Abstract

We present a method of data reduction and analysis that has been developed for a novel experiment to measure the spatial statistics of atmospheric turbulence in the tropopause. We took 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 isotropic models for the spatial power spectrum of the turbulence. In addition to examining the parameters of the von Kármán spectrum, we have allowed the so-called power law to be a parameter in the estimation algorithm. A maximum-likelihood-based approach is used to estimate the turbulence parameters from the measurements. Simulation results presented here show that, in the presence of the anticipated levels of measurement noise, this approach allows turbulence parameters to be estimated with good accuracy, with the exception of the inner scale.

© 2001 Optical Society of America

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References

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  1. M. C. Roggemann, B. M. Welsh, Imaging Through Turbulence. (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 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.
  4. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (Dover, New York, 1967).
  5. R. R. Beland, “Propagation through atmospheric optical turbulence,” in The Infrared and Electro-Optical Systems Handbook (SPIE, Bellingham, Wash., 1993).
  6. P. J. Gardner, M. C. Roggeman, 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]
  7. P. J. Gardner, M. C. Roggeman, 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 number,” Appl. Opt. 36, 2559–2576 (1997).
    [CrossRef] [PubMed]
  8. 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).
  9. J. Vernin, “Atmospheric turbulence profiles,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarski, A. Ishimaru, V. U. Zovorontny, eds., Vol. PM09 of SPIE Press Monographs (SPIE Press, Bellingham, Wash., 1992), pp. 248–260.
  10. D. K. Lilly, P. F. Lester, “Waves and turbulence in the stratosphere,” J. Atmos. Sci. 31, 800–812 (1974).
    [CrossRef]
  11. M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
    [CrossRef]
  12. C. E. Coulman, J. Vernin, Y. Coqueugniot, J. L. Caccia, “Outer scale of turbulence appropriate to modeling refractive-index structure profiles,” Appl. Opt. 27, 155–160 (1988).
    [CrossRef] [PubMed]
  13. V. P. Lukin, “Investigation of some peculiarities in the structure of large-scale atmospheric turbulence,” Atmos. Oceanic Opt. 5, 834–840 (1992).
  14. V. P. Lukin, E. V. Nosov, B. V. Fortes, “The efficient outer scale of atmospheric turbulence,” Atmos. Oceanic Opt. 10, 100–105 (1997).
  15. V. V. Voitsekhovich, C. Cuevas, “Adaptive optics and the outer scale of turbulence,” J. Opt. Soc. Am. A 12, 2523–2531 (1995).
    [CrossRef]
  16. 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]
  17. L. J. Otten, D. T. Kyrazis, D. W. Tyler, N. Miller, “Implication of atmospheric models on adaptive optics designs,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 201–211 (1994).
    [CrossRef]
  18. 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 June–1 July 1999 (American Institute of Aeronautics and Astronautics, New York, 1999).
  19. I. S. G., I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).
  20. A. Papoulis, Probability, Random Variables, and Stochiastic Processes (McGraw-Hill, New York, 1991).
  21. M. C. Roggemann, B. M. Welsh, D. Montera, T. A. Rhoadarmer, “Method for simulating atmospheric phase effects for multiple time slices and anisoplanatic conditions,” Appl. Opt. 34, 4037–4051 (1995).
    [CrossRef] [PubMed]
  22. M. A. Branch, A. Grace, MATLAB Optimization Toolbox (The Math Works, 24 Prime Park Way, Natick, Mass., 1996).
  23. H. D. Young, Statistical Treatment of Experimental Data (McGraw-Hill, New York, 1962).

1997 (2)

1996 (1)

1995 (3)

V. V. Voitsekhovich, C. Cuevas, “Adaptive optics and the outer scale of turbulence,” J. Opt. Soc. Am. A 12, 2523–2531 (1995).
[CrossRef]

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).

M. C. Roggemann, B. M. Welsh, D. Montera, T. A. Rhoadarmer, “Method for simulating atmospheric phase effects for multiple time slices and anisoplanatic conditions,” Appl. Opt. 34, 4037–4051 (1995).
[CrossRef] [PubMed]

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]

1992 (2)

V. P. Lukin, “Investigation of some peculiarities in the structure of large-scale atmospheric turbulence,” Atmos. Oceanic Opt. 5, 834–840 (1992).

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

1988 (1)

1974 (1)

D. K. Lilly, P. F. Lester, “Waves and turbulence in the stratosphere,” J. Atmos. Sci. 31, 800–812 (1974).
[CrossRef]

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).

Beland, R. R.

R. R. Beland, “Propagation through atmospheric optical turbulence,” in The Infrared and Electro-Optical Systems Handbook (SPIE, Bellingham, Wash., 1993).

Bester, M.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

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, 24 Prime Park Way, Natick, Mass., 1996).

Caccia, J. L.

Coqueugniot, Y.

Coulman, C. E.

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]

Cuevas, C.

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]

Danchi, W. C.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Degiacomi, C. G.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[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.

V. P. Lukin, E. V. Nosov, B. V. Fortes, “The efficient outer scale of atmospheric turbulence,” Atmos. Oceanic Opt. 10, 100–105 (1997).

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).

G., I. S.

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

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, 24 Prime Park Way, Natick, Mass., 1996).

Greenhill, L. J.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

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, “Implication of atmospheric models on adaptive optics designs,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 201–211 (1994).
[CrossRef]

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).

Lester, P. F.

D. K. Lilly, P. F. Lester, “Waves and turbulence in the stratosphere,” J. Atmos. Sci. 31, 800–812 (1974).
[CrossRef]

Lilly, D. K.

D. K. Lilly, P. F. Lester, “Waves and turbulence in the stratosphere,” J. Atmos. Sci. 31, 800–812 (1974).
[CrossRef]

Luke, T. E.

Lukin, V. P.

V. P. Lukin, E. V. Nosov, B. V. Fortes, “The efficient outer scale of atmospheric turbulence,” Atmos. Oceanic Opt. 10, 100–105 (1997).

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).

V. P. Lukin, “Investigation of some peculiarities in the structure of large-scale atmospheric turbulence,” Atmos. Oceanic Opt. 5, 834–840 (1992).

Miller, N.

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

Montera, D.

Nosov, E. V.

V. P. Lukin, E. V. Nosov, B. V. Fortes, “The efficient outer scale of atmospheric turbulence,” Atmos. Oceanic Opt. 10, 100–105 (1997).

Otten, L. J.

L. J. Otten, D. T. Kyrazis, D. W. Tyler, N. Miller, “Implication of atmospheric models on adaptive optics designs,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 201–211 (1994).
[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 June–1 July 1999 (American Institute of Aeronautics and Astronautics, New York, 1999).

Papoulis, A.

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

Rhoadarmer, T. A.

Roggeman, M. C.

Roggemann, M. C.

M. C. Roggemann, B. M. Welsh, D. Montera, T. A. Rhoadarmer, “Method for simulating atmospheric phase effects for multiple time slices and anisoplanatic conditions,” Appl. Opt. 34, 4037–4051 (1995).
[CrossRef] [PubMed]

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 June–1 July 1999 (American Institute of Aeronautics and Astronautics, New York, 1999).

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

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).

Ryzhik, I. M.

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

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).

Townes, C. H.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Tyler, D. W.

L. J. Otten, D. T. Kyrazis, D. W. Tyler, N. Miller, “Implication 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]

C. E. Coulman, J. Vernin, Y. Coqueugniot, J. L. Caccia, “Outer scale of turbulence appropriate to modeling refractive-index structure profiles,” Appl. Opt. 27, 155–160 (1988).
[CrossRef] [PubMed]

J. Vernin, “Atmospheric turbulence profiles,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarski, A. Ishimaru, V. U. Zovorontny, eds., Vol. PM09 of SPIE Press Monographs (SPIE Press, Bellingham, Wash., 1992), pp. 248–260.

Voitsekhovich, V. V.

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).

Young, H. D.

H. D. Young, Statistical Treatment of Experimental Data (McGraw-Hill, New York, 1962).

Appl. Opt. (4)

Astrophys. J. (1)

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Atmos. Oceanic Opt. (3)

V. P. Lukin, “Investigation of some peculiarities in the structure of large-scale atmospheric turbulence,” Atmos. Oceanic Opt. 5, 834–840 (1992).

V. P. Lukin, E. V. Nosov, B. V. Fortes, “The efficient outer scale of atmospheric turbulence,” Atmos. Oceanic Opt. 10, 100–105 (1997).

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. (2)

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]

D. K. Lilly, P. F. Lester, “Waves and turbulence in the stratosphere,” J. Atmos. Sci. 31, 800–812 (1974).
[CrossRef]

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

Other (12)

L. J. Otten, D. T. Kyrazis, D. W. Tyler, N. Miller, “Implication of atmospheric models on adaptive optics designs,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 201–211 (1994).
[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 June–1 July 1999 (American Institute of Aeronautics and Astronautics, New York, 1999).

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

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

J. Vernin, “Atmospheric turbulence profiles,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarski, A. Ishimaru, V. U. Zovorontny, eds., Vol. PM09 of SPIE Press Monographs (SPIE Press, Bellingham, Wash., 1992), pp. 248–260.

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).

R. R. Beland, “Propagation through atmospheric optical turbulence,” in The Infrared and Electro-Optical Systems Handbook (SPIE, Bellingham, Wash., 1993).

M. A. Branch, A. Grace, MATLAB Optimization Toolbox (The Math Works, 24 Prime Park Way, Natick, Mass., 1996).

H. D. Young, Statistical Treatment of Experimental Data (McGraw-Hill, New York, 1962).

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Figures (5)

Fig. 1
Fig. 1

Anemometer placement. Note that the probes are attached to a thin rigid structure, which is not shown here.

Fig. 2
Fig. 2

Location of the samples taken of the autocorrelation of the temperature fluctuations.

Fig. 3
Fig. 3

Theoretical ACF and sample-based ACF versus probe separation for l 0 = 0.0, 0.02, 0.04, and 0.06 m for 0 ≤ r ≤ 10 m. Note that the theoretical curves overlap on this vertical scale for the various values of l 0.

Fig. 4
Fig. 4

Autocorrelation function versus probe separation for l 0 = 0.0, 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 l 0 = 0; the three curves below are for l 0 = 0.02, 0.04, and 0.06 m, respectively.

Fig. 5
Fig. 5

Theoretical ACF and sample-based ACF versus probe separation for L 0 = 70, 150, and 230 m for α = 9/6. The upper curve is the ACF with L 0 = 230 m, the middle curve is for L 0 = 150 m, and the lowest curve is for L 0 = 70 m.

Tables (7)

Tables Icon

Table 1 Results of L 0 and C n 2 as a Function of SNR for the von Kármán Spectrum a

Tables Icon

Table 2 Mean and Variance of the Mean of the Diagonal Elements of the ACF

Tables Icon

Table 3 Parameter Estimation Results for the Case of Input α = 11/6 (1.83), Input P C = 1 × 10 -17 , and a SNR of 50 a

Tables Icon

Table 4 Parameter Estimation Results for the Case of Input α = 9/6 (1.50), Input PC = 1 × 10 -17 , and a SNR of 50 a

Tables Icon

Table 5 Parameter Estimation Results for the Case of Input α = 10/6 (1.7), Input PC = 1 × 10 -17 , and a SNR of 50 a

Tables Icon

Table 6 Parameter Estimation Results for the Case of Input α = 12/6 (2), Input PC = 1 × 10 -17 , and a SNR of 50 a

Tables Icon

Table 7 Parameter Estimation Results for the Case of Input α = 13/6 (2.17), Input PC = 1 × 10 -17 , and a SNR of 50 a

Equations (37)

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+k02α,
Bnr=4πr0PCk2+k02α k sin krdk,
Bnr=4πr1Γα20.5-αk01-απk0rα-0.5×K3/2-αk0r,
Bnr=exp-k0rπr4k0.
Bn=RRT,
ααT=IM,
ϕ=Rα,
ϕϕT=RαRαT=RααTRT.
ϕϕT=RααTRT=RααTRT=RIMRT=RRT=Bn,
a=signal strengthSNR,
ϕN=Rα+av,
ϕNϕNT=Rα+avRα+avT,
ϕNϕNT=RααTRT+RαavT+avαTRT+a2vvT.
RαavT=Rα vTa=0,
avαTRT=avαTRT=0.
a2vvT=a2IM.
ϕNϕNT=Bn+a2IM.
fdk=i=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-1Trk=1K Σ-1dkdkT,
2LK=-N ln PC-ln|Σ|-PC-1TrΣ-11Kk=1K dkdkT.
S=1Kk=1K dkdkT
SB=S-a2Im.
C=PC-1TrΣ-1SB+N ln PC+ln|Σ|,
CPC=-PC-2TrΣ-1SB+NPC.
P˜C=1NTrΣ-1SB.
C˜=N lnTrΣ-1SB+N ln P˜C+ln|Σ|,

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