Abstract

The hybrid matrix operator, Monte Carlo (HMOMC) method previously reported [Appl. Opt. 47, 1063–1071 (2008)] is improved by neglecting higher-order terms in the coupling of the matrix operators and by introducing a dual grid scheme. The computational efficiency for solving the vector radiative transfer equation in a full 3D coupled atmosphere–surface–ocean system is substantially improved, and, thus, large-scale simulations of the radiance distribution become feasible. The improved method is applied to the computation of the polarized radiance field under realistic surface waves simulated by the power spectral density method. To the authors’ best knowledge, this is the first time that the polarized radiance field under a dynamic ocean surface and the underwater image of an object above such an ocean surface have been reported.

© 2009 Optical Society of America

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2009 (1)

2008 (4)

D. E. Bates and J. N. Porter, “AO3D: a Monte Carlo code for modeling of environmental light propagation,” J. Quant. Spectrosc. Radiat. Transfer 109, 1802-1814 (2008).
[CrossRef]

J. Hedley, “A three-dimensional radiative transfer model for shallow water environments,” Opt. Express 16, 21887-21902(2008).
[CrossRef] [PubMed]

P.-W. Zhai, G. W. Kattawar, and P. Yang, “Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. II. The hybrid matrix operator-Monte Carlo method,” Appl. Opt. 47, 1063-1071(2008).
[CrossRef] [PubMed]

P.-W. Zhai, G. W. Kattawar, and P. Yang, “Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. I. Monte Carlo method,” Appl. Opt. 47, 1037-1047 (2008).
[CrossRef] [PubMed]

2006 (1)

2005 (2)

C. Davis, C. Emde, and R. Harwood, “A 3D polarized reversed Monte Carlo radiative transfer model for millimeter and submillimeter passive remote sensing in cloudy atmospheres,” IEEE Trans. Geosci. Remote Sensing 43, 1096-1101 (2005).
[CrossRef]

Y. Chen, K. N. Liou, and Y. Gu, “An efficient diffusion approximation for 3D radiative transfer parameterization: application to cloudy atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 92, 189-200 (2005).
[CrossRef]

2004 (2)

C. Emde, S. A. Buehler, C. Davis, P. Eriksson, T. R. Sreerekha, and C. Teichmann, “A polarized discrete ordinate scattering model for simulations of limb and nadir long-wave measurements in 1-D/3-D spherical atmospheres,” J. Geophys. Res. 109, D24207 (2004).
[CrossRef]

P. N. Reinersman and K. L. Carder, “Hybrid numerical method for solution of the radiative transfer equation in one, two, or three dimensions,” Appl. Opt. 43, 2734-2743 (2004).
[CrossRef] [PubMed]

2003 (1)

F. Schwenger and E. Repasi, “Sea surface simulation for testing of multiband imaging sensors,” Proc. SPIE 5075, 72-84 (2003).
[CrossRef]

2001 (1)

2000 (1)

K. Stamnes, S.-C. Tsay, W. Wiscombe, and I. Laszlo, “DISORT, a general-purpose Fortran program for discrete-ordinate-method radiative transfer in scattering and emitting layered media: documentation of methodology,” ftp://climate1.gsfc.nasa.gov/wiscombe/Multiple_Scatt/DISORTReport1.1.pdf (2000).

1998 (2)

K. F. Evans, “The spherical harmonic discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429-446 (1998).
[CrossRef]

D. H. Tofsted and S. G. O'Brien, “Physics-based visualization of dense natural clouds. I. Three-dimensional discrete ordinates radiative transfer,” Appl. Opt. 37, 7718-7728(1998).
[CrossRef]

1997 (1)

J. L. Haferman, T. F. Smith, and W. F. Krajewski, “A multi-dimensional dicrete-ordinates method for polarized radiative transfer. Part I: validation for randomly oriented axisymmetric particles,” J. Quant. Spectrosc. Radiat. Transfer 58, 379-398 (1997).
[CrossRef]

1996 (1)

1994 (1)

A. Sánchez, T. F. Smith, and W. F. Krajewski, “A three-dimensional atmospheric radiative transfer model based on the discrete-ordinates method,” Atmos. Res. 33, 283-308(1994).
[CrossRef]

1993 (1)

1991 (1)

L. G. Stenholm, H. Störzer, and R. Wehrse, “An efficient method for the solution of 3D radiative transfer problems,” J. Quant. Spectrosc. Radiat. Transfer 45, 47-56 (1991).
[CrossRef]

1988 (1)

1984 (2)

1973 (2)

1967 (1)

S. Twomey, H. Jacobowitz, and H. B. Howell, “Light scattering by cloud layers,” J. Atmos. Sci. 24, 70-79 (1967).
[CrossRef]

1966 (1)

S. Twomey, H. Jacobowitz, and H. B. Howell, “Matrix methods for multiple-scattering problems,” J. Atmos. Sci. 23, 289-298 (1966).
[CrossRef]

1954 (1)

1941 (1)

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Bates, D. E.

D. E. Bates and J. N. Porter, “AO3D: a Monte Carlo code for modeling of environmental light propagation,” J. Quant. Spectrosc. Radiat. Transfer 109, 1802-1814 (2008).
[CrossRef]

Buehler, S. A.

C. Emde, S. A. Buehler, C. Davis, P. Eriksson, T. R. Sreerekha, and C. Teichmann, “A polarized discrete ordinate scattering model for simulations of limb and nadir long-wave measurements in 1-D/3-D spherical atmospheres,” J. Geophys. Res. 109, D24207 (2004).
[CrossRef]

Cairns, B.

Carder, K. L.

Catchings, F. E.

Chami, M.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Chang, G.

T. Dickey and G. Chang, “The Radiance in a Dynamic Ocean (RaDyO) Program,” presented at Ocean Optics XVIII, Montreal, Canada, 9-13 Oct. 2006, http://www.opl.ucsb.edu/radyo/docs/DickeyOOPoster06.pdf.

Chen, Y.

Y. Chen, K. N. Liou, and Y. Gu, “An efficient diffusion approximation for 3D radiative transfer parameterization: application to cloudy atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 92, 189-200 (2005).
[CrossRef]

Chowdhary, J.

Cox, C.

Davis, A. B.

A. Marshak and A. B. Davis, eds., 3D Radiative Transfer in Cloudy Atmospheres (Springer, 2005).
[CrossRef]

Davis, C.

C. Davis, C. Emde, and R. Harwood, “A 3D polarized reversed Monte Carlo radiative transfer model for millimeter and submillimeter passive remote sensing in cloudy atmospheres,” IEEE Trans. Geosci. Remote Sensing 43, 1096-1101 (2005).
[CrossRef]

C. Emde, S. A. Buehler, C. Davis, P. Eriksson, T. R. Sreerekha, and C. Teichmann, “A polarized discrete ordinate scattering model for simulations of limb and nadir long-wave measurements in 1-D/3-D spherical atmospheres,” J. Geophys. Res. 109, D24207 (2004).
[CrossRef]

Dickey, T.

T. Dickey and G. Chang, “The Radiance in a Dynamic Ocean (RaDyO) Program,” presented at Ocean Optics XVIII, Montreal, Canada, 9-13 Oct. 2006, http://www.opl.ucsb.edu/radyo/docs/DickeyOOPoster06.pdf.

Dilligeard, E.

Emde, C.

C. Davis, C. Emde, and R. Harwood, “A 3D polarized reversed Monte Carlo radiative transfer model for millimeter and submillimeter passive remote sensing in cloudy atmospheres,” IEEE Trans. Geosci. Remote Sensing 43, 1096-1101 (2005).
[CrossRef]

C. Emde, S. A. Buehler, C. Davis, P. Eriksson, T. R. Sreerekha, and C. Teichmann, “A polarized discrete ordinate scattering model for simulations of limb and nadir long-wave measurements in 1-D/3-D spherical atmospheres,” J. Geophys. Res. 109, D24207 (2004).
[CrossRef]

Eriksson, P.

C. Emde, S. A. Buehler, C. Davis, P. Eriksson, T. R. Sreerekha, and C. Teichmann, “A polarized discrete ordinate scattering model for simulations of limb and nadir long-wave measurements in 1-D/3-D spherical atmospheres,” J. Geophys. Res. 109, D24207 (2004).
[CrossRef]

Evans, K. F.

K. F. Evans, “The spherical harmonic discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429-446 (1998).
[CrossRef]

Fischer, J.

Fry, E. S.

Gentili, B.

Gordon, H. R.

Grassl, H.

Greenstein, J. L.

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Gu, Y.

Y. Chen, K. N. Liou, and Y. Gu, “An efficient diffusion approximation for 3D radiative transfer parameterization: application to cloudy atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 92, 189-200 (2005).
[CrossRef]

Haferman, J. L.

J. L. Haferman, T. F. Smith, and W. F. Krajewski, “A multi-dimensional dicrete-ordinates method for polarized radiative transfer. Part I: validation for randomly oriented axisymmetric particles,” J. Quant. Spectrosc. Radiat. Transfer 58, 379-398 (1997).
[CrossRef]

Harwood, R.

C. Davis, C. Emde, and R. Harwood, “A 3D polarized reversed Monte Carlo radiative transfer model for millimeter and submillimeter passive remote sensing in cloudy atmospheres,” IEEE Trans. Geosci. Remote Sensing 43, 1096-1101 (2005).
[CrossRef]

Hedley, J.

Henyey, L. C.

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Howell, H. B.

S. Twomey, H. Jacobowitz, and H. B. Howell, “Light scattering by cloud layers,” J. Atmos. Sci. 24, 70-79 (1967).
[CrossRef]

S. Twomey, H. Jacobowitz, and H. B. Howell, “Matrix methods for multiple-scattering problems,” J. Atmos. Sci. 23, 289-298 (1966).
[CrossRef]

Hu, Y.

Jacobowitz, H.

S. Twomey, H. Jacobowitz, and H. B. Howell, “Light scattering by cloud layers,” J. Atmos. Sci. 24, 70-79 (1967).
[CrossRef]

S. Twomey, H. Jacobowitz, and H. B. Howell, “Matrix methods for multiple-scattering problems,” J. Atmos. Sci. 23, 289-298 (1966).
[CrossRef]

Jayaweera, K.

Jin, Z.

Kattawar, G. W.

Krajewski, W. F.

J. L. Haferman, T. F. Smith, and W. F. Krajewski, “A multi-dimensional dicrete-ordinates method for polarized radiative transfer. Part I: validation for randomly oriented axisymmetric particles,” J. Quant. Spectrosc. Radiat. Transfer 58, 379-398 (1997).
[CrossRef]

A. Sánchez, T. F. Smith, and W. F. Krajewski, “A three-dimensional atmospheric radiative transfer model based on the discrete-ordinates method,” Atmos. Res. 33, 283-308(1994).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, 2006).

Laszlo, I.

K. Stamnes, S.-C. Tsay, W. Wiscombe, and I. Laszlo, “DISORT, a general-purpose Fortran program for discrete-ordinate-method radiative transfer in scattering and emitting layered media: documentation of methodology,” ftp://climate1.gsfc.nasa.gov/wiscombe/Multiple_Scatt/DISORTReport1.1.pdf (2000).

Liou, K. N.

Y. Chen, K. N. Liou, and Y. Gu, “An efficient diffusion approximation for 3D radiative transfer parameterization: application to cloudy atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 92, 189-200 (2005).
[CrossRef]

Liu, Q.

Lucker, P. L.

Marshak, A.

A. Marshak and A. B. Davis, eds., 3D Radiative Transfer in Cloudy Atmospheres (Springer, 2005).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge U. Press, 2006).

Mobley, C. D.

Morel, A.

Munk, W.

O'Brien, S. G.

Petzold, T. J.

T. J. Petzold, “Volume scattering functions for selected ocean waters,” Technical Report SIO 72-78 (Scripps Institution of Oceanography,1972).

Plass, G. N.

Porter, J. N.

D. E. Bates and J. N. Porter, “AO3D: a Monte Carlo code for modeling of environmental light propagation,” J. Quant. Spectrosc. Radiat. Transfer 109, 1802-1814 (2008).
[CrossRef]

Reinersman, P.

Reinersman, P. N.

Repasi, E.

F. Schwenger and E. Repasi, “Sea surface simulation for testing of multiband imaging sensors,” Proc. SPIE 5075, 72-84 (2003).
[CrossRef]

Ruprecht, E.

Sánchez, A.

A. Sánchez, T. F. Smith, and W. F. Krajewski, “A three-dimensional atmospheric radiative transfer model based on the discrete-ordinates method,” Atmos. Res. 33, 283-308(1994).
[CrossRef]

Santer, R.

Schwenger, F.

F. Schwenger and E. Repasi, “Sea surface simulation for testing of multiband imaging sensors,” Proc. SPIE 5075, 72-84 (2003).
[CrossRef]

Smith, T. F.

J. L. Haferman, T. F. Smith, and W. F. Krajewski, “A multi-dimensional dicrete-ordinates method for polarized radiative transfer. Part I: validation for randomly oriented axisymmetric particles,” J. Quant. Spectrosc. Radiat. Transfer 58, 379-398 (1997).
[CrossRef]

A. Sánchez, T. F. Smith, and W. F. Krajewski, “A three-dimensional atmospheric radiative transfer model based on the discrete-ordinates method,” Atmos. Res. 33, 283-308(1994).
[CrossRef]

Sreerekha, T. R.

C. Emde, S. A. Buehler, C. Davis, P. Eriksson, T. R. Sreerekha, and C. Teichmann, “A polarized discrete ordinate scattering model for simulations of limb and nadir long-wave measurements in 1-D/3-D spherical atmospheres,” J. Geophys. Res. 109, D24207 (2004).
[CrossRef]

Stamnes, K.

Stavn, R. H.

Stenholm, L. G.

L. G. Stenholm, H. Störzer, and R. Wehrse, “An efficient method for the solution of 3D radiative transfer problems,” J. Quant. Spectrosc. Radiat. Transfer 45, 47-56 (1991).
[CrossRef]

Störzer, H.

L. G. Stenholm, H. Störzer, and R. Wehrse, “An efficient method for the solution of 3D radiative transfer problems,” J. Quant. Spectrosc. Radiat. Transfer 45, 47-56 (1991).
[CrossRef]

Teichmann, C.

C. Emde, S. A. Buehler, C. Davis, P. Eriksson, T. R. Sreerekha, and C. Teichmann, “A polarized discrete ordinate scattering model for simulations of limb and nadir long-wave measurements in 1-D/3-D spherical atmospheres,” J. Geophys. Res. 109, D24207 (2004).
[CrossRef]

Tofsted, D. H.

Travis, L. D.

Trepte, C. R.

Tsay, S.-C.

K. Stamnes, S.-C. Tsay, W. Wiscombe, and I. Laszlo, “DISORT, a general-purpose Fortran program for discrete-ordinate-method radiative transfer in scattering and emitting layered media: documentation of methodology,” ftp://climate1.gsfc.nasa.gov/wiscombe/Multiple_Scatt/DISORTReport1.1.pdf (2000).

K. Stamnes, S.-C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502-2509 (1988).
[CrossRef] [PubMed]

Twomey, S.

S. Twomey, H. Jacobowitz, and H. B. Howell, “Light scattering by cloud layers,” J. Atmos. Sci. 24, 70-79 (1967).
[CrossRef]

S. Twomey, H. Jacobowitz, and H. B. Howell, “Matrix methods for multiple-scattering problems,” J. Atmos. Sci. 23, 289-298 (1966).
[CrossRef]

Voss, K. J.

Wehrse, R.

L. G. Stenholm, H. Störzer, and R. Wehrse, “An efficient method for the solution of 3D radiative transfer problems,” J. Quant. Spectrosc. Radiat. Transfer 45, 47-56 (1991).
[CrossRef]

Wiscombe, W.

K. Stamnes, S.-C. Tsay, W. Wiscombe, and I. Laszlo, “DISORT, a general-purpose Fortran program for discrete-ordinate-method radiative transfer in scattering and emitting layered media: documentation of methodology,” ftp://climate1.gsfc.nasa.gov/wiscombe/Multiple_Scatt/DISORTReport1.1.pdf (2000).

K. Stamnes, S.-C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502-2509 (1988).
[CrossRef] [PubMed]

Yang, P.

P.-W. Zhai, G. W. Kattawar, and P. Yang, “Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. I. Monte Carlo method,” Appl. Opt. 47, 1037-1047 (2008).
[CrossRef] [PubMed]

P.-W. Zhai, G. W. Kattawar, and P. Yang, “Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. II. The hybrid matrix operator-Monte Carlo method,” Appl. Opt. 47, 1063-1071(2008).
[CrossRef] [PubMed]

Zhai, P.

Zhai, P.-W.

P.-W. Zhai, G. W. Kattawar, and P. Yang, “Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. II. The hybrid matrix operator-Monte Carlo method,” Appl. Opt. 47, 1063-1071(2008).
[CrossRef] [PubMed]

P.-W. Zhai, G. W. Kattawar, and P. Yang, “Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. I. Monte Carlo method,” Appl. Opt. 47, 1037-1047 (2008).
[CrossRef] [PubMed]

Appl. Opt. (13)

G. N. Plass, G. W. Kattawar, and F. E. Catchings, “Matrix operator theory of radiative transfer. 1: Rayleigh scattering,” Appl. Opt. 12, 314-329 (1973).
[CrossRef] [PubMed]

G. W. Kattawar, G. N. Plass, and F. E. Catchings, “Matrix operator theory of radiative transfer. 2: Scattering from maritime haze,” Appl. Opt. 12, 1071-1084 (1973).
[CrossRef] [PubMed]

K. Stamnes, S.-C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502-2509 (1988).
[CrossRef] [PubMed]

M. Chami, R. Santer, and E. Dilligeard, “Radiative transfer model for the computation of radiance and polarization in an ocean-atmosphere system: polarization properties of suspended matter for remote sensing,” Appl. Opt. 40, 2398-2416 (2001).
[CrossRef]

J. Chowdhary, B. Cairns, and L. D. Travis, “Contribution of water-leaving radiances to multiangle, multispectral polarimetric observations over the open ocean: bio-optical model results for case 1 waters,” Appl. Opt. 45, 5542-5567 (2006).
[CrossRef] [PubMed]

D. H. Tofsted and S. G. O'Brien, “Physics-based visualization of dense natural clouds. I. Three-dimensional discrete ordinates radiative transfer,” Appl. Opt. 37, 7718-7728(1998).
[CrossRef]

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Supplementary Material (8)

» Media 1: MPG (1280 KB)     
» Media 2: MPG (852 KB)     
» Media 3: MPG (1027 KB)     
» Media 4: MPG (596 KB)     
» Media 5: MPG (684 KB)     
» Media 6: MPG (962 KB)     
» Media 7: MPG (684 KB)     
» Media 8: MPG (484 KB)     

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Figures (12)

Fig. 1
Fig. 1

Schematic configurations of the discretizations in the horizontal dimension for the (a) single grid and (b) dual grid schemes. Starting at the top, the three planes consecutively represent the atmosphere, surface, and ocean layers.

Fig. 2
Fig. 2

Distributions of underwater downwelling (a) radiance and (b) ratio of Stokes parameters Q / I for a CAO system with a flat ocean surface computed from both single grid and dual grid models. The radiance is normalized to unit incident solar irradiance. Results from a direct Monte Carlo computation are included for comparison. The optical thickness between the surface and the detector is τ det = 1 . See text for specifics of the scattering system.

Fig. 3
Fig. 3

Same as Fig. 2 except that a 1D cosine surface wave is present. The detector is at x = 2.33 m , y = 0 m and the same level below the ocean surface. The scattering azimuthal angle is ϕ = 0 ° .

Fig. 4
Fig. 4

Model generated gravity waves for wind speeds (a) v = 5 m / s and (b) v = 10 m / s at time t = 0 . The simulation domain shown here is 21 m × 21 m . The height distribution is calculated for 255 × 255 regularly distributed points.

Fig. 5
Fig. 5

Normalized downward radiance distributions just below the ocean surface ( τ det = 0.001 ) when wind driven surface waves shown in Fig. 4 are present. The atmosphere and ocean are the same as that for Fig. 2, and the sun is at the zenith θ s = 0 ° . Symbols are averaged HMOMC results using the gravity wave fields; lines are Monte Carlo results using the Cox–Munk statistical wave model. Results corresponding to a flat ocean surface are also shown for comparison.

Fig. 6
Fig. 6

(a) Snapshot from a time series (Media 1) of gravity waves modulated by swells, modeled by the PSD approach; (b) locations of underwater detectors (circles) and nine of them for which the radiation fields will be shown (solid circles).

Fig. 7
Fig. 7

Snapshot from a time series (Media 2) of the angular distributions of the normalized downwelling radiance in units of W sr 1 m 2 at nine detectors just below the ocean surface with τ det = 0.001 , when the surface waves shown in Fig. 6a (Media 1) is present. The locations of the nine detectors in the horizontal dimensions are as illustrated in Fig. 6b.

Fig. 8
Fig. 8

Snapshots from time series of the angular distributions of downwelling Stokes vector components (a)  Q / I (Media 3) and (b)  U / I (Media 4) measured by the same array of detectors as for Fig. 7.

Fig. 9
Fig. 9

(Media 5) Same as Fig. 7 except that the detectors are halfway within the ocean layer at τ det = 5 .

Fig. 10
Fig. 10

(Media 6 and 7, Media 6 and 7) Same as Fig. 8 except that the detectors are halfway within the ocean layer at τ det = 5 .

Fig. 11
Fig. 11

(a) Illustration of the implementation of a disk with radius r = 3.5 m placed just above the ocean surface in the CASO system. (b) The simulated angular distribution of the normalized downwelling radiance in units of W sr 1 m 2 observed by a detector at the center of the computational domain at τ det = 2 below the surface. The mapping from the downwelling hemisphere to a round disk is the same as that for Fig. 7.

Fig. 12
Fig. 12

Same as Fig. 11b, but shows a snapshot from a time series (Media 8) of downwelling radiance distributions when the wave field shown in Fig. 6 (Media 1) is present.

Equations (18)

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D 0 d = D 2 d · T 12 · T 01 + D 2 d · T 12 · R 10 · R 12 · T 01 + D 2 d · R 21 · R 23 · T 12 · T 01 + D 2 d · R 21 · R 23 · T 12 · R 10 · R 12 · T 01 + D 2 d · T 12 · R 10 · R 12 · R 10 · R 12 · T 01 + D 2 d · R 21 · R 23 · R 21 · R 23 · T 12 · T 01 + D 2 d · T 12 · R 10 · T 21 · R 23 · T 12 · T 01 + ,
D 0 d = D 2 d · ( I + R 21 · R 23 ) · T 12 · ( I + R 10 · R 12 + ( R 10 · R 12 ) 2 + R 10 · T 21 · R 23 · T 12 ) · T 01 ,
R 12 = R 12 ( 0 ) + R 12 ( wave ) .
T 01 , eff = ( I + R 10 · R 12 ( 0 ) + ( R 10 · R 12 ( 0 ) ) 2 + R 10 · T 21 ( 0 ) · R 23 · T 12 ( 0 ) ) · T 01 ,
D 0 d = D 2 d · T 12 · T 01 , eff + D 2 d · R 21 · R 23 · T 12 · T 01 , eff ,
D 2 d = D 2 d ( 0 ) + D 2 d diff ,
D 0 d = D 2 d ( 0 ) · T 12 · T 01 , eff + D 2 d diff · T 12 · T 01 , eff + D 2 d ( 0 ) · R 21 · R 23 · T 12 · T 01 , eff + D 2 d diff · R 21 · R 23 · T 12 · T 01 , eff .
o , p = 1 N D 2 d diff ( r m ( r ) ; r o ( i ) ) · T 12 ( r o ( r ) ; r p ( i ) ) · T 01 ( r p ( r ) ; r l ( i ) ) ,
o , p = 1 N [ D 2 d diff ( r m ( r ) ; r o ( i ) ) · o 1 , p 1 = 1 N [ T 12 ( r [ o , o 1 ] ( r ) ; r [ p , p 1 ] ( i ) ) · T 01 ( r p ( r ) ; r l ( i ) ) ] ] ,
P ( μ , g ) = 1 g 2 ( 1 2 g μ + g 2 ) 3 / 2 ,
g = 1 4 π 4 π P ( θ ) cos ( θ ) d Ω .
Z ( r , t ) = k Z ( k ) e i ( k · r + ω ( k ) t + Φ k ) ,
ω ( k ) = G · k ,
Z g ( k ) = S z , g ( k ) cos 2 Θ ,
S z , g ( k ) = 0.00506 π k 4.5 e 2 G / k v 2 ,
Z s ( k ) = S z , s ( k ) ,
S z , s ( k ) = A 2 2 π σ k 2 e [ ( k k p ) · ( k k p ) ] / 2 σ k 2
σ k = 0.8494 × FWHM × k p G .

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