Abstract

We introduce a channel selection method for atmospheric remote-sensing problems described by a Fredholm integral equation of the first kind. Whether one set of channels (CH) is more suitable than another (CH′) can be judged by whether (1) the degree of predominance (DP) value of CH is larger than that of CH′, i.e., if the number of channels is the same and (2) the number of channels of CH is more than that of CH′, if the DP values of both are acceptable. One can calculate the DP of the unknown function f(y) for a set of remote-sensing channels by

DP=[1+(Rf˜a2-1)Rd2]-1/2,Rf˜a2=Rc2[Rb2+Ra2(1+Rb2)],

where R a, R b, R c, and (1 − R d 2)1/2 of this channel set represent the influences on the ability to recover the unknown function caused by various measurement errors, the noise parameter, the relativity of the kernel functions, and the blindness of remote sensing means, respectively. Our channel selection method can be simplified to a conventional method when there are no differences in the relative measurement errors, no blind components of the unknown function and no noise parameters in the kernel function.

© 1996 Optical Society of America

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References

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  1. S. Twomey, “Information content in remote sensing,” Appl. Opt. 13, 942–945 (1974).
  2. C. D. Rodgers, “Retrieval of atmospheric temperature and composition from remote measurements of thermal radiation,” Rev. Geophys. Space Phys. 14, 609–624 (1976).
  3. G. E. Shaw, “Inversion of optical scattering and spectral extinction measurements to recover aerosol size spectra,” Appl. Opt. 18, 988–993 (1979).
  4. R. Rizzi, R. Guzzi, R. Legnani, “Aerosol size spectra from the spectral extinction data: the use of a linear inversion method,” Appl. Opt. 21, 1578–1587 (1982).
  5. E. Thomalla, H. Quenzel, “Information content of aerosol optical properties with respect to their size distribution,” Appl. Opt. 21, 3170–3177 (1982).
  6. C. B. Smith, “Inversion of the anomalous diffraction approximation for variable complex index of refraction near unity,” Appl. Opt. 21, 3363–3366 (1982).
  7. C. D. Capps, R. L. Henning, G. M. Hess, “Analytic inversion of remote-sensing data,” Appl. Opt. 21, 3581–3587 (1982).
  8. G. Viera, M. A. Box, “Information content analysis of aerosol remote-sensing experiments using an analytic eigenfunction theory: anomalous diffraction approximation,” Appl. Opt. 24, 4525–4533 (1985).
  9. A. Ben-David, B. M. Herman, J. A. Reagan, “Inverse problem and the pseudoempirical orthogonal function method of solution. 1: Theory; 2: Use” Appl. Opt. 27, 1235–1254 (1988).
  10. G. Viera, M. A. Box, “Information content analysis of aerosol remote-sensing experiments using singular function theory. 1: Extinction measurements,” Appl. Opt. 26, 1312–1327 (1987).
  11. G. Viera, M. A. Box, “Information content analysis of aerosol remote-sensing experiments using singular function theory 2: Scattering measurements,” Appl. Opt. 27, 3262–3274 (1988).
  12. E. R. Westwater, O. N. Strand, “Statistical information content of radiation measurements used in indirect sensing,” J. Atmos. Sci. 25, 750–758 (1968).
  13. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977), p. 211.
  14. Q. Yin, “Remote sounding of maritime atmospheric aerosols from space,” Ph.D. dissertation (Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai, China, 1993).

1988

1987

1985

1982

1979

1976

C. D. Rodgers, “Retrieval of atmospheric temperature and composition from remote measurements of thermal radiation,” Rev. Geophys. Space Phys. 14, 609–624 (1976).

1974

1968

E. R. Westwater, O. N. Strand, “Statistical information content of radiation measurements used in indirect sensing,” J. Atmos. Sci. 25, 750–758 (1968).

Ben-David, A.

Box, M. A.

Capps, C. D.

Guzzi, R.

Henning, R. L.

Herman, B. M.

Hess, G. M.

Legnani, R.

Quenzel, H.

Reagan, J. A.

Rizzi, R.

Rodgers, C. D.

C. D. Rodgers, “Retrieval of atmospheric temperature and composition from remote measurements of thermal radiation,” Rev. Geophys. Space Phys. 14, 609–624 (1976).

Shaw, G. E.

Smith, C. B.

Strand, O. N.

E. R. Westwater, O. N. Strand, “Statistical information content of radiation measurements used in indirect sensing,” J. Atmos. Sci. 25, 750–758 (1968).

Thomalla, E.

Twomey, S.

S. Twomey, “Information content in remote sensing,” Appl. Opt. 13, 942–945 (1974).

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977), p. 211.

Viera, G.

Westwater, E. R.

E. R. Westwater, O. N. Strand, “Statistical information content of radiation measurements used in indirect sensing,” J. Atmos. Sci. 25, 750–758 (1968).

Yin, Q.

Q. Yin, “Remote sounding of maritime atmospheric aerosols from space,” Ph.D. dissertation (Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai, China, 1993).

Appl. Opt.

S. Twomey, “Information content in remote sensing,” Appl. Opt. 13, 942–945 (1974).

G. E. Shaw, “Inversion of optical scattering and spectral extinction measurements to recover aerosol size spectra,” Appl. Opt. 18, 988–993 (1979).

R. Rizzi, R. Guzzi, R. Legnani, “Aerosol size spectra from the spectral extinction data: the use of a linear inversion method,” Appl. Opt. 21, 1578–1587 (1982).

E. Thomalla, H. Quenzel, “Information content of aerosol optical properties with respect to their size distribution,” Appl. Opt. 21, 3170–3177 (1982).

C. B. Smith, “Inversion of the anomalous diffraction approximation for variable complex index of refraction near unity,” Appl. Opt. 21, 3363–3366 (1982).

C. D. Capps, R. L. Henning, G. M. Hess, “Analytic inversion of remote-sensing data,” Appl. Opt. 21, 3581–3587 (1982).

G. Viera, M. A. Box, “Information content analysis of aerosol remote-sensing experiments using an analytic eigenfunction theory: anomalous diffraction approximation,” Appl. Opt. 24, 4525–4533 (1985).

G. Viera, M. A. Box, “Information content analysis of aerosol remote-sensing experiments using singular function theory. 1: Extinction measurements,” Appl. Opt. 26, 1312–1327 (1987).

A. Ben-David, B. M. Herman, J. A. Reagan, “Inverse problem and the pseudoempirical orthogonal function method of solution. 1: Theory; 2: Use” Appl. Opt. 27, 1235–1254 (1988).

G. Viera, M. A. Box, “Information content analysis of aerosol remote-sensing experiments using singular function theory 2: Scattering measurements,” Appl. Opt. 27, 3262–3274 (1988).

J. Atmos. Sci.

E. R. Westwater, O. N. Strand, “Statistical information content of radiation measurements used in indirect sensing,” J. Atmos. Sci. 25, 750–758 (1968).

Rev. Geophys. Space Phys.

C. D. Rodgers, “Retrieval of atmospheric temperature and composition from remote measurements of thermal radiation,” Rev. Geophys. Space Phys. 14, 609–624 (1976).

Other

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977), p. 211.

Q. Yin, “Remote sounding of maritime atmospheric aerosols from space,” Ph.D. dissertation (Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai, China, 1993).

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Figures (2)

Fig. 1
Fig. 1

Kernel functions for different definitions of aerosol size distributions when the distributions are to be determined from multispectral aerosol optical depths: 1, f(r) = n(r); 2, f(r) = πr 2 n(r); 3, f(r) = (4/3)πr 3 n(r).

Fig. 2
Fig. 2

Signal-to-noise ratios of aerosol optical depths from a simulated space sensing of maritime atmospheric aerosols.14

Tables (2)

Tables Icon

Table 1 Simulated Channel Selections for Three Cases

Tables Icon

Table 2 Optimal Channel Set for the Determination of Aerosol Size Distribution from Multispectral Aerosol Optical Depths

Equations (60)

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DP = [ 1 + ( R f ˜ a 2 - 1 ) R d 2 ] - 1 / 2 , R f ˜ a 2 = R c 2 [ R b 2 + R a 2 ( 1 + R b 2 ) ] ,
g λ = y min y max k λ ( y , z ) f ( y ) d y ,
τ λ = 0 Q λ ( r , m ) π r 2 n ( r ) d r ,
g = y min y max k ( y , z ) f ( y ) d y ,
g = ( g 1 g 2 g M ) ,             k = [ k 1 ( y , z ) k 2 ( y , z ) k M ( y , z ) ] .
g ¯ = y min y max k ( y , z ¯ ) f ¯ ( y ) d y .
f ˜ ( y ) = f ( y ) - f ¯ ( y ) , k ˜ ( y , z ) = k ( y , z ) - k ( y , z ¯ ) , g ˜ = g - g ¯ ,
g ˜ = y min y max k ( y , z ) f ( y ) d y - y min y max k ( y , z ¯ ) f ¯ ( y ) d y = y min y max [ k ( y , z ) - k ( y , z ¯ ) ] f ¯ ( y ) d y + y min y max k ( y , z ¯ ) [ f ( y ) - f ¯ ( y ) ] d y = g z + g f ,
g f = y min y max k ( y , z ¯ ) f ˜ ( y ) d y ,
g z = y min y max k ˜ ( y , z ) f ¯ ( y ) d y ,
g * = ( I + E ) g ˜ = g f + g z + E ( g f + g z ) ,
I = ( 1 1 1 ) ,             E = ( 1 2 M ) ,
f ˜ ( y ) = f ˜ a ( y ) + f ˜ e ( y ) ,
y min y max k ( y , z ¯ ) f a ( y ) d y g f ,
y min y max k ( y , z ¯ ) f ˜ e ( y ) d y 0.
f ˜ a ( y ) = ϕ T ( y , z ¯ ) Ψ ,
C = y min y max k ( y , z ¯ ) k T ( y , z ¯ ) d y ,
C U = U Λ ,
U U T = I ,
ϕ ( y , z ¯ ) = Λ - 1 2 U T k ( y , z ¯ ) .
f ˜ a ( y ) = ϕ T ( y , z ¯ ) Ψ = k T ( y , z ¯ ) U Λ - 1 2 Ψ .
g f = y min y max k ( y , z ¯ ) f ˜ a ( y ) d y = y min y max k ( y , z ¯ ) k T ( y , z ¯ ) U Λ - 1 2 Ψ d y = C U Λ - 1 2 Ψ = U Λ 1 2 Ψ .
f ˜ a ( y ) = k T ( y , z ¯ ) U Λ - 1 U T g f .
β ( y ) = k T U Λ - 1 U T η ,
η = g z + E ( g f + g z ) .
f ˜ a 2 ¯ = Ψ 2 ¯ ,
g f 2 ¯ ( M - 1 i = 1 M λ i ) · Ψ 2 ¯ ,
β 2 ¯ ( M - 1 i = 1 M λ i - 1 ) · η 2 ¯ ,
η 2 ¯ g z 2 ¯ + ( M - 1 i = 1 M i 2 ¯ ) · ( g z 2 ¯ + g f 2 ¯ ) ,
F ( x ) = [ x min x max F 2 ( x ) d x ] 1 / 2 ,             V = V T V ,
f a 2 ¯ = y min y max ( k T U Λ - 1 2 Ψ ) T ( k T U Λ - 1 2 Ψ ) d y ¯ = y min y max Ψ T Λ - 1 2 U T kk T U Λ - 1 2 Ψ d y ¯ = Ψ T Λ - 1 2 U T C U Λ - 1 2 Ψ ¯ = Ψ T Ψ ¯ = Ψ 2 ¯ ,
g f 2 ¯ = ( U Λ 1 2 Ψ ) T ( U Λ 1 2 Ψ ) ¯ = Ψ T Λ 1 2 U T U Λ 1 2 Ψ ¯ = Ψ T Λ Ψ ¯ = i = 1 M λ i Ψ i 2 ¯ ( M - 1 i = 1 M λ i ) · Ψ 2 ¯ ,
β 2 ¯ = y min y max ( k T U Λ - 1 U T η ) T · ( k T U Λ - 1 U T η ) d y ¯ = y min y max ( U T η ) T Λ - 1 U T kk T U Λ - 1 ( U T η ) d y ¯ = ( U T η ) T Λ - 1 U T C U Λ - 1 ( U T η ) ¯ = ( U T η ) T Λ - 1 ( U T η ) ¯ ( M - 1 i = 1 M λ i - 1 ) · U T η ) 2 ¯ = ( M - 1 i = 1 M λ i - 1 ) · η 2 ¯ ,
η 2 ¯ = [ g z + E ( g f + g z ) ] T · [ g z + E ( g f + g z ) ] ¯ = [ g z T + ( g f T + g z T ) E ] · [ g z + E ( g f + g z ) ] ¯ = g z T g z ¯ + g z T E 2 g z ¯ + g f T E 2 g f ¯ g z 2 ¯ + ( M - 1 i = 1 M i 2 ¯ ) · ( g z 2 ¯ + g f 2 ¯ ) ,
R f ˜ a 2 = β 2 ¯ / f ˜ a 2 ¯ = ( M - 1 i = 1 M λ i - 1 ) · η 2 ¯ / Ψ 2 ¯ ( M - 1 i = 1 M λ i - 1 ) = [ g z 2 ¯ + ( M - 1 i = 1 M i 2 ¯ ) · ( g z 2 ¯ + g f 2 ¯ ) ] ( M - 1 i = 1 M λ i ) - 1 · g f 2 ¯ .
R a = ( M - 1 i = 1 M i 2 ¯ ) 1 2 ,
R b = ( g z 2 ¯ / g f 2 ¯ ) 1 2 ,
R c = ( M - 1 i = 1 M λ i · M - 1 i = 1 M λ i - 1 ) 1 2 ,
R f ˜ a 2 = R c 2 [ R b 2 + R a 2 ( 1 + R b 2 ) ] .
R f ˜ 2 = f e + β 2 ¯ / f ˜ 2 ¯ = ( f ˜ e 2 ¯ + β 2 ¯ ) / f ˜ 2 ¯ = ( f ˜ e 2 ¯ + R f ˜ a 2 f ˜ a 2 ¯ ) / f ˜ 2 ¯ .
f ˜ 2 ¯ = f ˜ a 2 ¯ + f ˜ e 2 ¯ .
R f ˜ 2 = 1 + ( R f ˜ a 2 - 1 ) R d 2 ,
R d = ( f a 2 ¯ / f 2 ¯ ) 1 2 .
R d 2 = ( M - 1 i = 1 M λ i ) - 1 · g f 2 ¯ f 2 ¯ = g f T C - 1 g f ¯ f ˜ 2 ¯ .
f a = k T U Λ - 1 U T g f = k T U ( U Λ ) - 1 g f = k T C - 1 g f .
f ˜ a 2 ¯ = y min y max ( k T C - 1 g f ) T · ( k T C - 1 g f ) d y ¯ = y min y max g f T E - 1 kk T C - 1 g f d y ¯ = g f T C - 1 g f ¯ .
g = y min y max k ( y , z ) f ( y ) d y ,
D P = ( f ˜ 2 ¯ f ˜ e 2 ¯ + β 2 ¯ ) 1 / 2 .
D P = I / R f ˜ ,
R f ˜ 2 = 1 + ( R f ˜ a 2 - 1 ) R d 2 ,
R f ˜ a 2 = R c 2 [ R b 2 + R a 2 ( 1 + R b 2 ) ] ,
R d 2 = f ˜ a 2 ¯ / f ˜ 2 ¯ = ( M - 1 i = 1 M λ i - 1 ) · g f 2 ¯ / f ˜ 2 ¯ = g f T C - 1 g f ¯ / f ˜ 2 ¯ ,
R c 2 = M - 1 i = 1 M λ i · M - 1 i = 1 M λ i - 1 ,
R b 2 = g z 2 ¯ / g f 2 ¯ ,
R a 2 = M - 1 i = 1 M i 2 ¯ .
D P - 1 = R c · R a = [ M - 1 i = 1 i = M λ i · M - 1 i = 1 i = M λ i - 1 ] 1 / 2 · [ M - 1 i = 1 i = M i 2 ¯ ] 1 / 2 .
f ( r ) = 4 3 π r 3 n ( r ) ,             y = r .
k λ ( r , m ) = 3 4 r Q λ ( r , m ) ,             z = m .
n ( r ) = { 1 r < r d ( r d / r ) c r > r d } ,
n ( r ) = 1 σ r exp { - 1 2 [ ln ( r / r g ) σ ] 2 } ,

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