Abstract

We discuss a method of data reduction and analysis that has been developed for a novel experiment to detect anisotropic turbulence in the tropopause and to measure the spatial statistics of these flows. The experimental concept is to make measurements of temperature at 15 points on a hexagonal grid for altitudes from 12,000 to 18,000 m while suspended from a balloon performing a controlled descent. From the temperature data, we estimate the index of refraction and study the spatial statistics of the turbulence-induced index of refraction fluctuations. We present and evaluate the performance of a processing approach to estimate the parameters of an anisotropic model for the spatial power spectrum of the turbulence-induced index of refraction fluctuations. A Gaussian correlation model and a least-squares optimization routine are used to estimate the parameters of the model from the measurements. In addition, we implemented a quick-look algorithm to have a computationally nonintensive way of viewing the autocorrelation function of the index fluctuations. The autocorrelation of the index of refraction fluctuations is binned and interpolated onto a uniform grid from the sparse points that exist in our experiment. This allows the autocorrelation to be viewed with a three-dimensional plot to determine whether anisotropy exists in a specific data slab. Simulation results presented here show that, in the presence of the anticipated levels of measurement noise, the least-squares estimation technique allows turbulence parameters to be estimated with low rms error.

© 2002 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  14. W. W. Brown, M. C. Roggemann, T. J. Schulz, T. C. Havens, J. T. Beyer, L. J. Otten, “Measurement and data processing approach for estimating the spatial statistics of turbulence-induced index of refraction fluctuations in the upper atmosphere,” Appl. Opt. 40, 1863–1871 (2001).
    [CrossRef]
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2001 (1)

1998 (1)

A. Mahalov, B. Nicolaenko, Y. Zhou, “Energy spectra of strongly stratified and rotating turbulence,” Phys. Rev. E 57, 6187–6190 (1998).
[CrossRef]

1997 (1)

1996 (1)

1995 (1)

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer: preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).

1994 (1)

F. Dalaudier, C. Sidi, M. Crochet, J. Vernin, “Direct evidence of sheets in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (1994).
[CrossRef]

1990 (1)

1982 (1)

1966 (1)

Antoshkin, L. V.

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer: preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).

Barakat, R.

Beland, R. R.

R. R. Beland, “Propagation through atmospheric optical turbulence,” in The Infrared and Electro-Optical Systems Handbook, Vol. PM 10 of the SPIE Press Monographs, J. S. Accetla, D. L. Shumaker, eds. (SPIE, Bellingham, Wash., 1993), Vol. 1, Chap. 2.

Beletic, J. W.

Beyer, J. T.

Botygina, N. N.

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer: preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).

Bowersox, R. D.

Branch, M. A.

M. A. Branch, A. Grace, MATLAB Optimization Toolbox (The Math Works, Natick, Mass., 1996).

Brown, W. W.

Cohn, D. L.

J. A. Melsa, D. L. Cohn, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

Crochet, M.

F. Dalaudier, C. Sidi, M. Crochet, J. Vernin, “Direct evidence of sheets in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (1994).
[CrossRef]

Dalaudier, F.

F. Dalaudier, C. Sidi, M. Crochet, J. Vernin, “Direct evidence of sheets in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (1994).
[CrossRef]

Emaleev, O. N.

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer: preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).

Fortes, B. V.

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer: preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).

Fried, D. L.

Gardner, P. J.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Grace, A.

M. A. Branch, A. Grace, MATLAB Optimization Toolbox (The Math Works, Natick, Mass., 1996).

Havens, T. C.

Kolmogorov, A. N.

A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluids for very large Reynolds’ numbers,” in Turbulence, Classic Papers on Statistical Theory, S. K. Friedlander, L. Topper, eds. (Wiley-Interscience, New York, 1961), pp. 151–155.

Kyrazis, D. T.

L. J. Otten, D. T. Kyrazis, D. W. Tyler, N. Miller, “Implications of atmospheric models on adaptive optics designs,” in Symposium on Astronomical Telescopes and Instrumentation for the 21st Century, M. A. Ealey, F. Merkle, eds. (SPIE, Bellingham, Wash, 1994), Vol. 2201, pp. 201–211.

Lavrinova, L. N.

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer: preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).

Luke, T. E.

Lukin, V. P.

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer: preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).

Mahalov, A.

A. Mahalov, B. Nicolaenko, Y. Zhou, “Energy spectra of strongly stratified and rotating turbulence,” Phys. Rev. E 57, 6187–6190 (1998).
[CrossRef]

Melsa, J. A.

J. A. Melsa, D. L. Cohn, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

Miller, N.

L. J. Otten, D. T. Kyrazis, D. W. Tyler, N. Miller, “Implications of atmospheric models on adaptive optics designs,” in Symposium on Astronomical Telescopes and Instrumentation for the 21st Century, M. A. Ealey, F. Merkle, eds. (SPIE, Bellingham, Wash, 1994), Vol. 2201, pp. 201–211.

Nicolaenko, B.

A. Mahalov, B. Nicolaenko, Y. Zhou, “Energy spectra of strongly stratified and rotating turbulence,” Phys. Rev. E 57, 6187–6190 (1998).
[CrossRef]

Otten, L. J.

W. W. Brown, M. C. Roggemann, T. J. Schulz, T. C. Havens, J. T. Beyer, L. J. Otten, “Measurement and data processing approach for estimating the spatial statistics of turbulence-induced index of refraction fluctuations in the upper atmosphere,” Appl. Opt. 40, 1863–1871 (2001).
[CrossRef]

L. J. Otten, D. T. Kyrazis, D. W. Tyler, N. Miller, “Implications of atmospheric models on adaptive optics designs,” in Symposium on Astronomical Telescopes and Instrumentation for the 21st Century, M. A. Ealey, F. Merkle, eds. (SPIE, Bellingham, Wash, 1994), Vol. 2201, pp. 201–211.

Roggeman, M. C.

Roggemann, M. C.

Rostov, A. P.

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer: preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).

Schulz, T. J.

Sidi, C.

F. Dalaudier, C. Sidi, M. Crochet, J. Vernin, “Direct evidence of sheets in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (1994).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (Dover, New York, 1967).

Tyler, D. W.

L. J. Otten, D. T. Kyrazis, D. W. Tyler, N. Miller, “Implications of atmospheric models on adaptive optics designs,” in Symposium on Astronomical Telescopes and Instrumentation for the 21st Century, M. A. Ealey, F. Merkle, eds. (SPIE, Bellingham, Wash, 1994), Vol. 2201, pp. 201–211.

Vernin, J.

F. Dalaudier, C. Sidi, M. Crochet, J. Vernin, “Direct evidence of sheets in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (1994).
[CrossRef]

Welsh, B. M.

Yankov, A. P.

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer: preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).

Zhou, Y.

A. Mahalov, B. Nicolaenko, Y. Zhou, “Energy spectra of strongly stratified and rotating turbulence,” Phys. Rev. E 57, 6187–6190 (1998).
[CrossRef]

Appl. Opt. (3)

Atmos. Oceanic Opt. (1)

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinova, V. P. Lukin, A. P. Rostov, B. V. Fortes, A. P. Yankov, “Investigation of turbulence spectrum anisotropy in the ground atmospheric layer: preliminary results,” Atmos. Oceanic Opt. 8, 993–996 (1995).

J. Atmos. Sci. (1)

F. Dalaudier, C. Sidi, M. Crochet, J. Vernin, “Direct evidence of sheets in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (1994).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Phys. Rev. E (1)

A. Mahalov, B. Nicolaenko, Y. Zhou, “Energy spectra of strongly stratified and rotating turbulence,” Phys. Rev. E 57, 6187–6190 (1998).
[CrossRef]

Other (9)

R. R. Beland, “Propagation through atmospheric optical turbulence,” in The Infrared and Electro-Optical Systems Handbook, Vol. PM 10 of the SPIE Press Monographs, J. S. Accetla, D. L. Shumaker, eds. (SPIE, Bellingham, Wash., 1993), Vol. 1, Chap. 2.

J. A. Melsa, D. L. Cohn, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

M. A. Branch, A. Grace, MATLAB Optimization Toolbox (The Math Works, Natick, Mass., 1996).

MATLAB Reference Guide (The Math Works, Natick, Mass., 1996).

L. J. Otten, D. T. Kyrazis, D. W. Tyler, N. Miller, “Implications of atmospheric models on adaptive optics designs,” in Symposium on Astronomical Telescopes and Instrumentation for the 21st Century, M. A. Ealey, F. Merkle, eds. (SPIE, Bellingham, Wash, 1994), Vol. 2201, pp. 201–211.

M. C. Roggemann, B. M. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluids for very large Reynolds’ numbers,” in Turbulence, Classic Papers on Statistical Theory, S. K. Friedlander, L. Topper, eds. (Wiley-Interscience, New York, 1961), pp. 151–155.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (Dover, New York, 1967).

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Figures (6)

Fig. 1
Fig. 1

Interpolated ACF for no noise (left column) and a SNR of 50 (right column); number of bins: (a) 1, (b) 2, (c) 3, and (d) 4. The parameter values are a x = a y = 1.0 and P C = 1.0 × 10-20.

Fig. 2
Fig. 2

Exploded view of interpolated noisy ACF of the isotropic index fluctuations in Fig. 1(b).

Fig. 3
Fig. 3

Interpolated ACF for no noise (left column) and a SNR of 50 (right column); a x :a y = 1.1:1, P C = 1.0 × 10-20, and number of bins is 2.

Fig. 4
Fig. 4

Interpolated ACF for no noise (left column) and a SNR of 50 (right column); a x :a y = 1.2:1, P C = 1.0 × 10-20, and number of bins is 2.

Fig. 5
Fig. 5

Interpolated ACF for no noise (left column) and a SNR of 50 (right column); a x :a y = 1.3:1, P C = 1.0 × 10-20, and number of bins is 2.

Fig. 6
Fig. 6

Interpolated ACF for no noise (left column) and a SNR of 50 (right column); a x :a y = 1.4:1, P C = 1.0 × 10-20, and number of bins is 2.

Tables (10)

Tables Icon

Table 1 Results for Estimates of ax, ay, θ, and PC as a Function of SNRa

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Table 2 Parameter Estimation Results for the Case of Isotropic Input ax = ay and PC/a = 50

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Table 3 Parameter Estimation Results for the Case of Input ax = 1.1, 1.2, 1.3, and 1.4; ay = 1.0, θ = 0; and PC /a = 50

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Table 4 Parameter Estimation Results for the Case of Input a x = 1.1, 1.2, 1.3, and 1.4; a y = 1.0; θ = π/6 (0.5236); and PC/a = 50

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Table 5 Parameter Estimation Results for the Case of Input a x = 1.1, 1.2, 1.3, and 1.4; a y = 1.0; θ = π/4 (0.7854); and PC/a = 50

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Table 6 Parameter Estimation Results for the Case of Input a x = 1.1, 1.2, 1.3, and 1.4; a y = 1.0; θ = π/3 (1.0472); and PC/a = 50

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Table 7 Parameter Estimation Results for the Case of Input a x = 1.1, 1.2, 1.3, and 1.4; a y = 1.0; θ = 3π/4 (2.3562); and PC/a = 50

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Table 8 Parameter Estimation Results for the Case of Input PC/a = 50, θ = 0, and Varying Values of a x and a y

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Table 9 Parameter Estimation Results with Different Numbers of Bins Useda

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Table 10 Parameter Estimation Results for the Modified Von Karman PSD Modela

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

n=1+77.61+7.52×10-3λ-2PT×10-6,
Δn=-77.61+7.52×10-3λ-2PT2×10-6ΔT.
Bnr=---ϕnkexp-jk · rd3k,
Fnkx, ky; z=-ϕnkx, ky, kzdkz.
Bnp; z=-- Fnk; zexp-jk · rd2k,
Bnr=PC · exp-Δxax2-Δyay2-Δzaz2,
Bnp; z=PC · exp-Δxax2-Δyay2,
ΔxΔy=cosθsinθ-sinθcosθΔxΔy.
P˜C=k=1N Bn, kkp; zN,
CLSB˜n, =k=1Nl=1NB˜n,kl-kl2,
Φnk=PCk2+2πL0211/6,

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