measurement and data-processing approach for estimating the spatial statistics of turbulence-induced...

$: Measurement and Data-Processing Approach for Estimating the Spatial Statistics of Turbulence-Induced Index of Refraction Fluctuations in the Upper Atmosphere$
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Measurement and data-processing approach forestimating the spatial statistics of turbulence-inducedindex of refraction fluctuations in the upper atmosphere

Wade W. Brown, Michael C. Roggemann, Timothy J. Schulz, Timothy C. Havens, Jeff T. Beyer,and L. John Otten

We present a method of data reduction and analysis that has been developed for a novel experiment tomeasure the spatial statistics of atmospheric turbulence in the tropopause. We took measurements oftemperature at 15 points on a hexagonal grid for altitudes from 12,000 to 18,000 m while suspended froma balloon performing a controlled descent. From the temperature data we estimate the index of refrac-tion and study the spatial statistics of the turbulence-induced index of refraction fluctuations. Wepresent and evaluate the performance of a processing approach to estimate the parameters of isotropicmodels for the spatial power spectrum of the turbulence. In addition to examining the parameters of thevon Karman spectrum, we have allowed the so-called power law to be a parameter in the estimationalgorithm. A maximum-likelihood-based approach is used to estimate the turbulence parameters fromthe measurements. Simulation results presented here show that, in the presence of the anticipatedlevels of measurement noise, this approach allows turbulence parameters to be estimated with goodaccuracy, with the exception of the inner scale. © 2001 Optical Society of America

OCIS code: 010.0010.

1. Introduction

Atmospheric turbulence affects the performance oflaser beam projection systems by causing randomfluctuations in the index of refraction of the atmo-sphere.1,2 Laser beam propagation through a turbu-ent medium results in a turbulence-inducedecorrelation of the wave front, broadening the spotize on the target or receiver, thereby decreasing theerformance of the system. The turbulent effectsccur at all altitudes and are stronger in the lowertmosphere. However, at higher altitudes, turbu-ence effects can also be strong if it is necessary toropagate over a sufficiently long path. For exam-le, in the case of the Airborne Laser system it isecessary to propagate a laser beam over a path of

W. W. Brown, M. C. Roggemann [email protected]!, T. J.Schulz, T. C. Havens, and J. T. Beyer are with the Department ofElectrical Engineering, Michigan Technological University, 1400Townsend Drive, Houghton, Michigan 49931. L. J. Otten is withthe Kestrel Corporation, 3815 Osuna, Albuquerque, New Mexico87109.

Received 6 July 2000; revised manuscript received 12 December2000.

0003-6935y01y121863-09$15.00y0© 2001 Optical Society of America

the order of several tens to a few hundred kilometersat altitudes exceeding 10 km. These path lengthsare sufficient to have a strong influence on the abilityto aim and to focus the beam on a moving target.

Turbulence in the lower atmosphere is well under-stood, and experiments agree with the predictions.The models developed for the turbulence derive fromthe seminal research done by Kolmogorov3 and Ta-tarskii.4 These models are statistical in nature andrely heavily on dimensional analysis.5 Another as-pect of these models is that they make strong as-sumptions about the nature of the turbulence,including the assumptions of homogeneity and isot-ropy. Heuristic extensions to Kolmogorov’s resultshave been made to take into account finite inner andouter scales of the turbulence.2

There is a significant body of experimental evi-dence from measurements of turbulence made athigh altitudes that conflicts with the assumptionsmade in the conventional statistical models. Anisot-ropy of turbulence has been demonstrated in labora-tory investigations, even at Reynolds numbers wellbeyond the critical Reynolds numbers for flow geom-etry.6,7 Because the tropopause is known to havefluid flow properties similar to the shear layers cre-ated in the laboratory, these studies cast doubt on the

20 April 2001 y Vol. 40, No. 12 y APPLIED OPTICS 1863

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correctness of the conventional assumption of isotro-pic turbulence through the entire atmosphere. Mea-surements taken near the ground have also shownanisotropic behavior of the turbulence.8 French re-searchers have collected data at high altitudes thatindicate the existence of anisotropic turbulence andnon-Kolmogorov behavior.9 Flight test studies haveshown the presence of stratified layers in the upperatmosphere at altitudes between 12,000 to 18,000 m,indicating that conventional models of how turbu-lence strength varies smoothly as a function of alti-tude can be in error.10 Experimental studiesdesigned to measure directly the power law of thespatial power spectral density of the turbulence-induced index of refraction fluctuations have showndepartures from the power law predicted by Kolmo-gorov theory within the inertial subrange.11 Theonventional assumption that the outer scale L0 of

the turbulence is typically much larger than theapertures of interest has been challengedexperimentally,12–14 and the resulting implications ofthis problem for adaptive optical systems have beenexamined.15 Finally, nearly all the nonflight testdata taken to date have been collected from sensorstowed behind a rising balloon. However, wake tur-bulence arising from the blunt body of the balloon isknown to corrupt these measurements and could bethe source of the surprisingly consistent turbulencelevels reported worldwide.16,17

The methods for simulation and data reductionpresented here were developed in conjunction with anexperiment that takes measurements of temperatureat altitudes from 12,000 to 18,000 m. The temper-ature measurements were taken with 13 anemome-ters placed on the perimeter of a hexagonal grid andone anemometer placed at the end of each of two armsextending from the hexagonal array, for a total of 15anemometers. Figure 1 shows the positions of theanemometers. The widest separation in the array is10 m, and the smallest separation of probes is 4 cm.

Fig. 1. Anemometer placement. Note that the probes are at-tached to a thin rigid structure, which is not shown here.

864 APPLIED OPTICS y Vol. 40, No. 12 y 20 April 2001

We chose the locations of the anemometers so that wecould obtain a reasonable even sample of the spacecontaining the autocorrelation of the temperaturefluctuations.18 The sample locations in the space ofthe autocorrelation function are shown in Fig. 2.Note that only one half of the symmetric sample lo-cations are shown here. The anemometer array issuspended below the balloon, and the data are col-lected during descent. The experiment was de-signed to reduce or eliminate turbulent effects thatare due to the wake of the balloon. The anemome-ters are operated in a constant current, low overheatmode, making them sensitive to temperature fluctu-ations. The instrument package has a pressureprobe, as the pressure varies significantly over therange of altitudes, and, as shown in Section 2, theindex of refraction is a function of both temperatureand pressure. The instrument package also in-cludes an attitude sensing package, allowing themeasurements to be registered in three-dimensionalspace during the balloon descent. Our processingapproach is to convert the measured temperatureand pressure fluctuations into index of refractionfluctuations. The important step of calibrating theanemometer probes to convert the measurements toactual temperatures of the surrounding air is beingtreated in the development of the experiment but isbeyond the scope of this paper.

In the processing approach developed here, we usesample-based estimates of the spatial autocorrelationof the index of refraction fluctuations for separationscorresponding to all possible pairs of anemometers byaveraging data gathered over thin slabs of the atmo-sphere. In the processing paradigm presented here,the index of refraction fluctuations are assumed ex-plicitly to be homogeneous and isotropic, and we en-force this assumption in the data processing bytreating the vector separations of the probes as sca-lars. We are aware of the possibility of anisotropicturbulence.6,7 Currently we are developing turbu-

Fig. 2. Location of the samples taken of the autocorrelation of thetemperature fluctuations.

o

ed

i

slii

l

mtnitiat

bim

wt

lence models and algorithms for estimating the pa-rameters that take into account the possibility ofanisotropy, but these are beyond the scope of thispaper.

The measurements are used in a maximum-likelihood-based nonlinear optimization routine toestimate the turbulence parameters of interest. Forthe von Karman spectrum, there are three parame-ters of interest: ~1! the outer scale L0, ~2! the innerscale l0, and ~3! a constant that describes the strengthf the fluctuations denoted by Cn

2 and referred to asthe structure constant of the turbulence. We imple-mented a non-von Karman model for the spatialpower spectrum of the index of refraction fluctuationsthat allows the exponent on the wave number in thedenominator to vary from 11y6 and that eliminatesthe exponential term that contains the inner scale.In the model developed here we denote this exponentby the symbol a and note that, when a Þ 11y6, theconstant in the numerator cannot be denoted prop-erly by Cn

2 because this constant would have differ-nt units and could not be inserted into equationserived from the von Karman model. Thus we in-

troduce the symbol PC to represent the proportional-ty constant when a Þ 11y6.

In this paper we describe the measurement andignal processing aspects of this approach to turbu-ence characterization and explore the noise sensitiv-ty of this technique. Results presented herendicate that we can estimate L0, PC, and a with good

accuracy when the measurements have a signal-to-noise ratio ~SNR! of 20 or greater. Extensive effortswere made to estimate l0 by use of the maximum-ikelihood-based approach; however, estimates of l0

were found to contain large errors, and efforts toestimate l0 with this approach were abandoned.This problem arises because, when the value of l0 isvaried over a reasonable range, it does not stronglyaffect the shape of the autocorrelation in the range ofbaselines that we sample.

The paper is organized as follows. The theoreticalbackground for the Kolmogorov and von Karman

odels for turbulence are given and discussed in Sec-ion 2. The random index screen and measurementoise simulation methods that are used are discussed

n Section 3. The estimation method that is used,he maximum-likelihood-based estimator, is derivedn Section 4. The results are presented in Section 5,nd the concluding remarks are provided in Sec-ion 6.

2. Theory

The index of refraction of atmosphere n at opticalfrequencies is predominately a function of tempera-ture2:

n 5 1 1 77.6~1 1 7.52 3 1023l22!PT

3 1026, (1)

where l is the wavelength of the light measured inmicrometers, P is the atmospheric pressure in milli-

ars, and T is the temperature in kelvins. Linear-zing Eq. ~1! in a Taylor-series expansion about the

ean temperature T# keeping the linear term yields

n < 1 1 77.6~1 1 7.52 3 1023l22!P

T3 1026

2 77.6~1 1 7.52 3 1023l22!P

T 2 3 1026DT, (2)

where DT 5 T 2 T. The fluctuations in the index ofrefraction can be written as Dn 5 n~T 1 DT! 2 n~T!;the result for Dn when we use the result from approx-imation ~2! is

Dn 5 277.6~1 1 7.52 3 1023l22!P

T 2 3 1026DT. (3)

Because the fluctuations in the index of refraction arelinear in DT and we assume that the distribution ofDT is Gaussian, the fluctuations in the index of re-fraction are also Gaussian. In the free atmosphere,fluctuations of the index of refraction are due primar-ily to inhomogeneities in the temperature structurebecause of differential heating of the Earth’s surface,2temperature-induced buoyancy, and mixing causedby wind action. It should be noted that at high Machnumbers or in strong acoustic fields, pressure fluctu-ations can arise that are sufficiently strong to domi-nate the fluctuations in the index of refraction.

The spatial autocorrelation function ~ACF! of theindex of refraction fluctuations Bn~r! can be writtenin terms of the power spectral density ~PSD! of theindex of refraction fluctuations Fn~r! as2

Bn~r! 5 *** Fn~k!exp~2jk z r!d3k, (4)

where r 5 ~Dx, Dy, Dz! is the vector separation ofpoints in space, k 5 ~kx, ky, kz! is the spatial wavenumber, and the integration is taken over all space.The three-dimensional Fourier transform can be re-duced to a one-dimensional integral when we inte-grate in spherical coordinates over the two angulardimensions for the case of isotropic turbulence to ob-tain2

Bn~r! 54p

r *0

`

Fn~k!k sin~kr!dk, (5)

here r 5 uru and k 5 uku. In Kolmogorov’s theoryhe PSD Fn~k! encompasses three distinct regions.2

In the low wave-number region where k , 2pyL0,where L0 is called the outer scale of the turbulence,the statistics of atmospheric turbulence are not pre-dicted by Kolmogorov’s theory. Little is knownabout the range of sizes of L0, although it is thoughtto be of the order of a few meters to several tens ofmeters. In the high wave-number region where k .2pyl0, the Kolmogorov spectrum disagrees with ob-servation because turbulence in this region is gov-erned by small-scale fluctuations and viscousdissipation. Near sea level l0 is known to be of the


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w

1

order of a few to several millimeters, and l0 increaseswith increasing altitude with the result that, for therange of altitudes where measurements will be made,l0 will be of the order of a few to several centimeters.The region in which Kolmogorov’s theory agrees wellwith experiment is termed the inertial subrange 2pyL0 , k , 2pyl0. Within the inertial subrange thePSD is given by2

Fn~k! 5 0.033Cn2k211y3. (6)

A significant theoretical problem with the Kolmo-gorov spectrum given in Eq. ~6! is that it is oftennecessary to integrate over all wave numbers.1 Thepectrum in Eq. ~6! has a nonintegrable pole at k 5 0,

yielding nonphysical results such as the index of re-fraction fluctuations having infinite variance. Toaccount for some of these difficulties, the von Karmanspectrum has been proposed2:

Fn~k! 50.033Cn

2

~k2 1 k02!11y6 expS2

k2

km2D , (7)

where km 5 5.92yl0 and k0 5 2pyL0. The vonarman spectrum rolls off at low and high spatial

requencies and has a finite variance.1In this experiment we process the data to estimate

the parameters of a modified form of Eq. ~7! given by

Fn~k! 5Pc

~k2 1 k02!a , (8)

where PC is used to represent the power constant,with PC 5 0.033Cn

2 when a 5 11y6. This PSDmodel has three parameters: ~1! the outer scale L0,~2! the proportionality constant PC, and the so-calledpower law a. It should be noted that in Eq. ~8! thereis no term containing the inner scale l0 because wehave not been able to obtain accurate estimates of l0,most likely because values of the von Karman PSD atwave numbers higher than 2pyl0 contribute little tothe spatial autocorrelation that is computed from themeasured data. The ACF corresponding to the PSDin Eq. ~8! is

Bn~r! 54p

r *0

` PC

~k2 1 k02!a k sin krdk, (9)

which can be solved analytically for noninteger val-ues of a19:

Bn~r! 54p

r F 1G~a!G @2~0.52a!k0

~12a!Îp~k0 r!~a20.5!

3 K3y22a~k0 r!#, (10)

here G~a! is the gamma function and K3y22a~br! is amodified Bessel function of the first kind of order3y2 2 a. We evaluated the performance for 9y6 # a13y6. Hence the only integer value of a we are in-


terested in is 2, and the resulting expression for theACF for a 5 2 is19

Bn~r! 5@exp~2k0r!pr#

4k0. (11)

To evaluate the performance of our approach to esti-mating PC, a, and L0 from the anemometer arraymeasurements, we developed a simulation. The de-tails of the simulation are presented in Section 3, andthe estimation approach is described in Section 4.

3. Simulation of Anemometer Array Measurements

To simulate the effects of atmospheric turbulence it isnecessary to make random draws of the index of re-fraction fluctuations on the grid of sample points hav-ing statistics that we can control; we refer to therandom draws as index screens. Because the covari-ance matrix Bn computed for all pairs of points on thesampled grid is real and positive definite, it can bewritten as a product of two matrices20:

Bn 5 RRT, (12)

where R is the Cholesky factor of Bn.1 Let the ran-dom vector f represent a sample function of the dis-crete Gaussian zero-mean random process that has acorrelation function Bn. To create f, we start with arandom vector of independent, zero-mean Gaussianrandom variables a. The covariance of the randomvector is given by

^aaT& 5 IM, (13)

where IM is the identity matrix. A random draw ofthe index screen f is formed by

f 5 Ra, (14)

where the covariance of f is given in21

^ff T& 5 ^Ra~Ra!T& 5 ^RaaTRT&. (15)

Realizing that R is a constant matrix, we can factor itout; and using the covariance of the random drawshown above, we obtain

^ff T& 5 ^RaaTRT& 5 R^aaT&RT 5 RIM RT

5 RRT 5 Bn, (16)

which is the desired input ACF.To replicate the expected experimental situation as

close as possible, we also added measurement noisewith SNR’s that matched the expected range of SNR’sfrom the experimental setup. For each randomdraw we add a noise term, which is a normally dis-tributed random vector with zero mean and a stan-dard deviation of a, to the random draw of the indexscreen. The standard deviation of the noise termwas chosen to match the desired SNR by the follow-ing rule:

a 5signal strength

SNR, (17)

w

T

t

t

o

tmorpb

w

w

wb

where the signal strength is taken as the square rootof the diagonal elements of the covariance matrix Bn.The square root of the diagonal elements is used be-cause it describes the root-mean-square ~rms!strength of the fluctuations of the index of refractionat each point. The random draw of an index screenthat includes the additive noise is given by

fN 5 Ra 1 av, (18)

where v is a zero mean and a unit variance Gaussianvector whose elements are uncorrelated. The co-variance of fN is given by

^fNfNT& 5 ^~Ra 1 av!~Ra 1 av!T&, (19)

hich can be expanded to obtain

^fNfNT& 5 ^RaaTRT& 1 ^RaavT& 1 âvaTRT&

1 a2^vvT&. (20)

he first term on the right-hand side of Eq. ~20! is thedesired quantity Bn. Because a and v are indepen-dent and â& 5 0 and ^v& 5 0, the second and thirderms of Eq. ~2! are evaluated as

^RaavT& 5 Râ& ^vT&a 5 0, (21)

âvaTRT& 5 a^v&âT&RT 5 0. (22)

The last term in Eq. ~20! can be simplified when werealize that ^vvT& is equal to IM, so that

a2^vvT& 5 a2IM. (23)

Combining all these specifications and reductionsyields the final result:

^fNfNT& 5 Bn 1 a2IM. (24)

Equation ~24! shows that ^fNfNT& is a biased estima-

or of Bn, and hence we must subtract a diagonalmatrix with elements equal to a2 from ^fNfN

T& tobtain an unbiased estimate of Bn.The maximum-likelihood-based estimation method

is derived for the estimation of PC, a, and L0 in thefollowing section, which details the results of the sim-ulations and estimations for a variety of conditions.

4. Estimation Approach for PC, a, and L0

We developed a parameter estimation techniquebased on the maximum-likelihood estimator tech-nique for the case in which no measurement noise ispresent. We then tested the performance of thistechnique in the presence of noise and now report theresults. This approach is justified in part by the factthat the experimental instrumentation is expected toprovide data with a SNR $ 50 and in part by the facthat the number of parameters that must be esti-ated by nonlinear optimization can be reduced by

ne, greatly reducing the number of computationsequired to process the large amount of data antici-ated. For the maximum-likelihood-estimator-ased method we assume noise-free signals from the

N sensors; for K realizations the probability densityfunction is given by

f @~dk!# 5 )i51

k

~2p!2Ny2 u PCSu21y2 exp@2dkT~PCS!21dk#,

(25)

here we defined

Bn 5 PcS (26)

to separate the power constant and spatial depen-dence of the autocorrelation. We understand thatthe assumption of noise-free signals is not correct inthis case, even after we subtracted the bias termalong the diagonal. At the end of this section wediscuss the effect of noise on the off-diagonal terms.To obtain the log-likelihood function L, we take thenatural logarithm of both sides of Eq. ~25! to obtain

L 5 (k51

K F2N2

ln~2p! 212

lnuPCSu 212

dkT~PCS!21dkG ,

(27)

here the first term in the summation in Eq. ~27! canbe ignored as it does not affect the optimization; and,because PCS is symmetric, the following result is ob-tained:

(k51

K

dkT~PCS!21dk 5 (

k51

K

Tr@~PCS!21dkdkT#. (28)

hen we use these results, the log-likelihood functionecomes

L 5 2K2

lnu PCSu 212

TrF(k51

K

~PCS!21dkdkTG . (29)

Factoring the power constant term PC from the sum-mation in Eq. ~29! results in

L 5 2K2

lnu PCSu 212

PC21 TrS(

k51

K

S21dkdkTD , (30)

and rearranging the terms in Eq. ~30! yields

2LK

5 2N ln PC 2 lnuSu 2 PC21 TrSS21 1

K (k51

K

dkdkTD .

(31)

The term in Eq. ~31! given by

S 51K (

k51

K

dk dkT (32)

is observed to be the sample-based estimate of theACF computed from the data. The measurementnoise bias term mentioned in the discussion of Eq.~24! is subtracted from S to obtain

SB 5 S 2 a2Im. (33)


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1

Table 1. Results of L and C 2 as a Function of SNR for the von

tv

1

We now define the objective function C as

C 5 PC21 Tr~S21SB! 1 N ln PC 1 lnuSu, (34)

hich could, in principle, be minimized by iterationver the turbulence parameters. However, as weow describe, an analytic convenience can be used toimplify the optimization.By taking the partial derivative of Eq. ~34! with

espect to PC and setting it equal to zero, we can finda solution for PC in terms of the other quantities;furthermore this solution would be an optimal solu-tion because the partial derivatives equal zero at thecritical points. The required partial derivative isgiven by

]C]PC

5 2PC22 Tr~S21SB! 1

NPC

. (35)

Setting this result equal to zero and solving for PC, weobtain the following:

PC 51N

Tr~S21SB!. (36)

Putting this result into the cost function given by Eq.~34! and calling this new function C, we obtain

C 5 N ln@Tr~S21SB!# 1 N ln PC 1 lnuSu, (37)

hich is the final cost function to minimize. Notehat by using the optimal solution for PC in Eq. ~34!

to obtain C, we reduced the number of fitting param-eters by one. After the iteration is complete, we sub-stitute the resulting estimates of L0 and a into Eq.~36! to obtain the next estimate of PC. This processis then repeated until the estimates of PC, L0, and atabilize. A quasi-Newton optimization techniqueased on the Broyden–Fletcher–Goldfarb–Shanno al-orithm was used to minimize C.22

5. Results

We now present the results of a series of computerruns designed to show the limits of performance ofthis technique. Although our parameter estimationprocess allows the power law to differ from 11y6, weanticipate that the range of departures from a 5 11y6will be small. Hence we present results for inputvalues of a in the range of 9y6 # a # 13y6. Althoughwe test the algorithm over the range of SNR’s 10 #SNR # 100, we expect the instrument to operate inthe range of a SNR of approximately 50. All theresults presented below were computed from simu-lated data sets of 500 realizations of the noisy mea-surement vector. The results are presented in a setof tables that are discussed below.

A significant issue in this measurement and pro-cessing paradigm is the sensitivity of the estimationapproach to measurement noise. In Table 1 weshow the results of the parameter estimation for L0and Cn

2 as a function of various SNR’s for the vonKarman spectrum with L0 5 10 m and Cn

2 5 1 30217 m22y3. Random guesses of L0 and Cn

2 were


used to start the algorithm. Inspection of Table 1shows that the estimates of L0 and Cn

2 are reason-ably close to the known input values over a widerange of SNR’s. We are unable to estimate l0 reli-ably with this approach. The inability to estimatethe inner scale for the von Karman spectrum is due tothe differences between the theoretical-based ACFfor the different values of l0 that are too small to bedetected in the sample-based ACF. Figures 3 and 4show plots of the ACF versus the probe separation forthe von Karman spectrum for values of l0 of 0.0, 0.02,0.04, and 0.06 m. Figure 3 also shows a plot of thesample-based ACF for l0 5 0. In Fig. 3 the differ-ence between the four theoretical ACF’s is barelydiscernible so we include Fig. 4 to show the differ-ences among the four ACF’s. By inspection of Fig. 3and 4 it can be seen that the sample-based ACF hastoo much noise to detect l0 reliably. Because theshape of the analytic ACF for the four values of l0shown in Fig. 4 differs only in the small sensor sep-aration region, combined with the fact that the sensorarray does not sample well for small r ~except for the

0 n

Karman Spectruma

SNR L0 Cn2

100 10.24 0.99 3 10217

90 10.75 0.95 3 10217

80 10.51 0.96 3 10217

70 9.98 1.00 3 10217

60 10.42 0.97 3 10217

50 10.16 0.98 3 10217

40 10.52 0.96 3 10217

30 10.15 0.98 3 10217

20 10.40 0.98 3 10217

10 9.83 1.02 3 10217

aInput values: L0 5 10 m and Cn2 5 1 3 10217 m22y3.

Fig. 3. Theoretical ACF and sample-based ACF versus probe sep-aration for l0 5 0.0, 0.02, 0.04, and 0.06 m for 0 # r # 10 m. Notehat the theoretical curves overlap on this vertical scale for thearious values of l0.

m

2

v

w

0

c

nput

Input

diagonal elements!, it implies that the average valueof the diagonal elements of the sample-based ACFwill strongly influence the estimation of l0. The

Fig. 4. Autocorrelation function versus probe separation for l0 5.0, 0.02, 0.04, and 0.06 m for 0 # r # 3 m. The curve with the

greatest height at the zero separation is with l0 5 0; the threeurves below are for l0 5 0.02, 0.04, and 0.06 m, respectively.

Table 2. Mean and Variance of the Mean of the Diagonal Elementsof the ACF

Number ofIndex Screens

Number ofSamples m s

500 500 3.31 0.13931000 500 3.31 0.0984

10,000 500 3.31 0.0311

Table 3. Parameter Estimation Results for the Case of I

L0 ~m! L0 fit ~m! sL0P

10 10.33 0.6624 0.9590 330 31.78 2.069 1.026 350 52.96 3.946 1.027 370 74.02 5.583 1.015 390 96.38 7.764 1.000 3

110 117.4 10.20 0.9560 3130 136.8 12.56 0.9367 3150 161.3 16.20 0.9026 3

aStarting guesses for L0 and a were L0 5 100 and a 5 1.

Table 4. Parameter Estimation Results for the Case of

L0 ~m! L0 fit ~m! sL0PC

10 8.180 4.130 1.241 330 29.86 6.560 0.9914 350 54.21 13.70 0.9546 370 74.34 21.32 0.9750 390 102.2 30.57 0.9548 3

110 120.6 39.90 0.9591 3130 148.7 75.68 0.9483 3150 157.6 72.69 0.8282 3


minimum difference in the analytic diagonal terms ofthe ACFyPC for adjacent values of l0 is 0.0040, im-plying that the standard deviation of the averagevalue of the diagonal element of the sample-basedACFyPC will need to be less than 0.0040y3 to esti-

ate l0. Table 2 shows the results of calculating thevariance of the means of the diagonal elements of 500sample-based ACFyPC with L0 5 70 m and a 5 11y6with a SNR of 50 generated with a varying number ofindex screens per each of the 500 samples. As ex-pected, the standard deviation of the mean of a set ofN measurements decreases at a rate of 1y=N ~Ref.3! where N in this case is the number of index

screens. From Table 2 the standard deviation with10,000 index screens is 0.0311, and we need a stan-dard deviation of less than 0.00133 to be able to es-timate the inner scale; hence to reduce the standarddeviation enough to estimate the inner scale wewould have to use over five million index screens.

We now present the results of a series of paramet-ric studies to explore the ability of the algorithm toestimate L0, PC, and a over a range of possible values.The results are presented in Tables 3–7. In thesetables the SNR was fixed at 50 and PC was fixed atPC 5 1 3 10217. In each table the input value of ais set in the range of 10y6 # a # 13y6, whereas L0 isaried in steps of 20 m in the range of 10 m # L0 #

150 m. In each case the starting guesses for a andL0 were a 5 1 and L0 5 100 m. The tables list thesample-based means and standard deviations of theparameters for 100 different runs. Inspection of theresults shows that this measurement and processingapproach provides estimations of PC and a that are

ithin 1 standard deviation of the actual value for all

a 5 11y6 ~1.83!, Input PC 5 1 3 10217, and a SNR of 50a

sPCa sa

17 0.0631 3 10217 1.825 0.010817 0.0631 3 10217 1.827 0.007417 0.0412 3 10217 1.827 0.006217 0.0320 3 10217 1.828 0.006717 0.0304 3 10217 1.827 0.006617 0.0265 3 10217 1.828 0.006417 0.0234 3 10217 1.830 0.005617 0.0244 3 10217 1.828 0.0061


sPCa sa

17 0.554 3 10217 1.509 0.023117 0.1579 3 10217 1.498 0.010517 0.1092 3 10217 1.496 0.049217 0.1204 3 10217 1.498 0.006717 0.0964 3 10217 1.496 0.007017 0.1100 3 10217 1.498 0.007017 0.1022 3 10217 1.498 0.007017 0.0881 3 10217 1.498 0.0070

C

102

102

102

102

102

102

102

102

102

102

102

102

102

102

102

102


Table 5. Parameter Estimation Results for the Case of Input a 5 10y6 ~1.7!, Input P 5 1 3 10217, and a SNR of 50a

Table 6. Parameter Estimation Results for the Case of Inpu

Table 7. Parameter Estimation Results for the Case of Input


values of a and L0. The estimations of PC for thelower values of L0 are within 1 standard deviation ofthe actual values, but for the larger values of L0 theestimations are consistently low. Inspection of Ta-ble 4 shows that, although the mean value of theestimates of L0 are within 1 standard deviation of theactual values, the standard deviations in each caseare much larger than the standard deviations for theother values of a. Figure 5 shows a plot of the ACFfor values of L0 5 70, 150, and 230 m and a sample-based ACF with L0 5 150 m for all cases of a 5 9y6.An inspection of Fig. 5 shows that the noise on thesample-based ACF makes the estimation techniqueinsensitive to L0 for values of a 5 9y6.

6. Conclusion

We have presented a measurement and processingparadigm to estimate the parameters of homoge-neous, isotropic turbulence in the tropopause. Thistechnique uses a maximum-likelihood-estimator-based algorithm to estimate turbulence parameters

C

sPCa sa

0.0955 3 10217 1.663 0.01060.0545 3 10217 1.662 0.00730.0369 3 10217 1.663 0.00570.0349 3 10217 1.664 0.00590.0351 3 10217 1.663 0.00490.0406 3 10217 1.664 0.00600.0331 3 10217 1.664 0.00520.0323 3 10217 1.664 0.0055

t a 5 12y6 ~2!, Input PC 5 1 3 10217, and a SNR of 50a

sPCa sa

0.0651 3 10217 1.988 0.01310.0346 3 10217 1.990 0.00850.0276 3 10217 1.992 0.00780.0235 3 10217 1.995 0.00660.0250 3 10217 1.994 0.00700.0218 3 10217 1.994 0.00680.0228 3 10217 1.995 0.00690.0199 3 10217 1.995 0.0066


sPCa sa

17 0.0589 3 10217 2.155 0.012717 0.0327 3 10217 2.158 0.008317 0.0282 3 10217 2.159 0.009317 0.0236 3 10217 2.159 0.008317 0.0216 3 10217 2.159 0.008217 0.0218 3 10217 2.159 0.008717 0.0216 3 10217 2.159 0.009817 0.0200 3 10217 2.159 0.0089

Fig. 5. Theoretical ACF and sample-based ACF versus probe sep-aration for L0 5 70, 150, and 230 m for a 5 9y6. The upper curveis the ACF with L0 5 230 m, the middle curve is for L0 5 150 m,and the lowest curve is for L0 5 70 m.


10 9.943 1.112 0.9848 3 10217

30 31.19 2.887 0.9763 3 10217

50 52.56 4.865 0.9051 3 10217

70 72.97 7.979 0.8704 3 10217

90 95.71 10.82 0.9782 3 10217

110 118.1 14.87 0.9791 3 10217

130 140.3 17.17 0.9389 3 10217

150 159.2 20.89 0.9030 3 10217



10 10.32 0.5319 0.9529 3 10217

30 31.38 1.587 0.9812 3 10217

50 52.52 3.141 0.9817 3 10217

70 73.13 4.616 0.9823 3 10217

90 95.52 6.754 0.9615 3 10217

110 116.7 8.430 0.9063 3 10217

130 137.6 10.30 0.8944 3 10217

150 158.6 11.10 0.9039 3 10217



10 10.23 0.4186 0.9619 3 102

30 30.82 14.24 0.9810 3 102

50 52.29 2.573 0.9703 3 102

70 73.58 4.071 0.9141 3 102

90 94.65 5.143 0.9003 3 102

110 115.9 7.679 0.8745 3 102

130 136.8 10.12 0.8501 3 102

150 159.3 11.06 0.8496 3 102


“Investigation of turbulence spectrum anisotropy in the

2

2

2

2

from a set of 15 anemometer probes arranged on agrid with a maximum probe separation of 10 m. Wehave shown that the processing approach allows us toestimate the power law, proportionality constant,and outer scale with reasonable accuracy in the rangeof SNR’s and input turbulence parameters antici-pated. The primary advantages of this approach areto make the first balloonborne measurements thatare free of wake turbulence effects and to make mea-surements that require a weaker set of assumptionsabout the turbulence parameters than have beenused previously.

In principle, it should also be possible to determineif the turbulence is homogeneous and isotropic fromthe type of measurements proposed here. This de-termination of course requires that the probe loca-tions be registered in three-dimensional space as theballoon descends. Efforts are under way to imple-ment the required calculations, but are beyond thescope of our research presented here.

This research was supported by the U.S. Air ForceOffice of Scientific Research under contract F-49620-99-1-00-88.

References1. M. C. Roggemann and B. M. Welsh, Imaging Through Turbu-

lence. ~CRC Press, Boca Raton, Fla., 1996!.2. J. W. Goodman, Statistical Optics ~Wiley, New York, 1985!.3. A. N. Kolmogorov, “The local structure of turbulence in incom-

pressible viscous fluids for very large Reynolds’ numbers,” inTurbulence, Classic Papers on Statistical Theory, S. K. Fried-lander and L. Topper, eds. ~Wiley InterScience, New York,1961!, pp. 151–155.

4. V. I. Tatarskii, Wave Propagation in a Turbulent Medium ~Do-ver, New York, 1967!.

5. R. R. Beland, “Propagation through atmospheric optical tur-bulence,” in The Infrared and Electro-Optical Systems Hand-book ~SPIE, Bellingham, Wash., 1993!.

6. P. J. Gardner, M. C. Roggeman, B. M. Welsh, R. D. Bowersox,and T. E. Luke, “Statistical anisotropy in free turbulence formixing layers at high Reynolds numbers,” Appl. Opt. 35,4879–4889 ~1996!.

7. P. J. Gardner, M. C. Roggeman, B. M. Welsh, R. D. Bowersox,and T. E. Luke, “Comparison of measured and computedStrehl ratios for light propagated through a channel flow of aHe–N2 mixing layer at high Reynolds number,” Appl. Opt. 36,2559–2576 ~1997!.

8. L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavri-nova, V. P. Lukin, A. P. Rostov, B. V. Fortes, and A. P. Yankov,

ground atmospheric layer: preliminary results,” Atmos. Oce-anic Opt. 8, 993–996 ~1995!.

9. J. Vernin, “Atmospheric turbulence profiles,” in Wave Propa-gation in Random Media ~Scintillation!, V. I. Tatarski, A. Ishi-maru, and V. U. Zovorontny, eds., Vol. PM09 of SPIE PressMonographs ~SPIE Press, Bellingham, Wash., 1992!, pp. 248–260.

10. D. K. Lilly and P. F. Lester, “Waves and turbulence in thestratosphere,” J. Atmos. Sci. 31, 800–812 ~1974!.

11. M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, andC. H. Townes, “Atmospheric fluctuations: empirical struc-ture functions and projected performance of future instru-ments,” Astrophys. J. 392, 357–374 ~1992!.

12. C. E. Coulman, J. Vernin, Y. Coqueugniot, and J. L. Caccia,“Outer scale of turbulence appropriate to modeling refractive-index structure profiles,” Appl. Opt. 27, 155–160 ~1988!.

13. V. P. Lukin, “Investigation of some peculiarities in the struc-ture of large-scale atmospheric turbulence,” Atmos. OceanicOpt. 5, 834–840 ~1992!.

14. V. P. Lukin, E. V. Nosov, and B. V. Fortes, “The efficient outerscale of atmospheric turbulence,” Atmos. Oceanic Opt. 10,100–105 ~1997!.

15. V. V. Voitsekhovich and C. Cuevas, “Adaptive optics and theouter scale of turbulence,” J. Opt. Soc. Am. A 12, 2523–2531~1995!.

16. F. Dalaudier, C. Sidi, M. Crochet, and J. Vernin, “Direct evi-dence of sheets in the atmospheric temperature field,” J. At-mos. Sci. 51, 237–248 ~1994!.

17. L. J. Otten, D. T. Kyrazis, D. W. Tyler, and N. Miller, “Impli-cation of atmospheric models on adaptive optics designs,” inAdaptive Optics in Astronomy, M. A. Ealey and F. Merkle, eds.,Proc. SPIE 2201, 201–211 ~1994!.

18. M. C. Roggemann and L. J. Otten, “Design of an anemometerarray for measurement of cn

2 and turbulence spatial powerspectrum,” paper AIAA-99-3620, presented at the ThirteenthLighter-than-Air Systems Technology Conference, Norfolk,Va., 28 June–1 July 1999 ~American Institute of Aeronauticsand Astronautics, New York, 1999!.

19. I. S. G. and I. M. Ryzhik, Table of Integrals, Series, and Prod-ucts ~Academic, New York, 1980!.

0. A. Papoulis, Probability, Random Variables, and StochiasticProcesses ~McGraw-Hill, New York, 1991!.

1. M. C. Roggemann, B. M. Welsh, D. Montera, and T. A. Rhoa-darmer, “Method for simulating atmospheric phase effects formultiple time slices and anisoplanatic conditions,” Appl. Opt.34, 4037–4051 ~1995!.

2. M. A. Branch and A. Grace, MATLAB Optimization Toolbox~The Math Works, 24 Prime Park Way, Natick, Mass., 1996!.

3. H. D. Young, Statistical Treatment of Experimental Data~McGraw-Hill, New York, 1962!.


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