Abstract

A two-dimensional (two-color) statistical structure is formulated that is applicable to pattern recognition, discrimination, and detection problems occurring in infrared signal-processing systems. The methodology relates physical quantities such as the temperature T of an object, its projected area A, emissivity , range R from the sensor, and noise equivalent flux density (NEFD) to the geometry of a local orthogonal coordinate system where the coordinate axes correspond to the apparent radiant intensity J in each micron bandwidth. The bivariate distribution, correlation, and transformation properties attendant to this framework are discussed in detail. Additional insight into the structure of the problem is achieved by investigating the two-color system in terms of a nonorthogonal local coordinate system. The various results presented in the paper may be extended to three-, four-, or five-color systems by direct analogies.

© 1974 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. W. Anderson, An Introduction to Multivariate Statistical Analysis (New York, Wiley, 1958).
  2. K. Fukunaga, Introduction to Statistical Pattern Recognition (New York, Academic Press, 1972).
  3. C. R. Rao, Linear Statistical Inference and Its Application (New York, Wiley, 1965).
  4. G. S. Sebestyen, Decision Making Processes in Pattern Recognition (New York, Macmillan Co., 1962).
  5. H. Solomon, Studies in Item Analysis and Prediction (Stanford, California, Stanford University Press, 1961).
  6. R. Clow, E. Hansen, F. McNolty, submitted for publication 1973.
  7. R. Clow, E. Hansen, F. McNolty, IEEE Trans. Aerospace Electron. Syst. AES-8(4), 428 (July1972).
    [CrossRef]
  8. R. Clow, E. Hansen, F. McNolty, IEEE Trans. Aerospace Electron. Syst. AES-8(4), 552 (July1972).
    [CrossRef]
  9. R. Clow, E. Hansen, F. McNolty, Proc. IEEE 62, No. 4 (April1974).
    [CrossRef]
  10. J. A. Jamieson, R. H. McFee, G. N. Plass, R. H. Grube, R. G. Richards, Infrared Physics and Engineering (New York, McGraw-Hill, 1963).

1974 (1)

R. Clow, E. Hansen, F. McNolty, Proc. IEEE 62, No. 4 (April1974).
[CrossRef]

1972 (2)

R. Clow, E. Hansen, F. McNolty, IEEE Trans. Aerospace Electron. Syst. AES-8(4), 428 (July1972).
[CrossRef]

R. Clow, E. Hansen, F. McNolty, IEEE Trans. Aerospace Electron. Syst. AES-8(4), 552 (July1972).
[CrossRef]

Anderson, T. W.

T. W. Anderson, An Introduction to Multivariate Statistical Analysis (New York, Wiley, 1958).

Clow, R.

R. Clow, E. Hansen, F. McNolty, Proc. IEEE 62, No. 4 (April1974).
[CrossRef]

R. Clow, E. Hansen, F. McNolty, IEEE Trans. Aerospace Electron. Syst. AES-8(4), 552 (July1972).
[CrossRef]

R. Clow, E. Hansen, F. McNolty, IEEE Trans. Aerospace Electron. Syst. AES-8(4), 428 (July1972).
[CrossRef]

R. Clow, E. Hansen, F. McNolty, submitted for publication 1973.

Fukunaga, K.

K. Fukunaga, Introduction to Statistical Pattern Recognition (New York, Academic Press, 1972).

Grube, R. H.

J. A. Jamieson, R. H. McFee, G. N. Plass, R. H. Grube, R. G. Richards, Infrared Physics and Engineering (New York, McGraw-Hill, 1963).

Hansen, E.

R. Clow, E. Hansen, F. McNolty, Proc. IEEE 62, No. 4 (April1974).
[CrossRef]

R. Clow, E. Hansen, F. McNolty, IEEE Trans. Aerospace Electron. Syst. AES-8(4), 552 (July1972).
[CrossRef]

R. Clow, E. Hansen, F. McNolty, IEEE Trans. Aerospace Electron. Syst. AES-8(4), 428 (July1972).
[CrossRef]

R. Clow, E. Hansen, F. McNolty, submitted for publication 1973.

Jamieson, J. A.

J. A. Jamieson, R. H. McFee, G. N. Plass, R. H. Grube, R. G. Richards, Infrared Physics and Engineering (New York, McGraw-Hill, 1963).

McFee, R. H.

J. A. Jamieson, R. H. McFee, G. N. Plass, R. H. Grube, R. G. Richards, Infrared Physics and Engineering (New York, McGraw-Hill, 1963).

McNolty, F.

R. Clow, E. Hansen, F. McNolty, Proc. IEEE 62, No. 4 (April1974).
[CrossRef]

R. Clow, E. Hansen, F. McNolty, IEEE Trans. Aerospace Electron. Syst. AES-8(4), 552 (July1972).
[CrossRef]

R. Clow, E. Hansen, F. McNolty, IEEE Trans. Aerospace Electron. Syst. AES-8(4), 428 (July1972).
[CrossRef]

R. Clow, E. Hansen, F. McNolty, submitted for publication 1973.

Plass, G. N.

J. A. Jamieson, R. H. McFee, G. N. Plass, R. H. Grube, R. G. Richards, Infrared Physics and Engineering (New York, McGraw-Hill, 1963).

Rao, C. R.

C. R. Rao, Linear Statistical Inference and Its Application (New York, Wiley, 1965).

Richards, R. G.

J. A. Jamieson, R. H. McFee, G. N. Plass, R. H. Grube, R. G. Richards, Infrared Physics and Engineering (New York, McGraw-Hill, 1963).

Sebestyen, G. S.

G. S. Sebestyen, Decision Making Processes in Pattern Recognition (New York, Macmillan Co., 1962).

Solomon, H.

H. Solomon, Studies in Item Analysis and Prediction (Stanford, California, Stanford University Press, 1961).

IEEE Trans. Aerospace Electron. Syst. (2)

R. Clow, E. Hansen, F. McNolty, IEEE Trans. Aerospace Electron. Syst. AES-8(4), 428 (July1972).
[CrossRef]

R. Clow, E. Hansen, F. McNolty, IEEE Trans. Aerospace Electron. Syst. AES-8(4), 552 (July1972).
[CrossRef]

Proc. IEEE (1)

R. Clow, E. Hansen, F. McNolty, Proc. IEEE 62, No. 4 (April1974).
[CrossRef]

Other (7)

J. A. Jamieson, R. H. McFee, G. N. Plass, R. H. Grube, R. G. Richards, Infrared Physics and Engineering (New York, McGraw-Hill, 1963).

T. W. Anderson, An Introduction to Multivariate Statistical Analysis (New York, Wiley, 1958).

K. Fukunaga, Introduction to Statistical Pattern Recognition (New York, Academic Press, 1972).

C. R. Rao, Linear Statistical Inference and Its Application (New York, Wiley, 1965).

G. S. Sebestyen, Decision Making Processes in Pattern Recognition (New York, Macmillan Co., 1962).

H. Solomon, Studies in Item Analysis and Prediction (Stanford, California, Stanford University Press, 1961).

R. Clow, E. Hansen, F. McNolty, submitted for publication 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 (4)

Fig. 1
Fig. 1

Cumulative distribution for total error in radiant intensity for a typical threat object. Sample size equals 5000.

Fig. 2
Fig. 2

Fraction of elliptical normal density contained within each constant probability contour for a typical threat object.

Fig. 3
Fig. 3

Ninety percent probability contours for a typical threat object.

Fig. 4
Fig. 4

Relationship of decorrelation lines, major axis, regression lines, and limiting boundary. Distorted angular scale.

Equations (49)

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

s ( t ) = - S ( f ) e j 2 π f t d f = - C ( f ) H ( f ) e j 2 π f t d f = K - C ( f ) 2 β ( f ) e - j 2 π f ( Δ - t ) d f = 1 K - N m ( ω ) e - j ω ( Δ - t ) · d ω 2 π
radiant intensity = J = ( R 2 · P i n ) / ( A c o l · L ) = R 2 S p / [ s ( Δ ) · A c o l L ] = R 2 · M · S p ,
H λ = watts / ( μ · cm 2 ) = [ η ( λ ) A F 2 H λ , terr / π R 2 ] + [ η ( λ ) A F 1 H λ , solar / π R 2 ] + [ ( λ ) A W λ ( T ) / π R 2 ]
G ( T , λ 1 , λ 2 ) = G ( T ) = λ 1 λ 2 W λ ( T ) d λ = C 1 λ 1 λ 2 { λ 5 [ exp ( C 2 / λ T - 1 ) ] } d λ ,
Q terr = λ 1 λ 2 H λ , terr d λ , Q solar = λ 1 λ 2 H λ , solar d λ = 1.82 × 10 - 5 · G ( 6000 K , λ 1 , λ 2 ) ,
F ( T , λ 1 , λ 2 ) = F ( T ) = d G ( T , λ 1 , λ 2 ) / d T , M ( T , λ 1 , λ 2 ) = M ( T ) = d F ( T , λ 1 , λ 2 ) d T .
F 1 = π N λ , solar / η ( λ ) H λ , solar ,             F 2 = π N λ , terr / η ( λ ) H λ , terr ,
( NEFD ) · L ( λ ) · A c o l = ( NEFD ) ( losses ) ( collector area ) = NEP ,
P c o l , λ = H λ · A c o l · L ( λ ) = watts / μ
λ 1 λ 2 P c o l , λ d λ = A c o l · L ( λ ) ¯ λ 1 λ 2 H λ d λ = A c o l · L · H = P i n ( watts ) ,
H = P i n / ( A c o l · L ) .
H = [ ( η · A F 2 Q terr ) / π R 2 ] + [ ( η · A F 1 Q solar ) / π R 2 ] + [ A G ( t ) / π R 2 ] , i . e . , H = f ( T , R , A , ) = K / π R 2 = A K / π R 2 ,
J ( λ 1 , λ 2 ) = J = R 2 · H = R 2 λ 1 λ 2 H λ d λ .
d f = f R Δ R + f A Δ A + f T Δ T + f Δ + 1 2 · 2 f R 2 ( Δ R ) 2 + 1 2 · 2 f A 2 ( Δ A 2 ) + 1 2 · 2 f T 2 ( Δ T ) 2 + 1 2 · ( 2 f / 2 ) ( Δ ) 2 + 2 f R A ( Δ R ) ( Δ A ) + 2 f R T ( Δ R ) ( Δ T ) + 2 f R ( Δ R ) ( Δ ) + 2 f A T ( Δ A ) ( Δ T ) + 2 f A ( Δ A ) ( Δ ) + 2 f T ( Δ T ) ( Δ ) + ,
( R + Δ R ) 2 · H ( T + Δ T , R , A + Δ A , + Δ ) = apparent radiant intensity J ,
( R + Δ R ) 2 · H ( R + Δ T , R , A + Δ A , + Δ ) - R 2 · H ( T , R , A , ) = d J
d J = 2 ( Δ R / R ) · ( K / π ) + [ ( Δ R ) 2 / R 2 ] · ( K / π ) + = x 1 ,
R 2 · d f = R 2 · d H = - 2 ( Δ R / R ) · ( K / π ) + [ 3 ( Δ R ) 2 / R 2 · ( K / π ) + = x 2 .
E ( x 1 ) = ( K / π R 2 ) σ R 2 + < E ( x 2 ) = ( 3 K / π R 2 ) σ R 2 + ,
Var ( x 1 ) = ( 2 K 2 σ R 2 / π 2 R 2 ) · ( 2 + σ R 2 ) + < Var ( x 2 ) = ( 2 K 2 σ R 2 / π 2 R 2 ) · ( 2 + 9 σ R 2 ) + ,
ν = R 2 · d f + R 2 · ( NEFD ) = a x + a 2 y + a 1 z + ( d / π ) w + b x 2 + b 1 z 2 + c 1 x y + c 2 x z - ( 2 d / π R ) x w + c 3 y z + ( d 2 / π ) y w + ( d 1 / π ) z w + γ ,
u = a x + a 2 y + a 1 z + b x 2 + b 1 z 2 + c 1 x y + c 2 x z + c 3 y z + γ
E ( ν ) = b σ R 2 + b 1 σ T 2 + ( d 1 / π ) σ T σ ρ T , E ( u ) = b σ R 2 + b 1 σ T 2 .
σ ν 2 = a 2 σ R 2 + a 2 2 σ A 2 + a 1 2 σ T 2 + ( d 2 / π 2 ) σ 2 + 2 b 2 σ R 4 + 2 b 1 2 σ T 4 + c 1 2 σ R 2 σ A 2 + c 2 2 σ R 2 σ T 2 + ( 4 d 2 / π 2 R 2 ) σ R 2 σ 2 + c 3 2 σ A 2 σ T 2 + ( d 2 2 / π 2 ) σ A 2 σ 2 + ( d 1 2 / π 2 ) σ T 2 σ 2 + ( d 1 2 / π 2 ) σ 2 σ T 2 ρ T 2 + σ N 2 + 2 a 1 ( d / π ) ρ T σ σ T + 2 ( c 3 d 2 / π ) σ A 2 σ σ T ρ T - ( 4 c 2 d / π R ) σ R 2 σ σ T ρ T + ( 4 b 1 d 1 / π ) σ T 3 σ ρ T
ν 1 = ( a ) 1 x + ( a 2 ) 1 y + ( a 1 ) 1 z + [ ( d ) 1 / π ] w + ( b ) 1 x 2 + ( b 1 ) 1 z 2 + ( c 1 ) 1 x y + ( c 2 ) 1 x z - [ 2 ( d ) 1 / π R ] x w + ( c 3 ) 1 y z + [ ( d 2 ) 1 / π ] y w + [ ( d 1 ) 1 / π ] z w + γ 1
ν 2 = ( a ) 2 x + ( a 2 ) 2 y + ( a 1 ) 2 + [ ( d ) 2 / π ] w + ( b ) 2 x 2 + ( b 1 ) 2 z 2 + ( c 1 ) 2 x y + ( c 2 ) 2 x z - [ 2 ( d ) 2 / π R ] x w + ( c 3 ) 2 y z + [ ( d 2 ) 2 / π ] y w + [ ( d 1 ) 2 / π ] z w + γ 2 ,
E [ ( ν 1 - ν ¯ 1 ) · ( ν 2 - ν ¯ 2 ) ] = ( a ) 1 ( a ) 2 σ R 2 + ( a 2 ) 1 ( a 2 ) 2 σ A 2 + ( a 1 ) 1 ( a 1 ) 2 σ T 2 + [ ( a 1 ) 1 ( d ) 2 / π ] ρ T σ σ T + [ ( d ) 1 ( a 1 ) 2 / π ] ρ T σ σ T + [ ( d ) 1 ( d ) 2 / π 2 ] σ 2 + 2 ( b ) 1 ( b ) 2 σ R 4 + 2 ( b 1 ) 1 [ ( d 1 ) 2 / π ] ρ T σ T 3 σ + 2 ( b 1 ) 1 ( b 1 ) 2 σ T 4 + ( c 1 ) 1 ( c 1 ) 2 σ R 2 σ A 2 + ( c 2 ) 1 ( c 2 ) 2 σ R 2 σ T 2 - [ 2 ( c 2 ) 1 ( d ) 2 / π R ] σ R 2 ρ T σ σ T - [ 2 ( c 2 ) 2 ( d ) 1 / π R ] σ R 2 ρ T σ σ T + [ 4 ( d ) 1 ( d ) 2 / π 2 R 2 ] σ R 2 σ 2 + ( c 3 ) 1 ( c 3 ) 2 σ A 2 σ T 2 + [ ( c 3 ) 1 ( d 2 ) 2 / π ] σ A 2 ρ T σ σ T + [ ( d 2 ) 1 ( c 3 ) 2 / π ] σ A 2 ρ T σ σ T + [ ( d 2 ) 1 ( d 2 ) 2 / π 2 ] σ A 2 σ 2 + 2 [ ( d 1 ) 1 ( b 1 ) 2 ρ T / π ] σ T 3 σ + [ ( d 1 ) 1 ( d 1 ) 2 / π 2 ] σ 2 σ T 2 + [ ( d 1 ) 1 ( d 1 ) 2 / π 2 ] ρ T 2 σ 2 σ T 2 .
h ( t ) d t = ( 1 / π σ A σ T ) K 0 ( t / σ A σ T ) d t ,
r 1 = ( 1 / ρ ) · ( σ 2 / σ 1 ) ,             r 2 = ρ · ( σ 2 / σ 1 ) .
u 1 = u 1 ,             u 2 = - ρ · ( σ 2 / σ 1 ) u 1 + u 2
u 1 = u 1 ,             u 2 = u 2 + ρ · ( σ 2 / σ 1 ) u 1
u 1 = u 1 - ρ ( σ 1 / σ 2 ) u 2 ,             u 2 = u 2
u 1 = u 1 + ρ ( σ 1 / σ 2 ) u 2 ,             u 2 = u 2
σ u l 2 = σ u 1 2 ,             σ u 2 2 = σ u 2 2 ( 1 - ρ 2 ) ,
σ u 1 2 = σ u 1 2 ( 1 - ρ 2 ) ,             σ u 2 2 = σ u 2 2 .
slope of l 1 = s 1 - F 2 ( T , λ 3 , λ 4 ) / F 1 ( T , λ 1 , λ 2 ) . slope of l 2 = s 2 = K 2 / K 1 ,
l 1 = - s 2 u 1 + u 2 ,             l 2 = ( s 1 u 1 - u 2 ) / ( s 1 - s 2 )
N 2 = S 2 S 1 S 2 - S 1 [ σ N 2 + 27 K 1 π 2 R 4 σ R 4 + 3 ( b 1 ) 1 2 σ T 4 + 4 K 1 2 π 2 R 2 σ R 2 σ A 2 + 4 R 2 ( a 1 ) 1 2 σ R 2 σ T 2 + ( a 1 ) 1 2 A 2 σ A 2 σ T 2 + 6 K 1 π R 2 ( b 1 ) 1 σ R 2 σ T 2 ] - S 1 + S 2 S 2 - S 1 { 27 K 1 K 2 π 2 R 4 σ R 4 + 3 ( b 1 ) 1 · ( b 1 ) 2 σ T 4 + 4 K 1 K 2 π 2 R 2 σ R 2 σ A 2 + 4 R 2 ( a 1 ) 1 ( a 1 ) 2 σ R 2 σ T 2 + ( a 1 ) 1 ( a 1 ) 2 A 2 σ A 2 σ T 2 + 3 π R 2 [ ( b 1 ) 2 K 1 + ( b 1 ) 1 K 2 ] σ R 2 σ T 2 } + 1 S 2 + S 1 [ σ N 2 + 27 K 2 2 π 2 R 4 σ R 4 + 3 ( b 1 ) 2 2 σ T 4 + 4 K 2 2 π 2 R 2 σ R 2 σ A 2 + 4 R 2 ( a 1 ) 2 2 σ R 2 σ T 2 + ( a 1 ) 2 2 A 2 σ A 2 σ T 2 + 6 K 2 π R 2 ( b 1 ) 2 σ R 2 σ T 2 ] + ( B J ) 1 · ( B J ) 2 ( S 1 + S 2 ) - S 1 S 2 ( B J ) 1 2 - ( B J ) 2 2 S 2 - S 1 ,
E ( l 1 2 ) = ( a 1 ) 1 2 ( s 1 - s 2 ) 2 σ T 2 + s 2 2 [ σ N 2 + 28 K 1 2 π 2 R 4 σ R 4 + 3 ( b 1 ) 1 2 σ T 4 + 4 K 1 2 π 2 R 2 σ R 2 σ A 2 + 4 R 2 ( a 1 ) 1 2 σ R 2 σ T 2 + ( a 1 ) 1 2 A 2 × σ A 2 σ T 2 + 6 K 1 π R 2 ( b 1 ) 1 σ R 2 σ T 2 ] - 2 S 1 { 27 K 1 K 2 π 2 R 4 σ R 4 + 3 ( b 1 ) 1 ( b 1 ) 2 σ T 4 + 4 K 1 K 2 π 2 R 2 σ R 2 σ A 2 + 4 R 2 ( a 1 ) 1 ( a 1 ) 2 σ R 2 σ T 2 + ( a 1 ) 1 ( a 1 ) 2 A 2 σ A 2 σ T 2 + 3 π R 2 [ ( b 1 ) 2 K 1 + ( b 1 ) 2 K 2 ] σ R 2 σ T 2 } + σ N 2 + 27 K 2 2 π 2 R 4 σ R 4 + 3 ( b 1 ) 2 2 σ T 4 + 4 K 2 2 π 2 R 2 σ R 2 σ A 2 + 4 R 2 ( a 1 ) 2 2 σ R 2 σ T 2 + ( a 1 ) 2 2 A 2 σ A 2 σ T 2 + 6 K 2 π R 2 ( b 1 ) 2 σ R 2 σ T 2 ,
E ( l 2 2 ) = 4 K 1 2 π 2 R 2 σ R 2 + K 1 2 π 2 σ A 2 + S 1 2 ( S 1 - S 2 ) 2 [ σ N 2 + 27 K 1 2 π 2 R 4 σ R 4 + 3 ( b 1 ) 1 2 σ T 4 + 4 K 1 2 π 2 R 2 σ R 2 σ A 2 + 4 R 2 ( a 1 ) 1 2 σ R 2 σ T 2 + ( a 1 ) 1 2 A 2 σ A 2 σ T 2 + 6 K 1 π ( b 1 ) 1 σ R 2 σ T 2 ] - 2 S 1 ( S 1 - S 2 ) 2 { 27 K 1 K 2 π 2 R 2 σ R 2 + 3 ( b 1 ) 1 ( b 1 ) 2 σ T 4 + 4 K 1 K 2 π 2 R 2 σ R 2 σ A 2 + 4 R 2 ( a 1 ) 1 ( a 1 ) 2 σ R 2 σ T 2 + ( a 1 ) 1 ( a 1 ) 2 A 2 σ A 2 σ T 2 + 3 π R 2 [ ( b 1 ) 2 K 1 + ( b 1 ) K 2 ] σ R 2 σ T 2 } + 1 ( S 1 - S 2 ) 2 [ 27 K 2 2 π 2 R 4 σ R 4 + 3 ( b 1 ) 2 2 σ T 4 + σ N 2 + 4 K 2 2 π 2 R 2 σ R 2 σ A 2 + ( a 1 ) 2 2 A 2 σ A 2 σ T 2 + 4 R 2 ( a 1 ) 2 2 σ R 2 σ T 2 + 6 K 2 π R 2 ( b 1 ) 2 σ R 2 σ T 2 ] ,
σ l 1 2 = E ( l 1 2 ) - [ ( B J ) 2 - S 2 ( B J ) 1 ] 2
σ l 2 2 = E ( l 2 2 ) - [ S 1 ( B J ) 1 - ( B J ) 2 ] 2 ( S 1 - S 2 ) 2
a = - b [ σ γ 1 2 / ( σ γ 2 2 + b 2 σ γ 1 2 ) ] ,
σ γ 2 / σ γ 1 = + [ ( b 1 / a ) ( 1 - a b 1 ) ] 1 / 2 ,
G ( T , λ 1 , λ 2 ) = C 1 n = 1 { e - ( n c 2 / λ 2 T ) × ( T n c 2 λ 2 3 + 3 T 2 n 2 c 2 2 λ 2 2 + 6 T 3 n 3 c 2 3 λ 2 + 6 T 4 n 4 c 2 4 ) - e - ( n c 2 / λ 1 T ) ( T n c 2 λ 1 3 + 3 T 2 n 2 c 2 2 λ 1 2 + 6 T 3 n 3 c 2 3 λ 1 + 6 T 4 n 4 c 2 4 ) } ,
F ( T , λ 1 , λ 2 ) = c 1 n = 1 { e - ( n c 2 / λ 2 T ) × ( 1 λ 2 4 T + 4 n c 2 λ 2 3 + 12 T n 2 c 2 2 λ 2 2 + 24 T 2 n 3 c 2 3 λ 2 + 24 T 3 n 4 c 2 2 ) - e - ( n c 2 / λ 1 T ) ( 1 λ 1 4 T + 4 n c 2 λ 1 3 + 12 T n 2 c 2 2 λ 1 2 + 24 T 2 n 3 c 2 3 λ 1 + 24 T 3 n 4 c 2 4 ) } ,
M ( T , λ 1 , λ 2 ) = c 1 n = 1 { e - ( n c 2 / λ 2 T ) × ( n c 2 λ 2 5 T 3 + 3 λ 2 4 T 2 + 12 n c 2 λ 2 3 T + 36 n 2 c 2 2 λ 2 2 + 72 T n 3 c 2 3 λ 2 + 75 T 2 n 4 c 2 4 ) - e - ( n c 2 / λ 1 T ) ( n c 2 λ 1 5 T 3 + 3 λ 1 4 T 2 + 12 n c 2 λ 1 3 T + 36 n 2 c 2 2 λ 1 2 + 72 T n 3 c 2 3 λ 1 + 72 T 2 n 4 c 2 4 ) } .
Q terr = λ 1 λ 2 H λ , terr d λ = ( r r 0 + h ) 2 { i = 1 m a ¯ i G [ T 1 , λ 1 + ( i - 1 ) δ λ , λ 1 + i δ λ ] + i = 1 m ( 1 - a ¯ i ) G [ T 2 , λ 1 + ( i - 1 ) δ λ , λ 1 + i δ λ ] } ,
a ¯ i = 1 / δ λ λ 1 + ( i - 1 ) δ λ λ 1 + i δ λ a ( λ ) d λ ,

Metrics