similarity of the Richardson–Kolmogorov phenomenological cascade picture.

Multifractal description of wind power fluctuations using arbitrary order Hilbert spectral analysis

The objectives are to study and model the aggregate wind power fluctuations dynamics in the multifractal framework. We present here the analysis of aggregate power output sampled at 1 Hz during three years. We decompose the data into several Intrinsic Mode Functions (IMFs) using Empirical Mode Decomposition (EMD). We use a new approach, arbitrary order Hilbert spectral analysis, a combination of the EMD approach with Hilbert spectral analysis (or Hilbert–Huang Transform) and the classical structure-function analysis to extract the scaling exponents or multifractal spectrum ζ(q): this function provides a full characterization of a process at all intensities and all scales. The application of both methods, i.e. structure-function and arbitrary-order Hilbert spectral analyses, gives similar results indicating that the aggregate power output from a wind farm, possesses intermittent and multifractal properties. In order to check this result, we generate stochastic simulations of a Multifractal Random Walk (MRW) using a log-normal stochastic equation. We show that the simulation results are fully compatible with the experimental results.

Empirical study of the scaling behavior of the amplitude–frequency distribution of the Hilbert–Huang transform and its application in sunspot time series analysis

Investigating long-range correlation by the Hurst exponent, H, is crucial in the study of time series. Recently, empirical-mode-decomposition-based arbitrary-order Hilbert spectral analysis (EMD-HSA) has been proposed to numerically obtain without proof a scaling relationship, generated from the amplitude–frequency distribution, related to H. We propose a formalism to empirically study EMD-HSA, to deduce its scaling exponent ξ(q) from the perspective of EMD-based arbitrary-order Hilbert marginal spectrum (EMD-HMS), and to numerically compare the results with the expected H. EMD-HSA and EMD-HMS experiments show that, by incompletely removing (quasi-)periodic trends, the sunspot series should have an H value around 0.12.

Application of Arbitrary-Order Hilbert Spectral Analysis to Passive Scalar Turbulence

In previous work [Huang et al., PRE 82, 26319, 2010], we found that the passive scalar turbulence field maybe less intermittent than what we believed before. Here we apply the same method, namely arbitrary-order Hilbert spectral analysis, to a passive scalar (temperature) time series with a Taylor's microscale Reynolds number Reλ ≃ 3000. We find that with increasing Reynolds number, the discrepancy of scaling exponents between Hilbert ξθ(q) and Kolmogorov-Obukhov-Corrsin (KOC) theory is increasing, and consequently the discrepancy between Hilbert and structure function could disappear at infinite Reynolds number.

Arbitrary-order Hilbert spectral analysis for time series possessing scaling statistics: Comparison study with detrended fluctuation analysis and wavelet leaders

In this paper we present an extended version of Hilbert-Huang transform, namely arbitrary-order Hilbert spectral analysis, to characterize the scale-invariant properties of a time series directly in an amplitude-frequency space. We first show numerically that due to a nonlinear distortion, traditional methods require high-order harmonic components to represent nonlinear processes, except for the Hilbert-based method. This will lead to an artificial energy flux from the low-frequency (large scale) to the high-frequency (small scale) part. Thus the power law, if it exists, is contaminated. We then compare the Hilbert method with structure functions (SF), detrended fluctuation analysis (DFA), and wavelet leader (WL) by analyzing fractional Brownian motion and synthesized multifractal time series. For the former simulation, we find that all methods provide comparable results. For the latter simulation, we perform simulations with an intermittent parameter μ=0.15. We find that the SF underestimates scaling exponent when q>3. The Hilbert method provides a slight underestimation when q>5. However, both DFA and WL overestimate the scaling exponents when q>5. It seems that Hilbert and DFA methods provide better singularity spectra than SF and WL. We finally apply all methods to a passive scalar (temperature) data obtained from a jet experiment with a Taylor's microscale Reynolds number Re(λ)≃250. Due to the presence of strong ramp-cliff structures, the SF fails to detect the power law behavior. For the traditional method, the ramp-cliff structure causes a serious artificial energy flux from the low-frequency (large scale) to the high-frequency (small scale) part. Thus DFA and WL underestimate the scaling exponents. However, the Hilbert method provides scaling exponents ξ(θ)(q) quite close to the one for longitudinal velocity, indicating a less intermittent passive scalar field than what was believed before.

Second-order structure function in fully developed turbulence

We relate the second-order structure function of a time series with the power spectrum of the original variable, taking an assumption of statistical stationarity. With this approach, we find that the structure function is strongly influenced by the large scales. The large-scale contribution and the contribution range are, respectively, 79% and 1.4 decades for a Kolmogorov -5/3 power spectrum. We show numerically that a single scale influence range, over smaller scales is about 2 decades. We argue that the structure function is not a good method to extract the scaling exponents when the data possess large energetic scales. An alternative methodology, the arbitrary order Hilbert spectral analysis which may constrain this influence within 0.3 decade, is proposed to characterize the scaling property directly in an amplitude-frequency space. An analysis of passive scalar (temperature) turbulence time series is presented to show the influence of large-scale structures in real turbulence and the efficiency of the Hilbert-based methodology. The corresponding scaling exponents ζ(θ)(q) provided by the Hilbert-based approach indicate that the passive scalar turbulence field may be less intermittent than what was previously believed.

Analysis of daily river flow fluctuations using empirical mode decomposition and arbitrary order Hilbert spectral analysis

In this paper we presented the analysis of two long time series of daily river flow data, 32 years recorded in the Seine river (France), and 25 years recorded in the Wimereux river (Wimereux, France). We applied a scale based decomposition method, namely Empirical Mode Decomposition (EMD), on these time series. The data were decomposed into several Intrinsic Mode Functions (IMF). The mean frequency of each IMF mode indicated that the EMD method acts as a filter bank. Furthermore, the cross-correlation between these IMF modes from Seine river and Wimereux river demonstrated correlation among the large scale IMF modes, which indicates that both rivers are likely to be influenced by the same events maritime climate event of Northern France. As a confirmation we found that the large scale parts have the same evolution trend. We finally applied arbitrary order Hilbert spectral analysis, a new technique coming from turbulence studies and time series analysis, on the flow discharge of Seine river. This new method provides an amplitude-frequency representation of the original time series, giving a joint pdf $p(\omega,\mathcal{A})$. When marginal moments of the amplitude are computed, one obtains an intermittency study in the frequency space. Applied to river flow discharge data from the Seine river, this shows the scaling range and characterizes the intermittent fluctuations over the range of scales from 4.5 to 60 days, between synoptic and intraseasonal scales.