Abstract

Simulating the effects of atmospheric turbulence on optical imaging systems is an important aspect of understanding the performance of these systems. Simulations are particularly important for understanding the statistics of some adaptive-optics system performance measures, such as the mean and variance of the compensated optical transfer function, and for understanding the statistics of estimators used to reconstruct intensity distributions from turbulence-corrupted image measurements. Current methods of simulating the performance of these systems typically make use of random phase screens placed in the system pupil. Methods exist for making random draws of phase screens that have the correct spatial statistics. However, simulating temporal effects and anisoplanatism requires one or more phase screens at different distances from the aperture, possibly moving with different velocities. We describe and demonstrate a method for creating random draws of phase screens with the correct space–time statistics for arbitrary turbulence and wind-velocity profiles, which can be placed in the telescope pupil in simulations. Results are provided for both the von Kármán and the Kolmogorov turbulence spectra. We also show how to simulate anisoplanatic effects with this technique.

© 1995 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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1994 (4)

1993 (5)

1992 (1)

1991 (1)

1990 (3)

J. D. Gonglewski, D. G. Voelz, J. S. Fender, D. C. Dayton, B. K. Spielbusch, R. E. Pierson, “First astronomical application of postdetection turbulence compensation: images of a Aurigae, n Ursae Majoris, and a Geminorum using self-referenced speckle holography,” Appl. Opt. 29, 4527–4529 (1990).
[CrossRef] [PubMed]

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1589–1608 (1990).
[CrossRef]

N. Roddier, “Atmospheric wave-front simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

1989 (2)

1987 (1)

1983 (2)

1979 (1)

1977 (1)

1976 (1)

Barakat, R.

Boeke, B. R.

Carlson, L.

T. Goldring, L. Carlson, “Analysis and implementation of non-Kolmogorov phase screens appropriate to structured environments,” in Nonlinear Optical Beam Manipulation and High Energy Beam Propagation through the Atmosphere, R. A. Fisher, L. E. Wilson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1060, 244–264 (1989).

Cleis, R. A.

Cochran, G.

G. Cochran, “Phase screen generation,” Tech. Rep. TR-663 (Optical Sciences Company, Placentia, Calif., 1985).

Dainty, J. C.

Dayton, D. C.

Ellerbroek, B. L.

Fender, J. S.

Flannery, B.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Fontanella, J. C.

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1589–1608 (1990).
[CrossRef]

Fried, D. L.

D. L. Fried, “Postdetection wave-front compensation,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 127–133 (1987).

Fugate, R. Q.

Gardner, C. S.

Goldring, T.

T. Goldring, L. Carlson, “Analysis and implementation of non-Kolmogorov phase screens appropriate to structured environments,” in Nonlinear Optical Beam Manipulation and High Energy Beam Propagation through the Atmosphere, R. A. Fisher, L. E. Wilson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1060, 244–264 (1989).

Gonglewski, J. D.

Goodman, J. W.

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

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Greenaway, A. H.

Greenwood, D. P.

Hanson, D.

M. Miller, P. Zieske, D. Hanson, “Characterization of atmospheric turbulence,” in Imaging Through The Atmosphere, J. C. Wyant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.75, 30–38 (1976).

Higgins, C. H.

Hufnagel, R. E.

R. E. Hufnagel, “Variations of atmospheric turbulence,” in Optical Propagation through Turbulence, OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1974), paper WA1.

Ishimaru, A.

A. Ishimaru, “The beam wave case and remote sensing,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978), Vol. 25, pp. 129–170.
[CrossRef]

Jelonek, M. P.

Lange, W. J.

Lohmann, A. W.

Matson, C. L.

Meinhardt, J. A.

Miller, M.

M. Miller, P. Zieske, D. Hanson, “Characterization of atmospheric turbulence,” in Imaging Through The Atmosphere, J. C. Wyant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.75, 30–38 (1976).

Moroney, J. F.

Niederhausern, R. N. V.

Nisenson, P.

Noll, R. J.

Oliker, M. D.

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965).

Paxman, R. G.

Peri, M. L.

Pierson, R. E.

Press, W.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Primot, J.

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1589–1608 (1990).
[CrossRef]

Rhoadarmer, T. A.

M. C. Roggemann, B. L. Ellerbroek, T. A. Rhoadarmer, “Widening the effective field-of-view of adaptive-optics telescopes using deconvolution from wave-front sensing: average and signal-to-noise ratio performance,” Appl. Opt. (to be published).

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. XIX, pp. 281–376.
[CrossRef]

Roddier, N.

N. Roddier, “Atmospheric wave-front simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Roggemann, M. C.

Rousset, G.

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1589–1608 (1990).
[CrossRef]

Ruane, R. E.

Sasiela, R. J.

R. J. Sasiela, J. D. Shelton, “Transverse spectral filtering and Mellin-transform techniques applied to the effect of outer scale on tilt and tilt anisoplanatism,” J. Opt. Soc. Am. A 10, 646–660 (1993).
[CrossRef]

R. J. Sasiela, J. D. Shelton, “Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations,” J. Math. Phys. N.Y. 34, 2572–2619 (1993).
[CrossRef]

R. J. Sasiela, “A unified approach to electromagnetic wave propagation in turbulence and the evaluation of multiparameter integrals,” Tech. Rep. TR 807 (Lincoln Laboratory, MIT, Cambridge, Mass., 1988).

Seldin, J. H.

Shelton, J. D.

R. J. Sasiela, J. D. Shelton, “Transverse spectral filtering and Mellin-transform techniques applied to the effect of outer scale on tilt and tilt anisoplanatism,” J. Opt. Soc. Am. A 10, 646–660 (1993).
[CrossRef]

R. J. Sasiela, J. D. Shelton, “Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations,” J. Math. Phys. N.Y. 34, 2572–2619 (1993).
[CrossRef]

Sindle, D. W.

Slavin, A. C.

Smithson, R. C.

Spielbusch, B. K.

Spinhirne, J. M.

Strang, G.

G. Strang, Linear Algebra and its Applications (Academic, New York, 1980).

Teukolsky, S.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Thelen, B. J.

Troxel, S. E.

Tyler, G. A.

G. A. Tyler, “Merging: a new method for tomography through random media,” J. Opt. Soc. Am. A 10, 409–425 (1993).

Vetterling, W.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Voelz, D. G.

Wallner, E. P.

Weigelt, G.

Welsh, B. M.

Wild, W. J.

Winker, D. M.

Wirnitzer, B.

Wynia, J. M.

Zieske, P.

M. Miller, P. Zieske, D. Hanson, “Characterization of atmospheric turbulence,” in Imaging Through The Atmosphere, J. C. Wyant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.75, 30–38 (1976).

Appl. Opt. (4)

J. Math. Phys. N.Y. (1)

R. J. Sasiela, J. D. Shelton, “Mellin transform methods applied to integral evaluation: Taylor series and asymptotic approximations,” J. Math. Phys. N.Y. 34, 2572–2619 (1993).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (11)

G. A. Tyler, “Merging: a new method for tomography through random media,” J. Opt. Soc. Am. A 10, 409–425 (1993).

M. C. Roggemann, C. L. Matson, “Power spectrum and Fourier phase spectrum estimation by using fully and partially compensating adaptive-optics and bispectrum postprocessing,” J. Opt. Soc. Am. A 9, 1525–1535 (1992).
[CrossRef]

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1589–1608 (1990).
[CrossRef]

R. Q. Fugate, B. L. Ellerbroek, C. H. Higgins, M. P. Jelonek, W. J. Lange, A. C. Slavin, W. J. Wild, D. M. Winker, J. M. Wynia, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, M. D. Oliker, D. W. Sindle, R. A. Cleis, “Two generations of laser-guide-star adaptive-optics experiments at the Starfire Optical Range,” J. Opt. Soc. Am. A 11, 310–314 (1994).
[CrossRef]

B. L. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric turbulence compensation in extended field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
[CrossRef]

S. E. Troxel, B. M. Welsh, M. C. Roggemann, “Off-axis optical transfer function calculations in an adaptive-optics system by means of a diffraction calculation for weak index fluctuations,” J. Opt. Soc. Am. A 11, 2100–2111 (1994).
[CrossRef]

P. Nisenson, R. Barakat, “Partial atmospheric correction with adaptive optics,” J. Opt. Soc. Am. A 4, 2249–2253 (1987).
[CrossRef]

R. C. Smithson, M. L. Peri, “Partial correction of astronomical images with active mirrors,” J. Opt. Soc. Am. A 6, 92–97 (1989).
[CrossRef]

B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive optics systems using slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
[CrossRef]

R. J. Sasiela, J. D. Shelton, “Transverse spectral filtering and Mellin-transform techniques applied to the effect of outer scale on tilt and tilt anisoplanatism,” J. Opt. Soc. Am. A 10, 646–660 (1993).
[CrossRef]

M. C. Roggemann, J. A. Meinhardt, “Image reconstruction by means of wave-front sensor measurements in closed-loop adaptive-optics systems,” J. Opt. Soc. Am. A 10, 1996–2007 (1993).
[CrossRef]

Opt. Eng. (1)

N. Roddier, “Atmospheric wave-front simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Opt. Lett. (1)

Other (15)

T. Goldring, L. Carlson, “Analysis and implementation of non-Kolmogorov phase screens appropriate to structured environments,” in Nonlinear Optical Beam Manipulation and High Energy Beam Propagation through the Atmosphere, R. A. Fisher, L. E. Wilson, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1060, 244–264 (1989).

A. Ishimaru, “The beam wave case and remote sensing,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978), Vol. 25, pp. 129–170.
[CrossRef]

R. J. Sasiela, “A unified approach to electromagnetic wave propagation in turbulence and the evaluation of multiparameter integrals,” Tech. Rep. TR 807 (Lincoln Laboratory, MIT, Cambridge, Mass., 1988).

IMSL “Math/library: special functions,” in IMSL Fortran Subroutines for Mathematical Applications (IMSL, 2500 Permian Tower, 2500 City West Boulevard, Houston, Tex. 77042-3020, 1991), pp. 97–142.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. XIX, pp. 281–376.
[CrossRef]

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

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

G. Cochran, “Phase screen generation,” Tech. Rep. TR-663 (Optical Sciences Company, Placentia, Calif., 1985).

M. Miller, P. Zieske, D. Hanson, “Characterization of atmospheric turbulence,” in Imaging Through The Atmosphere, J. C. Wyant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.75, 30–38 (1976).

R. E. Hufnagel, “Variations of atmospheric turbulence,” in Optical Propagation through Turbulence, OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1974), paper WA1.

M. C. Roggemann, B. L. Ellerbroek, T. A. Rhoadarmer, “Widening the effective field-of-view of adaptive-optics telescopes using deconvolution from wave-front sensing: average and signal-to-noise ratio performance,” Appl. Opt. (to be published).

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965).

G. Strang, Linear Algebra and its Applications (Academic, New York, 1980).

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes—The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

D. L. Fried, “Postdetection wave-front compensation,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 127–133 (1987).

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Figures (4)

Fig. 1
Fig. 1

Stack of three random time-sequential images in the case of atmospheric layers moving in the same direction and at the same speed (von Kármán spectrum): (a) surface plot of the phase screen, t = 0 s; (b) intensity image of (a); (c) surface plot of the phase screen, t = 0.01 s; (d) intensity image of (c); (e) surface plot of the phase screen, t = 0.02 s; (f) intensity image of (e).

Fig. 2
Fig. 2

Same as Fig. 1 but with the Kolmogorov spectrum.

Fig. 3
Fig. 3

Stack of three random time-sequential images in the case of atmospheric layers moving in different directions and at different speeds (von Kármán power spectrum): (a) surface plot of the phase screen, t = 0 s; (b) intensity image of (a); (c) surface plot of the phase screen, t = 0.01 s; (d) intensity image of (c); (e) surface plot of the phase screen, t = 0.02 s; (f) intensity image of (e).

Fig. 4
Fig. 4

Same as Fig. 3 but with the Kolmogorov power spectrum.

Tables (2)

Tables Icon

Table 1 Weights for a Four-Layer Atmospheric Modela

Tables Icon

Table 2 Wind Velocities of the Four Atmospheric Layers for the Phase Screens shown in Figs. 3 and 4

Equations (60)

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

Γ ( x 1 , t 1 ; x 2 , t 2 ) = ϕ ( x 1 , t 1 ) ϕ ( x 2 , t 2 ) ,
n ( x , z ; t ) = n 0 + n 1 ( x , z ; t ) ,
ψ ( x , t ) = k 0 h d z n 1 ( x , z ; t ) ,
ψ ( x , t ) = k 0 h d z n 1 ( x v ( z ) t , z ; 0 ) .
ϕ ( x , t ) = ψ ( x , t ) p d x W A ( x ) ψ ( x , t ) ,
d x W A ( x ) = 1 .
Γ ( x 1 , t 1 ; x 2 , t 2 ) = k 2 { 0 h d z 1 n 1 [ x 1 v ( z 1 ) t 1 , z 1 ; 0 ] p d x 0 h d z 1 W ( x ) n 1 [ x v ( z 1 ) t 1 , z 1 ; 0 ] } × { 0 h d z 2 n 1 [ x 2 v ( z 2 ) t 2 , z 2 ; 0 ] p d x 0 h d z 2 W ( x ) n 1 [ x v ( z 2 ) t 2 , z 2 ; 0 ] } .
0 h d z 1 0 h d z 2 n 1 [ x 1 v ( z 1 ) t 1 , z 2 ; 0 ] × n 1 [ x 2 v ( z 2 ) t 2 , z 2 ; 0 ] .
n 1 ( x , z ; t ) = i = 1 N n ˆ 1 ( x , z ; t ) δ ( z z i ) ,
0 h d z n 1 ( x , z ; t ) = i = 1 N n ˆ 1 ( x , z i ; t ) = i = 1 N n ˆ 1 [ x v ( z i ) t , z i ; 0 ] ,
ψ ( x , z i ; t ) = k n ˆ 1 ( x , z i ; t ) ,
Γ ( x 1 , t 1 ; x 2 , t 2 ) = i = 1 N j = 1 N ψ [ x 1 v ( z i ) t 1 , z i ; 0 ] ψ [ x 2 v ( z j ) t 2 , z j ; 0 ] p d x W ( x ) i = 1 N j = 1 N ψ [ x v ( z i ) t 1 , z i ; 0 ] × ψ [ x 2 v ( z i ) t 2 , z j ; 0 ] p d x W ( x ) i = 1 N j = 1 N ψ [ x 1 v ( z i ) t 1 , z i ; 0 ] × ψ [ x v ( z j ) t 2 , z j ; 0 ] + p 2 d x d x W ( x ) W ( x ) × i = 1 N j = 1 N ψ [ x v ( z i ) t 1 , z i ; 0 ] × ψ [ x v ( z j ) t 2 , z j ; 0 ] .
ψ ( x , z i , t ) ψ ( x , z j , t ) = 0 if i j ,
Γ ( x 1 , t 1 ; x 2 , t 2 ) = i = 1 N B ψ i [ | x 1 x 2 v ( z i ) ( t 1 t 2 ) | ] p d x W A ( x ) { i = 1 N B ψ i [ | x x 2 v ( z i ) ( t 1 t 2 ) | ] } p d x W A ( x ) { i = 1 N B ψ i [ | x 1 x v ( z i ) ( t 1 t 2 ) | ] } + p d x d x W A ( x ) W A ( x ) × { i = 1 N B ψ i [ | x x v ( z i ) ( t 1 t 2 ) | ] } ,
B ψ i [ | x 1 x 2 v ( z i ) ( t 1 t 2 ) | ] = ψ [ x 1 v ( z i ) t 1 , z i ; 0 ] ψ [ x 2 v ( z i ) t 2 , z i ; 0 ] .
Φ n V ( κ , z ) = 0 . 033 C n 2 ( z ) ( κ 2 + 4 π 2 / L 0 2 ) 11 / 6 ,
Φ n V ( κ , z ) = f ( z ) Φ 0 V ( κ ) ,
f ( z ) = 0 . 033 C n 2 ( z ) ,
Φ 0 V ( κ ) = 1 ( κ 2 + 4 π 2 / L 0 2 ) 11 / 6 .
B ψ ( ρ ) = ( 2 π ) 2 k 2 0 h d z 0 κ d κ J 0 ( κρ ) Φ n V ( κ , z ) = ( 2 π ) 2 k 2 0 h d z 0 κ d κ J 0 ( κρ ) Φ 0 V ( κ ) f ( z ) ,
C n 2 ( z ) = i = 1 N C ˆ n 2 ( z ) δ ( z z i ) ;
0 h d z f ( z ) = 0 . 033 0 h d z C n 2 ( z ) = 0 . 033 i = 1 N C ˆ n 2 ( z i ) .
B ψ ( ρ ) = 0 . 033 ( 2 π ) 2 k 2 [ i = 1 N C ˆ 2 2 ( z i ) 0 κ d κ J 0 ( κρ ) Φ 0 V ( κ ) ] .
B ψ i ( ρ ) = 0 . 033 ( 2 π ) 2 k 2 C ˆ n 2 ( z i ) 0 κ d κ J 0 ( κρ ) Φ 0 V ( κ ) .
Γ V ( x 1 , t 1 ; x 2 , t 2 ) = 0 . 033 ( 2 π ) 2 k 2 { i = 1 N C ˆ n 2 ( z i ) 0 κ d κ J 0 × [ κ | x 1 x 2 v ( z i ) ( t 1 t 2 ) | ] Φ 0 V ( κ ) } .
0 κ d κ J 0 ( κρ ) ( κ 2 + 4 π 2 / L 0 2 ) 11 / 6 = ( L 0 / 2 π ) 5 / 6 K 5 / 6 ( 2 πρ / L 0 ) ρ 5 / 6 2 5 / 6 Γ ( 11 / 6 ) ,
Φ n K ( κ , z ) = 0 . 033 C n 2 ( z ) κ 11 / 3 .
Φ n K ( κ , z ) = f ( z ) Φ 0 K ( κ ) ,
Φ 0 K ( κ ) = κ 11 / 3 .
Γ K ( x 1 , t 1 ; x 2 , t 2 ) = i = 1 N d x d x W A ( x ) W A ( x ) × { B ψ i [ | x 1 x 2 v ( z i ) ( t 1 t 2 ) | ] B ψ i [ | x x 2 v ( z i ) ( t 1 t 2 ) | ] B ψ i [ | x 1 x v ( z i ) ( t 1 t 2 ) | ] + B ψ i [ | x x v ( z i ) ( t 1 t 2 ) | ] } .
Γ K ( x 1 , t 1 ; x 2 , t 2 ) = 1 2 i = 1 N d x d x W A ( x ) W A ( x ) × { D ψ i [ | x 1 x 2 v ( z i ) ( t 1 t 2 ) | ] D ψ i [ | x x 2 v ( z i ) ( t 1 t 2 ) | ] D ψ i [ | x 1 x v ( z i ) ( t 1 t 2 ) | ] + D ψ i [ | x x v ( z i ) ( t 1 t 2 ) | ] } ,
D ψ i ( ρ ) = [ ψ i ( x ) ψ i ( x + ρ ) ] 2 = 6 . 88 ( ρ r 0 i ) 5 / 3 ,
Γ K ( x 1 , t 1 ; x 2 , t 2 ) = 6 . 88 i = 1 N r 0 i 5 / 3 [ 1 2 | x 1 x 2 v ( z i ) ( t 1 t 2 ) | 5 / 3 + 1 2 d x W A ( x ) | x x 2 v ( z i ) ( t 1 t 2 ) | 5 / 3 + 1 2 d x W A ( x ) | x 1 x v ( z i ) ( t 1 t 2 ) | 5 / 3 1 2 d x d x W A ( x ) W A ( x ) × | x x v ( z i ) ( t 1 t 2 ) | 5 / 3 ] ,
ψ ( x , θ ) = k 0 h d z n 1 ( x + z θ , z ) .
ϕ ( x , θ ) = ψ ( x , θ ) p d x W A ( x ) ψ ( x , θ ) ,
Γ ( x 1 , θ 1 ; x 2 , θ 2 ) = ϕ ( x 1 , θ 1 ) ϕ ( x 2 , θ 2 ) .
Γ ( x 1 , θ 1 ; x 2 , θ 2 ) = i = 1 N B ψ i [ | x 1 x 2 + z i ( θ 1 θ 2 ) | ] p d x W A ( x ) { i = 1 N B ψ i [ | x x 2 + z i ( θ 1 θ 2 ) | ] } p d x W A ( x ) { i = 1 N B ψ i [ | x 1 x + z i ( θ 1 θ 2 ) | ] } + p d x d x W A ( x ) W A ( x ) × { i = 1 N B ψ i [ | x x + z i ( θ 1 θ 2 ) | ] } .
B ψ i [ | x 1 x 2 + z ( θ 1 θ 2 ) | ] = ψ i ( x 1 , θ 1 ) ψ i ( x 2 , θ 2 ) .
0 h z m C n 2 ( z ) d z = i = 1 4 z i m C ˆ n 2 ( z i ) ,
W i = C ˆ n 2 ( z i ) I C ,
r 0 i = 0 . 185 [ λ 2 C ˆ n 2 ( z i ) ] 3 / 5 = 0 . 185 ( λ 2 W i I C ) 3 / 5 ,
r 0 5 / 3 = i = 1 4 r 0 i 5 / 3 .
C ˆ n , new 2 ( z i ) = a C ˆ n , old 2 ( z i ) ,
r 0 i , new = 0 . 185 ( λ 2 a W i I C ) 3 / 5 = a 3 / 5 r 0 i , old ,
a = ( r 0 i , new r 0 i , old ) 5 / 3 .
Γ new V , K = a Γ old V , K ,
Γ = R R T ,
αα T = I M ,
ϕ = R α .
ϕ ϕ T = R α ( R α ) T = R αα T R T .
R αα T R T = R αα T R T = R I M R T = R R T = Γ ,
Γ sim = ϕ ϕ T ,
2 = ( Γ V Γ sim ) 2 N arr 2 ,
Γ K ( x 1 , t 1 ; x 2 , t 2 ) = 6 . 88 i = 1 N r 0 i 5 / 3 × [ 1 2 | x 1 x 2 v ( z i ) ( t 1 t 2 ) | 5 / 3 + 1 2 d x W A ( x ) | x x 2 v ( z i ) ( t 1 t 2 ) | 5 / 3 + 1 2 d x W A ( x ) | x 1 x v ( z i ) ( t 1 t 2 ) | 5 / 3 1 2 d x d x W A ( x ) W A ( x ) × | x x v ( z i ) ( t 1 t 2 ) | 5 / 3 ] .
Γ K ( x 1 , t 1 ; x 2 , t 2 ) = 6 . 88 i = 1 N r 0 i 5 / 3 × { 1 2 | x 1 x 2 v ( z i ) ( t 1 t 2 ) | 5 / 3 + 1 2 R 5 / 3 F 1 [ x 2 R + v ( z i ) R ( t 1 t 2 ) ] + 1 2 R 5 / 3 F 1 [ x 1 R + v ( z i ) R ( t 1 t 2 ) ] 1 2 R 5 / 3 F 2 [ v ( z i ) R ( t 1 t 2 ) ] } ,
F 1 ( Ω ) = { 6 11 F 2 1 ( 11 6 , 5 6 ; 1 ; | Ω | 2 ) | Ω | 1 | Ω | 5 / 3 F 2 1 ( 5 6 , 5 6 ; 2 ; | Ω | 2 ) | Ω | 1 ,
2 F 1 ( a , b ; c ; z ) = Γ ( c ) Γ ( a ) Γ ( b ) n = 0 Γ ( a + n ) Γ ( b + n ) Γ ( c + n ) z n n ! ,
F 2 ( Ω ) = 2 π 0 1 K 1 ( ρ ) H ( ρ , Ω ) d ρ ,
K 1 ( ρ ) = { π | ρ | 0 2 cos 1 ( | ρ | 2 ) | ρ | [ 1 ( | ρ | 2 ) 2 ] 1 / 2 0 < | ρ | < 2 , 0 otherwise
H ( ρ , Ω ) = { | ρ | 8 / 3 2 F 1 ( 5 6 , 5 6 ; 1 ; | Ω | 2 | ρ | 2 ) | Ω | | ρ | | ρ | | Ω | 5 / 3 2 F 1 ( 5 6 , 5 6 ; 1 ; | ρ | 2 | Ω | 2 ) | Ω | | ρ | .

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