Sensing scaled scintillations

1074 J. Opt. Soc. Am. A/Vol. 7, No. 6/June 1990

Sensing scaled scintillations

Bruce J. West

Department of Physics, University of North Texas, P.O. Box 5368, Denton, Texas 76203

Received August 8, 1989; accepted January 9, 1990

We review some of the ways in which the fractal concept has found application in wave-propagation contexts. Thescaling properties of fractals in both geometrical and statistical situations are reviewed and the relation to inversepower laws discussed. The relationship among the self-similar scaling properties of fractals, L6vy distributions,and renormalized group theory is explored to provide a simple picture of wave propagation through multiscalemedia. Finally, the notion of using a wavelet transform in the processing of fractal time series is considered.

1. INTRODUCTION

The remote sensing of the environment is usually accom-plished through the decoding of information carried bywaves that have been scattered by distant objects of interest.This decoding is routinely done passively as in the detectionand processing of reflected sunlight or sound, but only alimited amount of information is obtainable from the sens-ing of such uncontrolled signals. A great deal more can belearned when the generation of the wave to be detected iscontrolled in both amplitude and frequency. The wavesthat one uses in the latter case are selected to probe particu-lar properties of systems, e.g., they can be electromagnetic asin the use of radar, 1 or acoustic as in sonar, 2 or even elastic asin the nondestructive testing of solids.3 In these cases thetransmitted wave is scattered from the object of interest,and in the scattering process there is an attendant modifica-tion in its amplitude and phase. These modifications con-stitute the data of interest, i.e., the signal in the receivedwave. A proper physical theory would be one in which thescattering data could be directly inverted to yield the char-acteristics of the scattering object. Of course such a com-plete theory does not at present exist, the dramatic accom-plishments in such areas as the inverse scattering transformnotwithstanding.4 Instead we must be satisfied with theless-exact procedure of constructing physical models of thescattering process and comparing the predictions based onmodel calculations with the observed data. This has beenthe traditional method of explaining wave data, e.g., LordRayleigh's correct description of the scattering of long-wave-length light by the molecules in the air, resulting in theblueness of the sky.5

The scattering of vector and scalar waves from smoothgeometric shapes in homogeneous isotropic media is wellunderstood.6 On the other hand the scattering from roughsurfaces or objects having complicated shapes is less wellunderstood. In fact the simple propagation of wavesthrough inhomogeneous media is an area of active research.In particular, when the inhomogeneities are due to fluctua-tions in the refractive index of the medium, the scatteredwave field becomes stochastic. The statistics of the trans-mitted wave contain information about the fluctuations inthe traversed medium. The distortion in the signal trans-

mitted from satellites to Earth,7 the rapid variation in acous-tic waves in the deep ocean,8 the interplanetary scintillationof quasar radio sources,9 and the twinkling of starlight'0 eachdepend on the irregularities in the intervening medium. Ifwe are interested in the unperturbed wave, then the fluctua-tions imposed by the medium are considered to be noise.Then we employ a number of signal-processing techniquesto suppress them. If, however, it is the medium that we wishto understand, then the fluctuations contain the informa-tion. The strategy then is to enhance these fluctuations inorder to understand the irregularities in the medium.

The theory of the propagation of waves in random media,and in particular of the effects of multiple scattering on thewave traversing the medium, had its first success with theinvestigations of Foldy.11 He was interested in such pro-cesses as the multiple scattering of sound waves by the waterdroplets in fog, for which interference effects may be impor-tant. He considered the first- and second-moment proper-ties of scalar waves traveling in a medium of randomly dis-tributed isotropic scatterers and calculated the index of re-fraction. Lax12 later generalized this self-consistent treat-ment to include, in addition to the random placement ofscatterers, such effects as inelastic scattering, partial or com-plete ordering of the scatterers, and the effects of anisotropy.

More recently these considerations have been extended tothe study of multiple scattering of ultrasonic waves in elasticmedia with randomly positioned scatterers.13 The motiva-tion for these latter studies is the use of ultrasonic waves asprobes into the structure of materials. Material properties,such as the distribution of grain sizes in polycrystalline ma-terials, the degree of homogeneity, the existence of macro-scopic cracks, inclusions, twin boundaries, and dislocations,all affect fracture micromechanics and fracture-controltechnology. The basis of the ultrasonic approach is theobservation that the amplitudes of low-frequency (long-wavelength) ultrasonic waves (of known amplitude and di-rection) are exponentially attenuated with distance; i.e., thewave intensity decays as exp[-a(w)z], where z is the line-of-sight distance between the source and the receiver and a(W)is the frequency-dependent attenuation factor.

The standard theories of wave attenuation partition a(U)into an absorption attenuation coefficient (from such effectsas dislocation damping as well as magnetoelastic and

0740-3232/90/061074-27$02.00 © 1990 Optical Society of America

Bruce J. West

Vol. 7, No. 6/June 1990/J. Opt. Soc. Am. A 1075

thermoelastic hysteresis) and a scattering attenuation coef-ficient [from Rayleigh (X >> Do) and stochastic (X S Do)scattering, where X is the wavelength of the incident waveand Do is the typical size of the scatterer]. Each of theseeffects gives rise to an a(X) of the forma(c)) -W5,(1.1)where s = 1, 4, or 2, respectively, for dislocation damping,Rayleigh scattering, and stochastic scattering. In typicalmetals one encounters a range of grain sizes so that neitherthe Rayleigh nor the stochastic scattering limits are appro-priate. Instead both types of scattering can be simulta-neously present. In practice the phenomenological expres-sion

a(w) = Bco, 1 < < 4 (1.2)

is used, where B and kt are frequency-independent constants.West and Shlesinger'4 have argued that A is a measure of thedensity of scatterers in the material.

The noninteger value of As in Eq. (1.2) suggests that thevolume of space occupied by the scatterers is also nonin-teger. West and Shlesinger'4 argue that the volume occu-pied by the scatterers is fractal and that u is a direct measureof the dimensionality of the volume; i.e., A is a measure of thenoninteger fractal dimension. Wu15 has given a generaldiscussion of these ideas in the context of seismic velocity orimpedance inhomogeneities in the lithosphere of the Earth,which is the hard shell of the Earth with a thickness ofapproximately 100 km.

In Section 2 we review the concept of a fractal'6 anddiscuss both its geometrical and statistical realizations. Inthis discussion the self-similar nature of fractal objects (pro-cesses) is shown to be related to the notions of renormaliza-tion group (RG) theory.17""8 A RG relation or scaling func-tional relation for a given measure of a fractal process isshown to yield the dominant features of the process, includ-ing the fractal dimension, in terms of the parameters in thescaling relation. These ideas are important for the under-standing of fractal time series, which is often what is ob-tained by processing a wave that has traversed a multiscaledmedium.

Waves can also be used as fractal time series by scatteringfrom single multiscale objects, such as radar waves backscat-tered from the surface of the ocean and acoustic waves back-scattered from the ocean bottom. Therefore we reviewmodels of fractal surfaces that have been used to representthese physical surfaces. We find that the dimension of amodel sea surface is d = 2.25, indicating that the free oceansurface in a stormy sea is a fractal.1 9 This conclusion stemsfrom the inverse power-law behavior of the sea surface spec-trum and is related to a much more general result. In gener-al a self-similar (fractal) time series (one without a charac-teristic scale of time) has a power spectrum [S(w)] that is aninverse power law, i.e., S(w) - 1/fa. In this spectrum theindex a can be related to the fractal dimension of the under-lying process. We discuss some of the differences that onemight expect from stochastic processes having an inversepower-law behavior from, say, a Gaussian random processwith finite second moments.

In Section 3 we consider the statistics of a scalar wave fieldjust after scattering from a fractal surface or just afteremerging from a second medium containing multiple-scale

irregularities in its index of refraction. For weak scatteringin which only the phase of the transmitted wave is affected,the scaling nature of the fluctuations induce a scaling behav-ior in the scintillation statistics. This scaling charactergives rise to the stable limit distribution of Levy20 "' for thestatistics of the phase fluctuations. By showing that theprocess of coarse graining followed by scaling leaves theL6vy form of the characteristic function invariant when thescaling is appropriately chosen, it is also established that theLevy distribution has a RG symmetry.

The propagation of a scalar wave in free space away from afractal surface that has induced a fluctuating phase havingL6vy statistics is considered. Near the boundary the fieldfluctuates in phase only, but, as the wave propagates away,an interference pattern develops. The larger the phase gra-dient, the greater is the angle through which the transmittedwave is refracted and the nearer the boundary is the subse-quent interference pattern. It is the small spatial structurein the fractal surface or fractal scattering medium that givesrise to the large spatial gradients in the phase. Thus we findthat self-similar irregularities strongly influence the ob-served wave field.

We also investigate the propagation of a scalar wavethrough a fractal medium and examine correlations generat-ed by the large scales and by the small scales. In this multi-scaled medium we find the spectra for spatial structure to begiven by an inverse power law, just as in the case of hydrody-namic turbulence.2 2 The index of the inverse power law isrelated to the scale sizes in the medium as well as to therelative frequency with which these scale sizes occur.

Of central interest to our overall discussion is how tomeasure the multiscale or fractal nature of the time seriesdefined by the above modified wave fields. The fractaldimension gives one measure. The solutions to the RGrelations provide a harmonic scale length as well as a fractaldimension. But what we would like is a processing tech-nique that would enable us to magnify specified regions of areceived signal, i.e., to emphasize any desired scale in afractal structure or time series by adjusting suitable parame-ters. This is precisely what the wavelet transform does,23 aswe review in Section 4. Unlike the Fourier transform of asignal, which provides us with the frequency content of atime series, the wavelet transform can be used to zoom in onhigher and higher frequency structures in the time series.

The wavelet transform has been compared to a mathemat-ical microscope2 3 in which one can select the position alongthe time series, the particular optics to be used, and theamplification factor for the observation. These propertiesare demonstrated by the application of the wavelet trans-form to fractal functions and in particular to multifractalssuch as nonuniform Cantor sets. The wavelet transform hasalso permitted new insights to be gained in turbulent-flowfield data, which we also discuss.

2. IRREGULAR SURFACES AND FRACTALS

A brief review of the concept of fractals and fractal dimen-sions is presented in this section, along with some discussionof how to construct fractal media and surfaces. Rather thanpresent the rigorous mathematical theory underlying theseideas, we proceed by way of examples and analogy in thehope that they will be more useful to the nonspecialist.

Bruce J. West


A. Fractal DimensionsAs we discussed in Section 1, a fractal object or process ischaracterized by a fractional dimension. Mandelbrot, theperson who introduced the term fractal into scientists' lexi-con, originally defined a fractal as follows16: "A fractal is bydefinition a set for which the Hausdorff-Besicovitch dimen-sion strictly exceeds the topological dimension."

As Feder2 4 points out, this definition requires the furtherdefinition of the terms set, Hausdorff-Besicovitch dimen-sion (D), and topological dimension (DT), the last-namedterm always being an integer. For the purposes of thisreview we avoid the formal mathematical discussion neces-sary to give a rigorous foundation for these concepts andadopt the more informal definition proposed by Mandelbrotsomewhat later2 5: "A fractal is a shape made of parts similarto the whole in some way." This definition encompassesmany of the physical phenomena excluded by the more re-strictive definition originally suggested. Given this some-what loose definition of a fractal, we further develop theconcept through a sequence of simple examples.

Compared with a smooth, classical geometrical form, afractal curve (surface) appears wrinkled. Furthermore, ifthe wrinkles of a fractal are examined through a microscope,more wrinkles become apparent. If these wrinkles are nowexamined at higher magnification, still smaller wrinkles(wrinkles or wrinkles on wrinkles) appear, and seeminglyendless levels of irregular structure emerge. The fractaldimension provides a measure of the degree of irregularity.A fractal as a mathematical entity has no characteristic scalesize, and so the emergence of irregularity proceeds to eversmaller scales. A physical fractal, on the other hand, alwaysends at some smallest scale as well as some largest scale, andwhether this is a useful concept depends on the size of theinterval over which the process appears to be scale free.

What then is the length of a fractal line? Clearly, therecan be no simply defined length for such an irregular curve,independent of the measure scale, since the smaller the rulerused to measure it, the longer the line appears to be. Forexample Richardson2 6 noted that the estimated length of anirregular coastline or boundary L(77), where q is the measur-ing unit, is given by

L(Q7) = Lo 7q -d. (2.1)

Here Lo is a constant with dimensions of length, and d is aconstant given by the slope of the linear log-log line

In L(7q) = In Lo + (1 - d)ln q. (2.2)

For a classical smooth line, d = DT = 1 and L(7n) = constant,independent of tq. For a fractal curve, such as an irregularcoastline, d is the fractal dimension d = D > DT = 1. In Fig. 1we see that the data for the apparent length of coastlines andboundaries fall on straight lines with slopes given by (d - 1).From these data we find that d - 1.3 for the coast of Britainand d = 1 for a circle, as expected. Thus we see that L(77) -

- as 7 - 0 for a fractal curve, since (1 - d) < 0. The self-similitude of the irregular curve results in the measuredlength's increasing without limit as the ruler size diminishes.

Mandelbrot' 6 investigated a number of curves having theabove property, i.e., curves whose lengths depend on the unitof measurement. One example is the triadic Koch curvedepicted in Fig. 2. The construction of this curve is initiated

4.0 ! CIRCLE

SOUTH AFRICAN COAST

CERRJAN3.35 _ _ _ _ _ L "N 0 FRO

3.0

LAND ~RON PrTUGAL

1.0 1.5 2.0 2.5 3.0 3.5

Fig. 1. Fractal plots of various coastlines in which the apparentlength L(n7 ) is graphed versus the measuring unit q26: plotted asloglo [total length (km)] versus loglo length of side (km).

with a line segment of unit length, L(1) = 1. A triangularkink is then introduced into the line, resulting in four sec-tions each of length 1/3, so that the total length of theprefractal, a term coined by Feder,2 4 is (4/3)1. If this processis repeated on each of the four line segments, the total lengthof the resulting curve is (4/3)2. Thus after n applications ofthis operation we have

L(77) = (4/3)n, (2.3)

where the length of each line segment is

17 = 1/3n.

Now the generation number n may be expressed in terms ofthe scale n as

n = -In 77/ln 3,

so that the length of the prefractal is

L(,7) = (4/3)-(In 1/n 3) = exp[- I (ln 4 - In 3)]

=77 .d-(2.4)

Comparing Eq. (2.4) with Richardson's equation [Eq. (2.1)],we obtain

d = In 4/In 3 - 1.2628 (2.5)

as the fractal (Hausdorff-Besicovitch) dimension of the tri-adic Koch curve. Furthermore we note that the number ofline segments at the nth generation is given by N(77 ) = 4n =

4-(ln 1/0n 3), yielding

N(7n) = - /d (2.6)

as the number of line segments necessary to cover an irregu-lar curve of fractal dimension d. (See Feder 24 for a morecomprehensive discussion.)

To study these relations experimentally, Underwood andBanerji2 7 undertook the numerical construction of a Kochquadric island and a Koch triadic island. In the plot of L(7 7)versus Xq reproduced in Fig. 3, they observed a periodicity inthe experimental points that they attribute to the method ofmeasurement. I suggest an alternative explanation basedon an assumed scaling relation for the perimeter:

Bruce J. West


L(i7) = 3 L(77/3).

The general solution to Eq. (2.7) is of the form

L(77) = A(n)/n',

where by direct substitution we find that

a = In 4/In 3 - 1 = d - 1,

which is in agreement with expression (2.5), and, since

A(q7 ) = A(n/3),

(2.7) L(77)

10

(2.8) 8

n65

4

A.

_0.01 0.05 0.1 0.2

A(q7 ) is a periodic function in In 71 with period In 3. This

modulated, inverse power law would clearly explain the ex-perimental pointsin Figure 3. Equation (2.7) is an exampleof a RG relation, about which more will be said subsequent-ly. The periodic coefficient A(X) in Eq. (2.8) is often set to a

constant value in which case the functional relation (2.7)reproduces, by iteration, the expression (2.3).

Consider a second example28 in which a given mass isdistributed in a spherical volume of radius r so that the totalmass is given by

n-0

Fig. 3. Experimental fractal plots for Koch quadric island (topcurve) and a Koch triadic island (bottom curve). The apparentlength L(77) is graphed versus the measuring unit -. 27 (Taken fromRef. 27; used with permission.)

M(r) = Mord. (2.9)

If the mass consists of particles of sand and the sphere is ahuge beach ball, then in the absence of other knowledge it isusually assumed that the matter is uniformly distributed inthe available space and that d is equal to the Euclideandimension of the space, d = E = 3. Let us suppose, however,that on closer inspection we observe that the sand is notuniformly distributed but instead is clumped in distinctspheres of radius r/b, each having a mass that is 1/a smallerthan the total mass.

Thus what we had initially visualized as a beach ball filleduniformly with sand turns out to resemble one filled withbasketballs, each of the basketballs being filled uniformlywith sand. We now examine one of these basketballs andfind that it consists of still smaller spheres, each of radius r/b2 and each having a mass 1/a2 smaller than the total mass.Now again the image changes, so that the basketballs appearto be filled with baseballs, and each baseball is uniformlyfilled with sand. If we assume that this procedure of con-structing spheres within spheres can be telescoped indefi-nitely, we obtain, after n scaling operations, the mass rela-tion

M(r) = M(r/bn)/an = Mord(an/bnd), (2.10)

where the characteristic size of the grains of sand is infinites-imally small. Relation (2.10) yields a finite value for thetotal mass in the limit of n becoming indefinitely large only ifan/bnd = 1, i.e.,

d = In a/In b. (2.11)

Fig. 2. On a line segment of unit length a kink is formed, giving riseto four line segments, each of length 1/3. The total length of thiswave is 4/3. On each of these line segments a kink is formed, givingrise to 16 line segments each of length 1/9. The total length of thiscurve is (4/3)2. This process is continued through n = 5.

This noninteger value for the dimension d indicates that themass is distributed through space in a nonsmooth or clus-tered way. At first, the mass seemed to be given by a singlelarge cluster. As one came closer, it was observed that thecluster is really composed of smaller clusters, such that, onapproaching further, one sees that each smaller cluster iscomposed of still smaller clusters. This self-similar cluster-ing in space is typical of fractal distributions. Here again,the fractional value for d is the Hausdorff dimension of themass distributed throughout the Euclidean volume of radiusr.

One cannot always prove that an irregular curve is or is not

Bruce J. West

.

jl�_'


a fractal. Take for example the function constructed by theGerman mathematician Karl Weierstrass to represent a con-tinuous, nondifferentiable curve. His function is a superpo-sition of harmonic terms: a fundamental with frequency woand unit amplitude, a second periodic term of frequency bwowith amplitude 1/a, a third, and so on. The resulting func-tion is the infinite series of periodic terms:

1W(t) = - cos(b'cwot), (2.12)

n=O

each term of which has a frequency that is a factor b largerthan the preceding term and an amplitude that is a factor 1/asmaller. For b > 1 in the limit of n terms, as n approachesinfinity the frequency bnwo goes to infinity, and there is nohighest-frequency contribution to the Weierstrass function.Of course, if one thinks in terms of periods rather thanfrequencies, then the shortest period contributing to theseries is zero.

Consider now what is implied by the lack of a smallestscale in period or equivalently by the lack of a largest scale infrequency in W(t). Imagine a continuous line on a two-dimensional Euclidean plane, and suppose that the line hasa fractal dimension greater than unity but less than two.How would such a curve appear? In this case we are super-posing smaller and smaller wiggles; the curve would look likethe irregular line on a map representing a rugged sea coast.Also, it would have the self-similarity property discussedabove.

The self-similarity property of the Weierstrass functioncan be obtained directly from Eq. (2.12) by reorganizing theterms in the series to obtain

WWt = I WMb) + cos w0 t.a

(2.13)

Thus, if we drop the harmonic term on the right-hand side ofEq. (2.13), we obtain a simple functional scaling relation forW(t). The dominant behavior of the Weierstrass function isthen expressed by the functional relation

W(t) - W(bt). (2.14)

[In Appendix A the solution to the exact RG Eq. (2.13) isgiven.] The interpretation of this relation is that, if oneexamines the properties of the function on the magnifiedscale bt, what is seen is the same function observed at thesmaller scale t but with an amplitude that is scaled by a.This is the self-similarity (self-affinity) property discussedabove. Expressions of the form of Eq. (2.13) and relation(2.14) are often called RG relations. Relation (2.14) can besolved by the substitution into it of the functional form

W(t) = A(t)ta, (2.15)

where the power-law index a must be related to the twoparameters a and b in the series expansion by

a = In a/In b, (2.16)

just as we discussed above. Also, the function A(t) must beequal to A(bt), so that it is periodic in the logarithm of thevariable t with period In b. The algebraic increase of W(t)with t is a consequence of the scaling property of the func-tion with time. The scaling Eq. (2.15) in itself does not

guarantee that W(t) is a fractal function, but Berry andLewis29 have studied similar functions and have concludedthat they are in fact fractal. Consider the function

X(t) = E' 1 (1 - cos b'.ot),= an (2.17)

first suggested by L6vy30 and later used by Mandelbrot.16

The function X(t) scales in the same way as does W(t). Thefractal dimension d of the curve generated by Eq. (2.17) isgiven by 2 - d = a so that

d = 2 - In a/In b, (2.18)

which, for the parameters a = 4 and b = 8, is d = 4/3.Mauldin and Williams31 examined the formal properties ofsuch functions and concluded that for b > 1 and 0 < a < b thedimension lies in the interval (2 - a - c/n b, 2 - a), where cis a positive constant and b is sufficiently large.

This analysis is not unlike the real-space RG transforma-tion for the free energy F,32

F(K) = 1 F(K') + G(K),1E

(2.19)

where K is the interaction parameter in the Hamiltonian forthe original system, K' = K'(K) is the interaction parameterin the transformed system, 1 is the decimation length, E is theEuclidean dimension (so that the transformed system is afactor of jE larger than the original system), and G is ananalytic function of K. Following Shlesinger and Hughes,3 2

we express Eq. (2.19) in terms of a scaling field u:

F(u) = - F(Xu) + G(u),1E

(2.20a)

with X a real, relevant eigenvalue greater than unity, wherewe assume that u is a regular function of K and G is a regularfunction of u. The functional relation (2.20a) can be iterat-ed N times to yield

F(u) 1 FNu)+E 1 G(u)n=0

(2.20b)

Since F(u) is assumed to be a bounded function of u in thelimit N - a, we have the boundary condition

lim 1 F(XNu) = 0,N-a, 1NE

and Eq. (2.20b) becomes

F(u) = >nE G(X'u).n=O

(2.21)

(2.20c)

Thus all singular behavior will be contained in the suminvolving G.

A solution for the singular part of F(u) is then obtainedfrom

Fsjng(U) = 1 Fsing(X)

which has a solution of the form

Fsing(U) = AMME11,

where

(2.22)

(2.23)

Bruce J. West


A = In X/In 1 (2.24)

and

A(u) = A(Xu) = E An exp[27rin(ln u/In A)J, (2.25)n

just as for Eq. (2.15). Shlesinger and Hughes3 2 note that therelevant eigenvalue A > 1, so gA > 0. The existence of theoscillatory solutions, the A(u), was noted first by Novikov33in his study of intermittency in turbulent fluid flow and laterin a critical-phenomena context by Nauenberg34 and by Nei-meijer and van Leeuwen35 and in a statistical context byJona-Lasino.3 6 In critical phenomena the eigenvalue X ischosen to depend on 1 in such a way that the exponent ,u isindependent of 1. The oscillatory terms are suppressed be-cause their period depends on the decimation scale I throughthe eigenvalue X(1), while the free energy should be 1-inde-pendent. These oscillations are quite prominent in otherphysical systems and in particular in a number of biomedicalsystems.3 7 -39

Mandelbrot's concept of a fractal liberates our ideas ofgeometric forms from the tyranny of straight lines, flatplanes, and regular solids and extends them into the realm ofthe irregular, disjoint, and singular. As rich as this idea is,we require one additional extension into the arena of fluctu-ations and probability, since it is usually in terms of averagesthat data sets and time series are understood. If we nowinterpret X(t) as a random function, then, by analogy withthe Weierstrass function, we assume that the probabilitydensity satisfies a scaling relation. Thus the scaling proper-ty that is present in the variable X(t) for the usual Weier-strass function is transferred to the probability distributionfor a stochastic variable. This transfer implies that, if theprocess X(t) is a random variable with a properly scaledprobability density, then the two stochastic functionsX(bllat) and bX(t) have the same distribution. This scalingrelation establishes that the irregularities in the stochasticprocess are generated at each scale in a statistically identicalmanner. Note that for a = 2 this is the well known scalingproperty of Brownian motion with the square root of thetime. Thus the self-similarity (self-affinity) that arises in astatistical context implies that the curve [the graph of thestochastic function X(t) versus t] is statistically equivalentat all scales rather than geometrically equivalent. We dis-cuss this idea more comprehensively below.

B. Fractal SurfacesThe extended Weierstrass function given by Eq. (2.17) is arestricted version of the one studied by Berry and Lewis29 :

X(t) = N M [1 - exp(ibw0 0t)]e'0", (2.26)n=--

where the phase On is arbitrary and we have set 2 - d = in a/In b, using Eq. (2.18), which results in the restriction 1 < d <2. In the following discussion we set w0 = 1 and draw fromthe analysis of Berry and Lewis those results that we finduseful for our present purposes.

We can choose the set of phases IMrn deterministically, aswe did in Subsection 2.A, or randomly as we do subsequent-ly. If S n is a random variable uniformly distributed on theinterval (0, 27r), then each choice of the set of values {¢0n)

constitutes a member of an ensemble for the stochastic func-tion X(t). If the phases are also independent and b - 1+,then X(t) is a Gaussian random function. The condition 1 <d < 2 is required to ensure the convergence of the sum in Eq.(2.26).

Ausloos and Berman40 further generalized the Weierstrassfunction to include space as well as time. In this generaliza-tion they impose the requirement that, like the originalWeierstrass function, its generalization must preserve theproperties of homogeneity and scaling. Consider the incre-ments of X(t):

AX(t, T) = X(t + T) - X(t)

= E b' n(2-d)exp(ibnt) - exp[ibn(t +r)]}ein,

n=--

(2.27)

and assume that the On are independent random variablesuniformly distributed on the interval (0, 27r). The mean-square increment is

C(T) = (IAX(t, r)|2)

= > b )2[1 - cos(b T)],

n=--

(2.28)

where the angle brackets denote an average over an ensem-ble of realizations of the 'On fluctuations. The right-handside of Eq. (2.28) is independent of t; i.e., it depends only onthe time difference r, SO that X(t) is a homogeneous (alsocalled stationary when t is the time) random process. Equa-tion (2.28) is identical in form to Eq. (2.17), which is the realpart of Eq. (2.26) with ,n set to zero and d = 2 - ln a/In b.

Like Eq. (2.17) Eq. (2.28) also leads to a scaling relation:

C(bT) = b2(2-d)C(T) (2.29)

so that the correlation of the increments of the extendedWeierstrass function is self-affine. If the sum that gives theextended Weierstrass function in Eq. (2.26) covered the in-terval n = 0 to -, then the scaling property expressed in Eq.(2.29) would be only approximately true. A two-variable,scalar, extended Weierstrass function X(r) was proposed byAusloos and Berman to have the properties

C(p) = (IAX(rp)12), (2.30)

where AX(r, p) is the analog of Eq. (2.27) and for all r and thecorrelation function satisfies the scaling relation

C(bp) = b2(3-d)C(p) (2.31)

where in this case 2 < d < 3.After discussing and discarding for physical reasons a

number of candidate functions, Ausloos and Berman decid-ed on one that is a random superposition of weighted ridge-like surfaces:

X(r) ( / Amm=l

X (kobn)(d-3)1 - exp[ibnkor cos(O - am)]Jeilm. (2.32)

n=--

Here ko, a wave number that can be used to scale the horizon-tal variation, replaces the frequency wo in Eq. (2.26). The

Bruce J. West


normalization factor (In b/M)1"2 is chosen to make the seriesconverge as b - 1+ and M - a. The angle am gives theorientation of the correlation of the surface having an ampli-tude Am, and the phases lqmn} are again defined as uniformrandom variables. It is a simple matter to verify that Eq.(2.32) has the scaling property of Eq. (2.31) and that C(p) istherefore spatially homogeneous.

Ausloos and Berman present a substantial number ofcomputer plots of the surfaces generated by the extendedWeierstrass function [Eq. (2.32)]. For deterministic phasesthe function is analogous to that studied by Berry and Lewis,for which they have shown that one can extract trends byusing the Poisson summation formula (when b - 1+ and thedensity of wave vectors is large). In the case of randomphases Ausloos and Berman demonstrate that the surfacecoincides with the fractional multidimensional Brownianmotion described by Mandelbrot.16 These surfaces are in-tended to represent undersea topography in order to mea-sure the effect of relatively small-length scales on long-rangeacoustic propagation in the ocean. The effect of such sur-faces on a vertically incident acoustic wave is discussed be-low.

Here we are also interested in the possible use of thisfunction to model dynamic surfaces such as wave motion onthe ocean surface. Therefore we examine some of the com-puter plots of Ausloos and Berman as if they were snapshotsof the sea surface. In Fig. 4 we see a profile that could beinterpreted as a two-dimensional sea surface consisting oflong ridges of random height and wavelength. This is notunlike the surface profiles shown in Ref. 41 (pp. 330 and331). In Fig. 5 two of these surfaces at right angles aresuperposed to give rise to a two-dimensional random surfacethat resembles that of a high-sea state.. To the unaided eyethe most convincing example of an ocean surface is given in

5.0 -

1.0 -

-3.0 -

Fig. 5. Surfaces for M = 2 with random phases (D = 2.5, b = 1.2).The upper surface is the sum of the two lower surfaces. (Takenfrom Ref. 40; used with permission.)

Fig. 6, in which we see the ever-present small-scale structureriding on top of a large-scale wave.

A different extended Weierstrass function, but one thatshares the properties of homogeneity and scaling in twodimensions, was considered by Falconer4 2:

X(r, t) = (lnb)l/ E Am E bn(3 d)m=l n=-X

X cos[kn(x cos Am + y sin Om) + 'mn]- (2.33)

If we interpret the phase as the time-dependent quantity

I'mn = 'Omn - Unt, (2.34)

where Wn is the frequency of a deep-water-gravity wave hav-ing the wave number kn = kobn,

n = = Ekobn2, (2.35)

PV 5.01

L- 0.2 L -0.51.0-

XN -3.0 - 1

L =-1.2

Fig. 4. The extended Weierstrass function [Eq. (2.32) with randomphases. L is the level of an artificial floor, which as it is loweredreveals more of the surface (M = 1, D = 2.5, b = 1.5). (Taken fromRef. 40; used with permission.)

and g is the acceleration due to gravity, then we have

X(r, t) =E d Amm=Am

X bn( 3 d) cos[kobn(x cos Am + y sin Om)n=O

+ qOmn - wobnl2t], (2.36)

in which a superposition of propagating deep-water-gravitywaves has been made to form X(r, t). The amplitudes of Amcan be chosen in such a way that the mean-squared value ofX(r, t) is the same as that of the sea surface. To do this wefollow Phillips4 3 and write the surface-wave spatial spec-trum as

T(k) = g'12Bu. G(Q) (2.37)

Bruce J. West


where the function G(0) accounts for the directionality of thewaves, u* is the wind-friction velocity, B - 0.012, and

J d2'k(k) = ( ') (2.38)

where t(r, t) is the surface displacement and (r2) is themean-squared level of the sea surface. Pierson4 4 proposedthat the random ocean surface be written as

t(r, t) = f J cos[k(x cos 0 + y sin 0) - Wnt + 4(k, 0)]

X [4I(k)kdkd0 J'l, (2.39)

where the wave vector is k = (k cos Ok sin 0), the linear deep-water-gravity wave frequency is ('Wk = igw, and 0(k, 0) is auniform phase on the interval (0, 27r). Stiassne'9 relates thetwo expressions by noting that we have selected a geometricseries in wave number so that we can replace kdkd0 by thediscrete version Om = mAO(AO = 27r/M) and dkn = kn in b and,by using Eq. (2.37), we can obtain

['I(k)kdkd0 ]/2 - [ f 2 (m)]b22. (2.40)

Thus we can write Eq. (2.39) as

t7rg"'1u.B in b 1 MP(r, t) = Mko0 )-2 ,E [G(0m)]"'

X b "/2'3 - cos[kobn(x cosOm + y sin Om) - wobt + 'kmnl.n=O

(2.41)

Comparing Eq. (2.41) with Eq. (2.36), we obtain

Am = [27rg91 2u.BG(0jm)' 1 2,

d = 4 - ,3/2.

(2.42)

(2.43)

Thus the surface represented by Eq. (2.41) has a fractaldimension d dependent on the index of the observed surface-wave spectrum. If we take /3 = 7/2 as suggested by Kitaigor-odskii,45 then the fractal dimension of the sea surface is d =2.25, indicating that the free water surface in a stormy sea isa fractal.1 9

As Stiassne further points out, surface tension introducesa short-wavelength cutoff of, say, k = ks, so that the actualsea surface is only approximately fractal. It has a fractal-like structure over length scales from 27r/ks, which corre-sponds to approximately 0.1-100 m in the open ocean.

Barenblatt and Leykin46 present a quite different argu-ment based on similarity principles to obtain a universal lawfor the dependence of the equilibrium spectrum on the de-gree of wave development. Glazman and Weichman 47 ex-press this law in the wave-number domain (in our notation)as

'F(k) = G(0)f(X)g2p UTM (2.44)

where ja is interpreted as the fractal codimension for a sur-face path. Here U is the mean wind speed at a convenientheight above the sea surface, say, 19.5 m, and f is a universalfunction of the nondimensional fetch x c (Co/U)1013 and Co isthe phase speed of the dominant wave on the sea surface.Comparing Eq. (2.44) with Eq. (2.43), we obtain

(2.45)/ = 4-2,

and

a =2+ ,u. (2.46)

L = -104.1 L = -479.2

Fig. 6. Four magnifications of the surface for D = 2.05, showingself-similarity (M = 8, b = 1.2). The upper-right-hand surface isthe fivefold magnification of a section of the upper-left-hand sur-face. Similarly, the lower-left-hand surface is a fivefold magnifica-tion of a piece of the upper-right-hand surface, and the lower-right-hand surface is a fivefold magnification of the lower-left-hand sur-face. The vertical extent is magnified by 5 3-D. (Taken from Ref.40; used with permission.)

Again assuming the value a3 = 7/2, we get y = 1/4, which is thevalue obtained by Zakharov and Filonenki4 8 and discussedby Glazman and Weichman. The latter authors discuss theabove concepts much more fully, and their paper is recom-mended reading.

C. Inverse Power LawsAs we have been discussing, many of the complex systems inwhich we are interested do not possess a characteristic scalelength. Static structures such as geological faults,49 thevariations in the plasma density of the ionosphere,5 0 and thetopography of the ocean bottom40 have a broad range ofspatial sizes that are scale-free. Dynamic structures such asthe ocean surface43 and the variability in air density of theatmospheres fall into the category of processes having asignificant range of temporal scales; i.e., they lack a charac-teristic time scale. The spectrum of the process, i.e., thedistribution of energy over frequency (dynamic process) or

L = -5.0 L = -23.9

Bruce J. West


over wave number (static structure), is one measure of itscomplexity. A process having a self-similar or self-affinebehavior has an inverse power-law spectrum.

Consider the extended Weierstrass function discussed inSubsection 2.B. Its correlation function is given by Eq.(2.28) and satisfies the scaling relation (2.29):

C(T) = b-2(2-d)Qbr) (2.47)

The solution to this renormalization equation is of course

C(r) = A(r)ra, (2.4

where

tion mechanism yielding a RG relation properly describesthis crossover behavior.

The noise spectrum of a purely random process is oneassociated with an autocorrelation function of the form C(t,T) = e-t/, where T is the relaxation time of the process. Thepower spectrum of such a random process is, at frequency f,

S(f, r) = 4 Re J C(t, r)exp(27rift)dt = r2)f ~~~~~1 + 2fT2

(2.51)

A complex system is characterized by a distribution of relax-8) ation times p(r), not just the single parameter in Eq. (2.51).

The power spectrum for the system is therefore given by

(2.52)S(f) = - 4rp(T)drJo 1 + (2.7rfr)2'

and in general A(r) is a harmonic function in the In T withperiod In b. If we assume that A(T) is essentially constant,we can apply a Tauberian theorem originally due to Hardy52

that states that a function f(tr) - Ta for small r has a Fouriertransform 1(W) (W-a-l for large w. Thus the spectrumassociated with the extended Weierstrass function is

Wa1S(cw) C(w) z +, (2.50)

an inverse power-law spectrum. Note that the index of theinverse power law is related to the fractal dimension of theunderlying process [Eq. (2.49)].

The systems in which such l/fa phenomena are observedrange from the physical to the biological to the social.Therefore one would not expect a detailed model designed toexplain its occurrence in one context to be transferable toanother disjoint context. Thus we do not focus herein onany one microscopic model proposed for its explanation, butrather we adopt the point of view that 1/fa noise is a conse-quence of a system's complexity, irrespective of the context.Complex systems consist of many interacting parts, and thevast number of interactions engenders a certain amount ofuncertainty. This uncertainty or erratic behavior is classi-fied as noise. The more the noise, the less certain one can beof the outcome of a process; the noise is unpredictable (ran-dom) variation in the phenomenon of interest. Thus, ratherthan knowing the evolution of a process with certainty, wecan only indicate its possible future states by a probabilityand the full span of possible future states by a probabilitydistribution function. The distribution function capturesthe system's complexity, and we are concerned with theuniversal limiting form of these distributions independentlyof details of possible mechanisms. It is the universality ofthe form of the distribution that explains the ubiquitousnature of 1/fa phenomena.

West and Shlesinger53 reviewed the arguments interrelat-ing the log-normal distribution, 1/fa noise, fractals, andbranching processes. They showed that, when a populationis engaged in tasks whose completion requires the successfulcompletion of many independent subtasks, the distributionfunction for success in the primary task is log normal. Also,when the dispersion of the log-normal distribution is large, itis mimicked by a 1/f distribution over a wide range of f.54 Itis commonly found that distributions that seem to be lognormal over a broad range (say, to the 95th percentile of apopulation) change into an inverse power-law distributionfor the last-few percentile. We show here that an amplifica-

Here we assume that the events determining p(T) are de-scribed by the successful completion of grand processeswhose progress is the consequence of the successful comple-tion of a number of independent subprocesses; i.e., it is of themultiplicative form P = PsP2 ... .PN, where pj, j = 1, 2, . .. , Nis the probability of completing the jth subtask and P theprobability of completing the grand task. If we interpret Pas the probability per unit time that a grand process occurs,then r = 1/P is the time required for a single grand success.The distribution function of T, weighting the many ways inwhich a success might be achieved, is obtained from In T =

ln(l/P) = E ,N ln(1/pj), and, since each pj is a randomvariable, so too is ln(1/pj), and for large N under appropriateconditions the distribution function become log normal.53The power spectrum in Eq. (2.52) then becomes

S ) 4i1 exp- [ln( r/r)]2/2,2d(J (2ra f1 .o, 1 + W r'/' k1 (2.53)

with T being the average relaxation time and w being thedimensionless frequency

w = 27rft. (2.54)

Then, defining z to be z = r/iT, Montroll and Shlesinger54obtain in the T - - limit

S(w, ff O ) -(2Xr 2)1/2o 1 + d2z2

- r 1/2 1= 8Tr2 W)

(2.55)

from which we see that the log-normal distribution in relax-ation times leads to a 1/f noise spectrum as a. -p , i.e., as thevariance of the log-normal distribution a. becomes infinite.Thus multiplicative processes can have a 1/f power spec-trum; however, high- and low-frequency cutoffs exist. Therange of 1/f behavior depends on the value of a, which in turndepends on the precise mechanisms involved. Differentsystems should have different cutoffs.

Mandelbrot 1 6 suggested a way to visualize such self-simi-lar time series by using sound. Consider a recording ofviolin music, for example. If the recording is played fasteror slower, the distinctive sound of the violin is distorted andlost, indicating that the music has a characteristic time scale.There is, however, a special class of sounds that when playedslower or faster than normal do not change; only an adjust-ment of the volume is required to have the new sound be-come identical with the old. Such sounds are called scaling

a = 2(2-d), (2.49)

Bruce J. West


noises, the simplest example of which is white noise havingan a = 0. This sound is the colorless hiss that is equally dullwhen played at any speed, for example, the shsh sound ofrunning water. The correlation function of white noise withitself is zero, indicating that fluctuations at one instant areindependent of those at any other instant. This noise is thestatic that one hears on the radio or the snow that is seen on atelevision set when there is no input signal.

There are a large number of phenomena that can be classi-fied as scale-invariant noise having the general spectral forml/fa. To understand this fact better, it is worth observingthat, over a large range of an independent variable, distribu-tions might be of a standard type such as normal or lognormal but then suffer a transition in the last-few percentileof a population into an inverse power law. Similar behavioris observed in correlation functions.5 3"54 Consider the basiccorrelation function C(t/r) = e-t/T used above, where r is thecorrelation (relaxation) time for the time series. With asmall probability p, let C(t/T) be amplified such that it has anew correlation time NT, i.e., C()dt - C(Q/N)d{/N, where t= t/l-. In the second stage of amplification, which we as-sume occurs with probability p2, the correlation time r be-comes N 2'r. The new correlation function c(t) containingcontinuing levels of amplification is

c(t) = (1 -P)[C() + - QC(/N) + p2 C(t/N2) +

(2.56)

where the factor (1 - p) arises to ensure that c(t) is properlynormalized. Equation (2.56) can be reordered into the formof the recursion relation

c(t) = N c(Q/N) + (1 - p)C(Q), (2.57)N

which as we have observed already can be solved by using themethod of Mellin transforms.

The singular part of the solution to Eq. (2.57) is of coursegiven by (as t - c)

c(t) = A(t)t"l, (2.58)

where a = -ln(1/p)/ln(N). In the present discussion eachscale r that contributes to c(t) corresponds to a shorterrelaxation time, so that N < 1 and a > 0. Thus we observethat, for slowly varying A(t) [recall Eq. (2.48)], we obtain thepower spectrum

S(fo 1/fa. (2.59)

Therefore, just as nature selects static structures with nofundamental length scales, so too it selects dynamic process-es with no characteristic time scale. So fractals can exist intime as well as in space.

3. NON-GAUSSIAN STATISTICS OF PHASEFLUCTUATIONS

Let us now turn to the question of how a scalar wave isdisrupted as it propagates through a finite region of spacehaving a fluctuating index of refraction. We assume thefluctuations to be weak, so the variations in the amplitude ofthe transmitted wave are negligible and only the vacillationsin the phase are important. This prototypical model has

proved to be of value in understanding the effect of a spatialspectrum of fluctuations on radio-wave and radar propaga-tion through layers of plasma turbulence in the ionosphere,50

on acoustic-wave propagation through fluid turbulence inthe deep ocean,8 and on laser-beam propagation throughatmospheric turbulence,5 ' to name just a few applications.We emphasize that this class of problems remains an activearea of research. Our intent here is to examine one smallpart of the overall problem of wave propagation in randommedia, that part concerned with the statistical properties ofthe fluctuations, and to explore some of the consequences ofself-similar statistics and the RG property.

If the scattering region is localized, it is assumed in thismodel that it can be replaced by a thin phase screen.5 0 Thecentral assumption in this replacement is that the individualscatterings that the wave undergoes as it propagates througha region of spatially fluctuating refractive index are notimportant. The physical observable outside the scatteringregion is the change in phase of the received wave from thatof a reference wave that would traverse the same region inthe absence of the scatterers. Thus, if one is interested onlyin the net effect of the scattering region, the entire intervalcan be characterized by a phase screen having the desiredbulk properties; i.e., the phase screen gives rise to the sameoverall phase shift as the scattering region. The originaltreatment15"50 assumed that the statistics of the phase fluc-tuations were Gaussian; we do not assume that here.

In this analysis the phase screen is segmented into a num-ber of sections. The number of segments, say, N, used torepresent the scattering region is arbitrary and is usuallyselected for computational ease. In order for this to be asatisfactory model of physical wave propagation, either thestatistical properties of the overall phase shift must be inde-pendent of N or any N dependence must vanish in the limitN - -. Using this simple scattering model, we find that thephase shifts resulting from the variations in the index ofrefraction are determined by a RG property. The distribu-tion of phase shifts is shown to yield statistical propertiesthat are independent of the phase-screen partitioning. Weestablish that the distribution of phase fluctuations is thestable distribution of L6vy,20

,2 and it is symmetric withrespect to a coarse graining of the phase screens followed bya scaling. This process of coarse graining and scaling is theprescription of Morris and Kadanoff17 for determining theRG properties of a bulk-spin system. Here we apply thistechnique to show that the Levy characteristic function is afixed point of this transformation and therefore has a RGsymmetry.

For the physical model that we consider here, we assumethat the refractive-index fluctuations are small and can betreated as perturbations and that the wavelength of theincident wave is much smaller than the smallest scale size ofthese fluctuations, i.e., smaller than the first Fresnel zone.Thus we can ignore polarization effects, and, for an electricfield component u(x, y, z) (or for an acoustic field) propagat-ing along the z axis, we can write

JV2 + k 2 [1 + nl(x)] 2 1v(x) = 0, (3.1)

where k is the free-space wave number and nl(x) is therefractive-index fluctuation. The high-frequency approxi-mation permits us to replace Eq. (3.1) by the parabolic equa-tion. If we ignore the effect of free-space diffraction, it can

Bruce J. West


be shown that the field u atz + Az is expressed in terms of thefield at z as

v(x, y, z + Az) = v0(x, y, z)exp[iT(x)].

The modulated phase I(x) due to the refractive-index fluc-tuation between z and z + Az is given by

NF(x) = k j nl(x, y, z')dz'. (3.2)

In our analysis we segment the interval Az into N pieces ofequal length, each contributing a phase Tj to Eq. (3.2), whichbecomes

N

1(x, t) = ' Fj(x, t).j=l

(3.3)

As mentioned, we are interested in determining the N-inde-pendent statistical properties of T(x, t), where kq!(x, t) is thephysical phase shift and k is the wave number of the incidentwave.

We restrict the line-of-sight propagation problem to theone-dimensional propagation of a monochromatic scalarwave through a region with a weakly spatially fluctuatingindex of refraction confined to the interval (0, Az) along the zaxis. The fluctuations are assumed to be stationary in time,as they would be for a disordered glassy material or forfrozen turbulence in the atmosphere or the deep ocean. Wedivide the region into N phase screens (intervals) each oflength 1, i.e., Az = NI, and examine the statistical propertiesof the wave emerging from the location z = Az. It is reason-able to assume that the statistics of the phase fluctuationsresulting from any given interval 1 are identical to the statis-tics resulting from any other interval. Any systematic devi-ation of the statistics could in principle be included in thedefinition of the reference field vo(x, t). Thus the overallphase shift of the transmitted wave consists of a sum ofidentically distributed random variables. This assumptionis usually accompanied by an argument involving the cen-tral-limit theorem with the inevitable conclusion that thephase shifts are normally distributed. We emphasize thatsuch is not the case here, and the most rigorous arguments,without making additional assumptions, lead one to con-clude that the distribution of phase-shift fluctuations is sta-ble but not necessarily normal.2 0'2 '

A. L6vy (Stable) DistributionHere we seek the probability distribution for the sum of Nrandom phase shifts in the limit N - -. In standard nota-tion55 we write the total phase shift for N phase screens as

NT(N) = B_ X, Ti - AN, (3.4)

where the parameters AN and BN are chosen to give theaverage phase shift AN and to normalize the sum with BNproperly to ensure convergence. The limit distribution ofsuch a sum of identically distributed random variables hasbeen studied extensively by L6vy20 and is discussed at lengthby Gnedenko and Kolmogorov.55

For clarity let us consider the separation of the scatteringregion (0, Az) into two equal parts. The phase shifts fromthe two regions are labeled T, and '2. The statistics of each

phase shift are determined by the same probability densitybut with different characterizing parameters. Then, if forthe positive numbers C, and C2 the variable

CN(2) = Cl,'l + C2 F2 , (3.5)

with C being a function of C, and C2, has the same probabilitydistribution as T, and 'T2 but with new characterizing pa-rameters, then the probability distribution is said to be sta-ble.2 ' Let P(4I)dV, be the probability that the dynamic vari-able T is the phase-space interval (,6, ,6 + dl). The charac-teristic function associated with the probability density P(ik)is

P(K) = J dP exp(iKq/)P(V/) (3.6)

and is the quantity for which an explicit analytic form can beobtained for stable statistical processes. It is worth goinginto some detail here because the argument provides insightinto the physical content of the form of the stable distribu-tion.

The characteristic function for the total phase shift T(2)

can be related to the characteristic functions of the individ-ual (independent) phase shifts T, and TI2. When two ran-dom variables are independent, their joint probability densi-ty P('1, N2) factors as

P(l'1, 42) = P1('PdP2(P2)-

The characteristic function for this joint process is

P(CK) = (exp(iCKx'( 2 )))

= J d', J dP2 exp[iK(C1f'1 + C20 2)]P1(6P)P2(4 2),

so that

P(CK) = Pl(CIK)P2(C2K).

(3.7)

(3.8)

(3.9)

For identically distributed random variables, P = P, = P2, sothat, for a stable process, Eq. (3.9) reduces to

ACK) = AC1K)P(C2K), (3.10)

and each characteristic function has the same functionalform for its argument.

To determine P(K), we find it convenient to introduce thelogarithm of the characteristic function

q(K) = log P5(K). (3.11)

The product form of Eq. (3.10) then enables us to write

q(CK) = q(ClK) + q(C 2x). (3.12)

Here we follow the inductive argument from Ref. 55 to indi-cate how such equations were solved before RG ideas. It isclear that if m is a positive integer, there exists a set ofnumbers (am) such that

(3.13)mq(K) = q(aMK)

or equivalently that

q(K/am) = - q(K).

Similarly for some other positive integer M

(3.14)

Bruce J. West


Mq(K) = q(aMK), (3.15)

so that, combining Eqs. (3.14) and (3.15), we obtain

M q(K) = q(Kamlaj). (3.16)m

Thus, for any rational number Mu = M/m, there is a number X= aM/am such that

Aq(K) = qG\K), (3.17)

and since the P(K) is a continuous function of K, so too is q(K).

Thus, for any positive number A, rational or otherwise, thereis a X(,u) such that the scaling relation (3.17) is satisfied, and

lim X(11) = 1. (3.18)IA1

The function q(K) is therefore different from zero every-where or identically zero, in which case P(K) = 1, correspond-ing to a deterministic phase shift; i.e., the probability densityis the delta function P(V#) = 6(V.

The functional Eq. (3.17) has a solution of the form

q(K) = AIKla (3.19)

W(a, K) = ftan(7ra/2)( 27r log K

if a $ 1if a = 1'

(3.26)

with y and /3 being arbitrary real constants. The probabilitydensity

P(46) = 2 dk exp(-iK46)exp{-iKIa[1 + i/3W(a, K)

(3.27)

is therefore referred to as the L6vy distribution.2 1

We now apply these arguments to the more general case ofN phase screens, each of length I - Az/N, representing thescattering interval (0, Az). We consider the sum of phaseshifts [Eq. (3.4)] with AN set equal to zero. The characteris-tic function of T(N) is determined by

N

j=1

(3.28)

which, if Eqs. (3.28) and (3.4) are compared, implies that Cj= 1 for all j. Thus

C = N-1a (3.29)

for all real K, and A is a complex constant. The harmoniccomponent of A is suppressed to satisfy the conditions out-lined below. Since

lim P(K) = 0,1K1 b <

(3.20)

we require the real part of A to be negative definite, and theconstant a is positive definite. Substituting the assumedsolution of Eq. (3.19) into Eq. (3.17), we again find that

the normalization constant BN in Eq. (3.4) is therefore Ni/a,and the total phase shift is

N

(N) = N 1 E jj=1

(3.30)

The characteristic function for a single phase screenwhose characteristics are described by a stable distributionis given by

a = ln i/ln X. (3.21)

The inverse Fourier transform of P(K) is the probabilitydensity P(41), and therefore it must satisfy certain auxiliaryconditions: (i) P(K) = P*(-K) from the reality requirementon P() and (ii) P(K = 0) = 1 from the normalization condi-tion of P(P). For the real constants oy and /, Eq. (3.19) cantherefore be rewritten as

q(K) = '(1 + i I .)IKIa, -y > O. (3.22)

Returning now to the two-variable case, we conclude thatthe functional relation between C and the constants Cl andC2 is

Ca = Cla + C2 a (3.23)

which is obtained by substituting

P(K) = exp[ -(1 + i/ )IKIa] (3.24)

into Eq. (3.10).L6vy20 examined the function q(K) in much greater detail

to determine all the required properties that it must possessin order that P(K) = exp[q(K)] be a proper characteristicfunction. He found that the most general form of q(K) is

q(K) = -,ylKI [1 + i/w(a, K) I-J, 0 < a < 2, (3.25)

where

P(K) = exp{ YIKIa[1 + i/W(a, K) a 5d 1. (3.31)

Then, using the product form of the individual characteris-tic functions given by Eq. (3.10) generalized to N phasescreens,

P(N(BNK) = [AK)]N, (3.32)

or, equivalently, using w(a, K) = tan(7ra/2) for a FF 1, we have

5(N)(K) = [P(K/N1/a)]N

= exp{ -TIKla[1 + iw(a, K) I (3.33)

Thus the statistics of the scattering region are independentof the number of phase screens used to represent that regionwhen the probability density for the phase fluctuation is ofthe Levy stable form.

Let us assume that the wave propagation takes place intwo dimensions and that the edge of the phase screen islocated at z = 0 and the wave propagates in the positive zdirection. In this configuration the phase is a function sole-ly of the transverse coordinate x. We assume '(x) to be arandom function with L6vy stable statistics. L6vy processesare represented by a homogeneous differential Markov pro-cess dB(x), where, given x0 < x, < ... < x,,, the differencesB(x,) - B(xo), B(X2) - B(x,), . . . B(xn) - B(xn,-,) are mutu-ally independent, random variables and the distribution ofB(x + A) - B(x) is independent of x. The statistics of a

Bruce J. West


symmetric L6vy process are specified in terms of the charac-teristic function

P(K, A) = exp(-,yJKIt), r > 0, (3.34)

where real y > 0 and 0 < a S 2. Thus the probability densitythat the random phase I(x) has a value in the phase-spaceinterval (A/, A + do) at position x is given by

P(V/, x) = | dK exp(-iKV/)exp(-7yKI'ixI). (3.35)

One can obtain closed-form expressions for the probabilitydensity P(V1, x) for only a limited number of parameter val-ues; among these are the Gaussian distribution with a = 2and the Cauchy distribution with a = 1 (see e.g., Ref. 21).

The L6vy distribution [Eq. (3.34)] or the more generalform [Eq. (3.31)] has a number of interesting properties.First, it satisfies the Bachelier-Smoluchowski-Chapman-Kolmogorov chain condition 2 '

P(62 - A11, J diPPk, - 'PI X)P(' - 1 , X)),

(3.36)

where P(V2 - 01, t) is the probability density that the valueof the process T(x) changes from A1 to 4/2 in the interval t.The form of the Bachelier-Smoluchowski-Chapman-Kol-mogorov chain condition given by Eq. (3.36) is quite generalfor translationally invariant homogeneous Markov process-es in continuous space. It is straightforward to prove thatEq. (3.36) is the probability equivalent of the scaling condi-tion from RG theory. Fourier transforming Eq. (3.36), weobtain

P(K, i) = P(K, -X)P(K, X), (3.37)

whose general solution is Eq. (3.31).Second, the L6vy distribution satisfies the scaling relation

P(rl/agz, rF) = r-l/ap(P, A) (3.38)

for an arbitrary parameter r. This relation can be estab-lished directly from the definition of the probability density[Eq. (3.35)]. It implies that if '(r) is a random variable withprobability density P(V, P) then the two stochastic functions

m(r-) and rl/ac(t) have the same distribution. Thisstretching of the argument of the random variable followedby a scaling in amplitude, resulting in the same statistics asfor the original random variable, is called self-affinity, as wenoted above. Note that a = 2 corresponds to one-dimen-sional Brownian motion of the scalar function I(t), and P(k,D) is then a zero-centered Gaussian distribution with vari-ance proportional to A, i.e., Einstein diffusion. 2'

The asymptotic form of the probability density is theinverse power law2l

distribution is quite different from the apparently uniquebehavior of the Gaussian distributions

B. Renormalization Group SymmetryThe RG symmetry properties of the phase-shift fluctuationscan be determined in a straightforward manner when theirstatistics are described by a L6vy distribution. We definean operator T that maps the statistical properties of (N - 1)phase screens into N phase screens:

P(N)(K) = TP(N-1)(K), (3.40)

where the length of the scattering region remains fixed as (0,Az). By successive application of the operator T and settingP(l)(K) = P(K), we obtain by induction

(N) (K) = TNP(K). (3.41)

The N-fold mapping indicated in Eq. (3.41) is equivalent tosegmenting the scattering interval into N phase screens.Thus, as we demonstrated above, for a Levy distribution theapplication of T is equivalent to the operation

TNP(K) = [P(K1N1/a)]N = P(K), (3.42)

where we have used Eq. (3.31) for P(K). Equation (3.42)shows that the L6vy distribution is a fixed point of the Ttransformation. This symmetry property of the L6vy char-acteristic function implies that P(K) is a member of a group,just as the invariance of a process under translation androtation implies the appropriate group structure for the pro-cess.

To interpret the T transformation we consider an explicitconstruction involving a regrouping of the original partition-ing of the scattering region from N groups to Ns, groups eachof length 15 = sl. We thereby coarse grain the original de-scription by grouping each sequence of s consecutive phasescreens into a single new one. The original phase shift

N

*(N) = W/; E" *= zis now replaced with

(N,) =- 1 m SN.1

(3.43)

(3.44)

where the phase shift I (s) is the total shift over a sequence ofs phase shifts:

Since the overall phase shift cannot depend on the detailsof the partitioning, we require the scaling relation

P(', D= wry|flr(s)sin(A-7r/2) as 1 1- 17Ii4l a+1 (3.46)'Q(N) = pS (N )

Equation (3.39) implies that the moment (1VIIA), where theangle brackets denote an average over an ensemble of L6vyfluctuations, is finite if and only if ,u < a and is infinite if A 2a for a < 2. For a Gaussian distribution a = 2, all momentsare finite since the inverse power-law form of Eq. (3.39) is nolonger applicable. Thus the long-tailed nature of the L6vy

In particular Xs is chosen such that each T (s) is scaled toensure that the contribution from each group of s phasescreens remains statistically unchanged.

The transformation indicated by Eq. (3.42) is now re-

placed by Ns factors of T9. The characteristic function forthe first s operations is

(3.39)

Bruce J. West

'F(S) =M �" qi.-

i.=1

(3.45)


T8P(K) = [P(X8 K)]J

- exp[-SYIXsKI(1 + i/3w(a, K) K (3.47)

where we have used the scaled phase shift [Eq. (3.46)]. ForP(K) to be a fixed point of Ts we require that the scale factorbe given by

As = 1/s (3.48)

and obtain

'IP(K) = P(K). (3.49)

Thus the process of coarse graining [Eq. (3.44)] followed byscaling [Eq. (3.46)] leaves the L6vy form of the characteristicfunction invariant when the scaling is appropriately chosen.The transformation T, defined by coarse graining followedby scaling, is a RG for the fluctuations in the phase shift.' 7

The fact that the L6vy stable distribution is a fixed point of aRG transformation seems to have been noted first by Jona-Lasino.36

C. L6vy Phase ScreenWe now investigate the statistical properties of a freelypropagating plane wave whose phase is specified on a trans-verse line at z = 0. The wave is propagating in the positive zdirection with constant frequency w and wavelength X(=27r/k) with amplitude v(x, z) such that

vo(x) = exp[ik'{(x)] at z = 0+, (3.50)

where the phase shift T(x) is only a function of the trans-verse coordinate x. The observed scintillations in the am-plitude and the phase of the received wave result from inter-ference of phase points along the wave front in the z = 0plane as the wave propagates away from the boundary. Theinterference pattern is determined by the assumed statisti-cal properties of i(x). Booker et al.50 were the first to use anexpression of the form of Eq. (3.50) to model the effect of alocalized scattering region on the propagation of a scalarwave as a thin phase screen.

Near a thin phase screen the field fluctuations are only inthe phase, but, as the wave propagates in the free space awayfrom the screen, an interference pattern develops. The dis-tance to this pattern is dependent on the density gradients inthe phase screen. The larger the phase gradient, the greateris the angle through which the transmitted wave is scatteredand the nearer to the screen is the subsequent interferencepattern. It is the small-scale spatial structure in the scatter-ing medium that produces these large spatial gradients.Thus we expect self-affine fluctuations to influence stronglythe propagated wave. For a scattering interval whose fluc-tuations are manifestations of some turbulent motion thereis a broad spectrum of scales associated with the irregular-ities in the medium. Thus waves of most frequencies en-counter irregularities on the same scale as their wavelength,and traditional propagation models, such as geometric op-tics, cannot fully describe the transmission of the wavethrough and beyond the phase screen; i.e., multiple scatter-ing effects can become important.5 6- 58 It is remarkabletherefore that weak-scattering theory has been so successfulin explaining the observed data from satellite signals trans-

mitted through the ionosphere.7'58 Note here that we do notmodel the propagation through the fluctuations but onlyaway from such a region.

The propagation of the wave field in free space can bedetermined by solving the wave equation (3.1) subject to theboundary condition of Eq. (3.50). The Helmholtz equation(3.1) is an elliptic, partial differential equation so that, inorder to find the field at a given point in space, one mustsolve the equation for the field at all points in space. Oneoften approximates the Helmholtz equation by a parabolicequation having the form of a one-dimensional Schrodingerequation:

(3.51)zi2k d + /2 +k2U(X,Z) = O.

which is also solved with the boundary condition of Eq.(3.50). Here v(x, t) = u(x, z)eikz/zlI 2 , and we neglect termsthat fall off faster than Z-1/2.8 The direction of propagation(z) in Eq. (3.51) plays the role of time in the actual Schro-dinger equation. In the parabolic approximation the nor-mal to the phase front of the field is assumed to remain closeto the direction of propagation, i.e., to the z axis in this case.For this reason the approximation is also referred to as theparaxial approximation. The conditions for the validity ofEq. (3.51) can often be expressed in terms of a correlationlength and will be discussed below.

Berry59 has applied the notion of a fractal in the presentcontext by introducing a new kind of wave called a diffractal.A diffractal is a wave that has encountered a fractal object,i.e., one that possesses structure on all scales. An incidentwave of wavelength X is fairly insensitive to structure onscales much smaller than X, and for structures much largerthan X the techniques of geometric optics can be employed.However, when this separation of effects cannot be made,the scattering object is a fractal, since the self-similar char-acter ensures that there will be structure on a range of scalesthat includes X, and thus no geometric-optics limit exists.Berry models diffractals by using Eq. (3.50) and assumingthat T(x) is a Gaussian, random fractal function with aninverse power-law spectrum having a fractal dimension dbetween 1 and 2. Physically, this boundary condition con-stitutes a phase-screen approximation for a wave reflectedby a fractal surface such as that considered in Subsection 2.Bor refracted by a slab of transparent material with an irregu-lar index of refraction and is emerging from the slab.

Mean Diffractal FieldThe solution to the parabolic equation (3.51), subject to theboundary condition of Eq. (3.50), is given by the elementarydiffraction integral

uxz eikz(~ k k\x/22u(x, z) = ei^Z( iZ ) | dx' exp[i+ (x-x') Ju0(x').

(3.52)

Equation (3.52) describes the diffraction of the phase frontaway from the phase-screen point (x', 0) to the observationpoint (x, z). The mean field detected at (x, z) is determinedby an average of Eq. (3.52) over an ensemble of realizationsof the phase fluctuations:

Bruce J. West


(UX, Z)) = e ikz,( k )/(u~~~x, z)) kk 2irxiz. (.53x f dx' exp[i 2z (x - xt)](uo(x')) (3.53)

Under the integral we have the average field emerging fromthe phase screen:

(uo(x)) = (u(x, 0k)) = (exp[ik'(x)]). (3.54)

We assume that the phase is a differential Markov processand write

'(x) = J d*(x') + C(0), (3.55)

where *(0) is the reference phase at x = 0, so that Eq. (3.54)becomes

(uO(x)) = exp[ikI(O)](exp[ik J d* (x)]). (3.56)

The reference phase 'I(O) is a random variable, which, for afractal, has an infinite variance, i.e., (exp[ikI(O)])o = 0, so

that the average field vanishes when averaged over an en-semble of fractal reference states:

(u(x, z))0 = 0. (3.57)

Equation (3.57) agrees with the strong diffractal result ob-tained by Berry59 by using his Gaussian, fractal phase func-tion.

Coherence of Diffractal AmplitudesThe coherence between the wave field at the point (x + ~, z)and at (x, z) is given by

k ' ,(u(x + A, z)u*(x, Z)) = 2r J dxl J dX2

X exp{i 2 [(x + - )2 - (X -X2)2}(UO(X1)UO*(X 2 ))-

(3.58)

The two-point correlation of the initial wave field for spa-tially homogeneous irregularities in the phase is evaluated inAppendix B to be

(u(x + , z)u*(x, z)) = | dxl dx2

X exp{i kZ [(X + x,)2 - (X - x2)2]}exp(-,ylk aJxJ - x21).

(3.59)

A simple change of variable allows us to integrate Eq. (3.59)directly, i.e., to choose q = X- X2 and t = (XI + X,)/2. Theintegration over t yields the delta function 2irzb(77 -)/k,

and the integration over -q yields the coherence function

(u(x + -, z)u*(x, z)) = exp(--yk"ki). (3.60)

Note that Eq. (3.60) is independent of how far the wave haspropagated (z), a result first obtained by Booker et al.50 for aphase screen with Gaussian statistics (a = 2). In the r = 0limit, we obtain unity for the wave intensity, as we shouldsince the propagation integral conserves energy.

The amplitude power spectrum of the wave field PU(K) isthe Fourier transform of the two-point coherence function:

Pa(K) = J d~eiKI(u(x + A, Z)U*(X, Z))

2yka

K 2+ ,z,2k2a'

Thus the power spectrum is a Cauchy (Lorentz) distributionhaving a K-2 power law as K - - and is constant as K - 0.The asymptotic inverse power-law behavior of Eq. (3.61) canbe used as an a posteriori justification for the paraxial ap-proximation [Eq. (3.51)].

The parabolic equation is an adequate description of thepropagation away from the phase screen when the scatteringangle, as measured by K/k, is small. When the scatteringangle is large, we require that the energy content of thepower spectrum be small; i.e., PJ(K) << P(O). From Eq.(3.61) we then have

(3.61)

PM(k) y2k2aP"(O) K2 + ,y2k2a

so that

K -k» >yka-1 = 0,

(3.62)

(3.63)

and 0 << 1 is then a measure of the scattering angle. Thequantity 0 replaces the parameter kL used by Berry,5 9 wherehis formalism is that L is a distance characteristic of thefluctuations. The present measure of the scattering angle 0increases as the wavelength of the incident wave decreases inthe parameter range 1 < a < 2 but decreases as the wave-length gets larger in the range 1/2 < a < 1. The validity ofthe paraxial approximation therefore depends on the pa-rameter regime of the phase fluctuations for a given incidentwavelength.

We can rewrite the spectrum [Eq. (3.61)] in terms of theparameter 0 as

kP,(K/k, 0) = 20 0

from which we construct the scaling law

P.(K/k, 0) = - Pa(K/0k, 1).0

(3.64)

(3.65)

Therefore, as 0 increases, indicating a large deformation ofthe phase front, features of the spectrum occur at largerangles, i.e., at larger K/k, independently of the distance fromthe phase screen.

DiscussionWe have examined some of the statistical properties of ascalar wave field freely propagating away from a fractalphase screen. We have not made the usual assumptions ofGaussian statistics and inverse power-law spectra of thewave front on the phase screen but have instead assumed thephase fluctuations to be described by a L6vy distribution.The emergent wave is defined as a weak diffractal to distin-guish it from Berry's diffractal whose phase is defined by aGaussian fractal function. The motivation for this investi-gation is to explore the nonuniqueness of the deduced prop-erties of the initial statistics of the wave field obtainedthrough an analysis of the received non-Gaussian signal.

The results that we have obtained from the weak diffrac-

Bruce J. West


tal are identical to those obtained by others for a thin phasescreen on which the emergent wave has Gaussian statisticsand a truncated linear correlation between phase fluctua-tions.6 0,6' The physical interpretation of these earlier re-sults was problematic, but the established second-orderequivalence of the present results with these earlier onessuggests one way to understand the nature of some of theearlier difficulties. For example T(x) is a Gaussian processonly if the irregularities in the medium that the phase screenrepresents are also Gaussian. If, however, the process lead-ing to the irregularities in the medium is turbulence, then, asis known, the statistics of the fluctuations cannot be Gauss-ian. Therefore either T(x) is not a Gaussian process or thesource of the inverse power-law behavior of the spectrum ofdensity fluctuations is not turbulence or both. Here wehave resolved this difficulty by using non-Gaussian (Levy)statistics, and yet we have obtained the standard results.Thus we conclude that the results are fairly insensitive to thestatistics of '{(x).

D. Wave Propagation in Multiple-Scale MediaLet us now turn our attention to a related but distinct prob-lem, that of the propagation of a scalar wave through amedium having fractal statistics. In a medium whose statis-tical properties are characterized by a single scale, the trans-mitted wave can have two quite distinct scintillation statis-tics. The first is due to unsaturated scattering in whichthere is a single ray describing the transmitted wave. Bydefinition a ray is a curve whose optical path from source toreceiver is an extremum.10 The number of such extremalpaths depends on the number of stationary points of theoptical path in the fluctuating medium. Therefore there is asingle stationary point in the optical-path length for unsatu-rated scattering. The second type of scattering is calledsaturated and results from multiple stationary points. Inthis situation there are an undetermined number of statisti-cally independent ray paths in the medium that result fromfluctuations. In the unsaturated regime, since essentiallyonly a single ray exists, the phase and the amplitude of thetransmitted wave are single quantities. Straightforwardcalculations indicate that the logarithm of the wave ampli-tude and the phase are linear in the medium fluctuationsand are therefore Gaussian variables. Thus the complexsignal u has log-normal statistics.' 0 "15 In the saturatedregime, the complex signal is a sum of a large number ofinterfering waves. Applying the central-limit theorem(with finite second moments) yields the result that the realand the imaginary components of the received signal areindependent, Gaussian random variables with zero meanand equal variances.

In a medium whose statistical properties depend on manydifferent scales, the above arguments become suspect. Thereason for this is that, for a fixed range L and wave number k,the smaller scales in the medium can give rise to a new set ofperturbed rays before the large scales do. This was shownby Dashen,'0 who used a scaling argument. The generationof perturbed rays by small-scale irregularities engenders asmearing of the boundary between the saturated and unsat-urated regimes, and a new region, called the partially satu-rated regime, develops. In the partially saturated regionany two rays can differ in phase sufficiently that they can bedistinguished but can still lie within some correlation dis-tance of each other. The detailed signal statistics in this

regime are less well understood than in the fully saturatedregime.

Questions regarding the statistics of scintillations pro-duced by multiple-scale isotropic media in the partially sat-urated regime remain largely unanswered. However, it hasbeen shown that solutions to the three-dimensional parabol-ic wave equation (z being the direction of propagation), withthe fluctuations in the refractive index being zero-centered,Gaussian random variables in space,'yield the correlationfunction between two receivers separated by a distance R =xl - x2 (Ref. 10):

C(R) = C 0(R)exp[- 1/2 D(IRI)]. (3.66)

Here Co(R) is the correlation that would be present if nofluctuations were present in the interval being traversed,and D(IRI) is the phase-structure function of first-order geo-metric optics and is defined by the integral of the two-pointcorrelation of fluctuations in the refractive index over theregion of fluctuations. Dashen has shown that the generalexpression (3.66) is valid as long as the parabolic wave equa-tion is valid.

Consider the situation in which the unperturbed waves areexactly correlated so that Co(R) = 1, and the phase-structurefunction is quadratic in the transverse coordinates. ThenEq. (3.66) reduces to' 5

C(R, a) = exp(-R 2/a2), (3.67)

where a. is the correlation length in the scattering medium.Equation (3.67) characterizes the scattering region by thescale length a.. In a multiple-scale medium there are a num-ber of such scale lengths, each with a different frequency ofoccurrence. Therefore we assume that the scattering regionhas a distribution of correlation lengths a.. We write P(a.)do.as the probability that the scale of the fluctuations has acorrelation length in the interval (a, a. + da). The quantityC(R, a) therefore is a random function in the stochasticquantity a, and the measured correlation function is givenby

C J(R) = JC(R, a)P(a.)da. (3.68)

One might interpret the ensemble average [Eq. (3.68)] asequivalent to a spatial averaging of the ray paths contribut-ing to the correlation function (ergodic theorem).

A related but distinct method of introducing multiplescales was studied by Bourgois.6 2 He considered a multi-Gaussian model in which a sequence of independent, iso-tropic, zero-centered, Gaussian-correlated, statistical pro-cesses was used to obtain the power spectrum of a wavepropagating through a multiple-scale medium. Although hedid not explicitly address the fractal concept, he did pointout that, for certain values of the index of the inverse power-law spectrum, the random function is not mean-square dif-ferentiable. He also noted that this is related to the conceptof the "degree-of-smoothness" introduced by Matheson andLittle. 63

We note that the measured correlation function that isgiven by Eq. (3.68) is not necessarily correct when a varietyof a. are simultaneously present in the medium. The inte-gral expression presupposes that multiple scattering effectsdo not correlate fields produced by fluctuations in the re-

Bruce J. West


fractive index from different scales. We know that this is infact not the situation for any but the unsaturated case.Thus Eq. (3.68) is, at most, valid as a first-order approxima-tion to the correlation function in a multiple-scale medium.Even with this restriction, however, we find the presentmodel to be a useful one.

Correlations Generated by Large ScalesThe correlation between two nearby ray paths is dependenton the scales of the fluctuations in the medium. The greaterthe scale length a, the larger is the distance over which theray paths are correlated. Thus larger a.'s should be weightedmore heavily in the probability density P(a.) than are smallerones. For a medium with an infinite number of scales (butnot a continuum), we follow the suggestions of the scalingstudies and write P(a.) as

(3.69)PL(O.) = a- 1E (- aL/bn),a n0, an

where a, b > 1 and aL is the largest correlation distance in thescattering region. Substituting Eq. (3.69) into Eq. (3.68)and integrating yields

CL(R) = a 1 1 exp(-b nR /L 2 ), (3.70)

so that the measured correlation function is a sum of contri-butions from each of the scales characterizing the scatteringregion.

The correlation function [Eq. (3.70)] is governed by a setof competing contributions. The contributions from fluctu-ations on a scale o.L/bn occur with relative frequency I/an, n= 0, 1, .... Because b > 1, each of the contributions arisesfrom fluctuations correlated over successively smaller re-gions in the scattering interval, just as successively smallereddies contribute to the velocity correlations in the inertialregime of homogeneous isotropic turbulence.22 In turn eachof these correlated fluctuations occurs with less frequencysince a > 1. This mechanism therefore emphasizes theunsaturated scattering of the transmitted wave.

Just as we did in Section 2, we determine the dominantbehavior of the correlation function [Eq. (3.70)] as a functionof the separation distance R. We shift the summation indexby unity and rewrite Eq. (3.70) as

CL( a =-exp(-R2/a.L2) +1 - (bR). (3.71)=L a ex(R +a U

The exact solution to the RG scaling relation (3.71) is con-tained in Appendix C, where we find that

CL(R) = (R/.L)aQL(R) + qL(R), (3.72)

with

QL(R) -2ab a r(-Y/2 + wim/In b)m=-In

(3.74)qL(R) - alE ( 1_2' 1 1q,(R) = a -a.L /n1-b2n/a'

and again a = In a/In b. The exponent therefore depends onthe size of the eddies, i.e., 1/b, and the relative number ofthese eddies in the scattering region, i.e., a. Here we see thatQL(R) is a periodic function in ln R with period In b and thatqL(R) is analytic in R. Thus the dominant behavior of thecorrelations in the fluctuations is given by the Ra term in Eq.(3.72).

The power-spectral density of the fluctuations F(K) is theFourier transform of the two-point correlation function.Therefore we write

FL(K) = J d2RL(R)eix R = a 1 2 exp( 22)'n=0

(3.75)

and, repeating the analysis in Appendix C, we obtain thespectral density

FL(K) = (.LK) a+2 + qL(K), (3.76)

where QL(K) is a periodic function in In k with period In b, i.e.,where

(a -i) ra.L2 a MrQL(K) = I- 2a+Z r 1-2+i

m=-X (

(3.77)PEIn b (L ]

and WL(K) is an analytic function of K:

qW(K) = a - E 12 _ -2/a) (_L / 2K2 4)m.am0' m!(1 - bm/a

(3.78)

Since both QL(K) and WL(K) are analytic functions, the domi-nant behavior of the spectrum is given by the inverse powerlaw K-a-2. Note the similarity to the argument presented inSubsection 2.C, which was based on a Tauberian theorem.

Correlations Generated by Small ScalesWe have argued that a superposition of Gaussian correlationfunctions for the media fluctuations provides a K-a-

2 spec-trum. This result, in conjunction with the simple propaga-tion model in Subsection 3.C, implies that the phase and log-amplitude spectra of the received signal have a similar be-havior at high frequencies. Certain data, however, suggestthat there is a crossover to a less steep spectrum at highfrequencies. 7 We model this effect by considering a secondsequence of contributions toP(a) in the preceding argument.

We assume that the effect of small-scale irregularities canbe modeled in the correlation function by writing

(3.79)Ps(0 = - 1 n- 6(a - ab"),an=

X cxp - 127im ln(R/oL) (3. 73)where a', b' > 1 and a, is the smallest correlation distance inthe scattering region. The length scale as. is analogous to theinner scale of homogeneous turbulence. Substituting Eq.(3.79) into Eq. (3.68) and integrating yieldsand

Bruce J. West


Cs(R) = a -1 EI exp(-R 2/ 2 b'2 ),a ' -, a"'

n=O

(3.80)

so that this correlation function is a sum of contributionsfrom sequentially larger scales in the scattering medium.Note that the scales in Eq. (3.80) eventually (in n) overlapthose of Eq. (3.70).

As in the earlier case the fluctuation correlation functionis governed by a set of competing correlations, the contribu-tions from fluctuations on a scale b"nan occurring with arelative frequency 1/a"n, n = 0, 1, .... Because b' > 1 eachof the contributions arises from irregularities correlated oversuccessively larger regions in the scattering interval, just assuccessively larger eddies contribute to the velocity correla-tions in the inertial regime of two-dimensional homogeneousturbulence. Each of these more correlated irregularitiesoccurs with less frequency since a' > 1. This mechanismtherefore emphasizes the saturated scattering of the trans-mitted wave. Thus, in the region of the overlap of scales inEqs. (3.70) and (3.80), one mechanism or the other is domi-nant.

The Fourier transform of Eq. (3.80) yields the power spec-tral density

F,(K) = a , lEba nra,2 exp(-b"IaTS2 K2 /4),

whchi qie ifeen ro E.(37). Weaai=nvk

so"w

10

3'

Z 10 9

'0

.J lo

I:

or

- so

0-

F to,

.'0.

05

-Ja

I..

1o0

lo2

(3.81)

which is quite different from Eq. (3.75). We again invokethe arguments from Appendix C and obtain for the spectrum

F,(K) (- )(-) + Qs(K), a' = ln a'/ln b > 0, (3.82)

where Q,(K) is periodic in In K with period In b' and 4(K) is ananalytic function of K. Note that although the spectrummaintains its inverse power-law form, the exponent haschanged, i.e., K-2- has become K-

2+1'. For a - a' and b b',we have a - a'. Consider the case a = a' - 1; then thespectrum changes from K-3 to K-' over some interval of wavenumbers, dependent on the relative weights of FL(K) andF 8(K) in the total power spectrum F(K).

7

We assume that both of the above cascade mechanisms arepresent in the irregularity spectrum, and, writing FL(K) =

70-aL2JL(K) and FJ(K) = 7ra 8s

2f.(K), we have the following for themeasured spectrum of irregularities:

F(K) = 7raL {fL(K) + ( f) fs(K)} (3.83)

Thus, although fL(K) decays more rapidly than does f,(K), ithas a substantially larger value at smaller wave numbers sothat the wave number at which the observed spectrumswitches from a K-2- to a K-2-a' character is determined bythe ratio of inner to outer scale lengths, i.e., by (.J,/a.L)

2.

In Fig. 7 we show the relative power-spectral densities forthe signal phase (St) and log amplitude (Sx fluctuationsgenerated by random plasma variations in the ionosphere)taken from Ref. 7. In this figure we observe a K-3 power lawover a significant wave-number interval, which was referredto in Ref. 7 as the high-frequency asymptote. However, thespectra apparently change slope at the higher frequencies.Crane 7 argues that the only possible cause for the flattenedregion in the spectrum is the addition of irregularities with

1001l _l'l. 10

FREOUENCY (Hz)

Fig. 7. Power-spectral density for the differential phase fluctua-tions (SO). The experimental and theoretical curves are taken fromRef. 7. The dashed curve has a K 1 slope, whereas the high frequen-cy asymptotes all have a K-3 slope.

an outer scale smaller than the size of the first Fresnel zone.This region corresponds to our second scaling region, and wesee from the figure that the flattened region does in fact havea K'1 power-law spectrum. Thus, although this does notprove the scaling argument presented here, it is the onlyquantitative explanation of the observed effect.

4. WAVELET TRANSFORMATION

In Section 2 we examined some of the properties of surfacesthat do not have a characteristic scale length. We demon-strated that such surfaces are self-affine fractals and mayperhaps be used to model such naturally occurring struc-tures as ocean-bottom topography or the surface of theocean. In Section 3 we discussed how one might expectscalar waves scattered from such surfaces to be modeled byusing a thin phase screen. We then investigated the proper-ties of the statistics of deep scattering regions made up ofmultiple phase screens. It was determined that L6vy statis-tics capture the fractal nature of the fluctuating phase of thewave propagating through such a tenuous scattering region.In this section we introduce a processing technique thatenables one to magnify specified regions of a received signalor, put differently, to emphasize any desired scale in a fractalstructure or time series by adjusting suitable parameters.Unlike the Fourier transform of a signal that provides uswith the frequency content of a time series, the wavelettransformation can be used to zoom in on higher and higherfrequency structure in the time series.

ifN. SX.400

LOW FREOUENCY ASYMPTOTE

HIGH FREQUENCY- ASYMPTOTE

l _

I-

95*o'Cfnt CONFIOENCEi LIMITS

0.14-Hz BANOWIOTH / \

HIGH FREOUENCYASYMPTOTE

Bruce J. West

I


The reciprocal integral relations of Fourier have been thetraditional bases of communication theory.64 If X(t) is atime series, then the Fourier transform of this signal is

g(t) = (1 - t2)e-t2/2. (4.7)

This transform is often called the Mexican hat because of itsshape.68 It has the Fourier transform

(4.1)X(w) = J eiw tX(t)dt,

with the inverse relation

X(t) = I | e~it9(c,)dt. (4.2)

(4.8)

which clearly satisfies the condition g(O) = 0. A secondtransform is the one considered by Morlet et al.6 5 for seismicdata:

g(t) = e-ift(e-t2/2 - Fe-2A/4e-t) (4.9)

If X(t) is real, then X(@) is complex, and the spectrum of X(t)is given in terms of IX(W)12. Although the method is mathe-matically unassailable, Gabor64 and subsequently many oth-ers have called into question the physical interpretation ofresults obtained by using it. The reason for this questioningis the mutually exclusive treatment of the frequency andtime representations of the signal, which occurs because thetime series is assumed to be infinitely long and each frequen-cy is defined for an infinite wave train. In the real world,however, time series are of finite duration, and frequencieschange over time.

These considerations led Gabor to propose expanding asignal in a sequence of wavelets rather than in the harmonicterms of a Fourier expansion. He selected a Gaussian enve-lope of the form

g(t, wo) = exp[-(2t/At) 2 In 2]exp(iw0t), (4.3)

where wo is the mean frequency and At is the duration de-fined as the time interval separating two points on the enve-lope at which the modulus drops to the value 1/2. Here theorigin of time has been taken to be zero, and g(t) is normal-ized to unit maximum amplitude. He then decomposed thesignal X(t) as

X(tl, oj) = J dtg*(t - ti, wj)X(t). (4.4)

This operation discretizes the time series on a grid of cells inthe two-dimensional phase space consisting of time indexedby i and frequency indexed byj. This technique was appliedby Morlet et al.65 to seismic reflection data. Subsequentinvestigation of this technique has led to the development ofthe wavelet transform2366-69 for representing one-dimen-sional signals as a function of both time and frequency.

The wavelet transform is more general than Gabor's trans-form in that the wavelets into which the signal is decom-posed are constructed from a simple function g(t) by meansof dilations and translations. A function is admissible as awavelet if the integral relation

c9 1 - 1gc~2 duo < co.5

is satisfied. Note that this condition implies that g(O) = 0,so that

J g(t)dt = J 5(w)R(c)dw = 9(0) = 0, (4.6)

which means that g(t) has a zero average value.For concreteness let us coilsider two examples of wavelet

transforms. One popular form is the second derivative ofthe Gaussian function:

whose Fourier transform for real Q is

g(co) = exp[-(w - U)2/2] - e-2/4 exp[-(cw -_ )2/4] (4.10)

and for which g(O) = 0. If Q is sufficiently large, then thesecond term in Eq. (4.10) can be neglected.

The wavelet transform Tg of a time series (bounded func-tion) X(t) with respect to the waveletg is a function over thehalf-plane parameterized by the real numbers a and b, with a> 0 (Ref. 23):

Tg(a, b)X =| (1C)'/ 2 g*[(t - b)/a]X(t)dt (4.11)

= a )1/2 J.g*(aw)eib`(9W)dw, (4.12)

since ag(aw)eiwb is the Fourier transform of g[(t - b)Ia]. It isclear from these two equations that the wavelet transformgives a time-frequency decomposition of the signal X(t) withthe filter g(aw) of constant relative resolution Aw/W.68 Theinverse of the wavelet transform is

X(t) = I ,,211 5/2 g[(t -b)/a]Tg(a, b)X,Cg"12 ff a5

/2 (4.13)

which can be verified by direct substitution of Eq. (4.12) intoEq. (4.13) and by using relation (4.5).

Here we have assumed that the signal X(t) is square inte-grable:

(4.14)

and therefore contains a finite amount of energy. The wave-let transform is therefore a linear transformation that pre-serves energy:

J dtlX(t)1 2 = dadb ITg(a, b)X12

, (4.15)

where again we have used Eq. (4.13), relation (4.5), and theFourier transform of g[(t - b)/a] to obtain the right-handside of Eq. (4.15).

As was pointed out in Ref. 23, this transformation is theequivalent of a mathematical microscope whose magnifica-tion is 1/a, whose position is b, and whose optics is given bythe choice of the specific wavelet function g. To see thismore clearly, let us apply the wavelet transform to a fractalfunction.

A. Wavelet Transform of Fractal FunctionsAs we discussed in Section 2, a fractal function is one thatdisplays the property of self-affinity. This concept can begeneralized by examining the local scaling properties of a

Bruce J. West

2 _W2 /29(w) = (.0 e

f dtlX(t) 12 < _


function at small-length scales; at an arbitrary point to thefunction remains the same up to a scale factor at differentlength scales. Consider

Xt0(t) - X(t + to) - X(t). (4.16)

Then self-affinity of X(t) at to means that, if we scale thelocal variable t with some parameter X > 0(t - Xt), then weobtain

Xto(At) = Xa(t.)Xto(t). (4.17)

The exponent a is the local scaling exponent at to. As wesaw in Section 2, a is the fractal dimension of the functionX(t). If in fact a is a function of to, then X(t) is called amultifractal; when it is independent of to, then X(t) is calleda homogeneous fractal.2 4

The scaling behavior [Eq. (4.17)] is retained by the wave-let transform, provided that the transform decays sufficient-ly fast at infinity:

Tg(Xa, Xb + to)X = C,/2 J dtg*[(t - to -- Xb)/Xa]X(t)

= ( aCg)12 J dtg*[(t - Xb/Xa)][X(t + to) - X(t)], (4.18)

where the second line follows, since g has zero mean. Insert-ing Eq. (4.16) into Eq. (4.18) and scaling yields

Tg(Xa, Xb + to)X = ( C ) / J dtg*[(t - b)/a]Xt,(Gt),

This argument can be generalized to all time series satisfy-ing the H6lder condition IX(t) - X(to)I < CIt - tol for t closeenough to to with C > 0 and a > 0. If Ha(t) denotes the conein the half-plane with opening angle r - 26:Ha(t) = 1(b,a)eHI6 < arg(b + a) < r - 61, where 6 > It - tol, then for 6small enough, T(')X = O(aa) and T°9X = O(am)(a - 0)uniformly in HA(to). Thus Holschneider 23 concludes thatthe wavelets perform as a filter with only the integrableexponents (a < m) being distinguishable by the wavelet,whereas the nonintegrable exponents (a 2 m) might give riseto the same scaling as TgX, owing to the decay of g atinfinity.

We have seen in previous sections that, if X(t) is a stochas-tic fractal, then its probability density also satisfies a scalingrelation just as X(t) does. If P(t) is the corresponding frac-tal measure, then we can define its wavelet transform as67

Tg(a, b)P = 1 J g[(t - b)/a]P(t)dt, (4.23)

where the normalization factor 1/an, with n an integer, canbe selected best to reveal the scaling structure of the fractalmeasure being considered. If the fractal measure displaysthe same scaling behavior as does the function X(t) at anarbitrary point to,

(4.24)P(Xt, to) Aa(ta)P(t, to),

then, integrating Eq. (4.23), we obtain6 8

Tg(Xa, Xb + to)P _a(to)n Tg(a, b + to)P, (4.25)

(4.19)

so that, using Eq. (4.16), we obtain

TgG(a, Xb + t0)X = Xa(to)+1/2Tg(a, b + t0)X. (4.20)

Holschneider 23 gives the following argument to show thatg must decay sufficiently rapidly at infinity in order for Eq.(4.20) to be satisfied by the wavelet transform. Let

0 fort S 0

X(t) =to for 0 < t < 1, (4.21)1 for t 2 1

so that X is approximately homogeneous around to = 0. Heassumes that the waveleg g behaves at t-m-l at infinity andexamines the behavior of the wavelet transform when theposition is fixed (b = 0) and the scales goes to zero (a - 0).In this case there is a local and a global contribution to TgX:

Tg(a, O)X = 1 g(ta)tdt + g(t/a)dt0,1] a J[1, a

= a" j g(t)t-dt + j g(t)dtf1O,l/a] fil/a,-]

= Tg)X + Tgg)X. (4.22)

The global term always behaves as T I-)X - am because of thedecay of g at infinity. The local term is different, however;for this term T (')X - a"(a < m) or - am(a 2 m), dependingon the relative size of a and m. If a 2 m, then Tg(O, a)Xbehaves as am, and the scaling relation for the wavelet trans-form [Eq. (4.20)] does not hold. Thus we see that the wave-let is able to reveal the local scaling exponent a < m but notthe others.

where a(to) is the same as in Eqs. (4.17) and (4.20).Halsey et al.70 suggest that, if a dynamic system is parti-

tioned into pieces of size 1, then the number of times theindex takes on a value a is

(4.26)

where f(a) is a continuous function that is often called thespectrum of singularities of strength a. In the limit I - 0,f(a) has been shown to be the Hausdorff-Besicovich dimen-sion of the subset of points of pointwise dimension a:

a(to) =_ lim In P(t, to)t-0 In t

(4.27)

The spectrum of singularities of the invariant measure isclosely related to the Renyi dimensions Dq used to describethe geometric and probabilistic features of strange attrac-tors. Halsey et al. show that the generalized fractal dimen-sion can be computed directly from f(a):

q = - f(a), (4.28)da

(4.29)Dq = 1 [qa - f(a)].

Alternatively we can write

ae = dq [(q - 1)Dq]- (4.30)

Measurements of the Dq's and the spectrum of singulari-ties provide global and statistical information on the scalingproperties of fractal measures. This information is not un-like the power-spectral density obtained from the Fourier

Bruce J. West

Nc (I) - -f(a)


transform of a time series that quantifies the relative contri-butions of the underlying frequencies. The function f(a)quantifies the relative contribution of the underlying singu-larities. Just as the Fourier transform does not keep trackin time of the frequencies contributing to a power spectraldensity, neither does f(a) denote the location of the singular-ities. Arneodo et al.67 point out that the wavelet transformis the appropriate tool to collect additional information con-cerning the hierarchy governing the temporal distribution ofthese singularities.

Relations (4.17) and (4.25) hold not for all real X but ratherfor an infinite sequence X)m , -m, where (3 is real and m iscomplex. This is the result found in our discussions aboveof RG theory. Thus in the following discussion the expo-nent a is complex, and the fractal manifests oscillatory scal-ing behavior. In the wavelet transform, oscillations of peri-od ln ( appear around an inverse power law of slope a whenlnITg(a, to)XI is graphed versus ln a, just as we discussed inSection 2. Similar periodic oscillations have been found inthe log-log plots of a number of physiological measure-ments37"38 as well as in the Renyi dimensions of highly self-similar Cantor sets,6 8 Koch triadic islands,27 and certainwell-known strange attractors.

B. Probability Measures on Cantor SetsFollowing Arneodo et al.,6 7'6 8 we consider the example of thestandard triadic Cantor set. This set is found by initiallyextracting the middle third from the interval [0, 1] to formthe two intervals [0, 1/3] and [2/3, 1]. The measure of the setis constructed by assigning the probabilities pi and P2, re-spectively, to the two intervals (p, > P2). This process ofdeletion and assignment of probabilities is carried out at thenext stage of construction for each of the two subintervalsand again with the four subintervals, and so on.

For a uniform Cantor set we assign equal probabilities toeach subinterval p = pi = P2 = 1/2. Halsey et al.70 show thatthe uniform Cantor set has the single dimension a = In p/lnlo = ln 2/ln 3, with the singularity spectrum f(a = ln 2/In 3) =ln 2/In 3, and therefore, from relation (4.29), Dq = ln 2/In 3.Figure 8(a) shows an overview of the wavelet transform ofthe uniform triadic Cantor set using the Mexican hat [Eq.(4.7)] and n = 2 in Eq. (4.23). The wavelet transform isshown in Fig. 8(b) to have an oscillatory inverse-power-lawbehavior with exponent &(b*) = ln 2/In 3 - 2, in agreementwith the theoretical prediction that all the singularities havethe same strength a = ln 2/In 3. This oscillatory behaviorwas discussed above in the context of the RG relation. Theperiod of the oscillations around the inverse-power-lawstraight line is ln lo-1 = ln 3, so that, up to a scaling factor, thefractal measure is invariant under dilation by a factor (3 = 3in the neighborhood of zero. This result is obtained at eachpoint of the uniform triadic Cantor set.

A less trivial example is obtained by constructing a two-scale triadic Cantor set. Halsey et al.70 use two rescalingparameters 11 and 12 with corresponding measure pi and P2 tosubdivide the unit interval self-similarly. The spectrum ofsingularities can be understood in this example by consider-ing the kneading sequences for the segments. In the firstlevel of construction the two subintervals 11 and 12 are la-beled L (left) and R (right), respectively. Four segments aregenerated at the next level; these are LL, LR, RL, and RR.Thus the kneading sequence of R's and L's forms an address

(a)

(b)

0

(c)

Fig. 8. Wavelet transform [sign(TgX)ITg(a, b)XI'12] of the triadicCantor set: (a) uniform measure pi = P2 = 1/2; (b) two distinctmeasures P1 = 3/4, P2 = 1/2 [note that the scales in (a) and (b) aredifferent]. A black (TgX < ") and a white (TgX 2 T) coding ofTg(a, b)X is also shown: (c) uniform Cantor set, (d) nonuniformCantor set. This coding is obtained by defining on each a = con-stant line a threshold P = 6 max Tg(a, b)X(6 > 0). In the limit a -0+, the white regions point to the singularities that are located at thepoints of the triadic Cantor set.23 The function g(x) is defined byEq. (4.7) and n = 2 in relation (4.23). (Taken from Ref. 68; usedwith permission).

that specifies both the measure and size of a particularsubinterval. For example, the size of segment is 1lml2n,where m and n are, respectively, the number of L's and R's inthe kneading sequence. It is clear that the sequence L L L... is associated with the strongest singularity &min = ln p1/In 11 that lies on one edge of the spectrum, while the weakestsingularity &max = ln p 2 /In 12 lies on the other edge of thespectrum. Other, more complicated kneading sequencesyield values of a between the extremes &min and &max (cf. Fig.9).

Arneodo et al.67' 68 calculate the wavelet transform of theabove multifractal measure with pi = 3/4 and P2 = 1/4 and 11= 12 = 1/3. Comparing Fig. 10(b) with Fig. 10(a), we canreadily see the difference between the uniform and nonuni-form Cantor sets. Arneodo et al. use a black-and-whitecoding of the wavelet transform in Figs. 8(c) and 8(d) todepict the imperfect pitchfork branchings as the signature ofthe nonequality of p, and P2. These numerical results con-firm the above theoretical prediction that the power-lawexponent in TgX is no longer unique but ranges through theinterval (&mrin, &max). In Fig. 10(b) a cut is made through thewavelet transform corresponding to the kneading sequenceb* = RRRRRRRR LL ... L, {-ln pl/In 3-2, -ln p2/ln 3 -21. A crossover is observed from the exponent &max at largescales to &rmin at smaller scales owing to the infinite tail of L'sin the kneading sequence.

Arneodo et al.68 point out that this example illustrateshow the wavelet transform can extract the scaling factors 11,12, .. . ik and the probabilities P1, P2, . .. Pk of the multifrac-tal measure's lying on a multiple-scale Cantor set. Thisillustration is done through the measurements of a(b) at a

Bruce J. West


finite number of points, provided that the number of scalesinvolved (k) is not too large.

C. Wavelet Analysis of TurbulenceArgoul et al.7" have applied the wavelet transform to theturbulent velocity field observed in a wind tunnel and havebeen able directly to verify the fractal nature of the flow fieldin physical space as opposed to phase space. In 1926 Rich-ardson72 concluded that it is unlikely that the turbulentvelocity field in the atmosphere could be represented by adifferential equation. He even proposed that the field bedescribed by a continuous but nondifferentiable functionsuch as the Weierstrass function. His efforts at such model-ing were not successful, but he did develop a picture ofturbulence that with the later refinements of Kolmogorov22

is still used today. This picture of fully developed homoge-

neous turbulence in three dimensions consists of energy be-ing supplied at the largest scales and cascading downwardthrough ever-smaller scales until dissipation takes over.The process consists of eddies within eddies within eddies,with the nonlinear hydrodynamic interaction coupling theeddies together and preferentially transferring energy fromthe large eddies to the small eddies. We previously used thispicture of turbulence in our discussion of wave propagationin random media (Subsection 3.D).

Kolmogorov 22 originally envisaged the cascade process asdistributing energy uniformly throughout the fluid such thatdissipation acts uniformly throughout the fluid as well.Mandelbrot73 contended that the cascade process describedby Richardson and Kolmogorov does not uniformly fill spacebut is intermittent so that the regions of activity in theturbulent fluid exist on a fractal set. He maintained that

-20 0 20 40

0.8

0.6

f(a)0.4

0.2

00

-to

0.5 1 1.5

q aFig. 9. (a) Renyi dimension Dq graphed versus q [relations (4.28) and (4.29)] for the uniform (long-dashed line) and the nonuniform (solidcurves) triadic Cantor set. (b) Spectrum of singularities f(a) is plotted versus the strength a for the uniform (dashed line) and the nonuniform(solid curve) triadic Cantor set. (Taken from Ref. 68; used with permission.)

-10 -5 0 _

0

-5

nin(a) -- n(&)

Fig. 10. (a) Logarithm of the wavelet transform graphed as a function of ln(a) for a uniform Cantor set, using an arbitrary scale. (b)Logarithm of the wavelet transform graphed as a function of ln(a) for the nonuniform Cantor set, where b* corresponds to the kneadingsequence RRRRRRRR LLL ... LL .... Here the wavelet function g(x) is defined by the second derivative of the Gaussian function [Eq. (4.7)]and n = 2 in relation (4.23). (Taken from Ref. 68; used with permission.)

1.5

D__.

1

Dq

0.5

0

ln2/1n3_ _ _ _ _

I . . . I . . . I . . . l l l l I

-40

.0z

..0

a

0

-5

-10

15

-IS

- . . . . - . . . � . . . . . . . .

Bruce J. West

. . I I I iI

_r5


the turbulent energy dissipation field must be regarded as anonhomogeneous fractal. In fact, it was recently verifiedexperimentally that the energy dissipation field is a multi-fractal74"75 in which the local behavior of the dissipation rateis described by a power law.'6

A number of models were recently developed to character-ize the fractal nature of turbulence. For example, the diffu-sion of a passive scalar in a turbulent flow field was found byRichardson to have a t3 behavior rather than the linear,secular behavior of molecular diffusion. An integral-equa-tion formalism based on a fractal random walk model wasdeveloped by Shlesinger et al.77 to describe the dynamics ofthis turbulent diffusion. The probability that a parcel ofpassive scalar, e.g., dye or smoke, be transported a givendistance is determined in this model by a L6vy distribution,thereby incorporating the spatial intermittency of the tur-bulent flow into the dynamics of the model.'8 Another mod-el that addresses the turbulent field directly is the (3 model,'79in which the flux of energy is transferred to only a fixedfraction (() of the smaller-scale eddies. All these modelshave a common disadvantage in that they cannot give anytopological information about the intermittent structures inreal space.

Argoul et al.7" use the wavelet transform on turbulencedata to provide the first visual evidence of the Richardsoncascade and demonstrate its fractal nature. The data set

4't

(a) (b)

consists of hot-wire-probe measurements of the velocityfield of wind-tunnel turbulence at high Reynolds numbers.They assumed that time variations are equivalent to spatialvariations, so that the time series obtained at the single pointin space could be interpreted in terms of a spatial series.This application of the Taylor hypothesis is for frozen turbu-lence. By using this hypothesis it was determined that thespatial structure in the turbulent field spanned a range ofscales from an integral scale lo = 15.0 m down to a dissipationscale td = 0.35 mm. The Reynolds number based on theTaylor microscale is Rx = 2720.

Just as the wavelet transform was applied to the Cantorset in Section 3, Argoul et al.7" applied it to the turbulentvelocity field and used the same coding for the transform.In Fig. 11 both the original time series and the transformeddata are depicted for a range of scales in a (2810, 10/i0 ).Figure 11(b) indicates the region in the vicinity of the arrowin Fig. 11(a) magnified by a factor of 20 in both position andscale. Figure 11(c) is related to Fig. 11(b) in the same way.A Mexican hat wavelet was used in the latter two figuresbecause it provides a (marginally) better visualization of theRichardson cascade. In the figures, the indicated transformwas restricted to positive values.

In Fig. 11(a) a so-called French top hat was used for theanalyzing wavelet; this is a piecewise constant g(x) that em-phasizes the conelike structure in (a, x) space. In this figure

x

(c)

a

X X XFig. 11. Wavelet analysis of fully developed turbulence from wind-tunnel data. The coordinates and intensity are the same as in Fig. 8. Theuppermost graphs show the signals being analyzed. (a) 852-m sample depicting the large scale (28lo-lo/lO) analyzed using a French top hatwavelet. (b) A Mexican hat wavelet for 20X magnification of the central portion is indicated by the arrow in the uppermost graph of (a). (c)Analogous to (b), again with the Mexican hat wavelet. The successive forkings reveal the fractal nature of the Richardson cascade. (Takenfrom Ref. 71; used with permission.)

Bruce J. West


large-scale eddies are revealed by the superposed conelikepatterns of different intensities, indicating structures onscales a < to. Argoul et al. point out that these large-scalestructures seem to be randomly distributed in space and arefairly space filling. On the other hand, the small-scale pat-tern depicted in Figs. 11(a) and 11(c) is seen to be spatiallyintermittent. It is argued that the small-scale structures arecreated by a process in which an eddy divides asymmetrical-ly (in intensity) into two smaller eddies, and this subdivisionis repeated eight to ten times before a scale at which they areeasily dissipated is achieved. These successive branchingsproduce a fractal structure in real space, not in phase space,in which small-scale activity becomes more and more sparse.Further, the ratio between the scales of the successive gener-ations can take different values, which is another indicationthat the turbulent field can be described as a multifractal.

APPENDIX A: SCALING OF THEWEIERSTRASS FUNCTION

The scaling property of the Weierstrass function [Eq. (2.12)]was determined in detail by Hughes et al.,80 and we followtheir argument here. The nonanalytic behavior of W(t) at t= 0 is revealed by writing the cosine in terms of its inverseMellin transform with respect to t:

1 ccoit r(s)cos(7rs/2)

0 < c = Re(s) < 1. (Al)

The Weierstrass function can then be written as

W(t) = 1 fC+ids r(s)cos(1rs/2) 1

27ri fC-i (@ot n=O (abs)n

Thus the Weierstrass function separates into a nonanalyt-ic part that carries the singular behavior of the functiont"Q(t), since Q(t) is an analytic function, and an analytic partdenoted by the series in Eq. (A4). Asymptotically, the sin-gular part of W(t) dominates, and one obtains the behaviorgiven by Eq. (2.15).

APPENDIX B: LEVY PHASE AVERAGES

To evaluate the average in Eq. (3.57),

(B1)

we replace the integral by a sum over N subintervals, each oflength Ax << x, and write

(B2)I = lim ( exp, i d(x') ])N-"' Li. IJI

A-0 j1 L E~X

Recall that for a differential Markov process the difference'4[(j + 1)Ax] - I(,jAx) is independent of jAx and that thedifferences are mutually independent from interval to inter-val. Therefore we can factor the average in Eq. (B2) andwrite

(B3)N-1

I = lim I| (explikT[(j + 1)Ax] - TI(AX)),A-0 j=o

and, using the ensemble definition of the angle brackets, weobtain

N-1

I = lim f | exp[ikAIj(Ax)]P(A''j, Ax)dWAIj.N-." J5 -'

j10 =0

(B4)

J 1 C+iO' r(s)cos(rs/2)

27ri LC-i (d t)s(- l/abl)'(A2)

where the series has been explicitly summed. Hughes et al.noted that the integral is a meromorphic function of s, withsingle poles at s = 0, -2, -4, . . . due to the factor r(s)cos(7rs/2) and at s =-A + 27rni/In b, n = 0 +1, A2 .... due to thezeros in the denominator, where

(A3)

If the contour of integration is translated to Re(s) = -- andthe residues of the poles over which the contour passes areproperly accounted for, they obtain

W(t) t"Q(t) + m ' (A4)m= 2m)!(1 - Vnla)

where

Q _t) = ' / b co[ n 2rni \ 17 2 1Q~~t) 0 r -a+ Icos~ -+ 2 lri ,r/2 IIn b I- n b I/ L n b I

X exp -27rni lnb ). (A5)

The analysis leading to Eqs. (A4) and (AS) can be justifiedrigorously when 1/2 < g < 2 and by introducing a conver-gence factor extended to 0 < ,u < 1/2 as well.

This integral is just the characteristic function for the smallphase shift A'j:

(B5)N-1

I = lim fl P(k,Ax),N-JA-0O j=O

where we have used Eq. (3.6). The characteristic functionP(k, Ax) is dependent only on the length of the spatial inter-val Ax and not on jlAx. In the continuum limit with

x = lim NAx = const.N-.AE -i

Eq. (B5) yields

I = exp(--yIkIzx), x > 0, (B6)

which is the L6vy characteristic function for a position-dependent process.2 '

The two-point correlation function of the wave field at theboundary is

(U0(X1)u0(X2)) = KexpikLJ| d"(x') - f dP(x')J}) B

(B7)

where we observe that the reference phase vanishes. Weagain replace the integrals by sums of statistically indepen-dent, differential Markov processes and factor the averageover each of the intervals of length Ax. This'procedureenables us to write

a = In a/In b.

Bruce J. West

I= exp, ik fo dif W) ,X


N-i r(i+1)Ax(U0(xd)u0(x2)) = lim 11 !exp[ik dF(X')

(B8)

and, again substituting in the probability density, we haveN-1

(Uo(X1)Uo(X2)) = lim I exp[ikAITj(Ax)]6X-0O j=M

N-i= lim rl P(k, Ax).

NM-0Ax-0 j=M

(B9)

I = - exp(-R2 /a.L2)2

( a2)-lu/2+(imxr/In b)+2 b m 2-/2Y Fn ) r(-y/2 + im7r/In b)

(C5)2 In b m-E-= E 1!(-,0/2 + I + imrn b)

we denote the first series in Eq. (C5) by (R/aL)QL(R)a/(a -1) to obtain Eq. (3.77). We denote the remaining terms byqL(R)a/(a - 1) so that

qL(R) = a -1 exp(-R 2/,L 2) 2 ln b

In the continuum limit,

X1 - lim NAx = const.r.,> X I Y 1![-M/2 + I, (-R /amrr)'l.M=-1=0 Flin0

and

X2 - lim MAx = const.,Ax-O

so that Eq. (B9) becomes

(U0 (Xd)U*(X2)) = exp[--ylk1(x1 - X2)],

Interchanging the summations in the double series by aproper limiting procedure, we obtain

v 1 v 2(1 - A/2)A -,/2 + I + im'r/In b __l (I- /2)2 + (imlr/In b)2

X1 2 X2. (B10)

Thus for a L6vy process the phase points along the boundaryare statistically homogeneous in space; i.e., they are transla-tionally invariant.

APPENDIX C: SOLVING THE SCALINGEQUATION

Consider the series taken from Eq. (3.73):

I = - exp(-b2 "R2/aL2) = g(n), (C1)n=a n=O

which we rewrite by using the Poisson summation formulaas2 1,37

g(n) = 2 g(0) + J g(x)dxn=O

+ 2 f J g(x)cos(27rmx)dx, (C2)mwle°

where

g(x) = exp(-x ln a)exp[(-a'/a. 42 )exp(2x In b)]. (C3)

Introducing the change of variables y = exp(2x In b), we getfor Eq. (C2)

I = I exp(-R 2 /a.L2 ) + 12 2lnb

X > y-l-/ 2 +(rim/In b) exp(-yR 2/aL.2 )dy, (C4)

mw-y

which yields

+ 1 = ln b coth[ln b(t - A12)],t - 2

so that

qL(R) = a {1 exp(-R aL 2)

1 y coth[In b(l-g/2)]2 1 1!

1=0

The series in Eq. (C8) can be rewritten by using (1 + x1)/(1 -xI) = 2/(1 - x) -1 to obtain

qL(R) = a - (-R 2 /L 2) !(1 - b21/a)' (C9)1=0

as used in Eq. (3.79).

ACKNOWLEDGMENT

The author thanks the U.S. Office of Naval Research forpartial support of this research (N00014 88c 0149).

REFERENCES

1. A. Ishimaru, Wave Propagation and Scattering Random Media(Academic, New York, 1978), Vols. 1 and 2.

2. Principles and Applications of Underwater Sound, reprint,NDRC Vol. 7 (U.S. Department of the Navy, Washington, D.C.,1946).

3. P. D. Edmonds, ed., Ultrasonics, Vol. 19 of Methods of Experi-mental Physics (Academic, New York, 1981).

4. M. J. Albowitz and H. Segur, Solitons and the Inverse Scatter-ing Transform (Society for Industrial and Applied Mathemat-ics, Philadelphia, Pa., 1981).

5. "Colors of sea and sky," in Scientific Papers by Lord Rayleigh,Vol. V, 1902-1910 (Dover, New York, 1964), pp. 540-546.

(C6)

(C7)

Bruce J. West

(-R 2/01 2)1 . (C8)


6. See, e.g., R. G. Newton, Scattering Theory of Waves and Parti-cles (McGraw-Hill, New York 1966).

7. R. K. Crane, "Ionospheric scintillation," Proc. IEEE 65, 180-199 (1977).

8. S. M. Flatt6, R. Dashen, W. H. Munk, K. M. Watson, and F.Zachariasen, Sound Transmission through a FluctuatingOcean (Cambridge U. Press, New York, 1979).

9. A. Hewish, P. F. Scott, and D. Wills, "Interplanetary scintilla-tion of small diameter radio sources," Nature (London) 203,1214-1217 (1964).

10. R. Dashen, "Path integrals for waves in random media," J.Math. Phys. 20, 894-920 (1979).

11. L. L. Foldy, "The multiple scattering of waves: I. Generaltheory of isotropic scattering by randomly distributed scatter-ers," Phys. Rev. 67, 107-119 (1945).

12. M. Lax, "Multiple scattering of waves: II. The effective fieldin dense systems," Phys. Rev. 85, 621-629 (1952).

13. J. E. Gubernatis and E. Domany, "Effects of microstructure onthe speed and attenuation of elastic waves in porous materials,"Wave Motion 6, 579-589 (1984).

14. B. J. West and M. F. Shlesinger, "The fractal interpretation ofthe weak scattering of elastic waves," J. Stat. Phys. 36, 779-786(1984).

15. R. S. Wu, "Heterogeneity spectrum, wave scattering response ofa fractal random medium and the rupture processes in themedium," J. Wave Mater. Int. 1, 79-96 (1986).

16. B. B. Madelbrot, Fractals, Form and Chance (Freeman, SanFrancisco, Calif., 1977).

17. H. J. Morris and L. Kadanoff, "Teaching the renormalizationgroup," Am. J. Phys. 46, 652-657 (1978).

18. K. G. Wilson, "Problems in physics with many scales of length,"Sci. Am. 241(2), 158-179 (1979).

19. M. Stiassne, Department of Applied Mathematics, CaliforniaInstitute of Technology, Pasadena, California 91125, "The frac-tal dimension of the ocean surface," preprint (1986).

20. P. Levy, Theorie de l'Addition des Variables Altatories(Gauthier-Vildars, Paris, 1937).

21. E. W. Montroll and B. J. West, "On an enriched collection ofstochastic processes," in Fluctuation Phenomena, E. W. Mon-troll and J. L. Lebowitz, eds. (North-Holland, Amsterdam,1976; 2nd ed., North-Holland, Amsterdam, 1987).

22. A. N. Kolmogorov C. R. Dokl. Acad. Sci. URSS 30, 301 (1941).23. M. Holschneider, "On the wavelet transformation of fractal

objects," J. Stat. Phys. 50, 963 (1988).24. J. Feder, Fractals (Plenum, New York, 1988).25. B. B. Mandelbrot, "Intermittent turbulence in self-similar cas-

cades: divergence of high moments and dimension of the car-rier," J. Fluid Mech. 62, 331-358 (1974).

26. L. F. Richardson, "The problem of continuity: an appendix ofstatistics of deadly quarrels," Gen. Syst. Yearbook 6, 139-187(1961).

27. E. E. Underwood and K. Banerji, "Fractals in fractography,"Mater. Sci. Eng. 80, 1-14 (1986).

28. B. J. West and A. L. Goldberger, "Physiology in fractal dimen-sions," Am. Sci. 75, 354-365 (1987).

29. M. V. Berry and Z. V. Lewis, "On the Weierstrauss-Mandelbrotfractal function," Proc. R. Soc. London Ser. A 370, 459-484(1980).

30. W. Feller, An Introduction to Probability Theory and Its Ap-plications, 2nd ed. (Wiley, New York, 1971), Vol. II.

31. R. D. Mauldin and S. C. Williams, "On the Hausdorff dimensionof some graphs," Trans. Am. Math. Soc. 298, 793-803 (1986).

32. M. F. Shlesinger and B. D. Hughes, "Analogs of renormalizationgroup transformations in random processes," Physica 109A,597-608 (1981).

33. E. A. Novikov, Soviet Phys. Dokl. 11, 497-501 (1966).34. M. Nauenberg, "Scaling representation for critical phenom-

ena," J. Phys. A Gen. Phys. 8, 925-928 (1975).35. T. Niemeijer and J. M. J. van Leeuwen, in Phase Transition and

Critical Phenomena, C. Domb and M. S. Green, eds. (Academic,London 1976), Vol. 6.

36. G. Jona-Lasinio, "The renormalization group: a probabilisticview," Nuovo Cimento 26B, 99-119 (1975).

37. B. J. West, V. Bhargava, and A. L. Goldberger, "Beyond the

principle of similitude: renormalization in the bronchial tree,"J. Appl. Physiol. 60, 1089-1097 (1986).

38. A. L. Goldberger and B. J. West, "Fractals: a contemporarymathematical concept with applications to physiology andmedicine," Yale J. Biol. Med. 60, 104-119 (1987).

39. A. L. Goldberger, V. Bhargava, B. J. West, and A. J. Mandel,"On a mechanism of cardiac electrical stability: the fractalhypothesis," Biophys. J. 48, 525-528 (1985).

40. M. Ausloos and D. H. Berman, "A multivariate Weierstrass-Mandelbrot function," Proc. R. Soc. London Ser. A 400, 331-350 (1985).

41. G. Neumann and W. J. Pierson, Jr., Principles of PhysicalOceanography (Prentice-Hall, Englewood Cliffs, N.J., 1966).

42. K. J. Falconer, The Geometry of Fractal Sets (Cambridge U.Press, Cambridge, 1985).

43. 0. M. Phillips, The Dynamics of the Upper Ocean, 2nd ed.(Cambridge U. Press, London, 1977).

44. W. J. Pierson, "Wind generated gravity waves," Adv. Geophys.2, 93-178 (1955).

45. S. Kitaigorodskii, "On the theory of the equilibrium range in thespectrum of wind-generated gravity waves," J. Phys. Oceanogr.13, 816-827 (1983).

46. G. I. Barenblatt and I. A. Leykin, "On the self-similar spectra ofwind waves in the high frequency range," Izv. Acad. Sci. USSRAtmos. Oceanic Phys. 17, 35-41 (1981).

47. R. E. Glazman and P. B. Weichman, "Statistical geometry of asmall patch in a developed sea," J. Geophys. Res. 94,4998-5010(1989).

48. V. E. Zakharov and N. N. Filonenki, "The energy spectrum forstochastic oscillation of a fluid's surface," Dokl. Akad. NaukSSSR 170, 1292-1295 (1966).

49. C. A. Aviles and C. H. Scholz, "Fractal analysis of characteristicfault segments of the San Andres fault system," Eos 66,314-321(1985).

50. H. G. Booker, J. A. Ratcliffe, and D. H. Shinn, "Diffraction froman irregular screen with applications to ionosphere problems,"Philos. Trans. R. Soc. London A 242, 579-591 (1950).

51. V. Tatarskii, The Effects of Turbulent Atmospheres on WavePropagation (National Technical Information Service, Spring-field, Va. 1971).

52. G. H. Hardy, Divergent Series (Oxford U. Press, London, 1949).53. B. J. West and M. F. Shlesinger, "On the ubiquity of 1/f-noise,"

Int. J. Mod. Phys. B 3, 795-820 (1989).54. E. W. Montroll and M. F. Shlesinger, "On 1/f-noise and distri-

butions with long tails," Proc. Natl. Acad. Sci. USA 79, 3380-3387 (1982).

55. V. B. Gnedenko and A. N. Kolmogorov, Limit Distributions forSums of Independent Random Variables, K. L. Chang, transl.(Addison-Wesley, Reading, Mass., 1968).

56. R. Uimeki, C. H. Lin, and K. C. Yeh, "Multifrequency spectra ofionospheric amplitude scintillations," J. Geophys. Res. 82,2752-2760 (1977).

57. C. L. Rino, "A power law phase screen model for ionosphericscintillations 1. Weak scatter," Radio Sci. 14, 1135-1146(1979); "2. Strong scatter," Radio Sci. 14, 1147-1155 (1979).

58. R. L. Fante, "Some physical insights into beam propagation instrong turbulence," Radio Sci. 15, 757-762 (1980).

59. M. V. Berry, "Diffractals," J. Phys. A 12, 781-797 (1979).60. M. Marians, "Computed scintillation spectra for strong turbu-

lence," Radio Sci. 10, 115-119 (1975).61. E. Jakeman and J. G. McWhirter, "Correlation function depen-

dence of scintillation behind a deep random phase screen," J.Phys. A 10, 1599-1643 (1977).

62. G. Bourgois, "About the ergodicity hypothesis in random propa-gation studies," Astron. Astrophys. 102, 218-222 (1981).

63. D. N. Matheson and L. T. Little, "Radio scintillation due toplasma irregularities with power law spectra: the interplane-tary medium," Planet. Space Sci. 19, 1615-1624 (1971).

64. D. Gabor, "Theory of communication," J. Inst. Electr. Eng.(London) 93, 429-457 (1946).

65. J. Morlet, G. Arens, E. Fourgeau, and D. Giord, "Wave propaga-tion and sampling theory-part I: complex signal and scatter-ing in multilayered media," Geophysica 47, 203-221 (1982);"part II: sampling theory and complex waves," Geophysica 47,222-236 (1982).

Bruce J. West


66. A. Grossmann and J. Morlet, "Decomposition of functions intowavelets of constant shape, and related transform," in Mathe-matics & Physics, L. Streit, ed. (World Scientific, Singapore,1985), p. 135.

67. A. Arneodo, G. Grasseau, and M. Holschneider, "Wavelet trans-form of multifractals," Phys. Rev. Lett. 61, 2281-2284 (1988).

68. A. Arneodo, F. Argoul, J. Elezgaray, and G. Gresseau, Centre deRecherche Paul Pascal, Domaine Universitaire, 33405 TalenceCedex, France, "Wavelet transform analysis of fractals: appli-cation to nonequilibrium phase transitions," (preprint).

69. I. Daubechies, AT&T Bell Laboratories, 600 Mountain Avenue,Murray Hill, New Jersey 07974, "The wavelet transform, time-frequency localization and signal analysis," (preprint).

70. T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Prococcia, and B.I. Shraiman, "Fractal measures and their singularities: thecharacterization of strange sets," Phys. Rev. A 33, 1141-1150(1986).

71. F. Argoul, A. Arneodo, G. Grasseau, Y. Gagne, E. J. Hopfinger,and U. Frisch, "Wavelet analysis of turbulence reveals themultifractal nature of the Richardson cascade," Nature (Lon-don) 338, 51-53 (1989).

72. L. F. Richardson, "Atmospheric diffusion shown on a distance-neighbor graph," Proc. R. Soc. London Ser. A 110, 709-725(1926).

73. B. B. Mandelbrot, "Intermittent turbulence in self-similar cas-cades: divergence of high moments and dimension of carrier,"J. Fluid Mech. 62, 331-358 (1974).

74. K. Sreenivasan and C. Meneveau, "The fractal facets of turbu-lence," J. Fluid Mech. 173, 357-386 (1986).

75. C. Meneveau and K. R. Sreenivasan, Center for Applied Mathe-matics, Yale University, New Haven, Connecticut 06520, "Themultifractal spectrum of the dissipation field in turbulentflows," (preprint).

76. P. L. Turcotte, "Fractals in fluid mechanics," Annu. Rev. FluidMech. 20, 5-16 (1988).

77. M. F. Shlesinger, B. J. West, and J. Klafter, "L6vy dynamics ofenhanced diffusion; application to turbulence," Phys. Rev. Lett.58, 1100-1103 (1987).

78. H. Takayasu, "Stable distribution and Levy process in fractalturbulence," Prog. Theor. Phys. 72,471-479 (1984).

79. U. Frisch, P. Sulem, and M. Nelkin, "A simple dynamical modelof intermittent fully developed turbulence," J. Fluid Mech. 87,719-736 (1978).

80. B. D. Hughes, M. F. Shlesinger, and E. W. Montroll, "Randomwalks with self-similar clusters," Proc. Nat. Acad. Sci. USA 78,3287-3291 (1981).

Bruce J. West

Documents

Sensing scaled scintillations