Stochastic interpolation of sparsely sampled time series via multi-point fractional Brownian bridges
MMulti-point fractional Brownian bridges and their applications
J. Friedrich, S. Gallon,
1, 2
A. Pumir, and R. Grauer Univ. Lyon, ENS de Lyon, Univ. Claude Bernard,CNRS, Laboratoire de Physique, F-69342, Lyon, France Institute for Theoretical Physics I, Ruhr-University Bochum,Universit¨atsstr. 150, D-44801 Bochum, Germany (Dated: May 18, 2020)We propose and test a method to interpolate sparsely sampled signals by a stochastic process witha broad range of spatial and/or temporal scales. To this end, we extend the notion of a fractionalBrownian bridge, defined as fractional Brownian motion with a given scaling (Hurst) exponent H and with prescribed start and end points, to a bridge process with an arbitrary number of prescribedintermediate and non-equidistant points. We demonstrate the validity of our method on a signalfrom fluid turbulence in a high Reynolds number flow. Furthermore, we discuss possible extensionsof the present work to include the non-self-similar character of the signal. The derived method couldbe instrumental within a variety of fields such as astrophysics, particle tracking, specific tailoring ofsurrogate data, and spatial planning. Many non-equilibrium phenomena in physics involverandom fluctuations with a wide range of spatial and/ortemporal scales [1, 2]. The theory of stochastic processesprovides a conceptual framework to describe such phe-nomena [3, 4]. The most emblematic example is pro-vided by Brownian motion, which results from randomuncorrelated collisions acting on a particle, and can bedescribed by a simple Wiener process [5]. By contrast,several complex systems in nature also involve long- orshort-range correlations, which require a description interms of fractional Brownian motion [6, 7]. Examples in-clude velocity fluctuations in turbulence [8–10], magneticfield fluctuations in the solar wind [11], random amoe-bid motion [12], or heart interbeat fluctuations [13]. Inthe following, we will restrict ourselves to temporal pro-cesses, or more generally to processes depending on onlyone variable. Fractional Brownian motion (fBm) X ( t )is a nonstationary centered Gaussian process , and thusentirely characterized by its covariance h X ( t ) X ( t ) i = 12 (cid:0) t H + t H − | t − t | H (cid:1) . (1)which implies that h ( X ( t ) − X ( t )) i = | t − t | H . TheHurst exponent H determines negative (antipersistent)correlations for 0 < H < /
2, whereas positive (persis-tent) correlations prevail for 1 / < H < X ( t = 0) of a fBm X ( t )is fixed and no further conditions can be imposed dur-ing the entire evolution, for 0 < t ≤ t , where t is thefinal time. Nevertheless, some applications such as thetracking of animal movement [14], information-based fi-nancial models [15], the number of neutrons involved inreactor diffusion [16, 17], or cosmic ray propagation inastrophysical turbulence [18] require that the stochasticprocess follows certain constraints .To be more specific, we focus on the important ex-ample of cosmic ray propagation in turbulent magneticfields [19, 20]. In order to overcome the overwhelming problem of resolving the wide range of scales involvedin these extremely high Reynolds number astrophysicalflows, several methods for generating synthetic turbulentfields were developed in the last decades [21–24]. Suchmethods are frequently implemented and used in majorcosmic ray propagation codes (see e.g. [18]). To capturelarge anisotropies due to the geometry of galaxies (spiralarms, outflow regions, bow shocks), synthetic turbulentfields must be embedded in large-scale magnetohydrody-namic (MHD) simulations of the turbulent interstellaror intergalactic plasma. Therefore, in this problem weface two challenges: i.) fBm, which is used to generatesynthetic fields, must be constrained to match the val-ues on the numerical grid of the MHD simulation and ii.)scaling properties, represented by the Hurst exponent offluctuations of the coarse-grained MHD simulation mustbe determined from sparse grid data, in order to allow foran “optimal stochastic interpolation” of sparsely-sampleddata. In this letter, we address precisely these two issues.We first present a method to construct a fBm X ( t ),which takes specific values X i at times t i , and discussits application as an optimal stochastic interpolation ofa sparsely-sampled real time signal. Accordingly, westart with the well-known notion of a fractional Brow-nian bridge [25, 26], which is defined as fBm startingfrom 0 at t = 0, ending at X at t = t , and possessingthe same statistical (including scaling (1)) properties as X ( t ). Such a fractional Brownian bridge (fBb) can beconstructed from X ( t ) according to X B ( t ) = X ( t ) − ( X ( t ) − X ) h X ( t ) X ( t ) ih X ( t ) i . (2)It is possible to generalize this ordinary fBb to an ar-bitrary number of prescribed intermediate grid points inthe following manner: First, we consider the n -times con- a r X i v : . [ phy s i c s . d a t a - a n ] M a y ditional moments h X ( t ) |{ X i , t i }i = h X ( t ) Q ni =1 δ ( X ( t i ) − X i ) i Q ni =1 h δ ( X ( t i ) − X i ) i , (3) h X ( t ) X ( t ) |{ X i , t i }i = h X ( t ) X ( t ) Q ni =1 δ ( X ( t i ) − X i ) i Q ni =1 h δ ( X ( t i ) − X i ) i . (4)We then demand that our bridge process X B ( t ) is con-ditional on X i at t i for i = 1 , . . . , n , which is equivalentto the process possessing the conditional moments (3-4).For a Gaussian process with zero mean and covariance h X ( t ) X ( t ) i , the conditional moments read h X ( t ) |{ X i , t i }i = h X ( t ) X ( t i ) i σ − ij X j , (5)and h X ( t ) X ( t ) |{ X i , t i }i = h X ( t ) X ( t ) i (6) − h X ( t ) X ( t i ) i h σ − ij − σ − ik X k X l σ − jl i h X ( t ) X ( t j ) i , where we implied summation over equal indices andwhere σ ij = h X ( t i ) X ( t j ) i denotes the covariance ma-trix. As shown in [27], the multi-point fractional Brown-ian bridge X B ( t ) = X ( t ) − ( X ( t i ) − X i ) σ − ij h X ( t ) X ( t j ) i , (7)possesses one- and two-point moments which are identicalto (5-6) and we thus conclude that X B ( t ) is the stochasticprocess X ( t ) conditioned on points X i at times t i . Weindeed obtain X B ( t k ) = X ( t k ) − ( X ( t i ) − X i ) σ − ij σ jk = X ( t k ) − ( X ( t i ) − X i ) δ ik = X k . Fig. 1(a) depicts themulti-point fBb (7) for three different Hurst exponentswith 16 equidistant fixed points (black).In order to check that the simulated bridge processespossess the desired properties (1) and (5-6), we have car-ried out numerical calculations of the second order struc-ture functions S ( τ ) = (cid:10) ( X B ( t + τ ) − X B ( t )) (cid:11) for threedifferent Hurst exponents ( H = 0 . , . , .
66) with a to-tal number of N = 32768 total grid points. From thesewe prescribed ˜ N = 32 equidistant points generated fromfBm (1) with a Hurst exponent ˜ H = 0 . S ( τ ) = | τ | H (dashed black lines)for small τ . For such a case, where the prescribed pointsalso follow fBm with Hurst exponent ˜ H , we can obtainan explicit formula for S ( τ ) from Eq. (7), namely S ( τ ) = | τ | H (8) − h δ τ X ( t ) X ( t i ) i h σ − ij − σ − ik ˜ σ kl σ − lj i h δ τ X ( t ) X ( t j ) i , where δ τ X ( t ) = X ( t + τ ) − X ( t ) and where ˜ σ ij = h X i X j i denotes the covariance matrix of the prescribed pointswith ˜ H . Consequently, H = ˜ H implies σ = ˜ σ , whichyields S ( τ ) = | τ | H . In other words, given a certain time . . . . . . t − . − . − . − . . . . . X B ( t ) ( a ) H = 0 . H = 0 . H = 0 . − − − − τ − − − − − − S ( τ ) ( b ) d t coarse H = 0 . H = 0 . H = 0 . . . . . . . . H . . . . . . | H − H e m p ( H ) | ( c ) sample 1sample 2sample 3 . . . . H . . . l s t s q FIG. 1. (a) Multi-point fractional Brownian bridge (7) fordifferent Hurst exponents H but identical grid points (black).The 16 equidistant grid points ( t i , X i ) were drawn randomlyfrom the interval X i ∈ [ − . , . S ( τ ) = h ( X B ( t + τ ) − X B ( t )) i calculated from 100realizations of the multi-point fractional Brownian bridge (7)for N = 32768 total grid points and ˜ N = 32 prescribed gridpoints. Dashed lines correspond to the explicit formula (8),whereas thin grey lines correspond to the ordinary scaling offBm | τ | H and extend up to the grid length of the coarse gridd˜ t = 1 /
32. The prescribed grid points were drawn as fBmwith ˜ H = 0 .
5. Accordingly, the fBb with H = 0 . optimal bridge that belongs to the prescribedpoints. (c) Cost curve H emp ( H ) − H for three under-sampledtime series ˜ N = 128 (fBm with ˜ H = 0 . , . , .
66 for sam-ples 1-3). Each time series is embedded into bridges (7) withvarying Hurst exponents H and resolution N = 4096. There-fore, the fit for H emp relies on the left part of (b). FBm wasgenerated by the Davies-Harte method [28]. series { X i , t i } that possess a self-similar part governedby ˜ H , the bridge with H = ˜ H can be considered as the optimal stochastic interpolation of this time series.Therefore, as already highlighted in the introduction,we are now in the position to describe an optimizationprocedure that allows us to estimate the Hurst exponentsfrom sparsely sampled time series. The basic idea is toembed a given time series { X i , t i } into fBbs (7) with vary-ing Hurst exponents H . For each of these bridges wedetermine the empirical Hurst exponent H emp as a func-tion of H by fitting the second order structure functionup to the smallest time scale of the time series d˜ t , (i.e.,fitting only the left part up to d t coarse in Fig. 1(b)). Thisprocedure ensures that we only measure deviations fromthe scaling | τ | H in the interpolated region (grey lines inFig. 1(b)) and are not directly contaminated by correla-tions contained in { X i , t i } . We have tested our optimiza- . . . . . . . x/L − . − . − . . . . . u ( x ) / √ σ ( a ) H=0.3786wind data . . . . . . . x/L − . − . − . − . . . . . u ( x ) / √ σ ( b ) H=0.3786rand. wind data
FIG. 2. (a) Turbulent velocity field measurements (blue) in avon K´arm´an experiment using normal Helium. The number ofpoints N = 16384 corresponds roughly to one integral lengthscale L . The corresponding fBb (orange) was constructedfrom ˜ N = 64 points of the signal (black) and possesses a res-olution of 1024 points. The smallest scale of the bridge pro-cess therefore corresponds approximately to the Taylor scaleof the flow which ensures that the fBb and the velocity field u ( x ) possess comparable inertial ranges. Small-scale turbu-lent fluctuations in the velocity field (blue) cannot be repro-duced by the fBb due to its restricting Gaussian properties.The Hurst exponent for the fBb H = 0 .
376 was determinedfrom randomized samples of the original turbulent signal suchas the one depicted in (b). Due to the self-similarity of thesesamples, an optimization procedure similar to the one de-picted in Fig. 1(c) could be applied. The bridge (orange) isin much better agreement with the self-similar signal (blue)in (b) than with the original signal (blue) in (a). tion for three different samples of fBm with Hurst expo-nents ˜ H = 0 . , . , .
66 and ˜ N = 128 grid points. Eachof the samples was embedded in fBbs with varying Hurstexponents H and N = 4096 grid points. Fig. 1(c) depictsthe minimization of H emp ( H ) for the three samples. Itcan be clearly seen that optimal Hurst exponents H opt are recovered with high accuracy, although slight varia-tions (∆ H opt ≈ .
015 from 20 different samples of thesame ˜ H ) between different samples can be observed [27].This effect can be attributed to finite sample sizes, andthe corresponding deviations remain rather small, whichis quite appealing given the fact that common methods(rescaled range analysis [29] or wavelets [30]) yield erro-neous results for such sparsely sampled time series. Theproposed method can thus roughly be considered as theextrapolation of self-similar properties of a given timeseries to finer scales.In the examples discussed so far, we have systemati-cally chosen the synthetic signal X i , as well as the pro-cess X ( t ) used in Eqs. (1,7), to be normalized in thesame manner. In order to apply our optimization toreal signals, we examine a turbulent velocity time se-ries obtained from hot wire anemometry in the super-fluid high Reynolds von K´arm´an experiment (SHREK) atCEA-Grenoble [31]. The particular experimental setupis a von K´arm´an cell with two-counter rotating disks (-0.12 Hz on top, +0.18 Hz on bottom) in normal Helium(see [32] for further specifications). The temporal reso-lution is 50kHz and the attained Taylor-Reynolds num-ber was Re λ = 2737. We applied Taylor’s hypothesis offrozen turbulence [33] to relate single-point velocity mea-surement at time t to scales x = h u i t , where h u i is themean velocity. Furthermore, a key prerequisite for theabove mentioned optimization procedure is the standard-ization of the signal by lim r → L p h ( u ( x + r ) − u ( x )) i = p h u ( x ) i = √ σ . This standardization ensures thecorrect large-scale limit of the second-order structurefunction in Fig. 1(b), which was necessarily fulfilled bythe synthetic samples of fBm (1) in the previous study.The blue curve in Fig. 2(a) depicts an extract of the ve-locity field u ( x ) standardized by σ of the total signal foraround one integral length scale L .By contrast to the above optimization procedure fordata from synthetic samples, the present analysis is com-plicated by: i.) the existence of different scaling regimesin the flow, namely a dissipative and integral range ofscales, and ii.) non-self similar (intermittent) features inthe signal. Turning to point i.), we chose sample sizesof length L and determined the subset of points in or-der to guarantee that their grid length lies within theinertial range of scales (here we choose ≈ η ). As faras point ii.) is concerned, intermittent features mani-fest themselves in form of strongly varying H opt for dif-ferent samples [27]. An example of such an intermit-tent fluctuation is well visible in the signal shown inFig. 2(a), at x ≈ . L . In this region our interpola-tion procedure does not work very well and the bridgeprocess (7) has to be generalized to a non-Gaussian pro-cess. In this letter, however, we are solely interested inthe self-similar part of the signal and thus perform a ran-domization of Fourier phases of the turbulent signal. Asnapshot of the resulting randomized signal is depictedin Fig. 2(b). Strikingly, and contrary to Fig. 2(a), our in-terpolation procedure leads to much better results thanfor the original signal. The randomization procedure haseffectively suppressed intermittency, and made the sig-nal essentially Gaussian. We fed several samples of thisrandomized signal into our optimization routine whichreduced fluctuations of the Hurst exponent to an ex-tend comparable to the ones observed in synthetic sig-nals [27]. Moreover, we obtain a Hurst exponent ofthe randomized signal H opt = 0 . ± . ζ n = n/ − µn ( n − /
18 for the scaling of velocitystructure functions h ( u ( x + r ) − u ( x )) n i ∼ | r | ζ n , where µ denotes the intermittency coefficient. Hence, the self-similar part in the model is given by H K = (2 + µ ) / µ = 0 . ± . µ = 0 . ± . multiscaling properties (i.e., which are non-self-similarand potentially possess a dissipative scale [36, 37]). Ageneralization of the bridge process (7) to an arbitrarynumber of dimensions is straight-forward and might beof potential interest for the construction of various syn-thetic fields in several physical contexts. In turbulence,the full spatio-temporal (though non-intermittent) Eule-rian velocity field u ( x , t ) can possibly be reconstructedfrom a set of Lagrangian trajectories X ( y , t ) where ˙X ( y , t ) = u ( X ( y , t ) , t ). Latter application could be ofconsiderable interest for particle tracking measurements,which sometimes require a certain knowledge of the flowfield in the vicinity of tracer particles [38]. Furthermore,the optimization procedure may help to shed light intothe ongoing discussion about the inertial range powerspectrum in the solar wind [11, 39]. Last, we mentionpossible applications to the widely different domainof urban decision making, where our method could be applied to model land price fields [40].We are grateful to the SHREK collaboration [31, 32]for providing us with their hot wire anemometrymeasurements. J.F. acknowledges funding from theHumboldt Foundation within a Feodor-Lynen fellowshipand also benefited from financial support of the ProjectIDEXLYON of the University of Lyon in the frameworkof the French program “Programme Investissementsd’Avenir” (ANR-16-IDEX-0005). S.G.’s visit to Lyonwas financially supported by the German AcademicExchange Service through a PROMOS scholarship. [1] H. Haken, Synergetics, An Introduction (Springer,Berlin, Heidelberg, 1983).[2] I. Prigogine, Eur. J. Oper. Res. , 97 (1987).[3] N. 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Wiese, Phys. Rev. E , 052105(2016).[26] T. Sottinen and A. Yazigi, Stochastic Processes Appl. , 3084 (2014).[27] See Supplemental Material for further references andproofs.[28] R. B. Davies and D. S. Harte, Biometrika , 95 (1987).[29] D. C. Caccia, D. Percival, M. J. Cannon, G. Raymond,and J. B. Bassingthwaighte, Physica A , 609 (1997).[30] I. Simonsen, A. Hansen, and O. M. Nes, Phys. Rev. E , 2779 (1998).[31] B. Rousset, P. Bonnay, P. Diribarne, A. Girard, J.-M. Poncet, E. Herbert, J. Salort, C. Baudet, B. Cas-taing, L. Chevillard, et al. , Rev. Sci. Instrum. , 103908(2014).[32] S. Kharche, M. Bon-Mardion, J.-P. Moro, J. Peinke,B. Rousset, and A. Girard, in iTi Conference on Turbu-lence (Springer, 2018) pp. 179–184.[33] A. S. Monin and A. M. Yaglom, Statistical Fluid Me- chanics: Mechanics of Turbulence (Courier Dover Publi-cations, 2007).[34] A. N. Kolmogorov, J. Fluid Mech. , 82 (1962).[35] A. M. Oboukhov, J. Fluid Mech. , 77 (1962).[36] L. Chevillard, C. Garban, R. Rhodes, and V. 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Neue Dimensionen der Mobilit¨at: Technischeund betriebswirtschaftliche Aspekte , edited by H. Proff(Springer Fachmedien Wiesbaden, Wiesbaden, 2020) pp.387–408. upplemental Material for “Multi-point fractional Brownian bridges and theirapplications”
J. Friedrich, S. Gallon,
1, 2
A. Pumir, and R. Grauer Univ. Lyon, ENS de Lyon, Univ. Claude Bernard,CNRS, Laboratoire de Physique, F-69342, Lyon, France Institute for Theoretical Physics I, Ruhr-University Bochum,Universit¨atsstr. 150, D-44801 Bochum, Germany (Dated: May 18, 2020)
I. CALCULATION OF CONDITIONAL MOMENTS FOR CENTERED GAUSSIAN PROCESS
Although the conditional moments of a multivariate Gaussian distribution are well known, in what follows, we wantto give a brief derivation. To this end, we consider the characteristic functional ϕ [ α ] = D e i R d tα ( t ) X ( t ) E . (1)For a centered Gaussian process X ( t ), the characteristic functional reduces to (see, for instance [1]) ϕ [ α ] = e − R d t R d t α ( t ) h X ( t ) X ( t ) i α ( t ) . (2)The n -point probability density function (PDF) f n ( X , t ; . . . ; X n , t n ) = n Y i =1 h δ ( X ( t i ) − X i ) i , (3)can thus be re-expressed according to f n ( X , t ; . . . ; X n , t n ) = Z d k π . . . d k n π e − i P ni =1 k i X i D e i P ni =1 k i X ( t i ) E = Z d k π . . . d k n π e − i P ni =1 k i X i ϕ " α ( t ) = n X i =1 k i δ ( t − t i ) = Z d k π . . . d k n π e − i P ni =1 k i X i exp " − n X i =1 n X i =1 k i k i h X ( t i ) X ( t i ) i . Therefore, we obtain a multivariate Gaussian distribution f n ( X , t ; . . . ; X n , t n ) = 1 p (2 π ) n det( σ ) e − X T σ − X , (4)where X = ( X , . . . , X n ) T , (5)and where σ ij = h X ( t i ) X ( t j ) i , (6)denotes the covariance matrix. Next, we can calculate the n + 1-point quantity, n Y i =1 h X ( t ) δ ( X ( t i ) − X i ) i = Z d k π . . . d k n π e − i P ni =1 k i X i D X ( t ) e i P ni =1 k i X ( t i ) E = Z d k π . . . d k n π e − i P ni =1 k i X i δϕ [ α ] δiα ( t ) (cid:12)(cid:12)(cid:12)(cid:12) α ( t )= P ni =1 k i δ ( t − t i ) = i Z d k π . . . d k n π e − i P ni =1 k i X i n X i =1 k i h X ( t ) X ( t i ) i e − P ni =1 P ni =1 k i k i h X ( t i ) X ( t i ) i = − n X i =1 h X ( t ) X ( t i ) i dd X i f n ( X , t ; . . . ; X n , t n ) , (7) a r X i v : . [ phy s i c s . d a t a - a n ] M a y as well as the n + 2-point quantity n Y i =1 h X ( t ) X ( t ) δ ( X ( t i ) − X i ) i = Z d k π . . . d k n π e − i P ni =1 k i X i D X ( t ) X ( t ) e i P ni =1 k i X ( t i ) E = Z d k π . . . d k n π e − i P ni =1 k i X i δ ϕ [ α ] δiα ( t ) δiα ( t ) (cid:12)(cid:12)(cid:12)(cid:12) α ( t )= P ni =1 k i δ ( t − t i ) = Z d k π . . . d k n π e − i P ni =1 k i X i " h X ( t ) X ( t ) i − n X i =1 n X i =1 k i k i h X ( t ) X ( t i ) i h X ( t ) X ( t i ) i e − P ni =1 P ni =1 k i k i h X ( t i ) X ( t i ) i = " h X ( t ) X ( t ) i + n X i =1 n X i =1 h X ( t ) X ( t i ) i h X ( t ) X ( t i ) i d d X i d X i f n ( X , t ; . . . ; X n , t n ) . , (8)The n -times conditional moments h X ( t ) |{ X i , t i }i = h X ( t ) Q ni =1 δ ( X ( t i ) − X i ) i Q ni =1 h δ ( X ( t i ) − X i ) i , (9) h X ( t ) X ( t ) |{ X i , t i }i = h X ( t ) X ( t ) Q ni =1 δ ( X ( t i ) − X i ) i Q ni =1 h δ ( X ( t i ) − X i ) i , (10)thus read h X ( t ) |{ X i , t i }i = σ − ij h X ( t ) X ( t i ) i X j , (11)and h X ( t ) X ( t ) |{ X i , t i }i = h X ( t ) X ( t ) i − h σ − ij − σ − ik X k X l σ − jl i h X ( t ) X ( t i ) i h X ( t ) X ( t j ) i . (12)where we made use of the symmetry of the covariance matrix and where we imply summation over equal indices. II. PROOF FOR MOMENTS OF MULTI-POINT FRACTIONAL BROWNIAN BRIDGE
In this section, we want to proof that the multi-point fBb X B ( t ) = X ( t ) − ( X ( t i ) − X i ) σ − ij h X ( t ) X ( t j ) i , (13)possesses identical one and two-points moments to the moments fBm process X ( t ) conditioned on { X i , t i } , i.e., eqs.(11-12). Therefore, we first calculate the mean of the fBb in eq. (13) (cid:10) X B ( t ) (cid:11) = h X ( t ) i | {z } =0 − ( h X ( t ) i | {z } =0 − X i ) σ − jk h X ( t ) X ( t j ) i = X i σ − jk h X ( t ) X ( t j ) i , (14)where we assumed that the fBm possesses zero mean. Next, the correlation function of the generalized fBb reads (cid:10) X B ( t ) X B ( t ) (cid:11) = h X ( t ) X ( t ) i − h X ( t ) X ( t i ) i σ − ij h X ( t ) X ( t j ) i + h X ( t k ) X ( t l ) i | {z } σ kl σ − ik h X ( t ) X ( t i ) ih X ( t ) X ( t j ) i σ − jl + σ − ik X k σ − jl X l h X ( t ) X ( t i ) i h X ( t ) X ( t j ) i = h X ( t ) X ( t ) i − σ − ij h X ( t ) X ( t i ) i h X ( t ) X ( t j ) i + σ − ik X k σ − jl X l h X ( t ) X ( t i ) i h X ( t ) X ( t j ) i , (15)where we made use of the identity σ kl σ − ki = δ il . .
275 0 .
300 0 .
325 0 .
350 0 .
375 0 .
400 0 .
425 0 . H opt P D F ( H o p t ) ( a ) .
275 0 .
300 0 .
325 0 .
350 0 .
375 0 .
400 0 .
425 0 . H opt . . . . . . . . . P D F ( H o p t ) ( c ) .
275 0 .
300 0 .
325 0 .
350 0 .
375 0 .
400 0 .
425 0 . H opt P D F ( H o p t ) ( c ) FIG. 1. (a) Histogram of the optimal Hurst exponent evaluated from 100 synthetic samples of fBm with ˜ H = 0 . III. VARIATIONS OF THE OPTIMAL HURST EXPONENT
An optimization procedure similar to Fig. 1(c) of the main paper for different synthetic samples drawn as fBmwith Hurst index ˜ H yields slightly different results for the optimal Hurst exponent H opt . In order to compare to theoptimization procedure for the turbulent signal later, we choose 100 synthetic samples of fBm with Hurst exponent˜ H = 0 . N = 128 points. Fig. 1(a) depicts the histograms for the optimal Hurstexponent obtained via optimization procedured as discussed in the main paper. The histogram is clearly peaked atthe prescribed values ˜ H and we obtain h H opt i = 0 . ± . − − δ r u/σ r − − − − − P D F ( δ r u ) ( a ) − − δ r u/σ r − − − − − − P D F ( δ r u ) ( b ) − − δ r u/σ r − − − − − − P D F ( δ r u ) ( c ) FIG. 2. (a) PDFs of velocity increments δ r u (log( r/L ) = − . , − . , − . , − . , − . , − . , − . , .
12 from top tobottom) of the randomized turbulent signal which has been obtained by random rotation of Fourier phases from the originalturbulent signal in (b). The PDFs in (a) exhibit self-similarity across different scales and the randomized signal can thus beused in the described optimization procedure via fBbs. (c) symmetrization of the PDFs in (b). The dashed curves correspondto the prediction of the K62 model of turbulence with µ = and are only shown for PDFs with scale separation r in the inertialrange. effects. Turning next to the turbulent signal in Fig. 2(a) of the main paper, we first perform a Fourier transformof the entire signal and then randomize Fourier phases in order to get rid of strong (intermittent) correlations. Theresulting signal possesses Gaussian statistics which can be verified by computing the PDF of velocity increments δ r u = u ( x + r ) − u ( x ) at different scales r , as it has been done in Fig. 2(a). For comparison, Fig. 2(b) depictsthe velocity increment PDF of the original turbulent signal for the same scale separations r as in (a). As to beexpected the PDFs develop pronounced tails at small scales, a key signature of small-scale intermittency. Due tothe self-similar property of the randomized signal in Fig. 2(a), the corresponding fluctuations can be treated by thesame optimization routine as the synthetic fBm. However, as explained in the main text, one has to take care of theexistence of a dissipation scale in both the turbulent and randomized signal. To this end, we consider plateaus in log-derivatives of structure functions and determine the small-scale cut-off to be approximately 96 η , where η denotes theKolmogorov microscale. We thus perform the optimization procedure for 100 different sub-samples of the randomizedsignal. The result is shown in Fig. 1(b) and shows many similarities to the one obtained from synthetic samples inFig. 1(a), although the standard deviation is slightly higher. In fact, we obtain h H opt i = 0 . ± . h ( δ r u ) n i ∼ | r | ζ n with scaling exponents ζ n = n − µ n ( n −
3) = an − bn . (16) H ε / σ ε ( a ) . . . . . . . . . ε/σ ε P D F ( ε ) ( b ) FIG. 3. (a) Joint histogram of energy dissipation rate and Hurst exponent evaluated from 320 different sub-samples of theturbulent signal. The axis of the energy dissipation has been divided by the standard deviation determined from all sub-samples.(b) Histogram of the energy dissipation rate.
The estimated value h H opt i implies that µ = 0 . ± . f ( δ r u ) = 12 πδ r vu √ ln r b Z ∞−∞ dx e − x exp − (cid:16) ln δ r ur a √ x (cid:17) b ln r , (17)This PDF is now used as a fit for the velocity increment PDFs in Fig. 2(b) after a symmetrization (the PDFs inFig. 2(b) are strongly skewed, a direct consequence of the turbulent energy transfer). The symmetrization is depictedin Fig. 2(c) and fitting the whole range of scales that were used for the optimization procedure from above, we obtain µ = 0 . ± . r are indicated as the dashed curvesin Fig. 2(c). Hence, the intermittency coefficient determined from the optimization procedure of the randomizedsignal and the intermittency coefficient from the turbulent signal agree fairly well.Finally, we applied the optimization procedure to the original turbulent signal in keeping the same range of scalesand number of sub-samples as for the randomized signal. The result is shown in Fig. 1(c). Compared to (a) and (b)the optimal Hurst exponent exhibits stronger sample-to-sample fluctuations. The mean value, however, agrees verywell with the Hurst exponent of the randomized signal and we obtain h H opt i = 0 . ± . ε = 2 ν (cid:16) ∂u ( x ) ∂x (cid:17) averaged over each sample. The result is depicted in Fig. 3 (a). Even with limited statistics, we observe a clearnegative correlation between energy dissipation and the determined value of the optimal Hurst exponent. [1] A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence (Courier Dover Publications, 2007).[2] A. N. Kolmogorov, J. Fluid Mech. , 82 (1962).[3] A. M. Oboukhov, J. Fluid Mech. , 77 (1962).[4] V. Yakhot, Phys. D215