Abstract

We report on the investigation and successful application of vertical profiling of the structure parameter Cm2, and of the outer scale L0 of absorption fluctuations and of the horizontal wind velocity (vector) during daytime by the analysis of solar transmission measurements. The method is relatively simple and straightforward so that the presented (or a similar) technique could be used in the routine remote sensing of daytime Cm2, L0, and wind profiles. It requires multiple beams pointing in different directions at the Sun. The retrieved profiles are consistent with the current knowledge of atmospheric physics. Simultaneous in situ wind velocity measurements agree with the retrieved wind velocity in the lowest 100 m above ground within the measurement uncertainties of less than ±2 m/s. The derived values of Cm2 at 200 m above ground are in good agreement (within a factor of 1.5) with the findings of an earlier investigation at the same test site. Finally, it is shown that irradiance fluctuations of millimeter and submillimeter waves are dominantly affected by humidity fluctuations, even at a dry and elevated site.

© 2005 Optical Society of America

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References

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    [CrossRef]
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  30. M. C. Wiedner, R. E. Hills, J. E. Carlstrom, O. P. Lay, “Interferometric phase correction using 183 GHz water vapor monitors,” Astrophys. J. 553, 1036–1041 (2001).
    [CrossRef]

2003 (1)

B. G. Kutuza, “Spatial and temporal fluctuations of atmospheric microwave emission,” Radio Sci. 38, 12.1–12.7 (2003), doi:.
[CrossRef]

2002 (2)

A. Lüdi, A. Magun, “Near horizontal line-of-sight mm-wave propagation measurements for the determination of outer length scales and anisotropy of turbulent refractive-index fluctuations in the lower troposphere,” Radio Sci. 37, 12.1–12.19 (2002), doi:.
[CrossRef]

J. E. R. Costa, A. V. R. Silva, A. Lüdi, A. Magun, “Beam profile determination by tomography of solar scans,” Astron. Astrophys. 387, 1153–1160 (2002), doi:.
[CrossRef]

2001 (2)

R. Avila, J. Vernin, L. J. Sáchez, “Atmospheric turbulence and wind profiles monitoring with generalized scidar,” Astron. Astrophys. 369, 364–372 (2001).
[CrossRef]

M. C. Wiedner, R. E. Hills, J. E. Carlstrom, O. P. Lay, “Interferometric phase correction using 183 GHz water vapor monitors,” Astrophys. J. 553, 1036–1041 (2001).
[CrossRef]

1998 (1)

J. P. Nieveen, A. E. Green, W. Kohsiek, “Using a large-aperture scintillometer to measure absorption and refractive index fluctuations,” Boundary-Layer Meteorol. 87, 101–116 (1998).
[CrossRef]

1994 (1)

A. Ziad, J. Borgnino, F. Martin, A. Agabi, “Experimental estimation of the spatial-coherence outer scale from a wavefront statistical analysis,” Astron. Astrophys. 282, 1021–1033 (1994).

1991 (1)

N. S. Nightingale, D. F. Buscher, “Interferometric seeing measurements at the La Palma Observatory,” Mon. Not. R. Astron. Soc. 251, 155–166 (1991).

1987 (1)

1977 (1)

1976 (1)

1974 (2)

1973 (1)

1972 (1)

1970 (1)

L. C. Shen, “Remote probing of atmosphere and wind velocity by millimeter waves,” IEEE Trans. Antennas Propag. AP-18, 493–497 (1970).
[CrossRef]

1969 (1)

R. W. Lee, J. C. Harp, “Weak scattering in random media, with applications to remote probing,” Proc. IEEE 57, 375–406 (1969).
[CrossRef]

1968 (1)

A. S. Gurvich, “Effect of absorption on the fluctuation in signal level during atmospheric propagation,” Radio Eng. Electron. Phys. 13, 1687–1694 (1968).

1966 (1)

R. W. Lee, A. T. Waterman, “A large antenna array for millimeter wave propagation studies,” Proc. IEEE 54, 454–458 (1966).
[CrossRef]

Agabi, A.

A. Ziad, J. Borgnino, F. Martin, A. Agabi, “Experimental estimation of the spatial-coherence outer scale from a wavefront statistical analysis,” Astron. Astrophys. 282, 1021–1033 (1994).

Arzner, K.

P. Kaufmann, J. E. R. Costa, E. Correia, A. Magun, K. Arzner, N. Kämpfer, M. Rovira, H. Levato, “Recent developments of the solar submm-wave telescope (SST),” in Coronal Physics from Radio and Space Observations, Lecture Notes in Physics, G. Trottet, ed. (Springer, Berlin, 1997), pp. 202–206.
[CrossRef]

Avila, R.

R. Avila, J. Vernin, L. J. Sáchez, “Atmospheric turbulence and wind profiles monitoring with generalized scidar,” Astron. Astrophys. 369, 364–372 (2001).
[CrossRef]

Azouit, M.

Borgnino, J.

A. Ziad, J. Borgnino, F. Martin, A. Agabi, “Experimental estimation of the spatial-coherence outer scale from a wavefront statistical analysis,” Astron. Astrophys. 282, 1021–1033 (1994).

Buscher, D. F.

N. S. Nightingale, D. F. Buscher, “Interferometric seeing measurements at the La Palma Observatory,” Mon. Not. R. Astron. Soc. 251, 155–166 (1991).

Caccia, J. L.

Carlstrom, J. E.

M. C. Wiedner, R. E. Hills, J. E. Carlstrom, O. P. Lay, “Interferometric phase correction using 183 GHz water vapor monitors,” Astrophys. J. 553, 1036–1041 (2001).
[CrossRef]

Clifford, S. F.

Correia, E.

P. Kaufmann, J. E. R. Costa, E. Correia, A. Magun, K. Arzner, N. Kämpfer, M. Rovira, H. Levato, “Recent developments of the solar submm-wave telescope (SST),” in Coronal Physics from Radio and Space Observations, Lecture Notes in Physics, G. Trottet, ed. (Springer, Berlin, 1997), pp. 202–206.
[CrossRef]

Costa, J. E. R.

J. E. R. Costa, A. V. R. Silva, A. Lüdi, A. Magun, “Beam profile determination by tomography of solar scans,” Astron. Astrophys. 387, 1153–1160 (2002), doi:.
[CrossRef]

P. Kaufmann, J. E. R. Costa, E. Correia, A. Magun, K. Arzner, N. Kämpfer, M. Rovira, H. Levato, “Recent developments of the solar submm-wave telescope (SST),” in Coronal Physics from Radio and Space Observations, Lecture Notes in Physics, G. Trottet, ed. (Springer, Berlin, 1997), pp. 202–206.
[CrossRef]

Cotton, M. G.

H. J. Liebe, P. W. Rosenkranz, G. A. Hufford, M. G. Cotton, “Propagation modeling of moist air and suspended water/ice particles below 1000 GHz,” (NASA Center for Aerospace Information, Linthicum Heights, Md., 1993).

Green, A. E.

J. P. Nieveen, A. E. Green, W. Kohsiek, “Using a large-aperture scintillometer to measure absorption and refractive index fluctuations,” Boundary-Layer Meteorol. 87, 101–116 (1998).
[CrossRef]

Gurvich, A. S.

A. S. Gurvich, “Effect of absorption on the fluctuation in signal level during atmospheric propagation,” Radio Eng. Electron. Phys. 13, 1687–1694 (1968).

Harp, J. C.

R. W. Lee, J. C. Harp, “Weak scattering in random media, with applications to remote probing,” Proc. IEEE 57, 375–406 (1969).
[CrossRef]

Hills, R. E.

M. C. Wiedner, R. E. Hills, J. E. Carlstrom, O. P. Lay, “Interferometric phase correction using 183 GHz water vapor monitors,” Astrophys. J. 553, 1036–1041 (2001).
[CrossRef]

Hufford, G. A.

H. J. Liebe, P. W. Rosenkranz, G. A. Hufford, M. G. Cotton, “Propagation modeling of moist air and suspended water/ice particles below 1000 GHz,” (NASA Center for Aerospace Information, Linthicum Heights, Md., 1993).

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).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.

Kämpfer, N.

P. Kaufmann, J. E. R. Costa, E. Correia, A. Magun, K. Arzner, N. Kämpfer, M. Rovira, H. Levato, “Recent developments of the solar submm-wave telescope (SST),” in Coronal Physics from Radio and Space Observations, Lecture Notes in Physics, G. Trottet, ed. (Springer, Berlin, 1997), pp. 202–206.
[CrossRef]

Kaufmann, P.

P. Kaufmann, J. E. R. Costa, E. Correia, A. Magun, K. Arzner, N. Kämpfer, M. Rovira, H. Levato, “Recent developments of the solar submm-wave telescope (SST),” in Coronal Physics from Radio and Space Observations, Lecture Notes in Physics, G. Trottet, ed. (Springer, Berlin, 1997), pp. 202–206.
[CrossRef]

Kohsiek, W.

J. P. Nieveen, A. E. Green, W. Kohsiek, “Using a large-aperture scintillometer to measure absorption and refractive index fluctuations,” Boundary-Layer Meteorol. 87, 101–116 (1998).
[CrossRef]

Kuntz, M.

M. Kuntz, “Bestimmung der Höhenverteilung stratosphärischer Spurengase aus Emissionsspektren eines bodengebundenen Millimeterwellen-Radiometers,” Ph.D. dissertation (Institut für Meteorologie und Klimaforschung, Forschungszentrum Karlsruhe, Germany, 1996), FZKA 5719.

Kutuza, B. G.

B. G. Kutuza, “Spatial and temporal fluctuations of atmospheric microwave emission,” Radio Sci. 38, 12.1–12.7 (2003), doi:.
[CrossRef]

Lawrence, R. S.

Lay, O. P.

M. C. Wiedner, R. E. Hills, J. E. Carlstrom, O. P. Lay, “Interferometric phase correction using 183 GHz water vapor monitors,” Astrophys. J. 553, 1036–1041 (2001).
[CrossRef]

Lee, R. W.

R. W. Lee, J. C. Harp, “Weak scattering in random media, with applications to remote probing,” Proc. IEEE 57, 375–406 (1969).
[CrossRef]

R. W. Lee, A. T. Waterman, “A large antenna array for millimeter wave propagation studies,” Proc. IEEE 54, 454–458 (1966).
[CrossRef]

Levato, H.

P. Kaufmann, J. E. R. Costa, E. Correia, A. Magun, K. Arzner, N. Kämpfer, M. Rovira, H. Levato, “Recent developments of the solar submm-wave telescope (SST),” in Coronal Physics from Radio and Space Observations, Lecture Notes in Physics, G. Trottet, ed. (Springer, Berlin, 1997), pp. 202–206.
[CrossRef]

Liebe, H. J.

H. J. Liebe, P. W. Rosenkranz, G. A. Hufford, M. G. Cotton, “Propagation modeling of moist air and suspended water/ice particles below 1000 GHz,” (NASA Center for Aerospace Information, Linthicum Heights, Md., 1993).

Lüdi, A.

J. E. R. Costa, A. V. R. Silva, A. Lüdi, A. Magun, “Beam profile determination by tomography of solar scans,” Astron. Astrophys. 387, 1153–1160 (2002), doi:.
[CrossRef]

A. Lüdi, A. Magun, “Near horizontal line-of-sight mm-wave propagation measurements for the determination of outer length scales and anisotropy of turbulent refractive-index fluctuations in the lower troposphere,” Radio Sci. 37, 12.1–12.19 (2002), doi:.
[CrossRef]

A. Lüdi, A. Magun, “Refractivity structure constant and length scales from mm-wave propagation in the stably stratified troposphere,” J. Atmos. Sol.-Terr. Phys. (to be published).

Magun, A.

A. Lüdi, A. Magun, “Near horizontal line-of-sight mm-wave propagation measurements for the determination of outer length scales and anisotropy of turbulent refractive-index fluctuations in the lower troposphere,” Radio Sci. 37, 12.1–12.19 (2002), doi:.
[CrossRef]

J. E. R. Costa, A. V. R. Silva, A. Lüdi, A. Magun, “Beam profile determination by tomography of solar scans,” Astron. Astrophys. 387, 1153–1160 (2002), doi:.
[CrossRef]

A. Lüdi, A. Magun, “Refractivity structure constant and length scales from mm-wave propagation in the stably stratified troposphere,” J. Atmos. Sol.-Terr. Phys. (to be published).

P. Kaufmann, J. E. R. Costa, E. Correia, A. Magun, K. Arzner, N. Kämpfer, M. Rovira, H. Levato, “Recent developments of the solar submm-wave telescope (SST),” in Coronal Physics from Radio and Space Observations, Lecture Notes in Physics, G. Trottet, ed. (Springer, Berlin, 1997), pp. 202–206.
[CrossRef]

Martin, F.

A. Ziad, J. Borgnino, F. Martin, A. Agabi, “Experimental estimation of the spatial-coherence outer scale from a wavefront statistical analysis,” Astron. Astrophys. 282, 1021–1033 (1994).

Nieveen, J. P.

J. P. Nieveen, A. E. Green, W. Kohsiek, “Using a large-aperture scintillometer to measure absorption and refractive index fluctuations,” Boundary-Layer Meteorol. 87, 101–116 (1998).
[CrossRef]

Nightingale, N. S.

N. S. Nightingale, D. F. Buscher, “Interferometric seeing measurements at the La Palma Observatory,” Mon. Not. R. Astron. Soc. 251, 155–166 (1991).

Ochs, G. R.

Rocca, A.

Roddier, C.

Roddier, F.

Roddier, R.

Rodgers, C. D.

C. D. Rodgers, Inverse Methods for Atmospheric Sounding. Theory and Practice Vol. 2 of Series on Atmospheric, Oceanic and Planetary Physics (World Scientific, Singapore, 2000).

Rosenkranz, P. W.

H. J. Liebe, P. W. Rosenkranz, G. A. Hufford, M. G. Cotton, “Propagation modeling of moist air and suspended water/ice particles below 1000 GHz,” (NASA Center for Aerospace Information, Linthicum Heights, Md., 1993).

Rovira, M.

P. Kaufmann, J. E. R. Costa, E. Correia, A. Magun, K. Arzner, N. Kämpfer, M. Rovira, H. Levato, “Recent developments of the solar submm-wave telescope (SST),” in Coronal Physics from Radio and Space Observations, Lecture Notes in Physics, G. Trottet, ed. (Springer, Berlin, 1997), pp. 202–206.
[CrossRef]

Sáchez, L. J.

R. Avila, J. Vernin, L. J. Sáchez, “Atmospheric turbulence and wind profiles monitoring with generalized scidar,” Astron. Astrophys. 369, 364–372 (2001).
[CrossRef]

Shen, L. C.

L. C. Shen, “Remote probing of atmosphere and wind velocity by millimeter waves,” IEEE Trans. Antennas Propag. AP-18, 493–497 (1970).
[CrossRef]

Silva, A. V. R.

J. E. R. Costa, A. V. R. Silva, A. Lüdi, A. Magun, “Beam profile determination by tomography of solar scans,” Astron. Astrophys. 387, 1153–1160 (2002), doi:.
[CrossRef]

Stix, M.

M. Stix, The Sun: An Introduction (Springer-Verlag, Berlin, 2002).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971).

Vernin, J.

Wang, T.

Waterman, A. T.

R. W. Lee, A. T. Waterman, “A large antenna array for millimeter wave propagation studies,” Proc. IEEE 54, 454–458 (1966).
[CrossRef]

Wheelon, A. D.

A. D. Wheelon, Electromagnetic Scintillation, Vol. 1 of Geometrical Optics (Cambridge U. Press, Cambridge, UK, 2001).
[CrossRef]

Wiedner, M. C.

M. C. Wiedner, R. E. Hills, J. E. Carlstrom, O. P. Lay, “Interferometric phase correction using 183 GHz water vapor monitors,” Astrophys. J. 553, 1036–1041 (2001).
[CrossRef]

Ziad, A.

A. Ziad, J. Borgnino, F. Martin, A. Agabi, “Experimental estimation of the spatial-coherence outer scale from a wavefront statistical analysis,” Astron. Astrophys. 282, 1021–1033 (1994).

Zirin, H.

H. Zirin, Astrophysics of the Sun (Cambridge U. Press, New York, 1988).

Appl. Opt. (5)

Astron. Astrophys. (3)

J. E. R. Costa, A. V. R. Silva, A. Lüdi, A. Magun, “Beam profile determination by tomography of solar scans,” Astron. Astrophys. 387, 1153–1160 (2002), doi:.
[CrossRef]

R. Avila, J. Vernin, L. J. Sáchez, “Atmospheric turbulence and wind profiles monitoring with generalized scidar,” Astron. Astrophys. 369, 364–372 (2001).
[CrossRef]

A. Ziad, J. Borgnino, F. Martin, A. Agabi, “Experimental estimation of the spatial-coherence outer scale from a wavefront statistical analysis,” Astron. Astrophys. 282, 1021–1033 (1994).

Astrophys. J. (1)

M. C. Wiedner, R. E. Hills, J. E. Carlstrom, O. P. Lay, “Interferometric phase correction using 183 GHz water vapor monitors,” Astrophys. J. 553, 1036–1041 (2001).
[CrossRef]

Boundary-Layer Meteorol. (1)

J. P. Nieveen, A. E. Green, W. Kohsiek, “Using a large-aperture scintillometer to measure absorption and refractive index fluctuations,” Boundary-Layer Meteorol. 87, 101–116 (1998).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

L. C. Shen, “Remote probing of atmosphere and wind velocity by millimeter waves,” IEEE Trans. Antennas Propag. AP-18, 493–497 (1970).
[CrossRef]

J. Opt. Soc. Am. (2)

Mon. Not. R. Astron. Soc. (1)

N. S. Nightingale, D. F. Buscher, “Interferometric seeing measurements at the La Palma Observatory,” Mon. Not. R. Astron. Soc. 251, 155–166 (1991).

Proc. IEEE (2)

R. W. Lee, J. C. Harp, “Weak scattering in random media, with applications to remote probing,” Proc. IEEE 57, 375–406 (1969).
[CrossRef]

R. W. Lee, A. T. Waterman, “A large antenna array for millimeter wave propagation studies,” Proc. IEEE 54, 454–458 (1966).
[CrossRef]

Radio Eng. Electron. Phys. (1)

A. S. Gurvich, “Effect of absorption on the fluctuation in signal level during atmospheric propagation,” Radio Eng. Electron. Phys. 13, 1687–1694 (1968).

Radio Sci. (2)

B. G. Kutuza, “Spatial and temporal fluctuations of atmospheric microwave emission,” Radio Sci. 38, 12.1–12.7 (2003), doi:.
[CrossRef]

A. Lüdi, A. Magun, “Near horizontal line-of-sight mm-wave propagation measurements for the determination of outer length scales and anisotropy of turbulent refractive-index fluctuations in the lower troposphere,” Radio Sci. 37, 12.1–12.19 (2002), doi:.
[CrossRef]

Other (11)

A. D. Wheelon, Electromagnetic Scintillation, Vol. 1 of Geometrical Optics (Cambridge U. Press, Cambridge, UK, 2001).
[CrossRef]

C. D. Rodgers, Inverse Methods for Atmospheric Sounding. Theory and Practice Vol. 2 of Series on Atmospheric, Oceanic and Planetary Physics (World Scientific, Singapore, 2000).

M. Kuntz, “Bestimmung der Höhenverteilung stratosphärischer Spurengase aus Emissionsspektren eines bodengebundenen Millimeterwellen-Radiometers,” Ph.D. dissertation (Institut für Meteorologie und Klimaforschung, Forschungszentrum Karlsruhe, Germany, 1996), FZKA 5719.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem, 1971).

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

A. Lüdi, A. Magun, “Refractivity structure constant and length scales from mm-wave propagation in the stably stratified troposphere,” J. Atmos. Sol.-Terr. Phys. (to be published).

H. J. Liebe, P. W. Rosenkranz, G. A. Hufford, M. G. Cotton, “Propagation modeling of moist air and suspended water/ice particles below 1000 GHz,” (NASA Center for Aerospace Information, Linthicum Heights, Md., 1993).

M. Stix, The Sun: An Introduction (Springer-Verlag, Berlin, 2002).
[CrossRef]

H. Zirin, Astrophysics of the Sun (Cambridge U. Press, New York, 1988).

P. Kaufmann, J. E. R. Costa, E. Correia, A. Magun, K. Arzner, N. Kämpfer, M. Rovira, H. Levato, “Recent developments of the solar submm-wave telescope (SST),” in Coronal Physics from Radio and Space Observations, Lecture Notes in Physics, G. Trottet, ed. (Springer, Berlin, 1997), pp. 202–206.
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.

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Figures (7)

Fig. 1
Fig. 1

Schematic view of our measurement configuration.

Fig. 2
Fig. 2

Measurement of solar transmission. The numbered circles indicate the antenna beams on the Sun with their center positions listed in Table 1. Beams 1–4 operate at 212 GHz (solid circles) and beam 5 (dashed circle) and 6 (coinciding with beam 1) operate at 405 GHz. EL, elevation; AZ, azimuth.

Fig. 3
Fig. 3

Time sections of a typical example (24 August 2000) of normalized signal variations during observations of the quiet Sun at 212 GHz (beams 1–4) and at 405 GHz (beam 5).

Fig. 4
Fig. 4

Median and the two quartiles (limits of horizontal bars) of the vertical C m 2 profiles at El Leoncito for June (dashed curve) and August (solid curve) 2000. The dashed–dotted and the dotted curves represent C m 2 ( h ) with a theoretical height dependence of C m 2 h - 2 / 3 and C m 2 h - 4 / 3, respectively, for the lower atmosphere (≲3 km above ground).

Fig. 5
Fig. 5

Median and the two quartiles (limits of horizontal bars) of the vertical L0 profiles at El Leoncito for June (dashed curve) and August (solid curve) 2000.

Fig. 6
Fig. 6

Retrieved vertical wind velocity profiles on 20 June 2000 for different times (in UTC, the local time is UTC −4.6 h). The solid and dashed curves represent the north–south and east–west wind components, respectively. In situ wind measurements from a nearby anemometer are shown by the horizontal bars at each time for north–south wind (upper bar) and east–west wind (lower bar). Positive values stand for wind from the north and east, respectively.

Fig. 7
Fig. 7

Variance σ T a 2 of the normalized signal fluctuations at 212 GHz versus σ T a 2 at 405 GHz. The horizontal error bars represent the maximal differences of σ T a 2 measured with the different beams at 212 GHz.

Tables (1)

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Table 1 Summary of the Beam Configurationa

Equations (17)

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T A = T a + T E = T B exp ( - t ) + T E ,
B a ( d T , d R ) = 4 ( 2 π ) 2 0 L d η 0 d κ κ J 0 ( κ d ) × H i 2 ( L - η , κ ) Φ m ( κ , η ) ,
Φ m ( κ , η ) = 0.033 C m 2 ( η ) [ L 0 ( η ) - 2 + κ 2 ] - 11 / 6 ,
H i ( L - η , κ ) = k cos [ ( L - η ) 2 k k 2 ] ,
d = ( 1 - η / L ) d T + ( η / L ) d R ,
B a ( α d T / d T , d R ) = 4 ( 2 π ) 2 0 d s 0 d κ × κ J 0 ( κ d ) H i 2 ( s , κ ) Φ m ( κ , s ) ,
d = s α ( d T / d T ) + d R ,
B a ( α d T / d T , d R , τ ) = 4 ( 2 π ) 2 0 d s 0 d κ κ × J 0 ( κ d + v τ ) H i 2 ( s , κ ) Φ m ( κ , s ) .
B a ( α d T / d T , d R , τ ) τ | τ = 0 = - 4 ( 2 π ) 2 0 d s 0 d κ × κ 2 J 1 ( κ d ) H i 2 ( s , κ ) × Φ m ( κ , s ) d d v ( s ) .
g = A C m 2 f C m 2 ,
f C m 2 = B C m 2 g ,
B C m 2 = S f C m 2 A C m 2 T ( A C m 2 S f C m 2 A C m 2 T + S g ) - 1
g = g + A L 0 ( f L 0 - f L 0 ) .
f L 0 , i + 1 = f L 0 , 0 + ( A L 0 , i T S g - 1 A L 0 , i + S f L 0 - 1 ) - 1 A L 0 , i T S g - 1 [ g - g i - A L 0 , i ( f L 0 , 0 - f L 0 , i ) ] .
C n 2 = A T 2 C T 2 + 2 A T A q C T q + A q 2 C q 2 ,
C m 2 = B T 2 C T 2 + 2 B T B q C T q + B q 2 C q 2 ,
σ T a , 405 2 σ T a , 212 2 ( k 405 k 212 ) 2 C m , 405 2 C m , 212 2 .

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