Abstract

Several aspects of the behavior of Fredholm integral equations are examined in this paper. It is shown that collocation methods are better in general than least squares methods in linear approaches. The amplification of random noise inherent to the numerical inversion of the equation puts an upper limit to the information content of an ill-conditioned system. An estimation based on the magnitude of SNR is proposed for a system that lacks statistical information to determine the information content and to reconstruct the solution profile. To reduce the numerical instability of matrix inversion, some specific kernel transformations are discussed. Illustrative examples are given and compared to results of other approaches. An alternative linear approach that orthonormalizes the kernels is also proposed. The linear approach was then employed in solving the radiative transfer equation with temperature-independent kernels. The necessary variable separation in linear inversions was examined. Iteration refinement was found necessary to accommodate the strong nonlinearity of high temperature sensing.

© 1975 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Y. Wang, “Theory and Applications of Inversion Techniques: A Review,” AA&ES 70-6, Purdue University, Lafayette, Ind. (1970).
  2. O. N. Strand, E. R. Westwater, SIAM J. Numer. Anal. 5, 287 (1968).
    [CrossRef]
  3. V. F. Turchin, V. Z. Nozik, Izv. Akad. Nauk SSR, Fiz. Atmos. Okeana 5, 14 (1969).
  4. D. L. Phillips, J. Assoc. Comp. Mach. 9, 84 (1962).
    [CrossRef]
  5. S. Twomey, J. Assoc. Comp. Mach. 10, 97 (1963).
    [CrossRef]
  6. H. H. Kagiwada, R. E. Kalaba, “Direct and Inverse Problems for Integral Equations via Inital Value Methods,” RM-5447-PR, The Rand Corp. (1967).
  7. M. T. Chahine, J. Opt. Soc. Am. 58, 1634 (1968).
    [CrossRef]
  8. J. I. F. King, J. Atmos. Sci. 21, 324 (1964).
    [CrossRef]
  9. J. Y. Wang, R. Goulard, “An Optimal Linear Solution in Thermal Remote Sensing,” AA&ES 72-1-1, Purdue University, Lafayette, Ind. (1972).
  10. S. Twomey, H. B. Howell, Appl. Opt. 6, 2125 (1967).
    [CrossRef] [PubMed]
  11. S. Twomey, Appl. Opt. 13, 942 (1974).
    [CrossRef] [PubMed]
  12. J. Y. Wang, R. Goulard, Appl. Opt. 13, 2467 (1974).
    [CrossRef] [PubMed]
  13. D. Q. Wark, H. E. Fleming, Mon. Weather Rev. 94, 351 (1966).
    [CrossRef]
  14. K. H. Bohm, Astrophys. J. 134, 264 (1961).
    [CrossRef]
  15. O. R. White, Astrophys. J. 152, 217 (1968).
    [CrossRef]
  16. R. Goulard, B. Singh, J. Y. Wang, “Remote Sensing in Flames: An Inversion Technique,” AA&ES 72-9-4, Purdue University, Lafayette, Ind. (1972).
  17. C. M. Chao, R. Goulard, “Nonlinear Inversion Techniques in Flame Temperature Measurements,” A&A 73-7-1, Purdue University, Lafayette, Ind. (1973).
  18. M. T. Chahine, J. Atmos. Sci. 27, 960 (1970).
    [CrossRef]
  19. B. Krakow, Appl. Opt. 5, 201 (1966).
    [CrossRef] [PubMed]

1974

1970

M. T. Chahine, J. Atmos. Sci. 27, 960 (1970).
[CrossRef]

1969

V. F. Turchin, V. Z. Nozik, Izv. Akad. Nauk SSR, Fiz. Atmos. Okeana 5, 14 (1969).

1968

O. N. Strand, E. R. Westwater, SIAM J. Numer. Anal. 5, 287 (1968).
[CrossRef]

M. T. Chahine, J. Opt. Soc. Am. 58, 1634 (1968).
[CrossRef]

O. R. White, Astrophys. J. 152, 217 (1968).
[CrossRef]

1967

1966

B. Krakow, Appl. Opt. 5, 201 (1966).
[CrossRef] [PubMed]

D. Q. Wark, H. E. Fleming, Mon. Weather Rev. 94, 351 (1966).
[CrossRef]

1964

J. I. F. King, J. Atmos. Sci. 21, 324 (1964).
[CrossRef]

1963

S. Twomey, J. Assoc. Comp. Mach. 10, 97 (1963).
[CrossRef]

1962

D. L. Phillips, J. Assoc. Comp. Mach. 9, 84 (1962).
[CrossRef]

1961

K. H. Bohm, Astrophys. J. 134, 264 (1961).
[CrossRef]

Bohm, K. H.

K. H. Bohm, Astrophys. J. 134, 264 (1961).
[CrossRef]

Chahine, M. T.

M. T. Chahine, J. Atmos. Sci. 27, 960 (1970).
[CrossRef]

M. T. Chahine, J. Opt. Soc. Am. 58, 1634 (1968).
[CrossRef]

Chao, C. M.

C. M. Chao, R. Goulard, “Nonlinear Inversion Techniques in Flame Temperature Measurements,” A&A 73-7-1, Purdue University, Lafayette, Ind. (1973).

Fleming, H. E.

D. Q. Wark, H. E. Fleming, Mon. Weather Rev. 94, 351 (1966).
[CrossRef]

Goulard, R.

J. Y. Wang, R. Goulard, Appl. Opt. 13, 2467 (1974).
[CrossRef] [PubMed]

R. Goulard, B. Singh, J. Y. Wang, “Remote Sensing in Flames: An Inversion Technique,” AA&ES 72-9-4, Purdue University, Lafayette, Ind. (1972).

J. Y. Wang, R. Goulard, “An Optimal Linear Solution in Thermal Remote Sensing,” AA&ES 72-1-1, Purdue University, Lafayette, Ind. (1972).

C. M. Chao, R. Goulard, “Nonlinear Inversion Techniques in Flame Temperature Measurements,” A&A 73-7-1, Purdue University, Lafayette, Ind. (1973).

Howell, H. B.

Kagiwada, H. H.

H. H. Kagiwada, R. E. Kalaba, “Direct and Inverse Problems for Integral Equations via Inital Value Methods,” RM-5447-PR, The Rand Corp. (1967).

Kalaba, R. E.

H. H. Kagiwada, R. E. Kalaba, “Direct and Inverse Problems for Integral Equations via Inital Value Methods,” RM-5447-PR, The Rand Corp. (1967).

King, J. I. F.

J. I. F. King, J. Atmos. Sci. 21, 324 (1964).
[CrossRef]

Krakow, B.

Nozik, V. Z.

V. F. Turchin, V. Z. Nozik, Izv. Akad. Nauk SSR, Fiz. Atmos. Okeana 5, 14 (1969).

Phillips, D. L.

D. L. Phillips, J. Assoc. Comp. Mach. 9, 84 (1962).
[CrossRef]

Singh, B.

R. Goulard, B. Singh, J. Y. Wang, “Remote Sensing in Flames: An Inversion Technique,” AA&ES 72-9-4, Purdue University, Lafayette, Ind. (1972).

Strand, O. N.

O. N. Strand, E. R. Westwater, SIAM J. Numer. Anal. 5, 287 (1968).
[CrossRef]

Turchin, V. F.

V. F. Turchin, V. Z. Nozik, Izv. Akad. Nauk SSR, Fiz. Atmos. Okeana 5, 14 (1969).

Twomey, S.

Wang, J. Y.

J. Y. Wang, R. Goulard, Appl. Opt. 13, 2467 (1974).
[CrossRef] [PubMed]

J. Y. Wang, “Theory and Applications of Inversion Techniques: A Review,” AA&ES 70-6, Purdue University, Lafayette, Ind. (1970).

R. Goulard, B. Singh, J. Y. Wang, “Remote Sensing in Flames: An Inversion Technique,” AA&ES 72-9-4, Purdue University, Lafayette, Ind. (1972).

J. Y. Wang, R. Goulard, “An Optimal Linear Solution in Thermal Remote Sensing,” AA&ES 72-1-1, Purdue University, Lafayette, Ind. (1972).

Wark, D. Q.

D. Q. Wark, H. E. Fleming, Mon. Weather Rev. 94, 351 (1966).
[CrossRef]

Westwater, E. R.

O. N. Strand, E. R. Westwater, SIAM J. Numer. Anal. 5, 287 (1968).
[CrossRef]

White, O. R.

O. R. White, Astrophys. J. 152, 217 (1968).
[CrossRef]

Appl. Opt.

Astrophys. J.

K. H. Bohm, Astrophys. J. 134, 264 (1961).
[CrossRef]

O. R. White, Astrophys. J. 152, 217 (1968).
[CrossRef]

Izv. Akad. Nauk SSR, Fiz. Atmos. Okeana

V. F. Turchin, V. Z. Nozik, Izv. Akad. Nauk SSR, Fiz. Atmos. Okeana 5, 14 (1969).

J. Assoc. Comp. Mach.

D. L. Phillips, J. Assoc. Comp. Mach. 9, 84 (1962).
[CrossRef]

S. Twomey, J. Assoc. Comp. Mach. 10, 97 (1963).
[CrossRef]

J. Atmos. Sci.

J. I. F. King, J. Atmos. Sci. 21, 324 (1964).
[CrossRef]

M. T. Chahine, J. Atmos. Sci. 27, 960 (1970).
[CrossRef]

J. Opt. Soc. Am.

Mon. Weather Rev.

D. Q. Wark, H. E. Fleming, Mon. Weather Rev. 94, 351 (1966).
[CrossRef]

SIAM J. Numer. Anal.

O. N. Strand, E. R. Westwater, SIAM J. Numer. Anal. 5, 287 (1968).
[CrossRef]

Other

J. Y. Wang, “Theory and Applications of Inversion Techniques: A Review,” AA&ES 70-6, Purdue University, Lafayette, Ind. (1970).

J. Y. Wang, R. Goulard, “An Optimal Linear Solution in Thermal Remote Sensing,” AA&ES 72-1-1, Purdue University, Lafayette, Ind. (1972).

H. H. Kagiwada, R. E. Kalaba, “Direct and Inverse Problems for Integral Equations via Inital Value Methods,” RM-5447-PR, The Rand Corp. (1967).

R. Goulard, B. Singh, J. Y. Wang, “Remote Sensing in Flames: An Inversion Technique,” AA&ES 72-9-4, Purdue University, Lafayette, Ind. (1972).

C. M. Chao, R. Goulard, “Nonlinear Inversion Techniques in Flame Temperature Measurements,” A&A 73-7-1, Purdue University, Lafayette, Ind. (1973).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

Construction of f(x) from Eq. (18) |j| = 0.

Fig. 2
Fig. 2

Construction of f(x) from Eq. (18) |j| ≤ 0.01.

Fig. 3
Fig. 3

An example of nonuniformly distributed kernels K(αj,x) = (xαj) exp[−2(xαj)]U(xαj).

Fig. 4
Fig. 4

Construction of f(x) from nonuniformly distributed kernels [Eq. (21)], without kernel transformation in the inversion, |j| ≤ 0.0005.

Fig. 5
Fig. 5

Construction of f(x) from nonuniformly distributed kernels [Eq. (21)], with transformation y = (x)1/2 in the inversion, |j| ≤ 0.0005.

Fig. 6
Fig. 6

Comparison of retrieved profiles through the least squares inversion (L) for M = 10 and the collocation inversion (C) with the original profile, from nonuniformly distributed kernels [Eq. (21)], |j| ≤ 0.0005.

Fig. 7
Fig. 7

Construction of a source function f(x) = 1 − exp(−x) from Eq. (30), with transformation y = (x)1/2 in the inversion, e = 0.001, and compared with results by other approaches.

Fig. 8
Fig. 8

Orthonormal kernels constructed from kernels in Eq. (30), N = 6.

Fig. 9
Fig. 9

Construction of a source function from orthonormal kernels with orthonormal expansion [Eq. (34)].

Fig. 10
Fig. 10

Comparison of m(ν,T) models in the 4.3-μm band range; in Chahine’s approach λ1 = 4.3 μm, λ2 = 4.5 μm.

Fig. 11
Fig. 11

Linear inversion of a Gaussian temperature profile with uniformly distributed kernels.

Fig. 12
Fig. 12

Combined linear inversion and iteration refinement of a Gaussian temperature profile with nonuniformly distributed kernels.

Tables (1)

Tables Icon

Table I Eigenvalues and Amplified Random Errors of the Solar Limb Darkening Equation (N = 7)

Equations (59)

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

g j + j = a b K ( α j , x ) f ( x ) d x ,
f ( x ) = i = 1 N C i ϕ i ( x ) .
A ( M , N ) x = b + M N ,
| j = 1 M ξ j g ¯ j | = | a b j = 1 M ξ j K ( α j , x ) f ( x ) d x | = | f M | | a b j = 1 M ξ j K ( α j , x ) d x |
j = 1 M ξ j 2 = 1 ,
g ¯ j = g j + j ,
| a b j = 1 M ξ j K ( α j , x ) d x | ( b a ) 1 / 2 { a b [ j = 1 M ξ j K ( α j , x ) ] 2 d x } 1 / 2
| j = 1 M ξ j g ¯ j | f M ( b a ) 1 / 2 ( Q ) 1 / 2 ,
Q = a b [ j = 1 M ξ j K ( α j , x ) ] 2 d x .
[ a b K ( α i , x ) K ( α j , x ) d x ] .
lim M det { [ a b K ( α i , x ) K ( α j , x ) d x ] } 0.
| j = 1 ξ j g ¯ j | = 0.
A T ( M , N ) A ( M , N ) x = A T ( M , N ) ( b + ) .
det [ A T ( N , N ) A ( N , N ) ] = { det [ A ( N , N ) ] } 2 .
x = U Λ 1 U T A T ( M , N ) ( b + ) ,
E = U Λ 1 U T A T ( M , N ) .
E j { [ A T ( M , N ) ] j } / λ j j = 1,2 ,…, N ,
E j e / [ ( λ j ) 1 / 2 ] j = 1,2 ,…, N ,
g j + j = 0 10 exp [ ( x α j ) 2 2 ] f ( x ) d x .
α j = 3 + { [ 4 ( j 1 ) ] / ( N 1 ) } j = 1,2 ,…, N ;
E j δ j = 1,2 ,…, K .
K ( α j , x ) = ( x α j ) exp [ 2 ( x α j ) ] U ( x α j ) ,
α j = ( j 1 ) / ( N 1 ) j = 1,2 ,…, N ,
g ¯ j = a b K ( α j , x ) h ( x ) f ( x ) h ( x ) d x .
x = T ( y ) ,
g ¯ j = a b K [ α j , T ( y ) ] f [ T ( y ) ] d T ( y ) d y d y .
K [ α j , T ( y ) ] { [ d T ( y ) ] / d y } = K ( α j , y )
f [ T ( y ) ] = f ( y ) ,
g ¯ j = a b K ( α j , y ) f ( y ) d y .
y = ( x ) 1 / 2
g ¯ j = 0 exp ( x α j ) 1 α j f ( x ) d x j = 1,2 ,…, N ,
f ( x ) = 1 exp ( x )
α j = α 1 + j 1 N 1 ( α N α 1 ) j = 1,2 ,…, N ,
K ( α j , x ) = k = 1 j γ j k ϕ k ( x ) j = 1,2 ,…, N .
f ( x ) = i = 1 N C i ϕ i ( x ) .
[ γ 11 0 0 . .. .. .0 γ 21 γ 22 0 . .. .. .0 γ 31 γ 32 γ 33 . .. .. .0 .. .. .. . .. .. .. .. .. .. . .. .. .. γ N 1 γ N 2 γ N 3 . .. .. γ N ] [ C 1 C 2 C 3 .. .. C N ] = [ g 1 g 2 g 3 .. .. g N ] + [ 1 2 3 .. .. N ] .
C i = 1 γ i i [ ( g i + i ) j = 1 i 1 γ i j C j ] , i = 1,2 ,…, N ,
det { γ i j } = i = 1 N γ i i .
f ( x ) = 1 [ ( x / 4 ) 1 ] 2 .
I ( ν ) + ( ν ) = a b B [ ν , T ( x ) ] τ ( ν ) x d x ,
B ( ν , T ) = ( C 1 ν 3 ) / [ exp ( C 2 ν / T ) 1 ]
m ( ν , T ) = B ( ν , T ) B ( ν r , T ) = ( ν ν r ) 3 exp ( C 2 ν r / T ) 1 exp ( C 2 ν / T ) 1 .
m ( ν , T ) = ( ν / ν r ) 3 exp ( C 2 Δ ν / T )
exp ( C 2 Δ ν / T ) = exp ( β 1 + β 2 Δ ν + β 3 / T ) .
ν 1 ν 2 T 1 T 2 [ C 2 Δ ν / T β 1 β 2 Δ ν β 3 / T ] 2 d T d ν = 0 ,
β 1 = C 2 { [ ln ( T 2 / T 1 ) ] / ( T 2 T 1 ) } [ ( Δ ν 12 ) / 2 ] ,
β 2 = C 2 { [ ln ( T 2 / T 1 ) ] / ( T 2 T 1 ) } ,
β 3 = C 2 [ ( Δ ν 12 ) / 2 ] ,
m ( ν , T ) = ( ν / ν r ) 3 exp ( β 1 + β 2 Δ ν + β 3 / T ) .
B ( ν , T ) = [ ( ν / ν r ) 3 exp ( β 1 + β 2 Δ ν ) ] [ exp ( β 3 / T ) B ( ν r , T ) ] ,
I ( ν ) + ( ν ) ( ν / ν r ) 3 exp ( β 1 + β 2 Δ ν ) = a b τ ( ν ) x [ exp ( β 3 / T ) B ( ν r , T ) ] d x .
f ( x ) = B [ ν r , T ( x ) ] exp [ β 3 / T ( x ) ] ,
T ( x ) = 2000 exp [ ( x 2.5 ) 2 / 4 ] , 0 x 5
K ( α j , x ) = exp [ ( x α j ) 2 ]
I obs ( ν ) = a b [ ( ν ν r ) 3 exp ( β 1 + β 2 Δ ν ) ] × τ ( ν ) x [ exp ( β 3 / T ) B ( ν r , T ) ] d x ,
I cal ( ν ) = a b B ( ν , T ) τ ( ν ) x d x ,
I 2 ( ν j ) I 1 ( ν j ) exp [ β 3 / T 2 ( x ) ] B [ ν r , T 2 ( x ) ] exp [ β 3 / T 1 ( x ) ] B [ ν r , T 1 ( x ) ] .
T ( x ) = 3000 exp [ ( x 5 ) 2 / 15 ] , 0 x 10 ,
I obs ( ν j ) ( ν j / ν r ) 3 exp ( β 1 + β 2 Δ ν j ) = 0 10 [ α j exp ( α j x ) ] × [ exp ( β 3 / T ) B ( ν r , T ) ] d x , i = 1,2 ,…, N ,

Metrics