Multifractal Processes Research Articles

Forecasting the solar modulation potential: Tests of time series models

This study analyzes the predictability of the solar modulation potential using time series models. Recently, new data sets for the modulation potential have become available, at daily, monthly, and annual resolutions. At lower frequencies, the data show the well-known 11-22-year cycle. Both the periodicity and amplitude vary over time. At higher resolutions, the probability distribution has heavy tails, while the data show the intermittent outliers characteristic of multifractal processes. Forecasting experiments are run using regressions in levels and differences, frequency domain methods, models with sinusoidal terms and neural networks. For the daily data, all the models achieve high degrees of accuracy at proximate horizons. As the horizon extends, accuracy falls away rapidly. At 27 days, corresponding to one solar rotation, a transfer function in differences achieves a more accurate forecast than either regressions or neural nets, since it is able to replicate the range of the data. At the annual resolution, both the regression and neural net predict well at horizons of 1 year. Again, forecast accuracy deteriorates sharply as the forecast horizon extends. At the monthly resolution, forecasting is problematic. The resolution is not low enough to bring out the low frequency cycles, but there is so much short-term dependence that the data are completely dominated by serial correlation. Any model incorporating proximate lags will generate inertial forecasts. Any model using lower frequency cyclical terms will be unable to pick up on near-term patterns. The forecasting skill of time series models appears limited to short horizons. The recommendation for forecasting over longer intervals is some combination of physics and statistical models.

Multifractional Hermite processes: Definition and first properties

We define multifractional Hermite processes which generalize and extend both multifractional Brownian motion and Hermite processes. It is done by substituting the Hurst parameter in the definition of Hermite processes as a multiple Wiener–Itô integral by a Hurst function. Then, we study the pointwise regularity of these processes, their local asymptotic self-similarity and some fractal dimensions of their graph. Our results show that the fundamental properties of multifractional Hermite processes are, as desired, governed by the Hurst function. Complements are given in the second order Wiener chaos, using facts from Malliavin calculus.

Exploring the Future of Rainfall Extremes Over CONUS: The Effects of High Emission Climate Change Trajectories on the Intensity and Frequency of Rare Precipitation Events

AbstractThe impacts of climate change on extreme rainfall characteristics at fine spatiotemporal scales are governed by substantial uncertainty, primarily due to the systematic error components inherited from conventional numerical prediction systems, and/or the intrinsic assumptions of the selected modeling schemes. Here, we attempt to robustly evaluate the effects of future climate scenarios on intensity‐duration‐frequency (IDF) curves over the entire Contiguous United States, while accounting for the nonstationary nature of the rainfall process across adequately fine spatiotemporal resolutions. To do so, we apply a parametric approach to statistically downscaled climate model outputs that reflect the Representative Concentration Pathway 8.5, which are offered by the North American Coordinated Regional Downscaling Experiment. Compared to traditional IDF estimation techniques, the employed framework is based on multifractal (MF) scaling arguments and assumes that the statistical structure of rainfall at interannual scales can be approximated by sequential realizations of a stationary MF process with parameters that vary slowly across (not within) realizations. The obtained results show that return period estimates exhibit significant downward trends over most of the domain, which slowly dampen with time, as the effects of climate change are more pronounced at lower exceedance probability levels. Given the observed rate of changes in the frequency and intensity of extreme rainfall for the remainder of the century, we argue that future infrastructure design should be strategically tailored to account for a wide range of potential outcomes.

ФРАКТАЛЬНА РАДІОФІЗИКА. Частина 2. ФРАКТАЛЬНИЙ І МУЛЬТИФРАКТАЛЬНИЙ АНАЛІЗ СИГНАЛІВ ТА ПРОЦЕСІВ

Subject and Purpose. The subject of this paper is to review the principal methods of fractal and multifractal analysis of signals and processes, in combination with a detailed consideration of the algorithms that can provide for a successful practical implementation of the methods described. Methods and Methodology. The results presented concern modeling of both deterministic and stochastic fractal and multifractal signals and processes. The corresponding practical methods of analysis are considered, with discussion of their essential features, advantages and disadvantages, as well as of the problems of application that may exist. Results. Several approaches have been discussed as to categorizing the signals and processes within the notion of fractality. A few tens of models of deterministic and stochastic fractal or multifractal signals and processes have been analyzed in detail. Over twenty methods of monofractal analysis have been analyzed, with identifi cation of their features, advantages or disadvantages, and limits of applicability. The expediency of resorting to complex methods of monofractal analysis has also been discussed. Those methods are not based upon application of fractal analysis techniques alone but rather combine them with linear and nonlinear integral time-frequency transforms. The effectiveness of the ten most popular multifractal analysis techniques has been confirmed, with consideration of their special features, advantages and drawbacks. Conclusion. The mathematical foundations have been presented which underlie modern methods of analysis and modeling of fractal and multifractal signals and processes. The methods discussed may allow revealing a great amount of unique hidden information on the world around us.

Multifractal analysis of model fractal and multifractal signals

One of the topical directions of modern fractal radio physics is the multifractal analysis of signals and processes of various origins. A set of deterministic and stochastic models of monofractal and multifractal signals and processes in the time domain is proposed. New multifractal signal characteristics, namely, the coefficient of asymmetry of the multifractal spectrum function, the relative width of the multifractal spectrum and the dimension of the multifractal support, are introduced, the necessity of their use is demonstrated on examples. Using Wavelet Transform Modulus Maxima Method and Multi-Fractal Detrended Fluctuation Analysis Method, a detailed multifractal analysis of model signals is performed. The features of multifractal analysis of monofractal, multifractal and non-fractal signals are established, the appropriate recommendations for practitioners are formulated. Convenient formats for presenting analysis results have been developed. It was found that during the transition of the multifractal signal to the monofractal regime, the function of the multifractal spectrum of the physical fractal does not collapse into a point, as it should happen in theory for a mathematical fractal. Threshold values of multifractal characteristics, which are indicators of the appearance of the monofractal, are proposed. It has been shown that multifractal analysis of non-fractal signals leads to the appearance of multifractal spectra with anomalous values of multifractal characteristics. The correction function method is modified for the methods of multifractal analysis of signals and processes. It is proved that its usage makes it possible to reduce the deviation of the obtained estimate of the generalized Hurst exponent from the true known value of the Hölder exponent of the analyzed signal from 5 – 90% to 3 – 8%.

A graph-theory-based multi-scale analysis of hierarchical cascade in molecular clouds

Context. The spatial properties of small star clusters suggest that they may originate from a fragmentation cascade starting from molecular cloud, of which there might be traces found at spatial scales up to a few tens of thousands of astronomical units (kAU). Aims. Our goal is to investigate the multi-scale spatial structure of gas clumps, to probe the existence of a hierarchical cascade over a range of characteristic spatial scales, and to evaluate its possible link with star production in terms of multiplicity. Methods. From the Berschel emission maps of NGC 2264 at [70, 160, 250, 350, 500] μm, clumps are extracted using getsf software at each of the associated spatial resolutions (respectively [8.4,13.5,18.2, 24.9,36.3]″). Using the spatial distribution of these clumps and the class 0/I young stellar object (YSO) from Spitzer data, we developed a graph-theoretic analysis to represent the multi-scale structure of the cloud as a connected network. This network is organised in levels, and each level represents a characteristic scale among the available spatial scales. A link is created between two nodes which could be either a clump or a YSO from two different levels if their footprints overlap with each other. A parent node is then associated with a child node from a lower scale. The way in which the network subdivides scale after scale is compared with a geometric model that we have developed. This model generates extended objects that have a particularity in that they are geometrically constrained and subdivide along the scales following a fractal law. This graph-theoretic representation allows us to develop new statistical metrics and tools aiming at characterising, in a quantitative way, the multi-scale nature of molecular clouds. Results. We obtain three classes of multi-scale structure in NGC 2264 according to the number of nodes produced at the deepest level (called graph-sinks): hierarchical (several graph-sinks), linear (a single graph-sink with at most a single parent at each level), and isolated (no connection to any other node). The class of structure is strongly correlated with the column density NH2 of NGC 2264. The hierarchical structures dominate the regions whose column density exceeds NH2 = 6 × 1022 cm−2. Although the latter are in the minority, namely 23% of the total number of structures, they contain half of the class 0/I YSOs, proving that they are highly efficient in producing stars. We define a novel statistical metric, the fractality coefficient F, corresponding to the fractal index that an equivalent population of clumps would have if they were generated by an ideal fractal cascade. For NGC 2264, over the whole range of spatial scales (1.4–26 kAU), this coefficient is globally estimated to be F = 1.45 ± 0.12 and its dispersion suggests that the cascade may depend on local physical conditions. However, a single fractal index is not the best fit for the NGC 2264 data because the hierarchical cascade starts at a 13 kAU characteristic spatial scale. Conclusions. Our novel methodology allows us to correlate YSOs with their gaseous environment which displays some degree of hierarchy for spatial scales below 13 kAU. We identify hierarchical multi-scale structures, which we associate with a hierarchical fragmentation process, and linear structures, which we associate with a monolithic fragmentation process. Hierarchical structures are observed as the main vectors of star formation. This cascade, which drives efficient star formation, is then suspected of being both hierarchical and rooted by the larger scale gas environment up to 13 kAU. We do not see evidence for any hierarchical structural signature of the cloud within the 13–26 kAU range, implying that the structure of the cloud does not follow a simple fractal law along the scales but instead might be submitted to a multi-fractal process.

Mathematical Modeling of Stress Using Fractal Geometry; The Power Laws and Fractal Complexity of Stress

In this study, we analyze the physiological data during real-world driving tasks to determine whether driver’s relative stress is mono-fractal or multi-fractal. We use the PhysioNet database including long term ECG recordings from 15 healthy volunteers, taken while they were driving on a prescribed route including city streets and highways in and around Boston, Massachusetts. The vibration analysis such as power spectral densities (PSD) analysis has been performed to estimate the exponent from realizations of these pro- cesses and to find out if the signal of interest exhibits a power-law PSD. Multifractal dynamics of heartbeat interval signals have been assessed by multifractal spectrum analysis to explore the possibility that ECG recordings belong to class of multi-fractal process for which a large number of scaling exponents are re- quired to characterize their scaling structures. We apply Higuchi algorithm to find the fractal complexity of each cardiac rhythm for different time intervals. According to our analysis, we investigate that driver’s ECG signals under relative stress follows fractal behavior unlike control healthy signals which are multi-fractal. Our findings provide a comprehensive framework for detect stress and differentiate people who experience stress with normal people without stress which is crucial in finding the best diagnostic and controlling strat- egy in fight against many health problems due to stress, such as high blood pressure, heart disease, obesity and diabetes. Moreover, being able to recognize stress can help us to manage it.

On the fractal geometry of different heart rhythms

In this study, we explore the possibility that ECG recordings belong to class of multifractal process for which a large number of scaling exponents are required to characterize their scaling structures. We use the BIDMC Congestive Heart Failure database including long term ECG recordings from 11 men, aged 22 to 71, and 4 women, aged 54 to 63 with severe congestive heart failure and the MIT-BIH Arrhythmia database that contains 48 half-hour excerpts of two-channel ambulatory ECG recordings, obtained from 47 subjects studied by the BIH Arrhythmia Laboratory between 1975 and 1979. We compare these two chronic heart diseases with the control people in the MIT-BIH Normal Sinus Rhythm database which includes 18 long-term ECG recordings of 5 men, aged 26 to 45, and 13 women, aged 20 to 50 without significant arrhythmia. The vibration analysis such as power spectral densities (PSD) analysis has been performed for differentiating the time series. Multifractal dynamics of heartbeat interval signals have been assessed by multifractal spectrum analysis to differentiate normal signals with arrhythmia and severe congestive heart failure patients. We apply Higuchi algorithm to find the fractal complexity of each cardiac rhythm and then compare the signals for different time intervals. According to our analysis, we investigate that neither the power spectral densities nor the Higuchi algorithm to find the fractal dimension alone were sufficient to separate different classes of patients and healthy people. However, when multifractal analysis and scaling exponent were used as a classifier, the three classes were well separated. In addition, multifractal analysis revealed that we have a narrow range of exponents for arrhythmia and congestive heart failure subjects and as a result, a clear loss of multifractality for them. It was of great significance to show that we have a narrower range of exponents for arrhythmia subject compared to congestive heart failure subject which is useful to recognize these two classes of patients with heart disease. Our findings provide a comprehensive framework for diagnostic and classifying different patients with cardiac disease such as arrhythmia and congestive heart failure and differentiate them with normal people without heart disease which is crucial in finding the best diagnostic and controlling strategy in fight against chronic heart disease.

Intelligent traffic monitoring and analysis system to detect attacks in software-defined networks

Today, the task of monitoring, security, management, traffic prioritization and optimization of the overall load on the network infrastructure is becoming extremely important not only for the corporate sector, but also for mobile and fixed-line operators. Current trends in Internet content services indicate that the requirements for traffic management are unpredictable. Therefore, service providers must implement new algorithms to monitor and analyze network traffic to detect attacks in today's networks, where software can be updated to support changing traffic management and control requirements as needed. That is why this paper developed a unique algorithm for monitoring and analyzing network traffic, which improved the efficiency of standard intrusion detection and prevention systems by using intelligent multifractal processes for analyzing incoming traffic. It has been experimentally proved that the use of the developed algorithms in the communication infrastructure allowed to reduce data losses up to 5 % if malicious traffic was present in the communication channels.

On the fractal geometry of gait dynamics in different neuro-degenerative diseases

Neuro-degenerative diseases influence significantly the gait behavior and the ability to move. To explore the etiology of neuro-degenerative disease, it would be useful to characterize gait dynamics. The purpose of this study is to classify different neuro-degenerative diseases using fractal geometry. We use Gait Dynamics in Neuro-Degenerative Disease Data Base including recordings from patients with Parkinson's disease (n = 15), Huntington's disease (n = 20), or amyotrophic lateral sclerosis (n = 13) and 16 healthy control subjects are also included (Hausdorff JM et al., 2000). The vibration analysis using power spectral densities (PSD) method has been carried out to discover whether some type of power-law scaling exists for various statistical moments at different scales of these databases. Using Discrete Wavelet Transform (DWT) and Wavelet Leader Multifractal (WLM) analysis, we explore the possibility that these recordings belong to the class of multifractal process for which a large number of scaling exponents are required to characterize their scaling structures. A non-linear analysis called the Fractal Dimension (FD) using Higuchi algorithm has been performed to quantify the fractal complexity of recordings. According to our results, we noticed that neither the power spectral densities nor the Higuchi algorithm to find the fractal dimension alone were sufficient to separate different classes of patients and healthy people. In addition, when multifractal analysis and scaling exponent were used as a classifier, the three classes could not be well separated. However, this study revealed that we have a wide range of exponents for some of the gait recordings which indicates they have multifractal structure and they need to be indexed by different exponents as we decompose them into different subsets. In other words, these multifractal subjects require much more exponents to characterize their scaling properties compared to monofractal gait recordings which their spectrum displays a narrow width of scaling exponent. Another important outcome from our multifractal analysis is recognizing obvious changes in the shape of D(h) curves for some of the gait recordings which is crucial in finding the best strategies to better controlling the gait mechanisms in different neuro-degenerative diseases. Although the vibration analysis, fractal dimension and multifractal analysis may not be able to classify gait recordings, however, they can be used as comprehensive frameworks to further analysis, characterize and compare the complexity and fractal behavior of gait recordings and data structures of different neuro-degenerative diseases in clinical database. Likewise, beside the Higuchi algorithm to find the fractal dimension as a complexity measure for the gait recordings, it will require much more efforts and further clinical analysis to find a specific threshold which make the fractal dimension to be considered as a biomarker and diagnosis tool for different neuro-degenerative diseases.

The Fractal Nature of Drought: Power Laws and Fractal Complexity of Arizona Drought

In this study, we explore the possibility that the Drought Monitor database belongs to class of fractal process which can be characterized using a single scaling exponent. The Drought Monitor map identifies areas of drought and labels them by intensity: D0 abnormally dry, D1 moderate drought, D2 severe drought, D3 extreme drought, and D4 exceptional drought. The vibration analysis using power spectral densities (PSD) method has been carried out to discover whether some type of power-law scaling exists for various statistical moments at different scales of this database. We perform multi-fractal analysis to estimate the multi-fractal spectrum of each group. We apply Higuchi algorithm to find the fractal complexity of each group and then compare them for different time intervals. Our findings reveal that we have a wide range of exponents for D0-D4. Therefore, D0-D4 belong to class of multi-fractal process for which a large number of scaling exponents are required to characterize the scaling structure.

On local path behavior of Surgailis multifractional processes

Multifractional processes are stochastic processes with non-stationary increments whose local regularity and self-similarity properties change from point to point. The paradigmatic example of them is the classical Multifractional Brownian Motion (MBM) { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} of Benassi, Jaffard, Lévy Véhel, Peltier and Roux, which was constructed in the mid 90’s just by replacing the constant Hurst parameter H {\mathcal {H}} of the well-known Fractional Brownian Motion by a deterministic function H ( t ) {\mathcal {H}}(t) having some smoothness. More than 10 years later, using a different construction method, which basically relied on nonhomogeneous fractional integration and differentiation operators, Surgailis introduced two non-classical Gaussian multifactional processes denoted by { X ( t ) } t ∈ R \{X(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t ∈ R \{Y(t)\}_{t\in \mathbb {R}} . In our article, under a rather weak condition on the functional parameter H ( ⋅ ) {\mathcal {H}}(\cdot ) , we show that { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} and { X ( t ) } t ∈ R \{X(t)\}_{t\in \mathbb {R}} as well as { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t ∈ R \{Y(t)\}_{t\in \mathbb {R}} only differ by a part which is locally more regular than { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} itself. On one hand this result implies that the two non-classical multifractional processes { X ( t ) } t ∈ R \{X(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t ∈ R \{Y(t)\}_{t\in \mathbb {R}} have exactly the same local path behavior as that of the classical MBM { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} . On the other hand it allows to obtain from discrete realizations of { X ( t ) } t ∈ R \{X(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t ∈ R \{Y(t)\}_{t\in \mathbb {R}} strongly consistent statistical estimators for values of their functional parameter.

From the Brownian motion to a multifractal process using the Lévy–Ciesielski construction

We give a construction of a multifractal process with prescribed Hölder exponents starting from the Lévy–Ciesielski construction of a Brownian motion. We also show that this method preserves the law of the iterated logarithm.

Monofractal analysis of functional magnetic resonance imaging: An introductory review

The following review will aid readers in providing an overview of scale‐free dynamics and monofractal analysis, as well as its applications and potential in functional magnetic resonance imaging (fMRI) neuroscience and clinical research. Like natural phenomena such as the growth of a tree or crashing ocean waves, the brain expresses scale‐invariant, or fractal, patterns in neural signals that can be measured. While neural phenomena may represent both monofractal and multifractal processes and can be quantified with many different interrelated parameters, this review will focus on monofractal analysis using the Hurst exponent (H). Monofractal analysis of fMRI data is an advanced analysis technique that measures the complexity of brain signaling by quantifying its degree of scale‐invariance. As such, the H value of the blood oxygenation level‐dependent (BOLD) signal specifies how the degree of correlation in the signal may mediate brain functions. This review presents a brief overview of the theory of fMRI monofractal analysis followed by notable findings in the field. Through highlighting the advantages and challenges of the technique, the article provides insight into how to best conduct fMRI fractal analysis and properly interpret the findings with physiological relevance. Furthermore, we identify the future directions necessary for its progression towards impactful functional neuroscience discoveries and widespread clinical use. Ultimately, this presenting review aims to build a foundation of knowledge among readers to facilitate greater understanding, discussion, and use of this unique yet powerful imaging analysis technique.

Girsanov theorem for multifractional Brownian processes

ABSTRACT In this article, we will present a new perspective on the variable-order fractional calculus, which allows for differentiation and integration to a variable order. The concept of multifractional calculus has been a scarcely studied topic within the field of functional analysis in the past 20 years. We develop a multifractional differential operator which acts as the inverse of the multifractional integral operator. This is done by solving the Abel integral equation generalized to a multifractional order. With this new multifractional differential operator, we prove a Girsanov's theorem for multifractional Brownian motions of Riemann–Liouville type. As an application, we show how Girsanov's theorem can then be applied to prove the existence of a unique strong solution to stochastic differential equations where the drift coefficient is merely of linear growth, and the driving noise is given by a non-stationary multifractional Brownian motion with a Hurst parameter as a function of time. The Hurst functions we study will take values in a bounded subset of . The application of multifractional calculus to SDEs is based on a generalization of the works of D. Nualart and Y. Ouknine [Regularization of differential equations by fractional noise, Stoch Process Appl. 102(1) (2002), pp. 103–116].

Moving average Multifractional Processes with Random Exponent: Lower bounds for local oscillations

In the last few years Ayache, Esser and Hamonier introduced a new Multifractional Process with Random Exponent (MPRE) obtained by replacing the Hurst parameter in a moving average representation of Fractional Brownian Motion through Wiener integral by an adapted Hölder continuous stochastic process indexed by the integration variable. Thus, this MPRE can be expressed as a moving average Itô integral which is a considerable advantage with respect to another MPRE introduced a long time ago by Ayache and Taqqu. Thanks to this advantage, very recently, Loboda, Mies and Steland have derived interesting results on local Hölder regularity, self-similarity and other properties of the recently introduced moving average MPRE and generalizations of it. Yet, the problem of obtaining, on a universal event of probability 1 not depending on the location, relevant lower bounds for local oscillations of such processes has remained open. We solve it in the present article under some conditions.

A Multiring Julia Fractal Chaotic System With Separated-Scroll Attractors

Fractal and chaos are two major phenomena in nonlinear systems. A few combinations of chaos with fractal for generating multiscroll attractors are reported. Thus, based on Chua multiscroll chaotic systems, a novel multiring chaotic system is proposed via Julia multifractal processes. The main purpose of this work is to make up for the deficiency of various fractal processes to further improve the performance of chaotic systems. A new class of multiscroll attractors is generated by combining different fractal processes, including symmetrical multiring attractor, separated multiring attractor, and nested multiring attractor. Furthermore, dynamic behaviors of the fractal chaotic system are analyzed by means of Lyapunov exponents and bifurcation diagram to indicate that the system exhibits chaotic characteristics. Experiments on the spectrum entropy (SE) complexity and image encryption performance demonstrate the good performance of the proposed fractal-based attractors. To verify the feasibility of multiring chaotic system, the various fractal-based attractors generated are implemented by micro controller unit (MCU), and the separated multiring attractors are observed on the digital oscilloscope.

Spectral Analysis for Some Multifractional Gaussian Processes

We study the small ball asymptotics problem in $L_2$ for two generalizations of the fractional Brownian motion with variable Hurst parameter. To this end, we perform careful analysis of the singular values asymptotics for associated integral operators.

Analysis of Volcanic Thermohaline Fluctuations of Tagoro Submarine Volcano (El Hierro Island, Canary Islands, Spain)

Temperature and conductivity fluctuations caused by the hydrothermal emissions released during the degasification stage of the Tagoro submarine volcano (Canary Islands, Spain) have been analysed as a robust proxy for characterising and forecasting the activity of the system. A total of 21 conductivity-temperature-depth time series were gathered on a regular high-resolution grid over the main crater of Tagoro volcano. Temperature and conductivity time series, as manifestations of stochastic events, were investigated in terms of variance and analysed by the Generalised Moments Method (GMM). GMM provides the statistical moments, the structure functions of a process whose shape is an indicator of the underlying stochastic mechanisms and the state of activity of the submarine volcano. Our findings confirm an active hydrothermal process in the submarine volcano with a sub-normal behaviour resulting from anti-persistent fluctuations in time. Its hydrothermal emissions are classified as multifractal processes whose structure functions present a crossover between two time scales. In the shorter time scale, findings point to the multiplicative action of two random processes, hydrothermal vents, which carries those fluctuations driving the circulation over the crater, and the overlying aquatic environment. Given that both temperature and conductivity fluctuations are nonstationary, Tagoro submarine volcano can be characterised as an open system exchanging energy to its surroundings.

Reexamining the framework for intermittency in Lagrangian stochastic models for turbulent flows: A way to an original and versatile numerical approach

The characterization of intermittency in turbulence has its roots in the refined similarity hypotheses of Kolmogorov, and if no proper definition is to be found in the literature, statistical properties of intermittency were studied and models were developed in an attempt to reproduce it. The first contribution of this work is to propose a requirement list to be satisfied by models designed within the Lagrangian framework. Multifractal stochastic processes are a natural choice to retrieve multifractal properties of the dissipation. Among them, we investigate the Gaussian multiplicative chaos formalism, which requires the construction of a log-correlated stochastic process X_{t}. The fractional Gaussian noise of Hurst parameter H=0 is of great interest because it leads to a log correlation for the logarithm of the process. Inspired by the approximation of fractional Brownian motion by an infinite weighted sum of correlated Ornstein-Uhlenbeck processes, our second contribution is to propose a stochastic model: X_{t}=∫_{0}^{∞}Y_{t}^{x}k(x)dx, where Y_{t}^{x} is an Ornstein-Uhlenbeck process with speed of mean reversion x and k is a kernel. A regularization of k(x) is required to ensure stationarity, finite variance, and logarithmic autocorrelation. A variety of regularizations are conceivable, and we show that they lead to the aforementioned multifractal models. To simulate the process, we eventually design a new approach relying on a limited number of modes for approximating the integral through a quadrature X_{t}^{N}=∑_{i=1}^{N}ω_{i}Y_{t}^{x_{i}}, using a conventional quadrature method. This method can retrieve the expected behavior with only one mode per decade, making this strategy versatile and computationally attractive for simulating such processes, while remaining within the proposed framework for a proper description of intermittency.