Abstract

We present results from an experiment to estimate the parameters of homogeneous, isotropic optical turbulence in the upper atmosphere. The balloon-borne experiment made high-resolution temperature measurements at seven points on a hexagonal grid for altitudes from 12,000 to 18,000 m. From the temperature data, we obtained index of refraction fluctuations that can be used to compute a sample-based estimate for a parameterized description of the spatial autocorrelation of the turbulence. The three parameters of interest were a proportionality constant P C, the power-law parameter α, and the outer scale L 0. The results obtained for P C are within the expected range and agree well with independent measurements made from a standard rising thermosonde measurement made approximately simultaneously with the data collection. Values for α were in the range 1.52 ≤ α ≤ 1.73 were observed, which are significantly less than the power law used in the Kolmogorov and von Karman models, α = 1.833. Values observed for L 0 were in the range 5 ≤ L 0 ≤ 190 m. Evidence that α may be consistently less than that used in the Kolmogorov and von Karman models likely has the most significant implications for systems that must work in or through the tropopause.

© 2003 Optical Society of America

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References

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  1. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  2. M. C. Roggemann, B. M. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996).
  3. A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluids for very large Reynolds’ numbers,” in Turbulence: Classical Papers on Statistical Theory, S. K. Friedlander, L. Topper, eds. (Wiley-Interscience, New York, 1961), pp. 151–155.
  4. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (Dover, New York, 1967).
  5. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [Crossref]
  6. D. L. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. 72, 52–61 (1982).
    [Crossref]
  7. R. R. Beland, “Propagation through atmospheric optical turbulence,” in The Infrared and Electro-Optical Systems Handbook, Vol. PM10 of the SPIE Press Monographs, J. S. Accetta, D. L. Shumaker, eds. (SPIE, Bellingham, Wash., 1993).
  8. P. J. Gardner, M. C. Roggemann, B. M. Welsh, R. D. Bowersox, T. E. Luke, “Statistical anisotropy in free turbulence for mixing layers at high Reynolds numbers,” Appl. Opt. 35, 4879–4889 (1996).
    [Crossref] [PubMed]
  9. P. J. Gardner, M. C. Roggemann, B. M. Welsh, R. D. Bowersox, T. E. Luke, “Comparison of measured and computed Strehl ratios for light propagated through a channel flow of a He/N2 mixing layer at high Reynolds numbers,” Appl. Opt. 36, 2559–2576 (1997).
    [Crossref] [PubMed]
  10. L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinove, 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).
  11. 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]
  12. L. J. Otten, D. T. Kyrazis, D. W. Tyler, N. Miller, “Implications of atmospheric models on adaptive optics designs,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 201–211 (1994).
    [Crossref]
  13. 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]
  14. G. Y. Jumper, E. A. Murphy, “Effect of balloon wake on thermosonde results,” paper AAIA-2001-2796, presented at the Thirty-Second Plasmadynamics and Lasers Conference, Anaheim, Calif., 11–14 June 2001 (American Institute of Aeronautics and Astronautics, Reston, Va., 2001).
  15. I. S. Gradshtegn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).
  16. G. Y. Jumper, R. R. Beland, P. Tracy, “Investigating sources of error in balloon-borne optical turbulence measurements,” paper AIAA-99-3618, presented at the Thirtieth Plasmadynamics and Lasers Conference, Norfolk, Va., 28 June–1 July 1999 (American Institute of Aeronautics and Astronautics, Reston, Va., 1999).
  17. M. C. Roggemann, L. J. Otten, “Design of an anemometer array for measurement of Cn2 and turbulence spatial power spectrum,” paper AIAA-99-3620, presented at the Thirteenth Lighter-Than-Air Systems Technology Conference, Norfolk, Va., 28–30 June 1999 (American Institute of Aeronautics and Astronautics, Reston, Va., 1999).
  18. L. J. Otten, A. Jones, D. Black, J. Lane, R. Hugo, M. C. Roggemann, J. Beyer, “Precision tropopause turbulence measurements,” in Propagation and Imaging through the Atmosphere, M. C. Roggemann, ed., Proc. SPIE4125, 33–40 (2000).
    [Crossref]
  19. S. M. Kay, Fundamental of Statistical Signal Processing: Estimation Theory (Prentice-Hall, Upper Saddle River, N.J., 1993).
  20. A. Papoulis, Probability, Random Variables, and Stochiastic Processes (McGraw-Hill, New York, 1991).
  21. M. A. Branch, A. Grace, MATLAB Optimization Toolbox User’s Guide, Math Works, 24 Prime Park Way, Natick, Mass., 1996.
  22. H. H. Bruun, Hot Wire Anemometry: Principles and Signal Analysis (Oxford U. Press, Oxford, UK, 1955), pp. 55–57.
  23. P. Parantheon, C. Petit, J. C. LeCordier, “The effects of the thermal prong-wire interaction in the response of a cold wire in gaseous flows (air, argon, helium),” J. Fluid Mech. 124, 457–473 (1982).
    [Crossref]

2001 (1)

1997 (1)

1996 (1)

1995 (1)

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinove, 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]

1982 (2)

P. Parantheon, C. Petit, J. C. LeCordier, “The effects of the thermal prong-wire interaction in the response of a cold wire in gaseous flows (air, argon, helium),” J. Fluid Mech. 124, 457–473 (1982).
[Crossref]

D. L. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. 72, 52–61 (1982).
[Crossref]

1966 (1)

Antoshkin, L. V.

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinove, 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).

Beland, R. R.

R. R. Beland, “Propagation through atmospheric optical turbulence,” in The Infrared and Electro-Optical Systems Handbook, Vol. PM10 of the SPIE Press Monographs, J. S. Accetta, D. L. Shumaker, eds. (SPIE, Bellingham, Wash., 1993).

G. Y. Jumper, R. R. Beland, P. Tracy, “Investigating sources of error in balloon-borne optical turbulence measurements,” paper AIAA-99-3618, presented at the Thirtieth Plasmadynamics and Lasers Conference, Norfolk, Va., 28 June–1 July 1999 (American Institute of Aeronautics and Astronautics, Reston, Va., 1999).

Beyer, J.

L. J. Otten, A. Jones, D. Black, J. Lane, R. Hugo, M. C. Roggemann, J. Beyer, “Precision tropopause turbulence measurements,” in Propagation and Imaging through the Atmosphere, M. C. Roggemann, ed., Proc. SPIE4125, 33–40 (2000).
[Crossref]

Beyer, J. T.

Black, D.

L. J. Otten, A. Jones, D. Black, J. Lane, R. Hugo, M. C. Roggemann, J. Beyer, “Precision tropopause turbulence measurements,” in Propagation and Imaging through the Atmosphere, M. C. Roggemann, ed., Proc. SPIE4125, 33–40 (2000).
[Crossref]

Botygina, N. N.

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinove, 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 User’s Guide, Math Works, 24 Prime Park Way, Natick, Mass., 1996.

Brown, W. W.

Bruun, H. H.

H. H. Bruun, Hot Wire Anemometry: Principles and Signal Analysis (Oxford U. Press, Oxford, UK, 1955), pp. 55–57.

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. Lavrinove, 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. Lavrinove, 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 User’s Guide, Math Works, 24 Prime Park Way, Natick, Mass., 1996.

Gradshtegn, I. S.

I. S. Gradshtegn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Havens, T. C.

Hugo, R.

L. J. Otten, A. Jones, D. Black, J. Lane, R. Hugo, M. C. Roggemann, J. Beyer, “Precision tropopause turbulence measurements,” in Propagation and Imaging through the Atmosphere, M. C. Roggemann, ed., Proc. SPIE4125, 33–40 (2000).
[Crossref]

Jones, A.

L. J. Otten, A. Jones, D. Black, J. Lane, R. Hugo, M. C. Roggemann, J. Beyer, “Precision tropopause turbulence measurements,” in Propagation and Imaging through the Atmosphere, M. C. Roggemann, ed., Proc. SPIE4125, 33–40 (2000).
[Crossref]

Jumper, G. Y.

G. Y. Jumper, R. R. Beland, P. Tracy, “Investigating sources of error in balloon-borne optical turbulence measurements,” paper AIAA-99-3618, presented at the Thirtieth Plasmadynamics and Lasers Conference, Norfolk, Va., 28 June–1 July 1999 (American Institute of Aeronautics and Astronautics, Reston, Va., 1999).

G. Y. Jumper, E. A. Murphy, “Effect of balloon wake on thermosonde results,” paper AAIA-2001-2796, presented at the Thirty-Second Plasmadynamics and Lasers Conference, Anaheim, Calif., 11–14 June 2001 (American Institute of Aeronautics and Astronautics, Reston, Va., 2001).

Kay, S. M.

S. M. Kay, Fundamental of Statistical Signal Processing: Estimation Theory (Prentice-Hall, Upper Saddle River, N.J., 1993).

Kolmogorov, A. N.

A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluids for very large Reynolds’ numbers,” in Turbulence: Classical 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 Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 201–211 (1994).
[Crossref]

Lane, J.

L. J. Otten, A. Jones, D. Black, J. Lane, R. Hugo, M. C. Roggemann, J. Beyer, “Precision tropopause turbulence measurements,” in Propagation and Imaging through the Atmosphere, M. C. Roggemann, ed., Proc. SPIE4125, 33–40 (2000).
[Crossref]

Lavrinove, L. N.

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinove, 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).

LeCordier, J. C.

P. Parantheon, C. Petit, J. C. LeCordier, “The effects of the thermal prong-wire interaction in the response of a cold wire in gaseous flows (air, argon, helium),” J. Fluid Mech. 124, 457–473 (1982).
[Crossref]

Luke, T. E.

Lukin, V. P.

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinove, 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).

Miller, N.

L. J. Otten, D. T. Kyrazis, D. W. Tyler, N. Miller, “Implications of atmospheric models on adaptive optics designs,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 201–211 (1994).
[Crossref]

Murphy, E. A.

G. Y. Jumper, E. A. Murphy, “Effect of balloon wake on thermosonde results,” paper AAIA-2001-2796, presented at the Thirty-Second Plasmadynamics and Lasers Conference, Anaheim, Calif., 11–14 June 2001 (American Institute of Aeronautics and Astronautics, Reston, Va., 2001).

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]

M. C. Roggemann, L. J. Otten, “Design of an anemometer array for measurement of Cn2 and turbulence spatial power spectrum,” paper AIAA-99-3620, presented at the Thirteenth Lighter-Than-Air Systems Technology Conference, Norfolk, Va., 28–30 June 1999 (American Institute of Aeronautics and Astronautics, Reston, Va., 1999).

L. J. Otten, A. Jones, D. Black, J. Lane, R. Hugo, M. C. Roggemann, J. Beyer, “Precision tropopause turbulence measurements,” in Propagation and Imaging through the Atmosphere, M. C. Roggemann, ed., Proc. SPIE4125, 33–40 (2000).
[Crossref]

L. J. Otten, D. T. Kyrazis, D. W. Tyler, N. Miller, “Implications of atmospheric models on adaptive optics designs,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 201–211 (1994).
[Crossref]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochiastic Processes (McGraw-Hill, New York, 1991).

Parantheon, P.

P. Parantheon, C. Petit, J. C. LeCordier, “The effects of the thermal prong-wire interaction in the response of a cold wire in gaseous flows (air, argon, helium),” J. Fluid Mech. 124, 457–473 (1982).
[Crossref]

Petit, C.

P. Parantheon, C. Petit, J. C. LeCordier, “The effects of the thermal prong-wire interaction in the response of a cold wire in gaseous flows (air, argon, helium),” J. Fluid Mech. 124, 457–473 (1982).
[Crossref]

Roggemann, M. C.

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]

P. J. Gardner, M. C. Roggemann, B. M. Welsh, R. D. Bowersox, T. E. Luke, “Comparison of measured and computed Strehl ratios for light propagated through a channel flow of a He/N2 mixing layer at high Reynolds numbers,” Appl. Opt. 36, 2559–2576 (1997).
[Crossref] [PubMed]

P. J. Gardner, M. C. Roggemann, B. M. Welsh, R. D. Bowersox, T. E. Luke, “Statistical anisotropy in free turbulence for mixing layers at high Reynolds numbers,” Appl. Opt. 35, 4879–4889 (1996).
[Crossref] [PubMed]

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

L. J. Otten, A. Jones, D. Black, J. Lane, R. Hugo, M. C. Roggemann, J. Beyer, “Precision tropopause turbulence measurements,” in Propagation and Imaging through the Atmosphere, M. C. Roggemann, ed., Proc. SPIE4125, 33–40 (2000).
[Crossref]

M. C. Roggemann, L. J. Otten, “Design of an anemometer array for measurement of Cn2 and turbulence spatial power spectrum,” paper AIAA-99-3620, presented at the Thirteenth Lighter-Than-Air Systems Technology Conference, Norfolk, Va., 28–30 June 1999 (American Institute of Aeronautics and Astronautics, Reston, Va., 1999).

Rostov, A. P.

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinove, 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).

Ryzhik, I. M.

I. S. Gradshtegn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

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).

Tracy, P.

G. Y. Jumper, R. R. Beland, P. Tracy, “Investigating sources of error in balloon-borne optical turbulence measurements,” paper AIAA-99-3618, presented at the Thirtieth Plasmadynamics and Lasers Conference, Norfolk, Va., 28 June–1 July 1999 (American Institute of Aeronautics and Astronautics, Reston, Va., 1999).

Tyler, D. W.

L. J. Otten, D. T. Kyrazis, D. W. Tyler, N. Miller, “Implications of atmospheric models on adaptive optics designs,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 201–211 (1994).
[Crossref]

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. Lavrinove, 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).

Appl. Opt. (3)

Atmos. Oceanic Opt. (1)

L. V. Antoshkin, N. N. Botygina, O. N. Emaleev, L. N. Lavrinove, 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. Fluid Mech. (1)

P. Parantheon, C. Petit, J. C. LeCordier, “The effects of the thermal prong-wire interaction in the response of a cold wire in gaseous flows (air, argon, helium),” J. Fluid Mech. 124, 457–473 (1982).
[Crossref]

J. Opt. Soc. Am. (2)

Other (15)

R. R. Beland, “Propagation through atmospheric optical turbulence,” in The Infrared and Electro-Optical Systems Handbook, Vol. PM10 of the SPIE Press Monographs, J. S. Accetta, D. L. Shumaker, eds. (SPIE, Bellingham, Wash., 1993).

L. J. Otten, D. T. Kyrazis, D. W. Tyler, N. Miller, “Implications of atmospheric models on adaptive optics designs,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 201–211 (1994).
[Crossref]

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

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

A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluids for very large Reynolds’ numbers,” in Turbulence: Classical 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).

G. Y. Jumper, E. A. Murphy, “Effect of balloon wake on thermosonde results,” paper AAIA-2001-2796, presented at the Thirty-Second Plasmadynamics and Lasers Conference, Anaheim, Calif., 11–14 June 2001 (American Institute of Aeronautics and Astronautics, Reston, Va., 2001).

I. S. Gradshtegn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

G. Y. Jumper, R. R. Beland, P. Tracy, “Investigating sources of error in balloon-borne optical turbulence measurements,” paper AIAA-99-3618, presented at the Thirtieth Plasmadynamics and Lasers Conference, Norfolk, Va., 28 June–1 July 1999 (American Institute of Aeronautics and Astronautics, Reston, Va., 1999).

M. C. Roggemann, L. J. Otten, “Design of an anemometer array for measurement of Cn2 and turbulence spatial power spectrum,” paper AIAA-99-3620, presented at the Thirteenth Lighter-Than-Air Systems Technology Conference, Norfolk, Va., 28–30 June 1999 (American Institute of Aeronautics and Astronautics, Reston, Va., 1999).

L. J. Otten, A. Jones, D. Black, J. Lane, R. Hugo, M. C. Roggemann, J. Beyer, “Precision tropopause turbulence measurements,” in Propagation and Imaging through the Atmosphere, M. C. Roggemann, ed., Proc. SPIE4125, 33–40 (2000).
[Crossref]

S. M. Kay, Fundamental of Statistical Signal Processing: Estimation Theory (Prentice-Hall, Upper Saddle River, N.J., 1993).

A. Papoulis, Probability, Random Variables, and Stochiastic Processes (McGraw-Hill, New York, 1991).

M. A. Branch, A. Grace, MATLAB Optimization Toolbox User’s Guide, Math Works, 24 Prime Park Way, Natick, Mass., 1996.

H. H. Bruun, Hot Wire Anemometry: Principles and Signal Analysis (Oxford U. Press, Oxford, UK, 1955), pp. 55–57.

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Figures (11)

Fig. 1
Fig. 1

Anemometer placement.

Fig. 2
Fig. 2

Sample location in the ACF space for 15 probes.

Fig. 3
Fig. 3

Locations of the seven probes.

Fig. 4
Fig. 4

Sample location in the ACF space for seven probes.

Fig. 5
Fig. 5

Vertical flight profile. IRIG, Inter-Range Instrumentation Group.

Fig. 6
Fig. 6

Vertical rate profile. IRIG, Inter-Range Instrumentation Group.

Fig. 7
Fig. 7

Two-parameter estimation results versus altitude: (a) P C , (b) α, (c) L 0.

Fig. 8
Fig. 8

Three-parameter estimation results versus altitude: (a) P C , (b) α, (c) L 0.

Fig. 9
Fig. 9

Two-parameter estimation results versus altitude at a fixed α = 11/6: (a) P C (b) L 0.

Fig. 10
Fig. 10

Three-parameter estimation results versus altitude at a fixed α = 11/6: (a) P C (b) L 0.

Fig. 11
Fig. 11

2 versus altitude for the four estimation routines.

Tables (5)

Tables Icon

Table 1 Three-Parameter Estimation Results for the Case of Input α = 11/6 (1.83), Input PC = 1 × 10 -18 , and a SNR of 2 a

Tables Icon

Table 2 Three-Parameter Estimation Results for the Case of Input α = 11/6 (1.83), Input PC = 1 × 10 -18 , and a SNR of 4 a

Tables Icon

Table 3 Two-Parameter Estimation Results for the Case of Input α = 11/6 (1.83), Input PC = 1 × 10 -18 , and a SNR of 2 a

Tables Icon

Table 4 dc Calibration Equations

Equations (46)

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

n=1+77.61+7.52×10-3λ-2PT×10-6,
n1+77.61+7.52×10-3λ-2PT¯×10-6-77.61+7.52×10-3λ-2PT¯2×10-6ΔT,
Δn=-77.61+7.52×10-3λ-2PT¯2×10-6ΔT.
Bnr=- Φnkexp-jk · rd3k,
Bnr= 4πr0 Φnkk sinkrdk,
Φnk=0.033Cn2k-11/3,
Φnk= 0.033Cn2k2+k0211/6exp- k2km2,
Φnk= PCk2+2πL02α,
Bnr= 4πr0PCk2+k02α k sin krdk,
Bnr=PC4πr1Γα×21/2-αk01-απk0rα-1/2K3/2-αk0r,
Bnr=PCexp-k0rπr4k0.
fdk=k=1K2π-N/2|PCΣ|-1/2 exp-dkTPCΣ-1dk,
Bn=PCΣ,
L=k=1K-N2ln2π- 12ln|PCΣ|- 12 dkTPCΣ-1dk,
k=1K dkTPCΣ-1dk=k=1KTrPCΣ-1dkdkT.
L=- K2ln|PCΣ|- 12Trk=1KPCΣ-1dkdkT.
L=- K2ln|PCΣ|- 12 PC-1 Trk=1K Σ-1dkdkT,
2LK=-N ln PC-ln|Σ|-PC-1 TrΣ-11Kk=1K dkdkT.
S= 1Kk=1K dkdkT
SB=S-a2Im,
C=PC-1TrΣ-1SB+N ln PC+ln|Σ|,
CPC=-PC-2 TrΣ-1SB+ NPC.
P˜C= 1NTrΣ-1SB.
C˜=N lnTrΣ-1SB+N ln P˜C+ln|Σ|,
fdk=i=1K2π-N/2 |PCΣ+a2Im|-1/2×exp-dkTPCΣ+a2Im-1dk.
L=k=1K-N2ln2π- 12ln |PCΣ+a2Im|- 12×dkTPCΣ+a2Im-1dk.
k=1K dkTPCΣ+a2Im-1dk=k=1KTrPCΣ+a2Im-1dkdkT.
L=- K2ln |PCΣ+a2Im|- 12Trk=1KPCΣ+a2Im-1dkdkT.
L=- K2ln |PCΣ+a2Im|- 12 PC-1Trk=1KΣ+PCa2Im-1dkdkT,
2LK=-ln |PCΣ+a2Im|-PC-1TrΣ+a2Im-11Kk=1K dkdkT.
Ĉ=PC-1 TrΣ+PCa2Im-1SB+ln|PCΣ+a2Im|,
mn=tn+γn,
mn-mn-1=tn-tn-1+γn-γn-1 γn-γn-1,
Emn-mn-12=Eγn-γn-12,
Eγn-γn-12=2a2,
2=i=1Nj=1NSij-Ŝij2,
ΔT= Cnts-offsetslope,
ρ= P287.053×T,
ν= 1.458×10-6×T3/2T+110.4ρ.
k= 4184×0.0000006325×T3/2T+245.4×10-12/T.
Re= Udwν,
Pr= 1006×νρk.
Nu=0.42×Pr1/5+0.57×Pr1/3Re1/2,
h= Nukdw.
Te= Tpr1- 12lwkdwh,
SNR= rmsΔn-aa,

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