Abstract

A method of interpreting data of multifrequency Raman lidar sensing is developed. An algorithm for separating aerosol layers with different scattering properties and subsequently estimating the average value of the lidar ratio and Ångström parameter within individual layers is suggested. The algorithm allows the error of reconstructing the backscattering coefficient from daytime observations to be at least halved. A well-posed numerical differentiation algorithm for determining the extinction coefficient is suggested for the interpretation of nighttime measurements based on the transformation of the range of allowable values that requires a solution of nonlinear equations. An iterative procedure envisaged for linearization improves the spatial resolution compared with the conventional methods. The methods can be successfully used to process routine lidar measurements under conditions of a priori uncertainty.

© 2008 Optical Society of America

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2008 (1)

2007 (1)

2005 (3)

2004 (5)

I. Veselovskii, A. Kolgotin, V. Griaznov, D. Muller, K. Franke, and D. M. Whiteman, “Inversion of multiwavelength Raman lidar data for retrieval of bimodal aerosol size distribution,” Appl. Opt. 43, 1180-1195 (2004).
[CrossRef]

C. Böckmann, U. Wandinger, A. Ansmann, J. Bösenberg, V. Amiridis, A. Boselli, A. Delaval, F. De Tomasi, M. Frioud, I. Videnov Grigorov, A. Hågård, M. Horvat, M. Iarlori, L. Komguem, S. Kreipl, G. Larcheveque, V. Matthias, A. Papayannis, G. Pappalardo, F. Rocadenbosch, J. A. Rodrigues, J. Schneider, V. Shcherbakov, and M. Wiegner, “Aerosol lidar intercomparison in the framework of the EARLINET project. 2. Aerosol backscatter algorithms,” Appl. Opt. 43, 977-989 (2004).
[CrossRef]

G. Pappalardo, A. Amodeo, M. Pandolfi, U. Wandinger, A. Ansmann, J. Bösenberg, V. Matthias, V. Amiridis, F. De Tomasi, M. Frioud, M. Iarlori, L. Komguem, A. Papayannis, F. Rocadenbosch, and X. Wang, “Aerosol lidar intercomparison in the framework of the EARLINET project: 3. Raman lidar algorithm for aerosol extinction, backscatter, and lidar ratio,” Appl. Opt. 43, 5370-5385 (2004).
[CrossRef]

A. P. Chaikovsky, A. I. Bril, V. V. Barun, O. Dubovik, B. N. Holben, P. Goloub, and P. Sobolewski, “Methodology and sample results of retrieving aerosol parameters by combined multiwavelength lidar and Sun-sky scanning measurements,” Proc. SPIE 5397, 146-157 (2004).

A. H. Omar, D. M. Winker, and J.-G. Won, “Aerosol models for the CALIPSO lidar inversion algorithms,” Proc. SPIE 5240, 153-164 (2004).

2003 (2)

S. V. Samoilova, Yu. S. Balin, and A. D. Ershov, “Stable procedure for retrieval of optical characteristics of aerosol from combination lidar sounding data,” Izv. Atmos. Oceanic Phys. 39, 395-404 (2003).

D. M. Winker, J. Pelon, and M. P. McCormick, “The CALIPSO mission: spaceborne lidar for observations of aerosols and clouds,” Proc. SPIE 4893, 1-11 (2003).

2001 (1)

T. Murayama, N. Sugimoto, I. Uno, K. Kinoshita, K. Aoki, N. Hagiwara, Z. Liu, I. Matsui, T. Sakai, T. Shibata, K. Arao, B.-J. Sohn, J.-G. Won, S.-C. Yoon, T. Li, J. Zhou, H. Hu, M. Abo, K. Iokibe, R. Koga, and Y. Iwasaka, “Ground-based network observation of Asian dust events of April 1998 in east Asia,” J. Geophys. Res. 106, 18345-18359 (2001).
[CrossRef]

1999 (2)

1998 (1)

J. Ackerman, “The extinction-to-backscatter ratio of tropospheric aerosol: a numerical study,” J. Atmos. Oceanic Technol 15, 1043-1050 (1998).

1997 (2)

1995 (1)

1994 (1)

M. V. Panchenko and S. A. Terpugova, “Annual behavior of the content of submicron aerosol in the troposphere over West Siberia,” Atmos. Oceanic Opt. 7, 553-557 (1994).

1993 (2)

V. A. Kovalev, “Lidar measurements of the vertical aerosol extinction profiles with range-dependent backscatter-to-extinction ratios,” Appl. Opt. 32, 6053-6065 (1993).

M. del Guasta, M. Morandi, L. Stefanutti, J. Brechet, and J. Piquad, “One year of cloud lidar data from d'Urville (Antarctica). 1. General overview of geometrical and optical properties,” J. Geophys. Res. 98, 18575-18587 (1993).
[CrossRef]

1992 (1)

1990 (1)

S. I. Kavkyanov and S. V. Samoilova “Numerical differentiation of experimental data by using the Fourier transform,” Atmos. Oceanic Opt. 3, 1102-1106 (1990).

1984 (1)

1976 (1)

G. V. Rosenberg, “Reconstruction of aerosol microphysical parameters from data of complex optical measurements,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 12, 1159-1167 (1976).

1974 (1)

1966 (1)

Abo, M.

T. Murayama, N. Sugimoto, I. Uno, K. Kinoshita, K. Aoki, N. Hagiwara, Z. Liu, I. Matsui, T. Sakai, T. Shibata, K. Arao, B.-J. Sohn, J.-G. Won, S.-C. Yoon, T. Li, J. Zhou, H. Hu, M. Abo, K. Iokibe, R. Koga, and Y. Iwasaka, “Ground-based network observation of Asian dust events of April 1998 in east Asia,” J. Geophys. Res. 106, 18345-18359 (2001).
[CrossRef]

Ackerman, J.

J. Ackerman, “The extinction-to-backscatter ratio of tropospheric aerosol: a numerical study,” J. Atmos. Oceanic Technol 15, 1043-1050 (1998).

Ackermann, J.

Amiridis, V.

Amodeo, A.

Ångström, A.

Ansmann, A.

C. Böckmann, U. Wandinger, A. Ansmann, J. Bösenberg, V. Amiridis, A. Boselli, A. Delaval, F. De Tomasi, M. Frioud, I. Videnov Grigorov, A. Hågård, M. Horvat, M. Iarlori, L. Komguem, S. Kreipl, G. Larcheveque, V. Matthias, A. Papayannis, G. Pappalardo, F. Rocadenbosch, J. A. Rodrigues, J. Schneider, V. Shcherbakov, and M. Wiegner, “Aerosol lidar intercomparison in the framework of the EARLINET project. 2. Aerosol backscatter algorithms,” Appl. Opt. 43, 977-989 (2004).
[CrossRef]

G. Pappalardo, A. Amodeo, M. Pandolfi, U. Wandinger, A. Ansmann, J. Bösenberg, V. Matthias, V. Amiridis, F. De Tomasi, M. Frioud, M. Iarlori, L. Komguem, A. Papayannis, F. Rocadenbosch, and X. Wang, “Aerosol lidar intercomparison in the framework of the EARLINET project: 3. Raman lidar algorithm for aerosol extinction, backscatter, and lidar ratio,” Appl. Opt. 43, 5370-5385 (2004).
[CrossRef]

D. Müller, U. Wandinger, and A. Ansmann, “Microphysical particle parameters from extinction and backscatter lidar data by inversion with regularization: theory,” Appl. Opt. 38, 2346-2357 (1999).
[CrossRef]

A. Ansmann, U. Wandinger, M. Riebesell, C. Weitkamp, and W. Michaelis, “Independent measurement of the extinction and backscatter profiles in cirrus clouds by using a combined Raman elastic-backscatter lidar,” Appl. Opt. 31, 7113-7131 (1992).

J. BösenbergA. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: a European aerosol research lidar network,” in Advances in Laser Remote Sensing, A.Dabas, C.Loth, and J.Pelon, eds. (Editions de L'Ecole Polytechnique, 2001), pp. 155-158.

Aoki, K.

T. Murayama, N. Sugimoto, I. Uno, K. Kinoshita, K. Aoki, N. Hagiwara, Z. Liu, I. Matsui, T. Sakai, T. Shibata, K. Arao, B.-J. Sohn, J.-G. Won, S.-C. Yoon, T. Li, J. Zhou, H. Hu, M. Abo, K. Iokibe, R. Koga, and Y. Iwasaka, “Ground-based network observation of Asian dust events of April 1998 in east Asia,” J. Geophys. Res. 106, 18345-18359 (2001).
[CrossRef]

Arao, K.

T. Murayama, N. Sugimoto, I. Uno, K. Kinoshita, K. Aoki, N. Hagiwara, Z. Liu, I. Matsui, T. Sakai, T. Shibata, K. Arao, B.-J. Sohn, J.-G. Won, S.-C. Yoon, T. Li, J. Zhou, H. Hu, M. Abo, K. Iokibe, R. Koga, and Y. Iwasaka, “Ground-based network observation of Asian dust events of April 1998 in east Asia,” J. Geophys. Res. 106, 18345-18359 (2001).
[CrossRef]

Arsenin, V. Y.

A. E. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).

B, J.

Baldasano, J. M.

J. BösenbergA. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: a European aerosol research lidar network,” in Advances in Laser Remote Sensing, A.Dabas, C.Loth, and J.Pelon, eds. (Editions de L'Ecole Polytechnique, 2001), pp. 155-158.

Balin, Yu. S.

S. V. Samoilova, Yu. S. Balin, M. M. Krekova, and D. M. Winker, “Method for reconstructing the atmospheric optical parameters from the data of polarization lidar sensing,” Appl. Opt. 44, 3499-3509 (2005).
[CrossRef]

S. V. Samoilova, Yu. S. Balin, and A. D. Ershov, “Stable procedure for retrieval of optical characteristics of aerosol from combination lidar sounding data,” Izv. Atmos. Oceanic Phys. 39, 395-404 (2003).

A. P. Chaikovsky, A. P. Ivanov, Yu. S. Balin, A. V. Elnikov, G. F. Tulinov, I. I. Plusnin, O. A. Bukin, and B. B. Chen, “CIS-LINET--lidar network for monitoring aerosol and ozone in CIS regions,” in Reviewed and Revised Papers Presented at the 23d ILRC, C. Nagasava and N. Sugimoto, eds. (Nara, 2006), pp. 671-672.

Balis, D.

J. BösenbergA. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: a European aerosol research lidar network,” in Advances in Laser Remote Sensing, A.Dabas, C.Loth, and J.Pelon, eds. (Editions de L'Ecole Polytechnique, 2001), pp. 155-158.

Barun, V. V.

A. P. Chaikovsky, A. I. Bril, V. V. Barun, O. Dubovik, B. N. Holben, P. Goloub, and P. Sobolewski, “Methodology and sample results of retrieving aerosol parameters by combined multiwavelength lidar and Sun-sky scanning measurements,” Proc. SPIE 5397, 146-157 (2004).

Böckmann, C.

Bohren, F. C.

F. C. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Boselli, A.

Bösenberg, J.

C. Böckmann, U. Wandinger, A. Ansmann, J. Bösenberg, V. Amiridis, A. Boselli, A. Delaval, F. De Tomasi, M. Frioud, I. Videnov Grigorov, A. Hågård, M. Horvat, M. Iarlori, L. Komguem, S. Kreipl, G. Larcheveque, V. Matthias, A. Papayannis, G. Pappalardo, F. Rocadenbosch, J. A. Rodrigues, J. Schneider, V. Shcherbakov, and M. Wiegner, “Aerosol lidar intercomparison in the framework of the EARLINET project. 2. Aerosol backscatter algorithms,” Appl. Opt. 43, 977-989 (2004).
[CrossRef]

J. BösenbergA. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: a European aerosol research lidar network,” in Advances in Laser Remote Sensing, A.Dabas, C.Loth, and J.Pelon, eds. (Editions de L'Ecole Polytechnique, 2001), pp. 155-158.

Brechet, J.

M. del Guasta, M. Morandi, L. Stefanutti, J. Brechet, and J. Piquad, “One year of cloud lidar data from d'Urville (Antarctica). 1. General overview of geometrical and optical properties,” J. Geophys. Res. 98, 18575-18587 (1993).
[CrossRef]

Bril, A. I.

A. P. Chaikovsky, A. I. Bril, V. V. Barun, O. Dubovik, B. N. Holben, P. Goloub, and P. Sobolewski, “Methodology and sample results of retrieving aerosol parameters by combined multiwavelength lidar and Sun-sky scanning measurements,” Proc. SPIE 5397, 146-157 (2004).

Buffon, J. L.

Bukin, O. A.

A. P. Chaikovsky, A. P. Ivanov, Yu. S. Balin, A. V. Elnikov, G. F. Tulinov, I. I. Plusnin, O. A. Bukin, and B. B. Chen, “CIS-LINET--lidar network for monitoring aerosol and ozone in CIS regions,” in Reviewed and Revised Papers Presented at the 23d ILRC, C. Nagasava and N. Sugimoto, eds. (Nara, 2006), pp. 671-672.

Calpini, B.

J. BösenbergA. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: a European aerosol research lidar network,” in Advances in Laser Remote Sensing, A.Dabas, C.Loth, and J.Pelon, eds. (Editions de L'Ecole Polytechnique, 2001), pp. 155-158.

Cavanaugh, J. F.

Chaikovsky, A.

J. BösenbergA. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: a European aerosol research lidar network,” in Advances in Laser Remote Sensing, A.Dabas, C.Loth, and J.Pelon, eds. (Editions de L'Ecole Polytechnique, 2001), pp. 155-158.

Chaikovsky, A. P.

A. P. Chaikovsky, A. I. Bril, V. V. Barun, O. Dubovik, B. N. Holben, P. Goloub, and P. Sobolewski, “Methodology and sample results of retrieving aerosol parameters by combined multiwavelength lidar and Sun-sky scanning measurements,” Proc. SPIE 5397, 146-157 (2004).

A. P. Chaikovsky, A. P. Ivanov, Yu. S. Balin, A. V. Elnikov, G. F. Tulinov, I. I. Plusnin, O. A. Bukin, and B. B. Chen, “CIS-LINET--lidar network for monitoring aerosol and ozone in CIS regions,” in Reviewed and Revised Papers Presented at the 23d ILRC, C. Nagasava and N. Sugimoto, eds. (Nara, 2006), pp. 671-672.

Chen, B. B.

A. P. Chaikovsky, A. P. Ivanov, Yu. S. Balin, A. V. Elnikov, G. F. Tulinov, I. I. Plusnin, O. A. Bukin, and B. B. Chen, “CIS-LINET--lidar network for monitoring aerosol and ozone in CIS regions,” in Reviewed and Revised Papers Presented at the 23d ILRC, C. Nagasava and N. Sugimoto, eds. (Nara, 2006), pp. 671-672.

Chudamani, S.

De Tomasi, F.

del Guasta, M.

M. del Guasta, M. Morandi, L. Stefanutti, J. Brechet, and J. Piquad, “One year of cloud lidar data from d'Urville (Antarctica). 1. General overview of geometrical and optical properties,” J. Geophys. Res. 98, 18575-18587 (1993).
[CrossRef]

Delaval, A.

Dubovik, O.

A. H. Omar, J.-G. Won, D. M. Winker, S.-C. Yoon, O. Dubovik, and M. P. McCormick, “Development of global aerosol models using cluster analysis of Aerosol Robotic Network (AERONET) measurements,” J. Geophys. Res. 110, D10S14, doi:10.1029/2004JD004874 (2005).
[CrossRef]

A. P. Chaikovsky, A. I. Bril, V. V. Barun, O. Dubovik, B. N. Holben, P. Goloub, and P. Sobolewski, “Methodology and sample results of retrieving aerosol parameters by combined multiwavelength lidar and Sun-sky scanning measurements,” Proc. SPIE 5397, 146-157 (2004).

Eichinger, W. E.

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Winker, D. M.

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[CrossRef]

S. V. Samoilova, Yu. S. Balin, M. M. Krekova, and D. M. Winker, “Method for reconstructing the atmospheric optical parameters from the data of polarization lidar sensing,” Appl. Opt. 44, 3499-3509 (2005).
[CrossRef]

A. H. Omar, D. M. Winker, and J.-G. Won, “Aerosol models for the CALIPSO lidar inversion algorithms,” Proc. SPIE 5240, 153-164 (2004).

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Won, J.-G.

A. H. Omar, J.-G. Won, D. M. Winker, S.-C. Yoon, O. Dubovik, and M. P. McCormick, “Development of global aerosol models using cluster analysis of Aerosol Robotic Network (AERONET) measurements,” J. Geophys. Res. 110, D10S14, doi:10.1029/2004JD004874 (2005).
[CrossRef]

A. H. Omar, D. M. Winker, and J.-G. Won, “Aerosol models for the CALIPSO lidar inversion algorithms,” Proc. SPIE 5240, 153-164 (2004).

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[CrossRef]

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A. H. Omar, J.-G. Won, D. M. Winker, S.-C. Yoon, O. Dubovik, and M. P. McCormick, “Development of global aerosol models using cluster analysis of Aerosol Robotic Network (AERONET) measurements,” J. Geophys. Res. 110, D10S14, doi:10.1029/2004JD004874 (2005).
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T. Murayama, N. Sugimoto, I. Uno, K. Kinoshita, K. Aoki, N. Hagiwara, Z. Liu, I. Matsui, T. Sakai, T. Shibata, K. Arao, B.-J. Sohn, J.-G. Won, S.-C. Yoon, T. Li, J. Zhou, H. Hu, M. Abo, K. Iokibe, R. Koga, and Y. Iwasaka, “Ground-based network observation of Asian dust events of April 1998 in east Asia,” J. Geophys. Res. 106, 18345-18359 (2001).
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Figures (7)

Fig. 1
Fig. 1

Variations of the Ångström parameters η σ ( λ i / λ j ) for the extinction coefficient versus the complex refractive index m = m R i m I . Results are presented for three aerosol models; the family of curves shows variations of η σ ( λ i / λ j ) for the fixed m R value.

Fig. 2
Fig. 2

Variations of the Ångström parameters η β ( λ i / λ j ) for the backscattering coefficient versus the complex refractive index m = m R i m I . Results are presented for three aerosol models; the family of curves shows variations of η β ( λ i / λ j ) at the fixed m R value.

Fig. 3
Fig. 3

To the problem of a priori uncertainty of the lidar ratio S a ( λ 0 i ) in the case of inversion of lidar equations without Raman signals. The model profiles of the extinction coefficient σ a ( λ 0 i , z ) (а) and the corresponding profiles of the Ångström parameters η σ ( λ 0 i / λ 0 j ) (с) are shown at the left of the figure. Results of reconstruction of σ a ( λ 0 i , z , S a ( λ 0 i ) ) (b) and η σ ( λ 0 i / λ 0 j , S a ( λ 0 i ) , S ( λ 0 j ) ) (d) for fixed values S a ( λ 0 i ) = const are shown at the left of the figure. Here black symbols are for S a ( λ 0 i ) = 40 sr , and white symbols are for S a ( λ 0 i ) = 80 sr .

Fig. 4
Fig. 4

Reconstruction of the optical parameters from daytime observations without Raman signals: Ångström parameters η σ ( λ 0 i / λ 0 j ) , (a) and (b), calculated for S a ( λ ) 0 i = const = 50 sr (a) and for the profiles S a ( λ 0 i , z ) estimated from lidar signals (b). Profiles of the lidar ratio are shown in (c). Errors of reconstructing S a ( λ 0 i , z ) (d) and σ a ( λ 0 i , z , S a ( λ 0 i ) ) , (e) and (f), were calculated from the formula δ [ g ( λ 0 i , z ) ] = [ g ( λ 0 i , z ) g ^ ( λ 0 i , z ) ] / [ g ( λ 0 i , z ) ] × 100 for S a ( λ = 01 355 nm ) = 50 , S ( λ = 02 532 nm ) = 70 , and S ( λ = 03 1064 nm ) = 80 sr (e) and for the S a ( λ 0 i , z ) profiles (f) shown in (c).

Fig. 5
Fig. 5

Altitude profiles of the extinction coefficient, (a) and (b), optical thickness, (c) and (d), and lidar ratio, (e) and (f), reconstructed by the linear methods for wavelengths of 355 (at the left of the figure) and 532 nm (at the right of the figure). Curves 1 show the model profiles of the parameters, and curves 2 show the profiles reconstructed by the optimal linear filtration method. Curves 3 in (c) and (d) show the differentiable function.

Fig. 6
Fig. 6

Optical parameters reconstructed by the method that considers the range of their allowable values. Curves 1 show the model profiles. For the extinction coefficient, (a) and (b), and lidar ratio, (e) and (f), curves 2 show the results of reconstruction by the optimal linear filtration method, and curves 3 show the results of reconstruction by the method that considers the range of their allowable values. For the backscattering coefficient, (c) and (d), curves 2 were calculated from the ratio of elastic and Raman scattering signals, and curves 3 were calculated from relation (20).

Fig. 7
Fig. 7

Influence of the accuracy of estimating the parameters of reconstructing filter (12) on the results of reconstruction of the extinction coefficient u ( z ) = σ a ( λ 0 i , z ) by the optimal linear filtration method. Curve 0 in (b) shows the model σ a ( λ 0 i , z ) profile for a wavelength of 355 nm . Curves 1 were calculated for exact values of the spectral power density of the solution R u ( w ) , and curves 2 were calculated for R u ( w ) estimated from Eqs. (A2, A3). Variations of the parameters versus the effective correlation radius are illustrated by curves 3 ( r ˜ eff = 2 r eff ) and 4 ( r ˜ eff = 0.5 r eff ).

Tables (1)

Tables Icon

Table 1 Typical Parameters of the Bimodal Distribution for the Different Aerosol Types [25]

Equations (31)

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P ( λ 0 i , z ) = A 0 z 2 [ β m ( λ 0 i , z ) + β a ( λ 0 i , z ) ] exp { 2 0 z [ σ m ( λ 0 i , z ) + σ a ( λ 0 i , z ) d z } ,
P ( λ R i , z ) = A R z 2 β ( λ R i , z ) exp { 0 z [ σ m ( λ 0 i , z ) + σ a ( λ 0 i , z ) + σ m ( λ R i , z ) + σ a ( λ R i , z ) ] d z } .
σ a ( λ 0 i , z ) = S a ( λ 0 i ) β a ( λ 0 i , z ) = S a ( λ 0 i ) [ β m ( λ 0 i , z ) + Ψ ( λ 0 i , z ) Ψ ( λ 0 i , z * ) β a ( λ 0 i , z * ) + β m ( λ 0 i , z * ) + 2 S a ( λ 0 i ) z z * Ψ ( λ 0 i , z ) d z ] ,
Ψ ( λ 0 i , z ) = P ( λ 0 i , z ) z 2 exp { 2 z 0 z [ S a ( λ 0 i ) S m ( λ 0 i ) ] β m ( λ 0 i , z ) d z } .
ln β a ( λ 0 j , z ) = ln β a ( λ 0 i , z ) + ln β a ( λ 0 i , z ) ln λ [ ln λ 0 j ln λ 0 i ] + 2 ln β a ( λ 0 i , z ) ( ln λ ) 2 [ ln λ 0 j ln λ 0 i ] 2 + ... , ln σ a ( λ , 0 j z ) = ln σ a ( λ 0 i , z ) + ln σ a ( λ 0 i , z ) ln λ [ ln λ 0 j ln λ 0 i ] + 2 ln σ a ( λ 0 i , z ) ( ln λ ) 2 [ ln λ 0 j ln λ 0 i ] 2 + ... .
Ψ i 1 ( λ 01 , λ 0 i , z ) = ln P ( λ 0 i , z ) P ( λ 01 , z ) = ln A 0 i A 01 + η β ( z ) λ 0 i λ 01 2 0 z σ ( λ 01 , z ) [ ( λ 0 i λ 01 ) η σ ( z ) 1 ] d z .
σ ( λ 01 , z ) [ ( λ 0 i λ 01 ) η σ ( z ) 1 ] = 1 2 [ d Ψ i 1 ( λ 01 , λ 0 i , z ) d z d η β ( z ) d z λ 0 i λ 01 ] .
F ( r ) = d V ( r ) d ln r = i = f , c V i 2 π ln σ i exp { ( ln r ln r i ) 2 2 ( ln σ i ) 2 } ,
Φ ( S a ( λ 01 ) , S a ( λ 02 ) , S a ( λ 03 ) ) = z k z k + 1 { η σ [ 355 / 1064 , S a ( 355 ) , S a ( 1064 ) ] η σ [ 532 / 1064 , S a ( 532 ) , S a ( 1064 ) ] 1 } 2 d z .
D f = u
A u z 0 z u ( z ) d z = f ( z ) .
z 0 z max [ A u f ( z ) ] 2 d z σ ε 2
A u h ( z z ) u ( z ) d z + ε ( z ) = f ( z ) ,
h ( z ) = { 1 , z [ z 0 , z max ] , u ( z ) = f ( z ) = ε ( z ) = 0 , z [ z 0 , z max ] ,
u ˜ ( w ) = f ˜ ( w ) h ˜ ( w ) K ( α , w ) = f ˜ ( w ) h ˜ ( w ) | h ˜ ( w ) | 2 | h ˜ ( w ) | 2 + α Q ( w ) ,
K ( α , w ) = K Wiener ( w ) = | h ˜ ( w ) | 2 | h ˜ ( w ) | 2 + R ε ( w ) / R u ( w ) ,
u min ( z ) u ( z ) u max ( z )
u ( z ) = u max ( z ) exp ( v ( z ) ) + u min ( z ) exp ( v ( z ) ) + u max ( z ) .
v ( z ) = ln [ u max ( z ) u ( z ) u min ( z ) u max ( z ) u ( z ) ]
Ψ α ( v ) = { h ( z z ) u ( v ( z ) ) d z f ( z ) } 2 d z + α Q ( v )
u ( z ) = u ¯ ( z ) × Δ u ( z ) ,
Δ u ( z ) 1 + D ( z ) Δ v ( z ) ,
D ( z ) = ( Δ u ( z ) ) ( Δ v ( z ) ) = exp ( v ¯ ) ( u max u min ) ( exp ( v ¯ ) + u min ) ( exp ( v ¯ ) + u max ) .
h ( z z ) u ¯ ( z ) D ( z ) Δ v ( z ) d z = f ( z ) h ( z z ) u ¯ ( z ) d z
Δ v ( z ) = g ( z ) u ¯ ( z ) D ( z ) ,
g ˜ ( w ) = [ f ˜ ( w ) h ˜ ( w ) u ¯ ˜ ( w ) ] | h ˜ ( w ) | 2 | h ˜ ( w ) | 2 + α Q ( w ) ,
β a ( λ 0 i , z ) = β m ( λ 0 i , z ) + P ( λ 0 i , z ) z 2 [ β m ( λ 0 i , z * ) + β a ( λ 0 i , z * ) ] P ( λ 0 i , z * ) z * 2 exp { 2 [ τ m ( λ 0 i , z * , z ) + τ ^ a ( λ 0 i , z * , z ) ] } .
σ min ( z ) = β a ( λ 0 i , z ) S min σ ^ a ( λ 0 i , z ) β a ( λ 0 i , z ) S max = σ max ( z ) ,
| h ˜ ( w ) | 2 R u ( w ) + R ε ( w ) = R f ( w ) .
R u ( w ) = σ u 2 2 α r α r 2 + w 2 = 1 k 1 + k 2 w 2 ,
k 1 = 1 p Δ ( w ) | h ˜ ( w ) | 2 d w k 2 q p , k 2 = 1 r q 2 / p Δ ( w ) | h ˜ ( w ) | 2 ( w 2 q p ) d w , p = Δ 2 ( w ) d w , q = Δ 2 ( w ) w 2 d w , r = Δ 2 ( w ) w 4 d w , Δ ( w ) = R f ( w ) R ε ( w ) .

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