Abstract

The differential absorption lidar (DIAL) at the Institut für Meteorologie und Klimaforschung has been upgraded for precise ozone and aerosol studies in the entire troposphere and the lower stratosphere. Its excellent technical performance offers the opportunity to apply improved data processing. The existing inversion algorithm is extended to derive the optical coefficients from the backscatter profiles for three wavelengths. Correlating the correction terms of the DIAL equation and the ozone concentration yields the wavelength dependence of the backscatter and extinction coefficients of the aerosol. Under some conditions, in particular if homogeneous layers are present, the backscatter-to-extinction ratio and the reference value can also be retrieved. We find the solutions by applying evolutionary strategies. From the optical coefficients obtained in this way the ozone concentration can be calculated with substantially reduced error.

© 2005 Optical Society of America

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2003 (3)

F. Immler, “A new algorithm for simultaneous ozone and aerosol retrieval from tropospheric DIAL measurements,” Appl. Phys. B 76, 593–596 (2003).
[CrossRef]

T. Trickl, O. R. Cooper, H. Eisele, P. James, R. Mücke, A. Stohl, “Intercontinental transport and its influence on the ozone concentrations over central Europe: three case studies,” J. Geophys. Res. 108 (D12), 8530, doi: (2003).
[CrossRef]

D. Müller, I. Mattis, U. Wandinger, A. Ansmann, D. Althausen, O. Dubovik, S. Eckhardt, A. Stohl, “Saharan dust over a central European EARLINET–AERONET site: combined observations with Raman lidar and Sun photometer,” J. Geophys. Res. 108 (D12), 4345, doi: (2003).
[CrossRef]

2002 (1)

W. Carnuth, U. Kempfer, T. Trickl, “Highlights of the tropospheric lidar studies at IFU within the TOR project,” Tellus Ser. B 54, 163–185 (2002).
[CrossRef]

2000 (2)

1999 (3)

G. J. Kunz, “Two-wavelength lidar inversion algorithm,” Appl. Opt. 38, 1015–1020 (1999).
[CrossRef]

J. Ackermann, “Analytical solution of the two-frequency lidar inversion technique,” Appl. Opt. 38, 7414–7218 (1999).
[CrossRef]

H. Eisele, H. E. Scheel, R. Sladkovic, T. Trickl, “High-resolution lidar measurements of stratosphere–troposphere exchange,” J. Atmos. Sci. 56, 319–330 (1999).
[CrossRef]

1998 (1)

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

1997 (2)

1996 (4)

1995 (2)

U. Gropengiesser, “The ground state of the spin glass: a comparison of various biologically motivated algorithms,” J. Stat. Phys. 79, 1005–1012 (1995).
[CrossRef]

S. A. Young, “Analysis of lidar backscatter profiles in optically thin clouds,” Appl. Opt. 34, 7019–7031 (1995).
[CrossRef] [PubMed]

1994 (2)

P. Piironen, E. W. Eloranta, “Demonstration of a high-spectral-resolution lidar based on an iodine absorption filter,” Opt. Lett. 19, 234–236 (1994).
[CrossRef] [PubMed]

U. Kempfer, W. Carnuth, R. Lotz, T. Trickl, “A wide-range ultraviolet lidar system for tropospheric ozone measurements: development and application,” Rev. Sci. Instrum. 65, 3145–3164 (1994).
[CrossRef]

1993 (4)

T. J. McGee, M. Gross, R. Ferrare, W. Heaps, U. Singh, “Raman DIAL measurements of stratospheric ozone in the presence of volcanic aerosols,” Geophys. Res. Lett. 20, 955–958 (1993).
[CrossRef]

F. J. Dentener, P. J. Crutzen, “Reaction of N2O5on tropospheric aerosols: impact on the global distributions of NOx,O3, and OH,” J. Geophys. Res. 98 (D4), 7149–7163 (1993).
[CrossRef]

D. Gutkowicz-Krusin, “Multiangle lidar performance in the presence of horizontal inhomogeneities in atmospheric extinction and scattering,” Appl. Opt. 32, 3266–3272 (1993).
[CrossRef] [PubMed]

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

1992 (1)

1991 (2)

W. B. Grant, E. V. Browell, N. S. Higdon, S. Ismail, “Raman shifting of KrF laser radiation for tropospheric ozone measurements,” Appl. Opt. 30, 2628–2633 (1991).
[CrossRef] [PubMed]

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, “Combined Raman elastic-backscatter LIDAR for vertical profiling of moisture, aerosol extinction, backscatter, and LIDAR ratio,” Appl. Phys. B 55, 18–28 (1991).
[CrossRef]

1990 (1)

1987 (2)

1986 (1)

1985 (2)

1984 (3)

F. G. Fernald, “Analysis of atmospheric lidar observations: some comments,” Appl. Opt. 23, 652–653 (1984).
[CrossRef] [PubMed]

R. Reiter, W. Carnuth, R. Sladkovic, “Determination of physical and chemical properties of the aerosol from 1972 to 1982 at a North-Alpine pure air station at 1780 m a.s.l., Part III,” Arch. Met. Geoph. Bioclim. B 35, 179–201 (1984).
[CrossRef]

H. W. M. Salemink, P. Schotanus, J. B. Bergwerff, “Quantitative lidar at 532 nm for vertical extinction profiles and the effect of relative humidity,” Appl. Phys. B. 34, 187–189 (1984).
[CrossRef]

1983 (3)

1980 (1)

J. D. Spinhirne, J. A. Reagan, B. M. Herman, “Vertical distribution of aerosol extinction cross section and inference of aerosol imaginary index in the troposphere by lidar technique,” J. Appl. Meteorol. 19, 426–438 (1980).
[CrossRef]

Ackermann, J.

Althausen, D.

D. Müller, I. Mattis, U. Wandinger, A. Ansmann, D. Althausen, O. Dubovik, S. Eckhardt, A. Stohl, “Saharan dust over a central European EARLINET–AERONET site: combined observations with Raman lidar and Sun photometer,” J. Geophys. Res. 108 (D12), 4345, doi: (2003).
[CrossRef]

Ancellet, G.

G. Ancellet, J. Bösenberg, “Multiwavelength techniques,” in Transport and Chemical Transformation of Pollutants in the Troposphere. Vol. 8, Instrument Development for Atmospheric Research and Monitoring, J. Bösenberg, D. Brassington, P. Simon, eds. (Springer-Verlag, Berlin, Germany, 1996), pp. 24–30.

G. Ancellet, J. Bösenberg, “Differential aerosol backscatter,” in Transport and Chemical Transformation of Pollutants in the Troposphere. Vol. 8, Instrument Development for Atmospheric Research and Monitoring, J. Bösenberg, D. Brassington, P. Simon, eds. (Springer-Verlag, Berlin, Germany, 1996), pp. 18–21.

Ancellet, G. M.

Ansmann, A.

D. Müller, I. Mattis, U. Wandinger, A. Ansmann, D. Althausen, O. Dubovik, S. Eckhardt, A. Stohl, “Saharan dust over a central European EARLINET–AERONET site: combined observations with Raman lidar and Sun photometer,” J. Geophys. Res. 108 (D12), 4345, doi: (2003).
[CrossRef]

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

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, “Combined Raman elastic-backscatter LIDAR for vertical profiling of moisture, aerosol extinction, backscatter, and LIDAR ratio,” Appl. Phys. B 55, 18–28 (1991).
[CrossRef]

Beheng, K. D.

G. Kramm, K. D. Beheng, H. Müller, “Modeling of the vertical transport of polydispersed aerosol particles in the atmospheric surface layer,” in Precipitation Scavenging and Atmosphere-Surface Exchange, Vol. 2, The Semonin Volume: Atmosphere-Surface Exchange Processes, S. E. Schwartz, W. G. N. Slinn, eds. (Hemisphere, Washington, D.C., 1992), pp. 1125–1140.

Bergwerff, J. B.

H. W. M. Salemink, P. Schotanus, J. B. Bergwerff, “Quantitative lidar at 532 nm for vertical extinction profiles and the effect of relative humidity,” Appl. Phys. B. 34, 187–189 (1984).
[CrossRef]

Bisson, S. E.

Bösenberg, J.

P. Völger, J. Bösenberg, I. Schult, “Scattering properties of selected model aerosols calculated at UV-wavelengths: implications for DIAL measurements of tropospheric ozone,” Beitr. Phys. Atmos. 69, 177–187 (1996).

G. Ancellet, J. Bösenberg, “Differential aerosol backscatter,” in Transport and Chemical Transformation of Pollutants in the Troposphere. Vol. 8, Instrument Development for Atmospheric Research and Monitoring, J. Bösenberg, D. Brassington, P. Simon, eds. (Springer-Verlag, Berlin, Germany, 1996), pp. 18–21.

V. Matthias, J. Bösenberg, V. Wulfmeyer, “Improvement of ozone measurements with DIAL by using an additional Raman channel,” in Proceedings of the EUROTRAC Symposium 1994, P. M. Borrell, P. Borrell, T. Cvitas, W. Seiler, eds. (SPB Academic, Den Haag, The Netherlands, 1994), pp. 326–329.

G. Ancellet, J. Bösenberg, “Multiwavelength techniques,” in Transport and Chemical Transformation of Pollutants in the Troposphere. Vol. 8, Instrument Development for Atmospheric Research and Monitoring, J. Bösenberg, D. Brassington, P. Simon, eds. (Springer-Verlag, Berlin, Germany, 1996), pp. 24–30.

Brenner, P.

P. Brenner, O. Reitebuch, K. Schäfer, T. Trickl, A. Stichternath, “A novel mobile vertical-sounding system for ozone studies in the lower troposphere,” in Advances in Atmospheric Remote Sensing with Lidar, Selected Papers of the 18th International Laser Radar Conference, A. Ansmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, Berlin, Germany, 1996), pp. 383–386.

Browell, E. V.

Carnuth, W.

W. Carnuth, U. Kempfer, T. Trickl, “Highlights of the tropospheric lidar studies at IFU within the TOR project,” Tellus Ser. B 54, 163–185 (2002).
[CrossRef]

W. Carnuth, T. Trickl, “Transport studies with the IFU three-wavelength aerosol lidar during the VOTALP Mesolcina experiment,” Atmos. Environ. 34, 1425–1434 (2000).
[CrossRef]

U. Kempfer, W. Carnuth, R. Lotz, T. Trickl, “A wide-range ultraviolet lidar system for tropospheric ozone measurements: development and application,” Rev. Sci. Instrum. 65, 3145–3164 (1994).
[CrossRef]

R. Reiter, W. Carnuth, R. Sladkovic, “Determination of physical and chemical properties of the aerosol from 1972 to 1982 at a North-Alpine pure air station at 1780 m a.s.l., Part III,” Arch. Met. Geoph. Bioclim. B 35, 179–201 (1984).
[CrossRef]

W. Carnuth, T. Trickl, “Entfernungsaufgelöste Messung von Extinktionskoeffizienten mit Lidar,” Final Report, Contract T/R320/Q0004/Q0300, RüT IV7 (Bundesministerium für Verteidigung, Germany, 1997), in German.

Claude, H.

H. Eisele, T. Trickl, H. Claude, “Lidar als wichtige Ergänzung zur Messung troposphärischen Ozons,” in Ozonbulletin des Deutschen Wetterdiensts, Ausgabe Nr. 44, Erscheinungstermin, 25August1997 (Deutscher Wetterdienst, Offenbach, Germany, 1997), in German; http://www.dwd.de/de/FundE/Observator/MOHP/hp2/ozon/bulletin.htm .

Cooper, O. R.

T. Trickl, O. R. Cooper, H. Eisele, P. James, R. Mücke, A. Stohl, “Intercontinental transport and its influence on the ozone concentrations over central Europe: three case studies,” J. Geophys. Res. 108 (D12), 8530, doi: (2003).
[CrossRef]

Crutzen, P. J.

F. J. Dentener, P. J. Crutzen, “Reaction of N2O5on tropospheric aerosols: impact on the global distributions of NOx,O3, and OH,” J. Geophys. Res. 98 (D4), 7149–7163 (1993).
[CrossRef]

d'Almeida, G. A.

G. A. d'Almeida, P. Koepke, E. P. Shettle, Atmospheric Aerosols, Global Climatology and Radiative Characteristics (Deepak Publishing, Hampton, Va., 1991).

de Leeuw, G.

Dentener, F. J.

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T. J. McGee, M. Gross, R. Ferrare, W. Heaps, U. Singh, “Raman DIAL measurements of stratospheric ozone in the presence of volcanic aerosols,” Geophys. Res. Lett. 20, 955–958 (1993).
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U. Kempfer, W. Carnuth, R. Lotz, T. Trickl, “A wide-range ultraviolet lidar system for tropospheric ozone measurements: development and application,” Rev. Sci. Instrum. 65, 3145–3164 (1994).
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Mattis, I.

D. Müller, I. Mattis, U. Wandinger, A. Ansmann, D. Althausen, O. Dubovik, S. Eckhardt, A. Stohl, “Saharan dust over a central European EARLINET–AERONET site: combined observations with Raman lidar and Sun photometer,” J. Geophys. Res. 108 (D12), 4345, doi: (2003).
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T. J. McGee, M. Gross, R. Ferrare, W. Heaps, U. Singh, “Raman DIAL measurements of stratospheric ozone in the presence of volcanic aerosols,” Geophys. Res. Lett. 20, 955–958 (1993).
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Müller, D.

D. Müller, I. Mattis, U. Wandinger, A. Ansmann, D. Althausen, O. Dubovik, S. Eckhardt, A. Stohl, “Saharan dust over a central European EARLINET–AERONET site: combined observations with Raman lidar and Sun photometer,” J. Geophys. Res. 108 (D12), 4345, doi: (2003).
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G. Kramm, K. D. Beheng, H. Müller, “Modeling of the vertical transport of polydispersed aerosol particles in the atmospheric surface layer,” in Precipitation Scavenging and Atmosphere-Surface Exchange, Vol. 2, The Semonin Volume: Atmosphere-Surface Exchange Processes, S. E. Schwartz, W. G. N. Slinn, eds. (Hemisphere, Washington, D.C., 1992), pp. 1125–1140.

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P. Brenner, O. Reitebuch, K. Schäfer, T. Trickl, A. Stichternath, “A novel mobile vertical-sounding system for ozone studies in the lower troposphere,” in Advances in Atmospheric Remote Sensing with Lidar, Selected Papers of the 18th International Laser Radar Conference, A. Ansmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, Berlin, Germany, 1996), pp. 383–386.

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H. Eisele, H. E. Scheel, R. Sladkovic, T. Trickl, “High-resolution lidar measurements of stratosphere–troposphere exchange,” J. Atmos. Sci. 56, 319–330 (1999).
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Singh, U.

T. J. McGee, M. Gross, R. Ferrare, W. Heaps, U. Singh, “Raman DIAL measurements of stratospheric ozone in the presence of volcanic aerosols,” Geophys. Res. Lett. 20, 955–958 (1993).
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H. Eisele, H. E. Scheel, R. Sladkovic, T. Trickl, “High-resolution lidar measurements of stratosphere–troposphere exchange,” J. Atmos. Sci. 56, 319–330 (1999).
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Stichternath, A.

P. Brenner, O. Reitebuch, K. Schäfer, T. Trickl, A. Stichternath, “A novel mobile vertical-sounding system for ozone studies in the lower troposphere,” in Advances in Atmospheric Remote Sensing with Lidar, Selected Papers of the 18th International Laser Radar Conference, A. Ansmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, Berlin, Germany, 1996), pp. 383–386.

Stohl, A.

T. Trickl, O. R. Cooper, H. Eisele, P. James, R. Mücke, A. Stohl, “Intercontinental transport and its influence on the ozone concentrations over central Europe: three case studies,” J. Geophys. Res. 108 (D12), 8530, doi: (2003).
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Y. Sasano, H. Nakane, S. Hayashida-Amano, N. Sugimoto, I. Matsui, “Multiple-wavelength DIAL and a new analysis technique to deduce the ozone profile without systematic errors due to aerosol effects,” in Ozone in the Atmosphere, R. D. Bojkov, P. Fabian, eds. (Deepak Publishing, Hampton, Va., 1989), pp. 743–746.

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T. Trickl, O. R. Cooper, H. Eisele, P. James, R. Mücke, A. Stohl, “Intercontinental transport and its influence on the ozone concentrations over central Europe: three case studies,” J. Geophys. Res. 108 (D12), 8530, doi: (2003).
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H. Eisele, T. Trickl, “Lidar sounding of tropospheric ozone at Garmisch-Partenkirchen,” in Atmospheric Ozone, Proceedings of the 18th Quadrennial Ozone Symposium, R. D. Bojkov, G. Visconti, eds. (International Ozone Commission, Genève, Switzerland, 1998), pp. 351–354.

P. Brenner, O. Reitebuch, K. Schäfer, T. Trickl, A. Stichternath, “A novel mobile vertical-sounding system for ozone studies in the lower troposphere,” in Advances in Atmospheric Remote Sensing with Lidar, Selected Papers of the 18th International Laser Radar Conference, A. Ansmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, Berlin, Germany, 1996), pp. 383–386.

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H. Eisele, T. Trickl, “Second Generation of the IFU Stationary Tropospheric Ozone Lidar,” in Advances in Atmospheric Remote Sensing with Lidar, Selected Papers of the 18th International Laser Radar Conference, A. Ansmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, Berlin, Germany, 1996), pp. 379–382.

H. Eisele, T. Trickl, H. Claude, “Lidar als wichtige Ergänzung zur Messung troposphärischen Ozons,” in Ozonbulletin des Deutschen Wetterdiensts, Ausgabe Nr. 44, Erscheinungstermin, 25August1997 (Deutscher Wetterdienst, Offenbach, Germany, 1997), in German; http://www.dwd.de/de/FundE/Observator/MOHP/hp2/ozon/bulletin.htm .

Tryon, P. J.

Völger, P.

P. Völger, J. Bösenberg, I. Schult, “Scattering properties of selected model aerosols calculated at UV-wavelengths: implications for DIAL measurements of tropospheric ozone,” Beitr. Phys. Atmos. 69, 177–187 (1996).

Wandinger, U.

D. Müller, I. Mattis, U. Wandinger, A. Ansmann, D. Althausen, O. Dubovik, S. Eckhardt, A. Stohl, “Saharan dust over a central European EARLINET–AERONET site: combined observations with Raman lidar and Sun photometer,” J. Geophys. Res. 108 (D12), 4345, doi: (2003).
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Weinman, J. A.

Weitkamp, C.

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C. Werner, scientific coordinator,“Improvement of lidar measurement technique for discrimination of polar stratospheric clouds and volcanic aerosols,” Final Report, European Union, Environment Research Programme Report Nr.:EV5V-CT02-0066 (European Union, Brussels, 1995).

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V. Matthias, J. Bösenberg, V. Wulfmeyer, “Improvement of ozone measurements with DIAL by using an additional Raman channel,” in Proceedings of the EUROTRAC Symposium 1994, P. M. Borrell, P. Borrell, T. Cvitas, W. Seiler, eds. (SPB Academic, Den Haag, The Netherlands, 1994), pp. 326–329.

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H. Eisele, “Aufbau und Betrieb eines Dreiwellenlängen-Lidars für Ozonmessungen in der gesamten Troposphäre und En-twicklung eines neuen Auswerteverfahrens zur Aerosol-korrektur,” Dissertation, Universität Tübingen, 1997 [published as Schriftenreihe des Fraunhofer-Instituts für At-mosphärische Umweltforschung, Vol. 55 (Verlag Ma-raun W. Dr., Frankfurt/Main, Germany, 1998), ISBN 3-932666-08-9, in German].

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H. Eisele, T. Trickl, “Second Generation of the IFU Stationary Tropospheric Ozone Lidar,” in Advances in Atmospheric Remote Sensing with Lidar, Selected Papers of the 18th International Laser Radar Conference, A. Ansmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, Berlin, Germany, 1996), pp. 379–382.

H. Eisele, T. Trickl, “Lidar sounding of tropospheric ozone at Garmisch-Partenkirchen,” in Atmospheric Ozone, Proceedings of the 18th Quadrennial Ozone Symposium, R. D. Bojkov, G. Visconti, eds. (International Ozone Commission, Genève, Switzerland, 1998), pp. 351–354.

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Figures (8)

Fig. 1
Fig. 1

Quality function of a PBL aerosol layer with BP = 0.04 sr−1 and βm = 3.2 × 10−6 sr−1 m−1.

Fig. 2
Fig. 2

Quality function Qδ of the two-wavelength method for a synthetic cloud with δ = γ = 0.5.

Fig. 3
Fig. 3

Quality function Qγ of the two-wavelength method for the synthetic cloud of Fig. 2 with δ = γ = 0.5.

Fig. 4
Fig. 4

Quality function Qγ of the three-wavelength method for the synthetic cloud of Fig. 2 with δ = γ = 0.5.

Fig. 5
Fig. 5

Ozone retrievals for a measurement taken at 2:00 CET on 1 June 1996. The application of the two-wavelength method yields an ozone distribution free of any of the gradients seen in the uncorrected ozone profile. The filled squares represent the corresponding in situ measurements at the Wank and Zugspitze stations of the IMK-IFU.

Fig. 6
Fig. 6

Ozone retrievals for a measurement taken at 15:20 CET on 9 April 1997. The aerosol correction is based on the three-wavelength method. If one takes into account the differences in smoothing the errors of the uncorrected profiles next to a 3100-m scale as a function of the DIAL wavelengths as known from the literature.2,43 Our result (solid curve) was confirmed by a calculation with the dual-DIAL method.

Fig. 7
Fig. 7

Measurement taken on 18 October 1996 at 11:00 p.m. CET in the presence of a contrail with an optical depth of τ = 0.28 at 8 km. The aerosol correction was performed with the two-wavelength method and its result is compared with a measurement taken without a contrail 1 h later.

Fig. 8
Fig. 8

Quality function Qn corresponding to the method proposed by Sasano et al.,28 again for the synthetic cloud of Fig. 2 with δ = γ = 0.5.

Tables (1)

Tables Icon

Table 1 Aerosol Parameters Retrieved from Measurements on 7 May 1996

Equations (31)

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Q ( x ) = min ! , x = ( x 1 , .. , x n ) n .
β K ( r ) = X ( r ) exp [ Y ( r ) ] { X ( r m ) β m + 2 r r m X ( r ) B P ( r ) exp [ Y ( r ) ] d r } ,
Y ( r ) = 2 r r m [ 1 B P ( r ) 1 B R ] β R ( r ) d r 2 r r m α O 3 ( r ) d r ,
β K = exp [ 2 ( α P + B P 1 β R ) ( r m r ) ] β m 1 + exp [ 2 ( α P + B P 1 β R ) ( r m r ) ] 1 B P α P + β P ,
α K , P = β K β R B P α P
β K ( r s ) = lim ɛ 0 [ β K ( r s + ɛ ) P ( r s ) P ( r s + ɛ ) ] .
α K ( r s ) = lim ɛ 0 [ α K ( r s + ɛ ) P ( r s ) P ( r s + ɛ ) B P ( r s + ɛ ) B P ( r s ) ] .
α P ( r ) = 1 2 d d r ln ( n ( r ) P ( r ) r 2 ) α R ( r ) + 1 2 d d r ln [ β ( r ) n ( r ) ] .
R L = β P ( r ) + β R ( r ) β R ( r ) ,
R L = B P B R { 1 2 α R ( r ) d d r ln [ n ( r ) P ( r ) r 2 ] 1 } + 1 ,
R ( r ) = R L 1 1 + C exp [ 2 K n ( r ) d r ] K ,
K = σ P , h + B P 1 d σ R d Ω ,
d d r ln [ P ( r , λ 1 ) P ( r , λ 2 ) ] = 2 n ( r ) F ( λ 1 , λ 2 ) ,
Q = layers r i = bottom top [ β P Klett ( r i ) / ( B P ) L α P slope ( r i ) ] 2 .
ln [ r 2 P ( r ) ] ln [ r 2 P ( r ) ] + 2 0 r α O 3 ( r ) d r ,
n O 3 = 1 2 Δ σ O 3 d d r ln [ P ( λ 1 , r ) P ( λ 2 , r ) ] 1 2 Δ σ O 3 d d r ln [ β ( λ 1 , r ) β ( λ 2 , r ) ] correction term 1 1 Δ σ O 3 [ α ( λ 2 , r ) α ( λ 1 , r ) ] correction term 2 .
α ( λ , r ) = α ( λ 0 , r ) ( λ λ 0 ) γ , β ( λ , r ) = β ( λ 0 , r ) ( λ λ 0 ) δ .
Q δ = L n O 3 ( r ) d d r ln [ β ( λ ! , r ) β ( λ 2 , r ) ] d r 1 l L n O 3 ( r ) d r L d d r ln [ β ( λ ! , r ) β ( λ 2 , r ) ] d r σ β 2 σ O 3 2 ,
Q γ = L n O 3 ( r ) [ α P ( λ 1 , r ) α P ( λ 2 , r ) ] d r 1 l L n O 3 ( r ) d r L [ α P ( λ 1 , r ) α P ( λ 2 , r ) ] d r σ α 2 σ O 3 2 .
σ β 2 = L { d d r ln [ β ( λ ! , r ) β ( λ 2 , r ) ] } 2 d r 1 l { L d d r ln [ β ( λ ! , r ) β ( λ 2 , r ) ] d r } 2 , σ O 3 2 = L [ n O 3 ( r ) ] 2 d r 1 l [ L n O 3 ( r ) d r ] 2 , σ α 2 = L [ α P ( λ 1 , r ) α P ( λ 2 , r ) ] 2 d r 1 l { L [ α P ( λ 1 , r ) α P ( λ 2 , r ) ] d r } 2 ,
d d r ln [ P ( λ 1 , r ) P ( λ 2 , r ) ] d d r ln [ β ( λ ! , r ) β ( λ 2 , r ) ] 2 [ α ( λ 2 , r ) α ( λ 1 , r ) ] ,
Q γ = | 1 l L n O 3 ( λ 1 , λ 3 ) d r 1 l L n O 3 ( λ 2 , λ 3 ) d r | ,
Q n = L [ n O 3 ( λ 1 , λ 3 ) n O 3 ( λ 2 , λ 3 ) ] 2 d r , Q β = L [ β P ( λ 1 , λ 3 ) β P ( λ 2 , λ 3 ) ] 2 d r .
d d r ln β 1 β 2 = d d r ln R 1 R 2
d d r ln β R ( λ 1 , r ) β R ( λ 2 , r ) = 0 .
ln R 1 = ln [ 1 + ( R 2 1 ) ( λ 1 λ 2 ) 4 δ ] ln R 2 ( 4 δ ) R 2 1 R 2 ( 1 λ 1 λ 2 ) .
d d r ln β 1 β 2 ( 4 δ ) ( 1 λ 1 λ 2 ) d d r R 2 1 R 2 .
L N t A t d r 1 l L N t d r L A t d r = 0 .
Δ α P = ( α 2 α 1 ) P = ( α 2 ) P [ 1 ( λ 1 λ 2 ) γ ] .
q = [ 1 ( λ 1 λ 2 ) γ ] / [ 1 ( λ 1 λ 2 ) γ t ] .
L N t ( Δ α P ) t d r 1 l L N t ( Δ α P ) t d r = 0 .

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