Abstract

A systematic analysis is presented of the extent to which the accuracy of a differential-absorption lidar (DIAL) measurement may be improved by using the combined effects of signal averaging and temporal cross correlation. Previous studies which considered these effects separately are extended by incorporating both effects into a single analytical framework. In addition, experimental results involving lidar returns from a diffusely reflecting target using a dual-CO2 laser DIAL system with both heterodyne and direct detection are presented. These results are shown to be in good agreement with the theoretical analysis and help establish the limits of accuracy achievable under various experimental conditions.

© 1985 Optical Society of America

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  1. D. K. Killinger, A. Mooradian, Eds., Optical and Laser Remote Sensing (Springer, Berlin, 1983).
  2. J. H. Shapiro, B. A. Capron, R. C. Harney, “Imaging and Target Detection with a Heterodyne-Reception Optical Radar,” Appl. Opt. 20, 3292 (1981).
    [CrossRef] [PubMed]
  3. P. Brockman, R. V. Hess, L. D. Staton, C. H. Bair, “DIAL with Heterodyne Detection Including Speckle Noise: Aircraft/Shuttle Measurements of O3, H2O and NH3 with Pulsed Tunable Lasers, NASA Tech. Paper 1725 (1980).
  4. R. M. Hardesty, “A Comparison of Heterodyne and Direct Detection CO2 DIAL Systems for Ground Based Humidity Profiling,” NOAA Tech. Memo. ERL WP-64 (1980).
  5. R. J. Hill, S. F. Clifford, R. S. Lawrence, “Refractive-Index and Absorption Fluctuations in the Infrared Caused by Temperature, Humidity, and Pressure Fluctuations,” J. Opt. Soc. Am. 70, 1192 (1980).
    [CrossRef]
  6. A. G. Kjelaas, P. E. Nordal, A. Bjerkestrand, “Scintillation and Multiwavelength Coherence Effects in a Long-Path Laser Absorption Spectrometer,” Appl. Opt. 17, 277 (1978).
    [CrossRef] [PubMed]
  7. J. F. Holmes, V. S. Rao Gudimetla, “Variance of Intensity for a Discrete Spectrum, Polychromatic Speckle Field after Propagation Through the Turbulent Atmosphere,” J. Opt. Soc. Am. 71, 1176 (1981).
    [CrossRef]
  8. J. W. Goodman, “Some Effects of Target-Induced Scintillation on Optical Radar Performance,” Proc. IEEE 53, 1688 (1965).
    [CrossRef]
  9. C. M. Mclntyre, M. H. Lee, J. H. Churnside, “Statistics of Irradiance Scattered from a Diffuse Target Containing Multiple Glints,” J. Opt. Soc. Am. 70, 1084 (1980).
    [CrossRef]
  10. T. Chiba, “Spot Dancing of the Laser Beam Propagated Through the Turbulent Atmosphere,” Appl. Opt. 10, 2456 (1971).
    [CrossRef] [PubMed]
  11. M. H. Lee, J. F. Holmes, “Effect of the Turbulent Atmosphere on the Autocovariance Function for Speckle Field Generated by a Laser Beam with Random Pointing Error,” J. Opt. Soc. Am. 71, 559 (1981).
    [CrossRef]
  12. G. Megie, R. T. Menzies, “Complementarity of UV and IR Differential Absorption Lidar for Global Measurements of Atmospheric Species,” Appl. Opt. 19, 1173 (1980).
    [CrossRef] [PubMed]
  13. C. S. Gardner, G. S. Micherle, “Speckle Noise in Direct Detection LIDAR Systems,” U. Illinois Radio Research Report 495 (1978).
  14. C. S. Gardner, A. M. Saleh, “Speckle Noise in Differential Absorption LIDAR Systems,” U. Illinois Radio Research Report 496 (1978).
  15. D. K. Killinger, N. Menyuk, W. E. DeFeo, “Experimental Comparison of Heterodyne and Direct Detection for Pulsed Differential Absorption,” Appl. Opt. 22, 682 (1983).
    [CrossRef] [PubMed]
  16. E. Jakeman, C. J. Oliver, E. R. Pike, “Optical Homodyne Detection,” Adv. Phys. 24, 349 (1975).
    [CrossRef]
  17. M. Elbaum, M. C. Teich, “Heterodyne Detection of Random Gaussian Signals in the Optical and Infrared: Optimization of Pulse Duration,” Opt. Commun. 27, 257 (1978).
    [CrossRef]
  18. N. Menyuk, D. K. Killinger, W. E. DeFeo, “Laser Remote Sensing of Hydrazine, MMH, and UDMH Using a Differential Absorption CO2 Lidar,” Appl. Opt. 21, 2275 (1982).
    [CrossRef] [PubMed]
  19. N. Menyuk, D. K. Killinger, C. R. Menyuk, “Limitations of Signal Averaging due to Temporal Correlation in Laser Remote-Sensing Measurements,” Appl. Opt. 21, 3377 (1982).
    [CrossRef] [PubMed]
  20. D. K. Killinger, N. Menyuk, “Effect of Turbulence-Induced Correlation on Laser Remote Sensing Errors,” Appl. Phys. Lett. 38, 968 (1981).
    [CrossRef]
  21. For simplicity, Eq. (2) ignores the effect of possible differences in target reflectivity or background absorption at the two frequencies. This omission does not affect later discussions.
  22. N. Menyuk, D. K. Killinger, “Temporal Correlation Measurements of Pulsed Dual CO2 Lidar Returns,” Opt. Lett. 6, 301 (1981).
    [CrossRef] [PubMed]
  23. B. Marthinsson, J. Johansson, S. T. Eng, “Air Pollution Monitoring with a Computer-Controlled CO2-Laser Long-Path Absorption System,” Opt. Quantum Electron. 12, 327 (1980).
    [CrossRef]
  24. R. M. Hardesty, R. J. Keeler, M. J. Post, R. A. Richter, “Characteristics of Coherent Lidar Returns from Calibration Targets and Aerosols,” Appl. Opt. 20, 3763 (1981).
    [CrossRef] [PubMed]
  25. J. L. Bufton, T. Itabe, D. A. Grolemund, “Dual-Wavelength Correlation Measurements with an Airborne Pulsed Carbon Dioxide Lidar System,” Opt. Lett. 7, 584 (1982).
    [CrossRef] [PubMed]
  26. D. K. Killinger, N. Menyuk, W. E. DeFeo, “Remote Probing of the Atmosphere Using a CO2 DIAL System,” IEEE J. Quantum Electron. QE-17, 1917 (1981).
    [CrossRef]
  27. J. F. Holmes, “The Effects of Target Induced Speckle, Atmospheric Turbulence and Beam Pointing Jitter on the Errors in Remote Sensing Measurements,” in Optical and Laser Remote Sensing, D. K. Killinger, A. Mooradian, Eds. (Springer, Berlin, 1983), pp. 164–169.
  28. B. J. Rye, “Power Ratio Estimation in Incoherent Backscatter Lidar: Heterodyne Receiver with Square Law Detection,” J. Climate Appl. Meteorol. 22, 1899 (1983).
    [CrossRef]
  29. N. Menyuk, P. F. Moulton, “Development of a High-Repetition-Rate Mini-TEA Laser,” Rev. Sci. Instrum. 51, 216 (1980).
    [CrossRef]
  30. J. W. Goodman, G. Parry, in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, New York, 1975), Chaps. 2 and 3, respectively.

1983 (2)

D. K. Killinger, N. Menyuk, W. E. DeFeo, “Experimental Comparison of Heterodyne and Direct Detection for Pulsed Differential Absorption,” Appl. Opt. 22, 682 (1983).
[CrossRef] [PubMed]

B. J. Rye, “Power Ratio Estimation in Incoherent Backscatter Lidar: Heterodyne Receiver with Square Law Detection,” J. Climate Appl. Meteorol. 22, 1899 (1983).
[CrossRef]

1982 (3)

1981 (7)

1980 (6)

N. Menyuk, P. F. Moulton, “Development of a High-Repetition-Rate Mini-TEA Laser,” Rev. Sci. Instrum. 51, 216 (1980).
[CrossRef]

C. M. Mclntyre, M. H. Lee, J. H. Churnside, “Statistics of Irradiance Scattered from a Diffuse Target Containing Multiple Glints,” J. Opt. Soc. Am. 70, 1084 (1980).
[CrossRef]

P. Brockman, R. V. Hess, L. D. Staton, C. H. Bair, “DIAL with Heterodyne Detection Including Speckle Noise: Aircraft/Shuttle Measurements of O3, H2O and NH3 with Pulsed Tunable Lasers, NASA Tech. Paper 1725 (1980).

R. J. Hill, S. F. Clifford, R. S. Lawrence, “Refractive-Index and Absorption Fluctuations in the Infrared Caused by Temperature, Humidity, and Pressure Fluctuations,” J. Opt. Soc. Am. 70, 1192 (1980).
[CrossRef]

G. Megie, R. T. Menzies, “Complementarity of UV and IR Differential Absorption Lidar for Global Measurements of Atmospheric Species,” Appl. Opt. 19, 1173 (1980).
[CrossRef] [PubMed]

B. Marthinsson, J. Johansson, S. T. Eng, “Air Pollution Monitoring with a Computer-Controlled CO2-Laser Long-Path Absorption System,” Opt. Quantum Electron. 12, 327 (1980).
[CrossRef]

1978 (2)

A. G. Kjelaas, P. E. Nordal, A. Bjerkestrand, “Scintillation and Multiwavelength Coherence Effects in a Long-Path Laser Absorption Spectrometer,” Appl. Opt. 17, 277 (1978).
[CrossRef] [PubMed]

M. Elbaum, M. C. Teich, “Heterodyne Detection of Random Gaussian Signals in the Optical and Infrared: Optimization of Pulse Duration,” Opt. Commun. 27, 257 (1978).
[CrossRef]

1975 (1)

E. Jakeman, C. J. Oliver, E. R. Pike, “Optical Homodyne Detection,” Adv. Phys. 24, 349 (1975).
[CrossRef]

1971 (1)

1965 (1)

J. W. Goodman, “Some Effects of Target-Induced Scintillation on Optical Radar Performance,” Proc. IEEE 53, 1688 (1965).
[CrossRef]

Bair, C. H.

P. Brockman, R. V. Hess, L. D. Staton, C. H. Bair, “DIAL with Heterodyne Detection Including Speckle Noise: Aircraft/Shuttle Measurements of O3, H2O and NH3 with Pulsed Tunable Lasers, NASA Tech. Paper 1725 (1980).

Bjerkestrand, A.

Brockman, P.

P. Brockman, R. V. Hess, L. D. Staton, C. H. Bair, “DIAL with Heterodyne Detection Including Speckle Noise: Aircraft/Shuttle Measurements of O3, H2O and NH3 with Pulsed Tunable Lasers, NASA Tech. Paper 1725 (1980).

Bufton, J. L.

Capron, B. A.

Chiba, T.

Churnside, J. H.

Clifford, S. F.

DeFeo, W. E.

Elbaum, M.

M. Elbaum, M. C. Teich, “Heterodyne Detection of Random Gaussian Signals in the Optical and Infrared: Optimization of Pulse Duration,” Opt. Commun. 27, 257 (1978).
[CrossRef]

Eng, S. T.

B. Marthinsson, J. Johansson, S. T. Eng, “Air Pollution Monitoring with a Computer-Controlled CO2-Laser Long-Path Absorption System,” Opt. Quantum Electron. 12, 327 (1980).
[CrossRef]

Gardner, C. S.

C. S. Gardner, G. S. Micherle, “Speckle Noise in Direct Detection LIDAR Systems,” U. Illinois Radio Research Report 495 (1978).

C. S. Gardner, A. M. Saleh, “Speckle Noise in Differential Absorption LIDAR Systems,” U. Illinois Radio Research Report 496 (1978).

Goodman, J. W.

J. W. Goodman, “Some Effects of Target-Induced Scintillation on Optical Radar Performance,” Proc. IEEE 53, 1688 (1965).
[CrossRef]

J. W. Goodman, G. Parry, in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, New York, 1975), Chaps. 2 and 3, respectively.

Grolemund, D. A.

Hardesty, R. M.

R. M. Hardesty, R. J. Keeler, M. J. Post, R. A. Richter, “Characteristics of Coherent Lidar Returns from Calibration Targets and Aerosols,” Appl. Opt. 20, 3763 (1981).
[CrossRef] [PubMed]

R. M. Hardesty, “A Comparison of Heterodyne and Direct Detection CO2 DIAL Systems for Ground Based Humidity Profiling,” NOAA Tech. Memo. ERL WP-64 (1980).

Harney, R. C.

Hess, R. V.

P. Brockman, R. V. Hess, L. D. Staton, C. H. Bair, “DIAL with Heterodyne Detection Including Speckle Noise: Aircraft/Shuttle Measurements of O3, H2O and NH3 with Pulsed Tunable Lasers, NASA Tech. Paper 1725 (1980).

Hill, R. J.

Holmes, J. F.

Itabe, T.

Jakeman, E.

E. Jakeman, C. J. Oliver, E. R. Pike, “Optical Homodyne Detection,” Adv. Phys. 24, 349 (1975).
[CrossRef]

Johansson, J.

B. Marthinsson, J. Johansson, S. T. Eng, “Air Pollution Monitoring with a Computer-Controlled CO2-Laser Long-Path Absorption System,” Opt. Quantum Electron. 12, 327 (1980).
[CrossRef]

Keeler, R. J.

Killinger, D. K.

Kjelaas, A. G.

Lawrence, R. S.

Lee, M. H.

Marthinsson, B.

B. Marthinsson, J. Johansson, S. T. Eng, “Air Pollution Monitoring with a Computer-Controlled CO2-Laser Long-Path Absorption System,” Opt. Quantum Electron. 12, 327 (1980).
[CrossRef]

Mclntyre, C. M.

Megie, G.

Menyuk, C. R.

Menyuk, N.

Menzies, R. T.

Micherle, G. S.

C. S. Gardner, G. S. Micherle, “Speckle Noise in Direct Detection LIDAR Systems,” U. Illinois Radio Research Report 495 (1978).

Moulton, P. F.

N. Menyuk, P. F. Moulton, “Development of a High-Repetition-Rate Mini-TEA Laser,” Rev. Sci. Instrum. 51, 216 (1980).
[CrossRef]

Nordal, P. E.

Oliver, C. J.

E. Jakeman, C. J. Oliver, E. R. Pike, “Optical Homodyne Detection,” Adv. Phys. 24, 349 (1975).
[CrossRef]

Parry, G.

J. W. Goodman, G. Parry, in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, New York, 1975), Chaps. 2 and 3, respectively.

Pike, E. R.

E. Jakeman, C. J. Oliver, E. R. Pike, “Optical Homodyne Detection,” Adv. Phys. 24, 349 (1975).
[CrossRef]

Post, M. J.

Rao Gudimetla, V. S.

Richter, R. A.

Rye, B. J.

B. J. Rye, “Power Ratio Estimation in Incoherent Backscatter Lidar: Heterodyne Receiver with Square Law Detection,” J. Climate Appl. Meteorol. 22, 1899 (1983).
[CrossRef]

Saleh, A. M.

C. S. Gardner, A. M. Saleh, “Speckle Noise in Differential Absorption LIDAR Systems,” U. Illinois Radio Research Report 496 (1978).

Shapiro, J. H.

Staton, L. D.

P. Brockman, R. V. Hess, L. D. Staton, C. H. Bair, “DIAL with Heterodyne Detection Including Speckle Noise: Aircraft/Shuttle Measurements of O3, H2O and NH3 with Pulsed Tunable Lasers, NASA Tech. Paper 1725 (1980).

Teich, M. C.

M. Elbaum, M. C. Teich, “Heterodyne Detection of Random Gaussian Signals in the Optical and Infrared: Optimization of Pulse Duration,” Opt. Commun. 27, 257 (1978).
[CrossRef]

Adv. Phys. (1)

E. Jakeman, C. J. Oliver, E. R. Pike, “Optical Homodyne Detection,” Adv. Phys. 24, 349 (1975).
[CrossRef]

Appl. Opt. (8)

Appl. Phys. Lett. (1)

D. K. Killinger, N. Menyuk, “Effect of Turbulence-Induced Correlation on Laser Remote Sensing Errors,” Appl. Phys. Lett. 38, 968 (1981).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. K. Killinger, N. Menyuk, W. E. DeFeo, “Remote Probing of the Atmosphere Using a CO2 DIAL System,” IEEE J. Quantum Electron. QE-17, 1917 (1981).
[CrossRef]

J. Climate Appl. Meteorol. (1)

B. J. Rye, “Power Ratio Estimation in Incoherent Backscatter Lidar: Heterodyne Receiver with Square Law Detection,” J. Climate Appl. Meteorol. 22, 1899 (1983).
[CrossRef]

J. Opt. Soc. Am. (4)

NASA Tech. (1)

P. Brockman, R. V. Hess, L. D. Staton, C. H. Bair, “DIAL with Heterodyne Detection Including Speckle Noise: Aircraft/Shuttle Measurements of O3, H2O and NH3 with Pulsed Tunable Lasers, NASA Tech. Paper 1725 (1980).

Opt. Commun. (1)

M. Elbaum, M. C. Teich, “Heterodyne Detection of Random Gaussian Signals in the Optical and Infrared: Optimization of Pulse Duration,” Opt. Commun. 27, 257 (1978).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

B. Marthinsson, J. Johansson, S. T. Eng, “Air Pollution Monitoring with a Computer-Controlled CO2-Laser Long-Path Absorption System,” Opt. Quantum Electron. 12, 327 (1980).
[CrossRef]

Proc. IEEE (1)

J. W. Goodman, “Some Effects of Target-Induced Scintillation on Optical Radar Performance,” Proc. IEEE 53, 1688 (1965).
[CrossRef]

Rev. Sci. Instrum. (1)

N. Menyuk, P. F. Moulton, “Development of a High-Repetition-Rate Mini-TEA Laser,” Rev. Sci. Instrum. 51, 216 (1980).
[CrossRef]

Other (7)

J. W. Goodman, G. Parry, in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, New York, 1975), Chaps. 2 and 3, respectively.

D. K. Killinger, A. Mooradian, Eds., Optical and Laser Remote Sensing (Springer, Berlin, 1983).

J. F. Holmes, “The Effects of Target Induced Speckle, Atmospheric Turbulence and Beam Pointing Jitter on the Errors in Remote Sensing Measurements,” in Optical and Laser Remote Sensing, D. K. Killinger, A. Mooradian, Eds. (Springer, Berlin, 1983), pp. 164–169.

C. S. Gardner, G. S. Micherle, “Speckle Noise in Direct Detection LIDAR Systems,” U. Illinois Radio Research Report 495 (1978).

C. S. Gardner, A. M. Saleh, “Speckle Noise in Differential Absorption LIDAR Systems,” U. Illinois Radio Research Report 496 (1978).

R. M. Hardesty, “A Comparison of Heterodyne and Direct Detection CO2 DIAL Systems for Ground Based Humidity Profiling,” NOAA Tech. Memo. ERL WP-64 (1980).

For simplicity, Eq. (2) ignores the effect of possible differences in target reflectivity or background absorption at the two frequencies. This omission does not affect later discussions.

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Figures (6)

Fig. 1
Fig. 1

Schematic of dual-CO2 laser differential-absorption lidar system with capability of simultaneous heterodyne and direct detection of backscattered laser signals.

Fig. 2
Fig. 2

Lidar signal return data from diffusely reflecting target at a 2.7-km range using direct detection during period of continuously decreasing atmospheric extinction: (a) standard deviation of returns from lasers 1 and 2 and their ratio as functions of the number of pulses averaged; (b) temporal autocorrelation and cross correlation of laser return signals as functions of time delay; and (c) comparison of measured and calculated values of signal-averaged cross-correlation coefficient as a function of the number of pulses averaged.

Fig. 3
Fig. 3

Lidar signal return data from diffusely reflecting target at a 2.7-km range using direct detection during the period of essentially constant average atmospheric extinction: (a) standard deviation of returns from lasers 1 and 2 and their ratio as functions of the number of pulses averaged; (b) temporal autocorrelation and cross correlation of laser return signals as functions of time delay; and (c) comparison of measured and calculated value of signal-averaged cross-correlation coefficient as a function of the number of pulses averaged.

Fig. 4
Fig. 4

Lidar signal return data from diffusely reflecting target at a 2.7-km range using heterodyne detection with both CO2 lasers radiating on the 10.6-μm P(20) laser transition (λ = λ′): (a) Standard deviation of returns from lasers 1 and 2 and their ratio as functions of the number of pulses averaged, showing the >n−1/2 variation for small n when the signals are properly averaged prior to taking ratios; (b) temporal autocorrelation of individual laser return signals and of their ratio as functions of time delay; and (c) measured signalaveraged cross-correlation coefficient variation with number of pulses averaged showing the near approach to unity with increasing n.

Fig. 5
Fig. 5

Comparison of measured and calculated values of σ as a function of the number of pulses averaged based on the data used in Fig. 4. Agreement is seen to be excellent for σny ≤ 0.5.

Fig. 6
Fig. 6

Standard deviation as functions of the number of pulses averaged for lidar signal returns from the diffusely reflecting target at a 2.7-km range using heterodyne detection with lasers 1 and 2 radiating on the P(20) and P(22) CO2 laser transitions of the 10.6-μm band, respectively (λ ≠ λ′).

Equations (30)

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P r = ( P t K ρ A / π R 2 ) exp [ 2 ( σ a N a + α ) R ] ,
N a = ln ( P / P ) / 2 ( σ a σ a ) R ,
σ ξ 2 = σ x 2 + σ y 2 2 ρ c σ x σ y ,
σ n = σ n [ 1 + 2 j = 1 n 1 ( 1 j / n ) ρ j ] 1 / 2 .
σ n ξ 2 = σ n x 2 + σ n y 2 2 ρ n c σ n x σ n y ,
ρ n c = σ x σ y n σ n x σ n y [ ρ c + 2 j = 1 n 1 ( 1 j / n ) ρ jxy ] ,
ρ jxy = 1 σ x σ y ( Γ j ) k = 1 Γ j I k x I ( k + j ) y .
ρ n c = [ ρ c + 2 j = 1 n 1 ( 1 j / n ) ρ jxy ] [ 1 + 2 j = 1 n 1 ( 1 j / n ) ρ j x ] 1 / 2 [ 1 + 2 j = 1 n 1 ( 1 j / n ) ρ j y ] 1 / 2 .
ρ n c = [ ρ c + 2 j = 1 n 1 ( 1 j / n ) ρ jxy ] [ 1 + 2 j = 1 n 1 ( 1 j / n ) ρ j x ] .
ρ n c = ρ c [ 1 + 2 ζ j = 1 n 1 ( 1 j / n ) ρ jxy ] [ 1 + 2 j = 1 n 1 ( 1 j / n ) ρ j x ] .
( ρ jxy + jxy ) / [ ( ρ j x + j x ) 1 / 2 ( ρ j y + j y ) 1 / 2 ] ,
d σ n ξ d ρ n c = σ n y / 2 ( 1 ρ n c ) 1 / 2 .
I k x = ( P k x P ¯ x ) / P ¯ x and I k y = ( P k y P ¯ y ) / P ¯ y ,
σ x 2 = ( I k x ) 2 = 1 Γ k = 1 Γ I k x 2 ,
ρ j x = 1 σ x 2 I x ( t k ) I x ( t k + j τ ) = 1 σ x 2 ( Γ j ) k = 1 Γ j I k x I ( k + j ) x ,
ρ c = σ x y σ x σ y I x ( t k ) I y ( t k + Δ t ) σ x σ y = 1 σ x σ y Γ k = 1 Γ I k x I k y .
ρ jxy = I x ( t k ) I y ( t k + Δ t + j τ ) σ x σ y = 1 σ x σ y ( Γ j ) k = 1 Γ j I k x I ( k + j ) y .
σ n i 2 = ( σ i 2 / n ) [ 1 + 2 j = 1 n 1 ( 1 j / n ) ρ j i ] ,
ρ n c = 1 σ n x σ n y [ 1 Γ / n ( ( I 1 x + I 2 x + + I n x n ) ( I 1 y + I 2 y + + I n y n ) + ( I ( n + 1 ) x + + I 2 n x n ) ( I ( n + 1 ) y + + I 2 n y n ) + + ( I [ Γ ( n + 1 ) ] x + + I Γ x n ) ( I [ Γ ( n 1 ) ] y + + I Γ y n ) ) ]
= 1 n Γ σ n x σ n y [ ( I 1 x I 1 y + I 2 x I 2 y + + I Γ x I Γ y ) + ( [ I 1 x I 2 y + + I ( n 1 ) x I n y ] + [ I 2 x I 1 y + + I n x I ( n 1 ) y ] + [ I ( n + 1 ) x I ( n + 2 ) y + + I ( 2 n 1 ) x I 2 n y ] + [ I ( n + 2 ) x I ( n + 1 ) y + + I 2 n x I ( 2 n 1 y ) y ] + + { I [ Γ ( n 1 ) x ] I [ Γ ( n 2 ) y ] + + I ( Γ 1 ) x I Γ y } + { I [ Γ ( n 1 ) y ] I [ Γ ( n 2 ) x ] + + I Γ x I ( Γ 1 ) y } ) + + ( { I 1 x I n y + I ( n + 1 ) x I 2 n y + + I [ Γ ( n 1 ) ] x I Γ y } + { I n x I l y + I 2 n x I ( n + 1 ) y + + I Γ x I [ Γ ( n 1 ) ] y } ) ] .
ρ n c = 1 n Γ σ n x σ n y { Γ ρ c σ x σ y + 2 ( Γ 1 ) σ x σ y ( n 1 n ) ρ 1 x y + + 2 [ Γ ( n 1 ) ] σ x σ y ( 1 n ) ρ ( n 1 ) x y } ,
ρ n c = σ x σ y n σ n x σ n y [ ρ c + 2 j = 1 n 1 ( 1 j / Γ ) ( 1 j / n ) ρ jxy ] .
ρ n c = σ x σ y n σ n x σ n y [ ρ c + 2 j = 1 n 1 ( 1 j / n ) ρ jxy ] .
z = f ( x , y ) = f ( x ¯ , y ¯ ) + f x | p ( x x ¯ ) + f y | p ( y y ¯ ) + 1 2 [ 2 f x 2 | p ( x x ¯ ) 2 + f y | p ( y y ¯ ) 2 + 2 2 f x y | p ( x x ¯ ) ( y y ¯ ) ] ,
z ¯ = f ( x ¯ , y ¯ ) + 1 2 { 2 f x 2 | p ( x x ¯ ) 2 + 2 f y 2 | p ( y y ¯ ) 2 + 2 2 f x y | p ( x x ¯ ) ( y y ¯ ) } = f ( x ¯ , y ¯ ) + 1 2 ( 2 f x 2 | p σ x 2 + 2 f x y | p σ x 2 + 2 2 f x y | p ρ c σ x σ y ) .
σ z 2 = ( z z ¯ ) 2 = ( f x ) 2 ( x x ¯ ) 2 + ( f y ) 2 ( y y ¯ ) 2 + 2 ( f x ) ( f y ) ( x x ¯ ) ( y y ¯ ) + { f x ( 2 f x 2 ) ( x x ¯ ) 3 + [ 2 ( f x ) ( 2 f x y ) + f y ( 2 f x 2 ) ] ( x x ¯ ) 2 ( y y ¯ ) + [ 2 ( f y ) ( 2 f y x ) + f x ( 2 f y 2 ) ] ( x x ¯ ) ( y y ¯ ) 2 + f y ( 2 f y 2 ) ( y y ¯ ) 3 } + 1 4 { ( 2 f y 2 ) 2 [ ( x x ¯ ) 4 σ x 4 ] + 4 ( 2 f x 2 ) ( 2 f x y ) [ ( x x ¯ ) 3 ( y y ¯ ) ρ c σ x 3 σ y ] + 4 ( 2 f x y ) 2 [ ( x x ¯ ) 2 ( y y ¯ ) 2 ρ c σ x 2 σ y 2 ] + 2 ( 2 f x 2 ) ( 2 f y 2 ) [ ( x x ¯ ) 2 ( y y ¯ ) 2 σ x 2 σ y 2 ] + 4 ( 2 f y 2 ) ( 2 f x y ) [ ( x x ¯ ) ( y y ¯ ) 3 ρ c σ x σ y 3 ] + ( 2 f y 2 ) 2 [ ( y y ¯ ) 4 σ y 4 ] } ) ,
f x | p = 1 ; y ¯ , 2 f x 2 | p = 0 ; f y | p = x ¯ y ¯ , 2 f y 2 | p = 2 x ¯ y ¯ 3 , and 2 f x y | p = 2 f x y | p = 1 y ¯ 2 .
( ξ ξ ¯ ) 2 ( x ¯ 2 / y ¯ 2 ) = [ ( x x ¯ ) 2 x ¯ 2 2 ( x x ¯ ) ( y y ¯ ) x y ¯ + ( y y ¯ ) 2 y ¯ 2 ] 2 [ ( x x ¯ ) 2 ( y y ¯ ) x ¯ 2 y ¯ 2 ( x x ¯ ) ( y y ¯ ) 2 x y ¯ 2 + ( y y ¯ ) 3 y ¯ 2 ] + [ ( x x ¯ ) 2 ( y y ¯ ) 2 x ¯ 2 y ¯ 2 2 ( x x ¯ ) ( y y ¯ ) 3 x y ¯ 3 + ( y y ¯ ) 4 y ¯ 4 ] σ y 2 ( σ x 2 + σ y 2 2 ρ c σ x σ y ) .
σ ξ 2 = ( 1 ( y y ¯ ) y ¯ ) 2 { ( x x ¯ ) 2 x ¯ + ( y y ¯ ) 2 y ¯ 2 ( x x ¯ ) ( y y ¯ ) x y ¯ } σ y 2 ( σ x 2 + σ y 2 2 ρ c σ x σ y ) .
( y y ¯ ) 2 y ¯ 2 .

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