Abstract

Advances in high spectral resolution sensors and in data handling capabilities are enabling development of greatly improved remote-sensing devices for resource monitoring, so that design trade-offs are required. A methodology for optimizing selection of spectral bands for multispectral instruments such as those on the LANDSAT series of satellites is described. The method is applied to a collection of laboratory and outdoor spectra of natural and artificial materials. These reflectance spectra represent the visible and near-infrared spectral ranges at high (0.01-μm) spectral resolution. For most natural materials 15–25 spectral bands appear to be sufficient to describe spectral variability, whereas description of minerals and some artificial substances may require double this number of bands.

© 1994 Optical Society of America

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References

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  1. G. T. Vane, “First results of the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS),” in Imaging Spectroscopy II, G. T. Vane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.834, 166–174(1987).
  2. L. J. Ricard, M. Landers, “HYDICE, an airborne system for hyperspectral imaging,” presented at the International Symposium on Spectral Sensing Research, Maui, Hawaii, 15–20 November 1992.
  3. D. J. Wiersma, D. A. Landgrebe, “Analytical design of multispectral sensors,” IEEE Trans. Geosci. Remote Sensing GE-18, 180–189 (1980).
    [CrossRef]
  4. C-C. T. Chen, D. A. Landgrebe, “A spectral feature design system for the HIRIS/MODIS era,” IEEE Trans. Geosci. Remote Sensing 27, 681–686 (1989).
    [CrossRef]
  5. F. Csillag, L. Pasztor, L. L. Biehl, “Spectral band selection for the characterization of salinity status of soils,” Remote Sensing Environ. 43, 231–242 (1993).
    [CrossRef]
  6. J. C. Price, “On the information content of soil reflectance spectra,” Remote Sensing Environ. 33, 113–121 (1990).
    [CrossRef]
  7. J. C. Price, “Variability of high resolution crop reflectance spectra,” Int. J. Remote Sensing 13, 2593–2610 (1992).
    [CrossRef]
  8. J. C. Price, “Information content of IRIS spectra,” J. Geophys. Res. 80, 1930–1936 (1975).
    [CrossRef]
  9. J. C. Price, “On the value of high spectral resolution measurements in the visible and near infrared,” in Proceedings of the Fifth International Colloquium—Physical Measurements and Signatures in Remote Sensing, Courchevel, France (European Space Agency, Paris, 1991), pp. 131–136.
  10. L. L. Biehl, C. S. T. Daughtry, M. E. Bauer, “Vegetation and soils field research data base: experiment summaries,” LARS Tech. Rep. 042382 (Purdue University, West Lafayette, Ind., 1984).
  11. C. Daughtry, L. Biehl (personal communication, 15February1992).
  12. E. R. Stoner, M. F. Baumgardner, L. L. Biehl, B. F. Robinson, “Atlas of soil reflectance properties,” Purdue Univ. Agric. Exp. St. Res. Bull. 962, 1–75.
  13. W. B. Cohen, “Response of vegetation indices to changes in three measures of leaf water stress,” Photogram. Eng. Remote Sensing 57, 195–202 (1991).
  14. W. B. Cohen, “Chaparral vegetation reflectance and its potential utility for assessment of fire hazard,” Photogram. Eng. Remote Sensing 57, 203–207 (1991).
  15. W. B. Cohen, “Temporal versus spatial variation in leaf reflectance under changing water stress conditions,” Int. J. Remote Sensing 12, 1865–1876 (1991).
    [CrossRef]
  16. M. B. Satterwhite, J. P. Henley, “Hyperspectral signatures (400–2500 nm) of vegetation, minerals, soils, rocks, and cultural features: laboratory and field measurements,” Rep. ETL-0573 (U.S. Army Corps of Engineers, Fort Belvoir, Va., 1991).
  17. D. Krohn, U.S. Geological Survey, Reston, Va. (personal communication, 26March1991).
  18. M. Viollier, “Teledetection des concentrations de seston et pigments chlorophylliens contenu dans l’ocean,” Ph.D. dissertation (1980).
  19. K. Ya Kondratyev, Radiation in the Atmosphere (Academic, New York, 1969).
  20. D. Tanre, C. Deroo, P. Duhaut, M. Herman, J. J. Morcrette, J. Perbos, P. Y. Deschamps, “Description of a computer code to simulate the satellite signal in the solar spectrum: the 5S code,” Int. J. Remote Sensing 11, 659–668 (1990).
    [CrossRef]
  21. H. R. Lang, M. J. Bartholomew, C. I. Grove, E. D. Paylor, “Spectral reflectance characterization (0.4 to 2.5 and 8.0 to 12.0 μm) of Phanerozoic strata, Wind River Basin and Southern Bighorn Basin Area, Wyoming,” J. Sediment. Petrol. 60, 504–524(1990).
  22. C. I. Grove, S. J. Hook, E. D. Paylor, “Laboratory reflectance spectra of 160 minerals, 0.4 to 2.5 micrometers,” JPL Rep. 92-2 (Jet Propulsion Laboratory, Pasadena, Calif., 1992).
  23. Sets, Inc., Spectral Catalog (Sets Inc., Milalani, Hawaii, 1990), Vols. 1–5.

1993 (1)

F. Csillag, L. Pasztor, L. L. Biehl, “Spectral band selection for the characterization of salinity status of soils,” Remote Sensing Environ. 43, 231–242 (1993).
[CrossRef]

1992 (1)

J. C. Price, “Variability of high resolution crop reflectance spectra,” Int. J. Remote Sensing 13, 2593–2610 (1992).
[CrossRef]

1991 (3)

W. B. Cohen, “Response of vegetation indices to changes in three measures of leaf water stress,” Photogram. Eng. Remote Sensing 57, 195–202 (1991).

W. B. Cohen, “Chaparral vegetation reflectance and its potential utility for assessment of fire hazard,” Photogram. Eng. Remote Sensing 57, 203–207 (1991).

W. B. Cohen, “Temporal versus spatial variation in leaf reflectance under changing water stress conditions,” Int. J. Remote Sensing 12, 1865–1876 (1991).
[CrossRef]

1990 (3)

D. Tanre, C. Deroo, P. Duhaut, M. Herman, J. J. Morcrette, J. Perbos, P. Y. Deschamps, “Description of a computer code to simulate the satellite signal in the solar spectrum: the 5S code,” Int. J. Remote Sensing 11, 659–668 (1990).
[CrossRef]

H. R. Lang, M. J. Bartholomew, C. I. Grove, E. D. Paylor, “Spectral reflectance characterization (0.4 to 2.5 and 8.0 to 12.0 μm) of Phanerozoic strata, Wind River Basin and Southern Bighorn Basin Area, Wyoming,” J. Sediment. Petrol. 60, 504–524(1990).

J. C. Price, “On the information content of soil reflectance spectra,” Remote Sensing Environ. 33, 113–121 (1990).
[CrossRef]

1989 (1)

C-C. T. Chen, D. A. Landgrebe, “A spectral feature design system for the HIRIS/MODIS era,” IEEE Trans. Geosci. Remote Sensing 27, 681–686 (1989).
[CrossRef]

1980 (1)

D. J. Wiersma, D. A. Landgrebe, “Analytical design of multispectral sensors,” IEEE Trans. Geosci. Remote Sensing GE-18, 180–189 (1980).
[CrossRef]

1975 (1)

J. C. Price, “Information content of IRIS spectra,” J. Geophys. Res. 80, 1930–1936 (1975).
[CrossRef]

Bartholomew, M. J.

H. R. Lang, M. J. Bartholomew, C. I. Grove, E. D. Paylor, “Spectral reflectance characterization (0.4 to 2.5 and 8.0 to 12.0 μm) of Phanerozoic strata, Wind River Basin and Southern Bighorn Basin Area, Wyoming,” J. Sediment. Petrol. 60, 504–524(1990).

Bauer, M. E.

L. L. Biehl, C. S. T. Daughtry, M. E. Bauer, “Vegetation and soils field research data base: experiment summaries,” LARS Tech. Rep. 042382 (Purdue University, West Lafayette, Ind., 1984).

Baumgardner, M. F.

E. R. Stoner, M. F. Baumgardner, L. L. Biehl, B. F. Robinson, “Atlas of soil reflectance properties,” Purdue Univ. Agric. Exp. St. Res. Bull. 962, 1–75.

Biehl, L.

C. Daughtry, L. Biehl (personal communication, 15February1992).

Biehl, L. L.

F. Csillag, L. Pasztor, L. L. Biehl, “Spectral band selection for the characterization of salinity status of soils,” Remote Sensing Environ. 43, 231–242 (1993).
[CrossRef]

L. L. Biehl, C. S. T. Daughtry, M. E. Bauer, “Vegetation and soils field research data base: experiment summaries,” LARS Tech. Rep. 042382 (Purdue University, West Lafayette, Ind., 1984).

E. R. Stoner, M. F. Baumgardner, L. L. Biehl, B. F. Robinson, “Atlas of soil reflectance properties,” Purdue Univ. Agric. Exp. St. Res. Bull. 962, 1–75.

Chen, C-C. T.

C-C. T. Chen, D. A. Landgrebe, “A spectral feature design system for the HIRIS/MODIS era,” IEEE Trans. Geosci. Remote Sensing 27, 681–686 (1989).
[CrossRef]

Cohen, W. B.

W. B. Cohen, “Response of vegetation indices to changes in three measures of leaf water stress,” Photogram. Eng. Remote Sensing 57, 195–202 (1991).

W. B. Cohen, “Chaparral vegetation reflectance and its potential utility for assessment of fire hazard,” Photogram. Eng. Remote Sensing 57, 203–207 (1991).

W. B. Cohen, “Temporal versus spatial variation in leaf reflectance under changing water stress conditions,” Int. J. Remote Sensing 12, 1865–1876 (1991).
[CrossRef]

Csillag, F.

F. Csillag, L. Pasztor, L. L. Biehl, “Spectral band selection for the characterization of salinity status of soils,” Remote Sensing Environ. 43, 231–242 (1993).
[CrossRef]

Daughtry, C.

C. Daughtry, L. Biehl (personal communication, 15February1992).

Daughtry, C. S. T.

L. L. Biehl, C. S. T. Daughtry, M. E. Bauer, “Vegetation and soils field research data base: experiment summaries,” LARS Tech. Rep. 042382 (Purdue University, West Lafayette, Ind., 1984).

Deroo, C.

D. Tanre, C. Deroo, P. Duhaut, M. Herman, J. J. Morcrette, J. Perbos, P. Y. Deschamps, “Description of a computer code to simulate the satellite signal in the solar spectrum: the 5S code,” Int. J. Remote Sensing 11, 659–668 (1990).
[CrossRef]

Deschamps, P. Y.

D. Tanre, C. Deroo, P. Duhaut, M. Herman, J. J. Morcrette, J. Perbos, P. Y. Deschamps, “Description of a computer code to simulate the satellite signal in the solar spectrum: the 5S code,” Int. J. Remote Sensing 11, 659–668 (1990).
[CrossRef]

Duhaut, P.

D. Tanre, C. Deroo, P. Duhaut, M. Herman, J. J. Morcrette, J. Perbos, P. Y. Deschamps, “Description of a computer code to simulate the satellite signal in the solar spectrum: the 5S code,” Int. J. Remote Sensing 11, 659–668 (1990).
[CrossRef]

Grove, C. I.

H. R. Lang, M. J. Bartholomew, C. I. Grove, E. D. Paylor, “Spectral reflectance characterization (0.4 to 2.5 and 8.0 to 12.0 μm) of Phanerozoic strata, Wind River Basin and Southern Bighorn Basin Area, Wyoming,” J. Sediment. Petrol. 60, 504–524(1990).

C. I. Grove, S. J. Hook, E. D. Paylor, “Laboratory reflectance spectra of 160 minerals, 0.4 to 2.5 micrometers,” JPL Rep. 92-2 (Jet Propulsion Laboratory, Pasadena, Calif., 1992).

Henley, J. P.

M. B. Satterwhite, J. P. Henley, “Hyperspectral signatures (400–2500 nm) of vegetation, minerals, soils, rocks, and cultural features: laboratory and field measurements,” Rep. ETL-0573 (U.S. Army Corps of Engineers, Fort Belvoir, Va., 1991).

Herman, M.

D. Tanre, C. Deroo, P. Duhaut, M. Herman, J. J. Morcrette, J. Perbos, P. Y. Deschamps, “Description of a computer code to simulate the satellite signal in the solar spectrum: the 5S code,” Int. J. Remote Sensing 11, 659–668 (1990).
[CrossRef]

Hook, S. J.

C. I. Grove, S. J. Hook, E. D. Paylor, “Laboratory reflectance spectra of 160 minerals, 0.4 to 2.5 micrometers,” JPL Rep. 92-2 (Jet Propulsion Laboratory, Pasadena, Calif., 1992).

Krohn, D.

D. Krohn, U.S. Geological Survey, Reston, Va. (personal communication, 26March1991).

Landers, M.

L. J. Ricard, M. Landers, “HYDICE, an airborne system for hyperspectral imaging,” presented at the International Symposium on Spectral Sensing Research, Maui, Hawaii, 15–20 November 1992.

Landgrebe, D. A.

C-C. T. Chen, D. A. Landgrebe, “A spectral feature design system for the HIRIS/MODIS era,” IEEE Trans. Geosci. Remote Sensing 27, 681–686 (1989).
[CrossRef]

D. J. Wiersma, D. A. Landgrebe, “Analytical design of multispectral sensors,” IEEE Trans. Geosci. Remote Sensing GE-18, 180–189 (1980).
[CrossRef]

Lang, H. R.

H. R. Lang, M. J. Bartholomew, C. I. Grove, E. D. Paylor, “Spectral reflectance characterization (0.4 to 2.5 and 8.0 to 12.0 μm) of Phanerozoic strata, Wind River Basin and Southern Bighorn Basin Area, Wyoming,” J. Sediment. Petrol. 60, 504–524(1990).

Morcrette, J. J.

D. Tanre, C. Deroo, P. Duhaut, M. Herman, J. J. Morcrette, J. Perbos, P. Y. Deschamps, “Description of a computer code to simulate the satellite signal in the solar spectrum: the 5S code,” Int. J. Remote Sensing 11, 659–668 (1990).
[CrossRef]

Pasztor, L.

F. Csillag, L. Pasztor, L. L. Biehl, “Spectral band selection for the characterization of salinity status of soils,” Remote Sensing Environ. 43, 231–242 (1993).
[CrossRef]

Paylor, E. D.

H. R. Lang, M. J. Bartholomew, C. I. Grove, E. D. Paylor, “Spectral reflectance characterization (0.4 to 2.5 and 8.0 to 12.0 μm) of Phanerozoic strata, Wind River Basin and Southern Bighorn Basin Area, Wyoming,” J. Sediment. Petrol. 60, 504–524(1990).

C. I. Grove, S. J. Hook, E. D. Paylor, “Laboratory reflectance spectra of 160 minerals, 0.4 to 2.5 micrometers,” JPL Rep. 92-2 (Jet Propulsion Laboratory, Pasadena, Calif., 1992).

Perbos, J.

D. Tanre, C. Deroo, P. Duhaut, M. Herman, J. J. Morcrette, J. Perbos, P. Y. Deschamps, “Description of a computer code to simulate the satellite signal in the solar spectrum: the 5S code,” Int. J. Remote Sensing 11, 659–668 (1990).
[CrossRef]

Price, J. C.

J. C. Price, “Variability of high resolution crop reflectance spectra,” Int. J. Remote Sensing 13, 2593–2610 (1992).
[CrossRef]

J. C. Price, “On the information content of soil reflectance spectra,” Remote Sensing Environ. 33, 113–121 (1990).
[CrossRef]

J. C. Price, “Information content of IRIS spectra,” J. Geophys. Res. 80, 1930–1936 (1975).
[CrossRef]

J. C. Price, “On the value of high spectral resolution measurements in the visible and near infrared,” in Proceedings of the Fifth International Colloquium—Physical Measurements and Signatures in Remote Sensing, Courchevel, France (European Space Agency, Paris, 1991), pp. 131–136.

Ricard, L. J.

L. J. Ricard, M. Landers, “HYDICE, an airborne system for hyperspectral imaging,” presented at the International Symposium on Spectral Sensing Research, Maui, Hawaii, 15–20 November 1992.

Robinson, B. F.

E. R. Stoner, M. F. Baumgardner, L. L. Biehl, B. F. Robinson, “Atlas of soil reflectance properties,” Purdue Univ. Agric. Exp. St. Res. Bull. 962, 1–75.

Satterwhite, M. B.

M. B. Satterwhite, J. P. Henley, “Hyperspectral signatures (400–2500 nm) of vegetation, minerals, soils, rocks, and cultural features: laboratory and field measurements,” Rep. ETL-0573 (U.S. Army Corps of Engineers, Fort Belvoir, Va., 1991).

Stoner, E. R.

E. R. Stoner, M. F. Baumgardner, L. L. Biehl, B. F. Robinson, “Atlas of soil reflectance properties,” Purdue Univ. Agric. Exp. St. Res. Bull. 962, 1–75.

Tanre, D.

D. Tanre, C. Deroo, P. Duhaut, M. Herman, J. J. Morcrette, J. Perbos, P. Y. Deschamps, “Description of a computer code to simulate the satellite signal in the solar spectrum: the 5S code,” Int. J. Remote Sensing 11, 659–668 (1990).
[CrossRef]

Vane, G. T.

G. T. Vane, “First results of the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS),” in Imaging Spectroscopy II, G. T. Vane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.834, 166–174(1987).

Viollier, M.

M. Viollier, “Teledetection des concentrations de seston et pigments chlorophylliens contenu dans l’ocean,” Ph.D. dissertation (1980).

Wiersma, D. J.

D. J. Wiersma, D. A. Landgrebe, “Analytical design of multispectral sensors,” IEEE Trans. Geosci. Remote Sensing GE-18, 180–189 (1980).
[CrossRef]

Ya Kondratyev, K.

K. Ya Kondratyev, Radiation in the Atmosphere (Academic, New York, 1969).

IEEE Trans. Geosci. Remote Sensing (2)

D. J. Wiersma, D. A. Landgrebe, “Analytical design of multispectral sensors,” IEEE Trans. Geosci. Remote Sensing GE-18, 180–189 (1980).
[CrossRef]

C-C. T. Chen, D. A. Landgrebe, “A spectral feature design system for the HIRIS/MODIS era,” IEEE Trans. Geosci. Remote Sensing 27, 681–686 (1989).
[CrossRef]

Int. J. Remote Sensing (3)

J. C. Price, “Variability of high resolution crop reflectance spectra,” Int. J. Remote Sensing 13, 2593–2610 (1992).
[CrossRef]

W. B. Cohen, “Temporal versus spatial variation in leaf reflectance under changing water stress conditions,” Int. J. Remote Sensing 12, 1865–1876 (1991).
[CrossRef]

D. Tanre, C. Deroo, P. Duhaut, M. Herman, J. J. Morcrette, J. Perbos, P. Y. Deschamps, “Description of a computer code to simulate the satellite signal in the solar spectrum: the 5S code,” Int. J. Remote Sensing 11, 659–668 (1990).
[CrossRef]

J. Geophys. Res. (1)

J. C. Price, “Information content of IRIS spectra,” J. Geophys. Res. 80, 1930–1936 (1975).
[CrossRef]

J. Sediment. Petrol. (1)

H. R. Lang, M. J. Bartholomew, C. I. Grove, E. D. Paylor, “Spectral reflectance characterization (0.4 to 2.5 and 8.0 to 12.0 μm) of Phanerozoic strata, Wind River Basin and Southern Bighorn Basin Area, Wyoming,” J. Sediment. Petrol. 60, 504–524(1990).

Photogram. Eng. Remote Sensing (2)

W. B. Cohen, “Response of vegetation indices to changes in three measures of leaf water stress,” Photogram. Eng. Remote Sensing 57, 195–202 (1991).

W. B. Cohen, “Chaparral vegetation reflectance and its potential utility for assessment of fire hazard,” Photogram. Eng. Remote Sensing 57, 203–207 (1991).

Purdue Univ. Agric. Exp. St. Res. Bull. (1)

E. R. Stoner, M. F. Baumgardner, L. L. Biehl, B. F. Robinson, “Atlas of soil reflectance properties,” Purdue Univ. Agric. Exp. St. Res. Bull. 962, 1–75.

Remote Sensing Environ. (2)

F. Csillag, L. Pasztor, L. L. Biehl, “Spectral band selection for the characterization of salinity status of soils,” Remote Sensing Environ. 43, 231–242 (1993).
[CrossRef]

J. C. Price, “On the information content of soil reflectance spectra,” Remote Sensing Environ. 33, 113–121 (1990).
[CrossRef]

Other (11)

J. C. Price, “On the value of high spectral resolution measurements in the visible and near infrared,” in Proceedings of the Fifth International Colloquium—Physical Measurements and Signatures in Remote Sensing, Courchevel, France (European Space Agency, Paris, 1991), pp. 131–136.

L. L. Biehl, C. S. T. Daughtry, M. E. Bauer, “Vegetation and soils field research data base: experiment summaries,” LARS Tech. Rep. 042382 (Purdue University, West Lafayette, Ind., 1984).

C. Daughtry, L. Biehl (personal communication, 15February1992).

G. T. Vane, “First results of the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS),” in Imaging Spectroscopy II, G. T. Vane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.834, 166–174(1987).

L. J. Ricard, M. Landers, “HYDICE, an airborne system for hyperspectral imaging,” presented at the International Symposium on Spectral Sensing Research, Maui, Hawaii, 15–20 November 1992.

M. B. Satterwhite, J. P. Henley, “Hyperspectral signatures (400–2500 nm) of vegetation, minerals, soils, rocks, and cultural features: laboratory and field measurements,” Rep. ETL-0573 (U.S. Army Corps of Engineers, Fort Belvoir, Va., 1991).

D. Krohn, U.S. Geological Survey, Reston, Va. (personal communication, 26March1991).

M. Viollier, “Teledetection des concentrations de seston et pigments chlorophylliens contenu dans l’ocean,” Ph.D. dissertation (1980).

K. Ya Kondratyev, Radiation in the Atmosphere (Academic, New York, 1969).

C. I. Grove, S. J. Hook, E. D. Paylor, “Laboratory reflectance spectra of 160 minerals, 0.4 to 2.5 micrometers,” JPL Rep. 92-2 (Jet Propulsion Laboratory, Pasadena, Calif., 1992).

Sets, Inc., Spectral Catalog (Sets Inc., Milalani, Hawaii, 1990), Vols. 1–5.

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Figures (2)

Fig. 1
Fig. 1

Data sets analyzed, as described in Appendix A, and the recommended spectral intervals of observation for the restricted spectral range that corresponds to data sets 1 and 2. Adjacent bands are offset for clarity.

Fig. 2
Fig. 2

Data sets analyzed and the recommended spectral intervals of observation for the full spectral range 0.4–2.5μm. Adjacent bands are offset for clarity.

Tables (2)

Tables Icon

Table 1 Spectral-Band Selection for the Restricted Range (0.55–1.34 μm, 1.48–1.80 μm, 2.03–2.31 μm)

Tables Icon

Table 2 Spectral-Band Selection for Full Range (0.40–2.50 μm)

Equations (17)

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

x α i = 1 M S i α φ i ( λ ) ,
S i α = 1 [ λ i ( max ) λ i ( min ) ] λ i ( min ) λ i ( max ) δ x i α d λ .
δ x i α d λ i = 1 [ λ i ( max ) λ i ( min ) ] λ i ( min ) λ i ( max ) δ x i α d λ ,
S i α = δ x i α d λ i .
φ i ( λ ) = δ x i S i / ( S i ) 2 ,
φ i d λ i = 1 [ λ i ( max ) λ i ( min ) ] × λ i ( min ) λ i ( max ) δ x i S i / ( S i ) 2 d λ = 1 ,
S i = δ x i d λ i = ( x j = 1 i 1 S j φ j ) d λ i .
b i j = φ j d λ i ,
S i = s i j = 1 i 1 b i j S j .
S i = j = 1 i d i j s j ,
S i = s i j = 1 i 1 b i j k = 1 j d j k s k .
S i = s i j = 1 i 1 k = j i 1 b i k d k j s j .
d i j = j = 1 i 1 b i k d k j .
φ i = ( x j = 1 i 1 S j φ j ) ( k = 1 i d i k s k ) / ( j = 1 i d i j s j ) ( k = 1 i d i k s k ) .
P i j = k = 1 i l = 1 i d i k d j l s k s l .
φ i = ( j = 1 i d i j x s j j = 1 i 1 P i j φ j ) / P i i .
R ( M ) = 100 % ( x i = 1 M S i φ i ) 2 d λ / ( x 2 ) d λ ,

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