ik-27-11-2150relations of the parameters of the /-k distribution for irradiance fluctuations to...

8/12/2019 IK-27-11-2150Relations of the parameters of the /-K distribution for irradiance fluctuations to physical parameters

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Relationsof the parameters of the /-K distribution forirradiance fluctuations to physical parameters ofthe turbulenceLarry C. Andrews, Ronald L. Phillips, and Bhimsen K. Shivamoggi

By using results from perturbation techniques for weak turbulence and the asymptotic theory for strongturbulence, we develop expressionsfor the parameters a and p of the I-K distribution in terms of the Rytovvariance for plane waves a2= 1.23C2 k7 16Llll6 and the inner scale of turbulence parameter lo. Comparisons ofthe resulting scintillation indexto experimental data and numerical results from the solution of the fourth-moment equation for 3-D propagation show good agreement.

1. IntroductionA great deal of progress has been made over the lastseveral years developing an understanding of the sta-tistical fluctuations that are induced by an opticalwavepropagating through atmospheric turbulence. -22A major goal of this work has been the construction of amathematical model which describes the probability

density function (PDF) of the irradiance (or intensity)fluctuations that evolve as the wave propagatesthrough the turbulence.2 ,3,5,9,10,13,171 8,21 ,22 In the earlyyears of research on this topic, the Rytov approxima-tion method (based on multiplicative perturbations),along with the Kolmogorovspectral model for the in-dex of refraction variations, led toan expression for thevariance of the log amplitude.23 Furthermore, theRytov method predicted that the log amplitude of theoptical wave obeys Gaussian statistics, and hence theamplitude (and alsothe irradiance) of the field is log-normally distributed. This conclusion led to the ex-pression o2 = 1.23C2k7/6L 11 /6 for the variance of the logintensity for plane waves,which is 4 times the varianceof the log amplitude. Here C is the refractive-indexstructure parameter, k is the wavenumber of the opti-cal wave,and L is the length of the propagation path.

Allauthors are withUniversity of Central Florida, Orlando,Flori-da 32816;R. L. Phillips is in the Department of Electrical Engineer-ing & CommunicationSciences, the other authors are in the Mathe-matics Department.Received 14 August 1987.0003-6935/88/112150-07$02.00/0. 1988 Optical Society of America.

The lognormal model and concomitant Rytov vari-ance o, provided a good fit to most of the data fromearly experiments, causing considerable optimismabout the range of validity of the model. However,theearly experimental data were taken over relativelyshort path lengths or through weak turbulence-in-duced index of refraction fluctuations for which crl 1.When experiments were conducted over longer pathlengths, and hence stronger conditions of turbulence,it became evident that lognormal statistics overesti-mated the statistical fluctuations of the irradiance 24Qualitatively,the lognormal model predicted spikesinthe irradiance fluctuations that should increase con-tinually as the propagation path length increased, butthis did not happen. In fact, the irradiance fluctua-tions increased only up to some peak value with pathlength (or Rytov variance), after which they steadilydecreased with still increasing path length (or Rytovvariance). Basically, the lognormal model was re-stricted to only short path lengths and/or weak turbu-lence conditions for which 2 < 0.3.4,6,8,2425While many other models have been proposed fordescribing irradiance fluctuations, it is perhaps thefamily of K distributions that has received the mostattention.3 -6 This general family has been useful inpredicting intensity statistics in a variety of experi-ments involving scattered radiation but is limited toconditions for which cr2> 1, where rI2s the normalizedvariance of intensity. The recently developed I-Kdistribution is a generalization of the K distributionthat is applicable in the same general conditions as theK distribution but alsoin conditions for which rI2< 1.18In weak fluctuation regimes the I-K and lognormalmodelspredict virtually identical statistics, and the I-K distribution is asymptotic to the K distribution in

2150 APPLIEDOPTICS / Vol. 27, No. 11 / 1 June 1988


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strong fluctuation regimes, approaching the negativeexponential distribution when saturation occurs.The I-K distribution arises from a treatment of theoptical wave fluctuations as a compound or doublystochastic random process. That is, the irradiance ofthe optical field is first described in terms of condition-al statistics that are representative of fluctuations oververy short time intervals. These statistics are thenaveraged over random fluctuations in the average in-tensity of the random field component that take placeover longer intervals of time. In essence, therefore,the I-K distribution is a two time-scale model thataccounts for both small scale and large scale turbu-lence effects.The I-K distribution has two parameters, the valuesof which determine the shape of the PDF curve. Pre-viously, these two parameters were selected by match-ing the first three statistical moments of the irradiancepredicted by the I-K model with the experimentallymeasured moments. The higher-order theoreticalmoments, completely determined by these two param-eters, were then compared with the higher-order ex-perimental moments. This commonly used procedurepermits a simple comparison of the theoretical PDFmodel to the experimental data.3 -6 18 However,high-er-order moments may require a large number of sam-ple values to keepthe scatter of the measured momentswithin acceptable bounds.26 Moreover, Consortiniand Conforti1 and Consortini et al.1 have pointed outthe possible effects of detector saturation on the high-er-order moments. In addition to the uncertainty as-sociated with measured higher-order moments, thisprocedure provides little direct insight into the rela-tionship of the parameters of a given theoretical modelwith the physical conditions of the propagation path.The purpose of this paper is to develop expressionsfor the parameters of the I-K distribution model interms of physical parameters of the turbulence itself,such as the refractive-index structure parameter, opti-cal wavenumber, and propagation path length, forhomogeneous conditions along the path. These rela-tions, developed first for both a plane wave and aspherical wave with the assumption of a vanishingsmall inner scale of turbulence, are based on resultsderived from the Rytov approximation method validfor weak turbulence conditions and an asymptotic the-ory that has been developed for strong conditions ofturbulence. In addition, suitable modifications due tothe effect of an inner scale are developed for the planewave case by introducing an inner scale parameter inthe spectrum model of a modified Kolmogorov spec-trum of the turbulence. These parameter relations forthe I-K model, based only on measured short pathsecond moment data (for calculating C2) and otherphysical parameters, permit us to predict the shape ofthe distribution and all higher-order statistical mo-ments-including second-order moments beyond thepoint of the short path measurements. The prelimi-nary comparisons presented here with experimentaldata and numerical models show favorable results.

II. I-K DistributionBy assuming the field of an optical wave propagatingthrough a turbulent medium can be expressed as thecoherent sum of a constant amplitude component anda randomly scattered component, we write the field ata given detection point and time asU(t) = exp(iwt)[A exp( iO) + R exp(io)]. (1)

Here A exp(iO) s the constant amplitude component,which physically represents the unscattered or averagecomponent of the optical field, and R exp(io) is thefluctuating or randomly scattered component. Thisgeneral formulation of the optical field is consistentwith that used in standard perturbation methods, suchas the Born approximation,' wherein the field is ex-pressed as a sum of the unperturbed field componentand one or more random perturbation terms due toscattering. A small but important modification of thisstandard model is what leads to the I-K distribution.Many statistical models for the irradiance fluctua-tions are based on the assumption that atmosphericturbulence is homogeneous, isotropic, and stationary.Even if homogeneity and isotropy are reasonable as-sumptions in many conditions of turbulence, the as-sumption of stationarity is probably valid only oververy short intervals of time during which the parame-ters of the turbulence remain essentially constant.Over longer periods of time, such as those normallyassociated with experimental measurements, theseturbulence parameters will likely fluctuate in a ran-dom fashion.2 3 27 Because of this situation, we believeit may be helpful to describe initially the statisticalproperties of a wave propagating through such a medi-um in terms of a compound statistical model. The I-Kdistribution evolves from a compound statistical mod-el whereby the conditional irradiance distribution isassumed to be the modified Rice-Nakagami PDF, aspredicted by the Born approximation. The effect ofrandom fluctuations in the turbulence parameters arethen modeled by allowing random variations in theaverage irradiance of the random component (or vari-ance) of the field described by Eq. (1). Thus, when theRice-Nakagami distribution is averaged over gammastatistics for the fluctuating variance of the field, weare led to the I-K distribution as the unconditional orabsolute PDF for the irradiance.18 Its functional formis given byIa A_

2a ,~A)Ka-1 2A

ia-1 (2A

I a-i

F) Ka-(2 b)(2I, I >A2,

(2)where a is a parameter generally associated with thenumber of scatterers forming the random componentof the optical field (1) and bo s the absolute mean valueof the intensity of the random component. The func-tions I(-) and K,(-) are modified Bessel functions ofthe first and second kind, respectively. The specialcase a = 1 has also been derived for the probability of

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single-pulsedetection of an N-glint target immersed inK-distributed clutter.2 8111. Scintillation Index for Zero Inner ScalePerhaps the most important measure of the irradi-ance fluctuations of a laser beam is the scintillationindex

2 = (2)-1,Ij -11)2

2

(3)where (2)/(I)2 is the second normalized moment ofthe irradiance. The scintillation index is often ex-pressed as a function of the Rytov variance a2. Forexample, based on the Rytov perturbation method forplane waves, the variance of the log intensity in weakfluctuation regimes is defined by2 3

a1 = (1.23 C 2k7 / 6 L11/6 )iFig. 1. Variation of the scintillation index with al for both planewaves and spherical waves with zero inner scale.

= 16 k JJ )(,) sin [ 2k ]Kdf,where q0,(K) is the refractive-index power spectrum.Assuming a pure Kolmogorov spectrum, i.e., kn K) =0.033C2K-113, this expression reduces to

aL2"= 1.23C'k 7 /6L 1/6 = a2 (5)Hence the scintillation index is simply23

1U=exp( 22)-10 j 1 + 0.5 ), al 1 (plane wave), (6)

where wehave retained only the first few terms of theseries for the exponential function. In the case of aspherical wave,the corresponding expression is1a exp(0.41o)-

0.41a2(1 + 0.5cr2),a2 >1 (plane wave), (8)aI2 1+ 8 a2>> 1 (spherical wave). (9)

A. Parameter ValuesThe normalized moments of the I-K distributionare' 8

I) n r(a + ) (p)' n = 1,2,3. 10)I)f anl + P)n(-ra + ) k

where p = A2/bo is a parameter denoting the powerratio of the mean intensity of the constant amplitudecomponent to the absolute mean intensity of the ran-

dom component of the optical field (1). Thus thescintillation index for the I-K distribution assumesthe form

+)2 2 a) (11)By comparing the scintillation index (11) with knownresults for both weak and strong turbulence regimes asdescribedby Eqs. (6)-(9), we can relate the I-K param-eters a and p to the Rytov variance a 2for both planeand spherical waves.Weak turbulence conditions are characterized in theI-K distribution by large values of the power ratio p.In this case we find that the scintillation index (11)assumes the approximate form2G_-, p>>l.p (12)By comparison of this expression with the similar con-ditions of weak turbulence described by Eqs. (6) and(7), we deduce that

ap = 2 2 , O.,


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relate the product ap to a-2n weak turbulence regimes.Although the validity of these expressionsis subject tosevere restrictions, we will examine the consequencesof using them to estimate a and p for all conditions ofturbulence. In this fashion we will have a model forthe scintillation index that agrees with known resultsin both weak and strong turbulence regimes but maybe less accurate in the vicinity of peak scintillations.Moreover,these relations permit us to predict the sec-ond-, third-, and higher-order moments based on theI-K model for all conditions of turbulence.In Fig. 1 we have plotted the scintillation index (11)for both plane and spherical waves as a function of thesquare root of the Rytov variance, where a and p aredefined in the plane wave case by

a = 2.33a1,(2

= aa2(1+ o.5aY)and in the spherical wave case by

"I

1-

0 2 4 6a91 = (1.23 2 k7/6 L1/6)i

Fig. 2. Variations of the square root of the scintillation index witha, for plane waves and zero inner scale. The solid curve is thatpredicted by the I-K distribution, and the experimental data arefrom Ref. 24. The dots represent measured data at a fixed propaga-tion distance of 1750 m, while the pluses are the same at 250 m.

a = 0.71a4',4.88= + 2)aa2,(1 + 0.2a2)

(19)

The predicted peak value of the scintillation index ishigher for the spherical wave than for the plane wave,and it also occurs at a larger value of al than that of theplane wave.In Fig. 2 we compare experimental data of Grachevaet al.24 for a plane wave with that predicted by the I-Kdistribution. A similar graph for the spherical wavecase using the data of Refs. 6 and 8 is shown in Fig. 3.In both figures we see that the general trend of thetheoretical curves follows that of the experimentaldata, but the maximum values are somewhat less thanmost of the data. There are at least two possibleexplanations for this discrepancy. First, our tech-nique for choosing the parameters a and p is onlyapproximate, especiallyin the regime of peak scintilla-tions where errors are most likely to occur. Second, wehave thus far ignored inner scale effects which areknownto lead to larger peak values in the scintillationindex."1,'9In Fig. 4 we compare our model for plane waves tonumerical results derived by Whitman and Beran,15which are based on the fourth-moment equation for 3-D propagation. The dashed curve is the numericalresult, while the solid curve is that of the I-K distribu-tion. The two curves agree quite well everywhereexcept in the vicinity of the scintillation peak wherethere is some small discrepancy.IV. Inner Scale Effect for Plane WavesUp to this point our analysis has been based on theassumption of a zero inner scale. However, a finiteinner scale can lead to larger values of the scintillationindex, particularly in the region of peak values of thescintillation index, than observed when the inner scaleis vanishingly small. By using a modified Kolmogorovspectrum which introduces an inner scale parameter,

0 2 4 6 8a1 (1.23 C 2k7/ 6 L11/6)

Fig. 3. Variation of the square root of the scintillation index with a1for spherical waves and zero inner scale. The solid curve is thatpredicted by the I-K distribution, and the experimental data arefrom Refs. 6 and 8. The dots are Ref. 8 data with variable propaga-tion path lengths up to 1500 m, while the pluses are Ref. 6 data takenat 1250 m.

0~

1

0 2 6a1 = (1.23 c2k7/ 6 L11/ 6 )A

Fig. 4. Comparison of the square root of the scintillation index forthe I-K distribution with numerical results from Ref. 15 based onfourth-moment equations for 3-D propagation.1 June 1988 / Vol. 27, No. 11 / APPLIEDOPTICS 2153


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GI

2 4 6a1 = (1.23 C2 k7/6 L11/6)

Fig. 5. Variation of the square root of the scintillationindex witha1and inner scale of turbulence loand comparisonwith the experimen-tal data of Ref. 24. The theoretical curves correspond to a propaga-tion path length of 1750 m.

Fantell developed an asymptotic theory for planewavesthat led to the scintillation index described byl _ + 2.03 , 2 >> 1,

wherea = 6.94a 17 16 .

(20)

(21)The parameter 3 n Eq. (21) is the square of the ratio ofthe size of the first Fresnel zone to inner scaleparame-ter, i.e.,

{ = XL (22)where X is the optical wavelength and lo s the innerscale parameter. Thus, by equating Eqs. (15) and(20),we see that the inner scale correction in a leads to

a = 0.985al/3= 136(a2 7/6)1/6. (23)To find a comparable expression for the parameter pwith inner scalecorrections, weneed to calculate a newexpression for the variance of the log intensity definedby Eq. (4) that is not based on a pure Kolmogorovspectrum. By assuming that

= 0.033C2 ex l(-K/ ' (24)K + K g)11where Km = 5.92/10and KOis generally the reciprocal ofthe outer scale of turbulence, and performing the re-sulting integrations in Eq. (4)we obtain the first-orderapproximation (seeAppendix)

'n = 1.23(C 2 )k7 16 L"16= 2-2 (25)

where&2 3.864 1 + 1 211/12\ 31.14/3J

X sin (124an-1(5.581) - 1.69 (26)A morecomplete expression for a2 involvingboth inner

and outer scaleparameters of turbulence isgiven in theAppendix. However,we found that outer scale effectsare generally negligible, and thus Eq. (26) is adequatefor nearly all conditions of turbulence. In the limit ofvanishingly small inner scale (3 - ), it can be shownthat 2 1.The inner scale correction factor in the log intensity(25) leads to calculated values of C2 hat are generallygreater than those calculated from a pure Kolmogorovspectrum model that assumes a zero inner scale. Thatis,if one were to ignorethe inner scaleand make a shortpath measurements of a-Ior a-Ln,, the quantity Cnc2represents the apparent value of the refractive-indexstructure parameter C2. If the inner scale parameterlo is known, however, the actual value of Cn can alwaysbe calculated by dividing C2&by the correction factor.2, hose value can be determined with the use of Eq.(26).Based on the result of Eq. (25),the inner scale cor-rection in the scintillation index for weak turbulenceleads to

I2 = a2&2(1 + 0.5a 2), a2


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ULn = 0.2647n2 k2 C(I-12),where

JLJf K exp-K /Km)(K + Ko) /

I2 = Kexp- /n) cos[K2(L dKdq.

Al)

(A2)

(A3)

& = 1.06 ( 2 U(1;1/6;/Krn) - 1272 k50 M ~~~2X E -(1)n(LK'/k)n 1 +k2 (n+l)/2(2)((+/6)n L2K4X sin [(n + 1) tan' (f K ]

The integral I, is readily evaluated to give (see Ref. 30,p. 303)+ 3.864 2 (11/6)(LK/k 1I 1/)n

I, = 2 LK- 3/U(1;1/6;K2/K2), (A4)where U(a;c;x) s the confluent hypergeometric func-tion of the second kind. Using Euler's formula for thecosine function, a similar evaluation of 12 yields

12= /4 Ko J [U(1;1/6;X iY)+ U(1;1/6;X + iY)Id7, (A5)

where i2 = -1 andXB siYc v aria bles, (A 6)

By suitable changes of variables, Eq. (5)ten in the more compact formI2 = h K 11/3a U(1;16;t)dt,

can be writ-

(A7)

wherea ib = K2 1 iL) (A8)

Introducing the identity30U(1;1/6;t)= 6 ,F,(l;1/6;t) + rI-5/6)t 5 I6et, (A9)5

where lFl(a;c;x) denotes the confluent hypergeometricfunction of the first kind, it can then be shown thatI2 = 1 K 3 [(a + ib) 2 F2 (1,1;2,1/6;a + ib)

- (a - ib)2F2(1,1;2,1/6;a ib)]3r(-5/6) kK-1113[(a ib)11/6F 1(11/6;17/6;a+ ib)22i

- (a - ib)1"/61F,(11/6;17/6;a - ib)], (A10)where 2F2 (a,b;c,d;x) is a generalized hypergeometricfunction.3 0Finally, by expressingthe hypergeometric functionsin Eq. (A10) by their series representations and com-bining I, and I2, we are led to the result

,L" = 1.23Cgk7/6L"'/6&2, (All)where

Xsin (n + 6 )tan- (Lk)] (A12)The Pochhammer symbol in Eq. (A12)is defined by

(a)n= r(a + n) n = 0,1,2 (A13)F(a)Equation (A12)provides a multiplicative correctionterm to the variance of log intensity defined by Eq.(All), which includes both inner scale and outer scaleeffects of turbulence. For most cases of interest, how-ever, the inner and outer scales differ by several ordersof magnitude, so that all terms of the series are negligi-ble except for the first term of each series. Retainingonly the n = 0 term in each series and usingthe approx-imation

1.06 ) U(1;1/6;Kg/K) 272-K)5LK2( Km+ 1.06r(-5/6) 2)16

o/K2


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10. W. A. Coles and R. G. Frehlich, "Simultaneous Measurements ofAngular Scattering and Intensity Scintillation in the Atmos-phere," J. Opt. Soc. Am. 72, 1042 (1982).11. R. L. Fante, "Inner-Scale Size Effect on the Scintillations ofLight in the Turbulent Atmosphere," J. Opt. Soc. Am. 73, 277(1983).12. L. R. Bissonnette, "Propagation Model of Laser Beams in Tur-bulence,' J. Opt. Soc. Am. 73, 262 (1983).13. R. Dashen, "Distribution of Intensity in a Multiply ScatteringMedium," Opt. Lett. 10, 110 (1984).14. A. Consortini and G. Conforti, "Detector Saturation Effect onHigher-Order Moments of Intensity Fluctuations in Atmo-spheric Laser Propagation Measurement," J. Opt. Soc. Am. A 1,1075 (1984).15. A. M. Whitman and M. J. Beran, "Two-Scale Solution for Atmo-spheric Scintillation," J. Opt. Soc. Am. A 2, 2133 (1985).16. A. Consortini, E. Briccolani, and G. Conforti, "Strong-Scintilla-tion-Statistics Deterioration due to Detector Saturation," J.Opt. Soc. Am. A 3, 101 (1986).17. R. Barakat, "Weak-Scatter Generalization of the K-DensityFunction with Applicationsto Laser Scattering in AtmosphericTurbulence," J. Opt. Soc. Am. A 3, 401 (1986).18. L. C. Andrews and R. L. Phillips, "Mathematical Genesis of theI-K Distributionfor Random Optical Fields,"J. Opt. Soc.Am.A

3, 1912 (1986).19. R. G. Frehlich, "Intensity Covariance of a Point Source in aRandom Medium with a KolmogorovSpectrum and an InnerScale of Turbulence," J. Opt. Soc. Am. A 4, 360 (1987).20. D. Link, R. L. Phillips, and L. C. Andrews, "Theoretical Model

for Optical-Wave Phase Fluctuations," J. Opt. Soc. Am. A 4,374(1987).21. J. H. Churnside and R. J. Hill, "Probability Density of Irradi-ance Scintillations for Strong Path-Integrated Refractive Tur-bulence," J. Opt. Soc. Am. A 4, 727 (1987).22. J. H. Churnside and S. F. Clifford, "Log-Normal Rician Proba-bility-Density Function of Optical Scintillations in the Turbu-lent Atmosphere," J. Opt. Soc. Am. A 4, 1923 (1987).23. V. I. Tatarski, Wave Propagation in a Turbulent Medium,translated by R. S. Silverman (McGraw-Hill, New York, 1961).24. M. E. Gracheva, A. S. Gurvich, and M. A. Kallistratova, "Mea-surements of the Variance of 'Strong' Intensity Fluctuations ofLaser Radiation in the Atmosphere," Izv. Vyssh. Uchebn.Zaved. Radiofiz. 13, 55 (1970).25. J. W. Strohbehn, T.-I. Wang, and J. P. Speck, "On the Probabili-ty Distribution of Line-of-Sight Fluctuations of Optical Sig-nals," Radio Sci. 10, 59 (1975).26. H. Cramer, Mathematical Methods of Statistics (PrincetonU.P., Princeton, NJ, 1966),p. 364.27. V. I. Tatarski and V. U. Zavorontnyi, "Wave Propagation inRandom Media with Fluctuating Turbulent Parameters," J.Opt. Soc. Am. A 2, 2069 (1985).28. R. Fante, "Detection of Multiscatter Targets in K-DistributedClutter," IEEE Trans. Antennas Propag. AP-32, 1358(1984).29. A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvili, and V. I.Shishov,"Laser Irradiance Propagation ina Turbulent Media,"Proc. IEEE 63, 790 (1975).30. L. C. Andrews, Special Functions for Engineers and AppliedMathematicians (Macmillan, New York, 1985).

Meetingsontinuedrompage 1261988June27-8 July 1st Int. School & Workshop in Photonics, Oaxtepec J.Ojeda-Castaneda, INAOE, Apdo. Postal 216, 72000Puebla Pue, Mexico

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2156 APPLIEDOPTICS / Vol. 27, No. 11 / 1 June 1988continued npage 213

ik-27-11-2150relations of the parameters of the /-k distribution for irradiance fluctuations to...

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