Abstract

A numerical investigation was carried out into the feasibility of deriving the aerosol size distribution from aerosol volume extinction and backscattering coefficient measurements by a multiwavelength laser radar. This study employs the regularization method for matrix inversion with the first-order B-spline function as basis functions to approximate the aerosol size distribution. The results of numerical simulations show that (1) the effects of round off errors in the numerical calculation are negligible and the approximation errors in the size distribution by the B-spline function are small, (2) the reconstruction errors in the size distribution at its peak are about twice as large as the relative measurement errors when the Lagrange multiplier, which determines the degree of smoothness in the reconstruction, is suitably chosen, and (3) the variation in the complex refractive index due to the humidity change does not produce large errors in the size distribution.

© 1989 Optical Society of America

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References

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  1. M. D. King, “Sensitivity of Constrained Linear Inversions to the Selection of the Lagrange Multiplier,” J. Atmos. Sci. 39, 1356–1369 (1982).
    [CrossRef]
  2. M. Tanaka, T. Nakajima, T. Takamura, “Simultaneous Determination of Complex Refractive Index and Size Distribution of Airborne and Water-Suspended Particles from Light Scattering Measurements,” J. Meteorol. Soc. Jpn. 60, 1259–1272 (1982).
  3. J. D. Spinhirne, J. A. Reagan, B. M. Herman, “Vertical Distribution of Aerosol Extinction Cross Section and Interference of Aerosol Imaginary Index in the Troposphere by Lidar Technique,” J. Appl. Meteorol. 19, 426–438 (1980).
    [CrossRef]
  4. J. D. Klett, “Stable Analytical Inversion Solution for Processing Lidar Returns,” Appl. Opt. 20, 211–220 (1981).
    [CrossRef] [PubMed]
  5. H. Salemink, P. Schitanus, J. Bergwerff, “Atmospheric Parameters at 532 and 1064 nm: Quantitative Vertical Profiles, Humidity and Wavelength,” at Twelfth International Laser Radar Conference, Aix-en-Provence (1984).
  6. F. G. Fernald, “Analysis of Atmospheric Lidar Observation: Some Comments,” Appl. Opt. 23, 652–653 (1984).
    [CrossRef] [PubMed]
  7. Y. Sasano, H. Nakane, “Significance of the Extinction/ Backscatter Ratio and the Boundary Value Term in the Solution for the Two-Component Lidar Equation,” Appl. Opt. 23, 11–13 (1984).
    [CrossRef]
  8. H. Nakane et al., “Obtaining Quantitative Aerosol Concentration Distribution for Wide Area with Large Laser Radar,” Research Report from the National Institute for Environmental Studies, Japan, No. 77, 75–98 (1985), in Japanese.
  9. J. D. Klett, “Lidar Inversion with Variable Backscatter/Extinction Ratios,” Appl. Opt. 24, 1638–1643 (1985).
    [CrossRef] [PubMed]
  10. Y. Sasano, E. V. Browell, “Wavelength Dependence of Aerosol Backscatter Coefficients Obtained by Multiple Wavelength Lidar Measurements,” at Thirteenth International Laser Radar Conference, Toronto (1986).
  11. Y. Sasano, E. V. Browell, S. Ismail, “Error Caused by Using a Constant Extinction/Backscattering Ratio in the Lidar Solution,” Appl. Opt. 24, 3929–3932 (1985).
    [CrossRef] [PubMed]
  12. J. F. Potter, “Two-Frequency Lidar Inversion Technique,” Appl. Opt. 20, 1250–1256 (1987).
    [CrossRef]
  13. D. L. Phillips, “A Technique for the Numerical Solution of Certain Integral Equations of the First Kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
    [CrossRef]
  14. A. N. Tikhonov, “On the Solution of Incorrectly Stated Problems and a Method of Regularization,” Dokl. Arad. Nauk SSSR. 151, 501–504 (1963).
  15. S. Twomey, “On the Numerical Solution of Fredholm Integral Equation of the First Kind by the Inversion of the Linear System Produced by Quadrature,” J. Assoc. Comput. Mach. 10, 97–101 (1963).
    [CrossRef]
  16. H. Müller, H. Quenzel, “Information Content of Multispectral Lidar Measurements with Respect to the Aerosol Size Distribution,” Appl. Opt. 24, 648–654 (1985).
    [CrossRef] [PubMed]
  17. J. Heintzenberg, H. Müller, H. Quenzel, E. Thomalla, “Information Content of Optical Data with Respect to Aerosol Properties: Numerical Studies with a Randomized Minimization-Search-Technique Inversion Algorithm,” Appl. Opt. 20, 1308–1315 (1981).
    [CrossRef] [PubMed]
  18. G. E. Shaw, “Inversion of Optical Scattering and Spectral Extinction Measurements to Recover Aerosol Size Spectra,” Appl. Opt. 18, 988–993 (1979).
    [CrossRef] [PubMed]
  19. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).
  20. G. Yamamoto, M. Tanaka, “Determination of Aerosol Size Distribution from Spectral Attenuation Measurements,” Appl. Opt. 8, 447–453 (1969).
    [CrossRef] [PubMed]
  21. C. E. Junge, “The Size Distribution and Aging of Natural Aerosols as Determined from Electrical and Optical Data in the Atmosphere,”J. Meteorol. 12, 13–25 (1955).
    [CrossRef]
  22. D. J. Hoffmann, J. M. Rosen, “Sulfuric Acid Droplet Formation and Growth in the Stratosphere after the 1982 Eruption of El Chicon,” Science 222, 325–327 (1983).
    [CrossRef]
  23. T. Takamura, M. Tanaka, T. Nakajima, “Effects of Atmospheric Humidity on the Refractive Index and the Size Distribution of Aerosols as Estimated from Light Scattering Measurements,” J. Meteorol. Soc. Jpn. 62, 573–582 (1984).

1987 (1)

J. F. Potter, “Two-Frequency Lidar Inversion Technique,” Appl. Opt. 20, 1250–1256 (1987).
[CrossRef]

1985 (3)

1984 (3)

Y. Sasano, H. Nakane, “Significance of the Extinction/ Backscatter Ratio and the Boundary Value Term in the Solution for the Two-Component Lidar Equation,” Appl. Opt. 23, 11–13 (1984).
[CrossRef]

T. Takamura, M. Tanaka, T. Nakajima, “Effects of Atmospheric Humidity on the Refractive Index and the Size Distribution of Aerosols as Estimated from Light Scattering Measurements,” J. Meteorol. Soc. Jpn. 62, 573–582 (1984).

F. G. Fernald, “Analysis of Atmospheric Lidar Observation: Some Comments,” Appl. Opt. 23, 652–653 (1984).
[CrossRef] [PubMed]

1983 (1)

D. J. Hoffmann, J. M. Rosen, “Sulfuric Acid Droplet Formation and Growth in the Stratosphere after the 1982 Eruption of El Chicon,” Science 222, 325–327 (1983).
[CrossRef]

1982 (2)

M. D. King, “Sensitivity of Constrained Linear Inversions to the Selection of the Lagrange Multiplier,” J. Atmos. Sci. 39, 1356–1369 (1982).
[CrossRef]

M. Tanaka, T. Nakajima, T. Takamura, “Simultaneous Determination of Complex Refractive Index and Size Distribution of Airborne and Water-Suspended Particles from Light Scattering Measurements,” J. Meteorol. Soc. Jpn. 60, 1259–1272 (1982).

1981 (2)

1980 (1)

J. D. Spinhirne, J. A. Reagan, B. M. Herman, “Vertical Distribution of Aerosol Extinction Cross Section and Interference of Aerosol Imaginary Index in the Troposphere by Lidar Technique,” J. Appl. Meteorol. 19, 426–438 (1980).
[CrossRef]

1979 (1)

1969 (1)

1963 (2)

A. N. Tikhonov, “On the Solution of Incorrectly Stated Problems and a Method of Regularization,” Dokl. Arad. Nauk SSSR. 151, 501–504 (1963).

S. Twomey, “On the Numerical Solution of Fredholm Integral Equation of the First Kind by the Inversion of the Linear System Produced by Quadrature,” J. Assoc. Comput. Mach. 10, 97–101 (1963).
[CrossRef]

1962 (1)

D. L. Phillips, “A Technique for the Numerical Solution of Certain Integral Equations of the First Kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[CrossRef]

1955 (1)

C. E. Junge, “The Size Distribution and Aging of Natural Aerosols as Determined from Electrical and Optical Data in the Atmosphere,”J. Meteorol. 12, 13–25 (1955).
[CrossRef]

Bergwerff, J.

H. Salemink, P. Schitanus, J. Bergwerff, “Atmospheric Parameters at 532 and 1064 nm: Quantitative Vertical Profiles, Humidity and Wavelength,” at Twelfth International Laser Radar Conference, Aix-en-Provence (1984).

Browell, E. V.

Y. Sasano, E. V. Browell, S. Ismail, “Error Caused by Using a Constant Extinction/Backscattering Ratio in the Lidar Solution,” Appl. Opt. 24, 3929–3932 (1985).
[CrossRef] [PubMed]

Y. Sasano, E. V. Browell, “Wavelength Dependence of Aerosol Backscatter Coefficients Obtained by Multiple Wavelength Lidar Measurements,” at Thirteenth International Laser Radar Conference, Toronto (1986).

Fernald, F. G.

Heintzenberg, J.

Herman, B. M.

J. D. Spinhirne, J. A. Reagan, B. M. Herman, “Vertical Distribution of Aerosol Extinction Cross Section and Interference of Aerosol Imaginary Index in the Troposphere by Lidar Technique,” J. Appl. Meteorol. 19, 426–438 (1980).
[CrossRef]

Hoffmann, D. J.

D. J. Hoffmann, J. M. Rosen, “Sulfuric Acid Droplet Formation and Growth in the Stratosphere after the 1982 Eruption of El Chicon,” Science 222, 325–327 (1983).
[CrossRef]

Ismail, S.

Junge, C. E.

C. E. Junge, “The Size Distribution and Aging of Natural Aerosols as Determined from Electrical and Optical Data in the Atmosphere,”J. Meteorol. 12, 13–25 (1955).
[CrossRef]

King, M. D.

M. D. King, “Sensitivity of Constrained Linear Inversions to the Selection of the Lagrange Multiplier,” J. Atmos. Sci. 39, 1356–1369 (1982).
[CrossRef]

Klett, J. D.

Müller, H.

Nakajima, T.

T. Takamura, M. Tanaka, T. Nakajima, “Effects of Atmospheric Humidity on the Refractive Index and the Size Distribution of Aerosols as Estimated from Light Scattering Measurements,” J. Meteorol. Soc. Jpn. 62, 573–582 (1984).

M. Tanaka, T. Nakajima, T. Takamura, “Simultaneous Determination of Complex Refractive Index and Size Distribution of Airborne and Water-Suspended Particles from Light Scattering Measurements,” J. Meteorol. Soc. Jpn. 60, 1259–1272 (1982).

Nakane, H.

Y. Sasano, H. Nakane, “Significance of the Extinction/ Backscatter Ratio and the Boundary Value Term in the Solution for the Two-Component Lidar Equation,” Appl. Opt. 23, 11–13 (1984).
[CrossRef]

H. Nakane et al., “Obtaining Quantitative Aerosol Concentration Distribution for Wide Area with Large Laser Radar,” Research Report from the National Institute for Environmental Studies, Japan, No. 77, 75–98 (1985), in Japanese.

Phillips, D. L.

D. L. Phillips, “A Technique for the Numerical Solution of Certain Integral Equations of the First Kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[CrossRef]

Potter, J. F.

J. F. Potter, “Two-Frequency Lidar Inversion Technique,” Appl. Opt. 20, 1250–1256 (1987).
[CrossRef]

Quenzel, H.

Reagan, J. A.

J. D. Spinhirne, J. A. Reagan, B. M. Herman, “Vertical Distribution of Aerosol Extinction Cross Section and Interference of Aerosol Imaginary Index in the Troposphere by Lidar Technique,” J. Appl. Meteorol. 19, 426–438 (1980).
[CrossRef]

Rosen, J. M.

D. J. Hoffmann, J. M. Rosen, “Sulfuric Acid Droplet Formation and Growth in the Stratosphere after the 1982 Eruption of El Chicon,” Science 222, 325–327 (1983).
[CrossRef]

Salemink, H.

H. Salemink, P. Schitanus, J. Bergwerff, “Atmospheric Parameters at 532 and 1064 nm: Quantitative Vertical Profiles, Humidity and Wavelength,” at Twelfth International Laser Radar Conference, Aix-en-Provence (1984).

Sasano, Y.

Schitanus, P.

H. Salemink, P. Schitanus, J. Bergwerff, “Atmospheric Parameters at 532 and 1064 nm: Quantitative Vertical Profiles, Humidity and Wavelength,” at Twelfth International Laser Radar Conference, Aix-en-Provence (1984).

Shaw, G. E.

Spinhirne, J. D.

J. D. Spinhirne, J. A. Reagan, B. M. Herman, “Vertical Distribution of Aerosol Extinction Cross Section and Interference of Aerosol Imaginary Index in the Troposphere by Lidar Technique,” J. Appl. Meteorol. 19, 426–438 (1980).
[CrossRef]

Takamura, T.

T. Takamura, M. Tanaka, T. Nakajima, “Effects of Atmospheric Humidity on the Refractive Index and the Size Distribution of Aerosols as Estimated from Light Scattering Measurements,” J. Meteorol. Soc. Jpn. 62, 573–582 (1984).

M. Tanaka, T. Nakajima, T. Takamura, “Simultaneous Determination of Complex Refractive Index and Size Distribution of Airborne and Water-Suspended Particles from Light Scattering Measurements,” J. Meteorol. Soc. Jpn. 60, 1259–1272 (1982).

Tanaka, M.

T. Takamura, M. Tanaka, T. Nakajima, “Effects of Atmospheric Humidity on the Refractive Index and the Size Distribution of Aerosols as Estimated from Light Scattering Measurements,” J. Meteorol. Soc. Jpn. 62, 573–582 (1984).

M. Tanaka, T. Nakajima, T. Takamura, “Simultaneous Determination of Complex Refractive Index and Size Distribution of Airborne and Water-Suspended Particles from Light Scattering Measurements,” J. Meteorol. Soc. Jpn. 60, 1259–1272 (1982).

G. Yamamoto, M. Tanaka, “Determination of Aerosol Size Distribution from Spectral Attenuation Measurements,” Appl. Opt. 8, 447–453 (1969).
[CrossRef] [PubMed]

Thomalla, E.

Tikhonov, A. N.

A. N. Tikhonov, “On the Solution of Incorrectly Stated Problems and a Method of Regularization,” Dokl. Arad. Nauk SSSR. 151, 501–504 (1963).

Twomey, S.

S. Twomey, “On the Numerical Solution of Fredholm Integral Equation of the First Kind by the Inversion of the Linear System Produced by Quadrature,” J. Assoc. Comput. Mach. 10, 97–101 (1963).
[CrossRef]

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).

Yamamoto, G.

Appl. Opt. (10)

G. Yamamoto, M. Tanaka, “Determination of Aerosol Size Distribution from Spectral Attenuation Measurements,” Appl. Opt. 8, 447–453 (1969).
[CrossRef] [PubMed]

G. E. Shaw, “Inversion of Optical Scattering and Spectral Extinction Measurements to Recover Aerosol Size Spectra,” Appl. Opt. 18, 988–993 (1979).
[CrossRef] [PubMed]

J. D. Klett, “Stable Analytical Inversion Solution for Processing Lidar Returns,” Appl. Opt. 20, 211–220 (1981).
[CrossRef] [PubMed]

J. Heintzenberg, H. Müller, H. Quenzel, E. Thomalla, “Information Content of Optical Data with Respect to Aerosol Properties: Numerical Studies with a Randomized Minimization-Search-Technique Inversion Algorithm,” Appl. Opt. 20, 1308–1315 (1981).
[CrossRef] [PubMed]

F. G. Fernald, “Analysis of Atmospheric Lidar Observation: Some Comments,” Appl. Opt. 23, 652–653 (1984).
[CrossRef] [PubMed]

J. D. Klett, “Lidar Inversion with Variable Backscatter/Extinction Ratios,” Appl. Opt. 24, 1638–1643 (1985).
[CrossRef] [PubMed]

Y. Sasano, E. V. Browell, S. Ismail, “Error Caused by Using a Constant Extinction/Backscattering Ratio in the Lidar Solution,” Appl. Opt. 24, 3929–3932 (1985).
[CrossRef] [PubMed]

H. Müller, H. Quenzel, “Information Content of Multispectral Lidar Measurements with Respect to the Aerosol Size Distribution,” Appl. Opt. 24, 648–654 (1985).
[CrossRef] [PubMed]

Y. Sasano, H. Nakane, “Significance of the Extinction/ Backscatter Ratio and the Boundary Value Term in the Solution for the Two-Component Lidar Equation,” Appl. Opt. 23, 11–13 (1984).
[CrossRef]

J. F. Potter, “Two-Frequency Lidar Inversion Technique,” Appl. Opt. 20, 1250–1256 (1987).
[CrossRef]

Dokl. Arad. Nauk SSSR. (1)

A. N. Tikhonov, “On the Solution of Incorrectly Stated Problems and a Method of Regularization,” Dokl. Arad. Nauk SSSR. 151, 501–504 (1963).

J. Appl. Meteorol. (1)

J. D. Spinhirne, J. A. Reagan, B. M. Herman, “Vertical Distribution of Aerosol Extinction Cross Section and Interference of Aerosol Imaginary Index in the Troposphere by Lidar Technique,” J. Appl. Meteorol. 19, 426–438 (1980).
[CrossRef]

J. Assoc. Comput. Mach. (2)

S. Twomey, “On the Numerical Solution of Fredholm Integral Equation of the First Kind by the Inversion of the Linear System Produced by Quadrature,” J. Assoc. Comput. Mach. 10, 97–101 (1963).
[CrossRef]

D. L. Phillips, “A Technique for the Numerical Solution of Certain Integral Equations of the First Kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[CrossRef]

J. Atmos. Sci. (1)

M. D. King, “Sensitivity of Constrained Linear Inversions to the Selection of the Lagrange Multiplier,” J. Atmos. Sci. 39, 1356–1369 (1982).
[CrossRef]

J. Meteorol. (1)

C. E. Junge, “The Size Distribution and Aging of Natural Aerosols as Determined from Electrical and Optical Data in the Atmosphere,”J. Meteorol. 12, 13–25 (1955).
[CrossRef]

J. Meteorol. Soc. Jpn. (2)

T. Takamura, M. Tanaka, T. Nakajima, “Effects of Atmospheric Humidity on the Refractive Index and the Size Distribution of Aerosols as Estimated from Light Scattering Measurements,” J. Meteorol. Soc. Jpn. 62, 573–582 (1984).

M. Tanaka, T. Nakajima, T. Takamura, “Simultaneous Determination of Complex Refractive Index and Size Distribution of Airborne and Water-Suspended Particles from Light Scattering Measurements,” J. Meteorol. Soc. Jpn. 60, 1259–1272 (1982).

Science (1)

D. J. Hoffmann, J. M. Rosen, “Sulfuric Acid Droplet Formation and Growth in the Stratosphere after the 1982 Eruption of El Chicon,” Science 222, 325–327 (1983).
[CrossRef]

Other (4)

H. Salemink, P. Schitanus, J. Bergwerff, “Atmospheric Parameters at 532 and 1064 nm: Quantitative Vertical Profiles, Humidity and Wavelength,” at Twelfth International Laser Radar Conference, Aix-en-Provence (1984).

H. Nakane et al., “Obtaining Quantitative Aerosol Concentration Distribution for Wide Area with Large Laser Radar,” Research Report from the National Institute for Environmental Studies, Japan, No. 77, 75–98 (1985), in Japanese.

Y. Sasano, E. V. Browell, “Wavelength Dependence of Aerosol Backscatter Coefficients Obtained by Multiple Wavelength Lidar Measurements,” at Thirteenth International Laser Radar Conference, Toronto (1986).

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).

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Figures (7)

Fig. 1
Fig. 1

Approximation of a (a) given aerosol size distribution using (b) the zeroth-order B-spline function and (c) the first-order B-spline function as the basis function.

Fig. 2
Fig. 2

Wavelength dependence of kernel function.

Fig. 3
Fig. 3

Results of reconstructed distribution without measurement errors.

Fig. 4
Fig. 4

The γ-dependence of the results of inversion with relative measurement errors of 5%. The solid and dashed lines in (a) denote Q 1 ¯ and Δ f ¯ and of the size distribution as solutions, respectively. The dashed and solid lines in (b) indicate the given size distribution and range of standard deviation solutions, respectively.

Fig. 5
Fig. 5

As in Fig. 4 except for refractive measurement errors of 1%.

Fig. 6
Fig. 6

As in Fig. 4 except for refractive measurement errors of 10%.

Fig. 7
Fig. 7

Variation of solutions with the complex refractive index of aerosols. The dashed lines indicate the given size distributions, and the solid lines are reconstructed solutions.

Tables (2)

Tables Icon

Table I Dependence of the Errors in Size Distribution on the Measurement Errors and Lagrange Multiplier

Tables Icon

Table II Dependence of the Complex Refractive Index on the Relative Humidity Measured, Using Polar Nephelometers in Sendai, Japan

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

α l ( R ) = 0 π r 2 Q E ( m , 2 π r / λ l ) n ( r ) d r l = 1 , 2 , 3 , 4 ,
β l ( R ) = 0 π r 2 Q B ( m , 2 π r / λ l ) n ( r ) d r l = 1 , 2 , 3 , 4 ,
g i = x a x b K ( x , λ i , m ) f ( x ) d x i = 1 , 2 , , 8 ,
n ( r ) = d N d r = C r ( η + 1 ) ,
f ( x ) = 4 π 3 r 3 d N d ln r ,
K ( x , λ i , m ) = 3 4 r Q L ( m , 2 π r / λ i ) , L = E or B .
f ( x ) = j = 1 8 f j B j ( x ) + d ( x ) ,
B j ( x ) = { 0 ( x x j 1 ) , 1 x j x x j x j 1 ( x j 1 < x x j ) , 1 x x j + 1 x j x ( x j < x x j + 1 ) , 0 ( x > x j + 1 ) .
g i = j = 1 8 A i j f j + i t , i = 1 , 2 , , 8 ,
A i j = x j 1 x j + 1 K ( x , λ i , m ) B j ( x ) d x .
g = Af + ,
f = ( A T A + γ H ) 1 A T g ,
H = [ 1 2 1 0 2 5 4 1 0 1 4 6 4 1 0 0 1 4 6 4 1 0 0 1 4 5 2 0 1 2 1 ]
f ( x ) = j = 1 8 f j B j ( x ) .
γ = γ 0 exp [ 1 8 i = 1 8 ln ( A T A ) i i ] , γ 0 = 10 3 , 10 2 , , 10 2 .
Q 1 e T S 1 e ,
e ( g g ) = ( g Af ) ,
Q 1 = Σ ( e i 2 / σ i i 2 ) .
Q 1 < P .
α i = α i ( 1 + α ) ,
β i = β i ( 1 + β ) ,
Δ f = { j = 1 8 ( f j f ̂ j ) 2 / j = 1 8 f ̂ j 2 } 1 / 2 ,
RH ¯

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