Abstract

In a critique of two-wavelength lidar inversion techniques, Kunz [Appl. Opt. 38, 1015 (1999)] presented mathematical arguments that such techniques cannot yield unique solutions for extinction profiles. Ackermann [Appl. Opt. 38, 7414 (1999)] presented an analytical solution for the extinction profile from a two-wavelength lidar and also attempted to refute Kunz’s mathematical arguments. However, the fundamental reasons why the authors of these two papers reached different conclusions were not fully uncovered. These previous papers are critically examined, and a new mathematical proof of uniqueness is provided. Further analyses are presented to explain how the technique works, along with comments on its limitations.

© 2001 Optical Society of America

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References

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  1. J. F. Potter, “Two-frequency lidar inversion technique,” Appl. Opt. 26, 1250–1256 (1987).
    [CrossRef] [PubMed]
  2. J. Ackermann, “Two-wavelength lidar inversion algorithm for a two-component atmosphere,” Appl. Opt. 36, 5134–5143 (1997).
    [CrossRef] [PubMed]
  3. G. J. Kunz, “Two-wavelength lidar inversion algorithm,” Appl. Opt. 38, 1015–1020 (1999).
    [CrossRef]
  4. J. Ackermann, “Analytical solution of the two-frequency lidar inversion technique,” Appl. Opt. 38, 7414–7418 (1999).
    [CrossRef]
  5. The literature is inconsistent on this point. In Refs. 1 and 4 (and in this comment) the subscript S denotes the wavelength with the smaller extinction coefficient. In Refs. 2 and 3, it denotes the shorter wavelength.
  6. J. D. Klett, “Stable analytical inversion solution for processing lidar returns,” Appl. Opt. 20, 211–220 (1981).
    [CrossRef] [PubMed]
  7. F. G. Fernald, B. M. Herman, J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11, 482–489 (1972).
    [CrossRef]
  8. A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter LIDAR for vertical profiling of moisture, aerosol extinction, backscatter, and LIDAR ratio,” Appl. Phys. B 55, 18–29 (1992).
    [CrossRef]
  9. A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, 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]

1999 (2)

1997 (1)

1992 (2)

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, 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, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter LIDAR for vertical profiling of moisture, aerosol extinction, backscatter, and LIDAR ratio,” Appl. Phys. B 55, 18–29 (1992).
[CrossRef]

1987 (1)

1981 (1)

1972 (1)

F. G. Fernald, B. M. Herman, J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11, 482–489 (1972).
[CrossRef]

Ackermann, J.

Ansmann, A.

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, 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, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter LIDAR for vertical profiling of moisture, aerosol extinction, backscatter, and LIDAR ratio,” Appl. Phys. B 55, 18–29 (1992).
[CrossRef]

Fernald, F. G.

F. G. Fernald, B. M. Herman, J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11, 482–489 (1972).
[CrossRef]

Herman, B. M.

F. G. Fernald, B. M. Herman, J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11, 482–489 (1972).
[CrossRef]

Klett, J. D.

Kunz, G. J.

Lahmann, W.

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter LIDAR for vertical profiling of moisture, aerosol extinction, backscatter, and LIDAR ratio,” Appl. Phys. B 55, 18–29 (1992).
[CrossRef]

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, 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]

Michaelis, W.

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, 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, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter LIDAR for vertical profiling of moisture, aerosol extinction, backscatter, and LIDAR ratio,” Appl. Phys. B 55, 18–29 (1992).
[CrossRef]

Potter, J. F.

Reagan, J. A.

F. G. Fernald, B. M. Herman, J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11, 482–489 (1972).
[CrossRef]

Riebesell, M.

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter LIDAR for vertical profiling of moisture, aerosol extinction, backscatter, and LIDAR ratio,” Appl. Phys. B 55, 18–29 (1992).
[CrossRef]

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, 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]

Voss, E.

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, 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, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter LIDAR for vertical profiling of moisture, aerosol extinction, backscatter, and LIDAR ratio,” Appl. Phys. B 55, 18–29 (1992).
[CrossRef]

Wandinger, U.

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter LIDAR for vertical profiling of moisture, aerosol extinction, backscatter, and LIDAR ratio,” Appl. Phys. B 55, 18–29 (1992).
[CrossRef]

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, 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]

Weitkamp, C.

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, 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, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter LIDAR for vertical profiling of moisture, aerosol extinction, backscatter, and LIDAR ratio,” Appl. Phys. B 55, 18–29 (1992).
[CrossRef]

Appl. Opt. (6)

Appl. Phys. B (1)

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter LIDAR for vertical profiling of moisture, aerosol extinction, backscatter, and LIDAR ratio,” Appl. Phys. B 55, 18–29 (1992).
[CrossRef]

J. Appl. Meteorol. (1)

F. G. Fernald, B. M. Herman, J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11, 482–489 (1972).
[CrossRef]

Other (1)

The literature is inconsistent on this point. In Refs. 1 and 4 (and in this comment) the subscript S denotes the wavelength with the smaller extinction coefficient. In Refs. 2 and 3, it denotes the shorter wavelength.

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Equations (30)

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LLR=KLβLRexp-2 0R αLrdr,
LSR=KSβSRexp-2 0R αSrdr,
αSR=kαLR where k is a constant,
βSR=αSR/SS,
βLR=αLR/SL.
αLR=LLR2 R0RF LLrdrTLR0, RF-2-1+2 RRF LLrdr,
αSR=LSR2 R0RF LSrdrTSR0, RF-2-1+2 RRF LSrdr,
TLR0, RF=exp-R0RF αLrdr,
TSR0, RF=exp-R0RF αSrdr.
LLR1LSR2LSR1LLR2=exp-2 R1R2 αSrdrexp-2 R1R2 αLrdr.
LLR0LSRFLSR0LLRF=TS2R0, RFTL2R0, RF.
LLR0LSR0LSRLLR=exp-2k-1R0RαLrdr.
exp-2k-1R0RαLrdr=exp-2k*-1R0RαL*rdr.
R0RαLrdrR0RαL*rdr=k*-1k-1.
αLR=k*-1k-1 αL*R.
2 R0RF LLrdrTLR0, RF-2-1+2 RRF LLrdr=LLRαLR,
2 R0RF LLrdrTL*R0, RF-2-1+2 RRF LLrdr=LLRαL*R.
2 R0RF LLrdr1TLR0, RF-2-1-1TL*R0, RF-2=LLRαLR1-k*-1k-1.
LLRαLR=KLSLexp-2 0R αLrdr,
LLRiLSRjLSRiLLRj=exp-21-kRiRj αLrdr
kij=122-lnLLRiLSRjLSRiLLRjRiRj αLrdr
LLR=KLαLRSLexp-2 0R αLrdr,
LSR=KSkαLRSSexp-2k 0R αLrdr.
LLRkLS=1kKLkKSSSSLkαLRk-1,
LLRLLR0kLSR0LSR=αLRαLR0k-1.
BL-BˆL=CBS-BˆS.
BL=R0RF LLrdr,
DLR=RRF LLrdr,
αSRαLR=LSRLLR1C1-CTL2R0, RF1-TL2R0, RF×BL+TL-2R0, RF-1DLRBS+1C TL-2R0, RF-1DSR,
αSR0αLR0=αSRFαLRF.

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